Mechanical Engineering Series Frederick F. Ling Editor-in-Chief
Mechanical Engineering Series J. Chakrabarty, Applied...

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Jagabandhu Chakrabarty

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Mechanical Engineering Series Frederick F. Ling Editor-in-Chief

Mechanical Engineering Series J. Chakrabarty, Applied Plasticity, Second Edition G. Genta, Vibration Dynamics and Control R. Firoozian, Servo Motors and Industrial Control Theory G. Genta and L. Morello, The Automotive Chassis, Volumes 1 & 2 F. A. Leckie and D. J. Dal Bello, Strength and Stiffness of Engineering Systems Wodek Gawronski, Modeling and Control of Antennas and Telescopes M. Ohsaki and KiyohiroIkeda, Stability and Optimization of Structures: Generalized Sensitivity Analysis A.C. Fischer-Cripps, Introduction to Contact Mechanics, 2nd ed. W. Cheng and I. Finnie, Residual Stress Measurement and the Slitting Method J. Angeles, Fundamentals of Robotic Mechanical Systems: Theory, Methods and Algorithms, 3rd ed. J. Angeles, Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms, 2nd ed. P. Basu, C. Kefa, and L. Jestin, Boilers and Burners: Design and Theory J.M. Berthelot, Composite Materials: Mechanical Behavior and Structural Analysis I.J. Busch-Vishniac, Electromechanical Sensors and Actuators J. Chakrabarty, Applied Plasticity K.K. Choi and N.H. Kim, Structural Sensitivity Analysis and Optimization 1: Linear Systems K.K. Choi and N.H. Kim, Structural Sensitivity Analysis and Optimization 2: Nonlinear Systems and Applications G. Chryssolouris, Laser Machining: Theory and Practice V.N. Constantinescu, Laminar Viscous Flow G.A. Costello, Theory of Wire Rope, 2nd ed. K. Czolczynski, Rotordynamics of Gas-Lubricated Journal Bearing Systems M.S. Darlow, Balancing of High-Speed Machinery W. R. DeVries, Analysis of Material Removal Processes J.F. Doyle, Nonlinear Analysis of Thin-Walled Structures: Statics, Dynamics, and Stability J.F. Doyle, Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms, 2nd Edition P.A. Engel, Structural Analysis of Printed Circuit Board Systems A.C. Fischer-Cripps, Introduction to Contact Mechanics A.C. Fischer-Cripps, Nanoindentation, 2nd ed. J. García de Jalón and E. Bayo, Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge W.K. Gawronski, Advanced Structural Dynamics and Active Control of Structures W.K. Gawronski, Dynamics and Control of Structures: A Modal Approach G. Genta, Dynamics of Rotating Systems (continued after index)

J. Chakrabarty

Applied Plasticity, Second Edition

123

J. Chakrabarty Visiting Professor Department of Mechanical Engineering Florida State University Tallahassee FL 32303 USA [email protected]

ISSN 0941-5122 ISBN 978-0-387-77673-6 e-ISBN 978-0-387-77674-3 DOI 10.1007/978-0-387-77674-3 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009934696 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Mechanical Engineering Series Frederick F. Ling Editor-in-Chief

The Mechanical Engineering Series features graduate texts and research monographs to address the need for information in contemporary mechanical engineering, including areas of concentration of applied mechanics, biomechanics, computational mechanics, dynamical systems and control, energetics, mechanics of materials, processing, production systems, thermal science, and tribology.

Advisory Board/Series Editors Applied Mechanics

F.A. Leckie University of California, Santa Barbara D. Gross Technical University of Darmstadt

Biomechanics

V.C. Mow Columbia University

Computational Mechanics

H.T. Yang University of California, Santa Barbara

Dynamic Systems and Control/ Mechatronics

D. Bryant University of Texas at Austin

Energetics

J.R. Welty University of Oregon, Eugene

Mechanics of Materials

I. Finnie University of California, Berkeley

Processing

K.K. Wang Cornell University

Production Systems

G.-A. Klutke Texas A&M University

Thermal Science

A.E. Bergles Rensselaer Polytechnic Institute

Tribology

W.O. Winer Georgia Institute of Technology

Series Preface

Mechanical engineering, an engineering discipline forged and shaped by the needs of the industrial revolution, is once again asked to do its substantial share in the call for industrial renewal. The general call is urgent as we face profound issues of productivity and competitiveness that require engineering solutions, among others. The Mechanical Engineering Series features graduate texts and research monographs intended to address the need for information in contemporary areas of mechanical engineering. The series is conceived as a comprehensive one that covers a broad range of concentrations important to mechanical engineering graduate education and research. We are fortunate to have a distinguished roster of consulting editors on the advisory board, each an expert in one of the areas of concentration. The names of the consulting editors are listed on the facing page of this volume. The areas of concentration are applied mechanics, biomechanics, computational mechanics, dynamic systems and control, energetics, mechanics of materials, processing, production systems, thermal science, and tribology. Austin, Texas

Frederick F. Ling

This book is humbly dedicated to the loving memory of MA INDIRA who continues to be the source of real inspiration to me.

Preface

The past few years have witnessed a growing interest in the application of the mechanics of plastic deformation of metals to a variety of engineering problems associated with structural design and technological forming of metals. Written several years ago to serve as a companion volume to the author’s earlier work under the title Theory of Plasticity, which comprehensively expounds the fundamentals of plasticity of metals, the present work seems to have stood the test of time and has established itself as a comprehensive reference work that is equally useful for classroom purposes. While the earlier work is mainly concerned with the application of the theory to the solution of elastic/plastic problems, limit analysis of framed structures, and problems in plane plastic strain involving slipline fields, several important areas of plasticity related to the analysis of multidimensional structures and various metal-forming processes had to be left out for obvious reasons. The present text is intended to fill this gap and to make available to the reader in a single volume a detailed account of a wide range of useful results that are scattered in numerous periodicals and other sources. The fundamentals of the mathematical theory of plasticity are discussed in Chapter 1 with sufficient details, in order to eliminate the need for frequent references to the author’s earlier volume. The theory of plane plastic stress and its applications to structural analysis and sheet metal forming are presented in Chapter 2. The axially symmetrical plastic state, as well as a few three-dimensional problems of plasticity, is treated in Chapter 3. The plastic behavior of plates and shells, mainly from the point of view of limit analysis, is discussed with several examples in Chapters 4 and 5. The plasticity of metals with fully developed orthotropic anisotropy and its application to the plastic behavior of anisotropic sheets are presented in Chapter 6. The generalized tangent modulus theory of buckling in the plastic range for columns, plates, and shells is treated in Chapter 7 from the point of view of the bifurcation phenomenon. Chapter 8 deals with a wide range of topics in dynamic plasticity, including the wave propagation, armor penetration, and structural impact in the plastic range. The fundamentals of the rigid/plastic finite element method, with special reference to its application to metal-forming processes, are presented in Chapter 9, where several examples are included for illustration. The publication of the revised second edition of Applied Plasticity is deemed necessary not only for the obvious need for updating the book but also for the purpose ix

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of making it more suitable for the teaching of appropriate courses on plasticity at the graduate level. During the preparation of the second edition, several parts of the text have been extensively revised in the light of the recent developments of the subject, and new references to the published literature have been made in appropriate places. The discussion of the finite element method in plasticity, previously relegated to an appendix in the first edition, has now been expanded into a new chapter to permit a more complete treatment of the subject. A new section has been added in Chapter 4 to discuss the yield line theory for plate bending, not only for the derivation of complete solutions but also for the estimation of upper bounds on the limit load. A set of homework problems has been included at the end of each chapter for the benefit of both the student and the instructor, many of these problems having been designed to supplement the text. The references to the published literature have now been collected together and placed at the very end of the book for the sake of the expected convenience of the reader. The book in its present form would be suitable for teaching advanced graduate level courses on plasticity and metal forming to students of mechanical and manufacturing engineering, as well as on structural plasticity to students of civil and structural engineering. The book will also be found useful for teaching courses on dynamic plasticity to both the mechanical and civil engineering students. Though intended primarily for research workers in the field of plasticity, senior undergraduate students and practicing engineers are also likely to benefit from this book to a large extent. I take this opportunity to express my gratitude to the late Professor J. M. Alexander, formerly of Imperial College, London, who not only stimulated my interest in plasticity but also encouraged me to undertake the task of writing this book. I am also grateful to Dr. Frederick F. Ling, the Editor-in-Chief of this Series, for his encouragement and support for the publication of the second edition of Applied Plasticity. It is a pleasure to offer my sincere thanks to Ms. Jennifer Mirski, the Assistant Engineering Editor of Springer for her helpful cooperation and support during the preparation of the manuscript. Finally, I am deeply indebted to my wife Swati, who gracefully accepted the hardship of many lonely hours to enable me to complete this work in a satisfactory manner. J. Chakrabarty

Contents

1 Fundamental Principles . . . . . . . . . . . . 1.1 The Material Response . . . . . . . . . . 1.1.1 Introduction . . . . . . . . . . . . 1.1.2 The True Stress–Strain Curve . . . 1.1.3 Empirical Stress–Strain Equations . 1.2 Basic Laws of Plasticity . . . . . . . . . . 1.2.1 Yield Criteria of Metals . . . . . . 1.2.2 Plastic Flow Rules . . . . . . . . . 1.2.3 Limit Theorems . . . . . . . . . . 1.3 Strain-Hardening Plasticity . . . . . . . . 1.3.1 Isotropic Hardening . . . . . . . . 1.3.2 Plastic Flow with Hardening . . . . 1.3.3 Kinematic Hardening . . . . . . . 1.3.4 Combined or Mixed Hardening . . 1.4 Cyclic Loading of Structures . . . . . . . 1.4.1 Cyclic Stress–Strain Curves . . . . 1.4.2 A Bounding Surface Theory . . . . 1.4.3 The Two Surfaces in Contact . . . 1.5 Uniqueness and Stability . . . . . . . . . . 1.5.1 Fundamental Relations . . . . . . . 1.5.2 Uniqueness Criterion . . . . . . . 1.5.3 Stability Criterion . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . .

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2 Problems in Plane Stress . . . . . . . . . . . . . 2.1 Formulation of the Problem . . . . . . . . . 2.1.1 Characteristics in Plane Stress . . . . 2.1.2 Relations Along the Characteristics . 2.1.3 The Velocity Equations . . . . . . . 2.1.4 Basic Relations for a Tresca Material 2.2 Discontinuities and Necking . . . . . . . . . 2.2.1 Velocity Discontinuities . . . . . . .

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2.2.2 Tension of a Grooved Sheet . . . . . 2.2.3 Stress Discontinuities . . . . . . . . 2.2.4 Diffuse and Localized Necking . . . 2.3 Yielding of Notched Strips . . . . . . . . . . 2.3.1 V-Notched Strips in Tension . . . . . 2.3.2 Solution for Circular Notches . . . . 2.3.3 Solution for Shallow Notches . . . . 2.4 Bending of Prismatic Beams . . . . . . . . . 2.4.1 Strongly Supported Cantilever . . . . 2.4.2 Weakly Supported Cantilever . . . . 2.4.3 Bending of I-Section Beams . . . . . 2.5 Limit Analysis of a Hollow Plate . . . . . . 2.5.1 Equal Biaxial Tension . . . . . . . . 2.5.2 Uniaxial Tension: Lower Bounds . . 2.5.3 Uniaxial Tension: Upper Bounds . . 2.5.4 Arbitrary Biaxial Tension . . . . . . 2.6 Hole Expansion in Infinite Plates . . . . . . 2.6.1 Initial Stages of the Process . . . . . 2.6.2 Finite Expansion Without Hardening 2.6.3 Work-Hardening von Mises Material 2.6.4 Work-Hardening Tresca Material . . 2.7 Stretch Forming of Sheet Metals . . . . . . 2.7.1 Hydrostatic Bulging of a Diaphragm 2.7.2 Stretch Forming Over a Rigid Punch 2.7.3 Solutions for a Special Material . . . 2.8 Deep Drawing of Cylindrical Cups . . . . . 2.8.1 Introduction . . . . . . . . . . . . . 2.8.2 Solution for Nonhardening Materials 2.8.3 Influence of Work-Hardening . . . . 2.8.4 Punch Load and Punch Travel . . . . 2.9 Ironing and Flange Wrinkling . . . . . . . . 2.9.1 Ironing of Cylindrical Cups . . . . . 2.9.2 Flange Wrinkling in Deep Drawing . Problems . . . . . . . . . . . . . . . . . . . . . .

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3 Axisymmetric and Related Problems . . . . . . . . 3.1 Basic Theory and Exact Solutions . . . . . . . . 3.1.1 Fundamental Relations . . . . . . . . . . 3.1.2 Swaging in a Contracting Cylinder . . . 3.1.3 Fully Plastic State in a Cylindrical Tube 3.1.4 Plastic Flow Through a Conical Channel 3.2 Slipline Fields and Indentations . . . . . . . . . 3.2.1 Relations Along the Sliplines . . . . . . 3.2.2 Indentation by a Flat Punch . . . . . . . 3.2.3 Indentation by a Rigid Cone . . . . . . .

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3.2.4 The Hardness of Metals . . . . . . . . . . . 3.3 Necking of a Cylindrical Bar . . . . . . . . . . . . 3.3.1 Stress Distribution in the Neck . . . . . . . 3.3.2 Initiation of Necking . . . . . . . . . . . . . 3.4 Compression of Short Cylinders . . . . . . . . . . . 3.4.1 Compression of Solid Cylinders . . . . . . . 3.4.2 Estimation of Incipient Barreling . . . . . . 3.4.3 Compression of a Hollow Cylinder . . . . . 3.5 Sinking of Thin-Walled Tubes . . . . . . . . . . . . 3.5.1 Solution Without Strain Hardening . . . . . 3.5.2 Influence of Strain Hardening . . . . . . . . 3.6 Extrusion of Cylindrical Billets . . . . . . . . . . . 3.6.1 The Basis for an Approximation . . . . . . . 3.6.2 Extrusion Through Conical Dies . . . . . . 3.6.3 Extrusion Through Square Dies . . . . . . . 3.6.4 Upper Bound Solution for Square Dies . . . 3.6.5 Upper Bound Solution for Conical Dies . . . 3.7 Mechanics of Wire Drawing . . . . . . . . . . . . . 3.7.1 Solution for a Nonhardening Material . . . . 3.7.2 Influence of Back Pull and Work-Hardening 3.7.3 Ideal Wire-Drawing Dies . . . . . . . . . . 3.8 Some Three-Dimensional Problems . . . . . . . . . 3.8.1 Indentation by a Rectangular Punch . . . . . 3.8.2 Flat Tool Forging of a Bar . . . . . . . . . . 3.8.3 Bar Drawing Through Curved Dies . . . . . 3.8.4 Compression of Noncircular Blocks . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Plastic Bending of Plates . . . . . . . . . . . . . . . . 4.1 Plastic Collapse of Circular Plates . . . . . . . . . 4.1.1 The Basic Theory . . . . . . . . . . . . . 4.1.2 Circular Plates Carrying Distributed Loads 4.1.3 Other Types of Loading of Circular Plates 4.1.4 Solutions Based on the von Mises Criterion 4.1.5 Combined Bending and Tension . . . . . . 4.2 Deflection of Circular Plates . . . . . . . . . . . . 4.2.1 Basic Equations . . . . . . . . . . . . . . 4.2.2 Deflection of a Simply Supported Plate . . 4.2.3 Deflection of a Built-In Plate . . . . . . . 4.3 Influence of Membrane Forces . . . . . . . . . . 4.3.1 Simply Supported Circular Plates . . . . . 4.3.2 Built-In Circular Plates . . . . . . . . . . 4.4 Plastic Collapse of Noncircular Plates . . . . . . . 4.4.1 General Considerations . . . . . . . . . . 4.4.2 Uniformly Loaded Rectangular Plates . . .

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4.4.3 Finite Element Analysis for Plate Bending 4.5 Plane Strain Analogy for Plate Bending . . . . . . 4.5.1 The Use of Square Yield Condition . . . . 4.5.2 Application to Rectangular Plates . . . . . 4.5.3 Collapse Load for Triangular Plates . . . . 4.6 Yield Line Theory for Plates . . . . . . . . . . . . 4.6.1 Basic Yield Line Theory . . . . . . . . . . 4.6.2 Elliptical Plate Loaded at the Center . . . 4.6.3 A Plate Under Distributed Loading . . . . 4.6.4 Yield Line Upper Bounds . . . . . . . . . 4.6.5 Examples of Upper Bounds . . . . . . . . 4.7 Minimum Weight Design of Plates . . . . . . . . 4.7.1 Basic Principles . . . . . . . . . . . . . . 4.7.2 Circular Sandwich Plates . . . . . . . . . 4.7.3 Solid Circular Plates . . . . . . . . . . . . 4.7.4 Elliptical Sandwich Plates . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Plastic Analysis of Shells . . . . . . . . . . . . . . . . . . 5.1 Cylindrical Shells Without End Load . . . . . . . . . 5.1.1 Basic Equations . . . . . . . . . . . . . . . . 5.1.2 Yield Condition and Flow Rule . . . . . . . . 5.1.3 Shell Under Uniform Radial Pressure . . . . . 5.1.4 Shell Under a Band of Pressure . . . . . . . . 5.1.5 Solution for a von Mises Material . . . . . . . 5.2 Cylindrical Shells with End Load . . . . . . . . . . . 5.2.1 Yield Condition and Flow Rule . . . . . . . . 5.2.2 Shell Under Radial Pressure and Axial Thrust 5.2.3 Influence of Elastic Deformation . . . . . . . 5.3 Yield Point States in Shells of Revolution . . . . . . . 5.3.1 Generalized Stresses and Strain Rates . . . . . 5.3.2 Yield Condition for a Tresca Material . . . . . 5.3.3 Approximations for a von Mises Material . . . 5.3.4 Linearization and Limited Interaction . . . . . 5.4 Limit Analysis of Spherical Shells . . . . . . . . . . 5.4.1 Basic Equations . . . . . . . . . . . . . . . . 5.4.2 Plastic Collapse of a Spherical Cap . . . . . . 5.4.3 Spherical Cap with a Covered Cutout . . . . . 5.4.4 Solution for a Tresca Sandwich Shell . . . . . 5.4.5 Extended Analysis for Deeper Shells . . . . . 5.5 Limit Analysis of Conical Shells . . . . . . . . . . . 5.5.1 Basic Equations . . . . . . . . . . . . . . . . 5.5.2 Truncated Shallow Shell Under Line Load . . 5.5.3 Shallow Shell Loaded Through Rigid Boss . . 5.5.4 Centrally Loaded Shell of Finite Angle a . . .

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5.6 Limit Analysis of Pressure Vessels . . . . . . . . . 5.6.1 Plastic Collapse of a Toroidal Knuckle . . . 5.6.2 Collapse of a Complete Pressure Vessel . . . 5.6.3 Cylindrical Nozzle in a Spherical Vessel . . 5.7 Minimum Weight Design of Shells . . . . . . . . . 5.7.1 Principles for Optimum Design . . . . . . . 5.7.2 Basic Theory for Cylindrical Shells . . . . . 5.7.3 Simply Supported Shell Without End Load . 5.7.4 Cylindrical Shell with Built-In Supports . . 5.7.5 Closed-Ended Shell Under Internal Pressure Problems . . . . . . . . . . . . . . . . . . . . . . . . . .

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379 379 382 385 389 389 391 393 397 400 402

6 Plastic Anisotropy . . . . . . . . . . . . . . . . . . . . 6.1 Plastic Flow of Anisotropic Metals . . . . . . . . . 6.1.1 The Yield Criterion . . . . . . . . . . . . . 6.1.2 Stress–Strain Relations . . . . . . . . . . . 6.1.3 Variation of Anisotropic Parameters . . . . . 6.2 Anisotropy of Rolled Sheets . . . . . . . . . . . . . 6.2.1 Variation of Yield Stress and Strain Ratio . . 6.2.2 Localized and Diffuse Necking . . . . . . . 6.2.3 Correlation of Stress–Strain Curves . . . . . 6.2.4 Normal Anisotropy in Sheet Metal . . . . . 6.2.5 A Generalized Theory for Planar Anisotropy 6.3 Torsion of Anisotropic Bars . . . . . . . . . . . . . 6.3.1 Bars of Arbitrary Cross Section . . . . . . . 6.3.2 Some Particular Cases . . . . . . . . . . . . 6.3.3 Length Changes in Twisted Tubes . . . . . . 6.3.4 Torsion of a Free-Ended Tube . . . . . . . . 6.4 Plane Strain in Anisotropic Metals . . . . . . . . . 6.4.1 Basic Equations in Plane Strain . . . . . . . 6.4.2 Relations Along the Sliplines . . . . . . . . 6.4.3 Indentation by a Flat Punch . . . . . . . . . 6.4.4 Indentation of a Finite Medium . . . . . . . 6.4.5 Compression Between Parallel Platens . . . 6.5 Anisotropy in Stretch Forming . . . . . . . . . . . 6.5.1 Basic Equations for Biaxial Stretching . . . 6.5.2 Plastic Instability in Tension . . . . . . . . . 6.5.3 Forming Limit Diagram . . . . . . . . . . . 6.6 Anisotropy in Deep Drawing . . . . . . . . . . . . 6.6.1 The Radial Drawing Process . . . . . . . . . 6.6.2 Use of the Linearized Yield Condition . . . 6.6.3 The Limiting Drawing Ratio . . . . . . . . . 6.6.4 Earing of Deep-Drawn Cups . . . . . . . . . 6.7 Anisotropy in Plates and Shells . . . . . . . . . . .

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405 405 405 407 408 410 410 412 414 416 420 424 424 427 428 430 432 432 434 437 438 440 444 444 445 447 452 452 456 460 464 466

xvi

Contents

6.7.1 6.7.2 6.7.3 Problems .

Bending of Circular Plates . . . . . Plastic Collapse of a Spherical Cap Reinforced Circular Plates . . . . . . . . . . . . . . . . . . . . . . . . .

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466 469 472 475

7 Plastic Buckling . . . . . . . . . . . . . . . . . . . . . 7.1 Buckling of Axially Loaded Columns . . . . . . . 7.1.1 Analysis for Bifurcation . . . . . . . . . . 7.1.2 Analysis for Instability . . . . . . . . . . . 7.2 Behavior of Eccentrically Loaded Columns . . . . 7.2.1 Moment-Curvature Relations . . . . . . . 7.2.2 Analysis for a Pin-Ended Column . . . . . 7.2.3 Solution for an Inelastic Beam Column . . 7.3 Lateral Buckling of Beams . . . . . . . . . . . . 7.3.1 Pure Bending of Narrow Beams . . . . . . 7.3.2 Buckling of Transversely Loaded Beams . 7.4 Buckling of Plates Under Edge Thrust . . . . . . 7.4.1 Basic Equations for Thin Plates . . . . . . 7.4.2 Buckling of Rectangular Plates . . . . . . 7.4.3 Rectangular Plates Under Biaxial Thrust . 7.4.4 Buckling of Circular Plates . . . . . . . . 7.5 Buckling of Cylindrical Shells . . . . . . . . . . . 7.5.1 Formulation of the Rate Problem . . . . . 7.5.2 Bifurcation Under Combined Loading . . 7.5.3 Buckling Under Axial Compression . . . . 7.5.4 Influence of Frictional Restraints . . . . . 7.5.5 Buckling Under External Fluid Pressure . 7.6 Torsional and Flexural Buckling of Tubes . . . . . 7.6.1 Bifurcation Under Pure Torsion . . . . . . 7.6.2 Buckling Under Pure Bending . . . . . . . 7.7 Buckling of Spherical Shells . . . . . . . . . . . . 7.7.1 Analysis for a Complete Spherical Shell . 7.7.2 Solution for the Critical Pressure . . . . . 7.7.3 Snap-Through Buckling of Spherical Caps Problems . . . . . . . . . . . . . . . . . . . . . . . . .

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479 479 480 484 487 487 489 494 499 500 505 508 508 511 516 519 522 522 524 527 530 533 537 537 541 546 546 550 554 557

8 Dynamic Plasticity . . . . . . . . . . . . . . . . . . . . . . . 8.1 Longitudinal Stress Waves in Bars . . . . . . . . . . . . 8.1.1 Wave Propagation Without Rate Effects . . . . . . 8.1.2 Simple Wave Solution with Application . . . . . . 8.1.3 Solution for Linear Strain Hardening . . . . . . . 8.1.4 Influence of Strain-Rate Sensitivity . . . . . . . . 8.1.5 Illustrative Examples and Experimental Evidence 8.2 Plastic Waves in Continuous Media . . . . . . . . . . . . 8.2.1 Plastic Wave Speeds and Their Properties . . . . .

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8.2.2 A Geometrical Representation . . . . . . . . . 8.2.3 Plane Waves in Elastic/Plastic Solids . . . . . 8.3 Crumpling of Flat-Ended Projectiles . . . . . . . . . 8.3.1 Taylor’s Theoretical Model . . . . . . . . . . 8.3.2 An Alternative Analysis . . . . . . . . . . . . 8.3.3 Estimation of the Dynamic Yield Stress . . . . 8.4 Dynamic Expansion of Spherical Cavities . . . . . . 8.4.1 Purely Elastic Deformation . . . . . . . . . . 8.4.2 Large Elastic/Plastic Expansion . . . . . . . . 8.4.3 Influence of Elastic Compressibility . . . . . . 8.5 Mechanics of Projectile Penetration . . . . . . . . . . 8.5.1 A Simple Theoretical Model . . . . . . . . . . 8.5.2 The Influence of Cavitation . . . . . . . . . . 8.5.3 Perforation of a Thin Plate . . . . . . . . . . . 8.6 Impact Loading of Prismatic Beams . . . . . . . . . . 8.6.1 Cantilever Beam Struck at Its Tip . . . . . . . 8.6.2 Rate Sensitivity and Simplified Model . . . . 8.6.3 Solution for a Rate-Sensitive Cantilever . . . . 8.6.4 Transverse Impact of a Free-Ended Beam . . . 8.7 Dynamic Loading of Circular Plates . . . . . . . . . . 8.7.1 Formulation of the Problem . . . . . . . . . . 8.7.2 Simply Supported Plate Under Pressure Pulse 8.7.3 Dynamic Behavior Under High Loads . . . . 8.7.4 Solution for Impulsive Loading . . . . . . . . 8.8 Dynamic Loading of Cylindrical Shells . . . . . . . . 8.8.1 Defining Equations and Yield Condition . . . 8.8.2 Clamped Shell Loaded by a Pressure Pulse . . 8.8.3 Dynamic Analysis for High Loads . . . . . . 8.9 Dynamic Forming of Metals . . . . . . . . . . . . . . 8.9.1 High-Speed Compression of a Disc . . . . . . 8.9.2 Dynamic Response of a Thin Diaphragm . . . 8.9.3 High-Speed Forming of Sheet Metal . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The Finite Element Method . . . . . . . . . 9.1 Fundamental Principles . . . . . . . . . 9.1.1 The Variational Formulation . . . 9.1.2 Velocity and Strain Rate Vectors . 9.1.3 Elemental Stiffness Equations . . 9.2 Element Geometry and Shape Function . 9.2.1 Triangular Element . . . . . . . 9.2.2 Quadrilateral Element . . . . . . 9.2.3 Hexahedral Brick Element . . . . 9.3 Matrix Forms in Special Cases . . . . . 9.3.1 Plane Strain Problems . . . . . .

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671 671 671 672 675 676 676 679 681 682 682

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Contents

9.3.2 Axially Symmetrical Problems . . . 9.3.3 Three-Dimensional Problems . . . . 9.4 Sheet Metal Forming . . . . . . . . . . . . . 9.4.1 Basic Equations for Sheet Metals . . 9.4.2 Axisymmetric Sheet Forming . . . . 9.4.3 Sheet Forming of Arbitrary Shapes . 9.5 Numerical Implementation . . . . . . . . . 9.5.1 Numerical Integration . . . . . . . . 9.5.2 Global Stiffness Equations . . . . . . 9.5.3 Boundary Conditions . . . . . . . . 9.6 Illustrative Examples . . . . . . . . . . . . . 9.6.1 Compression of a Cylindrical Block . 9.6.2 Bar Extrusion Through a Conical Die 9.6.3 Analysis of Spread in Sheet Rolling . 9.6.4 Deep Drawing of Square Cups . . .

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683 685 685 685 687 689 691 691 694 696 697 697 698 701 704

Appendix: Orthogonal Curvilinear Coordinates . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

709

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743

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751

Chapter 1

Fundamental Principles

1.1 The Material Response 1.1.1 Introduction In a single crystal of many metals, the main mechanism of plastic deformation is simple shear parallel to preferred planes and directions, which at ordinary temperatures coincides with those of the highest atomic density. Slip is initiated along a particular plane and in a given direction when the associated component of the shear stress attains a critical value under increasing external load. The amount of plastic deformation in a single crystal is specified by the glide strain, which is the relative displacement of two parallel slip planes at a unit distance apart. When there are several possible slip directions in a crystal lattice, the displacement of any point in the crystal due to simultaneous shears in the appropriate directions can be found from simple geometry. The mechanism of slip-induced plasticity in single crystals, governed by the glide motion of dislocations along corresponding slip planes, has been the subject of numerous investigations in the past. The change in shape of a single crystal requires, in general, the operation of five independent slip systems (von Mises, 1928). This is due to the fact that an arbitrary state of strain is specified by the six independent components of the symmetric strain tensor, while the sum of the normal strain components vanishes by the condition of constancy of volume of the plastic material. The existence of five independent slip systems in a single crystal is necessary for the material to be ductile in the polycrystalline form. Face-centered cubic metals, having 12 potential slip systems in each crystal grain, satisfy this requirement and are known to have high degrees of ductility, while hexagonal close-packed metals having relatively low symmetry are noted for limited ductility at room temperatures. The ductility of a polycrystalline metal also requires slip flexibility which enables the five independent slip systems to operate simultaneously within a small volume of the aggregate. Mathematical theories of slip-induced plasticity in single crystals have been developed by Hill (1966), Hill and Rice (1972), and Asaro (1983). Some attempts have been made in the past to relate the tensile yield stress of polycrystalline metals in terms of the shear yield stress of the corresponding single crystals. Assuming each crystal grain to undergo the same uniform strain as the polycrystalline metal, and by minimizing the sum of the magnitudes of a set of geometrically possible shears, Taylor (1938) determined the uniaxial stress–strain

J. Chakrabarty, Applied Plasticity, Second Edition, Mechanical Engineering Series, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77674-3_1,

1

2

1 Fundamental Principles

curve for an aluminum aggregate, in good agreement with the experimental curve. The selection of active slip planes from a very wide range of possible combinations, on the basis of the minimum principle, is not necessarily unique. Taylor’s approach has been generalized by Bishop and Hill (1951), who developed a method of deriving upper and lower bounds on the yield function for a polycrystalline metal under any set of combined stresses. The plastic behavior of polycrystalline aggregates in relation to that of single crystals, including the effect of elastic deformation, has been similarly examined by Lin (1957, 1971). A useful review of the recent developments of the micromechanics of polycrystal plasticity has been presented by Khan and Huang (1995). Taylor’s theoretical model ensures the compatibility of strains across the grain boundaries, but fails to satisfy the conditions of equilibrium across these boundaries. In order to satisfy both the conditions of compatibility and equilibrium, a self-consistent model has been proposed by Kröner (1961) and by Budiansky and Wu (1962). These authors approximated each individual crystal grain by a spherical inclusion embedded in an infinitely extended homogeneous elastic matrix. The relationship between the stresses and the strains in the individual grains and those applying to the aggregate has been obtained by an averaging process based on an elastic/plastic analysis for the inclusion problem. Berveiller and Zaoui (1979) modified the theoretical model by introducing a plastic accommodation factor based on the assumption of isotropy of the elastic and plastic responses of both the crystal grain and the aggregate. A self-consistent model in which the anisotropy of the material response is allowed for has been developed by Hill (1965) and Hutchinson (1970), who considered the individual crystal grain as an ellipsoidal inclusion embedded in an elastic/plastic matrix. Another extension of the self-consistent model based on equivalent body forces has been put forward by Lin (1984). The macroscopic theory of plasticity, which the present volume is concerned with, is based on certain experimental observations on the behavior of ductile metals beyond the elastic limit under relatively simple states of combined stress. The theory is capable of predicting the distribution of stresses and strains in polycrystalline metals, not only in situations where the elastic and plastic strains are of comparable magnitudes but also in situations where the plastic strains are large enough for the elastic strains to be disregarded. The mathematical formulation of the plasticity problem is essentially incremental in nature, requiring due consideration of the complete stress and strain history of the elements that have been deformed in the plastic range. An interesting theoretical model, based on the nucleation of voids in the deforming material including interaction of neighboring voids, has been developed by Gurson (1977) and Tvergaard (1982). A theory of plastic yielding and flow of porous materials has been advanced by Tsuta and Yin (1998). A practical method of predicting the macroscopic behavior of polycrystalline aggregates from microstructural data has been discussed by Lee (1993).

1.1

The Material Response

3

1.1.2 The True Stress–Strain Curve In order to deal with large plastic deformation of metals, it is necessary to introduce the concept of true stress and true strain occurring in a test specimen. The true stress σ defined as the applied load divided by the current area of cross section of the specimen, can be significantly different from the nominal stress s, which is the load divided by the original area of cross section. If the initial and current lengths of a tensile specimen are denoted l0 and l, respectively, then the engineering strain e is equal to the ratio (l – l0 )/l0 , while the true strain ε is defined in such a way that its increment dε is equal to the ratio dl/l, where dl is the corresponding increase in length. It follows that the total true strain produced by a change in length from l0 to l during the tensile test is ε = ln

l = ln (1 + e). l0

(1.1)

Similarly, in the case of simple compression of a specimen, whose height is reduced from h0 to h during the test, the engineering strain is of magnitude e = (h0 – h)/h0 , while the magnitude of the true strain is ε = ln

h0 h

= − ln (1 − e).

As the deformation is continued in the plastic range, the true stress becomes increasingly higher than the nominal stress in the case of simple tension and lower than the nominal stress in the case of simple compression. The true strain, on the other hand, is progressively smaller than the engineering strain in simple tension and higher than the engineering strain in simple compression, as the deformation proceeds. There is sufficient experimental evidence to suggest that the macroscopic stress– strain curve of a polycrystalline metal in simple compression coincides with that in simple tension when the true stress is plotted against the true strain. Figure 1.1(a) shows the true stress–strain curve of a typical engineering material in simple tension. The longitudinal true stress σ existing in a test specimen under an axial load is a monotonically increasing function of the longitudinal true strain ε. The straight line OA represents the linear elastic response with A denoting the proportional limit. The elastic range generally extends slightly beyond this to the yield point B, which marks the beginning of plastic deformation. The strain-hardening property of the material requires the stress to increase with strain in the plastic range, but the slope of the stress–strain curve progressively decreases as the strain is increased until fracture occurs at G. Since plastic deformation is irreversible, unloading from some point C on the loading curve would make the stress–strain diagram follow the path CE, where E lies on the ε-axis for complete unloading and represents the amount of permanent or plastic strain corresponding to C. On reloading the specimen, the stress–strain

4

1 Fundamental Principles

Fig. 1.1 True stress–strain curve in simple tension: (a) loading and unloading with reloading and (b) idealized stress–strain behavior

curve follows the path EFG forming a hysteresis loop of narrow width, where F is a new yield point and FG is virtually a continuation of BC. Following Prandtl (1928), the stress–strain curve may be idealized by neglecting the width of the hysteresis loop and assuming the unloading path to be a straight line parallel to OA. The idealized curve, shown in Fig. 1.1(b), implies that the reloading proceeds along the path ECG, passing through the point C where the previous unloading started. Furthermore, the proportional limit A is assumed to coincide with the initial yield point, the corresponding stress being denoted by Y. It follows from the idealized stress–strain curve that the recoverable elastic strain at any point of the curve is equal to σ /E, where E is Young’s modulus and σ the current stress. Any stress increment dσ is associated with an elastic strain increment of amount dσ /E and a plastic strain increment of amount dσ /H, where H is the plastic modulus representing the current slope of the curve for the stress against plastic strain. Since the total strain increment is dσ /T, where T is the tangent modulus denoting the slope of the (σ , ε)-curve, we obtain the relationship 1 1 1 = + . T E H

(1.2)

The difference between T and H decreases rapidly with increasing strain, the elastic strain increment being increasingly small compared to the plastic strain increment. The material is said to be nonhardening when H = T = 0, which is approximately satisfied by a material that is heavily prestrained. Each increment of longitudinal strain dε in a tensile specimen is accompanied by a lateral compressive strain increment of magnitude dε’, the ratio dε’/dε being known as the contraction ratio denoted by η. Over the elastic range of strains, the contraction ratio has a constant value equal to Poisson’s ratio v, but once the yield

1.1

The Material Response

5

point is exceeded, the contraction ratio becomes a function of the magnitude of the strain. For an isotropic material, the elastic and plastic parts of the lateral strain increment have the magnitudes vdεe and dε p /2, respectively, where dε p = dε – dσ /E denotes the plastic component of the longitudinal strain increment. Consequently, the total lateral strain increment is given by 2dε = dε – (1 – 2v) dσ /E, and the contraction ratio is therefore expressed as T 1 1 − (1 − 2v) . η= 2 E

(1.3)

Thus, η depends on the current value of T. As the loading is continued in the plastic range, the tangent modulus progressively decreases from its elastic value E and the contraction ratio rapidly increases from v to approach the fully plastic value of 0.5. It follows that the elastic compressibility of the material becomes negligible when the tangent modulus is reduced to the order of the current yield stress σ . For an incompressible material, the relationship between the nominal stress s and the true stress σ is easily shown to be s = σ exp ( ∓ ε), where the upper sign corresponds to simple tension and the lower sign to simple compression of the test specimen. The distinction between the behaviors in tension and compression, in relation to the engineering strain, is illustrated in Fig. 1.2.

Fig. 1.2 The stress–strain behavior of metals with respect to nominal stress, true stress, and engineering strain: (a) simple tension and (b) simple compression

The tensile test is unsuitable for obtaining the stress–strain curve up to large values of the strain, since the specimen begins to neck when the rate of work-hardening

6

1 Fundamental Principles

decreases to a critical value. At this stage, the applied load attains a maximum, and the specimen subsequently extends under decreasing load. Setting the differential of the axial load P = σ A to zero, where A is the current cross-sectional area of the specimen, we get dσ /σ = –dA/A. On the other hand, the condition of zero incremental volume change, which holds very closely at the onset necking, gives –dA/A = dl/l = dε. The condition for plastic instability in simple tension therefore becomes dσ = σ. de

(1.4)

As the extension continues beyond the point of necking, the plastic deformation remains confined in the neck, which grows rapidly under decreasing load leading to fracture across the minimum section. The true stress–strain curve cannot be continued beyond the point of necking without introducing a suitable correction factor (Section 3.3). A neck is also formed in a cylindrical specimen subjected to a uniform fluid pressure on the lateral surface, and the amount of uniform strain at the point of necking is exactly the same as that in uniaxial tension (Chakrabarty, 1972). The strain-hardening characteristic of materials covering a fairly wide range of strains is most conveniently obtained by the simple compression test, in which a solid cylindrical block is axially compressed between a pair of parallel platens. However, due to the presence of friction between the specimen and the platens, the deformation of material near the regions of contact is constrained, resulting in a barreling of the specimen as the compression proceeds. Since the compression of the cylinder then becomes nonuniform, the true stress–strain curve cannot be derived from the compression test without introducing a suitable correction factor (Section 3.4). Several methods have been proposed in the past to eliminate the effect of friction on the stress–strain curve, but none of them seems to be entirely satisfactory. On the other hand, a state of homogeneous compression is very nearly achieved by inserting thin sheets of ptfe (polytetrafluoroethylene) between the specimen and the compression platens. The compressed ptfe sheets not only act as an effective lubricant but also help to inhibit the barreling tendency by exerting a radially outward pressure on the material near the periphery. It is necessary to apply the axial load, on an incremental basis, and to replace the deformed ptfe sheets with new ones before each application of the load. The true stress and the true strain are obtained at each stage from the measurement of the applied load and the current specimen height, together with the use of the constancy of volume. Consider an annealed specimen which is loaded in simple tension past the yield point and is subsequently unloaded to zero stress so that there is a certain amount of residual strain left in the specimen. If an axial compressive load is now applied, the specimen will begin to yield under a stress that is somewhat lower than the original yield stress in tension or compression. While the yield stress in tension at the time of unloading is much greater than Y owing to strain hardening of the material, the yield stress in compression is usually found to be lower than Y. A similar lowering of the yield stress is observed if the specimen is loaded plastically in compression and then pulled in tension. This phenomenon is known as the Bauschinger effect,

1.1

The Material Response

7

which occurs whenever there is a reversal of stress in a plastically deformed element. The phenomenon is generally attributed to residual stresses in the individual crystal grains due to the presence of grain boundaries in a polycrystalline metal. In some metals, such as annealed mild steel, the load at the elastic limit suddenly drops from an upper yield point to a lower yield point, followed by an elongation of a few percent under approximately constant stress. At the upper yield point, a discrete band of deformed metal, known as Lüders band, appears at approximately 45◦ to the tensile axis at a local stress concentration. During the yield point elongation, several bands usually form at several points of the specimen and propagate to cover the entire length. At this stage, the load begins to rise with further strain and the stress– strain curve then continues in the usual manner as a result of strain hardening. The upper yield point depends on such factors as the rate of straining, eccentricity of the loading, and the rigidity of the testing machine, but its value is usually 10–20% higher than the lower yield point.

1.1.3 Empirical Stress–Strain Equations In the theoretical treatment of plasticity problems, it is generally convenient to represent the true stress–strain curve of the material by a suitable empirical equation that involves constants to be determined by curve fitting with the experimental curve. For sufficiently large strains, the simplest empirical equation frequently used in the literature is the simple power law proposed by Ludwik (1909), which is σ = Cεn ,

(1.5)

where C is a constant stress and n is n dimensionless constant, known as the strainhardening exponent, whose value is generally less than 0.5. Although (1.5) corresponds to an infinite initial slope, it does provide a reasonably good fit with the actual stress–strain curve over a fairly wide range of strains. Since dσ /dε is equal to nσ /ε according to (1.5), the true strain at the onset of necking in simple tension is ε = n in view of (1.4). A nonhardening material corresponds to n = 0 with C representing the constant yield stress of the material. When the material is assumed to be rigid/plastic having a distinct initial yield stress Y, the simple power law (1.5) needs to be suitably modified. One such modification, sometimes used in the solution of special problems, is σ = Y + Kεn , where K has the dimension of stress and n is an exponent. Although this equation predicts a nonzero initial yield stress, it does not provide a better fit with the actual stress–strain curve over the relevant range of strains. The preceding equation includes, as a special case, the linear strain-hardening law (n = 1), with K denoting the constant plastic modulus. A more successful empirical equation involving a definite yield point is the modified power law

8

1 Fundamental Principles

σ = C(m + ε)n ,

(1.6)

where C, m, and n are constants. The stress–strain curve defined by (1.6), which is due to Swift (1952), is essentially the Ludwik curve (1.5) with the σ -axis moved through a distance m in the direction of the ε-axis. The parameter m therefore represents the amount of initial prestrain with respect to the annealed state. If the same stress–strain curve is fitted by both (1.5) and (1.6), the value of n in the two cases will of course be different. It follows from (1.4) and (1.6) that the magnitude of the true strain at the point of tensile necking is equal to n – m for m ≤ n and zero for n ≥ m. Figure 1.3(a) shows the Swift curves for several values of n based on a typical value of m.

Fig. 1.3 Nature of empirical stress–strain equations. (a) Swift equation and (b) Voce equation

For certain applications, it is sometimes more convenient to employ a different type of empirical equation proposed by Voce (1948). The Voce equation, which also involves an initial yield stress of the material, may be expressed as σ = C(1 − me−nε ),

(1.7)

where C, m, and n are material constants, and e is the exponential constant. The stress–strain curve defined by (1.7) exhibits an initial yield stress equal to (1 – m)C and tends to become asymptotic to the saturation stress σ = C. The slope of the stress–strain curve varies linearly with the stress according to the relation dσ /dε – n(C – σ ). Thus, the initial state of hardening of the material is represented by m, while the rapidity of approach to the saturation stress is represented by n. The stress–strain curves defined by (1.7) for a given value of m and several values of n are displayed in Fig. 1.3(b).

1.1

The Material Response

9

The preceding stress–strain equations can be used for elastic/plastic materials, with comparable elastic and plastic strains, provided ε is replaced by the plastic component εp . Since the elastic component of the strain εe is equal to σ /E, the simple power law of type (1.5) relating the stress to the plastic strain furnishes the total strain in the form m−1 σ σ , (1.8) ε= 1+α E σ0 where m = 1/n, σ 0 is a nominal yield stress, and α is a dimensionless constant. The stress–strain curve defined by (1.8), which is due to Ramberg and Osgood (1943), bends over with an initial slope equal to E, the plastic modulus H associated with any stress a being given by E/H = m(σ /σ 0 )m-1 . It may be noted that the secant modulus of the stress–strain curve is equal to E/(l + α) at the nominal yield point σ = σ 0 according to (1.8). Figure 1.4(a) shows several curves of this type for constant values of σ 0 /E and α.

Fig. 1.4 True stress–strain curves for elastic/plastic materials: (a) Ramberg–Osgood equation and (b) modified Ludwik equation

In some cases, the elastic/plastic analysis can be considerably simplified by using a stress–strain relation that corresponds to the Ludwik curve with its initial part replaced by a chord of slope equal to E. The stress–strain law is then given by the pair of equations σ =

Eε, ε ≤ Y/E, Y(Eε/Y)n , ε ≥ Y/E.

(1.9)

10

1 Fundamental Principles

The material has a sharp yield point at σ = Y, and the slope of the stress–strain curve changes discontinuously from E to nE at the yield point, Fig. 1.4(b). The value of the tangent modulus T at any stress σ ≥ Y is equal to nE(Y/σ )m–l , where m = l/n. The discontinuity in slope at σ = Y can be eliminated, however, by modifying the stress–strain equation in the plastic range as σ =Y

1−n Eε − nY n

n , ε ≥ Y/E.

The tangent modulus according to this equation is easily shown to be T = E(Y/σ )m–l , where m = l/n as before. Since T = σ at the incipient necking of a bar under simple tension, the magnitude of the uniform true strain exceeds n by the amount (1 – n)Y/E, which is negligible for most metals. The rate of straining has a profound influence on the yield strength of metals, particularly at elevated temperatures. The mechanical response of materials to high strain rates and temperatures is generally established on the basis of uniaxial stress– strain curves obtained under constant values of the strain rate and temperature. For a given temperature, the combined effects of strain ε and strain rate ε˙ can be expressed by the empirical equation σ = Cεn ε˙ m ,

(1.10)

where C, m, and n are constants, the parameters m being known as the strain rate sensitivity which is generally less than 0.2 for most metals and alloys. The higher the value of m, the greater is the strain at the onset of tensile necking (Hart, 1967). Since most of the heat generated during a high-speed test remains in the specimen, leading to an adiabatic rise in temperature, the results for a given test must be adjusted appropriately so that they correspond to a constant temperature. In the hot working of metals, the working temperature is high enough for recovery and recrystallization to occur without significant grain growth. Since the rate of work-hardening is then exactly balanced by the rate of thermal softening, the yield stress is practically independent of the strain except for sufficiently small values of the strain. There are certain metals and alloys known as superplastic materials, which exhibit very large neck-free tensile elongations prior to failure (Backofen et al., 1964). These materials are characterized by high values of the strain rate sensitivity m, which is generally greater than 0.4. The tensile fracture in superplastic materials is caused by the evolution of cavities at grain boundaries rather than by the development of diffuse necks. Conventional superplasticity is observed at relatively low strain rates, usually ranging from 10–4 /s to 10–3 /s, although recent studies have revealed the existence of superplasticity at considerably higher strain rates in certain alloys and composite materials. The relationship between the flow stress σ and the strain rate ε˙ in superplastic materials is often expressed by the equation σ = K ε˙ m ,

1.2

Basic Laws of Plasticity

11

where m is the strain rate sensitivity and K is generally independent of the strain. The elongation to failure in a tensile specimen of superplastic material increases with increasing values of m. In fine-grained superplastic materials, an m value of about 0.5 is fairly common, and the deformation takes place mainly by grain boundary sliding. Materials which are rendered superplastic by the generation of internal stresses through thermal or pressure cycling can have a strain rate sensitivity as high as unity. The superplasticity of metals and alloys has been discussed at length in the books by Presnyakov (1976), Padmanabhan and Davies (1980), and Nieh et al. (1997).

1.2 Basic Laws of Plasticity 1.2.1 Yield Criteria of Metals The macroscopic theory of plasticity is based on certain experimental observations regarding the behavior of ductile metals. The theory rests on the assumption that the material is homogeneous and is valid only at temperatures for which thermal phenomena may be neglected. For the present purpose, it is also assumed that the material is isotropic and has identical yield stresses in tension and compression. The plasticity of metals with fully developed states of anisotropy will be discussed in Chapter 6. A further simplific`ation results from the experimental fact that the yielding of metals is unaffected by a moderate hydrostatic pressure (Bridgman, 1945; Crossland, 1954). A law governing the limit of elastic behavior, consistent with the basic assumptions, defines a possible criterion of yielding under any combination of the applied stresses. The state of stress in any material element may be represented by a point in a nine-dimensional stress space. Around the origin of the stress space, there exists a domain of elastic range representing the totality of elastic states of stress. The external boundary of the elastic domain defines a surface, known as the initial yield surface, which may be expressed in terms of the components of the true stress σ ij , as f (σij ) = constant. Since the material is initially isotropic, plastic yielding depends only on the magnitudes of the three principal stresses, and not on their directions. This amounts to the fact that the yield criterion is expressible as a function of the three basic invariants of the stress tensor. The yield function is therefore a symmetric function of the principal stresses and is also independent of the hydrostatic stress, which is defined as the mean of the three principal stresses. Plastic yielding therefore depends on the principal components of the deviatoric stress tensor, which is defined as sij = σij − σ δij ,

(1.11)

12

1 Fundamental Principles

where σ denotes the hydrostatic stress, equal to (σ 1 + σ 2 + σ 3 )/3. Since the sum of the principal deviatoric stresses is sij = 0, the principal components cannot all be independent. It follows that the yield criterion may be expressed as a function of the invariants J2 and J3 of the deviatone stress tensor, which are given by (Chakrabarty, 2006) ⎫ 1 2 1 (s1 + s22 + s23 ) = sij sij ,⎪ ⎬ 2 2 1 1 ⎪ ⎭ J3 = s1 s2 s3 = (s31 + s32 + s33 ) = sij sjk ski . 3 3

J2 = − (s1 s2 + s2 s3 + s2 s1 ) =

(1.12).

The absence of Bauschinger effect in the initial state implies that yielding is unaffected by the reversal of the sign of the stress components. Since J3 changes sign with the stresses, an even function of this invariant should appear in the yield criterion. In a three-dimensional principal stress space, the yield surface is represented by a right cylinder whose axis is equally inclined to the three axes of reference, Fig. 1.5. The generator of the cylinder is therefore perpendicular to the plane σ 1 + σ 2 + σ 3 = 0, known as the deviatoric plane. Since σ 1 = σ 2 = σ 3 , along the geometrical axis of the cylinder, it represents purely hydrostatic states of stress. Points on the generator therefore represent stress states with varying hydrostatic part, which does not have any influence on the yielding. The yield surface is intersected by the deviatoric plane in a closed curve, known as the yield locus, which is assumed to be necessarily convex. Fig. 1.5 Geometrical representation of yield criteria in the principal stress space

Due to the assumed isotropy and the absence of the Bauschinger effect, the yield locus must possess a six-fold symmetry with respect to the projected stress axes and the lines perpendicular to them, as indicated in Fig. 1.6(b). In an experimental

1.2

Basic Laws of Plasticity

13

Fig. 1.6 Deviatoric yield locus. (a) Tresca hexagon and von Mises circle and (b) general shape of the locus

determination of the initial yield locus, it is therefore only necessary to apply stress systems covering a typical 30◦ segment of the yield locus. This may be achieved by introducing the Lode (1926) parameter μ, which is defined as μ=

√ 2σ2 − σ3 − σ1 = − 3 tan θ , σ3 − σ1

σ1 > σ2 > σ3 ,

(1.13)

where θ is the counterclockwise angle made by the deviatoric stress vector with the direction representing pure shear. To obtain the shape of the yield locus, it is only necessary to apply stress systems varying between pure shear (μ = ±, θ = 0) and uniaxial tension or compression (μ = ±1, θ = ±π/6). The yield locus is called regular when it has a unique tangent at each point and singular when it has sharp corners. The simplest yield criterion expressed in terms of the invariants of the deviatoric stress tensor is J2 = k2 , suggested by von Mises (1913), where k is a constant. The yield function does not therefore involve J3 at all. In terms of the stress component referred to an arbitrary set of rectangular axes, the von Mises yield criterion may be written as ⎫ 2 2 2 ⎪ sij sij = s2x + s2y + s2z + 2(τxy + τyz + τzx ) = 2 k2 ⎪ ⎬ or (1.14) ⎪ ⎭ 2 2 2 2 2 2 2⎪ (σ − σ ) + (σ − σ ) + (σ − σ ) + 6(τ + τ + τ ) = 6 k . x

y

y

z

z

x

xy

yz

zx

The second expression in (1.14) follows from the first on subtracting the identically zero term (sx + sy + sz )2 /3 and noting the fact that sx – sy = σ x – σ y , etc. The constant k is actually the yield stress in simple or pure shear, as may be seen

14

1 Fundamental Principles

by setting σ x = σ and σ y = –σ as the only nonzero √ stress components. According to (1.14), the uniaxial yield stress Y is equal to 3 k, which is obtained by considering σ x = Y as the only nonzero stress. The von Mises yield surface is evidently a right circular cylinder having its geometrical axis perpendicular to the deviatoric plane. The principal deviatoric stresses according to the von Mises criterion may be expressed in terms of the deviatoric angle θ as s1 =

π

π 2 2 2 Y cos + θ , s2 = Y sin θ , s3 = − Y cos −θ . 3 6 3 3 6

(1.15)

In the case of plane stress, the actual principal stresses σ 1 and σ 2 may be expressed in terms of θ using the fact that the sum of these stresses is equal to –3s3 which ensures that σ 3 is identically zero. On the basis of a series of experiments involving the extrusion of metals through dies of various shapes, Tresca (1864) concluded that yielding occurred when the magnitude of the greatest shear stress attained a certain critical value. In terms of the principal stresses, the Tresca criterion may be written as σ1 − σ3 = 2 k,

σ1 ≥ σ2 ≥ σ3 ,

(1.16)

where k is the yield stress in pure shear, the uniaxial yield stress being Y = 2k according to this criterion. All possible values of the principal stresses are taken into account when the Tresca criterion is expressed by a single equation in terms of the invariants J2 and J3 , but the result is too complicated to have any practical usefulness. For a given uniaxial yield stress Y, the Tresca yield surface is a regular hexagonal cylinder inscribed within the von Mises cylinder. The Tresca yield surface is not strictly convex, but each face of the surface may be regarded as the limit of a convex surface of vanishingly small curvature. The deviatoric yield loci for the Tresca and von Mises criteria are shown in Fig. 1.6(a). The maximum difference between the two criteria occurs in √ pure shear, for which the von Mises criterion predicts a yield stress which is 2/ 3 times that given by the Tresca criterion. Experiments have shown that for most metals the test points fall closer to the von Mises yield locus than to the Tresca locus, as indicated in Fig. 1.7. If the latter is adopted for simplicity, the overall accuracy can be improved by replacing 2k in (1.16) by mY, where m is an empirical constant lying between 1 √ and 2/ 3.

1.2.2 Plastic Flow Rules The following discussion is restricted to an ideally plastic material having a definite yield point and a constant yield stress. The influence of strain hardening will be discussed in the next section. The yield locus for the idealized material retains its size and shape so that the material remains isotropic and free from the Bauschinger effect. Each increment of strain in the plastic range is the sum of an elastic part

1.2

Basic Laws of Plasticity

15

Fig. 1.7 Experimental verification of the yield criterion for commercially pure aluminum (due to Lianis and Ford, 1957)

which may be recovered on unloading, and a plastic part that remains unchanged on unloading. The elastic part of the strain increment is given by the generalized Hooke’s law, while the plastic part is governed by what is known as the flow rule. The consideration of the plastic deformation of a polycrystalline metal in relation to that of the individual crystals leads to the existence of a plastic potential that is identical to the yield function (Bishop and Hill, 1951). The plastic strain increment, regarded as a vector in a nine-dimensional space, is therefore directed along the outward normal to the yield surface at the considered stress point. Denoting the unit vector along the exterior normal by nij , the associated flow rule for a nonhardening material with a regular yield surface may be written as p

dεij = nij dλ, nij dσij = 0,

(1.17)

where dλ is a positive scalar representing the magnitude of the plastic strain increment vector. The condition nij dσ ij = 0 implies that the stress point must remain on the yield surface during an increment of plastic strain. When nij dσ ij < 0, indicating unloading from the plastic state, the plastic strain increment is identically zero. Since the components of nij are proportional to ∂f/∂σ ij , where f defines the yield function, of J2 and J3 , the principal axes of nij coincide with those of σ ij . The flow rule therefore implies that the principal axes of stress and plastic strain increment p coincide for an isotropic solid. The plastic incompressibility condition dεii = 0 is identically satisfied, while the symmetric tensor nij satisfies the relations nij nij = 1,

nii = 0.

For a singular yield criterion, (1.17) holds for all regular points of the yield surface. At a singular point of the yield surface, the normal is not uniquely defined, and the plastic strain increment vector may lie anywhere between the normals to the

16

1 Fundamental Principles

faces meeting at the considered edge. When nij dσ ij < 0, the element unloads from the plastic state, and the plastic strain increment vanishes identically. The relation p

dσij dεij = 0, which holds for both loading and unloading for a regular yield surface, may be assumed to hold even for a singular point of a yield surface when the material is nonhardening (Drucker, 1951). When the yield criterion is that of von Mises, f (σij ) = 12 sij sij , which gives √ ∂f /∂σij = sij . The deviatoric stress vector √ is then of magnitude 2k, and √ the unit normal to the yield surface is nij = sij / 2 k. Replacing the quantity dλ/ 2 k by dλ in (1.17), the associated flow rule may be expressed as p

dεij = sij dλ, sij dsij = 0.

(1.18)

The ratios of the components of the plastic strain increment are therefore identical to those of the deviatoric stress. This relationship was proposed independently by Levy (1870) and von Mises (1913), both of whom used the total strain increment instead of the plastic strain increment alone. It therefore applies to a hypothetical rigid/plastic material whose elastic modulus is infinitely large. The extension of the flow rule to allow for the elastic part of the strain is due to Prandtl (1924) in the case of plane strain and to Reuss (1930) in the case of complete generality. The increment of plastic work per unit volume according to (1.18) and (1.14) is p

dW p = σij dεij = sij sij dλ = 2 k2 dλ in view of (1.11). Since plastic work must be positive, dλ is seen to be necessarily positive for plastic flow. Using the generalized form of Hooke’s law, the elastic strain increment may be written as dεije =

dsij 1 − 2v + dσkk δij , 2G 3E

(1.19)

where G is the shear modulus and v is Poisson’s ratio. When a stress increment satisfying sij dsij = 0 is prescribed, the elastic strain increment is known, but the plastic strain increment cannot be found from the flow rule alone. The flow rule associated with the Tresca criterion furnishes ratios of the components of the plastic strain increment depending on the particular side or corner of the deviatoric yield hexagon. If we consider the side AB of the hexagon, Fig. 1.6(a), the yield criterion is given by (1.16) and the normality rule furnishes p

p

dε1 = −dε3 > 0,

p

dε2 = 0.

1.2

Basic Laws of Plasticity

17

When the stress point is at the corner B of the yield hexagon, defining the equal biaxial state σ 1 = σ 2 , the plastic strain increment vector can lie between the normals for the sides meeting at B, giving p

dε1 > 0,

p

dε2 > 0,

p

p

p

dε3 = −(dε1 + dε2 ).

Similar relations hold for the other sides and corners of the yield hexagon. In each case, the rate of plastic work per unit volume is 2k times the magnitude of the numerically largest principal plastic strain rate. An interesting feature of Tresca’s associated flow rule is that it can be written down in the integrated form whenever the stress point remains on a side, remains at a corner, or moves from a side to a corner, but not when it moves from a corner back to a side. Let ψ denote the counterclockwise angle made by the plastic strain increment vector with the direction representing pure shear in the deviatoric plane. Evidently, ψ depends on the nature of the plastic potential, which is a closed curve similar to the yield locus. For an experimental verification of the flow rule, it is convenient to introduce the Lode parameter v (not to be confused with Poisson’s ratio), which is defined as p

ν=

p

√ = − 3 tan ψ,

p

2dε2 − dε3 − dε1 p dε3

p − dε1

p

p

p

dε1 > dε2 > dε3 .

(1.20)

For a regular yield function and plastic potential, ν = 0 when μ = 0, and ν = – 1 when μ = – 1. In the case of the von Mises yield criterion and the associated Prandtl–Reuss flow rule, μ = ν or θ = ψ for all plastic states. Tresca’s yield criterion and its associated flow rule, on the other hand, correspond to ν = 0 for 0 ≥ μ ≥ –1, and 0 ≥ ν ≥ – 1 for μ = –1. The (μ, ν) relations corresponding to the Tresca and von Mises theories are shown in Fig. 1.8. The experimental results of Hundy and Green (1954), included in the figure, clearly support the Prandtl–Reuss rule for the plastic flow of isotropic materials. p Suppose that a plastic strain increment dεij is associated with σ ij stress satisfying the yield criterion, while σij∗ is any other plastic state of stress, so that f (σij∗ ) = p f (σij ) = constant. The work done by σij∗ on the given plastic strain increment dεij has a stationary value for varying σ ij , when ∂ ∗ p σij dεij − f (σij∗ )dλ = 0, ∗ ∂σij where dλ is the Lagrange multiplier (Hill, 1950a). Carrying out the partial differentiation, we have p

dεij = p

∂ ∗ f (σij ) dλ. ∂σij∗

Since dεij is associated with σ ij according to the normality rule, the above equation is satisfied when σij∗ equals σ ij apart from a hydrostatic stress. The rate of plastic

18

1 Fundamental Principles

Fig. 1.8 Experimental verification of the (μ, v) relation due to Hundy and Green (+) and due to Lianis and Ford (o)

work is then a maximum in view of the convexity of the yield function. The maximum work theorem, which is due to von Mises (1928), may therefore be stated as

p σij − σij∗ dεij ≥ 0.

(1.21)

From the geometrical point of view, the result is evident from the fact that the vector representing the stress difference forms a chord of the yield surface and consequently makes an acute angle with the exterior normal defined at the actual stress point. The identity of the yield function and the plastic potential has a special significance in the mathematical theory of plasticity. Further details regarding the plastic stress–strain relations with and without strain-hardening of the material have been presented by Chen and Han (1988).

1.2.3 Limit Theorems In an elastic/plastic body subjected to a set of external forces, yielding begins in the most critically stressed element when the load attains a critical value. Under increasing load, a plastic zone continues to spread while the deformation is restricted to the elastic order of magnitude due to the constraint of the nonplastic material. When the plastic region expands to a sufficient extent, the constraint becomes locally ineffective and large plastic strains become possible. For a material whose rate of hardening

1.2

Basic Laws of Plasticity

19

is of the order of the yield stress, only a slight increase in load can produce an overall distortion of appreciable magnitude. If the material is nonhardening, and the change in geometry is disregarded, the load approaches an asymptotic value which is generally known as the yield point load. The basic theorems for the approximate estimation of this load have been obtained by Hill (1951) using a rigid/plastic model and by Drucker et al. (1952) using an elastic/plastic model. The elastic/plastic asymptotic load, frequently referred to as the collapse load in the context of structural analysis, is very closely attained while the elastic and plastic strains are still comparable in magnitudes. During the collapse, the deformation may therefore be assumed to occur under a constant load while changes in geometry are still negligible. Let σ˙ ij and vj denote the stress rate and particle velocity, respectively, under a distribution of boundary traction rate T˙ j at the incipient collapse. If the elastic and plastic components of the associated strain rate are denoted by ε˙ ije and p ε˙ ij , respectively, then by the rate form of the principle of virtual work we have

T˙ j vj dS =

σ˙ ij ε˙ ij dV =

p σ˙ ij ε˙ ije + ε˙ ij dV,

(1.22)

where the integrals are taken over the entire surface enclosing a volume V of the considered body. The integral on the left-hand side vanishes at the instant of colp lapse, since T˙ j = 0, while the scalar product σ˙ ij ε˙ ij vanishes in an ideally plastic material in view of (1.17). It follows therefore from (1.22) that σ˙ ij ε˙ ije = 0 at the incipient collapse, which indicates that σ˙ ij and ε˙ ije individually vanish, in view of the elastic stress–strain relation. A stress field is regarded as statically admissible if it satisfies the equilibrium equations and the stress boundary conditions without violating the yield criterion. Let σij and ε˙ ij denote the actual stress and strain rate in the considered body, and σij∗ any other statically admissible state of stress. If Tj and Tj∗ denote the surface tractions corresponding to σij and σij∗ , respectively, then by the virtual work principle, ∗ ∗ k − τ ∗ [u] dSD ≥ 0, σij − σij ε˙ ij dV + Tj − Tj vj dS = where τ ∗ ≤ k is the magnitude of the shear component of σ ij along a surface SD that actually involves a tangential velocity discontinuity of magnitude [u]. The inequality follows from (1.21) and the fact that the strain rate is purely plastic during the collapse. The assumed stress field may involve stress discontinuities across certain internal surfaces, which are limits of thin elastic regions of rapid but continuous variations of the stress. Since Tj = Tj∗ over the part SF of the surface where the traction is prescribed, the above inequality becomes

Tj vj dSv ≥

Tj∗ vj dSv ,

(1.23)

20

1 Fundamental Principles

where Sv denotes the part of the surface over which the velocity is prescribed. The above inequality (1.23) constitutes the lower bound theorem, which states that the rate of work done by the actual surface tractions on Sv is greater than or equal to that done by the surface tractions associated with any statically admissible stress field. The theorem provides a lower bound on the load itself at the incipient collapse when the prescribed velocity is uniform over this part of the boundary. A velocity field is considered as kinematically admissible if it satisfies the plastic p incompressibility condition ε˙ ii = 0 and the velocity boundary conditions. Let σ ij and vj denote the actual stress and velocity, respectively, in a deforming body, and v∗j any other kinematically admissible velocity producing a strain rate ε˙ ij∗ . Since the rate of deformation is purely plastic during the collapse, the associated work rate σij∗ εij∗ is uniquely defined by the flow rule. By the virtual work principle, we have

Tj v∗j dS =

σij ε˙ ij∗ dV +

∗ τ u∗ dSD ,

where τ is the magnitude of the actual shear stress and [u∗] is the magnitude of ∗ . Since the tangential discontinuity in the virtual velocity along a certain surface SD τ ≤ k, and σij ε˙ ij∗ ≤ σij ε˙ ij∗ in view of (1.21), the elastic strain rate being zero, the preceding expression furnishes

Tj vj dSv ≤

σij ε˙ ij dV +

k u∗ dSD −

Tj v∗j dSF ,

where use has been made of the fact that v∗j = vj on Sv . This result constitutes the upper bound theorem of limit analysis. When the last term of (1.24) is zero, the theorem states that the rate of work done by the actual surface tractions on Sv is less than or equal to the rate of dissipation of internal energy in any kinematically admissible velocity field. When the prescribed velocity is uniform on Su , the theorem provides an upper bound on the load itself at the instant of collapse. In a rigid/plastic body, no deformation can occur before the load reaches the yield point value. Over the range of load varying between the elastic limit and the yield point, the body remains entirely rigid, even though partially plastic. Under given surface tractions over a part SF of the boundary, and given velocities over the remainder Sv , the state of stress at the yield point is uniquely defined in the region where deformation is assumed to occur. On the other hand, the mode of deformation at the incipient collapse is not necessarily unique for an ideally plastic material. When positional changes are disregarded, the physically possible mode compatible with the rate of hardening can be singled out by specifying the traction rate on SF as an additional requirement (Hill, 1956). If geometry changes are duly taken into account, the deformation mode is found to be unique so long as the rate of workhardening exceeds a certain critical value (Hill, 1957). The nature of nonuniqueness

1.3

Strain-Hardening Plasticity

21

associated with an ideally plastic body has been illustrated with an example by Hodge et al. (1986).

1.3 Strain-Hardening Plasticity 1.3.1 Isotropic Hardening The most widely used hypothesis for strain hardening assumes the yield locus to increase in size during continued plastic deformation without change in shape. The yield locus is therefore uniquely defined by the final plastic state of stress regardless of the actual strain path (Hill, 1950a). According to this postulate, the material remains isotropic throughout the deformation, and the Bauschinger effect continues to be absent. The state of hardening at any stage is therefore specified by the current uniaxial yield stress denoted by σ¯ . When √ the yield criterion is that of von Mises, the current radius of the yield surface is 2/3 times σ¯ , and we have 1/2 3 σ¯ = sij sij 2 1/2

2 2 1 2 2 2 =√ + τyz + τzx . σx − σy + σy − σz + (σz − σx )2 + 6 τxy 2 (1.25) The quantity σ¯ is known as the equivalent stress or effective stress, which increases with increasing plastic strain. To complete the hardening rule, it is necessary to relate σ¯ to an appropriate measure of the plastic deformation. As a first hypothesis, it is natural to suppose that σ¯ is a function of the total plastic work per unit volume expended in a given element. The work-hardening hypothesis may be therefore stated mathematically as σ¯ = φ

p σij dεij ,

(1.26)

where the integral is taken along the strain path. Thus, no hardening is produced by the hydrostatic part of the stress which causes only an elastic change in volume. The function φ can be determined from the true stress–strain curve in uniaxial tension, where σ¯ is exactly equal to the applied tensile stress σ , and the incremental plastic work per unit volume is σ times the longitudinal plastic strain increment equal to dε – dσ /E. The argument of the function φ is simply the area under the curve for σ plotted against the quantity ln(l/l0 ) – σ /E, up to the ordinate σ , where l/l0 is the length ratio at any stage of the extension. A second hypothesis, frequently used in the literature, assumes σ¯ to be a function of a suitable measure of the total plastic strain during the deformation. In analogy ρ to the expression for σ¯ , we introduce a positive scalar parameter dε , known as the equivalent or effective plastic strain increment, defined as

22

1 Fundamental Principles

2 p p 1/2 dε = dεij dεij 3 2 2

2 1/2 2 p 2 p 2 p 2 p = + 2 dγyzp + 2 dγzxp . dεx + dεy + dεz + 2 dγxy 3 (1.27) p

p

The above definition implies that in the case of a uniaxial tension, dε is equal to the longitudinal plastic strain increment, provided the yield function is regular. The strain-hardening hypothesis may now be stated mathematically as σ¯ = F

dε

p

,

(1.28)

where the integral is taken along the strain path of a given element. Thus, the amount of hardening depends on the sum total of all the incremental plastic strains and not merely on the difference between the initial and the final shapes of the element. Both (1.26) and (1.28) imply that the longitudinal tensile stress is the same function of ln(l/l0 ) in uniaxial tension as the compressive stress is of ln(h0 /h) in simple compression, where h0 /h is the associated height ratio of the specimen. For a work-hardening Prandtl–Reuss material, the quantity dλ appearing in the flow rule (1.18) can be directly related to the equivalent stress and plastic strain p increment. Since sij = 0, it follows from (1.18), (1.27), and (1.25) that dε = (2σ¯ /3) dλ, and consequently, p

σij dεij = sij sij dλ =

2 2 p σ¯ dλ = σ¯ dε , 3

(1.29)

indicating that in this case the two hypotheses (1.26) and (1.28) are completely equivalent. Inserting the value of dλ from (1.29), the Prandtl–Reuss flow rule may be written as p

p

dεij =

3dε 3dσ¯ sij = sij, 2σ¯ 2H σ¯

(1.30)

p

where H = dσ¯ /dε , representing the current rate of work hardening of the material. Another important result for a Prandtl–Reuss material, which follows from (1.18) and (1.25), is p

σij dεij = sij dsij dλ =

2 p σ¯ dσ¯ dλ = dσ¯ dε , 3

(1.31)

where dσ¯ must be positive for plastic flow. The side of (1.31) is also right-hand p p equal to H(dε )2 , where H is a given function of dε . Adding the elastic and plastic strain increments given by (1.19) and (1.30), we obtain the complete Prandtl–Reuss strain–strain relation in the incremental form.

1.3

Strain-Hardening Plasticity

23

When yielding occurs according to the Tresca criterion (1.16), it is necessary to replace 2k by the current uniaxial yield stress σ¯ . According to Tresca’s associated flow rule, the increment of plastic work per unit volume is σ¯ times the magnitude of the numerically largest principal plastic strain increment denoted by dερ . If the work-hardening hypothesis is adopted, it follows that σ¯ = F

p dε ,

where the integral is taken along the strain path. When the stress point remains on a side, remains at a corner, or moves from a side to a corner, and the principal axes of stress and strain increments do not rotate with respect to the element, the argument of the function F is equal to the magnitude of the numerically greatest principal plastic strain in the element.

1.3.2 Plastic Flow with Hardening For continued plastic flow of a work-hardening material, the stress increment vector must lie outside the current yield locus, so that dσ¯ > 0. When the yield locus is regular, having a unique normal at each point, the plastic strain increment may be written as dεij = h−1 (nkl dσkl ) nij , nkl dσkl ≥ 0, p

(1.32)

where nij is the unit normal to the yield surface in a nine-dimensional stress space, and h (equal to 2H/3) is a parameter representing the rate of hardening. The equality in (1.32) represents neutral loading since it implies that the stress point remains on the same yield locus. When nkl dσ kl < 0, the element unloads and no incremental plastic strain is involved. The scalar products of (1.32) with dσ ij and dεij in turn furnish the result 2 p p p dσij dεij = h−1 dσij nij = hdεij dεij ≥ 0.

(1.33)

The equality holds not only for neutral loading but also for unloading from the plastic state. Since (1.21) holds whether the material work-hardens or not, the basic inequalities for a work-hardening element that is currently in a plastic state may be stated as

p p (1.34) σij − σij∗ dεij ≥ 0, dσij dεij ≥ 0. In both cases, the equality holds for neutral loading and unloading. If inequalities (1.34) are taken as the basic postulates for the plastic flow of work-hardening materials, the normality rule and the convexity of the yield surface can be easily deduced, as has been shown by Drucker (1951).

24

1 Fundamental Principles

Let P1 and P2 be two arbitrary stress points located on the two sides of a singular point P, as shown in Fig. 1.9(a). According to the first inequality of (1.34), each of the vectors P1 P and P2 P must make an acute angle with the plastic strain increment vector at P. This condition is evidently satisfied if the direction of the plastic strain increment lies between the normals PN1 and PN2 corresponding to the meeting surfaces. A further restriction is imposed by the second inequality of (1.34), which states that the plastic strain increment vector must make an acute angle with the p stress increment vector. In Fig. 1.9(b), the vector dεij may therefore lie anywhere between the normals PN1 and PN2 so long as the vector dσ ij lies within the angle T1 PT2 formed by the tangents at P. If the loading condition is such that dσ ij lies p outside this angle, the direction of dεij coincides with the normal that makes an acute angle with dσ ij . The flow rule at a singular point has been discussed by Koiter (1953), Bland (1957), and Naghdi (1960).

Fig. 1.9 Geometrical representation of the plastic stress–strain relation at a singular point of the yield surface

In the stress–strain relations considered so far, the strain increment dε ij must be interpreted as ε˙ ij dt, where ε˙ ij is the true strain rate and dt is an increment of time scale. The stress increment dσ ij is similarly given by a suitable measure of the stress rate, which must be defined in such a way that it vanishes in the event of a rigidbody rotation of the considered element. The most appropriate stress rate in the ◦ theory of plasticity is the Jaumann stress rate σ ij , which is related to the material rate of change σ˙ ij of the true stress by the equation ◦

σ = σ˙ ij − σik ωjk − σjk ωik ,

(1.35)

1.3

Strain-Hardening Plasticity

25

where ωij is the rate of rotation of the considered element. The tensors ε˙ ij and ωij are the symmetric and antisymmetric parts, respectively, of the velocity gradient tensor ∂vi /∂xj and are given by ε˙ ij =

1 2

∂vj ∂vi + ∂xj ∂xi

, ωij =

1 2

∂vj ∂vi − ∂xj ∂xi

.

(1.36)

Equation (1.35), which is originally due to Jaumann (1911), has been rederived by several investigators including Hill (1958) and Prager (1961a). The Jaumann ◦ stress rate σ i j is the rate of change of the true stress σ ij referred to a set of axes ◦ which participate in the instantaneous rotation of the element. Both σ ij; and σ i j have the same scalar product with any tensor whose principal axes coincide with those of σ ij . For an isotropic material, therefore, the material rate of change of the yield function f(σ ij ) is ∂f ◦ ∂f σ ij . σ˙ ij = f˙ = ∂σij ∂σij Since ∂f /∂σij is in the direction of the unit normal nij , the Jaumann stress rate satisfies the condition that the yield function has a stationary value during the neutral loading of a plastic element. No other definition of the objective stress rate, vanishing in the event of a rigid-body rotation of the element, satisfies this essential requirement. The constitutive equation for an elastic/plastic solid relates the strain rate to the stress rate, considered in the Jaumann sense. Combining the elastic and the plastic parts of the strain rate, the incremental constitutive equation for an isotropic workhardening material may be written as dεij =

3 1 v δij dσkk + nij nkl dσkl dσij − 2G 1+v 2H

(1.37)

for nkl dσ kl ≥ 0 in an element currently stressed to the yield point, the yield surface being considered as regular. The scalar product of (1.37) with nij furnishes the result

nij dσij =

2GH nij dεij . 3G + H

It follows that nij dσ ij ≥ 0 for nij dσ ij ≥ 0 when H > 0. Equation (1.37) therefore has the unique inverse

v 3G δij dεkk − nij nkl dεkl dσij = 2G dεij + 1 − 2v 3G + H

(1.38)

whenever nkl dσ kl ≥ 0. Equation (1.38) holds equally well for a nonhardening material (H = 0), but the magnitude of the last term of (1.37) becomes indeterminate

26

1 Fundamental Principles

when H = 0. When nkl dσ kl < 0, or nkl dεkl < 0, implying unloading of the element from the plastic state, the last terms of (1.37) and (1.38) must be omitted. Some computational aspects of the work-hardening Prandtl–Reuss theory of plasticity have been examined by Mukherjee and Liu (2003). An interesting strain space formulation of the constitutive relations for elastic/plastic solids has been developed by Casey and Naghdi (1981). A generalized constitutive theory for finite elastic/plastic deformation of solids has been developed by Lee (1969) and Mandel (1972) and further discussed by Lubiner (1990). A critical review of the subject of finite plasticity has been made by Naghdi (1990). A simplified stress–strain relation, proposed by Hencky (1924), assumes each component of the total plastic strain in any element to be proportional to the corresponding deviatoric stress. Although physically unrealistic, the Hencky theory does provide useful approximations when the loading is continuous and the stress path does not deviate appreciably from a radial path. For a work-hardening material, when the yield surface develops a corner at the loading point, the Hencky theory satisfies Drucker’s postulates (1.34) over a certain range of nonproportional loading paths, as has been shown by Budiansky (1959) and Kliushnikov (1959). When the material is rigid/plastic, and strain hardens according to the Ludwik power law (1.5), the Hencky theory coincides with the von Mises theory even for nonproportional loading during an infinitesimal deformation of the element, as has been shown by Ilyushin (1946) and Kachanov (1971).

1.3.3 Kinematic Hardening The simplest hardening rule that predicts the development of anisotropy and the Bauschinger effect, exhibited by real metals, is the kinematic hardening rule proposed by Prager (1956b) and Ishlinsky (1954). It is postulated that the hardening is produced by a pure translation of the yield surface in the stress space without any change in size or shape. If the initial yield surface is represented by f(σ ij ) = k2 in a nine-dimensional space, where k is a constant, the subsequent yield surfaces may be represented by the equation f σij − αij = k2 ,

(1.39)

where α ij is a tensor specifying the total translation of the center of the yield surface at a generic stage, as indicated in Fig. 1.10(b). To complete the hardening rule, it is further assumed that during an increment of plastic strain, the yield surface moves in the direction of the exterior normal to the yield surface at the considered stress point. Following Shield and Ziegler (1958), we therefore write p

dαij = cdεij ,

(1.40)

where c is a scalar parameter equal to two-thirds of the current slope of the uniaxial stress–plastic strain curve of the material. When c is a constant, (1.40) reduces to the

1.3

Strain-Hardening Plasticity

27

Fig. 1.10 Geometrical representation of the hardening rule considered in the stress space. (a) Isotropic hardening and (b) kinematic hardening

p

integrated form αij = cεij , the deformation being assumed small. In general, c may be regarded as a function of the equivalent plastic strain, the increment of which is defined by (1.27) in terms of the components of the plastic strain increment tensor. Since the material becomes anisotropic during the hardening process, the principal axes of stress and plastic strain increment do not coincide, unless the principal axes remain fixed in the element as it deforms. The loading condition df = 0, which ensures that the stress point remains on the yield surface, furnishes

∂f p p = 0 = dσij − cdεij dεij dσij − dαij ∂σij

(1.41)

in view of (1.39) and (1.40) and the normality rule for the plastic strain increment vector. If the initial yield surface is that of von Mises, f σij = 12 sij sj , and the yield criterion at any stage of the deformation becomes sij − αij sij − αij = 2 k2 ,

(1.42)

where k is the initial yield stress in pure shear. The associated flow rule furnishes the plastic strain increment as p

dεij =

∂f dλ = sij − αij dλ, ∂σij

(1.43)

where dλ is a positive scalar. Combining (1.43) with (1.41), and using (1.42), it is easily shown that dλ =

1 (skl − αkl ) dσkl . 2ck2

(1.44)

28

1 Fundamental Principles

The plastic strain increment for the kinematic hardening is completely defined by (1.43) and (1.44). A modified form of Prager’s hardening rule has been proposed by Ziegler (1959), while other types of kinematic hardening have been examined by Baltov and Sawczuk (1965), Phillips and Weng (1975), and Jiang (1993). When the deformation is large, the stress increment entering into the constitutive equation must be carefully defined, and this question has been examined by Lee et al. (1983) and Naghdi (1990).

1.3.4 Combined or Mixed Hardening The concept of kinematic hardening has been extended to include an expansion of the yield surface along with a translation by Hodge (1957), Kadashevich and Novozhilov (1959), and Mröz et al. (1976). Equation (1.39) is then modified by replacing its right-hand side with a function of the total equivalent plastic strain, whose increment is given by (1.27). Assuming the von Mises yield criterion for the initial state, the combined hardening rule may be stated as 2 sij − αij sij − αij = σ¯ 2 , 3

(1.45)

where dα ij is still given by (1.40). The right-hand side of (1.45) is the square of the current radius of the displaced yield cylinder. The associated plastic strain increment p p dεij is given by (1.43) with dλ = 3dε /2σ¯ , as may be seen by substituting (1.43) into (1.27) and using (1.45). The plastic strain increment therefore becomes p

3dε dσ¯ p = sij − αij , dεij = sij − αij 2σ¯ hσ¯

(1.46)

where h is a measure of the isotropic part of the rate of hardening, the anisotropic part being represented by the parameter c. The differentiation of the yield criterion (1.45) gives 2 p σ¯ dσ¯ = (skl − αkl ) dσkl − cdεkl 3 in view of (1.40) and the fact that dskk = 0. Substituting from (1.46) and using (1.45), we obtain the relation

c 3 1+ dσ¯ = (skl − αkl ) dσkl . h 2σ¯

(1.47)

The flow rule corresponding to the combined hardening process is completely defined by (1.46) and (1.47), the loading condition being specified by dσ¯ > 0 for an element that is currently plastic. Following the early experimental work due to Naghdi et al. (1958), the distortion of the yield surface under continued plastic deformation has been subsequently examined by several investigators.

1.3

Strain-Hardening Plasticity

29

The resultant strain increment in any element deforming under the combined hardening rule from an initially isotropic state may be written down on the assumption that the material continues to remain elastically isotropic. Then the elastic strain increment, which is given by the generalized Hooke’s law, may be written as d εije

1 1 1 − 2v = dsij + dσkk δij , 2G 3 1+v

(1.48)

where G is the shear modulus and v is Poisson’s ratio. Taking the scalar product of the above equation with the tensor sij – α ij and using (1.47), we have sij − αij d εije =

c+h p σ¯ d ε . 2G

On the other hand, the scalar product of (1.46) with the same tensor sij – αij gives p p sij − αij d εij = σ¯ d ε in view of (1.45). The last two equations are added together to obtain the relation H p sij − αij d εij = 1 + σ¯ d ε , 3G

(1.49)

where H denotes the plastic modulus corresponding to the current state of stress and is defined as H=

3 (c + h) . 2

Thus, H is the slope of the uniaxial stress–plastic strain curve corresponding to a longitudinal plastic strain equal to the total equivalent plastic strain suffered by the given element. Further results related to the mixed hardening rule have been given by Mröz et al. (1976), Rees (1981), and Skrzypek and Hetnarski (1993). A micromechanical model for the development of texture with plastic deformation in polycrystalline metals has been considered by Dafalias (1993). Consider the special case of proportional loading in which the stress path is a radial line in the deviatoric plane. Let the state of stress at the initial yielding be denoted by s0ij , satisfying the yield criterion s0ij s0ij = 2Y 2 /3. Since the plastic strain increment tensor in this case may be written as

p d εij

3d ε =± 2Y

p

s0ij ,

where the upper sign corresponds to continued loading and the lower sign to any subsequent reversed loading in the plastic range, the deviatoric stress increment is

30

1 Fundamental Principles

dsij =

p cd εij

±

dσ¯ Y

s0ij

3H =± 2Y

s0ij d ε

p

(1.50)

by the simple geometry of the loading path and the assumption of simultaneous translation and expansion of the yield surface in the stress space. During the unloading of an element from the plastic state, followed by a reversal of the load, the components of the deviatoric stress steadily decrease in magnitude. Plastic yielding would occur under the reversed loading when the vector representing the deviatoric stress changes by a magnitude equal to the current diameter of the yield surface.

1.4 Cyclic Loading of Structures 1.4.1 Cyclic Stress–Strain Curves The investigations of low-cycle fatigue in mechanical and structural components have resulted in the development of considerable interest in the study of plastic behavior of materials under cyclic loading. In uniaxial states of stress involving symmetric cycles of stress or strain, an annealed material usually undergoes cyclic hardening, and the hysteresis loop approaches a stable limit as shown in Fig. 1.11(a). If the material is sufficiently cold-worked in the initial state, cyclic softening would occur and the hysteresis loop would again stabilize to a limiting state. Based on a family of stable hysteresis loops, obtained by the cyclic loading of a material with different constant values of the strain amplitude, we can derive a cyclic strainhardening curve, such as that shown in Fig. 1.11(b), which may be compared with the standard strain-hardening curve for the same material (Landgraf, 1970). If the cyclic loading is continued in the plastic range, the stable hysteresis loops are repeated and failure eventually occurs due to low-cycle fatigue. Under certain stress cycles with materials exhibiting cyclic softening, the plastic strain may continue to grow in a unidirectional sense, causing failure by the phenomenon of ratcheting. Let us suppose that a specimen that is first loaded in tension to a stress equal to σ is subsequently unloaded from the plastic state and then reloaded in compression. It follows from above that yielding would again occur when the magnitude of the applied compressive stress becomes 2σ¯ − σ , where σ¯ depends on the magnitude of the previous plastic strain. If we assume the relations h=

2 βH, 3

c=

2 (1 − β) H, 3

where β is a constant less than unity, then dσ¯ = βdσ , which gives σ¯ − Y = β (σ − Y). The initial yield stress σ ’ in compression during the reversed loading is therefore given by σ − Y = (2β − 1) (σ − Y) .

1.4

Cyclic Loading of Structures

31

Fig. 1.11 Cyclic loading curves in the plastic: (a) constant strain cycles and (b) cyclic stress–strain curves

It follows that σ ≷ Y according as β ≷ 12 , irrespective of the rate of hardening. If the specimen is subjected to a complete cycle of loading and unloading with the longitudinal plastic strain varying between the limits – ε∗ to ε∗ , then the magnitude of the final stress under a constant plastic modulus H exceeds σ by the amount 4βHε∗ , which vanishes only when the hardening is purely kinematic. The shear stress–strain curve of a material under cyclic loading can be derived from the experimental torque–twist curve for a solid cylindrical bar subjected to cyclic torsion in the plastic range. Let T denote the applied torque at any stage of the loading, and let θ be the corresponding angle of twist per unit length. Since the engineering shear strain at any radius r is γ = rθ , we have T = 2π 0

a

τ r2 dr =

2π θ3

aθ

τ γ 2 dγ ,

0

where a denotes the external radius of the bar, and τ = τ (γ ) is the shear stress at any radius. Multiplying both sides of the above equation by θ 3 , and differentiating it with respect to θ , it is easy to show that θ

dT + 3T = 2π a3 τ (aθ ) , dθ

(1.51)

where τ (aθ ) is the shear stress at the boundary r = a, corresponding to a shear strain equal to aθ . The preceding relation provides a means of obtaining the (τ , γ )-curve from an experimental (T, θ )-curve during the loading process (Nadai, 1950). Consider now the unloading and reversed loading of a bar that has been previously twisted in the plastic range by a torque T0 producing a specific angle of twist

32

1 Fundamental Principles

θ 0. For a given value of θ 0 , the shear stress acting at any radius r, when the specific angle of twist has decreased to θ may be expressed as τ (rθ ) = τ0 (rθ0 ) + f [r (θ − θ0 )] ,

(1.52)

where τ 0 denotes the local shear stress at the moment of unloading. The function f represents the change in shear stress caused by the unloading or reversed loading. The torque acting at any stage is T = T0 + 2π

a

r2 f [r (θ − θ0 )]dr.

0

Setting ξ = r(θ – θ 0 ), which gives dξ = (θ – θ 0 ) dr, the preceding relation can be expressed as T − T0 =

2π (θ − θ0 )3

a(θ−θ0 )

ξ f (ξ ) dξ .

0

Multiplying both sides of this equation by (θ – θ 0 )3 and differentiating the resulting expression partially with respect to θ , we have (θ − θ0 )

∂T + 3 (T − T0 ) = 2π a3 f [a (θ − θ0 )] , ∂θ

since T0 is a function of θ 0 only. Substituting for f[a(θ –θ 0 )] from the above equation into (1.52), the shear stress at r = a is finally obtained as τ (aθ ) = τ0 (aθ0 ) +

1 ∂T . + − θ 3 − T ) (θ ) (T 0 0 ∂θ 2π a3

(1.53a)

Since ∂T/∂θ is positive, both terms in the curly brackets of (1.53a) are negative during unloading and reversed loading. The residual shear stress at r = a at the end of the unloading process corresponds to T = 0, the corresponding residual shear strain being found directly from the given torque–twist curve. Suppose that the reversed loading in torsion is terminated when T = T1 and θ = θ 1 , the corresponding shear stress at r = a being denoted by τ 1 (aθ 1 ). If the bar is again unloaded, and then reloaded in the same sense as that in the original loading, an analysis similar to the above gives the shear stress at the external radius in the form 1 ∂T − θ τ (aθ ) = τ1 (aθ1 ) + + − T . (1.53b) (θ (T ) ) 1 1 ∂θ 2π a3 Equations (1.51) and (1.53) completely define the cyclic shear stress–strain curve based on an experimentally determined cyclic torque–twist curve Wu et al. (1996). The derivative ∂T/∂θ is piecewise continuous, involving a jump at each reversal of the applied torque, the correspondence between the various points in the two

1.4

Cyclic Loading of Structures

33

Fig. 1.12 The cyclic shear stress–strain curve derived from the cyclic torque–twist curve for a solid cylindrical bar

cyclic curves being indicated in Fig. 1.12. It may be noted that (1.53) can be directly obtained from (1.51) if we simply replace T, θ , and τ in this equation by the appropriate differences of the physical quantities.

1.4.2 A Bounding Surface Theory The anisotropic hardening rule described in the preceding section cannot be applied without modifications to predict the plastic behavior of materials under cyclic loading with relatively complex states of stress (Dafalias and Popov, 1975; Lamba and Sidebottom, 1978). Following an earlier work by Mröz (1967a), various types of theoretical model involving two separate surfaces in the stress space have been widely discussed in the literature, notably by Tseng and Lee (1983), McDowell (1985), Ohno and Kachi (1986), and Hong and Liou (1993), among others. The two-surface model assumes the existence of a bounding surface that encloses the current yield surface throughout the loading history. Both the yield surface and the bounding surface can expand and translate, and possibly also deform in the stress space, as the loading and unloading are continued in the plastic range. The general features of the two-surface theory are illustrated in Fig. 1.13(a), where the yield surface or the loading surface S and the bounding surface S’ are represented by circles with centers √ C and C’, respectively. The current radii of the surfaces S and S’ are denoted by 2/3 times σ¯ and τ¯ , respectively, while the position vectors of the centers C and C’ are denoted by α ij and α’ij , respectively. The equation for the loading surface is given by (1.35), while that of the bounding surface is expressed as

sij − αij

2 sij − αij = τ¯ 2 . 3

34

1 Fundamental Principles

Fig. 1.13 Yield surface S and bounding surface S’ in the deviatoric stress space. (a) Both surfaces are in translation and (b) only the yield surface is in translation

For each point P on the yield surface S, there is a corresponding image point P’ on the bounding surface, the distance between the two points P and P’, which are defined by the vectors sij and s’ij , respectively, is an important parameter that enters into the theoretical framework. There are several possible ways of relating the two stress points, the one suggested by Mröz being sij − αij = (τ¯ /σ¯ ) sij − αij . The variation of α ij and α’ij with continued loading must be defined by appropriate hardening rules, for which there are several possibilities. Since the two-surface theory is not without its limitations in predicting the material response under cyclic loading (Jiang, 1993, 1994), we shall describe in what follows the simplest theoretical model that is consistent with the basic purpose of the theory. It is assumed, for simplicity, that the bounding surface at each stage is a circular cylinder concentric with the initial yield surface, which is taken as the von Mises cylinder. The radius of the bounding surface S’ therefore increases with the amount of plastic deformation following the isotropic hardening rule. The yield surface S, on the other hand, undergoes simultaneous expansion and rotation according to the mixed hardening process, Fig. 1.13(b). At a generic stage of the loading, the deviatoric stresses sij and s’ij associated with the surfaces S and S’, respectively, satisfy the relations 2 sij − αij sij − αij = σ¯ 2 , 3

sij sij =

2 2 τ¯ , 3

(1.54)

where α ij is the back stress defining the center of the current yield surface. So long as the two surfaces are separated from one another, the translation of the yield surface

1.4

Cyclic Loading of Structures

35

is assumed to be governed by Prager’s kinematic hardening rule, which requires the yield surface to move in the direction of the plastic strain increment. Thus p

dαij = cd εij , αij αij ≤

2 (τ¯ − σ¯ )2 , 3

(1.55)

where c represents the kinematic part of the rate of hardening of the material. The inequality in (1.55) ensures that the yield surface is not in contact with the bounding p surface. The plastic strain increment d εij is given by the flow rule (1.46), which applies to the mixed hardening process, the equivalent plastic strain increment d ερ corresponding to a given strain increment dεij being found from (1.49). The plastic modulus H at any stage is given by the relation H=

3 (h + c) , 2

where h represents the isotropic part of the rate of hardening and is two-thirds of the current slope of the curve obtained by plotting σ¯ against ε¯ p . In the case of cyclic loading, the parameter c depends not only on the accumulated plastic strain ε¯ p but also on the distance between the loading point P and its image point P’ on the bounding surface. For simplicity, the image point is considered here as the point of intersection of the outward normal to the yield surface at P with the bounding surface, Fig. 1.14(a). If ψ denotes the included angle between the vectors representing the deviatoric stress sjj and the reduced stress sij – α jj , then by the geometry of the triangle OPP’, we have s¯ 2 + δ 2 + 2¯sδ cos ψ = τ¯ 2 ,

Fig. 1.14 Simplified two-surface model for cyclic plasticity. (a) Separate loading and bounding surfaces and (b) the two surfaces are in contact

36

1 Fundamental Principles

√ where s¯ and δ are 3/2 times the lengths of the vectors OP and PP’, respectively. The above equation immediately gives δ = −¯s cos ψ +

τ¯ 2 − s¯ 2 sin2 ψ.

(1.56)

The quantities ψ and s¯ appearing in (1.56) can be determined from the relations 1/2 3sij sij − αij 3 , s¯ = sij sij . cos ψ = 2sσ 2

(1.57)

The plastic modulus H evidently depends on both ε¯ p and δ. For practical purposes, H can be estimated by using the empirical relation H=

dτ¯ dε

p

γ δ exp β , τ¯

(1.58)

where β and γ are dimensionless constants to be determined from experimental data p on uniaxial stress cycles. Since δ/τ¯ and dτ¯ /d ε monotonically decrease during the process, (1.58) implies a fairly rapid decrease in the value of H. When δ = 0, the plastic modulus becomes identical to the slope of the curve for τ¯ against ε¯ p . The quantities σ¯ and τ¯ are functions of ε¯ p alone and can be expressed by the empirical equations σ¯ = σ0 1 − m exp (− n¯εp ) , τ¯ = τ0 1 − m exp (− n ε¯ p ) ,

(1.59)

where σ 0 and τ 0 are saturation stresses, while m, n, m’, and n’ are appropriate dimensionless constants. The hardening rate parameters h and c at any given stage follow from (1.58) and (1.59). Suppose that all the physical quantities have been found for a generic stage of the cyclic loading. During an additional strain increment dεij satisfying the inequalp ity (sij – αij ) dεij > 0, the equivalent plastic strain increment d ε is computed from p (1.49), and the associated plastic strain increment tensor d εij then follows from p (1.46). Since the elastic strain increment d εije is equal to dεij – d εij , the deviatoric stress increment dsij is obtained from (1.48), where the second term in the curly brackets is equal to 2G dεkk δij . The incremental displacement dαij of the yield surp face is determined from (1.55) and the fact that c = 23 (H − dσ¯ /d ε ). The new stress tensors sij and α ij , together with the updated values of σ¯ and τ¯ obtained from (1.59), enable us to compute the new values of δ/τ¯ and H using (1.58) and (1.59), thereby completing the solution to the incremental problem.

1.4

Cyclic Loading of Structures

37

1.4.3 The Two Surfaces in Contact When the yield surface comes in contact with the bounding surface, the position of the stress point P will generally require the two surfaces to remain in contact during the subsequent loading process. Consider first the situation where P coincides with the point of contact T between the two surfaces, and the coincidence is then maintained following the stress path. Since the back stress αij in this case is in the direction of the deviatoric stress sij , it follows from simple geometry that σ¯ sij , αij = 1 − τ¯

sij sij =

2 2 τ¯ . 3

(1.60)

The yield surface is now assumed to expand at the same rate as the bounding surface so that dσ¯ = dτ¯ during this loading phase. The plastic modulus H is therefore continuous when the contact begins, in view of (1.58), and the incremental translation of the center of the yield surface is given by the relation σ¯ dτ¯ sij . dsij − dαij = 1 − τ¯ τ¯

(1.61)

This expression implies that the motion of the yield locus consists of a radial expansion of the circle together with a rigid body sliding along the bounding surface. Suppose now that the stress point P on the yield surface lies between the points T and D, where CD is parallel to the common tangent to the two surfaces in contact, Fig. 1.14(b). This condition, together with the condition of contact can be stated mathematically as αij αij =

2 (τ¯ − σ¯ )2 , 3

2 2 2 τ¯ ≥ sij sij ≥ (τ¯ − σ¯ )2 + σ¯ 2 . 3 3

Assuming dσ¯ and dτ¯ to be equal to one another as before, thus allowing a slight discontinuity in the plastic modulus as the contact is established, the translation of the center of the yield surface may be written as dαij = λdsij − sij dμ,

(1.62)

where λ and dμ are scalar parameters. The modified hardening rule expressed by (1.62) is consistent with the experimental observation of Phillips and Lee (1979). It implies that the radial expansion of the yield surface is accompanied by its sliding and rolling over the bounding surface. The unknown parameters in (1.62) can be determined from the condition that dα ij is orthogonal to α ij , and the fact that the stress point remains on the yield surface. Thus αij dαij = 0,

2 sij − αij dsij − dαij = σ¯ dσ . 3

(1.63)

38

1 Fundamental Principles

Taking the scalar product of (1.62) with α ij , and using the first relation of (1.63), we get dμ/λ = αij dsij /(αkl skl ) .

(1.64)

Using the contact condition, the second condition of (1.63) is easily reduced to 2 sij dαij = sij − αij dsij − σ¯ dτ¯ . 3 The scalar product of (1.62) with sjj and the substitution from above lead to the expression p sij − αij dsij − 23 H σ¯ d ε λ= , dskk − (dμ/λ) skl skl p

where H is the plastic modulus equal to dτ¯ /d ε during this phase. Equations (1.64) p and (1.65) define the hardening parameters λ and dμ when d ε is known for a given dsij . In the special case when (1.60) are applicable, we get λ = 1 − σ¯ /τ¯ , and dμ = λ (dτ¯ /τ¯ ), and the hardening rule then reduces to (1.61). Assuming the strain increment dεij to be prescribed, the corresponding value p of d ε can be approximately estimated from (1.49), which is not strictly valid for the situation considered here. The associated deviatoric stress increment dsij is then obtained as before, and the parameters λ and dμ are determined from (1.64) and p (1.65). An improved value of d ε subsequently follows from the relation 1 H p 1+ sij dαij , σ¯ d ε = sij − αij d εij − 3G 2G

(1.66)

where dα ij is given by (1.61). Equation (1.66) is obtained by taking the scalar products of (1.46) and (1.48) with the tensor sjj – αij , and using (1.63). The computation may be repeated until the difference between successive values of the effective plastic strain increment becomes negligible. The quantity τ¯ is still given by (1.59), but the value of σ¯ over this range is obtained from the relation σ¯ = τ¯ + (σ¯ ∗ − τ¯ ∗ ), where the asterisk refers to the instant when the two surfaces first come in contact with one another during the loading. The theoretical treatment of cyclic plasticity based on a single-surface model has been examined by Eisenberg (1976) and Drucker and Palgen (1981). The constitutive modeling of large strain cyclic plasticity has been discussed by Chaboche (1986), Lemaitre and Chaboche (1989), and Yoshida and Uemori (2003), among other investigators. The plastic response of materials under cyclic loading has also been discussed in recent years on the basis of an interesting theory of plasticity that does not require the specification of a yield surface. The theory, which is essentially due to Valanis (1975, 1980), who called it the endochronic theory of plasticity, is based on the concept of an intrinsic time that depends on the deformation history, the relationship

1.5

Uniqueness and Stability

39

between the two quantities being regarded as a material property. The theory also introduces an intrinsic time scale which is a function of the intrinsic time, the rate of change of the various physical quantities being considered with respect to this time scale. For further details of the endochronic theory of plasticity, together with some physical applications, the reader is referred to Wu et al. (1995).

1.5 Uniqueness and Stability 1.5.1 Fundamental Relations Consider the quasi-static deformation of a conventional elastic/plastic body whose plastic potential is identical to the yield function, which is supposed to be regular and convex. The current shape of the body and the internal distribution of stress are assumed to be known. We propose to establish the condition under which the boundary value problem has a unique solution and examine the related problem of stability. When positional changes are taken into account, it is convenient to formulate the boundary condition in terms of the rate of change of the nominal traction, which is based on the configuration at the instant under consideration. When body forces are absent, the equilibrium equation and the stress boundary condition for the rate problem may be written in terms of the nominal stress rate ˙tij and the nominal traction rate F˙ j as ∂ ˙tij = 0, ∂xi

F˙ j = li ˙tij ,

(1.67)

where xi denotes the current position of a typical particle and li the unit exterior normal to a typical surface element. The relationship between the unsymmetric nominal stress rate ˙tij and the symmetric true stress rate σ˙ ij , referred to a fixed set of rectangular axes (Chakrabarty, 2006), may be written as ˙tij = σ˙ ij − σjk

∂vi ∂vk + σij , ∂xk ∂xk

(1.68)

where vi denotes the velocity of the particle. The constitutive equations, on the other ◦ hand, must involve the Jaumann stress rate σ ij , which is the material rate of change of the true stress σij with respect to a set of rotating axes, and is given by (1.35). The elimination of σ˙ ij between (1.35) and (1.68) gives ˙tij = σ˙ ij + σij ε˙ kk + σik ωjk − σjk ε˙ ik

(1.69)

This equation relates the nominal stress rate ˙tij directly to the Jaumann stress rate σ˙ ij . Using the interchangeability of dummy suffixes, the scalar product of (1.69) with ∂vj /∂xi can be expressed as

40

1 Fundamental Principles

∂v ˙tij j = ∂xi

◦

[−4pt] [−6pt]σ ij

+ ε˙ kk σij

∂vi ∂vk ε˙ ij − σij 2˙εik ωjk + ∂xk ∂xj

(1.70)

in view of the symmetry of the tensors σ ij and ε˙ ij . This relation is derived here for later use in the analysis for uniqueness and stability. The constitutive law for the conventional elastic/plastic solid is such that the strain rate is related to the stress rate by two separate linear equations defining the loading and unloading responses. For an isotropic solid, when an element is currently plastic, the constitutive equation for loading may be written down by using the rate form of (1.38). The stress rate is therefore given by v 3G ◦ σ ij = 2G ε˙ ij + ε˙ kk δij − ε˙ kl nkl nkl , ε˙ kl nkl ≥ 0, 1 − 2v 3G + H v ◦ σ ij = 2G ε˙ ij + ε˙ kk δij , ε˙ kl nkl ≥ 0. 1 − 2v

(1.71)

Consider now a fictitious solid whose constitutive law is given by the first equation of (1.71), whenever an element is currently plastic, regardless of the sign of ε˙ kl nkl . Such a solid may be regarded as a linearized elastic/plastic solid, in which ◦ the stress rate corresponding to a strain rate ε˙ ij is denoted by τ ij . The scalar product of (1.71) with ε˙ ij then furnishes ◦ ◦ σ ij ≥ σ ij ε˙ ij = 2G ε˙ ij ε˙ ij +

2 v 3G 2 , − ε˙ kk ε˙ ij nij 1 − 2v 3G + H

(1.72)

where the equality holds only in the loading part of the current plastic region. In contrast to the bilinear elastic/plastic solid, the linearized solid has identical loading and unloading responses for any plastic element. ◦ ◦ Let (σ ij , ε˙ ij ) and (σ ∗ij , ε˙ ij∗ ) denote two distinct combinations of stress and strain rates in an element of the actual elastic/plastic solid corresponding to a given state ◦ ◦ of stress. The stress rates for the linearized solid in the two states are τ ij and τ ∗ij , respectively. If the element is currently plastic, and the two strain rates do not both call for instantaneous unloading, the scalar product of ε˙ ij∗ with the appropriate equation of (1.71) shows that ◦ ◦ σ ij ε˙ ij∗ ≤ τ ij ε˙ ij∗ = 2G ε˙ ij ε˙ ij∗ +

v 3G ∗ ∗ − nij nkl . ε˙ ij ε˙ kk ε˙ ij ε˙ kl 1 − 2v 3G + H

(1.73)

The equality holds when εij∗ calls for further loading, whatever the nature of ε˙ ij∗ . ◦ ◦ The inequalities satisfied by σ ∗ij ε˙ ij∗ and σ ∗ij ε˙ ij∗ are similar to (1.72) and (1.73), respectively. Consequently,

σ ij − σ ∗ij ◦

◦

◦ ◦ ε˙ ij − ε˙ ij∗ ≥ τ ij − σ ∗ij ε˙ ij − ε˙ ij∗ .

(1.74)

1.5

Uniqueness and Stability

41

If the difference between the unstarred and the starred quantities is denoted by the prefix , then it follows from above that ◦

◦

σij ε˙ ij ≥ τij ε˙ ij = 2G ˙εij ˆεij +

v (˙εkk )2 1 − 2v

(1.75)

with equality holding for instantaneous loading produced by both ε˙ ij and ε˙ ij∗ . When ◦ both the states call for instantaneous unloading, the relationship between σ ij and ◦ ˙εij is given by the second equation of (1.71), while that between τ ij and ˙εij is ◦ ◦ given by the first equation of (1.71), leading to the inequality σ ij ˙εij ≥ τ ij ˙εij . For an element that is currently elastic, there is the immediate identity

v 2 σ ij ˙εij = τ ij ˙εij = 2G ˙εij ˙εij + (˙εkk ) . 1 − 2v ◦

◦

(1.76)

◦

◦

It follows, therefore, that the inequality σ ij ˙εij ≥ τ ij ˙εij holds throughout the elastic/plastic body and under all possible conditions of loading and unloading. This result will now be used for the derivation of the uniqueness criterion.

1.5.2 Uniqueness Criterion Consider the typical boundary value problem in which the nominal traction rate F˙ j is specified on a part SF of the current surface of the body, and the velocity vj on the remainder Sv . Suppose that there could be two distinct solutions to the problem, involving the field equations (1.67), (1.69), and (1.71), together with the prescribed boundary conditions. If the difference between the two possible solutions is denoted by the prefix , then in the absence of body forces, we have ∂ ˙tij = 0, ∂xi

F˙ j = li ˙tij ,

in view of (1.67). The application of Green’s theorem to integrals involving surface S and volume V gives

F˙ j vj dS =

li ˙tij vj dS =

˙tij

∂ ∂xi

∂ ˙tij vj dV. ∂xi

The integral on the left-hand side vanishes identically, since F˙ j = 0 on SF and vj = 0 on Sv by virtue of the given boundary conditions. The condition for having two possible solutions therefore becomes

˙tij

∂ vj dV = 0. ∂xi

42

1 Fundamental Principles

The left-hand side of the above equation must be positive for uniqueness (Hill, 1958). Using (1.70), expressed in terms of the difference of the two possible states, a sufficient condition for uniqueness may be written as ∂ ∂ ◦ σ ij ˙εij + σij ˙εkk ˙εij − 2˙εik ω˙ jk − (vi ) (vk ) dV > 0. ∂xk ∂xj (1.77) for the difference vj of every possible pair of continuous velocity fields taking prescribed values on Sv . For applications to physical problems, it is preferable to replace the above condition by a slightly over-sufficient criterion, using the fact that ◦ ◦ σ ij ˙εij ≥ τ ij ˙εij throughout the body. Uniqueness is therefore assured when ∂ ∂ ◦ τ ij ˙εij − σij 2˙εik ωjk + (vi ) (vk ) dV > 0 ∂xk ∂xj

(1.78)

for all continuous difference fields vj vanishing on Sv . The term in εkk has been neglected, since the contribution made by it is small compared to that arising from ◦ a similar term in the quantity τ ij ˙εij , which is given by (1.75) and (1.76) in the plastic and elastic regions, respectively. It follows from (1.77) and (1.78) that the condition for uniqueness for the linearized elastic/plastic solid also ensures uniqueness for the actual elastic/plastic solid (Hill, 1959). If the constraints are rigid, so that vj = 0 on Sv , every difference field vj is a member of the admissible field vj , and the uniqueness criterion reduces to

∂vi ∂vk τ ij ε˙ ij − σij 2 ε˙ ik ωjk + dV > 0 ∂xk ∂xj ◦

(1.79)

for all continuous differentiable fields vj vanishing at the constraints. Splitting the tensors ∂vi /∂xk and ∂vk /∂xj into their symmetric and antisymmetric parts, it is easily shown that ∂vi ∂vk = σij ε˙ ik ε˙ jk − ωik ωjk . σij ∂xk ∂xj The remaining two triple products cancel one another by the symmetry and antisymmetry properties of their factors. In view of the above identity, the uniqueness criterion (1.79) becomes ◦ τ ij ε˙ ij − σij 2 ε˙ ik ωjk + ε˙ ik ε˙ jk − ωik ωjk dV > 0. (1.80) The leading term in square brackets is given by (1.72) for the current plastic region and by the same equation with the last term omitted for the elastic region. In the treatment of problems involving curvilinear coordinates, it is only necessary to regard the components of the tensors appearing in (1.80) as representing the curvilinear components.

1.5

Uniqueness and Stability

43

In a number of important physical problems, a part Sf of the boundary is submitted to a uniform fluid pressure p, which is made to vary in a prescribed manner. In this case, the change in the load vector on a given surface element, whose future orientation is not known in advance, cannot be specified. It can be shown that the nominal traction rate for the pressure-type loading is ∂v ∂v ˙Fj = p˙ lj + p lk k − lj k , ∂xj ∂xk where p˙ is the instantaneous rate of change of the applied fluid pressure. When the boundary value problem has two distinct solutions under a given pressure rate p˙ so that ˙p = 0, then the preceding relation gives ∂ ∂ ˙ Fj = p lk (vk ) − lj (vk ) on Sf . ∂xj ∂xk

(1.81)

It is assumed that the remaining surface area of the body is partly under a prescribed nominal traction rate F˙ j and partly under a prescribed velocity vj . Since F˙ j = 0 on Sf , the bifurcation condition (1.77) must be modified by replacing the right-hand side of this equation with the surface integral ∂ ∂ (vk ) − lj (vk ) vj dSf . p lk ∂xj ∂xk The uniqueness criterion (1.78) is therefore modified by subtracting the same quantity from the left-hand side of the inequality. In particular, (1.80) is modified to (Chakrabarty, 1969b).

◦ τ ij ε˙ ij − σij 2˙εik ωjk + εˆ ik ε¨ jk − ωik ωjk dV − p

lk ε˙ kj + ωkj − lj ε˙ kk vj dSf > 0. (1.82)

If the functional in (1.82) vanishes for some nonzero field vj , bifurcation in the linearized solid may occur for any value of the traction rate on Sf and pressure rate on Sf . In the actual elastic/plastic solid, on the other hand, bifurcation will occur only for those traction rates which produce no unloading of the current plastic region. When the material is rigid/plastic, the admissible velocity field is incompressible ◦ (˙εkk = 0), and the scalar product τ ij ε˙ ij becomes equal to 23 H ε˙ ij ε˙ ij , while the triple produce σij ε˙ ik ωjk vanishes due to the coaxiality of the principal axes of stress and strain rate, leading to a considerable simplification of the problem. The condition for uniqueness in rigid/plastic solids has been discussed by Hill (1957), Chakrabarty (1969a), and Miles (1969).

1.5.3 Stability Criterion Consider an elastic/plastic body which is rigidly constrained over a part Sv of its external surface, while constant nominal tractions are maintained over the remain-

44

1 Fundamental Principles

der SF . The deformation of the body will be stable if the internal energy dissipated in any geometrically possible small displacement from the position of equilibrium exceeds the work done by the external forces. Since these two quantities are equal to one another when evaluated to the first order, it is necessary to consider secondorder quantities for the investigation of stability. The stress and velocity distributions throughout the body are supposed to be given in the current state, which is taken as the initial reference state for the stability analysis. At any instant during a small virtual displacement of a typical particle, its velocity is denoted by ωj and the associated true stress by sij . Then the instantaneous rate of dissipation of internal energy per unit mass of material in the neighborhood of the particle is (sij /ρ)(∂ωi /∂zj ), where zj is the instantaneous position and ρ the current density. The rate of change of this quantity following the particle is sij ∂wi ρ˙ ∂wi ∂wk ∂ ∂ ∂wi ∂w ˙i 1 + sij − + wk s˙ij − sij = , ∂t ∂zk ρ ∂zj ρ ρ ∂zj ∂zj ∂zk ∂zj where w˙ j is the instantaneous acceleration of the considered particle (Chakrabarty, 1969a). The operator appearing in the first parenthesis represents the material rate of change and may be denoted by D/Dt for compactness. If the initial true stress is σ ij , the initial velocity vj , and the initial stress rate σ˙ ij , the above expression considered in the initial state furnishes ρ0

D Dt

sij ∂wi ρ ∂zj

= σ˙ ij t=0

∂vi + σij ∂xj

∂ v˙ i ∂vk ∂vi ∂vi ∂vk + − ∂xj ∂xk ∂xj ∂xk ∂xj

(1.83)

in view of the compressibility condition ρ˙ = −ρ (∂vk /∂xk ) in the initial state. Since the rate of dissipation of internal energy per unit volume in the initial state is σij (∂vi /∂xj ), the internal energy dissipated per unit initial volume during an interval of time δt required by the additional displacement δuj may be written as δU = σij

∂vi D sij ∂wi 1 δt + ρ0 (δt)2 , ∂xj 2 Dt ρ ∂zj t=0

(1.84)

which is correct to the second order irrespective of the strain path. If the nominal traction and its rate of change in the initial state are denoted by Fj , and F˙ j , respectively, the work done by the surface forces during the virtual displacement is δW =

1 Fj + F˙ j δt δuj dS = Fj δuj dS 2

in view of the boundary conditions F˙ j = 0 on SF and vj = 0 on Sv . Expressing δuj in terms of the initial velocity vj and the initial acceleration v˙ j , we have δW = δt

1 Fj vj dS + (δt)2 2

Fj v˙ j dS

(1.85)

1.5

Uniqueness and Stability

45

to second order. Since the total internal energy dissipated during the additional displacement must exceed the work done by the external forces, a sufficient condition for stability is δUdV − δW > 0. Substituting from (1.83), (1.84), and (1.85), the left-hand side of this inequality can be written entirely as a volume integral, since

Ff vj dS =

σij

∂vi dV, ∂xj

Ff v˙ j dS

σij

∂ v˙ i dV, ∂xj

by the principle of virtual work, the nominal traction being the same as the actual traction in the initial state. We therefore have σ˙ ij

∂vi + σij ∂xj

∂vk ∂vi ∂vi ∂vk − ∂xk ∂xj ∂xk ∂xj

dV > 0

as the required condition for stability. Substituting for σ˙ ij , using (1.68), and introducing the true strain rate ε˙ ij and the spin tensor ωij in the initial state, the stability criterion is finally obtained as

∂vi ∂vk σ ij ε˙ ij + σij ε˙ kk ε˙ ij − 2˙εik ωjk − dV > 0 ∂xk ∂xj ◦

(1.86)

for all continuous differentiable velocity fields vj vanishing at the Since constraints. the expression in the curly brackets of (1.86) is equal to ˙tij ∂vj /∂xi in view of (1.70), the surface integral of the scalar product F˙ j vj must be positive for the stability of the elastic/plastic solid (Hill, 1958). In the special case of rigid/plastic solids, an analysis similar to that presented above has been given by Chakrabarty (1969a). The stability functional (1.86) may be compared with the uniqueness functional (1.77), which involves the difference field vj instead of the velocity field vj. Due to the nonlinearity of the material response, the difference between two possible solutions is not necessarily a solution itself, and consequently (1.86) is always satisfied when (1.77) is. It follows that a partially plastic state in which the boundary value problem has a unique solution is certainly stable. When the solution is no longer unique, the partially plastic state may still be stable, and a point of bifurcation is therefore possible before an actual loss of stability. At such a stable bifurcation, the load must continue to increase with further deformation in the elastic/plastic range.

46

1 Fundamental Principles

Problems 1.1 For a certain application involving an elastic/plastic material, the stress–strain curve in the plastic range needs to be replaced by a straight line defined by σ = Y + T ε. The actual strain-hardening curve can be represented by σ = Y (Eε/Y) .n. If the linear strain-hardening law predicts the same area under the stress–strain curve as that given by the power law curve, over the range 0 ≤ ε ≤ ε 0 , when both the hardening laws are extended backward to ε = 0, show that Eε0 = Y

1 + n 1/n , 1−n

2n T = E i−n

1 − n 1/n . 1+n

1.2 For an element of work-hardening material yielding according to the von Mises yield criterion under biaxial compression, show that the principal stresses can be expressed in terms of the polar angle θ of the deviatoric stress vector as 2σ¯ σx = − √ cos θ, 3

π 2σ¯ σy = − √ sin −θ , 6 3

where σ¯ is the equivalent stress. Show also that the components of the plastic strain increment, according to the Prandtl–Reuss flow rule, can be expressed in terms of the angle θ and the current plastic modulus H as p d εx = − cos

π 6

+θ

dσ¯ H

,

d ε y p = sin θ

dσ¯ H

.

1.3 For an element of von Miss material deforming under a plane a strain tension in the x-direction and a stress-free state in the y-direction, show that√the applied stress and √ the deviatoric angle at the initial yielding are given by σ e = Y/ c and 2cos θ e = 3/c, where c = 1 – ν + ν 2 . If a prismatic beam made of such a material having a depth 2 h is bent to an elastic/plastic curvature, so that the depth of the elastic core becomes 2c, prove that the bending couple M is given by √ a2 M = 2 + 2 3c (cos θ ) ξ dξ , Me h 1

ξ=

y , h

c/h

where M√ e is the bending moment at the elastic limit. Assuming a mean value of cos θ, equal to cos θe , obtain the moment–curvature relationship in the dimensionless form

κ 2 M e = m − (m − 1) , Me κ

m=

3 (3c)1/4 . 2

1.4 An ideally plastic bar of circular cross section is rendered completely plastic by the combined action of an axial force N and a twisting moment T. If the ratio of the rate of extension to the rate of twist at the yield point is denoted by aα/3, show that the normal and shear stress distributions over the cross section of the bar are given by σ α = , Y α 2 + 3r2 /a2

r/a τ = . Y α 2 + 3r2 /a2

Problems

47

Denoting the fully plastic values of the axial force and twisting moment by N0 and T0 , respectively, and setting λ = 3 + α 2 , obtain the interaction relationship in the parametric form N 2 = α (λ − α) , N0 3

√ T 2 3 = λ − α 2 (λ − α) . T0 3

1.5 A block of isotropic material is compressed in the x-direction by a pair of rigid smooth dies, while the deformation in the y-direction is completely suppressed by constraints. If the material strain hardens linearly with a constant tangent modulus T, show that the polar equation of the stress path in the deviatoric plane is given by σ¯ = Y

√ √

3 sin θ − (1 − 2ν) (T/E) cos θe

αT/E

3 sin θe − (1 − 2ν) (T/E) cos θ

βT , exp (θe − θ) E

where σ¯ is the equivalent stress, and θ is the deviatoric angle having a value θ e at the initial yielding, while α and β are dimensionless parameters defined as α=

3 + (1 − 2ν)2 (T/E)2 3 + (1 − 2ν)2 (T/E)

√ β=

3 (1 − 2ν) (1 − T/E)

3 + (1 − 2ν)2 (T/E)

.

1.6 The plastic modulus of a certain kinematically hardening material varies with the equivalent plastic strain according to the relation H = Kn exp −n¯ε p , where K and n are material constants. A specimen of this material is first pulled in tension until the longitudinal stress is equal to σ 0 and is then subjected to a complete loading cycle which involves a plastic strain amplitude of amount ε∗. Show that the longitudinal stress at the end of the loading cycle exceeds σ 0 by the amount σ = −K 1 − exp (−4nε∗ ) exp −nε ∗ . 1.7 For a material that hardens according to the combined hardening rule, the isotropic and kinematic parts of the plastic modulus H are assumed to be in the ration β/(1 – β), where β is a constant. Assumie the plastic modulus to be given by H = Kn exp −n¯ε p Considering a complete loading cycle of a specimen involving a constant strain amplitude of amount ε ∗ , following a stress equal to σ 0 applied by simple tension, show that the tensile stress at the end of the cycle exceeds σ 0 by the amount σ = K 1 + exp ( − 2nε ∗ ) (2β − 1) + exp −2nε ∗ exp −nε ∗ . 1.8 The plastic yielding and flow of a certain isotropic material can be predicted with sufficient accuracy by modifying the von Mises yield criterion in the form J2 1 −

9J32 4J23

1/3 = k2

48

1 Fundamental Principles

where k is the yield stress in pure shear. Show that the uniaxial yield stress according to this criterion is Y = 1.853 k. Considering a state of plane stress defined by σ 3 = 0 and denoting σ 2 /σ 1 by α, prove that the ratio of the two in-plane plastic strain increments according to the associated flow rule is given by p

d ε2

p =

d ε1

⎧ 2

⎫ 2 ⎪ ⎬ + (1 + α) (2 − α) 2α 2 − 2α − 1 ⎪ 2α − 1 ⎨ 6 1 − α + α 2−α ⎪ ⎭ ⎩ 6 1 − α + α 2 2 + (1 + α) (2α − 1) 2 − 2α − α 2 ⎪

Chapter 2

Problems in Plane Stress

In many problems of practical interest, it is a reasonable approximation to disregard the elastic component of strain in the theoretical analysis, even when the body is only partially plastic. In effect, we are then dealing with a hypothetical material which is rigid when stressed below the elastic limit, the modulus of elasticity being considered as infinitely large. If the plastically stressed material has the freedom to flow in some direction, the distribution of stress in the deforming zone of the assumed rigid/plastic body would approximate that in an elastic/plastic body, except in a transition region near the elastic/plastic interface where the deformation is restricted to elastic order of magnitude. The assumption of rigid/plastic material is generally adequate not only for the analysis of technological forming processes, where the plastic part of the strain dominates over the elastic part, but also for the estimation of the yield point load when the rate of work-hardening is sufficiently small (Section 1.2). In this chapter, we shall be concerned with problems in plane stress involving rigid/plastic bodies which are loaded beyond the range of contained plastic deformation.

2.1 Formulation of the Problem A plate of small uniform thickness is loaded along its boundary by forces acting parallel to the plane of the plate and distributed uniformly through the thickness. The stress components σ z , τ xz , and τ vz are zero throughout the plate, where the z-axis is considered perpendicular to the plane. The state of stress is therefore specified by the three remaining components σ x , σ y , and τ xy , which are functions of the rectangular coordinates x and y only. During the plastic deformation, the thickness does not generally remain constant, but the stress state may still be regarded as approximately plane provided the thickness gradient remains small compared to unity.

2.1.1 Characteristics in Plane Stress For greater generality, we consider a nonuniform plate of small thickness gradient, subjected to a state of generalized plane stress in which σ x, σ y , and τ xy denote the J. Chakrabarty, Applied Plasticity, Second Edition, Mechanical Engineering Series, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77674-3_2,

49

50

2

Problems in Plane Stress

stress components averaged through the thickness of the plate. In the absence of body forces, the equations of equilibrium are ∂ ∂ hτxy = 0, (hσx ) + ∂x ∂y

∂ ∂ hτxy + hσy = 0, ∂x ∂y

(2.1)

where h is the local thickness of the plate. If yielding occurs according to the von Mises criterion, the stresses must also satisfy the equation 2 σx2 − σx σy + σy2 + 3τxy = σ12 − σ1 σ2 + σ22 = 3k2 ,

(2.2)

principal stresses in the xy-plane, and k is the yield stress where σ 1 and σ 2 are the√ in pure shear equal to 1/ 3 times the uniaxial yield stress Y. In the (σ 1, σ 2 )-plane, (2.2) represents an ellipse having its major and minor axes bisecting the axes of reference. Suppose that the stresses are given along some curve C in the plastically deforming region. The thickness h is regarded as a known function of x and y at the instant under consideration. Through a generic point P on C, consider the rectangular axes ( X, y) along the normal and tangent, respectively, to C. If we rule out the possibility of a stress discontinuity across C, the tangential derivatives ∂σx /∂y,∂σy /∂y, and ∂τxy /∂y must be continuous. Since ∂h/∂x and ∂h/∂y are generally continuous, the equilibrium equation (2.1) indicates that ∂σx /∂x and ∂τxy /∂xare also continuous across c. For given values of these derivatives, the remaining normal derivative ∂σy /∂x can be uniquely determined from the relation ∂σx ∂σy ∂τxy + 2σy − σx + 6τxy =0 2σx − σy ∂x ∂x ∂x

(2.3)

obtained by differentiating the yield criterion (2.2), unless the coefficient 2σ y – σ x is zero. When σ x = 2σ y , the above equation gives no information about ∂σy /∂x, which may therefore be discontinuous across C. It follows that S is a characteristic when the stress components are given by σx = σ ,

σy =

1 σ, 2

τxy = τ ,

where σ is the normal stress, and τ the shear stress transmitted across C. In view of the yield criterion (2.2), the relationship between σ and τ is σ 2 + 4τ 2 = 4k2

(2.4)

which defines an ellipse E in the (σ , τ )-plane, as shown in Fig. 2.1. If ψ denotes the angle of inclination of the tangent to E with the σ -axis, reckoned positive when τ decreases in magnitude with increasing σ , then tan ψ = ∓

σx − σy σ dτ =± =± . dσ 4τ 2τxy

(2.5)

2.1

Formulation of the Problem

51

Fig. 2.1 Yield envelope for an element in a state of plane plastic stress including the associated Mohr circle for stress

The expression in the parenthesis is the cotangent of twice the counterclockwise angle made by either of the two principal stresses with the x-axis. It follows that there are two characteristic directions at each point, inclined at an angle π /4 + ψ/2 on either side of the algebraically greater principal stress direction. The two characteristics are identified as α- and β-lines following the convention indicated in Fig. 2.2, when σ 1 > σ 2 . The two characteristics coincide when ψ = ±π/2. The normal and shear stresses acting across the characteristics are defined by the points of contact of the envelope E with Mohr’s circle for the considered state of stress. The locus of the highest and lowest points of the circle is an ellipse representing the yield criterion (2.2), which may be written as

Fig. 2.2 Characteristics directions in plane stress (σ 1 >σ 2 >0), designated by α- and β-lines

52

2

Problems in Plane Stress

F (σ0 ,τ0 ) = σ02 + 3τ02 = 3k2 , where

(2.6) 1 3 σ0 = (σ1 + σ2 ) = σ , 2 4

1 τ0 = (σ1 − σ2 ) . 2

Thus, τ 0 is numerically equal to the maximum shear stress and σ 0 the mean normal stress in the plane. Since σ cannot exceed 2k in magnitude, the material being ideally plastic, σ 0 be numerically less than 3k/2 and τ 0 numerically greater than k/2 for the characteristics to be real. Since dτ /dσ 0 is equal to −rsin ψ in view of (2.6) and Fig. 2.1, the acute angle which the tangent to the yield locus makes with the σ -axis must be less than π /4 for the stress equations to be hyperbolic. When τ 0 is numerically less than k/2, there is no real contact between the Mohr circle and the envelope E, and the stress equations then become elliptic. In the limiting case of |τ 0 | = k/2, the characteristics are coincident with the axis of the numerically lesser principal stress, the values of the principal stresses being ±(2k, k) in the limiting state.

2.1.2 Relations Along the Characteristics When the equations are hyperbolic, it is convenient to establish equations giving the variation of the stresses along the characteristics. Following Hill (1950a), we take the axes of reference along the normal and tangent to a typical characteristic C at a generic point P as before. Setting 2σ y = σ x tan ψ in (2.3), we get ∂τxy ∂σx ± tan ψ = 0, ∂x ∂x where the upper sign corresponds to the β-line and the lower sign to the α-line, respectively. The substitution of ∂σ x /∂x from above into the first equation of (2.1) gives ∂τxy ∂τxy ∂h ∂h ± tan ψ = ± σx + τxy tan ψ. h ∂x ∂y ∂x ∂y Regarding the curve C as a β-line, we observe that the space derivatives along the σ and β-lines at P are ∂ ∂ ∂ − sin ψ , = cos ψ ∂sα ∂x ∂y

∂ ∂ = , ∂sβ ∂y

and the preceding equation therefore reduces to h

∂τxy ∂h ∂h = tan ψ + (σ tan ψ + τ ) sin ψ ∂sα ∂sα ∂sβ

(2.7)

2.1

Formulation of the Problem

53

since σ x = σ and τxy = τ at the considered point. To obtain the derivative ∂τ xy /∂sα at P, let φ β denote the counterclockwise angle made by the tangent to C with an arbitrary fixed direction. If the x-axis is now taken in this direction, then 1 τxy = − σ sin 2φβ − τ cos 2φβ . 4 Differentiating this expression partially with respect to sα and then setting φ β = π /2. we obtain ∂τxy ∂φβ 1 ∂φβ ∂τ ∂σ = σ − = tan ψ 2τ − ∂sα 2 ∂sα ∂sα ∂sα ∂sα in view of (2.5). Inserting in (2.7) and rearranging, the result can be expressed in the form d (hσ ) − 2 hτ dφβ = − (σ sin ψ + τ cos ψ)

∂h dsα ∂sβ

(2.8)

along an α-line. Similarly, considering the curve C to be an α-line, it can be shown that d (hσ ) + 2 hτ dφα = − (σ sin ψ + τ cos ψ)

∂h dsβ ∂sα

(2.9)

along a β-line, where φ α is the counterclockwise orientation of the α-line with respect to the same fixed direction. Evidently, dφ β - dφ α =dψ. If the thickness distribution is given, (2.8) and (2.9) in conjunction with (2.4) would enable us to determine the stress distribution and the characteristic directions in the hyperbolic part of the plastic region. When the thickness is uniform, (2.8) and (2.9) reduce to the relations dσ − 2τ dφβ = 0 along an α − line, dσ + 2τ dφα = 0 along an β − line,

(2.10)

which are analogous to the well-known Hencky equations in plane strain. In the solution of physical problems, it is usually convenient to express the yield criterion parametrically through the angle ψ, or a related angle θ such that tan θ =

√ 3 sin ψ,

−

π π ≤ψ ≤ , 2 2

−

π π ≤θ ≤ . 3 3

(2.11)

The angle θ denotes the orientation of the stress vector in the deviatoric plane with respect to the direction representing pure shear in the plane of the plate. Indeed, it follows from (2.4), (2.5), and (2.11) that

54

2

Problems in Plane Stress

⎫ 4 sin ψ 4 σ ⎪ ⎪ = = √ sin θ , ⎪ ⎪ k 3 2 ⎪ ⎬ 1 + 3 sin ψ ⎪ 4 cos ψ τ ⎪ = = 1 − sin2 θ .⎪ ⎪ ⎪ ⎭ k 3 2 1 + 3 sin ψ

(2.12)

√ Since 2(σ 1 +σ 2 ) = 3σ and 2(σ1 −σ2 ) = σ 2 + 16τ 2 , we immediately get σ1 +σ2 = √ 3k sin θ and σ 1 – σ 2 = 2k cos θ , and the principal stresses become

π , σ1 = 2k sin θ + 6

π σ2 = 2k sin θ − 6

(2.13)

It is also convenient at this stage to introduce a parameter λ which is defined in the incremental form dλ =

1 2

dσ − dψ τ

=2

dσ 1 tan ψ − dψ σ 2

which is assumed to vanish with ψ or θ . Substituting from (2.12) and integrating, we get 1 λ = tan−1 (2 tan ψ) − ψ 2 1 2 1 = sin−1 √ sin θ − sin−1 √ tan θ . 2 3 3

(2.14)

Evidently, –π /4 ≤ λ ≤ π /4 over the relevant range. If ω denotes the counterclockwise angle made by a principal stress axis with respect to a fixed axis, then

1 dω = d φα + ψ 2

1 = d φβ + ψ . 2

Dividing (2.8) and (2.9) by hσ throughout, and substituting for dσ /σ , σ /τ , dφ α , and dφ β , we finally obtain the relations ⎫ 1 + 3 sin2 ψ ∂h dh ⎪ ⎪ =− d (λ − ω) + 2 tan ψ dsα ,⎪ ⎪ ⎬ h 2h cos ψ ∂sβ ⎪ ⎪ 1 + 3 sin2 ψ ∂h dh ⎪ dsβ .⎪ =− d (λ − ω) + 2 tan ψ ⎭ h 2h cos ψ ∂sα

(2.15)

along the α- and β-lines, respectively. Similar equations may be written in terms of θ using (2.11). Numerical values of λ are given in Table 2.1 for the whole range of values of ψ and θ . When h is a constant, (2.15) reduces to

2.1

Formulation of the Problem

55

Table 2.1 Parameters for plane strees characteristics Ψ degrees

Ψ Radians

λ radians

θ degrees

Ψ degrees

λ radians

0 10 20 30 40 50 60 70 80 90

0 0.17453 0.34907 0.52360 0.89813 0.87266 1.04720 1.22173 1.39626 1.57080

0 0.25177 0.4570 0.59527 0.68435 0.73722 0.76616 0.77992 0.78473 0.78540

0 10 20 30 35 40 45 50 55 60

0 5.843 12.130 19.471 23.845 28.977 35.264 55.542 55.542 90.000

0 0.15089 0.30013 0.44556 0.51581 0.58352 0.64757 0.75558 0.75558 0.78540

λ − ω = constant along an α-line, λ − ω = constant along a β-line,.

(2.16)

In analogy with Hencky’s first theorem, we can state that the difference in values of both λ and ω between a pair of points, where two given characteristics of one family are cut by a characteristic of the other family, is the same for all intersecting characteristics. It follows that if a segment of one characteristic is straight, then so are the corresponding segments of the other members of the same family, the values of λ and ω being constant along each straight segment. If both families of characteristics are straight in a certain portion of the plastic region, the state of stress is uniform throughout this region.

2.1.3 The Velocity Equations Let (vx , vy ) denote the rectangular components of the velocity averaged through the thickness of the plate. The material is assumed as rigid/plastic, obeying the von Mises yield criterion and the Lévy–Mises flow rule. In terms of the stresses σ x , σ y , and τ xy , the flow rule may be written as ∂vy /∂x ∂vx /∂y + ∂vy /∂x ∂vx /∂x = = . 2σx − σy 2σy − σx 6τxy

(2.17)

Consider a curve C along which the stress and velocity components are given, and let the x- and y-axes be taken along the normal and tangent, respectively, to C at a typical point P. Assuming the velocity to be continuous across C, the tangential derivatives dvx /dy and dvy /dy are immediately seen to be continuous. From (2.17), the normal derivatives dvx /dx and dvy /dx can be uniquely determined unless 2σ y – σ x = 0, which corresponds to dvy /dy = 0. Thus, C is a characteristic for the velocity field if it coincides with a direction of zero rate of extension. There are two such directions at each point and they are identical to those of the stress characteristics.

56

2

Problems in Plane Stress

Since the characteristics are inclined at an angle π /4+ψ/2 to the direction of the algebraically greater principal strain rate ε˙ 1 , the condition ε˙ = 0 along a characteristic gives 1 ε˙ 1 + ε˙ 2 = sin ψ = ε˙ 1 − ε˙ 2 3

σ1 + σ2 σ1 − σ2

1 = √ tan θ . 3

(2.18)

The range of plastic states for which the characteristics are real corresponds to |σ 0 |≤3|τ 0 | and are represented by the arcs AB and CD of the von Mises ellipse shown in Fig. 2.3. The numerically lesser principal strain rate vanishes in the limiting states, represented by the extremities of these arcs, where the tangents make an acute angle of π /4 with the σ 0 -axis. Fig. 2.3 Plane stress yield loci according to Tresca and von Mises for a material with a uniaxial yield stress Y

The velocity of a typical particle is the resultant of its rectangular components vx and vy . The resolved components of the velocity vector along the α and β-lines, denoted by u and v, respectively, are related to the rectangular components as u = vx cos φα + vy sin (ψ + φα ) , v = −vx sin φα + vy cos (ψ + φα ) , where φ α denotes the counterclockwise angle made by the α-line with the x-axis. The preceding relations are easily inverted to give vx = [u cos (ψ + φα ) − v sin φα ] sec ψ, vy = [u sin (ψ + φα ) − v cos φα ] sec ψ.

(2.19)

Differentiating vx partially with respect to x, and using the fact that dvx /dx = 0 when φ α = −ψ, we obtain ∂φα ∂v − (u tan ψ + v sec ψ) = 0. ∂sα ∂sα

2.1

Formulation of the Problem

57

Similarly, equating dvy /dy to zero after setting φ α = −ψ in the expression for the partial derivative of vy with respect to y, we get ∂φβ ∂v + (u sec ψ + v tan ψ) =0 ∂sβ ∂sβ in view of the relation dφ β – dφ α = dψ The velocity relations along the characteristics therefore become du − (u tan ψ + v sec ψ) dφα = 0 along an α-line, dv + (u sec ψ + v tan ψ) dφβ = 0 along an β-line,

(2.20)

When the characteristic directions are known at each point of the field, the velocity distribution can be determined from (2.20). For ψ = 0, these equations reduce to the well-known Geiringer equations in plane plastic strain. Since the thickness strain rate has the same sign as that of –(σ +σ 2 ), a thinning of the sheet corresponds to σ 0 > 0 and a thickening to σ 0 < 0. If the rate of change of thickness following the element is denoted by h, it follows from the associated flow rule that σn + σ1 ∂ω 1 ∂h ∂h h˙ = +ω =− , h h ∂t ∂s 2σ1 − σn ∂s

(2.21)

where w is the speed of a typical particle, s is the arc length along the momentary flow line, and (σ n , σ t ) are the normal stress components along the normal and tangent, respectively, to the flow line. The change in thickness during a small interval can be computed from (2.21). Evidently, whenever there is a discontinuity in the velocity gradient, there is also a corresponding jump in ∂h/∂t, leading to a discontinuity in the surface slope of an initially uniform sheet.

2.1.4 Basic Relations for a Tresca Material If the material yields according to Tresca’s yield criterion with a given uniaxial yield stress Y = 2k, the yield locus is a hexagon inscribed in the von Mises ellipse. When the principal stresses σ 1 and σ 2 have opposite signs, the greatest shear stress occurs in the plane of the sheet, and the yield criterion becomes 2 2 = 4k2 , σx + σy ≤ 2k. (σ1 − σ2 )2 = σx − σy + 4τxy

(2.22)

As in the case of plane strain, the stress equations are hyperbolic, and the characteristics are sliplines bisecting the angles between the principal stress axes. Since the shear stresses acting across the characteristics are of magnitude k, the envelope

58

2

Problems in Plane Stress

of the Mohr’s circles then coincides with the yield locus. The variation of the normal stress σ along the characteristics is obtained from (2.8) and (2.9) by setting τ = k, ψ = 0, and dφ α = dφ β = dφ, the expression in each parenthesis being then equal to k. For a uniform sheet, these relations reduce to the well-known Hencky equations in plane strain. Since the thickness strain rate vanishes by the associated flow rule, the velocity equations are also hyperbolic and the characteristics are again the sliplines, the velocity relations along the characteristics being given by the familiar Geiringer equations in plane strain. When the principal stresses σ 1 and σ 2 have the same sign, the greatest shear stress occurs out of the plane of the applied stresses, and the numerically greater principal stress must be of magnitude 2k for yielding to occur. On the (σ 0 , τ 0 )plane, the yield criterion is defined by the straight lines σ 0 ±τ 0 = ±2k. In terms of the principal stresses, the yield criterion may be written as (σ1 + 2k) (σ2 ± 2k) = 0,

σ1 σ2 ≥ 0.

This equation can be expressed in terms of the (x, y) components of the stress, using the fact that σ1 +σ 2 = σ x + σ y and σ 1 σ 2 = σ x σ y − τ2 xy the result being 2 − σx σy + 2k σx + σy = 4k2 , 2k ≤ σx + σy ≤ 4k. τxy

(2.23)

The partial differentiation of (2.23) with respect to x reveals that the stress derivatives are uniquely determined unless σ x = ±2k. Hence, there is a single characteristic across which the normal stress is of magnitude 2k. In other words, the stress equations are parabolic with the characteristic coinciding with the direction of the numerically lesser principal stress, whose magnitude is denoted by σ . When the xand y-axes are taken along the normal and tangent, respectively, to the characteristic, we have

σ sin 2ω. (2.24) σx = ± 2k and σy = ±σ , τxy = ∓ k − 2 where ω denotes the counterclockwise angle made by the characteristic with a fixed direction which is temporarily considered as the x-axis. Inserting (2.24) into the equilibrium equation (2.1), and setting ω = π /2 after the differentiation, we get

σ ∂ω 1 ∂h ∂ hσ σ ∂ω 1− =− , =− 1− h , 2k ∂s h ∂n ∂s 2k 2k ∂n

(2.25)

where ds and dn are the line elements along the characteristic and its orthogonal trajectory, forming a right-handed pair of curvilinear axes. When the thickness is uniform, ω is constant along each characteristic, which is therefore a straight line defined by y = x tan ω+f (ω), where f(ω) is a function of ω to be determined from the stress boundary condition. The curvature of the numerically greater principal stress trajectory is

2.2

Discontinuities and Necking

59

∂ω ∂ω cos ω ∂ω = sin ω − cos ω = − . ∂n ∂x ∂y x + f (ω) cos2 ω

(2.26)

Inserting from (2.26) into the second equation of (2.25), and using the fact that ds = sec ω dx along a characteristic, the above equation is integrated to give (Sokolovsky, 1969), 1−

σ g(ω) , = 2k x + f (ω) cos2 ω

(2.27)

where g(ω) is another function to be determined from the boundary condition. Since the numerically lesser principal strain rate vanishes according to the associated flow rule, it follows that the velocity equations are also parabolic, and the characteristic direction coincides with the direction of zero rate of extension. The tangential velocity v remains constant along the characteristic, and the normal velocity u follows from the condition of zero ate of shear associated with these two orthogonal directions.

2.2 Discontinuities and Necking 2.2.1 Velocity Discontinuities In a nonhardening rigid/plastic solid, the velocity may be tangentially discontinuous across certain surfaces where the shear stress attains its greatest magnitude k. For a thin sheet of metal, it is also necessary to admit a necking type of discontinuity involving both the tangential and normal components of velocity. To be consistent with the theory of generalized plane stress, the strain rate is considered uniform through the thickness of the neck, whose width b is comparable to the sheet thickness h. Since plastic deformation is confined in the neck, the rate of extension vanishes along its length, and the neck therefore coincides with a characteristic. The relative velocity vector across the neck must be perpendicular to the other characteristic in order that the velocity becomes continuous across it. Localized necking cannot occur, however, when the stress state is elliptic. Let v denote the magnitude of the relative velocity vector which is inclined at an angle ψ to the direction of the neck, Fig. 2.4a. The rate of extension in the direction perpendicular to the neck is (v/b) sin ψ, and the rate of shear across the neck is (v/2b) cos ψ. The condition of the zero rate of extension along the neck therefore gives the principal strain rates as (Hill, 1952). ε˙ 1 =

v v v (1 + sin ψ) , ε˙ 2 = − (1 − sin ψ) , ε˙ 3 = − sin ψ, 2b 2b b

(2.28)

irrespective of the flow rule. These relations imply that the axis of ε˙ 1 is inclined at an angle π/4 + ψ/2 to the neck. For a von Mises material, ψ is related to the stress according to (2.18), while for a Tresca material, ψ = 0 when the stress state

60

2

Problems in Plane Stress

Fig. 2.4 Velocity and stress discontinuities in plane stress including the associated principal directions

is hyperbolic. In the case of a uniaxial tension, ε˙ 1 = −2˙ε2 for any regular yield inclination of the neck to function, giving ψ = sin–1 1/3 ≈ 19.47◦ , the angle of √ the direction of tension being β = π/4 + ψ/2 = tan−1 2 ≈ 54.74◦ . For a Tresca material, on the other hand, ψ can have any value between 0 and π/2 under a uniaxial state of stress. When the material work-hardens, a localized neck is able to develop only if the rate of hardening is small enough to allow an incremental deformation to remain confined in the incipient neck. For a critical value of the rate of hardening, the deformation is just able to continue in the neck, while the stresses elsewhere remain momentarily unchanged. Since the force transmitted across the neck remains momentarily constant, we have −

dσ dY H dh = = = d ε, h σ Y Y

where σ is the normal stress across the neck, Y the current yield stress, H the rate of hardening, and dε the equivalent strain increment. If the Lévy–Mises flow rule is adopted, then dh − = h

σ1 + σ2 2Y

dε =

3σ d ε. 4Y

The last two equations reveal that the critical rate of hardening is equal to 3σ /4 or (σ 1 +σ 2 )/2. Using (2.12), the condition for localized necking to occur may therefore be expressed as

2.2

Discontinuities and Necking

√ H 3 sin ψ σ1 + σ2 ≤ = √ . Y 2Y 1 + 3 sin2 ψ

61

(2.29)

As It is also necessary to have –1≤σ 1 /σ 2 ≤2 for the equations to be hyperbolic. √ ψ increases from 0 to π/2, the critical value of H/Y increases from 0 to 3/2. In the case of a uniaxial tension (3 sin ψ = 1), localized necking can occur only if H/Y ≤ 0.5. Thus, for a sheet of metal with a rounded stress–strain curve, a gradually increasing uniaxial tensile stress a produces a diffuse neck when H = σ , and eventually a localized neck when H = σ /2. A microstructural model for the shear band type of strain localization has been examined by Lee and Chan (1991).

2.2.2 Tension of a Grooved Sheet Consider a uniform rectangular sheet of metal whose thickness is locally reduced by cutting a pair of opposed grooves in an oblique direction across the width, Fig. 2.5a. The grooves are deep enough to ensure that plastic deformation is localized there when the sheet is pulled longitudinally in tension. The width of the sheet is large compared to the groove width b, which is slightly greater than the local sheet thickness h so that a uniform state of plane stress exists in the grooves. The material in the grooves is prevented from extending along its length by the constraint of the adjacent nonplastic material. The principal strain rates in the grooves are therefore given by (2.28) in terms of the angle of inclination of the relative velocity with which the sides of the grooves move apart. If the material is isotropic, the directions of the principal stresses σ 1 and σ 2 are inclined at angles π /4 + ψ/2 and π /4 – ψ/2, respectively, to the direction of the grooves, where σ 1 > σ 2, the principal axes of stress and strain rate being coincident.

Fig. 2.5 Necking of grooved and notched metal strips under longitudinal tension

Let φ denote the counterclockwise angle made by the prepared groove with the direction of the applied tensile force P. Then the normal and shear stresses acting across the grove are P sinφ/lh and P cos φ/lh, respectively, where l is the length of the groove. By the transformation relations for the stress, we have (σ1 + σ2 ) + (σ1 − σ2 ) sin ψ = (2P/lh) sin φ, (σ1 − σ2 ) cos ψ = (2P/lh) cos φ.

62

2

Problems in Plane Stress

These equations can be solved for the principal stresses in terms of the angles φ and ψ, giving σ1 =

P [sin (φ − ψ) + cos φ] P [sin (φ − ψ) − cos φ] , σ2 = . hl cos ψ hl cos ψ

(2.30)

From (2.28) and (2.30), Lode’s well-known stress and strain parameters are obtained as ⎫ 3 cos φ − sin (φ − ψ) ⎪ 2σ2 − σ1 − σ3 =− ,⎪ μ= σ1 − σ3 sin (φ − ψ) + cos φ ⎬ 3 (1 − sin ψ) 2˙ε2 − ε˙ 1 − ε˙ 3 ⎪ ⎪ ⎭ =− ν= . ε1 − ε3 (1 + 3 sin ψ)

(2.31)

Equation (2.31) form the basis for establishing the (μ, v) relationship for a material using the measured values of φ and ψ. The ends of the sheet must be supported in such a way that they are free to rotate in their plane to accommodate the relative movement necessary to permit the localized deformation to occur. The shape of the deviatoric yield locus may be derived from the fact that the length of the deviatoric stress vector is 1/2 2 2 P = σ1 − σ1 σ2 + σ22 s= 3 hl

2 sin2 (φ − ψ) + 3 cos2 ψ , √ 3 cos ψ

(2.32)

√ and the angle made by the stress vector is θ = tan−1 ( − μ/ 3) with the direction representing pure shear. If the yield locus and plastic potential have a sixfold symmetry required by the isotropy and the absence of the Bauschinger effect, it is only necessary to cover a 30◦ segment defined by the direction of pure shear (μ = v = 0) and that of uniaxial √ tension (μ = v = –1). This is accomplished by varying φ between 90◦ and tan−1 2 ≈ 54.7◦ , and measuring ψ for each selected value of φ. It may be noted that according to the Lévy–Mises flow rule (μ = v), the relationship tan φ = 4 tan ψ always holds. The preceding analysis, due to Hill (1953), is equally applicable to the localized necking caused by the tension of a sheet provided with a pair of asymmetrical notches as shown in Fig. 2.5b. If the notches are deep and sharp, and the rate of work-hardening is sufficiently low, plastic deformation is localized in a narrow neck joining the notch roots. This method may be used for the determination of the yield criterion and the plastic potential for materials with sufficient degrees of pre-strain, provided it is reasonably isotropic. The method has been tried with careful experiments by Hundy and Green (1954), and by Lianis and Ford (1957), using specimens which can be effectively tested to ensure that they are actually isotropic. These investigations have confirmed the validity of the von Mises yield criterion and the associated plastic potential for several engineering materials, as indicated in Fig. 1.8.

2.2

Discontinuities and Necking

63

2.2.3 Stress Discontinuities We begin by considering the normal and shear stresses acting over a surface which coincides with a characteristic. When the state of stress is hyperbolic, there is only one stress circle that can be drawn through the given point on the Mohr envelope without violating the yield criterion. Since the stress states on both sides of the characteristic are represented by the same circle, all components of the stress are continuous. When the stress state is parabolic, and the yield criterion is that of Tresca, the principal stress acting along the tangent to the characteristic can have any value between 0 and ±2 k, permitting a discontinuity in the numerically lesser principal stress. When the considered surface is not a characteristic, a stress discontinuity is always possible with two distinct plastic states separated by a line of stress discontinuity. Let σ 1 , σ 2 be the principal stresses on one side of the discontinuity (σ 1 ≥ σ 2 ), and σ 1 , σ 2 , those on the other side (σ 1 ≥ σ 2 ) . The angles of inclination of σ 1 and σ 2 with the line of discontinuity are denoted by θ and θ , respectively, reckoned positive as shown in Fig. 2.4b. Since the normal and shear stresses across the line of discontinuity must be continuous for equilibrium, we have (σ1 + σ2 ) − (σ1 − σ2 ) cos 2θ = σ1 + σ2 + σ1 − σ2 cos 2θ , (σ1 − σ2 ) sin 2θ = σ1 − σ2 sin 2θ .

(2.33)

If the von Mises yield criterion is adopted, the stresses on each side of the discontinuity must satisfy (2.2). Considering the stress components along the normal and tangent to the discontinuity, specified by n and t, respectively, and using the continuity conditions σ n = σ n and τ nt = r nl , it is easily shown from (2.2) that σ n – σ t = σ t giving 2 cot 2θ = cot 2θ +

σ1 + σ2 σ1 − σ2

cosec2θ .

(2.34)

The elimination of σ 1 – σ 2 between the two equations of (2.33), and the substitution from (2.34), lead to the relation 2 σ1 − σ2 = (σ1 + σ2 ) − 3 (σ1 − σ2 ) cos 2θ .

(2.35)

Since σ 1 – σ 2 is then given by the yield criterion, the principal stresses and their directions are known on one side of the discontinuity when the corresponding quantities on the other side are given. Considering the Tresca criterion for yielding, suppose that (σ 1 , σ 2 ) represents a hyperbolic state, so that σ 1 –σ 2 = 2k and |σ 1 +σ 2 | ≤ 2k. If the (σ 1 , σ 2 ) state is also hyperbolic, then σ 1 –σ 2 = 2k by the yield criterion, and the continuity conditions (2.33) furnish θ = θ , and

64

2

σ1 = σ1 − 2k cos 2θ ≥ 0,

Problems in Plane Stress

σ2 = σ2 − 2k cos 2θ ≤ 0.

If the (σ 1 , σ 2 ) state is parabolic, the yield criterion is either σ 1 = 2k(σ 2 ≥ 0) or σ 2 = –2k(σ 1 ≤ 0). In the first case, (2.33) gives

⎫ σ2 ⎪ tan θ = 1 + cos 2θ − co sec 2θ , ⎬σ k 2 ≤ cos 2θ . #

σ2 σ2 σ2 σ2 ⎪ 2k =− − cos 2θ 1 + cos 2θ − ,⎭ 2k k 2k k

(2.36)

The corresponding results for the second case are obtained from (2.36) by replacing σ 2 and σ 2 with –σ 1 and σ 1 , respectively, tan θ with cos θ , and reversing the sign of cos 2θ . Exceptionally, when the stress normal to the discontinuity is ±2k, the other principal stress on either side can have any value between 0 and ±2k. Such a discontinuity may be considered as the limit of a narrow zone of a continuous sequence of plastic states. When yielding occurs according to the von Mises criterion, all the stress components must be continuous across a line of velocity discontinuity. This is evident for a necking type of discontinuity (since the neck must coincide with a characteristic), across which the stress is necessarily continuous. For a shearing discontinuity, the shear stress across the line of discontinuity is of magnitude k, and since the normal stress is continuous for equilibrium, the remaining stress must also be continuous in view of the yield criterion. As a consequence of this restriction, the velocity must be continuous across a line of stress discontinuity. The rate of extension along a line of stress discontinuity, which is the derivative of the tangential velocity, is evidently continuous. Since the flow rule predicts opposite signs for this component of the strain rate on the two sides of the stress discontinuity, the rate of extension must vanish along its length. The discontinuity may therefore be regarded as the limit of a narrow zone of elastic material through which the stress varies in a continuous manner.

2.2.4 Diffuse and Localized Necking It is well known that the deformation of a bar subjected to a longitudinal tension ceases to be homogeneous when the rate of work-hardening of the material is less than the applied tensile stress. At the critical rate of hardening, the load attains its maximum, and the subsequent extension of the bar takes place under a steadily decreasing load. Plastic instabilities of this sort, leading to diffuse local necking, also occur when a flat sheet is subjected to biaxial tension in its plane. The strainhardening characteristic of the material is defined by the equivalent stress σ , and the equivalent total strain ε, related to one another by the true stress–strain curve in uniaxial tension. Using the Lévy–Mises flow rule in the form

2.2

Discontinuities and Necking

65

d ε1 dε d ε2 d ε3 = =− = , 2σ1 − σ2 2σ2 − σ1 σ1 + σ2 2σ

(2.37)

and the differential form of the von Mises yield criterion (2.2) where 3k2 is replaced by σ 2 , it is easily shown that the stress and strain increments in any element must satisfy the relation d σ1 d ε1 + d σ2 d ε2 = d σ d ε. Consider a rectangular sheet whose current dimensions are b1 and b2 along the directions of σ 1 and σ 2 , respectively. If the applied loads hb2 σ 1 and hb2 σ 2 attain stationary values at the onset of instability, where h denotes the current thickness, then d σ1 db1 = = d ε1 , σ1 b1

d σ2 db2 = = d ε2 , σ2 b2

in view of the constancy of the volume hb1 b2 of the sheet material. Combining the preceding two relations, we have dσ dε

= σ1

d ε1 dε

2

+ σ2

d ε2

2

dε

.

Substituting from (2.37), and using the expression for σ 2 , and setting σ 2 /σ 1 = p, the condition for plastic instability is obtained as (Swift, 1952; Hillier, 1966) (1 + ρ) 4 − 7ρ + 4ρ 2 H = 3/2 , σ 4 1 − ρ + ρ2

(2.38)

where H = dσ /dε denotes the critical rate of hardening. The quantity on the lefthand side of (2.38) is the reciprocal of the subtangent to the generalized stress–strain curve. The variation of the critical subtangent with stress ratio ρ is shown in Fig. 2.6 for both localized and diffuse necking. If the stress ratio is maintained constant throughout the loading, the total equivalent strain at instability is obtained directly from (2.38), if we adopt the simple power law σ = C εn , which gives H/σ = n/ε. In the case of variable stress ratio, the instability strain will evidently depend on the prescribed loading path. In the biaxial tension of sheet metal, failure usually occurs by strain localization in a narrow neck following the onset of instability. The phenomenon can be explained by considering the development of a pointed vertex on the yield locus, which allows the necessary freedom of flow of the plastic material in the neck. When both the principal strains are positive, experiments seem to indicate that the neck coincides with the direction of the minimum principal strain ε2 in the plane of the sheet. The incremental form of the Hencky stress–strain relations may be assumed to hold at the incipient neck, where the subsequent deformation remains

66

2

Problems in Plane Stress

Fig. 2.6 Critical subtangent to the effective stress–strain curve as a function of the stress ratio

confined (Stören and Rice, 1975). The principal surface strains in the neck may therefore be written as 2σ1 − σ2 2σ2 − σ1 ε1 = ε , ε2 = ε , 2σ 2σ where σ and ε are the equivalent stress and total strain, respectively. The elimination of σ 2 and σ 1 in turn between these two relations gives the principal stresses σ1 = (2ε1 + ε2 )

2σ 2σ , σ2 = (2ε2 + ε1 ) . 3ε 3ε

Assuming the power law σ = C εn for the generalized stress–strain curve, the incremental form of the first equation above at the inception of the neck is found as d σ1 2d ε1 + d ε2 dε , = − (1 − n) σ1 ε (2 + α) ε1

(2.39)

where α ≥ 0 denotes the constant strain ratio ε2 /ε 1 , prior to the onset of neck2 ing. 2The quantity2 de is the increment of the equivalent total strain ε. Since ε = 4 3 ε1 + ε1 ε2 + ε2 according to the Hencky theory, we get dε (2 + α) d ε1 + (1 + 2α) d ε2 . = ε 2 1 + α + α 2 ε1 The neck is characterized by a discontinuity in dε1 , but dε2 must be regarded as continuous across the neck. Since the material outside the neck undergoes neutral loading at its inception, we set dε 2 = 0. Combining the last two equations, and using

2.3

Yielding of Notched Strips

67

the fact that dσ 1 /σ 1 = dε1 for the load across the neck to be stationary, the condition for localized necking is obtained as

2 (1 − n) (2 + α) − ε1 + 2 2 + α 2 1+α+α

d ε1 = 0. ε1

Since dε1 = 0 for the development of the neck, the expression in the curly brackets must vanish, giving the limit strain over the range 0 < α < 1 in the form (2 − ρ) (2ρ − 1)2 + 3n 3α 2 + n (2 + α)2 = , ε1 = 2 (2 + α) 1 + α + α 2 6 1 − ρ + ρ2

(2.40)

where ρ is the stress ratio σ 2 /σ 1 , equal to (1 + 2α)/(2 + α). The value of ε 1 given by (2.40) may be compared with that predicted by (2.38) for a material with a given value of n. The limit strain is seen to be higher than the instability strain except when α = 0, for which both the conditions predict ε 1 = n. For α < 0,√the neck forms along the line of zero extension, which is inclined at angle tan−1 −α to the direction of the minimum principal strain in the plane of the sheet. The strain ratio then remains constant during the incremental deformation, and the onset of necking is given by (2.29), with Y = a, the limit strain being easily shown to be 2−ρ n ε1 = n , − 12 ≤ α ≤ 0. = 1+ρ 1+α The curve obtained by plotting ε 1 against ε 2 corresponding to localized necking in a given material is called the forming limit diagram, which represents the failure curve in sheet stretching. This will be discussed in Section 6.5 on the basis of a different physical model including anisotropy of the sheet metal.

2.3 Yielding of Notched Strips 2.3.1 V-Notched Strips in Tension Consider the longitudinal extension of a rectangular strip having a pair of symmetrical V-notches of included angle 2α in the plane of the strip. The material is assumed to be uniformly hardened, obeying the von Mises yield criterion and the associated Lévy–Mises flow rule. When the load attains the yield point value, the region of incipient plastic flow extends over the characteristic √ field shown in Fig. 2.7. The triangular region OAB is under a uniaxial tension 3 k parallel to the √ notch face, and the characteristics are straight lines inclined at an angle β = tan−1 2 to the notch face. Within the fan OBC, one family of characteristics are straight lines passing through the notch root, and the state of stress expressed in polar coordinates (r, φ) is given by

68

2

Problems in Plane Stress

Fig. 2.7 Characteristic field in a sharply notched metal strip under longitudinal tension (α ≥ 70.53)

σr = k cos φ, σφ = 2k cos φ, τrφ = k sin φ,

(2.41)

so that the equilibrium equations and the yield criterion are identically satisfied. It follows from (2.5) that cos ψ = 2tan √ φ within the fan. Along the line OB, ψ = sin−1 31 , giving φ = β = tan−1 2. The baseline from which φ is measured therefore makes an angle 2β with the notch face. The curved characteristics in OBC are given by r (dφ/dr) = − cos ψ = −2 tan φ, or

r2 sin φ = constant.

The curved characteristics approach the baseline asymptotically, if continued, and are inflected where φ = π/2 – β. The region OCO is uniformly stressed, and the principal stress axes coincide with the axes of symmetry. From geometry, angle COD is equal to φ 0 – (2β + α – π ), where φ 0 is the value of φ along OC, the corresponding value of ψ being denoted by ψ 0 . Since the algebraically lesser principal stress direction in OCO is parallel to 00, angle COD is also equal to π /4 – ψ 0 /2. The relation cot ψ0 = 2tan φ 0 therefore gives 2 tan φ0 + tan 2 (α + 2β − φ0 ) = 0,

(2.42)

which can be solved for φ 0 when α lies between π − 2β and π /2, the limiting values of φ 0 being 0 and β, respectively. Since σ 1 + σ 2 = 3k cos φ 0 and σ1 − σ2 = k 1 + 3 sin2 φ0 within OCO in view of (2.41), the constraint factor is

2.3

Yielding of Notched Strips

σ1 f =√ = 3k

3 cos φ0 +

69

1 + 3 sin2 φ0 , π − 2β ≤ α ≤ π/2. √ 2 3

(2.43)

The field of Fig. 2.7, which is due to Hill (1952), has been extended by Bishop (1953) in a statically admissible manner to show that the solution is in fact complete. As α decreases from √ π/2, the constraint factor increases from unity to reach its highest value of 2/ 3 when α = π – 2β ≈ 70.53◦ . The field in this case shrinks to a coincident pair of characteristics along the transverse axis of symmetry. In general, the constraint factor is closely approximated by the empirical formula

π − α , π − 2β ≤ α ≤ π/2. f = 1 + 0.155 sin 4.62 2

(2.44)

For all sharper notches, the characteristic field and the constraint factor are identical to those for α = π – 2β. Indeed, by the maximum work principle, the constraint factor cannot decrease when material√is added to reduce the notch angle, while the value of f certainly cannot exceed 2/ 3 since no stress component can exceed 2k in magnitude. The yield point load can be associated with a deformation mode consisting of localized necking along both characteristics through the center D of the minimum section. If the ends of the strip are moved longitudinally with a unit speed relative to D, the particles on the transverse axis of symmetry must move inward with a speed equal to tan(π /4 – ψ/2). The vector representing the relative velocity of particles across the neck is then perpendicular to the other characteristic as required. Strictly speaking, there is no opportunity for the deformation to occur outside the localized necks.

2.3.2 Solution for Circular Notches Consider, now, the longitudinal tension of a strip with symmetrical circular notches of radius c, the roots of the notch being at a distance 2α apart. For sufficiently small values of the ratio a/c, the characteristic field is radially symmetric as shown in Fig. 2.8, and the stress distribution is defined by (2.13) where σ 2 and σ 1 represent the radial and circumferential stresses denoted by σ r and σ φ, respectively. The substitution into the equation of radial equilibrium gives σφ − σr 2k cos θ dθ 2 d σr , = = , or r =√ dr r r dr 3 + tan θ where r is the radius of a generic point in the field. The boundary condition σ r = 0 at the notch surface is equivalent to θ = π/6 at r = c, and the integration of the above equation results in

70

2

Problems in Plane Stress

Fig. 2.8 Characteristic field in a circularly notched strip under longitudinal tension (a/c ≤ 1.071)

√ √ r2 3 π π π = 3 0 − sec θ exp ≤θ ≤ . , 2 2 6 6 3 c

(2.45)

The characteristics coincide when θ = π /3, and this corresponds to r/c ≈ 2.071. The angular span of the circular root covered by the field in this limiting case is 4β – π ≈ 38.96◦ , which is obtained from (2.14) as twice the difference between the values of λ corresponding to θ = π /3 and θ = π /6. The characteristic field is easily constructed using (2.45) and (2.14), and the fact that the polar angle φ measured from the transverse axis is, by (2.16), equal to the decrease in the value of λ from that on the transverse axis. The resultant longitudinal force per unit thickness across the minimum section is P=2 c

a+c

σφ dr = 2

c

a+c

π d , (rσr ) dr = 4k (a + c) sin θ0 − dr 6

where θ 0 is the value of θ at the center of the minimum section where r = a + c, and is directly given by (2.45). The constraint factor is f =

P c 2 π a = √ 1+ sin θ0 − , 0 ≤ ≤ 1.071. √ a 6 c 2 3ka 3

(2.46)

The value of f computed from (2.46) exceeds unity by the amount 0.226a/(a + c) to a close approximation. For higher values of a/c, the characteristics coincide along a central part of the transverse axis, and the longitudinal force per unit thickness is 4k(a – 1.071c) over the central part, and 2k(2.071c) over the remainder of the minimum section, giving the constraint factor

2.3

Yielding of Notched Strips

71

1 c f = √ 2 − 0.071 , a 3

a ≥ 1.071. c √ As a/c increases, f approaches its asymptotic value of 2/ 3, which is the ratio of the maximum shear stresses in pure shear and simple tension. Localized necking would occur at the yield point along the characteristics through C if the rate of work-hardening of the material is not greater than the uniaxial yield stress of the material. Consider, now, the solution for Tresca’s yield criterion and its associated flow rule. Since no stress can exceed 2k, which is now equal to the uniaxial yield stress, the constraint factor f cannot exceed unity. On the other hand, f is unity for a strip of width 2a. Hence, the actual constraint factor is f = 1, whatever the shape of the notch. A localized neck forms directly across the minimum section when the yield point is attained, provided the rate of work-hardening is not greater than 2k.

2.3.3 Solution for Shallow Notches The preceding solutions hold only when the notches are sufficiently deep. In the case of shallow notches, the deformation originating at the notch roots spreads across to the longitudinal free edges. Considering a sharply notched bar with an included angle 2α, a good approximation to the critical notch depth may be obtained by extending the characteristic field further into the specimen. With reference to Fig. 2.9, which shows one-quarter of the construction, the solution involves the extension ABCEFG of the basic field OABC. The extended field is bounded by a stress-free boundary AG generated from a point on the notch face, the material lying beyond AG being assumed unstressed. The construction begins with the consideration of the curvilinear triangle CBE, which is defined by the β-line CB and the conditions of symmetry along CE. Since cotφ = 2 tan ψ along CB, the boundary conditions are

Fig. 2.9 Critical width of a V-notched strip subjected to longitudinal tension

72

2

λ=

π −φ− 2

1 2

tan−1

1 2

tan φ , ω = −γ + φ +

1 2

Problems in Plane Stress

tan−1

1 2

tan φ ,

along CB in view of (2.14), where ω denotes the counterclockwise angle made by the algebraically greater principal stress with the longitudinal axis of symmetry, while γ = σ + 2β – 3 π /4. Since ω = 0 along CE by virtue of symmetry, the values of λ and ω are easily obtained throughout the field CBE using the characteristic relations (2.16). Starting with the known coordinates of the nodal points along CB, and using the fact that φα = ω −

π ψ + 4 2

,

φβ = ω +

π ψ + 4 2

,

(2.47)

where ψ is obtained from (2.14), the coordinates of each point of the field can be determined numerically by the mean slope approximation for small arcs considered along the characteristics, Since AB is a straight characteristic, all the β-lines in the field ABEF are also straight, though not of equal lengths. The angles ψ, ω, and λ. along AF are therefore identical to those along BE. The known values of φ β and φ α along BE and AF furnish the coordinates of the nodal points of AF by simple geometry and the tangent approximation. Since all characteristics meet the stress-free boundary OAG at a constant angle β = 54.74◦ , we have the boundary conditions ψ = 19.47◦ and λ = 25.53◦ along AG. Starting from point A, where ω = π /2 – α, and using (2.16), the values of λ and ω throughout the field AFG are easily determined. The angles φ α and φ β at the nodal points of the field are then computed from (2.47), and the rectangular coordinates are finally obtained by the mean slope approximation. The stress-free boundary AG generated as a part of the construction has a maximum height ω∗ , which is the critical semiwidth of the strip. The numerical computation carried out by Ewing and Spurr (1974) suggests the empirical formula

π a = 1 − 0.286 sin 4.62 −α , ∗ ω 2

(2.48)

which is correct to within 0.5% over the range 70.5◦ ≤ α ≤ 90◦ . This formula actually provides an upper bound on the critical semiwidth, since all specimens wide enough to contain the extended field are definitely not overstressed, as may be shown by arguments similar to those used for the corresponding plane strain problem (Chakrabarty, 2006). When the semiwidth w of the notched bar is less than w∗ , we can find an angle α ∗ > α such that w∗ (α ∗ ) = w. Then, the corresponding constraint factor f(α ∗ ) calculated from (2.46) would provide a lower bound on the yield point load. Indeed, the yield point load cannot be lowered by the addition of material required to reduce the notch angle from 2α ∗ to the actual value 2α. The constraint factor for subcritical widths is closely approximated by the lower bound value, which may be expressed empirically as

2.4

Bending of Prismatic Beams

73

a , a ≤ w ≤ w∗ . f = 1 + 0.54 1 − w In the case of an unnotched bar, the above formula reduces to f = 1. The deformation mode then consists of a localized neck inclined at an angle β = 54.74◦ to the tension axis. Such a neck is also produced in a tensile strip with either a single notch or a symmetric central hole (Fig. 2.10). The tension of single-notched strips has been investigated by Ewing and Richards (1973), who also produced some experimental evidence in support of their theoretical prediction.

Fig. 2.10 Initiation of a localized neck in a flat sheet inclined at an angle β to the direction of tension

2.4 Bending of Prismatic Beams 2.4.1 Strongly Supported Cantilever A uniform cantilever of narrow rectangular cross section carries a load kF per unit thickness at the free end, just sufficient to cause plastic collapse, the weight of the cantilever itself being negligible. The material is assumed to obey the von Mises yield criterion and the Lévy–Mises flow rule. Consider first the situation where the cantilever is rigidly held at the built-in end. If the ratio of the length l of the beam to its depth d is not too large, the characteristic field in the yield point state will be that shown in Fig. 2.11. The deformation mode at the incipient collapse consists of rotation of the rigid material about a center C on the longitudinal axis of symmetry. The solution involves localized necking along EN and localized bulging along NF, together with a simple shear occurring at N. The vector representing the relative velocity of the material is inclined to EF at an angle ψ which varies along the discontinuity. Although a local bulging can only occur in a strain-softening material, the solution may be accepted as a satisfactory upper bound on the collapse load (Green, 1954a) for ideally plastic materials. In the triangular regions ABD and GHK, the state of stress √ is a uniform longitu3 k, the characteristics dinal tension and compression, respectively, of magnitude √ being straight lines inclined at an angle β = tan−1 2 to the free edge. The region ADE is an extension of the constant stress field round the singularity A, the corresponding stresses being given by (2.41), where the polar angle φ is measured from a baseline that is inclined at an angle 2β to the free edge AB. The curved characteristics in ADE have the equation r2 sin φ = constant, while the relation cot ψ = 2 tan φ holds for the characteristic angle ψ. The normal stresses vanish at N, and the stress components along the curve ENF in plane polar coordinates are

74

2

Problems in Plane Stress

Fig. 2.11 Characteristic fields for an end-loaded cantilever with strong support (i/d ≤ 5.65)

σr = k sin θ , σθ = 2k sin θ , τrθ = −k cos θ ,

(2.49)

so that the equilibrium equations and the yield criterion are identically satisfied, the polar angled being measured counterclockwise with respect to the longitudinal axis. The relation cot ψ = 2 cot θ immediately follows from (2.5) and (2.49). The polar equation to the curve ENF, referred to C taken as the origin, is given by r (dθ/dr) = cot ψ = 2 cot θ ,

r 2 cos θ = constant

or

Since the characteristic directions are everywhere continuous, the value of ψ at E may be written as −1

ψ0 = tan

1 cot φ0 2

−1

= tan

1 tan α , 2

2.4

Bending of Prismatic Beams

75

where φ 0 and α are the values of φ and θ , respectively, at E. It follows that φ 0 = π /2 – α. From geometry, angle AEC is 2β – φ 0 + α, which must be equal to π /2 + ψ 0, and substituting for φ 0 and ψ 0 in terms of α, we obtain 2α − tan−1

1 2

tan α = π − 2β.

(2.50)

The solution to this transcendental equation is α ≈ 51.20◦ , which gives ψ 0 ≈ 31.88◦ . The fan angle EAD is δ = α + β– π /2 = ψ 0 /2. It is interesting to note that the state of stress in the deforming region in plane stress bending varies from pure tension at the upper edge to pure shear at the center, whereas in plane strain bending the stress state is pure shear throughout the region of deformation. The geometry of the field is completely defined by the dimensions b and R, representing the lengths AE and CE, respectively. The ratios b/d and R/d depend on the given ratio l/d, and are determined in terms of F/d from the conditions: (a) the sum of the vertical projections of AE and CE is equal to d/2; and (b) the resultant vertical force transmitted across AENFK per unit thickness is equal to kF. The resultant force acting across ENF is most conveniently obtained by regarding CENF as a fully plastic region, the normal and shear stresses across CE and CF being of magnitudes σ and τ directed as shown. The pair of conditions (a) and (b) furnishes the relations R sin α + b cos λ = d/2, R (σ cos α − τ sin α) + b (τ sin λ − σ cos λ) = kF/2,

(2.51)

where σ = 2k sin α and τ = k cos α in view of (2.49), while λ = α + 2β – π /2 ≈ 70.68◦ . Substituting for σ and τ , and inserting the values of σ and λ, equations of (2.51) are easily solved for b/d and R/d as F R F b = 0.6075 − 0.9696 , = −0.0941 + 1.1741 . d d d d

(2.52)

The ratio F/d at the yield point is finally determined from the condition that the resultant moment of the forces acting on AENFK about the center of rotation C is equal to the moment of the applied force about the same point. Thus

KF (l + R cos α − b cos λ) = σ R2 + b2 + 2Rb sin ψ0 + 2τ Rb cos ψ0 . Substituting for σ and τ , using the values of σ , λ, and ψ 0 inserting the expressions for b/d and R/d from (2.52), the above equation may be rearranged into the quadratic

76

2 Problems in Plane Stress

0.4342 −

1 F F2 − 0.2600 − 0.5288 2 = 0, d d d

(2.53)

when F/d has been calculated from (2.53) for a given value of l/d, the ratios b/d and R/d follow from (2.52). Since R = 0 when F/d ≈ 0.080, the proposed field applies only for l/d ≤ 5.65. For higher l/d ratios, the characteristic field is modified in the same way as that in the corresponding plane strain problem, the collapse load being closely approximated by the empirical formula 1 1 d = 0.20 + 2.18 , ≥ 5.65. F d d The values of F/d, b/d, and R/d corresponding to a set of values of l/d < 5.65 are given in Table 2.2. The plane stress values of F/d are found to be about 14% lower than the corresponding plane strain values (Chakrabarty, 2006) over the Table 2.2 Results for an end-loaded cantilever Strong support

Work support

l/d

F/d

b/d

R/d

l/d

F/d

b/d

δ

1.33 1.62 2.00 2.55 3.36 4.72 5.65

0.346 0.287 0.233 0.182 0.137 0.086 0.80

0.272 0.329 0.382 0.431 0.475 0.514 0.530

0.313 0.243 0.180 0.120 0.067 0.019 0

0.328 0.275 0.255 0.177 0.134 0.095 0.079

0.492 0.482 0.477 0.477 0.482 0.492 0.498

0.536 0.531 0.527 0.523 0.518 0.515 0.512

54.74 49.06 43.26 37.53 31.82 26.07 23.49

whole range of values of l/d. The yield point load for a tapered cantilever has been discussed by Ranshi et al. (1974), while the influence of an axial force has been examined by Johnson et al. (1974). Let M denote the bending moment at the built-in end under the collapse load ktF, where t is the thickness of the cantilever. Since the fully plastic moment under pure √ √ bending is M0 = 3 ktd2 , the ratio M/M0 is equal to 4Fl/ 3 d2 , and (2.53) may be written in the form F F M ≈ 1 + 1.23 0.49 − M0 d d

(2.54)

which is correct to within 0.3% for F/d ≤ 0.62. The elementary theory of bending assumes M/M0 ≈1 irrespective of the shearing force. The results for the end-loaded cantilever are directly applicable to a uniformly loaded cantilever if we neglect the effect of surface pressure on the region of deformation. Since the resultant vertical load now acts halfway along the beam, the collapse load for a uniformly loaded cantilever of length 2l is identical to that for an end-loaded cantilever of length l.

2.4

Bending of Prismatic Beams

77

2.4.2 Weakly Supported Cantilever Consider a uniform cantilever of depth d, which fits into a horizontal-slot in a rigid vertical support. The top edge of the beam is clear of the support, so that the adjacent plastic region is able to spread into the slot under the action of a load kF per unit thickness at the free end. The length of the cantilever is l, measured from the point where the bottom edge is strongly held. The characteristic field, due to Green (1954b), is shown in Fig. 2.12. It consists of a pair of triangles ABN and CEF, under uniaxial tension and compression, respectively, and a singular field CEN where one family of characteristics is straight lines passing through C. The√characteristics in the uniformly stressed triangles are inclined at an angle β − tan−1 2 to the respective free edges. The curved characteristics in CEN are given by the polar equation r2 sin φ = constant, where φ is measured from a datum making an angle 2β to the bottom edge CF. The stresses in this region are given by (2.41) with an overall reversal of sign.

Fig. 2.12 Characteristic field for an end-loaded cantilever with weak support (l/d ≥ 1.33)

For a given depth of the cantilever, the field is defined by the angle δ at the stress singularity, and the length b of the characteristic CN. The height of the triangle ABN is a = d − b sin (β + δ) .

(2.55)

The stress is discontinuous across the neutral point N, about which the cantilever rotates as a rigid body at the incipient collapse. The normal and shear stresses over CN are

78

2

Problems in Plane Stress

σ = −2 k cos (β − δ) , τ = sin (β − δ) . √ Since the tensile stress across the vertical plane DN is 3 k parallel to AB, the condition of zero resultant horizontal force across CND and the fact that the resultant vertical force per unit thickness across this boundary is equal to kF furnish the relations √ b [τ cos (β + δ) − σ sin (β + δ)] = 3 ka, b [τ sin (β + δ) + σ cos (β + δ)] = kF. Substituting for σ and τ , and using the value of β, these relations can be simplified to F = sin2 δ, d

√ √ a 1 F 2 2 3 = + − tan (β + δ) b 3 3 b

(2.56)

When F/b and a/b have been calculated from (2.56) for a selected value of δ, the ratio d/b follows from (2.55), while l/b is obtained from the condition of overall moment equilibrium. Taking moment about N of the applied shearing force kF and also of the tractions acting over CND, we get $ √ 2% 3a l b . = cos (β + δ) + cos (β + δ) + b F 2b2

(2.57)

Numerical values of l/d, F/d, b/d, and a/d for various values of δ are given in Table 2.2. As l/d decreases, δ increases to approach the limiting value equal to β. The angle between the two characteristics CN and EN decreases with decreasing l/d, becoming zero in the limit when the triangle CEF shrinks to nothing. The ratios F/d and l/d attain the values 0.328 and 1.332, respectively, in the limiting state. √ The ratio M/M0 at the built-in section (through C), which is equal to 4Fl/ 3d2 , can be calculated from the tabulated values of l/d and F/d. The results can be expressed by the empirical formula F M F ≈ 1 + 1.45 0.34 − , M0 d d

(2.58)

which is correct to within 0.5% for F/d < 0.33. Equations (2.54) and (2.58) are represented by solid curves in Fig. 2.13, the corresponding relations for plane strain bending being shown by broken curves. Evidently, M exceeds M0 over the whole practical range, indicating that the constraining effect of the built-in condition outweighs the weakening effect of the shear except for very short cantilevers. In the case of a strong support, the maximum value of M/M0 is about 1.121 in plane strain and 1.074 in plane stress, corresponding to F/d equal to about 0.28 and 0.25, respectively.

2.4

Bending of Prismatic Beams

79

Fig. 2.13 Influence of transverse shear on the yield moment in the plane stress and plane strain bending of beams

2.4.3 Bending of I-Section Beams Consider an I-beam, shown in Fig. 2.14a, whose transverse section has an area Aω for the web and Af for each flange including the fillets. The depth of the web is denoted by d, and the distance between the centroids of the flanges is denoted by h. It is assumed that the flanges yield in simple tension or compression, while the

Fig. 2.14 Yield point states for I-section beams subjected to combined bending and shear

80

2

Problems in Plane Stress

web yields under√combined bending and shear. The bending moment carried by the flanges is Mf = 3 khAf , since √ one flange is in tension and the other in compression with a stress of magnitude 3 k. For a uniform cantilever of length l, carrying a load ktF at the free end, where t is the web thickness, the bending moment existing at the fixed end is M = ktFl. Hence the bending moment shared by the web is

√ Mω = M − Mf = k tFl − 3hAf giving 4Fl 4h Mω = √ − 2 M0 d 3d

Af Aw

,

where M 0 is the √ fully plastic moment of the web under pure bending, equal to √ 2 3 kd t/4 = 3 kAωd/4. When the cantilever is strongly supported at the builtin end, Mω/M 0 is given by the right-hand side of (2.54), and the collapse load is given by 1.23

l F2 F 4h Af − 1 = 0. + 2.31 − 0.60 − d2 d d d Aw

(2.59a)

For a weak end support, Mω/M 0 is given by the right-hand side of (2.58), and the equation for the collapse load becomes 1.45

l F2 F 4h Af − 1 = 0. + 2.31 − 0.49 − d2 d d d Aw

(2.59b)

Equation (2.59), due to Green (1954b), is certainly valid over the practical range of values of l/d for standard I-beams. Since the flanges do not carry any shearing load, Mf = M0 – M 0 , where M0 is the fully plastic moment of the I-beam under pure bending. The relation Mf = M – Mw therefore gives M =1+ M0

M0 Mw − 1 , M0 M0

M0 4h =1+ M0 d

Af Aw

.

The relationship between the bending moment and the shearing force at the yield point state of an I-beam is now obtained on substitution from (2.54) and (2.58), where M/M0 is replaced by Mw /M 0 . Considering the strong support, for instance, we have & F F 4h Af M = 1 + 1.23 0.49 − 1+ . (2.60) M0 d d d Aω The variation of M/M0 with F/d is displayed in Fig. 2.14b. A satisfactory lower bound solution for M/M0 at the yield point of an I-beam has been derived by Neal

2.5

Limit Analysis of a Hollow Plate

81

(1961). The influence of axial force on the interaction relation, based on the characteristic field in plane stress, has been investigated by Ranshi et al. (1976).

2.5 Limit Analysis of a Hollow Plate 2.5.1 Equal Biaxial Tension A uniform square plate, whose sides are of length 2α, has a circular hole of radius c at its center. The plate is subjected to a uniform normal stress σ along the edges in the plane of the plate. As the loading is continued into the plastic range, plastic zones spread symmetrically outward from the edge of the hole and eventually meet the outer edges of the plate when the yield point is reached. We begin by considering the von Mises yield criterion and its associated Lévy–Mises flow rule. For a certain range of values of c/a, the characteristic field would be that shown in Fig. 2.15a. The stress distribution within the field is radially symmetrical with the radial and circumferential stresses given by (2.13), where σ 1 = σ φ and σ 2 = σ r , the spatial distribution of the angle θ being given by (2.45). Assuming θ = α at the external boundary r = a, where ar = a, we obtain

√ 2 σ π c2 2 π = √ sin α − , 2 = √ cos α exp − 3 α − . Y 6 6 a 3 3

(2.61)

The deformation mode at the incipient collapse consists of localized necking along the characteristics through A, permitting the rigid corners to move diagonally

Fig. 2.15 Equal biaxial tension of a square plate with a central circular hole. (a) 0.483 ≤ c/a ≤ and (b)0.143 ≤ c/a ≤ 0.483

82

2

Problems in Plane Stress

outward. Since the characteristics of stress and velocity exist only over the range π /6 ≤ α ≤ π /3, the solution is strictly valid for 0.483 ≤ c/a ≤ 1. When α = π /3, the two characteristics coincide at A, and σ attains the value 0.577 Y. For c/a ≤ 0.483, (2.61) provides a lower bound on the yield point load, since the associated stress distribution is statically admissible in the annular region between the hole and the broken circle of radius a, shown in Fig. 2.15b. The remainder of the plate is assumed to be stressed below the yield limit under balanced biaxial stresses of magnitude σ , a discontinuity in the circumferential stress being allowed across the broken circle. On the other hand, an upper bound solution is obtained by extending the localized necks as straight lines from r = c∗ = 2.071c to r = a, permitting the same mode of collapse as in (a). Since σ φ =2σ r = 2k along the straight part of the neck, the longitudinal force per unit thickness across BAE is

a

aσ = c

σφ dr =

c∗ c

d c∗ 2Y , (rσr )dr + 2k a − c∗ = √ a − dr 2 3

where the second step follows from the equation of stress equilibrium. The upper bound therefore becomes 2Y c σ = √ 1 − 1.035 , a 3

0.143 ≤

c ≤ 0.483. a

(2.62a)

For c/a ≤ 0.143, a better upper bound is provided by the assumption of a homogeneous deformation mode in which the rate of plastic work per unit volume is 2U/a, where U is the normal velocity of each side of the square. Since the rate of external work per unit plate thickness is 8aσ U, we obtain the upper bound π c2 σ =Y 1− 2 , 4a

0≤

c ≤ 0.143. a

(2.62b)

The difference between the lower and upper bounds, given by (2.61) and (2.62), respectively, is found to be less than 3% over the whole range of values of c/a. When the material yields according to the Tresca criterion, a lower bound solution is obtained from the stress distribution σ r = Y(1– c/r), σ φ = Y in the annulus c ≤ r ≤ a, and σ r = σ φ = σ in the region r = a. The continuity of the radial stress across r = a gives the lower bound a = Y(1 – c/a). To obtain an upper bound, we assume localized necking along the axes of symmetry normal to the sides of the square, involving a diagonally outward motion of the four rigid corners with a relative velocity v perpendicular to the necks. For a unit plate thickness, the rate of internal work in the necks is 4(a – c)vY, while the rate of external work is 4aσ v, giving the upper bound σ = Y(1 – c/a). Since the upper and lower bounds coincide, it is in fact the exact solution for the yield point stress.

2.5

Limit Analysis of a Hollow Plate

83

2.5.2 Uniaxial Tension: Lower Bounds Suppose that the plate is brought to the yield point by the application of a uniform normal stress σ over a pair of opposite sides of the square. The effect of the circular cutout is to weaken the plate so that the yield point value of σ is lower than the uniaxial yield stress Y. A lower bound solution for the yield point stress may be obtained from the stress discontinuity pattern of Fig. 2.16a, consisting of four uniformly stressed regions separated by straight lines, across which the tangential stress is discontinuous. The material between the circular hole of radius ρa the inner square of side ρa is assumed stress free. The conditions of continuity of the normal and shear stresses across each discontinuity are given by (2.33). If the principal stresses are denoted by the symbols s and t where s ≤ t, the stress boundary conditions require t1 = σ , s3 = 0, t4 = 0, where the subscripts correspond to the numbers used for identifying the regions of uniform stress. Let α denote the counterclockwise angle which the direction of the algebraically lesser principal stress in region 2 makes with the vertical. Then the clockwise angles made by this principal axis with the discontinuities bordering regions 1, 3, and 4 are θ1 = α − θ1 , θ3 =

π π + α − θ3 , θ4 = − −α . 2 4

Fig. 2.16 Uniaxial tension of a square plate with a circular hole. (a) Stress discontinuities and (b) graphical representation of yields equalities

84

2

Problems in Plane Stress

From geometry, the counterclockwise angles made by the algebraically greater principal stress axis with the line of discontinuity in regions 1, 3, and 4 are given by π tan θ1 = 1 − ξ , tan θ3 = ρ/ (ξ − ρ) , θ4 = − . 4 Considering the discontinuity between regions 4 and 2, and using (2.33), the quantities (s2 + t2 )/s4 and (s2 – t2 )/s4 can be expressed as functions of α. The consideration of the discontinuity between regions 3 and 2 then furnishes α and the ratio t3 /s4 . Finally, the continuity conditions across the boundary between regions 1 and 2 furnish s4 and s1 in terms of the applied stress σ . The results may be summarized in the form ⎫ 2ρ ρσ ⎪ ⎪ , tan 2α = , s1 = ⎪ ⎪ ⎪ (1 − ρ) (1 − ξ ) ξ ⎪ ⎪ ⎬ 2 2 σ (ξ − 2ρ) σ ξ + 4ρ (2.63) , s2 − t2 = , s2 + t 2 = ⎪ ξ (1 − ρ) ξ (1 − ρ) ⎪ ⎪ ⎪ ⎪ ρσ σ ⎪ ⎪ , s4 = . −t3 = ⎭ (1 − ρ) (ξ − ρ) 1−ρ Since regions 3 and 4 are in uniaxial states of stress, the magnitudes of t3 and s4 must not exceed the yield stress Y. For the von Mises criterion, the required inequalities in regions 1 and 2 follow from (2.2) and (2.63). When ρ is sufficiently small, the greatest admissible value of a is that for which region 4 is at the yield limit, giving the lower bound σ = Y (1 − ρ) ,

0 ≤ ρ ≤ 0.204.

For higher values of ρ, region 4 is not critical, and we need to examine the following yield inequalities for the estimation of the lower bound: σ (1 − ρ) (1 − ξ ) ≤ 1 2 , Y (1 − ρ)2 (1 − ξ )2 − ρ (1 − ρ) (1 − ξ ) + ρ 2 / ξ (1 − ρ) σ ≤ 1 2 , Y ξ 2 − ρξ + 4ρ 2 /

(1 − ρ) (ξ − ρ) σ ≤ , Y ρ

(2.64a)

(2.64b,c)

The parameter ξ must be chosen in the interval 0 ≤ ξ ≤ 1 such that the inequalities (2.64) admit the greatest value of σ . If the right-hand sides of these inequalities are plotted as functions of ξ for a given ρ, the greatest admissible value of σ /Y is the largest ordinate of the region below all the curves, as indicated in Fig. 2.16b. Thus, a/Y is given by the point of intersection of the curves (a) and (b), if this point is below the line (c), and by the point of intersection of (a) and (c), if curve (b) passes above this point. It turns out that the former arises for 0.204 ≤ ρ ≤ 0.412, and the latter for 0.412 ≤ p ≤ 1.

2.5

Limit Analysis of a Hollow Plate

85

When the Tresca criterion is adopted, the inequalities corresponding to regions 1 and 2 only are modified, the stress distribution being statically admissible if s1 ≤ Y, s2 − t2 ≤ Y, − t3 ≤ Y, s4 ≤ Y. Since s2 – t2 is greater than s4 over the whole range in view of (2.63), it is only necessary to consider the first three of the above inequalities. To obtain the best lower bound, the first two conditions should be taken as equalities for relatively small values of ρ, while the first and third conditions should be taken as equalities for relatively large values of ρ. Using (2.63), the results may be expressed as ρ=

ξ (1 − ξ ) σ (1 − ξ ) (1 − ρ) = , √ , ρ (3ξ − 2) 2 − ξ Y

1+ρ , ξ= 2

σ (1 − ρ)2 = , Y 2ρ

0.44 ≤ ρ ≤ 1.

⎫ 0 ≤ ρ ≤ 0.44, ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(2.65)

The lower and upper bound solutions given in this section are essentially due to Gaydon and McCrum (1954), Gaydon (1954), and Hodge (1981). All the bounds discussed here apply equally well to the uniaxial tension of a square plate of side 2a, containing a central square hole with side 2c, where c = ρa.

2.5.3 Uniaxial Tension: Upper Bounds An upper bound on the yield point stress is derived from the velocity field involving straight localized necks which run from the edge of the hole to the stress-free edges of the plate, as shown in Fig. 2.17a. The rigid triangles between the two pairs of neck move away vertically toward each other, while the rigid halves of the remainder of the plate move horizontally at the incipient collapse. Let v denote the velocity of one side of the neck relative to the other, and ψ the angle of inclination of the neck to the relative velocity vector.For a von Mises material, the rate of plastic work per unit volume in the neck is kν 1 + 3 sin2 ψ/b in view of (2.28), where b is the width

Fig. 2.17 Velocity discontinuity patterns for the plastic collapse of a uniaxially loaded square plate with a circular hole

86

2

Problems in Plane Stress

of the neck. Since the length of each neck is a(1 – ρ) cosec β, where β is the angle of inclination of the neck to the free edge of the plate, the rate of internal work in the necks per unit plate thickness is W = 4kva (1 − ρ) cos ecβ 1 + 3 sin2 ψ.

(2.66)

Equating this to the rate of external work, which is equal to 4σ aU = 4σ avcos (β – ψ) per unit thickness, we obtain the upper bound (1 − ρ) cos ecβ 1 + 3 sin2 ψ σ . = √ Y 3 cos (β − ψ) Minimizing σ with respect to β and ψ, it is found that the best upper bound corresponds to β = π /4 + ψ/2 and ψ = sin−1 31 , giving σ = Y (1 − ρ) .

(2.67)

Since the upper bound coincides with the lower bound in the range 0 ≤ ρ ≤ 0.204, the exact yield point stress is Y(l – ρ) for a von Mises material over this range. For a Tresca material, the rate of plastic work per unit volume in a neck is kv(l + sin ψ)/b in view of (2.28), whatever the state of stress in the neck. The upper bound solution is therefore modified to σ (1 − ρ) cos ecβ (1 + 3 sin ψ) = . Y 2 cos (β − ψ) This has a minimum value for β = ψ = π /2, and the best upper bound is precisely that given by (2.67). The necks coincide with the vertical axis of symmetry, and the two halves of the plate move apart as rigid bodies at the incipient collapse. For relatively large values of ρ, a better upper bound is obtained by assuming that each quarter of the plate rotates as a rigid body with an angular velocity ω about a point defined by the distances ξ a and ηa as shown in Fig. 2.17b. The deformation mode involves localized necking and bulging, with normal stresses of magnitude 2k acting along the horizontal and vertical axes of symmetry. Since the normal component of the relative velocity vector is of magnitude ω|x – ξ a| along the x-axis, and to ω|y – ηa| along the y-axis, the rate of internal work per unit thickness of the quarter plate is W = 2 kω

a ρa

|x − ξ a |dx+ |

a

ρa

|y − ηa| dy .

Carrying out the integration, the result may be expressed as W = 2 kωa2

1 + ρ2 − (1 + ρ) (ξ + η) + ξ 2 + η2 .

(2.68)

2.5

Limit Analysis of a Hollow Plate

87

The rate of external work on the quarter plate is σ aU, where U = aω(η − 12 ) is the normal component of the velocity of the center of the loaded side. Equating the rates of external and internal work done, and setting to zero the partial derivatives of σ with respect to ξ and η, we get 2ξ = 1 + ρ,

2η = (1 + ρ) +

σ , 2k

for the upper bound to be a minimum. The best upper bound on a is therefore given by

σ 2 2k

+ 2ρ

σ 2k

− 2 (1 − ρ)2 = 0.

It is easily verified that the conditions for a minimum σ are the same as those required for equilibrium of the quarter plate under the tractions acting along the neck and the external boundaries. The solution to the above quadratic is σ = −ρ + 2k

2 − 4ρ + 3ρ 2 ,

(2.69)

√ where k = Y/ 3 for a von Mises material, and k = Y/2 for a Tresca material. Evidently, (2.67) should be used in the range 0 ≤ ρ ≤ 0.42, and (2.69) in the range of 0.42 ≤ ρ ≤ 1 for a von Mises material. The ranges of applicability of (2.67) and (2.69) for a Tresca material are modified to 0 ≤ ρ ≤ 13 and 13 ≤ ρ ≤ 1, respectively. The upper and lower bound solutions are compared with one another in Fig. 2.18a and b, which correspond to the von Mises and Tresca materials, respectively.

Fig. 2.18 Bounds on the collapse load for a square plate with a circular hole. (a) von Mises material and (b) Tresca material

88

2

Problems in Plane Stress

2.5.4 Arbitrary Biaxial Tension Suppose that a uniform normal stress λσ (0 ≤ λ ≤ 1) is applied to the horizontal edges of the plate, in addition to the stress σ acting on the vertical edges. For a given value of λ, let σ be increased uniformly to its yield point value. A graphical plot of λσ against σ at the yield point state defines a closed interaction curve such that points inside the curve represent safe states of loading. Since such a curve must be convex, a simple lower bound may be constructed by drawing a straight line joining the points representing the lower bounds corresponding to uniaxial and equal biaxial tensions acting on the plate. Upper bounds on the yield point stress for an arbitrary λ can be derived on the basis of the velocity discontinuity patterns of Fig. 2.17. Considering mode (a), the rate of external work done by the stress λσ is obtained from the fact that the speed of the rigid triangle is equal to –v sin(β – ψ), the length of its base being 2a (1 – ρ)cot β. Equating the total external work rate to the internal work rate given by (2.66), we get (1 − ρ) 1 + sin2 ψ

σ = Y sin β cos (β − ψ) − λ (1 − ρ) cos β sin (β − ψ) for a von Mises material. The upper bound has a minimum value when β = π /4 + ψ/2 and 3 sin ψ = (1 + z)/(l – z), where z = λ (l – ρ), the best upper bound for the considered deformation mode being σ 1−ρ , λ (1 − ρ) ≤ 0.5. = Y (1 − λ) (1 − ρ) + λ2 (1 − ρ)2

(2.70)

When λ(l–ρ) ≥ 0.5, the best configuration requires β = π /2, giving σ = 2Y(l − √ ρ)/ 3 as the upper bound. For a Tresca material, the quantity 1 + 3 sin2 ψ in (2.66) must be replaced by (1 + sin ψ), and the upper bound is then found to be σ = Y(1– ρ), on setting β = ψ = π /2 for all values of λ. If the rotational mode (b) is considered for collapse, the rate of external work per unit thickness is σ a2 ω(η − 12 ) due to the horizontal stress, and −λσ a2 ω(ξ − 12 ) due to the vertical stress. Equating the total external work rate to the internal work rate given by (2.68), and minimizing σ /2k with respect of ξ and η, we get 2ξ = (1 + ρ) −

λσ , 2k

2η = (1 + ρ) +

σ , 2k

and the best upper bound on the yield point stress is then given by σ 2

σ

+ 2ρ (1 − λ) − 2 (1 − ρ)2 = 0 1 + λ2 2k 2k

(2.71)

for both the von Mises and Tresca materials with the appropriate value of k. When λ = 0, E (2.70) and (2.71) coincide with (2.67) and (2.69), respectively. When

2.6

Hole Expansion in Infinite Plates

89

λ = 1, these bounds do not differ appreciably from those obtained earlier for the von Mises material, while coinciding with the exact solution for the Tresca material. For sufficiently small values of ρ, a better bound for the von Mises √ material is obtained by dividing the right-hand side of (2.62b) by the quantity 1 − λ + λ2 , which is appropriate for an arbitrary λ. The bounds on the collapse load for a square plate with reinforced cutouts have been discussed by Weiss et al. (1952), Hodge and Perrone (1957), and Hodge (1981).

2.6 Hole Expansion in Infinite Plates 2.6.1 Initial Stages of the Process An infinite plate of uniform thickness contains a circular hole of radius a, and a gradually increasing radial pressure ρ is applied round the edge of the hole, Fig. 2.19. While the plate is completely elastic, the radial and circumferential stresses are the same as those in a hollow circular plate of infinite external radius and are given by σr = −

ρa2 , r2

σθ =

ρa2 . r2

Fig. 2.19 Elastic and plastic regions around a finitely expanded circular hole in an infinite plate

Each element is therefore in a state of pure shear, the magnitude of which is the greatest at r = a. Yielding therefore begins at the edge of the hole when ρ is equal to the shear yield stress k. If the pressure is increased further, the plate is rendered plastic within some radius c, the stresses in the nonplastic region being σr = −

kc2 kc2 , σ = , θ r2 r2

r ≥ c.

90

2

Problems in Plane Stress

Since the stresses have opposite signs, the velocity equations must be hyperbolic in a plastic region near the boundary r = c, with the characteristics inclined at an angle π /4 to the plastic boundary. Over a range of values of c/a, the plastic material would be entirely rigid. The equation of equilibrium is σθ − σr ∂σr = . ∂r r If the von Mises criterion is adopted, the stresses may be expressed in terms of the deviatoric angle φ as σr = −2k sin

π

+φ ,

6

σθ = 2k sin

π 6

−φ ,

(2.72)

where φ = 0 at r = c. Inserting the relation σ θ – σ r > = 2k cos φ into the equilibrium equation, we have cos

π

+φ

6

∂φ ∂r

=−

cos φ . r

Using the boundary condition at r = c, this equation is readily integrated to give the radial distribution of φ as √ c2 = e 3φ cos φ. 2 r

The relationship between the applied pressure ρ and the plastic boundary radius c is given parametrically in the form ρ = 2k sin

π 6

+α ,

√ c2 3α = e cos α, a2

(2.73)

where α is the value of φ at r = a. As α increases from zero, ρ increases from k. The pressure attains its greatest value 2k when α = π /3, giving c = a

1 π/√3 e 2

1/2

≈ 1.751.

The characteristics at this stage envelop the edge of the hole, which coincides with the direction of the numerically lesser principal stress, equal to -k. If the hole is further expanded, the plastic boundary continues to move outward, and the inner radius p of the rigid part of the plastic material is such that c/ρ = 1.751 throughout the expansion. Since an ideally plastic material cannot sustain a stress greater than 2k in magnitude, the plate must thicken to support the load which must increase for continued expansion. If the material yields according to the Tresca criterion, σ θ – σ r = 2k in a plastic annulus adjacent to the boundary r = c. Substituting in the equilibrium equation and integrating, we obtain the stress distribution

2.6

Hole Expansion in Infinite Plates

91

c σr = −k 1 + 2In , r

c σθ = k 1 − 2In . r

(2.74)

in view of the continuity of the√ stresses across r = c. The applied pressure attains its greatest value 2k when c/a = e ≈ 1.649, and the corresponding circumferential stress vanishes at the edge of the hole. A further expansion of the hole must involve thickening of the plate, while a rigid annulus of plastic material exists over the region ρ ≤ r≤ c, where c/p = 1.649 at all stages of the continued expansion.

2.6.2 Finite Expansion Without Hardening A solution for the finite expansion of the hole will now be carried out on the basis of the Tresca criterion, neglecting work-hardening. The circumferential stress then changes discontinuously across r = ρ from zero to – k, the value required by the condition of the zero circumferential strain rate in the presence of thickening. The angle of inclination of the velocity characteristics to the circumferential direction √ changes discontinuously from cot−1 2 to zero across r = ρ. In the plastic region defined by a≤ r≤ p, the equations defining the stress equilibrium and the Lévy– Mises flow rule are ⎫ ∂ 2σr − σθ v ⎪ h ∂v ⎪ = ,⎪ (hσr ) = (σθ − σr ) , ∂r r ∂r 2σθ − σr r ⎬ (2.75) σr + σθ v ∂h 1 ∂h ⎪ +v =− , ⎪ ⎪ ⎭ h ∂ρ ∂r 2σθ − σr r where h is the local thickness and v the radial velocity with ρ taken as the time scale. These equations must be supplemented by the yield criterion which becomes σ r = –2k in the region r ≤ p. The set of equations (2.75) is hyperbolic with characteristics dρ = 0 and dr – vdρ = 0 in the (r, ρ)-plane. The solution for a hole expanded from a finite radius may be obtained from that expanded from zero radius by discarding the part of the solution which is not required. Indeed, it is immaterial whether the pressure at any radius is applied by an external agency or by the displacement of an inner annulus (Hill, 1949). Since the plate is infinite, the stress and velocity at any point must be functions of a single parameter ξ = r/p, so that ρ

d ∂ = , ∂r dξ

ρ

∂ d = −ξ . ∂ρ dξ

Setting σ r = –2k and σ θ = –2ks in (2.75), where s is a dimensionless stress variable; they are reduced to the set of ordinary differential equations dh h = (s − 1) , dξ ξ

2−s v dv =− , dξ 1 − 2s ξ

1+s h ξ dh 1− = . v dξ 1 − 2s ξ (2.76)

The elimination of dh/dξ between the first and third equations of (2.76) leads to

92

2

v (1 − s) (1 − 2s) , = ξ 2 1 − s + s2

Problems in Plane Stress

dv (1 − s) (2 − s) . =− dξ 2 1 − s + s2

Eliminating v between the above pair of equations, we have ξ

d dξ

2s − s2 1 − s + s2

=

3 (1 − s)2 . 1 − s + s2

(2.77)

In view of the boundary condition s = 12 when ξ = 1, the integration of (2.77) results in 1 1 3 (1 − s)2 1 1 − 2s 1 −1 1 − 2s + tan . (2.78) In =− + In √ √ ξ 3 1−s 2 1 − s + s2 3 3 The elimination of ξ between (2.77) and the first equation of (2.76) gives d h dh

2s − s2 1 − s + s2

=−

3 (1 − s) . 1 − s + s2

If the initial thickness of the plate is denoted by h0 , then h = h0 when s = 12 , and the above equation is integrated to h 1 − 2s 2 . = [2 (1 − s)]−1/3 exp √ tan1 √ h0 3 3

(2.79)

If r0 denotes the initial radius to a typical particle, the incompressibility of the plastic material requires h0 r0 dr0 = hr dr at any given stage of the expansion. Since r0 = ρ when r = p, we obtain the relation r2 1− 2 =2 ρ

1

ξ

h ξ dξ , h0

(2.80)

where the integration is carried out numerically using the relations (2.78) and (2.79). As ξ decreases from unity, s decreases from 0.5 and becomes zero at r = ρ ∗ , which is given by In

π ρ 1 1 = − + (In 3) + √ ≈ 0.518, ∗ ρ 3 2 6 3

c ρast

≈ 2.768,

in view of (2.78). The thickness ratio at r = p∗ is h∗ /h0 ≈ 1.453 in view of (2.79). The solution is therefore complete for c/a ≤ 2.768, where a is the current radius of the hole. For larger expansions of the hole, σ θ becomes positive when r/c ≤ 0.361, and the yield criterion reverts to σ θ – σ r = 2k. Equation (2.76) is then modified in such a way that an analytical solution is no longer possible. A numerical solution furnishes a/c ≈ 0.280 when the hole is expanded from zero radius, the value of h/h0 at the edge of the hole being 3.84 approximately. This is in close agreement with the value obtained experimentally by Taylor (1948a).

2.6

Hole Expansion in Infinite Plates

93

2.6.3 Work-Hardening von Mises Material When the material work-hardens and obeys the von Mises yield criterion with the Lévy–Mises flow rule, (2.75) must be supplemented by the yield criterion which is written parametrically through an auxiliary angle φ as

π 2σ σr = − √ sin +φ , 6 3

π 2σ σθ = − √ sin −φ , 6 3

(2.81)

where σ is the current yield stress in uniaxial tension or compression. We suppose that the uniaxial stress–strain curve is represented by the equation σ = σ0 1 − me−ne ,

(2.82)

where σ 0 and n are the empirical constants. The initial yield stress is Y = (1− m)σ 0 , and the current slope of the stress–strain curve is H = n(σ 0 – σ ), which decreases linearly with increasing stress. The material work-hardens only in the region r ≤ ρ, where ρ = 0.571c, since the plastic material beyond this radius remains rigid. It is convenient to define the dimensionless quantities ξ=

r , ρ

η=

h , h0

s=

σ . σ0

We consider the expansion of a hole from zero radius, so that the stresses and strains in any element depend only on ξ . On substitution from (2.81), the last two equations of (2.75) become √ dv 3 + tan φ v =− √ , dξ 3 − tan φ ξ

ξ −v η

2 tan φ dη v =− √ . dξ ξ 3 − tan φ

(2.83)

To obtain the differential equation for s, we write the expression for the circumferential strain rate in terms of the material derivative by using the Lévy–Mises flow rule. Thus v 2σθ − σr ∂σ ∂σ = +v . r 2Hσ ∂ρ ∂r Substituting for σ r , σ θ , and H, and introducing the parameter ξ = r/ρ, this equation is reduced to ξ − v ds 2n sec φ v =− √ . (2.84) 1 − s dξ ξ 3 − tan φ Inserting the expressions for σ r and σ θ from (2.81) into the first equation of (2.75) gives

94

2

Problems in Plane Stress

π dφ 2 1 dη 1 ds + +φ +ξ = −√ tan . η dξ s dη 6 dξ 3 − tan φ

Eliminating dη/dξ and ds/dξ from the above equation by means of (2.83) and (2.84), we finally obtain the differential equation for φ in the form dφ

√ 3 − tan φ dξ

π √ v √ 1−s = 1 + 3 tan φ sec + φ − 2. 3 sec φ + n ξ s 6

(ξ − v)

(2.85)

Equations (2.84), (2.85), and the first equation of (2.83) must be solved simultaneously for the three unknowns v, s, and φ, using the boundary conditions v = 0, s = 1 – m, and φ = π /3 when ξ = 1. The second equation of (2.83) can be subsequently solved for η under the boundary condition η = 1 when ξ = 1. The expressions for all the derivatives given by (2.83) to (2.85) become indeterminate at ξ = 1, but the application of L’Hospital’s rule furnishes √ 3 dφ dv dη 1 ds λ = , = =− , =− , ξ = 1, dξ 2 dξ dξ 2 (1 + λ) dξ 2 (1 + λ) √ where λ = (2/ 3)mn/(1−m). The first derivatives of all the physical quantities are therefore discontinuous across ξ = 1. For a nonhardening material, it is easy to see that the stress gradients ∂σ r /∂r and ∂σ θ /∂ r have the values zero and −3k/2ρ, respectively, just inside the radius r = ρ. Once the thickness distribution has been found, the initial radius ratio r0 /ρ to a typical element can be calculated from (2.80) for any assumed value of ξ . The current radius σ for a hole expanded from an initial radius σ 0 is obtained from the relation 1 a20 1 ηξ dξ = 1− 2 . 2 ρ a/ρ The stress distribution is shown graphically in Fig. 2.20 for a material with m = 0.60 and n = 9.0. The thickness distribution is displayed in Fig. 2.21 for both work-hardening and nonhardening materials. The selected material is similar to that used by Alexander and Ford (1954), who analyzed the corresponding elastic/plastic problem using the Prandtl–Reuss theory. A rigid/plastic analysis for a plate of variable thickness has been given by Chern and Nemat-Nasser (1969). From the known variations of r0 /ρ, s, and φ with ξ , the internal pressure necessary to expand a circular hole from an internal radius a0 to a final radius a is found by using the first equation of (2.81). On releasing an amount q of the expanding pressure, the plate is left with a certain distribution of residual stresses. This is obtained by adding the quantities q(a2 /r2 ) and −q(a2 /r2 ) to the values of σ r , and σ θ , respectively, at the end of the expansion, so long as the unloading is elastic. For sufficiently large values of q, secondary yielding would occur on unloading, and the

2.6

Hole Expansion in Infinite Plates

95

Fig. 2.20 Stress distribution in the finite expansion of a circular hole in an infinite plate of workhardening material

Fig. 2.21 Thickness variation in an infinite plate containing a finitely expanded circular hole

analysis becomes quite involved. A complete analysis of the unloading process for c/a ≤ 1.751, taking secondary yielding into account, has been given by Alexander and Ford (1954) and by Chakrabarty (2006).

96

2

Problems in Plane Stress

2.6.4 Work-Hardening Tresca Material Suppose, now, that the material obeys Tresca’s yield criterion and its associated flow rule, the stress–strain curve of the material is given by (2.82). For ρ ≤ r ≤ c, where p = 0.607c, the stress distribution is still given by (2.74), where k = Y/2, since there is no strain hardening in this region. Indeed, the flow rule corresponding to the yield condition σ θ – σ r – Y implies the thickness strain to be zero, and the velocity vanishes identically in view of the incompressibility condition and the boundary condition. For r ≤ ρ, the stresses over a certain finite region would be σ r = – σ and σ θ = 0, corresponding to a corner of the yield hexagon. The associated flow rule gives ε˙ r < 0,

ε˙ θ > 0,

ε˙ z > 0.

The sum of the three strain rates must vanish by the condition of plastic incompressibility. The rate of plastic work per unit volume is σ ε˙ , where ε = – εr , indicating that the relationship between σ and ε is the same as that in uniaxial tension or compression. It follows from the plastic incompressibility equation written in the integrated form that hr = eε = h0 r 0

1−s m

−1/n (2.86)

in view of (2.82), with s = σ /σ 0 . Since σ θ = 0, the first equation of (2.75) reveals that hrσ = h0 ρY in view of the boundary conditions at r = ρ. Equation (2.86) therefore gives Y r0 = e−ε = ρ σ

1−m s

1−s m

1/n .

(2.87)

Differentiating (2.87) partially with respect to r, and noting the fact that ε = ln(∂r0 /∂r), we have −

1−m s

1 1 + s n (1 − s)

1−s m

2/n

1 ∂s = . ∂r ρ

Integration of this equation under the boundary condition s = 1− m at r = ρ, and using the integration by parts, furnishes the result ξ=

1−m s

1−s m

2/n

1−m + mn

s

1−s m

2/n−1

ds s

(2.88)

1−m

where ξ = r/p. The integral can be evaluated exactly for n = 2 and n = 4, but the numerical integration for an arbitrary value of n is straightforward. The thickness change can be calculated from the relation

2.7

Stretch Forming of Sheet Metals

97

ρY 1−m h = = . h0 rσ ξs

(2.89)

This solution will be valid so long as the thickness strain rate is positive. Since the strain is a function of ξ only, this condition is equivalent to dh/dξ < 0, which gives -ds/dξ < s/ξ in view of (2.89). Using the expression for ∂ s /∂r, the condition for the validity of the solution may be written as ξ≤

1−m s

s 1+ n (1 − s)

1−s m

2/n .

This condition will be satisfied for most engineering materials √ for all values of r0 /ρ ≥ 0. For a nonhardening material, h0 /h = ξ and r0 /ρ = 2ξ − 1, giving a/c = 0.303 for a hole expanded from zero radius. For a work-hardening material with m = 0.6 and n = 9.0, it is found that h/h0 ≈ 1.69 and r/c ≈ 0.143 at the edge of the hole when its initial radius is zero. The computed results for the Tresca theory are plotted as broken curves in Figs. 2.20 and 2.21, which provide a visual comparison with the results corresponding to the von Mises theory. The Tresca theory for a hypothetical material with an exponentially rising stress– strain curve has been discussed by Prager (1953), Hodge and Sankaranarayanan (1958), and Nemat-Nasser (1968). A rigid/plastic analysis for the hole expansion under combined radial pressure and twisting moment has been given by Nordgren and Naghdi (1963). An elastic/plastic analysis for the finite expansion of a hole in a nonhardening plate of variable thickness has been presented by Rogers (1967). An elastic/plastic small strain analysis for a linearly work-hardening Tresca material has been given by Chakrabarty (1971).

2.7 Stretch Forming of Sheet Metals 2.7.1 Hydrostatic Bulging of a Diaphragm A uniform plane sheet is placed over a die with a circular aperture and is firmly clamped around the periphery. A gradually increasing fluid pressure is applied on one side of the blank to make it bulge through the aperture. If the material is isotropic in the plane of the sheet, the bulge forms a surface of revolution, and the radius of curvature at the pole can be estimated at any stage from the measurement of the length of the chord to a neighboring point and its corresponding sagitta. The polar hoop strain can be estimated from the radial expansion of a circle drawn from the center of the original blank. The stress–strain curve of the material under balanced biaxial tension obtained in this way is capable of being continued up to fairly large strains before instability. The process has been investigated by Hill (1950b), Mellor (1954), Ross and Prager (1954), Weil and Newmark (1955), Woo (1964), Storakers (1966), Wang and Shammamy (1969), Chakrabarty and Alexander (1970), Ilahi et al. (1981), and Kim and Yang (1985a), among others.

98

2

Problems in Plane Stress

Let r denote the current radius to a typical particle, and r0 the initial radius, with respect to the vertical axis of symmetry. The local thickness of the bulged sheet is denoted by t, and the inclination of the local surface normal to the vertical by φ, as shown in Fig. 2.22. The ratio of the initial blank thickness t0 to the blank radius a is assumed small enough for the bending stress to be disregarded. The circumferential and meridional stresses, denoted by σ θ and σ φ , respectively, must satisfy the equations of equilibrium which may be written in the form ∂ rtσφ = tσθ , ∂r

σφ sin φ =

pr , 2t

(2.90)

where ρ is the applied fluid pressure. If the meridional and circumferential radii of curvature are denoted by ρ φ and ρ θ , respectively, the equations of normal and tangential equilibrium may be expressed as σφ p σθ + = , ρθ ρφ t

σφ P = , ρθ 2t

(2.90a)

where ρθ = r cos ecφ,

ρφ =

∂r sec φ. ∂φ

The elimination of p/t between the two equations of (2.90a) immediately furnishes

Fig. 2.22 Bulging of a circular diaphragm by the application of a uniform fluid pressure

2.7

Stretch Forming of Sheet Metals

99

σθ ρθ =2− . σφ ρφ This equation indicates that σθ ≶ σ φ for ρ θ ≷ ρ θ . The principal surface strains εθ ,εφ and the thickness strain εt at any stage are

r εθ = ln r0

∂r t , εφ = ln . sec φ , εt = ln ∂r0 t0

(2.91)

The condition for incompressibility requires ε t = – (εθ + ε φ ). If the radial velocity is denoted by v the components of the strain rates may be expressed as ˙t v ∂v + φ˙ tan φ, ε˙ t = , ε˙ θ = , ε˙ φ = r ∂r t where the dot denotes rate of change following the particle. Eliminating v between the first two of the above relations, we obtain the equation of strain rate compatibility ∂ (r˙εθ ) = ε˙ φ − φ˙ tan φ. ∂r

(2.92)

It will be convenient to take the initial radius r0 as the independent space variable, and the polar compressive thickness strain ε 0 as the time scale, to carry out the analysis. Introducing an auxiliary angle ψ, representing the angle made by the deviatone stress vector with the direction representing pure shear, the von Mises yield criterion and the associated Lévy–Mises flow rule can be simultaneously satisfied by writing σθ = √2 σ sin π6 + ψ , σφ = √2 σ cos ψ, 3 3 ε˙ θ = ε˙ sin ψ, ε˙ φ = ε˙ cos π6 + ψ ,

(2.93)

where σ and ε˙ are the equivalent stress and strain rate, respectively. Introducing dimensionless variables ξ=

σ pa r0 , s= , q= , a c t0 C

where C is a constant stress, and using the fact that ∂r/∂r0 = cosφ exp(εφ ), r = rφ exp(εφ ), and t = t0 exp(ε1 ) in view of (2.91), (2.90) and (2.92) can be combined with (2.93) to obtain the set of differential equations

π ∂ + ψ exp (−εθ ) , ξ s cos ψ exp −εφ = s cos φ sin ∂ξ 6 √ 3 s sin φ = qξ sec ψ exp 2εθ + εφ , 4

(2.95)

100

2

Problems in Plane Stress

π ∂ + ψ − φ˙ sin φ exp −εφ , ξ ε˙ sin ψ exp (εθ ) = ε˙ cos φ cos ∂ξ 6

(2.96)

These equations must be supplemented by the strain-hardening law σ = Cf(ε), where ε is the equivalent total strain. Since σ θ = σ φ = σ at the pole (ξ = 0), while ε˙ = 0 at the clamped edge (ξ = 1), the boundary conditions may be written as ψ=

π and ε˙ = 1 at ξ = 0; 6

ψ = 0 at ξ = 1.

When the distributions of the relevant physical parameters have been found for any given polar strain ε0 , the shape of the bulge can be determined by the integration of the equation ∂z ∂r = − tan φ = − sin φ exp εφ . ∂r0 ∂r0 Using (2.95), this equation may be written in the more convenient form √ 3 q ∂ z =− ξ sec ψ exp 2εθ + 2εφ , ∂ξ a 4 s

(2.97)

which must be solved numerically under the boundary condition z = 0 at ξ = 1. The polar height h is finally obtained as the value of z at ξ = 0. The polar radius of curvature ρ is given by ρ = a

2σ0 t0 pa

exp (−ε0 ) =

2s0 q

exp (−ε0 ) ,

where σ 0 is the value of σ θ or σ φ the pole ξ = 0. Plastic instability occurs when the pressure attains a maximum, and this corresponds to dρ/ρ being equal to dσ 0 /σ 0 – dε 0 during an incremental deformation of the bulge. This condition can be used to establish the point of tensile instability in the bulging process. Suppose that the values of s, ψ, and φ are known at each point for the mth stage of the bulge. In order to continue the solution, we must find the corresponding distribution of ε˙ . Using the boundary condition ε˙ = 1 at ξ = 0, (2.96) is therefore solved numerically for ε˙ , the quantity φ being found from the values of φ in the previous and current stages of the bulge. The values of ε θ and ε φ for the (m + l)th stage are then obtained from their increments, using (2.93) and an assigned change in ε0 . It is convenient to adopt the power law of hardening σ = Cεn . Since s is a known function of ε, (2.94) can be solved for ψ, assuming a value of q for the (m + l)th stage, and using (2.95). The correct value of q is obtained when the boundary conditions ψ = 0 at ξ = 1 and ψ = π /6 at ξ = 0 are both satisfied.

2.7

Stretch Forming of Sheet Metals

101

If, for a certain stage of the bulge, ε˙ is found to vanish at ξ = 1, indicating neutral loading of the clamped edge, the condition ε˙ = 0 at ξ = 1 must be satisfied at all subsequent stages. The material rate of change of (2.95) gives φ˙ =

π n q˙ √ 3 sin + +ψ − ε˙ + ψ˙ tan ψ tan φ q 6 ε

in view of (2.93). The ratio q˙ /q in this case should be found by substituting the above expression for φ into (2.96), and integrating it under the boundary conditions ε˙ = 1 at ξ = 0, and ε˙ = 0 at ξ = 1, the quantity ψ˙ being given by the previous and current values of ψ. √ Initially, however, it is reasonable to assume 3 tanψ ≈ 1 – nξ 2 as a first approximation, which is appropriate over the whole bulge except at ξ = 1 (when n = 0). Since changes in geometry are negligible, (2.94) reduces in this case to √ 1 ∂ 1 − 3 tan ψ (s cos ψ) = 0. (s cos ψ) + ∂ξ 2ξ This equation is readily integrated under the boundary condition s = s0 at ξ = 0, resulting in √

n

n 3 qξ s0 exp − ξ 2 , φ = s cos ψ = exp ξ 2 , 2 4 2s0 4

(2.98)

in view of (2.95). The power law of hardening permits the Lévy–Mises flow rule to be replaced by the Hencky relations, so that the strain rates in (2.93) are replaced by the strains themselves. Substituting for the strain ratio εφ /εθ into the equation of strain compatibility, obtained by eliminating r between the first two relations of (2.91), we get √ √ 3 ∂εθ φ2 − cot ψ − 3 εθ = − ∂ξ 2ξ 2ξ

(2.99)

to a sufficient accuracy. Inserting the expressions for cot ψ and φ, this equation can be integrated under the boundary condition εθ = 0 at ξ = 1. Since εθ = εθ /2 at ξ = 0, we obtain −1/2

nx 1 ε0 q 3/4 = 2 2ε0 ≈ [8] . dx (1 − nx) exp s0 2 8−n 0 Using the above expression for q, the hoop strain εθ = ε sin ψ can be expressed as a function of εθ and ξ . It is sufficiently accurate to put the result in the form

−3/4 nξ 2 . 1 − nξ 2 ε sin ψ ≈ 12 ε0 1 − ξ 2 1 − 8−n

(2.100)

102

2

Problems in Plane Stress

Since s/s0 = (ε/ε0 )n , the distributions of s, ε, ψ, and φ can be determined from (2.98) and (2.100) for any small value of ε0. The shape of the bulge and the polar height are obtained from the integration of the equation ∂z/∂ξ = -a φ, the result being n q z 1 − ξ2 1 + ≈ 1 + ξ2 , a 4s0 32

h 1 ≈ a 2

1/2 16 + n . ε0 8−n

The initial shape of the bulge is therefore approximately parabolic for all values of n. Since φ changes at the rate φ = φ/2ε0 in view of (2.98), the solution for the von Mises material can proceed by integrating (2.96) as explained earlier. Figure 2.23 shows the surface strain distribution for various values of ε0 in a material with n = 0.2, the variation of ε0 with the polar height being displayed in Fig. 2.24. The theoretical predictions are found to agree reasonably well with available experimental results on hydrostatic bulging. A detailed analysis for the initial deformation of the diaphragm has been given by Hill and Storakers (1980).

Fig. 2.23 Distribution of circumferential and meridional true strains in the hydrostatic bulging process when n = 0.2 (after Wang and Shammamy, 1969)

2.7.2 Stretch Forming Over a Rigid Punch A flat sheet of metal of uniform thickness t0 is clamped over a die with a circular aperture of radius a, and the material is deformed by forcing a rigid punch with a hemispherical head. The axis of the punch passes through the center of the aperture and is normal to the plane of the sheet. The deformed sheet forms a surface of revolution with its axis coinciding with that of the punch. Due to the presence of friction between the sheet and the punch, the greatest thinning does not occur at the

2.7

Stretch Forming of Sheet Metals

103

Fig. 2.24 Variation of the polar thickness strain with polar height during the bulging of a circular diaphragm

pole but at some distance away from it, and fracture eventually occurs at this site. In a typical cupping test, known as the Erichsen test, a hardened steel ball is used as the punch head, and the height of the cup when the specimen splits is regarded as the Erichsen number, which is an indication of the formability of the sheet metal. The process has been investigated experimentally by Keeler and Backofen (1963) and theoretically by Woo (1968), Chakrabarty (1970a), Kaftanoglu and Alexander (1970), and Wang (1970), among others. Finite element methods for the analysis of the stretch-forming process have been discussed by Wifi (1976), Kim and Kobayashi (1978), and Wang and Budiansky (1978). In the theoretical analysis of the forming process, the coefficient of friction μ will be taken as constant over the entire surface of contact. The radius of the punch head, denoted by R, is somewhat smaller than the radius of the die aperture, Fig. 2.25. Let t denote the local thickness of an element currently at a radius r and at an angular distance φ from the pole, the initial radius to the element being denoted by r0 . Over the region of contact, the equations of tangential and normal equilibrium are ∂ rtσφ = tσθ + μpR tan φ, t σθ + σφ = pR, ∂r

(2.101)

where p is the normal pressure exerted by the punch, and r = R sin φ. The elimination of pR between the above equations gives ∂ rtσφ = tσθ (1 + μ tan φ) + μtσφ tan φ. ∂r If the material obeys the von Mises yield criterion, the stresses are given by (2.93). In terms of the dimensionless variables ξ = r0 /a and s = σ /C, where C is a constant stress, the above equation becomes

104

2

Problems in Plane Stress

Fig. 2.25 Stretch forming of a circular blank of sheet metal over a hemispherical-headed punch

∂ ξ s cos ψ exp −εφ ∂ξ

π = s (cos φ + μ sin φ) sin + ψ + μ sin φ cos ψ exp (−εθ ) , 6 where sin φ = (a/R)ξ exp (εθ ). In view of (2.93) and the second equation of (2.101), the pressure distribution over the punch head is given by

√ pR = s sin ψ + 3 cos ψ exp −εθ − εφ . t0 C

(2.103)

The geometrical relation r = R sin φ furnishes φ˙ = ε˙ θ tan φ over the contact region. Substituting into the compatibility equation (2.92), and using (2.93) for the components of the strain rate, we get π ∂ + ψ − tan2 φ sin ψ exp εφ . ξ ε˙ sin ψ exp (εθ ) = ε˙ cos φ cos ∂ξ 6 When the stresses and strains are known at the mth stage of the process, (2.104) can be solved using the condition ε˙ = 1 at ξ = 0, the polar compressive thickness strain ε0 being taken as the time scale. The computed distribution of ε˙ and an assigned increment of ε0 furnish the quantities εθ , εφ , and φ, while s follows from the given stress–strain curve. Equation (2.102) is then solved for ψ at the (m + l)th stage under the boundary condition ψ = π/6 at ξ = 0, and the stresses are finally obtained from (2.93).

2.7

Stretch Forming of Sheet Metals

105

Over the unsupported surface of unknown geometry, the equation of meridional equilibrium is given by (2.101) with p = 0. Since the circumferential and meridional curvatures at any point are sin φ/r and (∂φ/∂r) cos φ, respectively, the equation of normal equilibrium is ∂φ σθ sin φ + σφ cos φ = 0. r ∂r Using the relations (2.91) and (2.93), this may be rewritten as r0 cos ec φ

√ ∂φ 1 = − 1 + 3 tan ψ exp εφ − εθ . ∂r0 2

If the angle of contact is denoted by β, then ξ = ξ ∗ = (R/a) sin β exp(−εθ∗ ) at φ = β, where the asterisk refers to the contact boundary. The integration of the above equation results in ln

tan (φ/2) tan (β/2)

=−

1 2

ξ ξ∗

√ dξ 1 + 3 tan ψ exp εφ − εθ . ξ

(2.105)

The remaining equilibrium equation and the compatibility equation in the dimensionless form are identical to (2.94) and (2.96), respectively. To continue the solution from a known value of β at the mth stage, and the corresponding distributions of ψ and s, (2.96) is solved numerically for ε using the condition of continuity across ξ = ξ ∗ , the distribution of φ˙ being obtained from the previous values of φ. The distribution of φ for the (m + l)th stage is then obtained from (2.105), assuming a value of β and the previous distribution of ψ. The correct value of β is that for which the continuity condition for ψ across ξ = ξ ∗ is satisfied, when (2.94) is solved for ψ with the boundary condition ψ = 0 at ξ = 1. The total penetration h of the punch at any stage can be computed from the formula h R = (1 − cos β) + a a

1 ξ∗

sin φ exp εφ dξ .

(2.106)

in view of the relation ∂z/∂ξ = −a tan φ, where φ is given by (2.105) over the unsupported region. The resultant punch load is P = 2l Rt∗ σφ∗ σ £ sin2 β, and the substitution for t∗ and σφ∗ , furnishes

π P = s∗ sin2 β sin + ψ ∗ exp − εθ∗ + εφ∗ . 2π Rt0 C 6

(2.107)

Equations (2.106) and (2.107) define the load–penetration relation parametrically through β. When the load attains a critical value, a local neck is formed at the thinnest section (leading to fracture) due to some kind of instability of the biaxial stretching.

106

2

Problems in Plane Stress

If the strain hardening is expressed by the power law s = ε n√ , the Hencky theory may be used for the solution of the initial problem. Assuming 3tanψ ≈ 1 − n ξ 2 as a first approximation, and omitting the negligible friction terms in (2.102), it is found that s is given by the first equation of (2.98) throughout the deformed sheet. Furthermore, φ = a ξ /R for 0 ≤ ξ ≤ ξ ∗ , and n aξ ∗2 exp ξ 2 − ξ ∗2 , ξ ∗ ≤ ξ ≤ 1, ξR 4

φ=

in view of (2.105). The strain compatibility equation (2.99), which is the same as that for the stretch-forming process, gives 3/4 3/4 φ2 ∂ sin ψ = − . ε 1 − nξ 2 1 − nξ 2 ∂ξ 2ξ

(2.108)

Substituting for φ, and using the conditions of continuity of ε and ψ across ξ = ξ ∗ , we obtain the expression for the polar thickness strain as β2 ε0 = 2

1+

1 ξ ∗ 2 x∗

x

(1 − nx)

3/4

n ∗ x − x dx , exp 2

(2.109)

where x = ξ 2 and x∗ = ξ ∗2 = (RB/a)2 . The distribution of ψ is now obtained by the integration of (2.108). By (2.106), the polar height is h=

2 1 2 Rβ

1+

1 x∗

x − x∗ dx exp n , 4 x

(2.110)

while the punch load is easily found from (2.107). Starting with a sufficiently small value of ε 0 , for which a complete solution has just been derived, the analysis can be continued by considering (2.104) and (2.96) as explained before. The distribution of thickness strain and the load–penetration relationship are shown graphically in Figs. 2.26 and 2.27 for a material with n = 0.2, assuming R = a and μ = 0.2. The theory seems to be well supported by available experimental results. It is found that the punch load required for a given depth of penetration is affected only slightly by the coefficient of friction.

2.7.3 Solutions for a Special Material The solution to the stretch-forming problem becomes remarkably simple when the material is assumed to have a special strain-hardening characteristic. From the practical point of view, such a solution is extremely useful in understanding the physical behavior of the forming process and in predicting certain physical quantities with reasonable accuracy. The stress–strain curve for the special material is represented by

2.7

Stretch Forming of Sheet Metals

107

Fig. 2.26 Distribution of thickness strain in the stretch-forming process using a hemispherical punch with R = a (after N.M. Wang, 1970)

Fig. 2.27 Dimensionless load–penetration behavior in the punch stretching process using R = a (n = 0.2, μ = 0.2)

108

2

Problems in Plane Stress

σ = Y exp (ε) , where Y is the initial yield stress. The stress–strain curve is unlike that of any real metal, but the solution based on it should provide a good approximation for sufficiently prestrained metals (Hill, 1950b). Considering the hydrostatic bulging process, it is easy to see that the assumed strain-hardening law requires the bulge to be a spherical cap having a radius of curvature ρ, which is given by simple geometry as ρ=

h2 + a2 = a cos ec α, 2h

(2.111)

where α is the semiangle of the cap. Indeed, it follows from (2.90a) that σθ = σφ = σ = pρ/2t when ρθ = ρφ = ρ, indicating that tσ is a constant at each stage. Since t = t0 exp(–ε), we recover the assumed stress–strain curve. Substituting from (2.91) into the relation εθ = εφ = ε/2 given by the flow rule, and using the fact that r = ρ sin φ, we get r0

∂φ = sin φ ∂r0

or

r0 tan (φ/2) = a tan (α/2)

in view of the condition r0 = a at φ = α. The strain distribution over the bulge is therefore given by ε = 2 ln

r 1 + cos φ hz = 2 ln = 2 ln 1 + 2 . r0 1 + cos α a

(2.112)

It follows from (2.112) that ε = 0 at φ = α, indicating that there is no straining at the clamped edge, which merely rotates to allow the increase in bulge height. The magnitude of the polar thickness strain is ε0 = 4 ln sec

α h2 = 2 ln 1 + 2 . 2 a

(2.113)

This relation is displayed by a broken curve in Fig. 2.24 for comparison. To obtain the velocity distribution, we consider the rate of change of (2.112) following the particle, as well as that of the geometrical relation r = a (sin φ/ sin α) , taking α as the time scale. The resulting pair of equations for v and φ˙ may be solved to give v cos φ − cos α z = = , r sin α a

φ˙ =

sin φ r = . sin α a

The rate of change of the above expression for z and the substitution for φ˙ furnish the result z˙ = v cot φ, which shows that the resultant velocity of each particle is along the outward normal to the momentary profile of the bulge.

2.7

Stretch Forming of Sheet Metals

109

The relationship between the polar strain and the polar radius of curvature obtained for the special material provides a good approximation for a wide variety of metals. From (2.111) and (2.113), it is easily shown that

a ≈ 2ε0 exp − 38 ε0 . ρ to a close approximation. When the pressure attains a maximum, the parameter tσ /ρ has a stationary value at the pole, giving 1 1 dσ0 1 dρ 11 − =1+ ≈ σ0 dε0 ρ dε0 8 2ε0 in view of the preceding expression for ρ as a function of εθ . For the simple power law σ0 = Cε0n , the polar strain at the onset of instability therefore becomes ε0 =

4 11

(+2n) .

4 The instability strain is thus equal to 11 for a nonhardening material (n = 0). This explains the usefulness of the bulge test as a means of obtaining the stress– strain curve of metals for large plastic strains. The special material is also useful in deriving an analytical solution for stretch forming over a hemispherical punch head, provided friction is neglected (Chakrabarty, 1970). The stress–strain curve is then consistent with the assumption of a balanced biaxial state of stress throughout the deforming surface. We begin with the unsupported region, for which the equilibrium in the normal direction requires

ρφ σφ 1 ∂r tan φ =− =− σθ ρθ r ∂φ in view of the first equation of (2.90a) with p = 0. The assumption σ θ = σ φ therefore implies ρ θ = –ρφ over the unsupported region, giving ∂r = −r cot φ ∂φ

or

r sin α = , a sin φ

(2.114)

in view of the boundary condition φ = α at r = a. Since r = R sin β at φ = β, the angles β and α are related to one another by the equation sin α =

R 2 sin β a

(2.115)

It may be noted that the meridional radius of curvature changes discontinuously from –R to R across φ = β. The shape of the unsupported surface is given by the differential equation

110

2

Problems in Plane Stress

sin α ∂r ∂z = − tan φ =a , ∂φ ∂φ sin φ which is integrated under the boundary condition z = 0 at φ = α to obtain z tan (φ/2) = sin α ln . a tan (α/2)

(2.116)

The unsupported surface is actually a minimal surface since the mean curvature vanishes at each point. It is known from the geometry of surfaces that the only minimal surface of revolution is the catenoid. Indeed, the elimination of φ between (2.114) and (2.116) leads to the geometrical relation

z α r = sin α cosh cos ecα + ln tan a a 2 which is the equation of a catenoid. The flow rule requires εθ = εφ = ε/2, and the substitution from (2.91) and (2.114) gives r0

∂φ = − sin φ ∂r0

or

r0 tan (α/2) = , a tan (φ/2)

in view of the boundary condition r0 = a at φ = α. The expressions for r and r0 furnish the compressive thickness strain as 1 + cos α r ε = 2 ln = 2 ln , R sin β ≤ r ≤ a. r0 1 + cos φ

(2.117)

Since tσ is constant at each stage due to the assumed strain-hardening law, and the fact that ε = ln (t0 /t) , the first equation of (2.90) is identically satisfied. Over the region of contact, ρθ = ρφ = R, and equilibrium requires σθ = σφ = σ in the absence of friction, giving p = 2σ t/R. In view of the relations εθ = ε φ and r = R sinφ, the initial radius r0 to a typical particle is given by r0

∂φ = sin φ ∂r

or

r0 tan (α/2) tan (φ/2) , = a tan2 (β/2)

in view of the condition of continuity across φ = β. The compressive thickness strain therefore becomes r (1 + cos α) (1 + cos φ) , 0 ≤ r ≤ R sin β, (2.118) = 2 ln ε = 2 ln r0 (1 + cos β)2 The continuity of the strains evidently ensures the continuity of the stresses across the contact boundary. The thickness has a minimum value at the pole when there is no friction between the material and the punch head. The total penetration of the punch at any stage is obtained from (2.116) and the fact that the height of the pole above the contact boundary is equal to R(1–cos β).

2.8

Deep Drawing of Cylindrical Cups

111

Hence tan (β/2) h = (1 − cos β) + sin2 β ln R tan (α/2)

(2.119)

in view of (2.115). Available experimental results indicate that the relationship between h/R and β is practically independent of the material properties when the punch is well lubricated. It is therefore a good approximation to assume that the relationship between h/R and P/2Rt0 C is independent of the strain-hardening characteristic. Then, for the simple power law σ = Cε n , the load–penetration relationship is given parametrically through β by (2.119) and the formula P = 2π Rt0 C

1 + cos β 1 + cos α

2

1 + cos α n sin2 β 2 ln . 1 + cos β

(2.120)

This expression is obtained from the fact that t∗ σ ∗ is equal to t0 C(ε ∗ )n exp(–ε∗ ), where ε ∗ is given by (2.117) with φ = β. It is easily shown that the punch load given by (2.120) does not have a stationary value in the interval 0 < β 2.76. = 4 tan + cot β + 2M0 α α α β2 + 1 (4.115) For 1 ≤ α ≤ 2.76, Equation (4.113) gives a lower value of the collapse load for a simply supported plate. As α tend to infinity, the ratio β/α tends to unity, and the right-hand side of (4.115) tends to the asymptotic value of 2 + π . A graphical plot of the complete upper bound solution, furnished by (4.113) and (4.115), is included as broken lines in Fig. 4.25 for a visual comparison with the exact solution. The collapse load for a simply supported elliptical plate which carries a uniformly distributed load of intensity p may be obtained in a similar manner. In the case of complete collapse of the plate, the rate of internal energy dissipation is the same as that for the centrally loaded plate, but the rate of external work done is now becomes (pw0 /3)(π a b). The work equation therefore furnishes the upper bound solution as 2 b2 α +1 pb2 = 3 1 + . (4.116) =3 M0 α2 a2 The upper bound solution defined by (4.116) does not differ significantly from the exact solution (Sawczuk, 1989), which is known only over the range 1 ≤ α ≤ 3.52. As indicated earlier, the upper bound load would be exactly doubled if the plate were fully clamped. The yield line upper bound for a uniformly loaded rectangular plate has been given in Section 4.5 (ii). When a part of the boundary of the plate is unsupported, a realistic collapse mode requires the center of the yield line fan to be located outside the plate. It is necessary in this case to introduce the factor (1 − ρ1 /ρ) in the last integral of (4.109), where ρ 1 denotes the length of the radius vector to the point of intersection of a generic

4.7

Minimum Weight Design of Plates

293

yield line with the free edge of the plate. Consider, for example, the plastic collapse of a semi-circular plate which is simply supported along the curved edge and is loaded by a uniform line load q per unit length along the straight edge AB, which is unsupported. Assuming the yield line pattern shown in Fig. 4.28b, in which the center O of the fan is taken at a distance a from the free edge, we have π/4 D = M0 w0 −π/4

ρ1 1− ρ

π/4 2ρ 2 ρ 4 1+ 2 − dφ = M0 w0 2 − sec2 φ sec2 φ dφ = M0 w0 . ρ 3 ρ −π/4

(4.117)

in view of the relations ρ1 = a sec φ, ρ = 2a cos φ. If the distance of a generic point of the free edge from its center is denoted by x, then the rate of external work done is a W=

qwdx = qw0 −a

a a 2 ρ1 1 dx = qaw0 1− 1 − sec2 φ sec2 φ dφ = qaw0 . ρ 2 3

−a

−π/4

The work equation W = D finally gives the collapse load q = 2M0 /a, which is essentially due to Johansen (1943). Since the acute angle between the normal to the semi-circular edge and the radius vector nowhere exceeds π/4, the solution is also statically admissible, and the yield line load is therefore the actual collapse load for the considered plate.

4.7 Minimum Weight Design of Plates 4.7.1 Basic Principles Consider the problem of design of a flat plate which is just at the point of collapse under given conditions of loading and support, the thickness of the plate being allowed to vary in such a way that the total volume of the plate is a minimum. The material is assumed to be homogeneous so that the design for minimum volume is identical to that for minimum weight. Let M1 , M2 be the principal bending moments at any point in the plate that collapses under a distribution of transverse pressure p. If w denotes the rate of deflection of the plate whose middle surface is of area A, then pwdA = (M1 κ˙ 1 + M2 κ˙ 2 ) dA = M0 κ˙ 0 dA, where κ˙ 1 ,κ˙ 2 are the principal curvature rates of the middle surface, while κ˙ 0 is an effective curvature rate that depends on the yield function and is given by √ 1/2 2/ 3 κ˙ 12 + κ˙ 1 κ˙ 2 + κ˙ 22 Mises . κ˙ 0 = (1/2) [|κ˙ 1 | + |κ˙ 2 | + |κ˙ 1 + κ˙ 2 |] , Tresca

(4.118)

294

4 Plastic Bending of Plates

The plastic moment M0 depends on the local plate thickness, which is denoted by h in the optimum design. If M1∗ , M2∗ denote the principal moments in any other design which is capable of supporting the given distribution of pressure p, then by the principle of virtual work,

pwdA =

∗ M1 κ˙ 1 + M2∗ κ˙ 2 dA ≤

M0∗ κ˙ 0 dA,

where M0∗ is the plastic moment in the second design, characterized by a thickness ∗ distribution h . The inequality arises from the fact that (κ˙ 1 , κ˙ 2 ) need not be associated with M1∗ , M2∗ . It follows from the preceding relations that

M0 − M0∗ κ˙ 0 dA ≤ 0.

(4.119)

Consider first a sandwich plate which has a light-weight core of constant thickness H between two identical face sheets of variable thickness h made of the given material (h 0, the velocity field is kinematically admissible. For the analysis of the spherical vessel, it is instructive to consider the differential equation of equilibrium involving the shearing force Q. Setting r1 = R, r = R sin φ and pr = –p in the second equation of (5.47) and normalizing the forces by the yield force YH, we have ds + s cot φ + nθ + nφ = 2q. dφ Eliminating s cot φ by means of (5.72), where –q must be written for q, leads to the simple differential equation ds + nθ = q. dφ Introducing the yield condition nθ = 1, and using the boundary condition s = 0 at φ = α, where mφ is a maximum, the above equation is readily integrated to give s = (1 − q) (α − φ) , nθ = 1, α ≤ φ ≤ β. nφ = q − (1 − q) (α − φ) cot φ,

(5.135)

For simplicity, the hexagonal yield condition for the bending moment will be replaced by the square yield condition defined by mθ = ±1 and mφ = ±1. Since κ˙ θ is expected to be positive in view of the assumed mode of collapse, we set mθ = 1 in the second equation of (5.66) to obtain the differential equation d s mφ sin φ = cos φ + sin φ. dφ k Substituting from (5.135), this equation is integrated under the boundary condition mφ = 1 at φ = α to give mφ = 1 −

1−q sin α + (α − φ) cot φ , β ≤ φ ≤ α. 1− k sin φ

(5.136)

The continuity condition mφ = –h2 /H2 at junction φ = β, where sin β = a/R, furnishes the relationship between q and a as ⎫ ⎧ ' ⎨ 2 aH h2 a a⎬ = 1+ 2 . (5.137) (1 − q) (α − β) − 1 − 2 − sin α + ⎩ R ⎭ 4R2 R H Since q < 1 due to the weakening effect of the nozzle, the stress distribution predicted by (5.135) and (5.136) is statically admissible. The velocity distribution

388

5 Plastic Analysis of Shells

can be found from the associated flow rule, which gives λ˙ φ = κ˙ φ = 0, the corresponding velocity equations being du − w = 0, dφ

dw + u = c, dφ

in view of (5.67), where c is a constant velocity. Using the boundary condition u = w = 0 at φ = α, the solution is obtained as w = −c sin (α − φ) ,

u = c [1 − cos (α − φ)] .

(5.138)

It follows from (5.67) and (5.138) that λ˙ θ > 0 and κ˙ θ > 0 when c is positive, indicating that the velocity field is kinematically admissible. The continuity of the velocity vector across φ = β requires w0 = c (cos β − cos α) ,

u0 = c (sin α − sin β) .

The velocity distribution throughout the region of deformation is therefore completely determined in terms of a single constant c, which is of arbitrary magnitude at the incipient collapse. The collapse pressure p must be determined from the condition that the resultant of the meridional and shearing forces is continuous across the interface φ = β (Fig. 5.26). Resolving horizontally and vertically, we get Nβ cos β − Qβ sin β = Q0 ,

Nβ sin β + Qβ cos β = pa/2,

where Q0 is the value of Q in the cylinder at x = 0 and the subscript β refers to quantities at φ = β. The second equation of the above pair is equivalent to (5.72), considered at φ = β, with the necessary sign change of q. Since Nβ cos β − Qβ sin β = YH q cos β − (1 − q) (α − β) cosecβ in view of (5.135), while Q0 is the value of Yhs at ξ = 0 given by (5.134), the collapse pressure is given by ⎧ ⎫ ⎨ ' ⎬ 2 a R h a 1 − ηq, q 1 − 2 − (1 − q) (α − β) = ⎭ H h⎩ a R

(5.139)

where β = sin–1 (a/R). For given values of the ratios h/a, H/R, and a/R, (5.137) and (5.139) can be solved simultaneously for q and a. In Fig. 5.27, the parameters q and α – β at the incipient collapse are plotted against a/R for different values of H/R in the special case of η = 1. The collapse pressure is actual for the assumed yield condition, but provides only an upper bound for a non-hardening Tresca material.

5.7

Minimum Weight Design of Shells

389

Fig. 5.27 Collapse pressure and the extent of the associated plastic region in a spherical pressure vessel with a projecting cylindrical nozzle

5.7 Minimum Weight Design of Shells 5.7.1 Principles for Optimum Design The problem of optimum design considered here involves the requirement of minimizing the weight of the material of the shell that is capable of supporting given loads. For a homogeneous material, this is equivalent to finding the minimum volume of shell, which will provide us with a basis for comparison with any actual design. Since the shell carries both direct forces and bending moments, it is convenient to discuss the design criterion in terms of the rate of plastic energy dissipation D per unit area of the middle surface. The dissipation rate is a monotonically increasing function of the local shell thickness h and depends also on the velocity vj of the middle surface. For a shell of variable thickness h which is just at the point of collapse under a boundary traction Tj and a velocity field vj , we have

D vj dA =

Tj vj dA

(5.140)

390

5 Plastic Analysis of Shells

in the absence of body forces, the integrals being extended over the entire area of the middle surface. Consider now a neighboring thickness distribution h∗ = h + δh for the same geometry of the middle surface, where δh is an infinitesimally small variation in thickness. The dissipation rate for the shell of thickness h∗ in the deformation mode vj may be written as ∂ D∗ vj = D vj + D vj δh ∂h on neglecting higher-order terms, provided D(vj ) is a continuously differentiable function of h. Since the corresponding distribution of the generalized stresses in the shell of thickness h∗ is not necessarily in equilibrium, it follows from the upper bound theorem of limit analysis that

D∗ vj dA ≥

Tj vj dA.

Substituting for D∗ (vj ), and using (5.140), the preceding inequality is reduced to

∂ D vj δhdA ≥ 0. ∂h

(5.141)

An immediate consequence of inequality (5.141) is that if δD/δh is a positive constant for the shell of thickness distribution h, then

δhdA = δ

hdA ≥ 0.

This inequality implies that the volume of the shell of thickness h, which is designed to collapse under the given loads, is a relative minimum when δD/δh has a constant value over the middle surface. The minimum volume criterion established here, which is due to Shield (1960b), is an extension of that given earlier for the bending of plates (Section 4.5). A stronger result can be obtained for the ideal sandwich shell which consists of a core of thickness H between two thin identical face sheets each of thickness h/2. The membrane forces and bending moments across any section are carried by the face sheets, while the core carries only the shearing force. The strain rate in each of the face sheets may be considered as constant, so that the dissipation rate D is proportional to h. The core thickness H is assumed to be prescribed, and the face sheet thickness is to be determined for minimum volume so that the shell is just at the point of collapse under the given loads Tj . To obtain the condition for minimum volume of the sandwich shell, consider any sheet thickness distribution h∗ for which the shell is at or below collapse under the loads Tj and in the velocity pattern vj . Since the dissipation rate for this shell is D∗ (vj ) = (h∗ /h)D(vj ), we have

5.7

Minimum Weight Design of Shells

391

∗ h /h D vj dA ≥

Tj vj dA

by the upper bound theorem of limit analysis. The elimination of the right-hand side of the above inequality by means of (5.140) results in

∗ h − h D vj /h dA ≥ 0.

(5.142)

If the shell of thickness h is such that D(vj )/h is a positive constant, then (5.142) reduces to

∗

h dA ≥

hdA.

Thus, a sandwich shell designed to collapse in a mode that makes the ratio D/h a constant over the middle surface provides an absolute minimum for its volume under the prescribed loads (Shield, 1960). The condition D/h = constant for the optimum design of a sandwich shell is independent of the design thickness due to the linear dependence of D on h. For a solid shell, on the other hand, the condition ∂D/∂h = constant for the optimum design involves the thickness h, which renders the problem more complicated. The question of uniqueness of the optimum design has been examined by Hu and Shield (1961), and some criteria for minimum cost design have been discussed by Prager and Shield (1967).

5.7.2 Basic Theory for Cylindrical Shells Consider the particular case of a circular cylindrical shell of sandwich construction under axially symmetrical loading. When the applied load is an internal pressure p, which may vary with the axial coordinate x, the axial force Nx is a constant, and equilibrium also requires dMx = Q, dx

dQ Nθ =p− . dx a

These equations follow from (5.3) with a change in sign for p. The shear force Q is easily eliminated between the above equations to give ∂ 2 Mx Nθ = p. + 2 a ∂x

(5.143)

The circumferential bending moment Mθ is a passive or induced moment that arises from the fact that the corresponding curvature rate κ˙θ vanishes because of symmetry. The rate of dissipation of internal energy per unit area of the middle surface is

392

5 Plastic Analysis of Shells

D = Mx κ˙ x + Nθ λ˙ θ + Nx λ˙ x ,

(5.144)

where κ˙ x is the axial curvature rate, and λ˙ θ ,λ˙ x are the circumferential and axial extension rates of the middle surface. They are given by κ˙ x =

d2 w , dx2

λ˙ θ =

w , a

λ˙ x =

du , dx

(5.145)

where u and w are the axial and radially outward velocities of the middle surface. Suppose that the material of the face sheets is ideally plastic obeying Tresca’s yield criterion and the associated flow rule. The corresponding yield conditions in terms of the generalized stresses have been discussed in Sections 5.1 and 5.2. When the axial force is absent (Nx = 0), the condition D/h = constant for minimum volume restricts the stress point to be at one of the corners of the yield hexagon shown in Fig. 5.28a. Indeed, D vanishes along the sides BC and EF of the hexagon and is a function of x along the remaining sides of the hexagon. For the stress point lying at the corners, the condition D/h = constant may be written as λ˙ = constant, λ˙ ≥ H κ, ˙ corners A and D, λ˙ + H κ˙ = constant, λ˙ ≤ H κ, ˙ points B, C, E, F,

(5.146)

where λ˙ and κ˙ denote the absolute values of λ˙ θ and κ˙ x , respectively. The above relations are directly obtained by setting the values Nθ and Mx in (5.144) for the considered stress point and using the fact that N0 = Yh and M0 = YHh/2 for the sandwich shell having a uniaxial yield stress Y.

Fig. 5.28 Yield condition for a cylindrical sandwich shell. (a) Yield locus for no axial force and (b) a part of the yield surface for nonzero axial force

When the axial force is present (Nx = 0), the part of the yield surface for which the membrane forces N0 and Nx are positive is shown in Fig. 5.28b. It can be shown that only the stress states represented by points on the edges AG, AH, BG, FH, GK, HK, and KL, and also those on the plane AGKH can be associated with a rate of

5.7

Minimum Weight Design of Shells

393

deformation for which the condition D/h = constant is satisfied. Considering, for instance, stress points on the edges GK and HK, the condition D/h = constant is found to imply 2λ˙ + H κ˙ = constant,

λ˙ θ ≥ 0,

2λ˙ x = H κ. ˙

(5.147)

Similar relations can be established for the other plastic regimes relevant to the optimum design. The preceding results, as well as their applications to be discussed later in this section, are essentially due to Shield (1960). When the material yields according to the von Mises criterion, the stress point has the freedom to move along the yield locus, which for zero axial load is an ellipse circumscribing the Tresca hexagon. By the flow rule (5.9) associated with the yield condition (5.8), the dimensionless stress resultants can be expressed in terms of the generalized strain rates. Thus nθ = N0 λ˙ θ /D,

mx = 4M0 κ˙ x /3D,

3 n2θ + m2x = 1. 4

Since D/h is a constant for the optimum design, we write D = chY, where c is an arbitrary positive constant that can be associated with the mode of collapse. The above relations then give nθ = λ˙ θ /c,

mx = 2H κ˙ x /3c,

λ˙ 2θ + (H/3) λ˙ 2x = c2 .

(5.148)

In view of (5.145), the quantities nθ and mx are obtained from (5.148) as functions of the velocity w, the spatial distribution of which is determined by the numerical integration of the last equation of (5.148). The bending moment distribution can be found by the integration of (5.143) after eliminating Nθ by means of the relation Nθ /Mx = N0 nθ /M0 mx = 3λ˙ θ /H 2 κ˙ x . The design thickness h at each section is finally obtained from the ratio of Mx to mx . Applications of the basic theory to the von Mises material have been discussed by Reiss and Megarefs (1969). The optimum design of cylindrical shells based on the calculus of variation and the Tresca theory has been considered by Freiberger (1956), following an earlier work by Onat and Prager (1955).

5.7.3 Simply Supported Shell Without End Load As a first example illustrating the preceding theory, consider an open-ended cylindrical shell of length 2l loaded by a uniform internal pressure p. If the shell is simply supported at both ends, which correspond to x = 0 and x = 2l, the bending moment Mx and the radial velocity w must vanish at these sections. For relatively short shells, the mode of collapse should involve bending of the entire shell, and the stress point F in Fig. 5.28a will therefore apply throughout the shell. Using the second equation

394

5 Plastic Analysis of Shells

of (5.146), where λ˙ = λ˙ θ and κ˙ = κ˙ x , the differential equation for w in the optimum design is obtained as aH

d2 w − w = −w0 dx2

(5.149)

in view of (5.145), the quantity w0 being a positive constant velocity. Integrating, and using the boundary conditions w = 0 at x =0 and x = 2l, the solution is found as cosh (α − ξ ) w = w0 1 − , cosh α √ √ where ξ = x/ aH and α = l/ aH. The velocity field will be associated with the stress state at corner F if the first inequality of (5.146) is also satisfied. This gives w ≤ −d2 w/dξ 2

or

cosh α ≤ 2 cosh (α − ξ ) .

This condition will be satisfied throughout the shell if it holds at the central section ξ = α. Hence α ≤ cosh−1 2 or

√ l ≤ 1.317 aH.

The thickness distribution in the optimum design, when the shell is sufficiently short, is obtained by setting Mx = –YHh/2 and Nθ = Yh/2 in the equilibrium equation (5.143), the resulting differential equation for h being aH

2pa d2 h . =− 2 Y dx

(5.150)

Since the bending moment must vanish at the ends of the shell, the boundary conditions are h = 0 at x = 0 and x = 2l, and the solution is hY cosh (α − ξ ) =2 1− , α ≤ 1.317. pa cosh α

(151)

It is interesting to note that the variation of hY/pa with ξ in the case of short shells is identical to that of w/w0 with ξ , except for a scale factor. For longer shells (α ≥ 1.317), a central portion of the shell is stressed by Nθ alone, corresponding to corner A of the yield locus, Fig. 5.28a. The stress state √ in the remainder of the shell corresponds to point F and covers a length d = δ aH on either side of the central portion, where δ is a constant. Over the central portion, the condition λ˙ θ = constant for minimum volume requires w to be constant in the region. Considering the end portion 0 ≤ x ≤ d, the velocity distribution is easily determined from the differential equation (5.149) and the boundary conditions w = 0 at x = 0 and dw/dx = 0 (and hence continuous) at x = d, the result being

5.7

Minimum Weight Design of Shells

395

cosh (δ − ξ ) w = w0 1 − , 0 ≤ ξ ≤ δ. cosh δ

(5.152)

The constant δ is obtained from the requirement λ˙ = H κ, ˙ or w = –d2 w/dξ 2 at ξ = δ, giving δ = cosh−1 2 = 1.317. By the condition of continuity at ξ = δ, the velocity in the central portion of the shell is w = w0 /2. The inequality w ≤ –d2 w/dξ 2 in the end portion and the inequality w ≥ –d2 w/dξ 2 in the central portion are identically satisfied. The thickness distribution in each portion of the shell for minimum volume is determined from the equilibrium equation and the appropriate yield condition. In the end portion 0 ≤ x ≤ δ, the resulting differential equation is (5.150), which must be solved under the boundary conditions h = 0 at x = 0 and x = d, the bending moment at these sections being zero. The solution is easily shown to be hY 1 = 2 1 − √ [sinh ξ + sinh (δ − ξ )] , pa 3

0 ≤ ξ ≤ δ.

(5.153a)

In the central portion of the shell, the stress point corresponds to Mx = 0 and N0 = Yh, and (5.143) immediately furnishes hY/pa = 1,

δ ≤ ξ ≤ α,

(5.153b)

only one-half of the shell being considered because of symmetry. The thickness changes discontinuously from 0 to pa/Y at x = d. Since the thickness gradient dh/dx is zero in the central portion, but is nonzero in the end portion, the shearing force Q = dMx /dx is also discontinuous at x = d. The discontinuity in Q is removed, however, by adding a flange of vanishingly small width but of finite area of cross section at x = d. The total volume V of the face sheets in the optimum design is now determined by integration over the area of the middle surface, using the expression

l

V = 4π a 0

√ hdx = 4π a aH

a

hdξ .

(5.154)

0

Substituting from (5.151) and (5.153) for short and long shells, respectively, we obtain ⎧ ⎨ 8π a2 l (p/Y) (1 − tanh α/α) α ≤ 1.317,

√ (5.155) V= ⎩ 8π a2 l (p/Y) 1 − 3 − δ /α , α ≥ 1.317, √ A term equal to 1/ 3α has been included in the square brackets of (5.155) to take account of the flanges so that volume is continuous at α = δ = 1.317.

396

5 Plastic Analysis of Shells

It is instructive to compare the volume for the optimum design with that for the constant thickness design for the sandwich shell. To this end, we express the equilibrium equation (5.143) in the dimensionless form d 2 mx + 2nθ = 2q, dξ 2 where q = pa/Yh0 , with h0 /2 denoting the thickness of each face sheet. For a sufficiently short shell, the state of stress involves mx < 0 and nθ > 0, so that side AF of the yield hexagon applies throughout the shell. Using the yield condition 2nθ – mx = 2 to eliminate nθ from the equilibrium equation, we get d 2 mx + mx = 2 (q − 1) . dξ 2 In view of the boundary conditions mx = 0 at ξ = a and dmx /dξ = 0 at ξ = a, the solution becomes cos (α − ξ ) π (5.156) , α≤ . mx = 2 (q − 1) 1 − cos α 2 The velocity equation associated with the yield condition gives w = w0 sin ξ (ξ ≤ a) in view of the boundary condition w = 0 at ξ =0. The velocity slope dw/dξ is discontinuous at ξ = α < π /2, giving rise to a hinge circle, where mx = – 1. By (5.156), this condition furnishes q=

π 2 − cos α , α≤ . 2 − 2 cos α 2

(5.157)

For longer shells (α ≥ π/2), the plastic state of stress is represented by corner A of the yield hexagon. Then mx = 0 and nθ = 1, giving q = 1 for plastic collapse of the simply supported shell, the associated velocity field being w = w0 sin ξ (ξ ≤ π/2) ,

w = w0 (ξ ≥ π/2) .

Since the total volume of the face sheet material in the constant thickness design is V0 = 4π alh0 , the ratio of the face sheet volumes for the optimum and constant thickness designs may be written from (5.157) as 2q (1 − tanh α/α) V , α ≤ 1.317, √ = V0 3 − δ /α α ≥ 1.317. q 1−

(5.158)

The parameter q is given by (5.157) when α ≤ π /2 and is equal to unity when α ≥ π /2. The ratio V/V0 is plotted as a function of α in Fig. 5.29 on the basis of (5.158) for the simply supported shell. The discontinuities in slope of the curves at α = 1.317 and α = 1.571 are due to the change in character of the solution for the

5.7

Minimum Weight Design of Shells

397

Fig. 5.29 Ratio of face sheet volume in the optimum design to that in the constant thickness design of a cylindrical shell under uniform internal pressure

optimum design and constant thickness design, respectively. The saving of material effected by the minimum volume design is quite appreciable for short shells.

5.7.4 Cylindrical Shell with Built-In Supports √ A circular cylindrical shell of semilength l = α ah is provided with built-in support at its open ends and is subjected to a uniform internal pressure p. The condition D/h = constant for the minimum volume design requires the velocity slope to vanish at the clamped edges √ when the shell is at the point of collapse. A portion of the shell of length b = β ah exists at each end where the state of stress is represented by point B in Fig. 5.28a. If the shell is sufficiently short, the remainder of the shell would correspond to point F of the yield hexagon. By the second equation of (5.146), the velocity equation for the optimum design is aH

d2 w ± w = ±w0 , dx2

398

5 Plastic Analysis of Shells

where the upper sign applies to the end portions and the lower sign to the central portion of the shell. Considering only one-half of the shell and using the boundary conditions w = dw/dx = 0 at x = 0,

dw/dx = 0 at x = l,

and the condition of continuity of w at x = b, the solution for the velocity at the incipient collapse is obtained as 0 ≤ ξ ≤ β, w = w0 (1 − cos ξ ) , w = w0 1 − cos β cosh (α − ξ )/cosh (α − β) ,

β ≤ ξ ≤ α.

(5.159)

Since dw/dx must be continuous at ξ = β for the minimum volume design, the relationship between β and α is tan β = tanh (α − β) .

(5.160)

The compatibility of the velocity field with the stress field associated with corners B and F requires w≤

d2 w (0 ≤ ξ ≤ β) , dξ 2

w≤−

d2 w (β ≤ ξ ≤ α) . dξ 2

In view of (5.160), the above inequalities will be satisfied throughout the respective portions of the shell if 2 cos β ≥ 1,

cosh (α − β) ≤ 2 cos β.

It turns out that the first inequality is satisfied when the second inequality is. Combining the second inequality with (5.160), it is easily shown that α−β ≤

1 cosh−1 4, 2

β ≤ 0.659,

α ≤ 1.691.

(5.161)

The thickness distribution in the shell for α ≤ 1.691 is given by the solution of the differential equation aH

d2 h 2pa ±h=± , 2 dx Y

(5.162)

which is obtained by setting mx = ±YHh/2 and Nθ = Yh in (5.143). The upper sign applies to 0 ≤ x ≤ b and the lower sign to b ≤ x ≤ a. Since the bending moment must be continuous at x = b, it must vanish at this section. Further, symmetry requires the shearing force to vanish at x = l. Hence the boundary conditions are

5.7

Minimum Weight Design of Shells

h = 0 at ξ = β,

399

dh/dξ = 0 at ξ = α.

These conditions, together with the fact that dh/dξ changes sign at ξ = β while retaining its numerical values for the shearing force to be continuous, furnish the solution hY/pa = 2 1 − (cos ξ/ cos β) + 2 tan β sin (β − ξ ) , hY/pa = 2 1 − cosh (α − ξ ) / cosh (α − β) ,

0 ≤ ξ ≤ β, β ≤ ξ ≤ α.

(5.163)

For longer shells with α ≥ 1.691, the minimum volume condition D/h = constant can only be satisfied by introducing a central region of constant radial velocity associated with corner A in Fig. 5.28a. The stress points B and F of the yield hexagon then correspond to the regions 0 ≤ ξ ≤ γ and γ ≤ ξ ≤ δ, respectively, where γ and δ are dimensionless constants. The velocity distribution in these two regions can be written as w = w0 (1 − cos ξ ) , 0 ≤ ξ ≤ γ, w = w0 1 − cos γ cosh (δ − ξ ) / cosh (δ − γ ) ,

(5.164)

γ ≤ ξ ≤ δ.

The velocity at ξ = γ and the velocity slope at ξ = δ are automatically made continuous. The condition of continuity of dw/dξ at ξ = γ, and the requirement w = –d2 w/dξ 2 at ξ = δ furnish tan γ = tanh (δ − γ ) , 2 cos γ = cosh (δ − γ ) , in view of (5.164). These two equations are easily solved for γ and δ to give δ−γ =

1 cosh−1 4, 2

γ = 0.659,

δ = 1.691.

The velocity in the region δ ≤ ξ ≤ a is evidently w = w0 /2 by the condition of continuity. The thickness distribution over the region 0 ≤ ξ ≤ δ is determined by solving (5.162) under the conditions h = 0 at ξ = γ and ξ = δ, so that the bending moment is continuous, and the fact that dh/dξ merely changes its sign at ξ = γ. The result is hY = 2 1 − 1.265 cos ξ + 1.249 sin (γ − ξ ) , pa $ % hY 2 {sinh (ξ − γ ) + sinh (δ − ξ )} , =2 1− pa 3

⎫ ⎪ 0 ≤ ξ ≤ γ ,⎪ ⎪ ⎬ ⎪ ⎪ γ ≤ ξ ≤ δ, ⎪ ⎭

. (5.165a)

Since 2 sin γ = sinh (δ − γ ) = 32 . In the remainder of the considered half of the shell, the state of stress is given by Mx = 0, Nθ = Yh and (5.143) furnishes

400

5 Plastic Analysis of Shells

hY/pa = 1,

δ ≤ ξ ≤ α.

(5.165b)

Since dh/dx is discontinuous at ξ = δ, a flange of vanishingly small width but of finite area of cross section must be added at this section to maintain continuity of the shearing force Q. To obtain the total volume V of the face sheets of the shell designed for minimum volume, it is only necessary to insert in (5.154) the expressions for h in (5.163) and (5.165) for short and long shells, respectively, and to evaluate the integral. Including the volume of the flanges appropriately, so that the results match when α = 1.691, the minimum volume can be written in the dimensionless form 2q (1 − 2 sin β/α) V , √ = V0 6 − δ /α, q 1−

α ≤ 1.619, α ≥ 1.619,

(5.166)

where V0 = 4πalh0 represents the volume of a shell of constant face sheet thickness h0 , and q denotes the quantity pa/h0 Y. The relationship between q and a for the sandwich shell is directly obtained from (5.12) and (5.16) with w replaced by α. When a is very small, q ≈ 2/α 2 , while α ≈ 2 tan β in view of (5.160), so that V/V0 = 12 in the limit α = 0. The ratio V/V0 is plotted as a function of a in Fig. 5.29, which indicates that the saving of material effected by the optimum design is higher for the clamped shell than for the simply supported shell.

5.7.5 Closed-Ended Shell Under Internal Pressure Suppose now that a circular cylindrical shell of length 2l is closed at the ends by rigid plates and is subjected to a uniform internal pressure p. The rigid plates not only produce an axial force Nx = pa/2 per unit circumference but also give rise to clamped edge conditions at the ends of the shell. As in the case of a shell of constant thickness, the yield condition requires Nθ = N0 throughout the shell, only stress states represented by the edges GK and HK of Fig. 5.28b being involved in the minimum volume design. Hence the relationship between the axial force and the bending moment is pa Nx YH h− , =± Mx = ±M0 1 − N0 2 2Y

(5.167)

where the upper sign holds for the line GK and the lower sign for the line HK. The former applies to two identical outer portions of the shell, each having a √ length b =√β aH, and the latter applies to a central portion of length 2 (l − b) = 2 (α − β) aH. The condition D/h = constant, which is equivalent to (5.147) with λ˙ and κ˙ denoting λ˙ θ and ±κ˙ x , respectively, leads to the differential equation

5.7

Minimum Weight Design of Shells

aH

401

d2 w ± 2w = ±2w0 , dx2

where w0 is a positive constant. These two equations cover the entire shell for all values of l and are subject to the boundary conditions w = dw/dx = 0 at x = 0,

dw/dx = 0 at x = l.

Further, w and dw/dx must be continuous at the interface x = b. The solution for √ the velocity field is given by (5.159) with a factor of 2 for each of the quantities ξ , β, and α, the relationship between β and α being tan

√

√ 2β = tanh 2 (α − β) .

(5.168)

Inserting (5.167) into the equilibrium equation (5.143), and setting Nθ = Yh, the differential equation for the thickness h is obtained as aH

2pa d2 h . ± 2h = ± Y dx2

Since Mx must vanish at x = b by the condition of continuity, and the shearing force Q must vanish at x = l because of symmetry, we have h=

pa at ξ = β, 2Y

dh = 0 at ξ = α. dξ

In addition, dh/dξ must change sign at ξ = β without changing its absolute value for the shearing force to be continuous. The solution to the above equation therefore becomes ⎫

√ √ ⎬ hY/pa = 1 − 12 (cos ξ/cos β) + tan 2β sin 2 (β − ξ ) , 0 ≤ ξ ≤ β,⎪ √ √ ⎭ β ≤ ξ ≤ α. ⎪ hY/pa = 1 − 12 cosh 2 (α − ξ ) / cosh 2 (α − β) , (5.169) In Fig. 5.30, the thickness distribution for a closed-ended shell is compared with √ that for an open-ended shell when α = 2. The ratio of the volume of the face sheets for the optimum design to that for the constant thickness design is obtained from (5.154) and (5.169) as ⎧ ⎨

V =q 1− ⎩ V0

sin

√

⎫ 2β ⎬

√ 2α

⎭

=

⎧

2 + α2 ⎨ 1 + α2

⎩

1−

sin

√

⎫ 2β ⎬

√ 2α

⎭

,

(5.170)

where the last step follows from (5.33a) with ω replaced by α. When α tends to zero, β/a tends to 12 and V/V0 tends to unity. The variation of V/V0 with α for the closed-ended shell, computed from (5.168) and (5.170), is included in Fig. 5.29.

402

5 Plastic Analysis of Shells

Fig. 5.30 Thickness distribution in a clamped cylindrical sandwich shell designed for minimum weight under uniform internal pressure

The presence of the axial force appreciably reduces the saving of material caused by the optimum design. The minimum weight design of cylindrical shells based on the von Mises yield criterion has been presented by Reiss and Magarefs (969). The optimum design of closed pressure vessel heads has been discussed by Hoffman (1962) and by Save and Massonnet (1972). The minimum weight design of conical shells has been considered by Reiss (1974, 1979). The plastic design of shells of revolution for constant strength has been examined by Ziegler (1958), Issler (1964), and Dokmeci (1966). The minimum weight design of membrane shells of revolution subjected to uniform external pressure and vertical load has been investigated by Richmond and Azarkhin (2000).

Problems 5.1 A short cylindrical shell of mean radius a and wall thickness h is under a uniform internal pressure p, and is simply supported at both ends, the length of the shell being denoted by 2l. Adopting the linearized yield condition for a Tresca sandwich shell, show that the dimensionless collapse pressure and the associated velocity field are given by. pa 2 − cos ω = , Yh 2 (1 − cos ω)

sin (ω − ξ ) w = w0 sin ω

√ where ξ = x 2/ah, with x denoting the axial distance measured from the central section, ω is the value of ξ at x = l, and w0 is the deflection rate at the center.... 5.2 A cylindrical shell of length l, thickness h, and mean radius a is clamped at one end and is free at the other. The shell is subjected to a uniform radial pressure p to reach the point of plastic collapse. Using the yield condition for a Tresca sandwich shell, show that the intensity of the pressure and the velocity distribution are given by

Problems

403

pa 2 cosh ω − 1 = , N0 2 (cosh ω − 1)

sinh (ω − ξ ) w = w0 sinh ω

where ξ and ω denote the same quantities as those in the preceding problem, and w0 is the deflection rate at the free end of the shell. 5.3 A cylindrical shell of wall thickness h and mean radius a is free at both ends and is subjected to a radially outward √ ring load P per unit circumference applied at x = 0. Introducing the parameter ξ = x 2/ah, the ends of the shell are defined by ξ = – α and ξ = β. Assuming α and β to be sufficiently small, so that the shell can collapse without the formation of a hinge circle, and using the square yield condition defined by mx = ±1 and nθ = ±1, show that the hoop force changes sign at ξ = λ, where 2λ2 = α 2 + β 2 , and that the collapse load is given by P = N0

h β −α , α2 + β 2 − √ a 2

α ≤ 1,

β≤

√

2+

1 + α2

5.4 For higher values of β in the preceding problem, show that a hinge circle must form at ξ = 1 + α 2 , the outer portion of the shell remaining rigid, the associated collapse load being given by the modified expression P = N0

h α + 1 + α2 , 2a

α < 1,

β>

√ 2 + 1 + α2

√ Over the range α > 1 and β > 2 2, prove that the collapse mode involves a hinge circle at ξ = 0, and the corresponding collapse load is given by P = N0

√ h α+β , 1 G when X > Y, together with two similar inequalities for the first three parameters. Evidently, (6.1) reduces to the von Mises criterion when L = M = N = 3F = 3G = 3H = 3/2Y2 , where Y is the uniaxial yield stress of the isotropic material. For an arbitrary orthotropic material, the first three relations of (6.2) give 2F = Y −2 + Z −2 − X −2 , 2G = Z −2 + X −2 − Y −2 , 2H = X −2 + Y −2 − Z −2 . The state of anisotropy in an element is, therefore, specified by the directions of the three principal axes of anisotropy, and the values of the six independent yield stresses X, Y, Z, R, S, and T, which depend on the degree of previous cold work as well as on the subsequent heat treatment (Hill, 1948). The yield criterion is expressed in the form (6.1) only when the principal axes of anisotropy are taken as the axes of reference. For an arbitrary set of rectangular axes, the form of the yield criterion must be changed by the appropriate transformation of the stress components. When the state of anisotropy is rotationally symmetric about the z-axis, the form of the yield criterion must be independent of the choice of x-and y-axes. Since X = Y and R = S for such a symmetry, it is obviously necessary to set F = G and L = M. To obtain additional conditions for the rotational symmetry, we rewrite (6.1) in the form

2 2 (F + H) σx + σy + 2Fσz σz − σx − σy − 2 (F + 2H) σx σy − τxy

2 2 2 + 2 (N − F − 2H) τxy + τzx = 1. + 2L τyz Since the first four terms of the above expression are invariants for a fixed z-axis, the coefficient of the last term must be identically zero for the yield criterion

6.1

Plastic Flow of Anisotropic Metals

407

to be unaffected by any rotation of the x- and y-axes. The necessary and sufficient conditions for the anisotropy to be rotationally symmetric about the z-axis therefore becomes N = F + 2H = G + 2H, L = M. The number of independent parameters defining a rotationally symmetrical state of anisotropy is therefore reduced to three, which may be taken as the uniaxial yield stresses along and perpendicular to the axis of symmetry and the shear yield stress with respect to these two directions.

6.1.2 Stress–Strain Relations To derive the relations between the stress and the strain increments for an anisotropic material, we adopt the usual normality rule of plastic flow, assuming the plastic potential to be identical to the yield function. Referred to the principal axes of anisotropy, when the elastic strains are disregarded, the strain increment tensor for an orthotropic material may be written as dεij =

∂f dλ, ∂σij

where 2f(σ ij; ) denotes the expression on the left-hand side of (6.1), and dλ is necessarily positive for plastic flow in an element that is stressed to the yield point. Setting 2τ 2 yz = τ 2 yz +r2 zy , etc., in the field function (6.1), and treating all nine components of the stress tensor as independent, we obtain the strain increment relations ⎫ dεx = H σx − σy + G (σx − σz ) dλ, dγxy = Nτxy dλ, ⎪ ⎬ dεy = F σy − σz + H σy − σx dλ, dγyz = Lτyz dλ, , ⎪ ⎭ dεz = G (σz − σx ) + F σz − σy dλ, dγzx = Mτzx dλ,

(6.3)

which constitute a generalization of the Lévy–Mises flow rule. The sum of the three normal strain components is seen to be zero, satisfying the condition of plastic incompressibility. When the principal axes of stress coincide with those of anisotropy, the principal axes of the strain increment also occur in the same directions. In general, however, the principal axes of stress and strain increment do not coincide for an anisotropic material. The preceding results are due to Hill (1948), although similar relations have been given by Jackson et al. (1948) and Dorn (1949). The ratios of the anisotropic parameters can be determined by carrying out tensile tests on specimens cut at suitable orientations with respect to the principal axes of anisotropy. It is, of course, necessary for this purpose that the anisotropy is uniformly distributed through a volume of sufficient extent in order to allow the preparation of the specimens. For a tensile specimen cut parallel to the x-axis of anisotropy, the ratios of the principal strain increments are

408

6 Plastic Anisotropy

dεx :dεy :dεz = G + H: − H: − G.

(6.4)

A longitudinal extension is therefore accompanied by a contraction in each transverse direction, unless the yield stresses differ so much that one of the parameters G and H is negative. The magnitude of the incremental transverse strain is greater in the direction of the lesser yield stress. Tensile tests carried out on specimens cut parallel to the y- and z-axes of anisotropy similarly furnish the ratios F/H and GIF, respectively, providing an immediate test on the theory in view of the identity (HIG)(G/F)(FIH) = 1. Where the theory is applicable, the measurement of strain ratios in the appropriate tensile specimens provide an indirect method of finding the ratios of the yield stresses along the three principal axes of anisotropy.

6.1.3 Variation of Anisotropic Parameters It is assumed at the outset that the material has a very pronounced state of anisotropy, and that further changes in anisotropy during cold work are negligible over the considered range of strains. The yield stresses of the material in the different directions then increase in strict proportion as the material deforms, the factor of proportionality being denoted by a parameter h which increases monotonically from unity to represent the amount of hardening. Thus X = hXo, Y = hYo, etc., where the subscript zero denotes the initial value, giving F = Fo/h2 , G= Go/h2 , etc. The anisotropic parameters therefore decrease in strict proportion, and their ratios remain constant during the deformation. The scalar parameter h is a dimensionless form of the equivalent stress σ¯ which may be defined as '

3 3 =h , σ¯ = h 2 (F0 + G0 + H0 ) 2c

c = F0 + G6 + H0 ,

so that h is equal to σ¯ /Y for an isotropic material with an initial yield stress Y. The substitution for F = F0 /h2 , etc., in (6.1) furnishes h2 in terms of the initial values of the anisotropic parameters, and the expression for the equivalent stress becomes 2 2 3 F0 σy − σz + G0 (σz − σx )2 + H0 σx − σy σ¯ = 2c (6.5) 1/2 2 2 2 +2L0 τyz + 2M0 τzx + 2N0 τxy . As in the case of isotropic solids, σ¯ may be regarded as a function of an equivalent strain whose increment may be defined according to the hypothesis of strain equivalence as dε¯ =

2 3

2 dεx2 + dεy2 + dεz2 + 2dγxy + 2dγxz2

1/2

.

(6.6)

6.1

Plastic Flow of Anisotropic Metals

409

When a uniaxial tension X is applied in the x-direction, the ratios of the nonzero components of the strain increment are given by (6.4), and the equivalent stress and strain increments become ' 3 G0 + H0 X, σ¯ = 2 F0 + G0 + H0

2 G20 + G0 H0 + H02 dε = √ dεx . 3 (G0 + H0 )

(6.7)

Similar expressions for σ¯ and dε are obtained for uniaxial tensions Y and Z applied in the y- and z-directions, respectively. A comparison of the stress–stress curves along the three principal axes of anisotropy provides a direct means of testing the hypothesis. Consider the alternative hypothesis in which σ¯ is assumed to be a function of the total plastic work per unit volume of the element. This has been proposed by Jackson et al. (1948) and was later followed by Hill (1950a). The increment of plastic work per unit volume is dW = σij dεij = σij

∂f dλ = 2fdλ = dλ ∂σij

by Euler’s theorem of homogeneous functions and by the yield criterion expressed as 2 f = 1. From (6.3), we have Gdεy − Hdεz = (FG + GH + HF) σy − σz dλ, together with two similar relations obtained by cyclic permutation. The substitution for the normal stress differences into the yield criterion (6.1) then gives 1

Gdεy − Hdεz 2 2dγyz2 = (dλ)2 . + F FG + GH + HF L

Since dW = σ¯ d¯ε , where d¯ε is the equivalent strain increment according to the hypothesis of work equivalence, we have 1/2 2 dλ dλ = . (F dε¯ = 0 + G0 + H0 ) σ¯ 3 h Substituting for dλ, and using the relations F = F0 /h2 , etc., the equivalent strain increment according to the work-hardening hypothesis is finally obtained as

dε¯ =

1/2 2 (F0 + G0 + H0 ) 3 1/2 2 2γyz2 G0 dεy + H0 dεz × F0 ··· + + ··· . F 0 G0 + G 0 H0 + H 0 F 0 L0

(6.8)

410

6 Plastic Anisotropy

For a uniaxial tension X parallel to the x-axis of anisotropy, the expressions for σ¯ and d¯ε are ' 3 G0 + H0 X, σ¯ = 2 F0 + G 0 + H0

' 2 F0 + G0 + H0 dε¯ = dεx . 3 G0 + H 0

For an isotropic material, the equivalent strain increments defined by (6.6) and (6.8) are identical, and the two hypotheses for the hardening process are therefore equivalent. For an anisotropic material, the two hypothesis are distinct, and the predicted stress–strain curves along one axis of anisotropy derived from another will generally be different in the two cases. The choice of the appropriate expression for the equivalent strain increment for a particular material must be decided by experiment. The more general case of hardening of an orthotropic material, involving both expansion and translation of the yield surface in the stress space, has recently been discussed by Kojic et al. (1996).

6.2 Anisotropy of Rolled Sheets 6.2.1 Variation of Yield Stress and Strain Ratio In a rolled sheet of metal, the principal axes of anisotropy are along the rolling, transverse, and through-thickness directions at each point of the sheet. Let the axes of reference be so chosen that the x-axis coincides with the direction of rolling, the y-axis with the transverse direction in the plane, and the z-axis with the normal to the plane. If the sheet is subjected to forces in its plane, the only nonzero stress components are σ x , σ y ,and τ xy , and the yield criterion (6.1) reduces to 2 = 1. (G + H) σx2 − 2Hσx σy + (H + F) σY2 + 2 Nτxy

(6.9)

Let σ denote the uniaxial yield stress of the sheet metal in a direction making a counterclockwise angle α with the rolling direction. The stress components corresponding to a uniaxial tension σ applied in the α-direction are σx = σ cos2 α,

σy = σ sin2 α,

τxy = σ sin α cos α.

(6.9a)

The substitution of (6.9a) into the yield criterion (6.9) furnishes σ as a function of α for any given state of anisotropy, the result being −1/2 . (6.10) σ = F sin2 α + G cos2 α + H + (2 N − F − G − 4H) sin2 α cos2 α The uniaxial yield stress σ can be shown to have maximum and minimum values along the anisotropic axes, and also in the directions α = ±α0 , where

6.2

Anisotropy of Rolled Sheets

411

−1

α0 = tan

N − G − 2H . N − F − 2H

(6.11)

When N is greater than both F + 2H and G +2H, the yield stress has maximum unequal values in the x- and y-directions, and minimum equal values in the α0 -directions. If N is less than both F + 2H and G+2H, the yield stress has minimum unequal values in the x- and y-directions, and maximum equal values in the α0 -directions. When N lies between F + 2H and G + 2H, there is no real α0 , and σ is a maximum in the x-direction and a minimum in the y-direction if F > G, and vice versa if F where α0 is given by (6.11) and the two possible necks are then equally inclined to the direction of the applied tension. A localized neck is able to develop only if the rate of work-hardening of the material is lower than a certain critical value, for which it is exactly balanced by the rate of reduction of thickness in the neck. Since the normal stresses transmitted across the neck are proportional to the applied tension a, we have dε dσ dσ σ = −dεz = , or = , σ 1+R dε 1+R

(6.15)

where dε is the longitudinal strain increment, and R is given by (6.13). The critical subtangent to the appropriate stress–strain curve for localized necking is therefore (1+ R) times that for diffuse necking. Hence, a localized neck can be expected to form on a superimposed diffuse neck as in the case of an isotropic sheet metal. In the case of a plane sheet subjected to biaxial stresses σ1 and σ2 in the rolling and transverse directions, respectively, the condition of the zero rate of extension along the localized neck, which makes an angle ß with the direction of σ1, furnishes

tan2 β = −

dε1 (G + H) σ1 − Hσ2 = , dε2 Hσ1 − (F + H) σ2

in view of the flow rule (6.3). Assuming σ 1 > σ 2 , the range of stress ratios for which the necking can occur is given by σ 2 /σ 2 ≤ H/(F + H), which ensures dε2 ≤ 0. An analysis for localized necking based on the total strain theory along with the assumption of a yield vertex has been presented by Storen and Rice (1975),

414

6 Plastic Anisotropy

the analysis being similar to that given in Section 2.1. The development of localized necks as a result of void growth has been considered by Needleman and Triantafyllidis (1978). Consider now the initiation of diffuse necking in a thin sheet under biaxial tensile stresses σ 1 and σ 2 along the rolling and transverse directions, respectively. As in the case of isotropic sheets, the uniform deformation mode becomes unstable when the rate of hardening becomes critical. In terms of the initial values of the anisotropic parameters, the yield criterion (6.9) may be written as (G0 + H0 ) σ12 − 2H0 σ1 σ2 + (F0 + H0 ) σ22 = h2 .

(6.16)

It follows from the stress–strain relations and the differentiated form of (6.16) that dσ1 dε1 + dσ2 dε2 = (dh/h) dλ = (dσ¯ /σ¯ ) dλ. If the applied loads simultaneously attain their maximum at the onset of instability, then dσ 1 /σ 1 = dε1 and dσ 2 /σ 2 = dε 2 , and the preceding relation gives dε1 2 dε2 2 1 dσ¯ + σ2 = σ1 σ¯ dλ dλ dλ = (G + H)2 σ13 − H [(H + 2G) σ1 + (H + 2F) σ2 ] σ1 σ2 + (F + G)2 σ23 . The hypothesis of √ work equivalence will be adopted here for simplicity. Using the relation dλ = h 3/2c dε on the left-hand side, substituting for G, H, and F on the right-hand side, and denoting the stress ratio σ 2 /σ1 by ρ, the instability condition is finally obtained in the form

3 (G0 + H0 )2 − H0 (H0 + 2G0 ) ρ − H0 (H0 + 2F0 ) ρ 2 + (F0 + H0 )2 ρ 3 3/2 2 (F0 + G0 + H0 )1/2 (G0 + H0 ) − 2H0 ρ + (F0 + H0 ) ρ 2 (6.17) in view of (6.16). The expression on the right-hand side, which is due to Moore and Wallace (1964), can be evaluated for any given p using the measured r-values in the rolling and transverse directions. An instability condition similar to (6.17) follows for the hypothesis of strain equivalence. The physical significance of the instability condition (6.17), which reduces to (2.38) when the material is isotropic, has been discussed by Dillamore et al. (1972). 1 dσ¯ = σ¯ dε

6.2.3 Correlation of Stress–Strain Curves Consider a uniaxial tension σ applied to a specimen cut of an angle a to the direction of rolling. When the hypothesis of strain equivalence is adopted, the equivalent strain increment is most conveniently obtained by using the property of its invariance. Thus, by taking the x-axis temporarily along the axis of the specimen, we get

6.2

Anisotropy of Rolled Sheets

dεx = dε,

415

dεy = −

R dε, 1+R

dεz = −

1 dε, 1+R

where r is given by (6.13). The substitution from above into (6.6) then gives the equivalent strain increment d¯ε , while the equivalent stress σ¯ is directly obtained by inserting (6.9a) in (6.5), the results being ' σ¯ =

3 (1 + R) F0 ξ σ, 2 (F0 + G0 + H0 )

√ 2 1 + R + R2 dε¯ = dε, √ 3 (1 + R)

where ξ = sin2 α + (G0 /F0 ) cos2 α. Let the uniaxial stress–strain curve in the rolling direction be defined by the equation σ = f(ε). In view of (6.7), the equivalent stress– strain curve is given by ⎧ ⎫ ' ⎨ √ 3 3 (G0 + H0 ) ε¯ ⎬ G0 + H0 f σ¯ = . (6.18) 2 F 0 + G 0 + H 0 ⎩ 2 G2 + G H + H 2 ⎭ 0 0 0 0 Substituting for σ and ε into (6.18), and introducing the R-values in the rolling and transverse directions, respectively, the equation for the stress–strain curve in the direction α is obtained as ' ' R y 1 + Rx 1 + Rx 1 + R + R2 f ε . (6.19) σ = ξ Rx 1 + R 1 + R 1 + Rx + R2x The stress–strain curve transverse to the rolling direction is obtained by setting ξ = 1 and R = Ry in (6.19). When the hypothesis of work equivalence is adopted, the equivalent strain increment corresponding to a uniaxial tension σ is equal to (σ/σ¯ )dε, and (6.19) is then replaced by ' σ =

Ry ξ Rx

' Ry 1 + Rx 1 + Rx f ε . 1+R ξ Rx 1 + R

It is evident that the stress–strain curve predicted by this relation will be generally different from that predicted by (6.19), except when the sheet is isotropic in its plane. The effective stress–strain behavior of anisotropic sheet metals has been investigated by Wagoner (1980) and Stout et al. (1983). Consider now a state of balanced biaxial tension σ in the plane of the sheet, which is equivalent to a uniaxial compression σ normal to the plane. If the increment of the compressive thickness strain is denoted by dε, then dεx =

G0 F0 + G 0

dε,

dεy =

F0 F0 + G 0

dε,

dεz = −dε,

on setting σ x = σ y = σ in the stress–strain relations. The expressions for the equivalent stress σ¯ and the equivalent strain increment dε therefore become

416

6 Plastic Anisotropy

' 3 F0 + G 0 σ, σ¯ = 2 F0 + G0 + H0

G0 + H0 dε = F0 + G0

'

F02 + F0 G0 + G20 G20 + G0 H0 + H02

dε,

in view of (6.5) and (6.6). The substitution into (6.18) shows that the stress–strain curve in the through-thickness direction according to the hypothesis of strain equivalence is given by ' σ =

Ry

⎫ ⎧ ' ⎨ 2 + R R + R2 ⎬ R 1 + Rx 1 + Rx x y x y ε . f Rx + R y ⎩ Rx + R y 1 + Rx + R2x ⎭

(6.20)

The hypothesis of work equivalence, on the other hand, leads to the equation for the through-thickness stress–strain curve as ' σ =

Ry

' 1 + Rx 1 + Rx Ry f ε . Rx + Ry Rx + Ry

Experimentally, such a curve is most conveniently obtained by the bulge test, in which a thin circular blank of sheet metal is clamped round the periphery and deformed by uniform fluid pressure applied on one side. Due to the symmetry of the loading, a state of balanced biaxial tension exists at the pole of the bulge, where the compressive thickness strain e is equal to the sum of the two orthogonal surface strains. In Fig. 6.2, the stress–strain curves in the thickness direction given by the above equations are compared with the bulge test curve obtained by Bramley and Mellor (1966). The derived curves are based on their measured R-values and the experimental stress–strain curve in the rolling direction. The hypothesis of strain equivalence is evidently in better agreement with experiment, at least for the materials used in this investigation, as has been shown by Chakrabarty (1970b). Further experimental results available in the literature tend to suggest that the strain-hardening hypothesis is preferable to the work-hardening hypothesis for most engineering materials. A crystallographic method of predicting the anisotropic behavior of sheet metals has been developed by Chan and Lee (1990).

6.2.4 Normal Anisotropy in Sheet Metal In many applications, the anisotropy in the plane of the sheet is small and can be disregarded by considering a state of planar isotropy with a uniform mean R-value. This provides a radical simplification to the problem, since the yield criterion and the flow rule then become independent of the choice of coordinate axes in the plane of the sheet. Since H/F = H/G = R and N/F =1+2R, when the anisotropy is rotationally symmetric about the z-axis, the yield criterion and the flow rule become

6.2

Anisotropy of Rolled Sheets

417

Fig. 6.2 Equibiaxial stress–strain curves for anisotropic sheets, (a) Experimental curve, (b) theoretical curve based on the uniaxial curve in the rolling direction and the hypothesis of strain equivalence, (c) theoretical curve derived from the uniaxial curve on the basis of work equivalence

σx2 −

2R 1 + 2R 2 = Y 2, σx σy + σy2 + 2 τxy 1+R 1+R

dεy sdγxy dλ dεx = = = , (1 + R) σx − Rσy (1 + R) σy − Rσx (1 + 2R) τxy (1 + R) Y

(6.21) (6.22)

where Y is the uniaxial yield stress of the material in the plane of the sheet. For the present purpose, it is convenient to take the equivalent stress σ¯ as identical to the current value of the planar yield stress and redefine the equivalent strain increment dε¯ according to the hypothesis of strain equivalence as 1+R dε = 2

1/2 2 2 2 3 dεx + dεy + dεx − dεy + 4dγxy 1 + R + R2

,

(6.23a)

so that it reduces to the longitudinal strain increment for a uniaxial tension applied in the plane of the sheet. The corresponding expression for dε according to the hypothesis of work equivalence takes the form dε =

1/2 1+R 2 . (1 + R) dεx2 + dεy2 + 2Rdεx dεy + 2dγxy 2

(6.23b)

418

6 Plastic Anisotropy

Expressed in terms of the principal stresses (σ 1 , σ 2 ) in the plane of the sheet, (6.21) represents an ellipse whose major and minor axes coincide with those of the von Mises ellipse. For R > 1 , the effect of anisotropy is to elongate the ellipse along the major axis, and slightly contract it along the minor axis for a given planar yield stress μY, Fig. 6.3 (a). The extremities of the√major axis represent states of balanced biaxial stress of magnitude μY, where μ = (1 + R)/2, as may be seen on setting σ x = σ y = σ and τ xy = 0 in (6.21). A piecewise linear approximation to the nonlinear yield criterion is achieved by replacing the ellipse with a hexagon obtained by elongating the inclined sides of the Tresca hexagon (Chakrabarty, 1974). The new yield locus is defined by σ1 = ±μY, σ2 = ±μY, σ1 − σ2 = ±Y.

Fig. 6.3 Biaxial yield loci for sheet metals with normal anisotropy. (a) Quadratic yield function and its linearization for R 1, (b) quadratic and nonquadratic yield functions for different R-values

It may be noted that the hexagon meets the ellipse not only at points representing uniaxial and biaxial states of stress, but also at points where the principal stress ratio is μ2 (μ2 − 1), which is equal to (R + l)/(R − l). The flow rule associated with the hexagonal yield locus requires the strain increment vector to be directed along the exterior normal when the stress point is on one of the sides of the hexagon. At a corner of the hexagon, the strain increment vector may assume any position between the normals to the two sides meeting in the corner. The increment of plastic work per unit volume is uniquely defined for all points of the hexagon. The preceding theory, though generally adequate for R > 1, does not account for an anomalous behavior observed in materials with R < 1. Indeed, Woodthorpe and Pearce (1970) have found that the yield stress in balanced biaxial tension for rolled aluminum, having an R-value lying between 0.5 and 0.6, is significantly higher than the uniaxial yield stress in the plane of the sheet. A modification of the theory is therefore necessary for dealing with such materials. Following Hill (1979),

6.2

Anisotropy of Rolled Sheets

419

the modified yield criterion in terms of the principal stresses σ 1 and σ 2 may be taken as |σ1 + σ2 |1+m + (1 + 2R) |σ1 − σ2 |1+m = 2 (1 + R) Y 1+m ,

(6.24)

where m is an additional parameter that depends on the state of normal anisotropy. It is easy to show that (6.24) reduces to (6.21) when m = 1. The parameter R in (6.24) is indeed the customary strain ratio in simple tension, as may be seen from the plastic flow rule which may be expressed in the form |dε1 − dε2 | |dε1 + dε2 | dλ = = . m m |σ1 + σ2 | (1 + 2R) |σ1 − σ2 | (1 + R) Y m

(6.25)

The signs of dε1 +dε2 and dε1 −dε2 are the same of those of σ1 +σ2 and σ1 −σ2 , respectively. The parameter dλ in (6.25) is a positive scalar equal to the equivalent strain increment dε defined by the hypothesis of work equivalence. Indeed, it follows from (6.24) that 2dW = (σ1 + σ2 ) (dε1 + dε2 ) + (σ1 − σ2 ) (dε1 − dε2 ) = 2Ydλ. Eliminating σ1 + σ2 and σ1 − σ2 between (6.24) and (6.25), the corresponding expression for the equivalent strain increment is obtained as

dε

1+n

=

1 (1 + R)n |dε1 + dε2 |1+n + (1 + 2R)−n |dε1 − dε2 |1+n , 2

where n = 1/m. For m = 1, this expression reduces to (6.23b) when the principal components alone are considered. The equivalent strain increment defined by the hypothesis of strain equivalence is obtained directly from (6.23a). For a workhardening material, it is only necessary to replace the planar yield stress Y by the equivalent stress σ¯ , which is a given function of the total equivalent strain. In the case of an equal biaxial tension σ , the yield criterion (6.24) gives (σ /Y)1+m = (1 + R)/2m , which indicates that a can be greater than Y for R < 1 with suitable values of m. Parmar and Mellor (1978b) derived M-values of 0.5 for rim steel (R ≈ 0.44), 0.7 for soft aluminum (R ≈ 0.63), and 0.8 for software brass (R ≈ 0.86), on the basis of their experimental results using the bulge test and the hole expansion test. Similar experimental results for different materials have been examined by Kobayashi et al. (1985). Since the m-value depends on the R-value, these two parameters cannot be chosen independently for any given material. Other types of nonquadratic yield function for normal anisotropy of sheet metal have been considered by Logan and Hosford (1980), Vial et al. (1983), and Dodd and Caddell (1984).

420

6 Plastic Anisotropy

6.2.5 A Generalized Theory for Planar Anisotropy The inadequacy of the quadratic yield function in predicting the ratio of the equibiaxial and uniaxial yield stresses for materials with R-values less than unity has already been noted. In the case of planar anisotropy, similar discrepancies are expected when the average R-value of the sheet is less than unity. The difficulty can be overcome by considering a suitable extension of the nonquadratic yield criterion in (6.24) to include the effect of planar anisotropy. The simplest yield criterion of this type, when the x- and y-axes are taken along the rolling and transverse directions, respectively, may be taken in the form σx + σy v + 2a |σx |v − σy v + b 2τxy v v/2 2 2 + c σx + σy + 4τxy = (2Z)v ,

(6.26)

where a, b, c, and v > 1 are the dimensionless anisotropic parameters, while Z is the through-thickness yield stress of the material (Chakrabarty, 1993a). A state of planar isotropy requires a = b = 0, and (6.26) then reduces to (6.24) with v = 1 + m and c = 1 + 2R. When v = 2, the above yield criterion becomes identical to (6.9) if we set G−F N − 2H 4H a= , b=2 − 1, c = 1 + , G+F F+G F+G and invoke the relation f + g = 1/Z2 . The anisotropic parameters appearing in (6.26) can be determined from the measurement of the R-values in the rolling, transverse, and 45◦ directions, as well as by the estimation of the ratio of the equibiaxial yield stress to the uniaxial yield stress in one of these directions. Using (6.26) the anisotropic coefficients can be expressed as 4a =

v v ⎫ 2Z v 2Z 2Z ⎪ − , 2 (1 + c) = + ,⎪ ⎬ Y X Y v v v ⎪ 2Z 2Z 2Z ⎪ ⎭ 2b = 2 − − , S X Y

2Z X

v

(6.27)

where X, Y, and S denote the uniaxial yield stresses in the rolling, transverse, and 45◦ directions, respectively, in the plane of the sheet. The uniaxial yield stress σ in any direction making an angle α with the rolling direction is obtained by inserting (6.9a) into (6.26), the result being

2Z σ

v = 1 + c + b (sin 2α)v + 2a

cos2 α

v

v . − sin2 α

(6.28)

Setting dσ /dα = 0, the stationary values of the yield stress are found to occur along the anisotropic axes, as well as in the directions α = ±α0 , where α0 is given by the expression

6.2

Anisotropy of Rolled Sheets

421

(tan α0 )2−v + (tan α0 )v = 2v−2 1 − tan2 α0

b . a

Thus, α0 depends only on the parameters v and a/b. When a = 0, we have α0 = π/4 irrespective of the value of v, a result which is in accord with experiment. Other types of nonquadratic yield functions for planar anisotropy have been discussed by Barlat and Lian (1989), and Hill (1990). Taking the plastic potential to be identical to the yield function (6.26), the components of the strain increment in the plane of the sheet can be written down from the normality rule. It is convenient to express the stress–strain relations as ⎫ v−2 v−2 + (σx + σy ) σx + σy dμ,⎪ dεx + dεy = a σx |σx |v−2 − σy σy ⎪ ⎪ ⎪ ⎬ v−2 v−2 v−2 + c(σx − σy )(2τ ) dμ, + σy σy dεx + dεy = a σx |σx | ⎪ ⎪ ⎪ v−2 ⎪ v−2 ⎭ dμ, + cτxy (2τ ) dγxy= bτwy 2τxy (6.29) where τ denotes the magnitude of the maximum shear stress in the plane of the sheet, and dμ denotes a positive scalar factor of proportionality. For a uniaxial tension σ acting at an angle α to the rolling direction, the transverse strain increment is given by the expression 1 1 dεx + dεy − dεx − dεy cos 2α − dγxy sin 2α 2 2 v−1 1 v−1 c − 1 + b (sin 2α)v − 2α cos2 α sin2 α =− σ 2

v−1 − sin2 α cos2 α dμ,

dεw =

which is obtained on substitution from (6.9a) and (6.29). The ratio of the transverse strain increment dεω to the thickness strain increment dεz = −(dεx + dεy ) is the ft-value in the considered direction and is the R-value in the considered direction and is found as v−1 2 v−1 sin α − sin2 α cos2 α (c − 1) + b (sin 2α)v − 2a cos2 α R= . (6.30) v−1 2 v−1 2 + 2a cos2 α − sin α It is interesting to note that the R-value is independent of v when α = 0, π/4, and π/2. Denoting these three R-values by Rx , Rs , and Ry , respectively, and using (6.30), we obtain the relations a=

Ry − Rx , Ry + Rx

2Rx Ry b = 2 Rs − , Rx + R y

c=1+

4Rx Ry , Rx + Ry

(6.31)

422

6 Plastic Anisotropy

which are the same as those for the quadratic yield function. The remaining parameter v is given by the three independent relations

2Z S

ι

= 2 (1 + Rs ) ,

2Z X

ν

4Ry (+Rx ) = , Rx + R y

2Z Y

ι

4Rx 1 + Ry = , Rx + R y (6.31a)

which are easily obtained by suitable combinations of (6.27) and (6.31). The first expression in (6.31a) is generally the most appropriate one. It follows from (6.26) and (6.29) that the plastic work increment per unit volume is given by 2dW = σx + σy dεx + dεy + σx − σy dεx − dεy + 4τxy dγxy = (2Z)v dμ, which immediately indicates that dμ is a measure of the equivalent strain increment based on the hypothesis of work equivalence. If the equivalent stress σ¯ is defined in such a way that it reduces to the applied stress when a uniaxial tension acts in the rolling direction, then in view of (6.28) and (6.31), we can express it in the form (2Z/σ¯ )v = 1 + 2a + c = 4Ry (1 + Rx ) / Rx + Ry . For a uniaxial tension σ applied in any direction α in the plane of the sheet, the ratio σ¯ /σ is obtained by combining (6.33) with (6.28). The equivalent strain increment dε, based on the hypothesis of strain equivalence, is given by (6.23a) with R replaced by Rx . The results for the rolling, transverse, and 45◦ directions are v ⎫ Rx + Ry Ry − Rx σ¯ 1+R ⎪ ⎪ +η ,⎪ = ⎪ ⎬ σ 1 + Rx 2Ry 2Ry ' ⎪ dε 1 + R x 1 + R + R2 ⎪ ⎪ ⎪ , = ⎭ dε 1 + R 1 + Rx + R2x

(6.32)

where R is the strain ratio associated with the given direction; dε is the longitudinal strain increment; and η is a parameter whose value is 1 for α = 0, is –1 for α = π /2, and is 0 for α = –π /4. In the case of an equibiaxial tension σ , the equivalent stress directly follows from (6.33) with Z = σ . The equivalent strain increment is obtained from (6.23a), using the fact that dεx + dεy = dε, dε x – dεy = β a dε and dγxy = 0, where dε is the compressive thickness strain increment at any stage of the loading, and β = 2 ν-2 . In Figs. 6.4 and 6.5, some comparison has been made of the theoretical predictions with the experimental data reported by Naruse et al. (1993) and Lin and Ding (1995). An interesting explanation of the observed anomaly in biaxial tension has been advanced by Wu et al. (1997) with the help of kinematic hardening along with the quadratic yield function.

6.2

Anisotropy of Rolled Sheets

423

Fig. 6.4 Theoretical and experimental variations of the R-value and yield stress ratio in the plane of the sheet for annealed aluminum

Fig. 6.5 Theoretical and experimental yield loci for cold rolled aluminum sheet exhibiting planar anisotropy

424

6 Plastic Anisotropy

Fig. 6.6 Plastic torsion of an anisotropic bar. (a) Direction of shear stress in relation to that of the characteristic, components of shear stress on a transverse section

Better agreement with experimental data can be achieved by considering a higher order yield function, such as the biquadratic yield function proposed by Gotoh (1977). Taking the axes of reference along the principal axes of anisotropy, the yield. criterion may be expressed as 2 4 +Kτxy = 1. (6.33) Aσx4 +Bσx3 σy +Cσx2 σy2 +Dσx σy3 +Eσy4 +(Fσx2 +Gσx σy +Hσy2 )τxy

The nine anisotropic coefficients appearing in (6.33) can be uniquely determined from the measurement of the uniaxial yield stresses and R-values in directions mak◦ ◦ ◦ ing angles of 0◦ , 22.5◦ (or 67.5 ), 45 , and 90 with the direction of rolling, in addition to that of the through-thickness yield stress. The R-value of the sheet metal corresponding to a uniaxial tension in any given direction follows from the flow rule associated with the yield criterion. Evidently, the distribution of the yield stresses and R-values, according to (6.33), is defined by curves which pass through all the experimental points included in Figs. 6.6 and 6.7. An isotropic material corresponds to B = D = – 2A, C = 3A, E = A, F = –G = H = 6A, K = 9A, and the yield function (6.33) then becomes a perfect square of the von Mises yield function.

6.3 Torsion of Anisotropic Bars 6.3.1 Bars of Arbitrary Cross Section A prismatic bar of arbitrary cross section is twisted by terminal couples about an axis parallel to the generator. The material is considered as rigid/plastic, and the applied torque is assumed sufficient to render the bar fully plastic. The state of anisotropy is orthotropic with the z-axis of anisotropy parallel to the generator. Choosing the x- and y-axes in an end section of the bar, the components of the velocity may be taken as

6.3

Torsion of Anisotropic Bars

425

u = −yz,

v = xz,

w = w (x, y) ,

the rate of twist per unit length being taken as unity. The nonzero components of the strain rate corresponding to the above velocity field are given by 2γ˙xy =

∂w − y, ∂x

2γ˙yz =

∂w + x. ∂y

The nonzero stress components are τxz and τyx which must satisfy the equilibrium equation and the yield criterion, these equations being ∂τyz ∂τxz + = 0, ∂x ∂y

2 τxz

k12

+

2 τyz

k22

= 1,

(6.34)

where k1 = (2M)–/1/2 and k2 = (2L)–/1/2 are the yield stresses in simple shear in the x- and y-directions, respectively. Differentiating the second equation of (6.34) partially with respect to x and y, and using the first equation, we get τyz ∂τxz τxz ∂τxz − 2 = 0, 2 k2 ∂x k1 ∂y

τyz ∂τyz τxz ∂τyz − 2 = 0. 2 k2 ∂x k1 ∂y

These equations are seen to be hyperbolic, and the characteristic through a generic point P in the (x, y)-plane is in the direction 2 k2 τxz dy = tan ψ = − , dx k1 τyz

(6.35)

where ψ is the counterclockwise angle made by the tangent to the characteristic with the positive x-axis. Using the preceding relations, it is easy to show that dτxz = dτyz = 0 along a characteristic (Hill, 1954). It follows that the characteristics are straight, and the resultant shear stress r along each one of them is constant in direction and magnitude. Since the lateral surface of the bar is stress free, the resultant shear stress must be directed along the tangent to the boundary S at the point where it is cut by the characteristic. The magnitude of the shear stress along any characteristic can be calculated from (6.35) and the second equation of (6.34). The characteristics meet the boundary at varying angles when the bar is anisotropic. The resultant shear strain rate at a point of the cross section is orthogonal to the characteristic, as may be seen from the flow rule associated with the yield function.

426

6 Plastic Anisotropy

Indeed, the components of the rate of shear are in the ratio γyz = γyz

k1 k2

2

τyz = − cot ψ τxz

in view of (6.35), establishing the condition of orthogonality of the strain rate. The component of the rate of shear in the characteristic direction is therefore zero. The substitution for γ˙xz and γ˙yz above relation gives

∂w ∂w − y cos ψ + − x sin ψ = 0. ∂x ∂x

This equation is hyperbolic with characteristics identical to (6.35). The variation of the rate of warping along a characteristic is given by ∂w ∂w ∂w = cos ψ + sin ψ = y cos ψ − x sin ψ, ∂s ∂x ∂y where ds is a line element along the characteristic. If w0 denotes the value of w at any given point on the characteristic, the integration of the above equation furnishes ω = ω0 +

(ydx − xdy) = ω0 +

pds,

(6.36)

where p is the perpendicular distance from the origin to the considered element, reckoned positive when the vector ds has a clockwise moment about the z-axis, Fig. 6.6(a). The integration begins from the point of intersection of the characteristic with a stress discontinuity G, formed by the intersection of characteristics. Since the shear strain rate vanishes along G, (6.36) also holds along G. Any projecting corner of S is the source of one of the branches of G, while a reentrant corner is a point of singularity that generates a fan of characteristics. For the evaluation of the torque T in the fully plastic state it is convenient to introduce a stress function φ such that the shear stress components are given by τxz =

∂φ , ∂y

τyz = −

∂φ , ∂x

(6.37)

satisfying the equilibrium equation identically. The lines of constant φ are the shearing stress trajectories in the (x, y)-plane. Theexternal boundary C is also a contour line of φ, and we may choose φ = 0 along this boundary. Referring to Fig. 6.6(b), the applied torque is found to be

6.3

Torsion of Anisotropic Bars

T=

427

xτyz − yτxz dxdy = 2

φdxdy.

(6.38)

The second expression of (6.38) follows on substituting from (6.37), integrating by parts, and using the boundary condition φ = 0. If we now introduce the transformation k2 x = kξ , k1 y = kη, where k is an arbitrary constant having the dimension of stress, the yield criterion expressed by the second equation of (6.34) becomes

∂φ ∂ξ

2 +

∂φ ∂η

2 = k2 .

In the (ξ , η)-plane, the stress function therefore satisfies the same equation as that for an isotropic bar with a shear yield stress k. The characteristics in the (ξ , η)-plane are normal to the transformed contours of φ, and correspond to the characteristics in the (x, y)-plane. Equation (6.38) now becomes T=

2 k2 k1 k2

φdξ dη =

k2 k1 k2

T∗,

(6.39)

where T∗ denotes the fully plastic torque for an isotropic bar having the transformed cross section. The stress function for the anisotropic bar is obtained directly from 1/2 the fact that ∂φ/∂s along a characteristic is of magnitude k12 sin2 ψ + k22 cos2 ψ which is easily obtained from (6.35), (6.37), and (6.34).

6.3.2 Some Particular Cases As a simple application of the theory, let us consider a bar of elliptical cross section whose semiaxes are in the ratio k1 :k2 , the equation of the ellipse being y2 x2 + = 1, a2 b2

a k1 = . b k2

The corresponding contour in the (ξ , η)-plane becomes the circle ξ 2 + η2 = c2 , where kc = k1 b = k2 a. Since T ∗ = 23 π kc3 for the isotropic bar, (6.39) furnishes the fully plastic torque for the anisotropic bar as 2 T= π 3

k 3 c3 k1 k2

=

2 2 π k1 ab2 = π k2 a2 b. 3 3

428

6 Plastic Anisotropy

Since the lines of shearing stress in the (ξ, η)-plane are concentric circles, the stress trajectories in the (x, y)-plane are concentric ellipses. The characteristics are radial lines in both planes, and the warping is absent. When the cross section of the anisotropic bar is a circle of radius c, the transformed contour is an ellipse with semiaxes a = ck2 /k and b = ck1 /k. When k1 >k2 , a stress discontinuity extends along the η-axis over a length 2b(1–a2 /b2 ), which is the distance between the centers of curvatures at the extremities of the major axis. The corresponding discontinuity along the y-axis covers a length 2c(1 − k22 /k12 ). If the degree of anisotropy is sufficiently small, it is a good approximation to take the yield point torque for the transformed section as the mean of those for circular sections of radii a and b. Thus 3 3 3 π k π 3 3 3 k1 + k 2 . a +b = c T≈ 3 k1 k2 3 k1 k2 It follows from (6.36), and the nature of the characteristic field, that the warping is positive in the first and third quadrants, and negative in the others. As a final example, consider a bar whose cross section is a rectangle having sides of lengths a and b parallel to the x- and y-axes, respectively. The transformed contour is also a rectangle with sides α = ak2 /k and β = bk1 /k, parallel to the ξ and η-axes, respectively. Since T∗ = kβ 2 (3α–β)/6 for α ≥ β, the actual yield point torque is obtained from (6.39) as 1 T= 6

k3 β 2 k1 k2

(3α − β) =

1 3 k1 b 6

3a k1 − b k2

,

k1 a ≥ b k2

(6.40)

The solution for a/b ≤ k1 /k2 is obtainedby merely interchanging a, b and k1 , k2 in (6.40). The stress discontinuities emanating from the corners of the rectangle meet on the x- or y-axes accordingly as a/b is greater or less than k1 /k2 , the axial discontinuity being of length |ak2 —bk1 |/k. The warping of the cross section can be calculated from (6.36) in the same way as that employed for an isotropic bar (Chakrabarty, 2006).

6.3.3 Length Changes in Twisted Tubes A thin-walled cylindrical tube is twisted in the plastic range, the ends of the tube being supported in such a way that it is free to extend or contract in the axial direction. We begin with the situation where the tube may be assumed to remain isotropic during the deformation. Consider a small element of the tube formed by cross sections at a unit distance apart, so that the displacement of a particle on one edge due to its rotation relative to the other is equal to the engineering shear strain γ . Thus, a typical material line element OP, inclined at an angle ψ to the direction of

6.3

Torsion of Anisotropic Bars

429

Fig. 6.7 Torsion of an anisotropic tube. (a) Geometry of finite shear deformation, (b) stress acting on an element of the tube wall

shear, rotates to a new position OQ such that PQ = γ . If the lengths OP and OQ are denoted by l0 and l, respectively, Fig. 6.7(a), then from simple geometry,

l2 = l02 + 2l0 γ cos ψ + γ 2 = l02 1 + 2γ sin ψ cos ψ + γ 2 sin2 ψ in view of the relation l0 sin ψ = 1. Differentiating this expression with respect to ψ, it is found that the ratio l/l0 has a maximum value when − 2 cot 2ψ = γ , l = l0 tan ψ, l cos ψ = 1.

It follows that the angles of inclination of OP and OQ to the direction of shear are complementary when i/iq is a maximum. The same conclusion holds for the line elements OP’ and OQ’ in the initial and final states, respectively, for which the length ratio is a minimum. The directions OR and OR’, which correspond to zero resultant change in length, must be each inclined at an angle 2ψ to the direction of shear, since R’N = RN = γ /2, where ON is parallel to the tube axis. Suppose, now, that the tube becomes progressively anisotropic during the torsion, and that the anisotropic axes coincide at each stage with the directions of the greatest relative extension and contraction. Since the axial strain ε is small compared to the shear strain y, these directions may be assumed identical to those for an isotropic tube. Let the x- and y-axes of anisotropy, which correspond to OQ’ and OQ, respectively, at any given stage, make an angle φ with the direction of twist and the axis of the tube, respectively, Fig. 6.7(b). The nonzero components of the stress are σx = −σy = −τ sin 2φ,

τxy = τ cos 2φ,

where τ denotes the applied shear stress. The substitution into the yield criterion (6.9) then furnishes

430

6 Plastic Anisotropy

−1 τ 2 = 2 N + (F + G + 4H − 2 N) sin2 2φ . The ratios of the components of the strain increment associated with the yield criterion are obtained from (6.3) as dεy dγxy dεx dεz = = = tan 2φ. G + 2H F + 2H G−F N The axial strain increment dε and the engineering shear strain increment dγ produced by the torsion are dε = dεx sin2 φ + dεy cos2 φ − 2dγxy sin φ cos φ, dγ = dεy − dεx sin 2φ + 2dγxy cos 2φ. Substituting for the strain increment components appearing on the right-hand side of these relations, we obtain the ratio of dε to dγ in the form (N − G − 2H) sin2 φ − (N − F − 2H) cos2 φ sin 2φ dε = . dγ 2 N + (F + G + 4H − 2 N) sin2 φ

(6.41)

The value of dε/dγ is initially zero, when the tube is isotropic (φ = π /4). For small angles of twist, φ is slightly greater than π /4, and dε has the same sign as that of F – G, or equivalently of X–Y. With increasing angle of twist, φ eventually approaches π /2, and dε is finally positive if N > G + 2H. For many metals, the anisotropic parameters vary in such a way that the tube lengthens continuously during the torsion (Swift, 1947; Bailey et al. 1972). In exceptional cases, however, the length of the tube may progressively decrease with increasing angle of twist (Toth et al., 1992). As the tube becomes increasingly anisotropic during the torsion, the ratios F/H, G/H, and N/3H vary continuously from their common initial value of unity to approach some limiting values in an asymptotic manner. These limiting ratios can be found by direct measurements after subjecting the tube to a sufficiently large angle of twist. Each ratio of the anisotropic parameters at a generic stage of the torsion depends on its asymptotic value and the magnitude of the shear strain, according to a mathematical function which may be assumed to be the same for all three ratios. A suitable choice of such a function enables us to determine the length change at any stage by the integration of (6.41). Conversely, an actual measurement of dε and dγ for each increment of torque provides a means for determining the variation of the ratios of the anisotropic parameters under increasing strain.

6.3.4 Torsion of a Free-Ended Tube The change in length that takes place during the free-end twisting of a thin-walled tube should be taken into consideration for the derivation of the shear stress–strain

6.3

Torsion of Anisotropic Bars

431

curve from the torsion test. Let a0 and t0 denote the initial mean radius and wall thickness, respectively, of a thin-walled tube which is twisted in the plastic range with a torque T and an angle of twist θ per unit length of the tube. If the current internal and external radii of the tube are denoted by a1 and a2 , respectively, then the applied torque is T = 2π

a2

τ r2 dr =

a1

2π θ3

a2 θ

a1 θ

τ γ 2 dγ .

Multiplying both sides of this equation by θ 3 differentiating with respect to θ , and considering a mean shear stress τ¯ through the thickness of the tube, we get 3T + θ

d d dT = 2π τ¯ a22 (a2 θ) − a21 (a1 θ ) . dθ dθ dθ

Since the wall thickness is small compared to the mean radius a and remains practically unchanged during the torsion, it is reasonable to introduce the approximations da1 /a1 ≈ da2 /a2 ≈ −dε, a32 − a31 ≈ 3t0 a2 ≈ 3t0 a20 e−2ε dT t0 dτ¯ 1 θ τ2 − τ1 ≈ θ ≈ , τ1 + τ2 ≈ 2τ¯ a dθ 2π a3 dθ where ε denotes the axial strain, which is approximately equal to the magnitude of the hoop strain, the elastic deformation being disregarded. The expression for the mean shear stress is therefore closely approximated by the formula τ¯ ≈

T 2π t0 a20

dε 1 + 2ε + θ dθ

(6.42)

to a sufficient accuracy. The evaluation of the mean shear stress τ¯ requires the measurement of T,θ , and ε simultaneously at each stage of the loading. The corresponding mean shear strain γ¯ is approximately equal to α0 θ , the change in radius being small. A more elaborate expression for the shear stress in the free-end torsion of a thin-walled tube has been given by Wu et al. (1997). The shear stress–strain curve obtained from the torque-twist curve in the free-end torsion of a thin-walled tube, using (6.42), is found to coincide with that derived from the fixed-end torsion of the tube. The fixed-end torsion naturally gives rise to an axial compressive stress that suppresses the axial strain, while producing small amounts of circumferential and thickness strains in the twisted tube. Figure 6.8 shows the observed variation of the axial and hoop strains with the shear strain during the free-end torsion of an extruded aluminum tube, obtained experimentally by Wu (1996). The accumulation of axial strain in the finite torsion of tubular specimens with both free and fixed ends has been investigated by Wu et al. (1998).

432

6 Plastic Anisotropy

Fig. 6.8 Development of axial and hoop strains in finitely twisted thin-walled tubes in the plastic range (after H.C. Wu, 1996)

6.4 Plane Strain in Anisotropic Metals 6.4.1 Basic Equations in Plane Strain Consider the class of problems in which the plastic flow is restricted in the plane perpendicular to the z-axis of anisotropy, which coincides with a principal axis of the strain rate. The condition γ˙xz = γ˙yz = 0 requires τ xz = τ yz = 0 by the flow rule, indicating that σz is a principal stress. Using the plane strain condition dε z = 0 in the third equation of (6.3), we get σz =

Gσx + Fσy . G+F

In the special case when σy = 0, the above relation indicates that σz is greater or smaller than σx /2 according as G is greater or smaller than F. Substituting for σz in (6.1), and setting τ xz = τ yz = 0, the yield criterion is reduced to 2 FG 2 = 1. σx − σy + 2 Nτxy H+ F+G

6.4

Plane Strain in Anisotropic Metals

433

It is convenient at this stage to introduce a dimensionless parameter c defined as c=1−

N (F + G) (−∞ < c < 1) . 2 (FG + GH + HF)

(6.43)

Thus, c is positive when N is less than both F + 2H and G + 2H, and negative when N is greater than both F + 2H and G + 2H. If the anisotropy is rotationally symmetrical about the z-axis, c = 0. In view of (6.43), the yield criterion for anisotropic materials is expressed as 2 σx − σy 2 = k2 , + τxy 4 (1 − c)

(6.44)

where k = (2 N) –1/2 is the yield stress in pure shear associated with the x- and yaxes. The yield stress σ in plane strain tension in the direction making an angle α with the x-axis is obtained by inserting (6.9a) into (6.44), the result being 1/2 1−c σ = 2k . 1 − c sin2 2α Since σ has equal values in the directions ±α and ±(π /2–α), the angular variation of σ is symmetrical about the x- and y-axes, √ as well as about their bisectors. When c is positive, σ has a minimum value of 2 k 1 − c along the anisotropic axes, and a ◦ maximum value √ of 2 k in the 45 directions. When c is negative, σ has a maximum value of 2 k 1 − c along the anisotropic axes, and a minimum value of 2 k in the 45◦ . A more general theory for plane strain has been discussed by Rice (1973). Let vx and vy denote the components of velocity of a typical particle with respect to the anisotropic axes. The components of the strain rate in the plane of plastic flow referred to the anisotropic axes are ∂vy ∂vx ∂vv 1 ∂vx , ε˙ y = , γ˙xy = + . ε˙ x = ∂x ∂y 2 ∂y ∂x It follows from the flow rule associated with the yield criterion (6.44) that 2τxy 2γ˙xy . = (1 − c) ε˙ x − ε˙ y σx − σy

(6.45)

If ψ and ψ denote the angles of inclination of a principal stress direction and the corresponding principal strain rate direction, respectively, with respect to the x-axis, then 2ψ = (1 − c) tan 2ψ in view of (6.45). When c =0, it follows that ψ = ψ only for ψ = 0, π/4, and π/2, as expected. In terms of the velocity gradients, (6.45) can be expressed as ∂vx ∂vy ∂vy ∂vx + − = 2 (1 − c) τxy . σx − σy ∂y ∂x ∂x ∂y

434

6 Plastic Anisotropy

The last equation, together with the incompressibility condition, the equilibrium equations, and the yield criterion, constitutes a set of five equations for the three stress components and the two velocity components. The parameters k and c in the theoretical framework can be experimentally determined by carrying out the plane strain compression test at 0◦ and 45◦ to one of the axes of anisotropy. As in the case of isotropic solids, the governing equations are hyperbolic with characteristics in the directions of maximum rate of shear at each point of the deforming zone. The characteristics for the stress are identical to those for the velocity, but these curves do not generally coincide with the trajectories of the maximum shear stress in the plane of plastic flow. Due to the incompressibility of the material, the rate of extension vanishes along the characteristics, which are known as the sliplines.

6.4.2 Relations Along the Sliplines The two orthogonal families of sliplines will be designated by α and β following the convention that the acute angle made by the algebraically greater principal stress in the considered plane is measured counterclockwise with respect to the α-direction. If φ denotes the counterclockwise angle made by a typical α-line with the x-axis, as shown in Fig. 6.9(a), then dy/dx = tan φ along an α-line and dy/dx = –cot φ along a β-line. Since the left-hand side of (6.45) is equal to –cot 2φ, we have σx − σy cos 2φ + 2 (1 − c) τxy sin 2φ = 0.

(6.46)

Fig. 6.9 Orientation of sliplines, and the stresses acting across them in a curvilinear element

It follows from (6.44) and (6.46) that the outward drawn normal to the yield locus, obtained by plotting τxy against (σ x –σ y )/2, makes a counterclockwise angle of 2φ with respect to the τ xy -axis. This result actually holds for any convex yield function, under condition of plane strain, as has been shown by Rice (1973).

6.4

Plane Strain in Anisotropic Metals

435

Equation (6.46) can be solved simultaneously with the yield criterion (6.44) to give τxy cos 2φ = , k 1 − c sin2 2φ

σx − σy 2 (1 − c) sin 2φ = − . k 1 − c sin2 2φ

(6.47)

Let σα , σ β , and τ αβ denote the stress components referred to the local sliplines taken as a pair of curvilinear axes in the plane, as shown in Fig. 6.9(b). Then σα − σβ = σx − σy cos 2φ + 2τxy sin 2φ, ταβ = σx − σy sin 2φ − 2τxy cos 2φ. Substituting from (6.47), and setting p = –(σα + σβ )/2, which is the mean compressive stress in the plane of plastic flow, the (α, β) components of the stress can be expressed as ⎫ ds ds ⎪ σβ = −p + k , ταβ = 2ks, ⎪ σα = −p − k , ⎬ dφ dφ (6.48) ⎪ ⎪ ⎭ 2 2s = 1 − c sin 2φ. If the radii of curvature of the α- and β-lines at a generic point are denoted by Rα and Rβ , respectively, the equilibrium equations in the curvilinear coordinates (α, β) may be written as σα − σβ ∂ταβ 2ταβ ∂σα + + − = 0, ∂sα Rβ ∂sβ Rα σα − σβ ∂ταβ 2ταβ ∂σβ + + + = 0, ∂sβ Rα ∂sα Rβ where sα and sβ are the arc lengths along the respective sliplines. Since the curvatures 1/Rα and 1/Rβ are equal to ∂φ/∂Sα and ∂φ/∂Sβ , respectively, while the normal stress difference σ α –σ β is equal to = –dταβ /dφ in view of (6.48), the second and third terms of each of the above equations cancel one another, while the remaining terms give ∂ ds ∂φ p+k = 0, + 4ks ∂sα dφ ∂sα ∂φ ∂ ds + 4ks p−k = 0, ∂sβ dφ ∂sβ in view of (6.48). These equations can be expressed in the integrated form (Hill, 1949) to obtain the characteristic relations

436

6 Plastic Anisotropy

p + 2 kω = constant along an α line, p − 2 kω = constant along a β line,

(6.49)

The parameter ω depends on the angle φ and the anisotropic parameter c according to the relation 2ω =

ds +4 dφ

φ 0

√ c sin 2φ cos 2φ + E 2φ, c , sdφ = − 1 − c sin2 2φ

(6.50)

where E denotes the standard elliptic function of the second kind and is defined as

α

E (α, m) =

1 − m2 sin2 θ dθ .

0

For an isotropic material, ω = φ, and (6.49) reduces to the well-known Hencky equations for plane strain. If the components of the velocity along the α- and β-lines are denoted by u and v, respectively, the condition of zero rate of extension along the sliplines can be easily reduced to du − vdφ = 0 along an α−line, (6.51) dv + udφ = 0 along a β−line, These relations are the same as the Geiringer equations for isotropic solids. In analogy to Hencky’s first theorem, it is easily shown that the difference in the values of ω or p between a pair of points, where two given sliplines of one family are intersected by a slipline of the other family, remains constant along their lengths. It follows that if one segment of a slipline is straight, the corresponding segments of sliplines of the same family are also straight, and the straight segments are consequently of equal lengths.

Fig. 6.10 Indentation of the plane surface of a semi-infinite anisotropic medium by a rigid flat punch

6.4

Plane Strain in Anisotropic Metals

437

6.4.3 Indentation by a Flat Punch Consider, as an example, the indentation of the plane surface of a large block of metal by a rigid flat punch. At the incipient plastic flow, the sliplines emanating from the corners A and A of the punch reach the free surface of the block. The slipline field shown in Fig. 6.10 is an extension of the Prandtl field for isotropic solids. In the triangular regions ABD and A B D , the state of stress is a uniform compression 2p0 parallel to the surface, where p0 is the value of p in these regions. If ψ denotes the angle of inclination of the plane surface with the x-axis of anisotropy, it follows from (6.44) that p0 = k

1/2

1−c

.

1 − c sin2 2ψ

(6.52)

If λ denotes the acute angle made by the α-lines with the plane surface, then φ = λ – ψ along AB and A B . Since the direction of the algebraically greater principal stress is normal to the plane, making an angle π /2 – ψ with the x-axis, (6.46) gives cot 2 (λ − ψ) + (1 − c) tan (π − 2ψ) = 0, or

(6.53) λ=ψ+

1 2

−1

cot

[(1 − c) tan 2ψ] .

√ √ It is easily shown that λ lies between cot−1 1 − c and π/2 cot−1 1 − c whatever the value of ψ. Since the state of anisotropy cannot change appreciably during the deformation preceding the yield point, we are justified in treating c as a constant throughout the plastic region. It is assumed that the material beneath the punch is uniformly stressed with the α sliplines inclined at an angle λ to the punch face, regardless of the frictional condition. This is compatible with a rigid-body motion of the plastic triangle ACA which is attached to the punch. The slipline field is completed by introducing the 90◦ centered fans ACD and A CD , where the sliplines are radial lines and circular arcs. From (6.49), the value of p in ACD increases from p0 on AD to p0 + 2kE on AC, where

π √ π/2 , c = E=E 1 − c sin2 2θ dθ . 2 0 The principal compressive stresses in the region ACA , directed normal and parallel to the punch face, are 2(p0 + kE) and 2kE, respectively. The normal pressure q on the punch face is therefore given by q p0 = +E = 2k k

1−c 1 − c sin 2ψ 2

1/2 +E

π √ , c 2

(6.54)

438

6 Plastic Anisotropy

in view of (6.52). When the degree of anisotropy is small, it is convenient to expand P0 /k and E in ascending powers of c, and neglect all terms of order c2 and above. Then the result becomes

q π c π ≈ 1+ − + cos2 2ψ . 2k 2 2 4 For an isotropic solid (c = 0), the punch pressure reduces to the Prandtl value 2k(1+π /2). The punch pressure for an anisotropic solid is less than this when c is positive and greater than this when c is negative. It may be noted that the punch pressure is the same for orientations ψ and π /2 – ψ of the axes of anisotropy, which is a consequence of the symmetry of the anisotropy about the directions equally inclined to the anisotropic axes. The incipient deformation mode consists of plastic flow parallel to the sliplines CDB and CD B , the material below these boundaries being held rigid. The velocity is of magnitude U sin λ parallel to CDB, and of magnitude U cos λ parallel to CD’B’, where U is the downward speed of the punch. These are also the magnitudes of the velocity discontinuities across A C and AC, respectively. In an alternative solution, valid only for a smooth punch (Hill, 1950a), the bounding sliplines meet at a point S on the punch face. The magnitudes of the velocity discontinuity initiated at this point are then modified to U cosec λ and U sec λ, while the punch pressure is still given by (6.54).

6.4.4 Indentation of a Finite Medium Consider the indentation of the plane surface of a medium of finite depth h resting on a rigid smooth foundation. When the ratio h/a is sufficiently small, where a is the semiwidth of the punch, the plastic zone spreads downward from the punch face to reach the foundation at the yield point state. The slipline field shown in Fig. 6.11 (a) is developed from the triangle ABC, which is not an isosceles triangle unless AB coincides with an anisotropic axis. For an arbitrary angle ψ made by the x-axis of anisotropy with the free surface, the centered fans ACE and BCD are of unequal radii, their angular spans being θ and δ, respectively. The angle λ which the β-line makes with the punch face is given by (6.53). Since the foundation is smooth, the bounding β-line BDF meets the foundation at the same angle λ. The existence of an incipient velocity field at the yield point can be easily established following the method used for isotropic materials. By the analogue of Hencky’s first theorem, which is obtainable from (6.49), we have ωD = ωF + ωC − ωE , where the subscripts refer to the points in the slipline field. The right-hand side of the above equation can be evaluated for any assumed angle θ, using (6.50) and the relations

6.4

Plane Strain in Anisotropic Metals

439

Fig. 6.11 Indentation of a block of anisotropic material of finite depth. (a) Slipline field, (b) variation of punch pressure with block thickness

φ F = φC = −

π 2

−λ+ψ ,

φC = φE = θ .

The other fan angle δ is then computed from (6.50) by a trial-and-error procedure, using the fact that φ D –φ D –φ C = δ. The slipline field is generated from the given intersecting sliplines CD and CE by the usual small arc process of approximation. The construction is similar to that for isotropic solids (Chakrabarty, 1987), except that the value of φ at each unknown nodal point must be determined from the computed slipline variable ω. The mean compressive stress pF at F must be determined from the condition of zero horizontal force transmitted across the bounding sliplines AF and BF, Considering the α-line AEF, and a pair of rectangular axes (ξ , η) through the midpoint of AB, the resultant horizontal thrust exerted on AEF by the rigid material on its left may be written as F=−

τ dξ +

σ dη = 2 k

sdξ +

pdη − k

ds dφ

dη,

where τ = τ αβ and σ = –σβ , denoting the tangential stress and normal pressure on a slipline element, and the integrals are taken along the entire slipline AEF. Integrating the second term on the right-hand side by parts, and noting that dp = –2 k dω along

440

6 Plastic Anisotropy

an α-line, the boundary condition F = 0, which holds when point F lies on the foundation, is reduced to h

p F

2k

=

F A

1 sdξ + 2

F A

ds dφ

dη −

F

ηdω

(6.55)

E

where s(φ) is given by (6.48). The integrals can be evaluated numerically for a selected value of θ since the values of φ, ω, ξ , and η are known along the slipline, to obtain a value of pF using (6.55). As h increases to a critical value h∗ , the pressure pF decreases to the value – p0 , where p0 is given by (6.52). The normal pressure on the foundation then vanishes at F, an element at F being under a uniaxial tension 2p0 parallel to the foundation. For h > h∗ , it is necessary to introduce a triangle FGH of height t with the inclined sides tangential to the sliplines at F. The triangle contributes a horizontal tensile force 2p0 t, and the condition of zero resultant force across the boundary AEFH again leads to (6.55), which furnishes h (and hence t) for a given θ . The indentation pressure q, uniformly distributed over the punch face, is given by q pC + p0 pF + p0 = = + (ωF + ωC − 2ωE ) . 2k 2k 2k The first term on the right-hand side is positive for h 0

(7.53)

to a close approximation, the integral being extended over the entire volume of the material of the beam. The uniqueness is guaranteed when (7.53) is satisfied for all possible velocity fields consistent with (7.52). A sufficiently wide class of admissible velocity fields, representing simultaneous bending in two orthogonal planes together with nonuniform twisting and warping of the cross sections, would be too complicated for practical purposes. On the other hand, the velocity gradients appearing in (7.53) can be determined with sufficient accuracy from the relatively simple field

502

7

˙ υy = υ − φx ˙ , υx = u − φy,

Plastic Buckling

du dυ υz = x + y , dz dz

(7.54)

where (u, υ) denotes the velocity of the axis of the beam, and φ˙ denotes the rate of twist about this axis, all these quantities being functions of z only. It follows from (7.54) that 2 ⎫ d2 υ ∂υz d u ⎪ ⎪ =− x 2 +y 2 , ⎬ ∂z dz dz ∂υx du dφ ∂υy dυ dφ˙ ⎪ ⎪ = −y , = + x .⎭ ∂z dz dz ∂z dz dx

(7.55)

The assumed velocity field (7.54) is consistent with the fact that ε˙ xy = 0 throughout the beam, but it does not give a nonzero value of ε˙ xx = ε˙ yy. The field is also unsuitable for calculating the shear rates ε˙ xz and ε˙ yz , since the rate of warping of the cross section has been disregarded. It is reasonable, however, to assume the relations ε˙ xz = 0,

ε˙ yz = x

dφ˙ , dz

(7.56)

which apply to the elastic torsion of bars of narrow rectangular cross section. The usefulness of such simplified velocity fields with appropriate adjustments has been noted by Pearson (1956) in the context of elastic buckling. The substitution from (7.55) and (7.56) into (7.53) gives

2 2 d2 υ dφ˙ d2 u + 4G x (T − σ ) x 2 + y dz dz dz $ 2 2 % ˙ dφ dφ˙ du dυ −y +x dx dy dz > 0, + σ + dz dz dz dz

where x varies between –b and b, y varies between –h and h, and z varies between 0 and l. The evaluation of the above integral is greatly simplified by the fact that σ (–y) = –σ (y) and T(–y) = T(y), giving

h −h

σ dy =

h

−h

σ y dy = 2

h −h

Tydy = 0.

Since σ and T are independent of x, and the stress distribution across each vertical section is statically equivalent to a bending moment equal to M, the condition for uniqueness becomes

7.3

Lateral Buckling of Beams

l 0

αEIy

du dz

503

2 + βEIx

dυ dz

2 + GJ

dφ˙ dz

2 + 2M

du dφ˙ dz dz

dz > 0, (7.57)

where Ix and Iy are the principal moments of inertia of the cross section about the x- and y-axes, respectively, and GJ is the torsional rigidity with J = 4Iy , the constants α and β being defined as 1 αEh = 2

h

3 Tdy = Rσ0 , βEh = 2 −h 3

h −h

Ty2 dy,

where R is the radius of curvature of the bent axis at the incipient buckling, and σ 0 the magnitude of the numerically largest stress that occurs at y = ±h. The deformation prior to buckling is assumed to be small, so that the longitudinal strain is equal to –y/R throughout the elastic/plastic bending. The minimum value of the functional in (7.57) with respect to the variations of u, υ, and φ˙ must vanish at the point of bifurcation. The Euler–Lagrange differential equations characterizing this variational problem are easily established as ⎫ d2 φ˙ d4 υ d4 u ⎪ ⎪ − M = 0, = 0, ⎬ dz4 dz2 dz4 ⎪ d2 φ˙ d2 u ⎪ ⎭ GJ 2 + M 2 = 0. dz dz

αEIy

(7.58)

The ends of the beam are assumed to be supported in such a way that they cannot rotate about the z-axis. Then the curvature rate in the xz-plane also vanishes at these sections, and the boundary conditions become φ˙ =

d2 u = 0 at z = 0 and z = l. dz2

Due to the symmetry of the loading, the curvature rate in the yz-plane must have identical values at the sends of the beam. The first two equations of (7.58) may therefore be integrated twice to give αEIy =

d2 u d2 υ − M φ˙ = 0, = constant. 2 dz dz2

(7.59)

The second equation of (7.59) merely states that the axis of the beam continues to bend into a circular arc at the point of bifurcation. The elimination of d2 u/dz2 between the first equation of (7.59) and the last equation of (7.58) results in d2 φ˙ MI + k2 φ˙ = 0, k = EIy dξ 2

1+ν , 2α

(7.60)

504

7

Plastic Buckling

where ξ = z/l and ly = 4bh3 /3. The solution to this differential equation, subject to the boundary conditions φ = 0 at ξ = 0 and ξ = 1, is readily obtained as φ˙ = A sin π ξ , k = π , where A is a constant. Over the elastic range, we √ have σ 0 = Eh/R giving α = 1, and the critical moment is given by Ml/EIy = π 2/ (1 + υ), which is in agreement with the well-known result for elastic buckling of narrow rectangular beams. When the buckling occurs in the plastic range, α depends on the magnitude of the critical moment through the strain-hardening characteristic of the material. Assuming the power law σ /Y = (Eε/Y)n for the uniaxial stress–strain curve in the plastic range, the stress distribution for y ≥ 0 may be written as y σ = − (0 ≤ y ≤ c) , Y c

y n σ =− (c ≤ y ≤ h) . Y c

The radius of curvature of the bent axis, when the elastic/plastic boundary is at y = c, is R = Ec/Y. The bending moment across any section is

h

M = −4b 0

n h 3 4 2 1 − n c 2 . σ ydy = bh Y − 3 2+n c 2+n h

Introducing the initial yield moment Me = 4bh2 Y/3, and using the fact that h/c is equal to the curvature ratio k/ke , the moment–curvature relationship may be expressed in the dimensionless form M 3 = Me 2+n

κ κe

n −

1 − n κ e 2 . 2+n κ

(7.61)

Since the greatest bending stress at any stage is σ o = Y(h/c)n during the elastic/plastic bending, we have α=

Rσ0 = (c/h)1−n = (κe /κ)1−n . Eh

Substituting for α and M in (7.60) and setting k = π , the equation for the critical curvature ratio for lateral buckling is obtained as 3 2+n

κ κe

(1+n)/2

1 − n κe (3+n)/2 π Eh − = 2+n κ Yl

2 = λ (say) . 1+ν

(7.62)

When κ/κ e is computed from (7.62) for any given values of n and Eh/Yl, the critical bending moment follows from (7.61). This solution is valid for κ/κ e ≥ 1, which is equivalent to the condition λ ≥ 1. In the case of λ ≤ 1, the buckling will occur in the elastic range when M/Me = λ, obtained by setting n = 1 in (7.62). Figure 7.8 shows the variation of the dimensionless critical bending moment with the parameter μ for several values of n.

7.3

Lateral Buckling of Beams

505

Fig. 7.8 Dimensionless critical bending couple for the lateral buckling in the plastic range as a √ function of a parameter λ (Eh/Yl) 2/(1 + ν)

Equations (7.59) are equally applicable to thelateral buckling of beams of arbitrary doubly symmetric cross sections J = 4Iy subjected to terminal couples in the yz-plane, which is considered as the plane of maximum flexural rigidity. The last equation of (7.58) must be modified, however, by the inclusion of the term ˙ 4 on the left-hand side to take account of the warping rigidity, the −ECw d4 φ/dz warping constant Cw being identical to that for elastic buckling (Timoshenko and Gere, 1961). The governing differential equation (7.60) is then replaced by a fourthorder equation for φ˙ , which can be solved in terms of trigonometric and hyperbolic functions.

7.3.2 Buckling of Transversely Loaded Beams We begin with a cantilever beam of narrow rectangular cross section carrying a terminal load P applied in the yz-plane, as shown in Fig. 7.9. As the load is increased to its critical value, the beam can buckle laterally in the xz-plane, and this is accompanied by the rotation of cross sections by varying degrees along the beam. Following the customary treatment for the elastic buckling, the basic equations for the plastic analysis under transverse loads are assumed to be the same as those in pure bending, provided M is regarded as the local bending moment equal to –P(l – z). The basic differential equations involving u and φ˙ therefore become αEIy

d2 u + P (l − z) φ˙ = 0, dz2

GJ

d2 φ˙ d2u − P − z) = 0. (l dz2 dz2

506

7

Plastic Buckling

Fig. 7.9 Lateral plastic bucking of an end-loaded cantilever of narrow rectangular cross section

Eliminating d2 u/dz2 between these two relations and setting z = l(1–ξ ), we obtain the differential equation for φ˙ as d2 φ˙ Pl2 α 2 + β 2 ξ 2 φ˙ = 0, β = EIy dξ

1+ν , 2

(7.63)

where α is a function of ξ in the elastic/plastic portion of the beam. Evidently, α = 1 over the part of the beam that is entirely elastic. If the length of the elastic portion of the beam is denoted by a, then α=1

(0 ≤ ξ ≤ a/l) , α = ρ −(1−n)

(a/l ≤ ξ ≤ 1) ,

where ρ denotes the absolute value of the curvature ratio κ/κ e , the variation of ρ with ξ in the elastic/plastic portion of the beam being given by (7.61) with the substitution Plξ for M. Since Pa = Me , the result is l s= ξ = a

1−n 3 n ρ − ρ −2 . 2+n 2+n

(7.64)

It is convenient at this stage to change the independent variable from ξ to s. Since Me = 4bh2 Y/3 and Iy = 4hb3 /3, the differential equation (7.63) is transformed into d2 φ˙ ˙ + k2 λ2 φ, ds2

Yah k= Eb2

1+v , 2

(7.65)

7.3

Lateral Buckling of Beams

λ=

507

3 1−n ρ (1+n)/2 − ρ −(3+n)/2 . 2+n 2+n

(7.66)

Since elastic bending corresponds to n = 1, it is only necessary to set λ = s in the above differential equation over the elastic portion (0 ≤ s ≤ 1). At the built-in end of the beam, the angle of twist vanishes, requiring φ˙ = 0, while ˙ at the free end of the beam, the torque is zero requiring dφ/ds = 0. The boundary conditions may therefore be written as dφ˙ =0 ds

at s = 0, φ˙ = 0 at s =

l . a

Over the elastic portion of the beam, where λ = s, the solution to the differential equation (7.65) can be expressed in terms of Bessel functions. In view of the first ˙ boundary condition dφ/ds = 0at s = 0, the solution is easily shown to be φ˙ =

√ sJ−1/4

1 2 ks , 2

0 ≤ s ≤ 1.

(7.67a)

The constant of integration has been arbitrarily set to unity for convenience. By a well-known formula for the first derivative of the Bessel functions, we have dφ˙ = −ks3/ 2 J3/4 ds

1 2 ks , 2

0 ≤ s ≤ 1.

(7.67b)

Over the elastic/plastic portion of the beam (1 ≤ s ≤ l/a) the relationship between λ and s is given by (7.64) and (7.66) parametrically through ρ. For a given b/h and a selected value of a/h, the differential equation (7.65) can be integrated numerically ˙ starting from s = 1, the values of φ˙ and d φ/ds at this section being those given by (7.67). The remaining boundary condition φ˙ = 0 at s = l/a finally gives the value l/a, furnishing the solution for a definite value of l/h. The corresponding value of the critical load is obtained from the relation hIy h 4 Me . (7.68) = bhY =Y P= a 3 a ab2 The greatest value of hl/b2 for which the elastic/plastic analysis is applicable corresponds to a = l, for which φ˙ = 0 at s = 1 giving J–1/4 (k/2) = 0 or k = 4.013. Buckling will therefore occur in the plastic range if hl E ≤ 4.013 2 b Y

2 . 1+ν

For higher values of hl/b2 , buckling will occur in the elastic range, and the critical load is then obtained by setting a = l in the last expression of (7.68) and using the greatest value of hl/b2 defined by the above inequality.

508

7

Plastic Buckling

In the case of a simply supported beam of length 2 l bent in the yz-plane by a concentrated load 2P acting at the midspan, the governing equations for lateral buckling are still (7.64), (7.65), and (7.66), provided z is measured from the center of the beam and a denotes the length of the elastic part on each side of the central section. Assuming that the ends of the beam are prevented from rotation about the z-axis by appropriate constraints, the boundary conditions may be written as φ˙ = 0 at s = 0,

l dφ˙ = 0 at s = . ds a

The solution for the elastic portion on each side is obtained by setting λ = s in the differential equation (7.65) and using the boundary condition φ˙ = 0 at s = 0. The rate of twist and its derivative over the elastic region (0 ≤ s≤ 1) therefore become φ˙ =

√

sJ1/4

1 2 ks , 2

dφ˙ = ks3/2 J−3/4 ds

1 2 ks . 2

(7.69)

The integration of the differential equation (7.65) over the elastic/plastic lengths (1 ≤ s ≤ l/a) can be carried out in the same way as that indicated in the case of the cantilever, using the continuity conditions k dφ˙ = kJ−3/ 4 at s = 1, ds 2 ˙ in view of (7.69). The symmetry condition dφ/ds = 0 at s = –l/a finally gives the ratio l/h for any assumed value of a/h, while the critical value of P follows from (7.68). When a = l, the plastic zones disappear, and (7.69) holds for the entire beam. ˙ Then dφ/ds must vanish at s = 1, requiring J–3/4 (k/2) = 0, which gives k = 2.117. It follows that the elastic/plastic analysis applies only over the range φ˙ = J1/4

k , 2

E hl ≤ 2.117 2 b Y

2 . 1+v

Outside this range, buckling would occur while the beam is still elastic, and the corresponding value of P is obtained by inserting in the last expression of (7.68) the value of h/ab2 that corresponds to a = l. The solution to the plastic buckling problem for other kinds of loading may be obtained in a similar manner. The analysis can be extended to the buckling of beams of arbitrary doubly symmetric cross sections in the manner indicated for pure bending.

7.4 Buckling of Plates Under Edge Thrust 7.4.1 Basic Equations for Thin Plates Consider a thin plate of arbitrary shape in which the material is bounded between the planes z = ±h/2, and the middle surface z = 0 is bounded by a closed curve

7.4

Buckling of Plates Under Edge Thrust

509

that defines the edge of the plate. The bounding planes are unstressed, while uniform compressive stresses of magnitudes σ 1 and σ 2 act in the x-and y-directions, respectively, to represent the plastic state. If the transverse shear rates on the incipient deformation mode at bifurcation are disregarded, the admissible velocity field may be written as υx = u − z

∂w , ∂x

υy = υ − z

∂w , ∂y

υz = w,

(7.70)

where u, ν, and w are functions of x and y, representing the components of the velocity of the middle surface. The velocity field (7.70) is adequate for expressing all the strain rate components except the through-thickness one, which follows from the relation ε˙ zz = −η ε˙ xx + ε˙ yy , where η is the contraction ratio in the current state. If the yield surface is assumed to be that of von Mises, √ the relevant components of√the associated unit normal are nxx = − (2σ1 − σ2 )/ 6σ¯ and nyy = − (2σ1 − σ2 )/ 6σ¯ , where σ¯ is the equivalent stress defined as σ¯ 2 = σ12 − σ1 σ2 + σ22 . Using the rate form of the constitutive law defined by (1.37), and considering the linearized solid for which the Jaumann stress rate is denoted by τ˙ij we obtain the relations 3σ22 T τ ˙ = 1− − η− xx 4σ¯ 2 H 3σ12 T = 1− τ˙yy − η − 4σ¯ 2 H

T ε˙ xx T ε˙ yy

3σ1 σ2 T 4σ¯ 2 H 3σ1 σ2 T 4σ¯ 2 H

τ˙yy , (7.71)

τ˙xx ,

where T is the tangent modulus related to the plastic modulus H by T = EH/(E+ H), while the contraction ratio η satisfies the relation 1 – 2η = (1 – 2v)T/E. The constitutive relations (7.71) are readily inverted to express the stress rates in terms of the strain rates, the result being τ˙xx = E α ε˙ xx + β ε˙ yy ,

τ˙yy = E β ε˙ xx + γ ε˙ yy ,

τ˙xy = 2G˙εxy ,

where the last relation follows from the fact that the shear strain rate is purely elastic, the parameters α, β, and γ being given by

510

7

Plastic Buckling

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

2 α = ρ −1 4 − 3 (1 − T/E) σ12 / σ¯ , β = ρ −1 2 − 2 (1 − 2v) T / E − 3 (1 − T / E) σ1 σ2 /σ¯ 2 , γ = ρ −1 4 − 3 (T / E) σ22 /σ¯ 2 ,

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2⎭ ρ = (5 − 4v) − (1 − 2v) T / E − 3 (1 − 2v) (1 − T / E) σ1 σ2 /σ¯ .

(7.72)

These quantities evidently depend on the current state of stress. It follows from the preceding expressions for the stress rates that

2 2 2 + 4G˙εxy + 2β ε˙ xx ε˙ yy + γ ε˙ yy . τ˙ij ε˙ ij = E α ε˙ xx

(7.73)

To obtain the condition for bifurcation of the plate in the elastic/plastic range, consider the uniqueness criterion in the form (7.5). Since the only nonzero components of the stress tensor are σ xx = –σ 1 and σ yy = –σ 2 , which are small compared to the modulus of elasticity E, the condition for uniqueness becomes $

τ˙ij ε˙ ij − σ1

∂υy ∂x

2 +

∂υz ∂x

2 %

$ − σ2

∂υx ∂y

2 +

∂υz ∂y

2 % dV > 0.

Inserting (7.73) into the above inequality and substituting for the strain rates and the velocity gradients which are given by ∂ 2w ∂ 2w ∂ 2w ∂u ∂υ ∂u ∂υ − z 2 , ε˙ yy = −z 2, + − 2z , 2˙εxy = ∂x ∂y ∂y ∂x ∂x∂y ∂x ∂y ∂υy ∂υ ∂ 2w ∂υx ∂u ∂ 2w = −z , = −z , ∂x ∂x ∂x∂y ∂y ∂y ∂x∂y

ε˙ xx =

in view of (7.70), and integrating through the thickness of the plate, the condition for uniqueness is reduced to 2 2 ∂υ 2 ∂u ∂υ ∂ 2u ∂ 2υ G ∂υ + α dx dy + 2β 2 2 + γ + ∂x ∂y E ∂y ∂x ∂x ∂y 2 2 2 2 2 ∂ w ∂ 2w ∂ 2w 4G ∂ 2 w ∂ w h2 + 2β 2 2 + y + α + dx dy 12 E ∂x∂y ∂x2 ∂x ∂y ∂y2 $ 2 % $ 2 2 % ∂w ∂w ∂υ ∂u 2 σ1 σ2 + + − dx dy > 0, E ∂x ∂x E ∂y ∂y with a minor approximation that is perfectly justified for thin plates. All the integrals appearing in the above expression extend over the middle surface of the plate. The left-hand side of the above inequality is seen to have a minimum value when

7.4

Buckling of Plates Under Edge Thrust

511

u = v = 0, and the Euler–Lagrange differential equation, associated with the minimization with respect to arbitrary variations of w, is easily shown to be α

∂ 4w ∂ 4w 12 ∂ 4w + 2 (β + μ) 2 2 + γ 4 = − 2 4 ∂x ∂x ∂y ∂y h

σ1 ∂ 2 w σ2 ∂ 2 w , + E ∂x2 E ∂y2

(7.74)

where μ = 1/(1 + v). The solution to the bifurcation problem is therefore reduced to the solution of the differential equation (7.74) under appropriate boundary conditions. When the bifurcation occurs in the elastic range (T = E), we have α = β = μ = γ = 1/(1–v2 ) and (7.74) reduces to the well-known governing equation for elastic buckling. In order to establish the static boundary conditions in terms of w, it is convenient to take the components of the nominal stress rate s˙ij as approximately equal to those of τ˙ij . This is justified by the fact that the stresses at bifurcation will be small compared to the elastic and plastic moduli. Then the rates of change of the resultant bending and twisting moments per unit length are given by 2 ⎫ ∂ w Eh2 ∂ 2w ⎪ ˙x = ⎪ α 2 + β 2 ,⎪ M τ˙xx zdz = − ⎪ ⎪ 12 ∂x ∂y ⎪ −h/2 ⎪ ⎪ h/2 ⎬ 3 2 2 ∂ w Eh ∂ w ˙ β 2 +γ 2 , τ˙yy zdz = − My = ⎪ 12 ∂x ∂y −h/2 ⎪ ⎪ ⎪ h/2 ⎪ 2 2 ⎪ Eh ∂ w ⎪ ⎪ ˙ xy = τ˙xy zdz = − M , ⎭ 12 (1 + v) ∂x∂y −h/2

h/2

(7.75)

in view of the relations (7.72) and (7.70) with u = v = 0. These expressions furnish ˙ nt across a typical ˙ n and the twisting moment rate M the bending moment rate M ˙ n must vanish along boundary element by the rule of tensor transformation. While M ˙ n + ∂M ˙ nt /∂S is required to vanish along a a simply supported edge, the quantity Q ˙ n is the transverse shear force rate per unit length and ds is an arc free edge, where Q element of the boundary.

7.4.2 Buckling of Rectangular Plates To illustrate the preceding theory, consider a rectangular plate whose sides are of lengths a and b, the origin of coordinates being taken at one of the corners of the rectangle as shown in Fig. 7.10(a). The plate is assumed to be simply supported along two opposite sides x = 0 and x = a, while different edge conditions may apply to the remaining two sides y = 0 and y = b. The plate is compressed by equal and opposite forces in the direction perpendicular to the simply supported sides. ˙ x must vanish along the Since the deflection rate w and the bending moment rate M simply supported edges, the boundary conditions are

512

7

Plastic Buckling

Fig. 7.10 Buckling of uniformly compressed rectangular flat plates. (a) Uniaxial edge thrust, (b) biaxial edge thrust

w = 0,

∂ 2w =0 ∂x2

on x = 0 and x = a.

It may be noted that w = 0 implies ∂ 2 w/∂y2 = 0 along these sides. Assuming that the plate buckles in m sinusoidal half-waves, the deflection rate is taken in the form

mπ x w = f (y) sin , a where f(y) is an unknown function of y. The boundary conditions along x = 0 and x = a are identically satisfied, and the substitution for w in the differential equation (7.74) with σ 1 = σ and σ 2 = 0 results in the ordinary differential equation d2 f 2φ d2 f ψf − 2 2 − 4 = 0, 4 dy b dy b

(7.76)

where φ and ψ are dimensionless parameters expressed by the relations ⎫ ⎪ 7 − 2v mπ b 2 1 − 2v T ⎪ ⎪ −3 , ⎪ ⎬ a 1+v 1+v E $ % 2 2 2 ⎪ mπ b 12ρσ b 3T 1 mπ b ⎪ ,⎪ − 1+ ψ= ⎪ ⎭ 4 a E h E a

1 φ= 4

(7.77)

in view of (7.72). The physical constraints along the edges y = 0 and y = b are usually such that ψ > 0 at bifurcation. The solution to (7.76) may then be expressed in terms of two parameters k1 and k2 which are defined as k1 =

−φ +

φ2

+ ψ,

k2 =

φ+

φ 2 + ψ.

(7.78)

7.4

Buckling of Plates Under Edge Thrust

513

The general solution to the above fourth-order differential equation can be written in the form k1 y k2 y k2 y k1 y + B sin + C cosh + D sinh , (7.79) f (y) = A cos b b b b where A, B, C, and D are constants of integration, the ratios of which can be determined from the boundary conditions on y = 0 and y = b, which also furnish the critical stress σ for bifurcation. An incremental theory for the plastic buckling of plates has been discussed earlier by Pearson (1950). The analysis given here is essentially due to Sewell (1963, 1964). As a first example, consider the situation where the sides y = 0 and y = b are also simply supported, so that all four sides of the plate have identical supports. Then the additional boundary conditions are w = 0,

∂ 2 w/ ∂y2 = 0 on y = 0 and y = b.

In view of the assumed expression for w, these conditions are equivalent to f = d2 f/dy2 = 0 along y = 0 and y = b. They are satisfied by taking A = C = D = 0 in the general solution (7.79) and by setting k1 = nπ where n is an integer. The first relation of (7.78) therefore gives ψ − 2π 2 n2 φ − π 4 n4 = 0, and the substitution from (7.77) then furnishes the critical compressive stress for bifurcation as 2 σ 3T 3 1 − 2ν mb π 2 h2 a 2 T 2+ 1+ , + = 1− + 2 E 3ρb E 2a 2 1+ν E mb (7.80) ρ = 3 + (1 − 2ν) 2 − (1 − 2ν) T/E where we have set n = 1 to minimize σ . It remains to choose the value of m for given a/b, h/b, and T/E ratios, so that the right-hand side of (7.80) is a minimum. When the ratio a/b is not too large, it is natural to expect the bifurcation mode to involve a√single half-wave in the direction of compression (m = 1), which requires 2a2 /b2 ≤ 1 + 3T/E. For sufficiently large values of a/b, the critical stress is closely approximated by π 2 h2 σ = E 3ρb2

2+

3T 3 1+ + E 2

1 − 2ν 1+ν

T 1− E

,

(7.81)

√ which is obtained by setting 2a2 /m2 b2 = 1 + 3T/E in (7.80), although the corresponding value of m is not generally an integer. The graphical plot in Fig. 7.11

514

7

Plastic Buckling

Fig. 7.11 Dimensionless critical stress for buckling of rectangular plates under unidirectional compression. The parameter c denotes the exponent of the Ramberg–Osgood stress–strain law

shows how the critical stress varies with the ratio a/b, when the stress-strain curve is represented by (7.9) with m being replaced by c. The critical stress is considerably lowered by using the rate form of the Hencky stress-strain relation, as has been shown by Shrivastava (1979), Durban and Zuckerman (1999), and Wang et al. (2001). As a second example, suppose that the sides y = 0 and y = b are rigidly clamped so that these edges are prevented from rotation. The boundary conditions along these edges therefore take the form w = 0,

∂w/ ∂y = 0

on y = 0 and y = b.

These conditions are evidently equivalent to f = df/dy = 0 along y = 0 and y = b where f(y) is given by (7.79). The consideration of the side y = 0 indicates C = –A and D = –(k1 − k2 ) B, and the function f then becomes

7.4

Buckling of Plates Under Edge Thrust

k1 y f (y) = A cos b

515

k2 y − cosh b

k1 y + B sin b

k2 y k1 − sinh . k2 b

The application of the boundary conditions to the remaining side y = b leads to the pair of equations k1 A (cos k1 − cosh k2 ) + B sin k1 − sinh k2 = 0, k2 − A (k1 sin k1 + k2 sinh k2 ) + Bk1 (cos k1 − cosh k2 ) = 0,

which can be satisfied by nonzero values of A and V if the determinant of their coefficients is zero. The relationship between k1 and k2 at the point of bifurcation therefore becomes k2 k1 sin k1 sinh k2 . 2 (1 − cos k1 cosh k2 ) = − (7.82) k2 k1 For a given stress–strain curve and the ratio a/b, we may assume a value of σ /E, guided by the corresponding elastic solution (Timoshenko and Gere, 1961). The parameters k1 and k2 are then computed from (7.77) and (7.78) with an appropriate value of m. If the computed values do not satisfy (7.82), the initial assumption must be altered and the procedure repeated until a consistent value is obtained for the critical stress. As a final example, we consider the buckling of a rectangular plate in which the sides x = 0, x = a, and y = 0 are simply supported, while the remaining side y = b is free. The boundary conditions along the edges y = 0 and y = b may be written as w = 0, ∂ 2 w/∂y2 = 0 along y = 0 ˙ y + ∂M ˙ y = 0, Q ˙ xy /∂x = 0 along y = b M The condition of moment equilibrium of a typical element of the plate, the shear ˙ y , is given by the differential equation force rate Q ˙y ˙ xv ∂M ∂M ∂ 2w ∂ 2w Eh3 ∂ ˙ Qy = + =− (β + μ) 2 + γ 2 , ∂x ∂y 12 ∂y ∂x ∂y in view of (7.75). Using (7.72), the free-edge boundary conditions may be expressed in terms of the deflection rate w in the form ∂ 2w ∂ ∂ 2w + η = 0, ∂y ∂y2 ∂x2

∂ 2w ∂ 2w + ξ ∂y2 ∂x2

= 0 on y = b,

(7.83)

where η is the contraction ratio, and ξ = (1 + η) (2 − ν)/(1 + ν). Due to the simply supported edge conditions along y = 0, it is necessary to set A = C = 0 in (7.79), and the function f(y) then becomes

516

7

k1 y f (y) = B sin b

Plastic Buckling

k2 y + D sinh . b

The remaining boundary conditions (7.83), corresponding to the free edge y = b, furnish the relations m 2 π 2 b2 m 2 π 2 b2 2 2 − B k1 + η sin k1 + D k2 − η sin k2 = 0, a2 a2 m 2 π 2 b2 m2 π 2 b2 2 + Dk − ξ − Bk1 k12 + ξ cos k k cosh k2 = 0. 1 2 2 a2 a2 Setting the determinant of the coefficients of A and B in the above equations to zero, which is required by nonzero values of these constants, the relationship between k1 and k2 is obtained as m 2 π 2 b2 m2 π 2 b2 2 k tan k1 − ξ k2 k12 + η 2 a2 a2 m 2 π 2 b2 m2 π 2 b2 2 2 k1 + ξ tanh k2 . = k2 k2 − η a2 a2

(7.84)

For specified values of the ratios a/b and h/b, and a given stress–strain curve, the least value of the critical compressive stress that satisfies (7.84) can be determined by a trial-and-error procedure, following the same method as explained before. The direction of the yield surface normal has a marked influence on the critical stress, as has been shown by Sewell (1973).

7.4.3 Rectangular Plates Under Biaxial Thrust A rectangular plate, which is simply supported along all its edges, is subjected to compressive stresses σ 1 and σ 2 uniformly distributed along the sides perpendicular to the x- and y-axes, respectively, Fig. 7.10(b). All the boundary conditions are identically satisfied by taking the deflection rate in the form w = w0 sin

mπ x a

sin

nπ y b

,

where w0 is a constant. The substitution for w into the differential equation (7.74) furnishes 2

2 σ

2 π 2 h2 2 2 σ1 2 a 4 b 2 2 4 a + 2 (β + μ) m n + γ n m +n = αm . E b E a b 12b2 (7.85) This is the required relationship between the applied stresses σ 1 and σ 2 at the point of bifurcation. The integers m and n should be such that for a given value of one of these stresses, the other one is a minimum. When the tangent modulus is

7.4

Buckling of Plates Under Edge Thrust

517

independent of the stress, as in the case of an elastic material, the critical combination of stresses defined by (7.85) lie on a concave polygon in the (σ 1 , σ 2 )-plane, as has been shown by Timoshenko and Gere (1961). Consider any sequence of states satisfying (7.85) and lying in the neighborhood of the state σ 2 = 0, so that the mode of bifurcation corresponds to n = 1. Then for a given value of σ 2 , the critical value of σ 1 is established by the equation ⎧ ⎫ ⎛ ⎞ ⎪ ⎪ ⎪ ⎪ 2 ⎨

2σ ⎟ 2⎬ ⎜ σ1 mb 12b π 2 h2 a 2 ⎜γ − ⎟ α + 2 + μ) + = (β ⎝ E 12b2 ⎪ a π 2 h2 E ⎠ mb ⎪ ⎪ ⎪ ⎩ ⎭

(7.86)

for an appropriate value of m that minimizes the right-hand side of this equation. When σ 2 /E ≤ γ (π 2 h2 /12b2 ), the range of values of a/b, for which m = 1 is applicable, is given by a2 ≤ b2

α , λ

λ=γ −

12b2 σ2 ≥ 0. π 2 h2 E

For higher values of the ratio a/b, the critical state corresponds to m = 2. When a/b is sufficiently large, the minimization is very closely achieved by set√ ting (a/mb)2 = α/λ. The critical state is then independent of a/b and is given by ⎧' ⎫ 2 2 πh ⎨ πh⎬ σ1 γπ h 3σ2 = + (β + μ) . α − E 3b ⎩ E 2b ⎭ 4b2

(7.87)

This expression is an immediate generalization of (7.81) for bifurcation under biaxial compression satisfying the condition λ ≥ 0. For λ < 0, the bifurcation state will correspond to n = 1 only for a certain range of values of the aspect ratio a/b. For arbitrary combinations of σ 1 and σ 2 , the conditions under which the critical state corresponds to m = n = 1, so that the bifurcation mode involves only one halfwave in both directions of compression, can be established by using (7.85). The bifurcation state in this case is given by σ1 a 2 σ2 π 2 h2 + = E b E 12b2

a 2 b 2 α . + 2 (β + μ) + γ a b

(7.88)

The validity of (7.88) requires that for m = 2, n = 1 and for m = 1, n = 2 the equality sign in (7.85) must be replaced by an inequality in which the left-hand side is less than the right-hand side. Thus,

518

7

Plastic Buckling

2

a 2 b + 8 (β + μ) + γ 16α , a b

a 2

a 2 σ b 2 π 2 h2 σ1 2 α . + 8 (β + μ) + 16γ +4 ≤ E b E 12b2 a b

σ1 a 2 σ2 π 2 h2 4 + ≤ E b E 12b2

These inequalities together with (7.88) lead to the necessary restrictions on σ 1 and σ 2 for which the bifurcation state is defined by (7.88), the continued inequalities to be satisfied by the stress σ 1 being

α − 4y

a 4 b

≤

a 2 12a2 σ1 ≤ 5α + 2 + μ) . (β b π 2 h2 E

The value of σ 1 defined by the lower limit and the value of σ√2 furnished by the 2 2 upper √ limit in the above inequalities are negative when 2(a/b) > α/γ and (a/b) < 2 α/γ , respectively. When the state of stress is an equibiaxial compression defined by σ 1 = σ 2 = σ , the parameters ρα, ρ(β +μ), and ργ are each equal to 1+3T/E in view of (7.72) with σ¯ = σ , and the expression for the critical stress then becomes σ π 2 h2 = E 12a2

(1 + 3T / E) 1 + a2 / b2 . 2 (1 + v) [1 + (1 − 2ν) T / E]

(7.89)

For a given aspect ratio a/b, the bifurcation stress in equibiaxial compression is found to be considerably lower than that in unidirectional compression. Over the elastic range of buckling (T = E), the equibiaxial critical stress for a square plate (a = b) is exactly one-half of the unidirectional critical stress. The bifurcation stress predicted by (7.89) is plotted in Fig. 7.12 as a function of the aspect ratio, assuming the stress–strain curve to be given by (7.9) with different values of the exponent m, which is here replaced by c. The influence of edge restraints produced by friction on the critical stress has been investigated by Gjelsvik and Lin (1985) and Tugcu (1991). Solutions for the critical stress based on the Hencky stress–strain relations have been presented by Illyushin (1947), Bijlaard (1949, 1956), Gerard (1957), and El-Ghazaly and Sherbourne (1986). The buckling loads predicted by the total strain theory, despite its physical shortcomings, are found to be in better agreement with experiments, whereas those based on the incremental theory are found to be significantly higher than the experimental ones. This apparent paradox is partly due to the presence of geometrical and other imperfections which are not considered in the theory. It is also possible for some kind of non-associated flow rule, resembling the Hencky relations, to apply at the point of bifurcation. The plastic buckling of plates based on the slip theory of plasticity has been discussed by Batdorf (1949) and by Inoue and Kato (1993). A detailed investigation of the plastic buckling of relatively thick plates has been carried out by Wang et al. (2001).

7.4

Buckling of Plates Under Edge Thrust

519

Fig. 7.12 Variation of critical stress with aspect ratio for the buckling of rectangular plates under equibiaxial compression

7.4.4 Buckling of Circular Plates A circular plate of thickness h and radius a is submitted to a radial compressive stress σ uniformly distributed around the periphery. The incipient deformation mode at the point of bifurcation is assumed to be such that the deformed middle surface is a surface of revolution in which the deflection rate w is a function of the radius r only. Since the radial velocity of particles on the middle surface may be set to zero in the investigation for bifurcation, the velocity field may be written as υr = −z

d2 w , dr2

υθ = 0,

υz = w,

(7.90a)

in cylindrical coordinates (r, θ , z). The shear strain rates are identically zero, and the relevant components of the velocity gradient are d2 w ∂υr = −x 2 , ∂r dr

dυz dw ∂υr =− =− . ∂z dr dr

(7.90b)

520

7

Plastic Buckling

The radial and circumferential components of the linearized constitutive relation, which are similar to (7.71), can be written as τ˙rr = E (α ε˙ rr + β ε˙ θθ ) ,

τ˙θθ = E (β ε˙ rr + γ ε˙ θθ ) ,

where α, β, and γ are given by (7.72) with σ1 = σ2 = σ , the result being ⎫ 1 ⎪ , γ = α,⎪ β =α− ⎬ 1+ν ⎪ T ⎪ ⎭ ρ = 4 (1 + ν) (1 − η) , 2η = 1 + (1 − 2ν) . E

1 α= ρ

3T 1+ , E

(7.91)

Since the radial and circumferential strain rates are ε˙ rr = ∂νr /∂r and ε˙ θθ = νr /r, the criterion for uniqueness in this case takes the form % $

υ 2 ∂υz 2 υr ∂υr ∂υr 2 r −σ dV > 0, + 2β +γ E α ∂r r ∂r r ∂r where the integral extends over the entire volume of the plate. Substituting from (7.90) and integrating through the thickness, we get

a 0

2 2 d2 w 1 dw 2 12σ dw 2 1 dw d w α rdr > 0. +γ + 2β − 2 dr2 r dr dr2 r dr h E dr

The deflection rate w should be such that the left-hand side of the above inequality is a minimum. The associated variational problem is characterized by the Euler– Lagrange differential equation α

d2 dr2

2 1 dw 2 12σ d d w dw r 2 −γ + 2 r = 0. r dr dr dr h E dr

Since the coefficients of this equation have constant values throughout the plate at the point of bifurcation, the first integral of the above equation can be immediately written down, and the constant of integration can be set to zero. Introducing the notation 2r 3σ dw , ξ= , φ= dr h αE the governing differential equation for the occurrence of bifurcation is easily shown to be ξ2

d2 φ dφ 2 + ξ − 1 φ = 0, + ξ dξ dξ 2

(7.92)

which is recognized as Bessel’s differential equation having the general solution

7.4

Buckling of Plates Under Edge Thrust

521

φ = AJ1 (ξ ) + BY1 (ξ ) ,

(7.93)

where J1 (ξ ) and Y1 (ξ ) are Bessel functions of the first order, and of the first and second kinds, respectively. Since φ must be zero at the center of the plate ξ = 0, we must set B = 0 in (7.93). The critical stress for bifurcation evidently depends on the condition of support of the circular edge r = a. As a first application of the preceding theory, consider the plastic buckling of a circular plate which is fully clamped around its edge. Then the boundary condition is dw/dr = 0 along r = a, which is equivalent to φ = 0 along ξ = k, and the critical stress is then given by k2 h2 σ αk2 h2 = = E 12a2 12a2

1 + 3T/E , 2 (1 + ν) 1 + (1 − 2ν) T/E

(7.94)

in view of (7.91). It follows from the boundary condition applied to (7.93), where B is identically zero, that k is the smallest root of the equation J1 (k) = 0. Since T/E is a function of σ /E for any given stress–strain curve of the material, the solution must be found by a trail-and-error procedure. In the case of elastic buckling, we have T = E and α = 1/(1 – v2 ), giving k ≈ 3.832 and (1 – v2 )σ /E ≈1.224h2 /a2 at the point of bifurcation. As a second example, let the circular plate be simply supported along its edge, so that the rate of change of the radial bending moment vanishes along r = a. Since the bending moment rate is very closely given by the expression ˙r = M

h/2 −h/2

τ˙rr zdz = −

Eh3 12

2 d w β dw α 2 + , r dr dr

˙ r = 0 can be written down as the simply supported edge condition M α

φ dφ + β = 0 or dr r

dφ βφ + =0 dξ αξ

at r = a or ξ = k, where φ is given by (7.93) with B = 0. Using the well-known derivative formula ξ J0 (ξ ) − J1 (ξ ) = J1 (ξ ) and applying the boundary condition to the above expression for φ, the bifurcation state is given by β 4 (1 − η) 2a kJ0 (k) =1− = , k= J1 (k) α 1 + 3T/E h

3σ , αE

(7.95)

in view of (7.91). Since the right-hand sides of these equations are functions of σ /E, the critical stress must be computed by trial and error, using a table of Bessel functions. When the bifurcation occurs in the elastic range, T = E, and

522

7

Plastic Buckling

k2 = 12(1 – v2 )a2 σ /h2 E, and the critical stress is then given by (1 – v2 ) σ /E ≈ 0.35h2 /a2 when v = 0.3. The bifurcation stress is therefore strongly dependent on the edge condition. A detailed investigation of the plastic buckling of circular plates has been carried out by Hamada (1985). The influence of the transverse shear on the critical stress has been examined by Wang et al. (2001). If a radially compressed circular plate has a concentric circular hole, which is assumed as stress free, a uniform radial compressive stress applied around the outer edge produces a nonuniform distribution of stress within the plate. The differential equation (7.92) is therefore modified, and the critical stress for bifurcation then depends on the ratio b/a, where b denotes the radius of the hole. The plastic buckling of annular plates under pure shear has been discussed by Ore and Durban (1989). An analysis for the plastic buckling of relatively thick annular plates under uniform compression has been presented by Aung et al. (2005).

7.5 Buckling of Cylindrical Shells 7.5.1 Formulation of the Rate Problem Consider a circular cylindrical shell of uniform thickness h and mean radius a, subjected to the combined action of an axial compressive stress σ and a uniform lateral pressure p. The cylinder is supported in such a way that it is free to expand or contract radially during the loading. For certain critical values of the applied loads, a point of bifurcation is reached, and the shell no longer retains its cylindrical form. Let (x, θ , r) be a right-handed system of cylindrical coordinates, in which the x-axis is taken along the axis of the shell, the origin of coordinates being taken at one end of the shell. At a generic point in the material of the shell, situated at a radially outward distance z from the middle surface, the components of the velocity vector may be written as υx = u + zωθ ,

υθ = υ − zωx ,

υx = w,

(7.96)

where (u, ν, w) are the velocities at the middle surface, and ωx , ωθ are the rates of rotation of the normal to the middle surface about the positive x- and θ -axes, respectively. Within the framework of thin-shell theory, the latter quantities are directly obtained from the fact that the through-thickness shear rates ε˙ rx and ε˙ rθ are identically zero. Denoting the remaining component of the spin vector by ωr , we have

ωx =

1 a

∂w −υ , ∂θ

ωθ = −

∂w , ∂x

ωr =

1 2

∂υ 1 ∂u − . ∂x a ∂θ

(7.97)

The nonzero components of the anti-symmetric spin tensor ω ij are related to the components of the spin vector as

7.5

Buckling of Cylindrical Shells

− ωxθ = ωθx = ωr ,

523

− ωrx = ωxr = ωθ ,

− ωθr = ωrθ = ωx .

The nonzero components of the strain rate, except the through-thickness one, that are associated with the velocity field (7.96), may be written as ε˙ xx = λ˙ x − zκ˙ x , ε˙ θθ = λ˙ θ − zκ˙ θ , ε˙ xθ = λ˙ xθ − zκ˙ xθ ,

(7.98)

where λ˙ x , λ˙ θ , and λ˙ xθ are the rates of extension and shear of the middle surface, while κ˙ x , κ˙ θ, and κ˙ xθ are the rates of change of curvature and twist of the middle surface. It is easily shown that λ˙ x =

∂u , ∂x

κ˙ x =

∂ 2w , ∂x2

⎫ 1 ∂υ ∂υ 1 ∂u ⎪ ⎪ ˙ +w , λxθ = + , ⎪ ⎬ ∂θ 2 ∂x a ∂θ (7.99) ⎪ 1 ∂ ∂w 1 ∂ ∂w ⎪ ⎪ κ˙ θ = 2 −υ , κ˙ xθ = − υ .⎭ a ∂θ ∂θ a ∂x ∂θ

λ˙ θ =

1 a

The strain rates given by (7.98) and (7.99) are consistent with the customary thin-shell approximation and are adequate for the investigation of bifurcation. The material is assumed to obey the von Mises yield criterion and the associated Prandtl–Reuss flow rule. Since the nonzero components of the current stress tensor σ ij are σ xx = –σ and σ θθ = –pa/h, the outward drawn unit normal nij to the yield surface has the nonzero components nxx = −

2σ − pa/ h 2pa/ h − σ , nθθ = − √ , √ 6σ¯ 6σ¯

nrr =

σ + pa/ h , √ 6σ¯

where σ¯ is the equivalent stress, which is equal to the current yield stress in simple compression, and is given by σ¯ 2 = σ 2 − σ (pa/h) + (pa/h)2 . Introducing the linear comparison solid, for which the Jaumann stress rate is denoted by τ˙ij , the constitutive equation may be written as ε˙ ij =

3 1 (1 + v) τ˙ij − vτ˙kk δij + E 2

1 1 − τ˙kl nkl nij , T E

(7.100)

where T is the tangent modulus in the current state of hardening, E is Young’s modulus, and v is Poisson’s ratio for the material. Since τ˙rr is identically zero, the axial and circumferential components of the rate of extension are given by vT 3 1 T pa 2 T 3σ pa τ˙xx − T ε˙ xx = 1 − + 1− 1− 1− τ˙θθ , 4 E σ¯ h E 2 E 2σ¯ 2 h vT 3 1 T 3σ pa T σ 2 T ε˙ θθ = + 1 − + 1− 1− 1 − τ ˙ τ˙θθ . xx E 2 E 4 E σ¯ 2σ¯ 2 h

524

7

Plastic Buckling

The nonzero shear strain rate ε˙ xθ is purely elastic and is equal to (1 + v) τ˙xθ /E by Hooke’s law. The preceding pair of equations can be solved for τ˙xx and τ˙θθ to give τ˙xx =

E E E˙εxθ , (α ε˙ xx + β ε˙ θθ ) , τ˙θθ = (β ε˙ xx + γ ε˙ θθ ) , τ˙xθ = 1+v 1+v 1+v (7.101)

where ⎫

⎪ α = ρ −1 (1 + v) 4 − 3 (1 − T / E) σ / σ¯ 2 , ⎪ ⎪ ⎪

⎪ ⎪ ⎪ −1 2 ⎪ β = ρ (1 + v) 2 − 2 (1 − 2ν) T / E − 3 (1 − T / E) σ pa/ σ¯ h , ⎪ ⎬

2 (7.102) ⎪ ⎪ , γ = ρ −1 (1 + v) 4 − 3 (1 − T / E) pa/ σ¯ 2 h ⎪ ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ρ = (5 − 4v) − (1 − 2v)2 T / E − 3 (1 − 2v) (1 − T / E) σ pa/ σ¯ 2 h ,⎭ The parameters α, β, γ , and ρ are easily calculated for any given state of stress and rate of hardening. Using (7.101) for the nonzero stress rates, we get τ˙ij ε˙ ij =

E 2 2 2 . + 2˙εxθ α ε˙ ij + 2β ε˙ xx ε˙ θθ + γ ε˙ θθ 1+v

(7.103)

The complete rate problem also involves the rate equations of equilibrium in which geometry changes are duly allowed for. A set of equilibrium equations in terms of the rate of change of the stress resultants have been developed by Batterman (1964). Such equation are, however, not required in the present analysis, which involves a variational principle based on the appropriate criterion for uniqueness.

7.5.2 Bifurcation Under Combined Loading A sufficient condition for uniqueness of the deformation in an elastic/plastic body under the combined action of incremental dead loading and uniform fluid pressure is given by inequality (1.82). Since the plastic modulus H would be large compared to the applied stresses at the onset of buckling, the quantity σij ε˙ jk ε˙ ik may be omitted in the uniqueness criterion, which therefore becomes

τ˙ij ε˙ ij + σij ωik ωjk − 2˙εik ωjk dV − p

lk ε˙ kj + ωkj υj dSf > 0,

(7.104)

for all continuous velocity fields vanishing at the constraint. The volume integral extends throughout the material of the body, while the surface integral extends over the boundary that is submitted to fluid pressure. Using (7.103), and remembering that the only nonzero stress components are σ xx = – σ and σ θθ = – pa/h, the condition for uniqueness is reduced to

7.5

Buckling of Cylindrical Shells

525

2 2 2 α ε˙ xx dx dθ dz + 2β ε˙ xx ε˙ θθ + γ ε˙ θθ + 2˙εxθ σh ωr2 + ωθ2 + 2ωr λ˙ xθ dx dθ − (1 + v) E pa ωx2 + ωθ2 − 2ωθ λ˙ xθ dx dθ − (1 + v) E p λ˙ x + λ˙ θ w + uωθ − νωx dx dθ > 0, + (1 + v) E with sufficient accuracy, since σ /E and p/E are small compared to unity. Substituting from (7.97), (7.98), and (7.99) and introducing the dimensionless parameters s = (1 + v)

σ , E

q = (1 + v)

pa , Eh

k=

h2 , 12a2

ξ=

x , a

and integrating through the thickness of the shell, we obtain 2 ∂u 1 ∂υ ∂υ ∂u 2 ∂u ∂υ α dξ dθ +w +γ +w + + + 2β ∂ξ ∂ξ ∂θ ∂θ 2 ∂ξ ∂θ 2 2 2 2 ∂ w ∂ 2 w ∂ 2 w ∂υ ∂ w ∂υ +k α + 2β 2 − − +γ ∂ξ 2 ∂ξ ∂θ 2 ∂θ ∂θ 2 ∂θ 2 2 2 2 ∂υ ∂υ ∂ w ∂w − dξ dθ − s dξ dθ +2 + ∂ξ ∂θ ∂ξ ∂ξ ∂ξ 2 2 ∂w ∂u ∂w ∂u + −w +w +u dξ dθ > 0, −q ∂θ ∂θ ∂ξ ∂ξ where use has been made of the fact that the velocities are single-valued functions of θ . The terms containing the quantities sλ˙ 2x˙ θ and qλ˙ 2x˙ θ have been neglected in the last two integrals to be consistent with the basic approximation. The occurrence of bifurcation is marked by the vanishing of the above functional, which must be minimized with respect to the admissible velocities u, υ, and w. The Euler–Lagrange differential equations associated with this variational problem are easily found to be α

∂ 2u 1 ∂ 2u 1 ∂ 2υ ∂w ∂ 2 u ∂w = 0, + + β + + β + q − ∂ξ 2 2 ∂θ 2 2 ∂ξ ∂θ ∂ξ ∂ξ ∂θ 2

2 2 ∂ υ ∂w ∂ u ∂ 2υ 1 1 ∂ 2υ + γ + − s β+ + 2 ∂ξ 2 ∂θ 2 ∂ξ 2 ∂θ ∂θ 2 ∂ξ 2 2 2 ∂ υ ∂ 3w ∂ υ ∂ 3w = 0, − + 2) +k 2 2 +γ − (β ∂ξ ∂θ 2 ∂θ 3 ∂ξ 2 ∂θ

526

β

7

∂u +γ ∂ξ

Plastic Buckling

∂u ∂υ ∂ 2υ ∂ 2w +w +s 2 +q +w+ 2 ∂θ ∂ξ ∂ξ ∂θ 3 4 ∂ υ ∂ υ ∂ 4w ∂ 4w + k − (β + 2) 2 − γ 4 + 2 (β + 1) 2 2 + γ 4 = 0. ∂ξ ∂θ ∂θ ∂ξ ∂θ ∂θ (7.105)

In the case of elastic buckling, we have α = γ = 1/(1 – ν) and β = v/(l – v), and (7.105) reduce to those given by Timoshenko and Gere (1961), except for the coefficients of certain small-order terms, the effects of which are insignificant in the final result. The above equations provide a systematic generalization of the eigenvalue problem when buckling occurs in the plastic range (Chakrabarty, 1973). The class of admissible velocity fields for the investigation of bifurcation, which is characterized by a nonuniform mode of deformation, may be considered as that in which the radial velocity vanishes at the ends of the shell. Denoting the length of the shell by l, the solution of (7.105) may therefore be sought in the form u = U cos λξ cos mθ , υ = V sin λξ sin mθ , w = W sin λξ cos mθ ,

(7.106)

where λ = rπ a/1, and m and r are integers, while U, V, and W are arbitrary constant velocities. The virtual velocity field (7.106) evidently satisfies the boundary conditions w = ∂ 2 w/ ∂ξ 2 = 0

at

ξ = 0 and ξ = 1,

which correspond to a shell with simply supported edges. For sufficiently long shells, the results based on (7.106) can be applied to other types of edge condition without appreciable error. The velocity field (7.106) implies that the generator of the shell is subdivided into r half-waves and the circumference into 2m half-waves at the onset of bifurcation. Substituting (7.106) into (7.105), these equations are found to be satisfied everywhere in the shell if

⎫ 1 1 2 ⎪ ⎪ −q m U− + β λmV − (β + q) λW = 0, αλ + ⎪ ⎪ 2 2 ⎪ ⎪ ⎪

⎪ ⎪ 1 1 2 ⎪ 2 2 2 2 ⎪ V − + β λmU + λ + γ m − λ s + k 2λ + γ m ⎪ ⎪ 2 2 ⎬ + γ m + km (2 + β) λ2 + γ m2 W = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ − (β + q) λU + γ m + km (2 + β) λ + γ m V ⎪ ⎪ ⎪

⎪ ⎪ 2 2 4 2 2 4 W = 0.⎭ + γ − λ s − m − 1 q + k αλ + 2 (1 + β) λ m + γ m (7.107)

2

This is a system of three linear homogeneous equations for the unknown velocities U, V, and W. For nontrivial solutions to these quantities, the determinant of

7.5

Buckling of Cylindrical Shells

527

their coefficients must vanish. It is interesting to note that the matrix of this determinant is symmetric. Expanding the determinant, and neglecting the small-order terms involving the squares and products of s, q, and k, the result may be expressed in the form A + Bk = Cs + Dq,

(7.108)

where δ = αγ − β 4 = (4/ ρ) (1 + v)2 (T / E) , A = δλ4 , B = αλ4 + 2 (δ − β) λ2 m2 + γ m4 αλ4 + 2 (1 + β) λ2 m2 + γ m4 − 2m2 (2 + β) λ2 + γ m2 (2δ − β) λ2 + γ m2 ,

C = λ2 αλ4 + 2 (δ − β) λ2 m2 + γ m2 + λ2 2δλ2 + γ m2 ,

D = m2 − 1 αλ4 + 2 (δ − β) λ2 m2 + γ m4 + λ2 2βλ2 − γ m2 .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (7.109)

For a given material and shell geometry, and with selected values of r and m, (7.108) defines the relationship between the critical values of s and q for bifurcation. Using a constant value of r and successive values of m, a series of curved segments may be obtained in a graphical plot of s against q. The value of m appropriating over a certain range is that which provides the smallest ordinate for a given abscissa. A set of curves may be constructed in this way for various values of λ = rπ a/l, similar to those presented by Flügge (1932) for the buckling of an elastic shell.

7.5.3 Buckling Under Axial Compression Consider the special case where the lateral pressure is absent and the axial compressive stress σ is increased to a critical value to cause buckling in the plastic range. The critical stress is then obtained by setting q = 0 in (7.108) and using the values of λ and m which correspond to a minimum value of s. When the cylindrical shell is relatively short, we may expect it to buckle into short longitudinal waves, so that λ2 is sufficiently large. Retaining only the first terms in the expressions for B and C in (7.109), the critical stress may be written in the simplified form k αλ4 + 2 (1 + β) λ2 m2 + γ m4 δλ2 + , (7.110) S= αλ4 + 2 (δ − β) λ2 m2 + γ m4 λ2 where δ = 4(1 + v)2 (T/ρE), the coefficients α, β, γ , and ρ being given by (7.102) with σ¯ = σ and ρ = 0. The value of λ2 which makes the right-hand side of (7.110) a minimum is found to be given by

528

7

αλ4 −

√

Plastic Buckling

δ / k − 2 (δ − β) m2 λ2 + γ m4 = 0,

and the corresponding expression for the critical stress for buckling then becomes s=2

kg + (1 − δ + 2β) m2 .

Since δ – 2β ≤ 1 for 0 < T/E < 1, the smallest value of s for plastic buckling corresponds to m = 0, which represents a symmetrical mode of buckling. The preceding relations therefore reduce to αλ2 =

√

√ s = 2 δk.

δ / k,

Substituting for k, α, and δ, the critical values of σ /E and λ2 for plastic buckling are finally obtained as 2h σ = E a

' T , 3ρE

4a 3T −1 3ρT λ = 1+ h E E 2

(7.111)

This is the true tangent modulus formula for the plastic buckling of relatively short cylindrical shells under axial compression. The same formula has been obtained by Batterman (1965) using a different method. Setting T/E = 1 and ρ = 4(1 – v2 ) reduces (7.111) to the well-known formulas for elastic buckling in which case m need not be zero. The influence of a singular yield criterion in lowering the critical stress has been examined by Ariaratnam and Dubey (1969). Plastic buckling formulas based on the total strain theory together with some experimental results have been given by Bijlaard (1949) and also by Jones (2009). The effect of an initial imperfection on the critical stress based on the total strain theory has been investigated by Lee (1962) and by Bardi et al. (2006). Since the eigenmode is symmetrical, the second equation of (7.107) is identically satisfied, and either of the two remaining equations furnishes W/U = αλ/β. The eigenfield may therefore be written in terms of an arbitrary constant velocity u0 as u = u0 β cos λξ ,

υ = 0,

w = u0 αλ sin λξ .

Since a uniform axial compression is always a possible mode, the nonzero components of the actual velocity at bifurcation may be expressed as u = −u0

λx x + cβ 1 − cos , l l

w = u0

λx ηa + cαλ sin , l l

(7.112)

where c is a constant and η is the contraction ratio, the axial velocity being assumed to vanish at the end x = 0. The condition of no instantaneous unloading of the plastically compressed cylinder may be written as ε˙ xx < 0, and it follows from (7.98) and (7.112) that

7.5

Buckling of Cylindrical Shells

529

cλl z δ 1+ β− sin λξ > 0 a a k √ in view of the result αλ2 = δ/k. This inequality will be satisfied throughout the shell if ' a 3T a − < β + 2 (1 + v) λc < , l ρE l where β and ρ are given by (7.102). The buckling may therefore occur in a range of possible modes, when the axial stress σ attains the value given by (7.111), with the load increasing as the shell continues to deform in the postbuckling range. In the case of long cylindrical shells, the generators are expected to buckle into relatively long waves in the longitudinal direction, so that λ is sufficiently small. We may then neglect in the expressions for B and C all powers of λ higher than the second to obtained the result

C = γ λ2 m2 m2 + 1 , A = δλ4 ,

2 γ m2 + 2 (1 + δ) λ2 . B = γ m2 m2 − 1 Substituting in (7.108), where q is set to zero, the critical stress for buckling may be expressed as 2 k m2 − 1 δλ2 γ m2 + s= + 2 (1 + δ) . m2 + 1 λ2 γ m2 m2 + 1

(7.113)

Minimizing this expression with respect to λ, the associated values of λ2 and s are found to be √

k/ δ, λ2 = γ m2 m2 − 1

m2 − 1 √ 2 s=2 kδ + k + δ) − 1 . m (1 m2 + 1 Evidently, the smallest value of the critical stress corresponds to m = 2, giving the final results ' ' 4 h 3E 6h h 1+δ T σ 2 = + , (7.114) , λ = E 5a 3ρE 4a 1 + ν a ρT where δ is given by the first equation of (7.109), and ρ by the last equation of (7.102) with the last term set to zero. For an elastic material, T = E,δ = (1 + v)/(1–ν), and

530

7

Plastic Buckling

ρ = 4(1 – v2 ), reducing the above expressions to those given by Timoshenko and Gere (1961). A limit of applicability of (7.114) is marked by the buckling of the shell as a column, the critical stress then being given by the formula σ = π 2 a2 T/2l The plastic buckling of axially compressed cylindrical shells has also been considered by Gellin (1979), Shrivastava (1979), and Reddy (1980). The preceding discussion is based on the assumption that the shell is thick enough for the critical stress σ to exceed the yield stress of the material. The results for the whole range of values of h/a may, however, be presented on the basis of the plastic buckling formula if we adopt the relations σ ε= E

3 1+ 7

σ σ0

n−1 ,

3n E =1+ T 7

σ σ0

n−1 ,

(7.115)

for the uniaxial stress–strain curve, where σ 0 and n are empirical constants. The assumed curve has an initial slope E, but the tangent modulus T steadily decreases as the stress increases from zero. Inserting (7.115) into the critical stress formula (7.111) or (7.114), a relationship between σ /E and h/a is obtained for any given values of σ 0 /E and n. The relationship between λ and h/a then follows from the corresponding formula for λ2 . The results are displayed in Fig. 7.13 for several values of n, assuming v = 0.3 and σ 0 /E = 0.002. Due to geometrical imperfections and other uncertainties, the experimental buckling loads are found to be appreciably lower than the theoretical ones. A useful review of the subject has been reported by Babcock (1983). The buckling problem for a square-section tube, based on the total strain theory, has been investigated by Li and Reid (1992).

7.5.4 Influence of Frictional Restraints When a cylindrical shell is axially compressed between a pair of rigid platens, lateral displacement of the elements in contact with the platens is prevented by friction, and the shell is therefore subjected to simultaneous bending and compression right from the beginning of the loading. The bending moment at the crests of the longitudinal waves nearest to the platens rapidly increases with increasing load until the yield point state is reached. The first convolution that appears at one end of the cylinder, where the frictional restraint is more predominant, then collapses under decreasing load. When the first convolution nearly flattens out, a second convolution begins to form on top of the other one, and the load starts to increase again with further compression. If the shell is relatively thick, the result is a concertina-type of deformed shape, for which an average buckling load has been given by Alexander (1960b). For thinner shells, there is generally a polygonal-type of final configuration treated by Pugsley and Macualay (1960), and Pugsley (1979). An experimental investigation on the energy absorption of tubular structures during buckling has been made by Balen and Abdul-Latif (2007).

7.5

Buckling of Cylindrical Shells

531

Fig. 7.13 Critical stress for plastic bucking of an axially compressed cylindrical shell as a function of the wall thickness

Consider the concertina-type of collapse in which each convolution is approximated by two identical conical surfaces as shown in Fig. 7.14. In this simplified model, the convolutions are taken to be purely external to the cylinder, though in actual practice they are formed partly outward and partly inward. The formation of each convolution is associated with three circular hinges which allow rotation of the conical bands, each having a constant slant height b. For simplicity, the small change in thickness that occurs in the formation of convolution will be disregarded. Neglecting elastic strains and work-hardening, and denoting the current semivertical angle of the conical bands by ψ the increment of plastic energy dissipation in the hinge circles during a small change in angle dψ is found as b sin ψ dψ, dW1 = 4π bM0 (2a + b sin ψ) dψ = 2π ah2 Y 1 + 2a the influence of the meridional force on the fully plastic moment being disregarded. Since the mean circumferential strain in the material between the hinge circles during the incremental change in angle dψ is dεθ = b cosψdψ(2a + bsinψ), the increment of plastic work done during the circumferential stretching of the material is

532

7

Plastic Buckling

Fig. 7.14 Concertina-type of plastic collapse of a circular cylindrical shell under axial compression with frictional restraint

dW2 = 2π bN0 (2a + b sin ψ) dεθ = 2π hb2 Y cos ψdψ. This expression involves the approximate yield condition Nθ ≈ N0 and the fact that the meridional strain increment is assumed to be zero. The total plastic work done in collapsing a typical convolution, as ψ increases from 0 to π/2, is W=

(dW1 + dW2 ) = π h2 (π a + b) Y + 2π b2 hY.

(7.116)

The applied compressive stress decreases from an initial value σ 0 during the collapse of a convolution, the stress at a generic stage being denoted by σ . Available theoretical and experimental evidences (Chakrabarty and Wasti, 1971) tend to suggest that the variation of the stress may be written as

σ = σ0 cos ψ + 2

h 2 sin ψ . a

The total work done by the external load must be equal to W. Since the amount of axial compression corresponding to a change dψ in the semi-vertical angle is equal to 2b sinψ, dψ, we have W = 4π ahbσ0 0

π/2

cos ψ + 2

h 2 sin ψ sin2 ψdψ. a

7.5

Buckling of Cylindrical Shells

533

Integrating, and inserting the expression (7.116) for W, the critical stress is found to be given by σ0 Y

h 2b 3 h πa +1+ 1+2 = . a 4a b h

The quantity b√is still an unknown but can be determined by minimizing σ 0 the result being b = π ah/2 . The critical stress formula therefore becomes 3h σ0 =3 Y 4a

πa 2b +1+ b h

5

h . 1+2 a

(7.117)

The critical stress for the initiation of the first convolution would be somewhat smaller than (7.117) since the hinge circle at the base of the deformation zone is absent during its formation. The analysis has practical importance in the design of buffers bringing moving bodies to a stop without appreciable damage. Further results on the axial crushing of cylindrical shells have been given by Mamalis and Johnson (1983). The influence of the meridional stress on the magnitude of the buckling load has been examined by Wierzbicki and Abramowicz (1983).

7.5.5 Buckling Under External Fluid Pressure We begin with the situation where a circular cylindrical shell is submitted to a uniform external pressure acting on the lateral surface only. The critical hoop stress at the incipient buckling is then directly given by (7.108), where s must be set to zero. The parameters α, β, γ , and δ appearing in (7.109) are obtained from (7.102) on setting σ = 0 and pa/h = σ¯ . If the length of the cylinder is greater than about twice its diameter, the ratio λ2 /m2 will be a small fraction, and we may omit all terms containing λ2 and λ4 in the expressions for B and D in (7.109). Then the result for the critical pressure is easily shown to be q≈

δλ4 + kγ m2 − 1 . γ m4 m2 − 1

(7.118)

Since q increases with λ, the least value of the critical hoop stress corresponds to r = 1, giving λ = π a/l. Substituting for γ , δ, λ, and k into the above expression, we finally obtain the buckling formula 2 h / 12a2 (1 + 3T / E) m2 − 1 (4T / E) (π a/ l)4 pa + = . Eh (1 + 3T / E) m4 m2 − 1 (5 − 4ν) − (1 − 2ν)2 T / E

(7.119)

In the case of elastic buckling (T = E), this formula reduces to that originally given by Southwell (1913). Equation (7.119), which is due to Chakrabarty (1973), represents a rigorous extension of the result when buckling occurs in the plastic

534

7

Plastic Buckling

range. For a very long tube, the first term on the right-hand side of (7.119) may be disregarded, and the least value of the critical stress then corresponds to m = 2. In a wide range of values of l/a and h/a would generally require m = 3 for a lower value of the critical stress, and the condition for this to happen is 27 h2 π 4 a4 (E/ T) (1 + 3T / E)2 . > l4 5a2 (5 − 4ν) − (1 − 2ν)2 T / E For shorter tubes, the critical stress corresponding to m = 4 would be lower than that given by m = 3. For exceptionally short tubes, however, the approximation leading to (7.118) is no longer valid. For the numerical evaluation of the critical stress, T/E may be eliminated from (7.119) by using (7.115), to obtain the relationship between the critical hoop stress pa/h and the shell parameters. The resulting equation can be most conveniently solved by selecting a value of pa/Eh for a given value of l/a, and calculating the corresponding value of h2 /a2 required for bifurcation. The computation may be carried out with a suitable value of m, changing it if necessary to obtain the least value of the critical stress. Figure 7.15 shows the results of the computation based on v = 0.3, n = 3.0, and σ 0 /E = 0.001. The curves for plastic buckling are represented by the solid lines, while those for elastic buckling are indicated by the broken lines. Only two of the equations in (7.107) are independent when U, V, and W correspond to the eigenfield. Setting s = 0 in the last two equations of (7.107), and using (7.118), it is easily shown that U = λw0 ,

V = mw0 ,

W = −m2 w0 ,

to a close approximation, where w0 is an arbitrary constant velocity. Introducing a constant parameter c, the components ν and w of the actual velocity at bifurcation may be written as ⎫

πx ⎪ sin mθ , υ = mw0 c sin ⎬ l

πx ⎭ w = −m2 w0 1 + c sin cos mθ ,⎪ l

(7.120)

the radial velocity at each end of the shell being taken as equal to –m2 wo. The substitution from (7.120) into the second equation of (7.98) furnishes the hoop strain rate ε˙ θθ = −

πx m2 w0 cz 2 1+ m − 1 sin cos mθ , a a l

which must be negative for no incipient unloading at the point of bifurcation. Since the value of the second term in the curly brackets of the above expression varies between –(m2 – 1)(ch/2a), and (m2 – 1)(ch/2a), the loading condition ε˙ θθ < 0 gives

7.5

Buckling of Cylindrical Shells

535

Fig. 7.15 Critical stress for plastic buckling of a circular cylindrical shell under uniform external pressure on the lateral surface

−

2a 2 2a < m −1 c< . h h

The nonlinear elastic/plastic solid may therefore buckle in a range of possible ways, and the external pressure must continue to increase from its critical value (7.118) as the deformation continues in the post-buckling range. Consider now the situation where a circular cylindrical shell is closed at both ends by rigid plates, and is submitted to an all-round external pressure of intensity √ ρ. The uniform deformation that precedes buckling involves σ = pa/2 h = σ¯ / 3, and (7.102) become α = (1 + ν) (3 + T / E) ρ −1 , γ = (1 + ν) (4T / ρE) = δ / (1 + ν),

⎫ β = ν (1 + ν) (4T / ρE) ,⎬

ρ = 3 + 1 − 4ν 2 T / E.⎭

(7.121)

536

7

Plastic Buckling

The critical pressure for buckling is obtained by setting s = q/2 in (7.108), the coefficient of q in the resulting expression being D + C/2. Neglecting small-order terms, and using the fact that δ = (1 + v)γ and δ – β = γ in view of (7.121), we obtain the simplified formula k m2 − 1 γ m2 + 2 (1 + δ) λ2 λ4 pa + = 2 2 Eh m m − 1 m2 + 2.5λ2 (1 + ν) m2 + 2.5λ2

(7.122)

to a close approximation, where λ = π a/l. For given l/a and h/a ratios, the value m that makes the right-hand side of (7.122) a minimum should be used for calculating the critical pressure. In Fig. 7.16, the results for a closed-ended cylinder under an all-round hydrostatic pressure are compared with those for an open-ended cylinder under a radial pressure alone, using (7.115) with n = 3 and σ 0 /E = 0.002. For relatively thick tubes, the critical values of the equivalent stress in the two cases are found to be approximately the same. The plastic buckling of an initially imperfect cylindrical shell under internal pressure and axial compression, based on the total strain theory, has been considered by Paquette and Kyriakides (2006).

Fig. 7.16 Comparison of critical external pressure for the plastic buckling of cylindrical shell with open and closed end conditions

7.6

Torsional and Flexural Buckling of Tubes

537

7.6 Torsional and Flexural Buckling of Tubes 7.6.1 Bifurcation Under Pure Torsion In the case of a thin cylindrical shell subjected to equal and opposite twisting moments at its ends, the deformation mode ceases to be uniform when the applied shear stress τ attains a certain critical value. The thickness h and length l in relation to the mean radius a are assumed to be such that the bifurcation occurs beyond the elastic limit. Since the only nonzero stress at the onset of buckling is σ xθ = τ , the rates of extension in the longitudinal and circumferential directions are purely elastic, and the corresponding normal components of the stress rate are τ˙xx =

E (˙εxx + ν ε˙ θθ ) , 1 − ν2

τ˙θθ =

E (˙εθθ + ν ε˙ θθ ) . 1 − ν2

The only nonzero rate of shear is ε˙ xθ whose plastic component is (3/2H) τ˙xθ and consequently, ε˙ xθ =

1 3 + 2G 2H

τ˙xθ =

3 1 − 2ν − 2T 2E

τ˙xθ ,

where T is the tangent modulus at the current state of hardening of the material. The shear stress rate may therefore be written as τ˙xθ

2Eα = ε˙ xθ , 1 − ν2

1 − ν2 T/ E α= . 3 − (1 − 2ν) T / E

(7.123)

During the uniform twisting, α progressively decreases from the elastic value (1 – v) / 2 with increasing plastic strain. It follows from the above expressions that τ˙ij ε˙ ij =

E 2 2 2 ε˙ xx + 2ν ε˙ xx ε˙ θθ + ε˙ θθ . + 4α ε˙ xθ 2 1−ν

The condition for uniqueness of the deformation mode is given by inequality (7.104) with ρ = 0. Since σ xθ = τ , while all other components are identically zero, the inequality becomes 2 2 2 ε˙ xx + 2ν ε˙ xx ε˙ θθ + ε˙ θθ + 4α ε˙ xθ τ

+2 1 − ν 2 (ωxr ωθr + ε˙ θθ ωθx ) dx dθ dz > 0 E

(7.124)

to a close approximation, the possibility of sideways buckling of the tube being excluded. In terms of the velocities of the middle surface, the components of the strain rate are given by (7.98) and (7.99), while those of the rate of spin are given by (7.97), in view of the relations ωxr = ωθ , ωθr = −ωx , and ωθx = ωr . Substituting into (7.124), and integrating through the shell thickness, we obtain

538

7

∂u ∂ξ

2

Plastic Buckling

2 ∂υ ∂u ∂υ ∂υ ∂u 2 +w + +w +α + ∂ξ ∂θ ∂θ ∂ξ ∂θ $ 2 2 2 ∂ 2w ∂ 2 w ∂ 2 w ∂υ ∂ w ∂υ +k + 2ν − − + ∂ξ 2 ∂ξ 2 ∂θ 2 ∂θ ∂θ 2 ∂θ % 2 2 ∂w ∂w ∂ w ∂υ + 2φ + 4α − −υ ∂ξ ∂θ ∂ξ ∂ξ ∂θ ∂υ ∂υ + +w dξ dθ > 0, ∂ξ ∂θ

+ 2ν

on neglecting certain small-order terms and on introducing the dimensionless quantities τ

, φ = 1 − ν2 E

k=

h2 , 12a2

ξ=

x . a

The bifurcation would occur when the functional on the left-hand side of the above inequality vanishes for some distribution of the velocities u, υ, and w that makes the functional a minimum. The Euler–Lagrange differential equations associated with this variational problem are easily shown to be ∂ 2u ∂ 2u ∂ 2υ ∂w + α + + ν) +υ = 0, (α ∂ξ 2 ∂θ 2 ∂ξ ∂θ ∂ξ 2 ∂ 2υ ∂ 2υ ∂ υ ∂w ∂w ∂ 2υ + α + + ν) + + 2φ + (α ∂θ 2 ∂ξ 2 ∂ξ ∂θ ∂θ ∂ξ ∂θ ∂ξ (7.125) 2 2 3 3 ∂ w ∂ υ ∂ w ∂ υ − = 0, + 4α − + ν) +k (4α ∂θ 2 ∂ξ 2 ∂ξ 2 ∂θ ∂θ 3 ∂υ ∂υ ∂u ∂ 2υ +υ + w + 2φ − ∂θ ∂ξ ∂ξ ∂ξ ∂θ 4 4 ∂ 3w ∂ w ∂ 4w ∂ 3w ∂ w + 2υ 2 2 + 2 − (4α + ν) 2 − 3 = 0. +k ∂ξ 2 ∂ξ ∂θ ∂θ ∂ξ ∂θ ∂θ In the case of an elastic material, (7.125) reduce to those given by Timoshenko and Gere (1961), except for certain small-order terms which do not affect the final results significantly. The problem of the buckling of a cylindrical shell under pure torsion is reduced to the integration of (7.125), using the boundary conditions. In each of the three equations of (7.125), there are both odd- and even-order derivatives of a given velocity component with respect to the same independent variable. These equations cannot therefore be satisfied by assuming solutions in the form of products of sines and cosines of angles involving ξ and θ . When the cylinder is sufficiently large, so that the critical stress is practically independent of the edge constraints, we may assume the simple velocity field

7.6

Torsional and Flexural Buckling of Tubes

u = A cos (λξ − mθ) , υ = B cos (λξ − mθ ) , W = C sin (λξ − mθ ) ,

539

(7.126)

where λ = rπ a/l, with r denoting the number of longitudinal waves. The mode of buckling involves m circumferential waves which run helically along the cylinder. The substitution of (7.126) into (7.125) results in

− λ2 + αm2 A + (α + ν) λmB + νλC = 0, (α + ν) λmA − (1 + k) m2 + (1 + 4 k) αλ2 − 2λmφ B − m + km3 + k (4α + ν) λ2 m − 2λφ C = 0, νλA − m + km3 + k (4α + ν) λ2 m − 2λφ B

− 1 + k λ4 + 2νλ2 m2 + m4 − 2λmφ C = 0. This system of linear homogeneous equations can have nontrivial solutions for A, B, and C only if the determinant of their coefficients vanishes. Equating the determinant to zero, and neglecting terms containing k2 , kφ, φ 2 , etc., the result may be expressed as Tφ = R + kQ,

(7.127)

where R, Q, and T are dimensionless parameters given by the expressions

R = 1 − ν 2 αλ4 ,

2

Q = m2 m2 − 1 αm2 + 1 + 4α 2 − ν 2 λ2 + 2λ2 m4 1 − ν 2 (α + ν) λ2 − 2α 2 m2 ,

T = 2λm m2 − 1 αm2 + 1 − 2αν − ν 2 λ2 + αλ4 . The terms involving higher powers of λ have been omitted in the above expression for Q. Substituting for R, Q, and T into (7.127), the expression for the critical shear stress may be written with sufficient accuracy as km m2 − 1 λ3 τ + , = E 2m m2 − 1 m2 + βλ2 2λ 1 − ν 2

(7.128)

where β = (1 – v2 )/α – 2v. The value of λ that minimizes the critical stress may be approximately taken as that for elastic buckling, in which case βλ2 is negligible in comparison with m2 . The optimum value of λ is then given by

540

7

m2 m2 − 1 h λ = √ . 6 1 − ν2 a 2

Plastic Buckling

(7.129)

The critical stress formula for plastic buckling, obtained by omitting the quantity βλ2 in the first term of (7.128) and substituting from (7.129), may be written as τ = E

−3/4 h 3/2 3 2 1−ν 2 3a

where we have set m = 2 to minimize τ . The critical stress for plastic buckling can be computed from (7.128) and (7.129) by setting m = 2 and using the relation ⎧ √ n−1 ⎫ ⎬ ⎨ 9n 3τ , α = 1 − ν 2 / 2 (1 + ν) + ⎩ ⎭ 7 σ0

(7.130)

√ which is obtained from (7.123) and (7.115) with the substitution σ = 3τ . The results for plastic buckling are displayed as solid curves in Fig. 7.17, using v = 0.3,

Fig. 7.17 Critical shear stress for plastic buckling of a long cylindrical tube under pure torsion

7.6

Torsional and Flexural Buckling of Tubes

541

σ 0 /E = 0.002, and two different values of n. The broken curve represents the elastic buckling formula which is valid only for very small thicknesses. In the case of relatively short cylindrical shells, the conditions of support at the ends of the cylinder must be taken into account for the estimation of the critical stress. For the elastic range of buckling, an approximate analysis of the problem has been discussed by Donnell (1933), who introduced several simplifications into the basic differential equations (7.125). By applying appropriate boundary conditions on v and w, while permitting some axial motion of the ends of the cylinder, Donnell obtained buckling formulas that were found to be in good agreement with experiment. An empirical extension of the elastic buckling formula to cover the plastic range of loading has been suggested by Gerard (1962). Assuming the cylindrical shell to be clamped at both ends, the critical stress given by Gerard may be written as

−5/8 τ = 0.82 1 − η2 S

5/4 h a 1/2 , a l

(7.131)

where S is the secant modulus at the onset of buckling and η is the contraction ratio given by the relation 1 – 2η = (1 – 2v)S/E. For a given h/a ratio, (7.131) should be used for the range of values of l/a, which corresponds to a lower value of the critical stress than that given by (7.128). Adopting the Ramberg–Osgood equation for the stress–strain curve, the values S and η are easily computed by using (7.115). A modified buckling formula that includes the minor effect of preventing the axial strain in an approximate manner has been proposed by Rees (1982), who also produced some experimental evidence in support of the theory.

7.6.2 Buckling Under Pure Bending When an initially straight cylindrical tube is subjected to a gradually increasing bending moment, the circular cross section becomes increasingly oval until the applied moment starts to decrease after attaining a maximum, which constitutes a state of collapse due to buckling of the tube. This phenomenon is particularly significant for sufficiently long tubes, and the maximum compressive stress at the point of buckling is found to be considerably lower than the critical stress in an axially compressed cylindrical shell. The problem has been investigated by Brazier (1926) for the elastic range of buckling, and by Ades (1957), Gellin (1980), and Zhang and Yu (1987) for the plastic range of buckling. The effect of an internal pressure on the plastic collapse of a bent tube has been examined by Corona and Kyriakides (1988). Figure 7.18 shows the deformed configuration of the cross section of the middle surface, in which the position of a typical particle is specified by the angular coordinate θ measured from the crown of the circle. The circumferential and radial components of the displacement of any particle will be denoted by v and w, respectively, and the curvature of the neutral surface of the bent tube will be denoted by

542

7

Plastic Buckling

Fig. 7.18 Buckling of a cylindrical tube under pure bending. (a) Overall shape of the bent tube, (b) cross section before and after buckling

v, which is measured positive as indicated in the figure. By simple geometry, the distance of any point on the deformed middle surface from the neutral plane is y = (a + w) cos θ − υ sin θ ,

(7.132)

where α is the mean radius of the cylindrical tube. If the deformation of the middle surface is assumed to be inextensional in the circumferential direction, and the deformation and rotation are everywhere small, the longitudinal and circumferential components of the strain are ⎫ εx = −yκ, εθ = −zκθ , ⎪ ⎬ dw 1 d υ− ,⎪ κθ = − 2 ⎭ dθ a dθ

(7.133)

where κ θ is the circumferential curvature of the middle surface, and z is the radially outward distance from the middle surface. The velocity components υ and w are related to one another through the equation dυ/dθ + w = 0, which is the condition of zero circumferential strain at z = 0. Since the variation of the stress ratio during the prebuckling deformation is expected to be sufficiently small, the total strain theory of Hencky may be used without introducing significant error. Denoting the axial and circumferential stresses by σ x and σ θ , respectively, the stress–strain relations may be written as Sεx = σx − ησθ ,

Sεθ = σθ − ησx ,

where S is the secant modulus of the effective stress–strain curve corresponding to the current effective stress σ¯ , and

7.6

Torsional and Flexural Buckling of Tubes

η=

S 1 1 − (1 − 2ν) , 2 E

543

E =1+ S

m−1 3 σ¯ , 7 σ0

(7.134)

when the stress–strain curve is represented by the Ramberg–Osgood equation (7.115). The stress–strain relations are readily inverted to express the stresses as

σx =

S (εx + ηεθ ) , 1 − η2

σθ =

S (εθ + ηεx ) , 1 − η2

(7.135)

The significant components of the resultant force and moment are the axial force Nx and the circumferential bending moment Mθ acting per unit perimeter. They are defined as Nx =

h/2 −h/2

σx dz, Mθ = −

h/2 −h/2

σθ z dz,

(7.136)

where h is the uniform wall thickness of the bent tube. The variation of the internal virtual work per unit length of the shell may be written to a close approximation in the form δU ≈ a

2π

0

(Nx δεx + Mθ δκθ ) dθ ,

(7.137)

where δε x and δκ θ are the variations of εx and κ θ , respectively. The contributions to the work from the remaining stress resultants Nθ and Mx are neglected. For a prescribed curvature κ of the neutral surface, the displacements v and w should be such that δU = 0, which is a variational statement of equilibrium of the bent tube. While evaluating Nx , and Mθ , it would be a good approximation to assume a constant value of S equal to that on the middle surface. Then, in view of (7.135) and (7.136), and the fact that εθ = –zkθ, we have Nx =

Sh 1 − η2

εx ,

Mθ =

Sh 1 − η2

h2 κθ . 12

Inserting in (7.137), and setting δU = 0, the variational relation is reduced to the compact form

2π 0

S h2 2 κ dθ = 0. y δy + δκ κ θ θ 1 − η2 12

(7.138)

To evaluate the integral, the secant modulus S must be determined as a function of θ . Since σ x = Sεx / (1–η2 ) and σ θ = ησ x at z = 0, we have S 1 − η + η2 2 2 |εx | σ¯ = σx − σx σθ + σθ = 1 − η2

544

7

Plastic Buckling

at z = 0. The substitution for σ¯ and η in terms of S/E into the above relation, using (7.134), leads to the equation

μ2 S 3E +μ− 4S E

3 μ2 S2 + 2 4 E

1/(m−1) −1/2 Eaκ 7 E = −1 3 S σ0

y , (7.139) a

where μ = 12 − v, and y/a is a known function for any given displacement field. The applied bending moment M increases with the curvature κ in the prebuckling stages, the magnitude of the bending couple being given by

2π

M = −a

2π

Nx y dθ = a3 hk

0

0

S y dθ , 1 − η2 a

(7.140)

where S/(1—η2 ) corresponds to z = 0 as before. The critical moment Mc is the maximum value of M for varying κ during the bending. The components of the displacement of the middle surface, satisfying the inextensibility condition dυ/dθ = –w and the fourfold symmetry condition, may be expressed in the general form

w=−

N 1

wm cos 2mθ ,

m=1

υ=

N 1

(wm /2m) sin 2mθ .

(7.141)

m=1

The analysis for the buckling problem will be given here for the special case of N = 2, which is a good approximation for practical purposes. We therefore write w = − (w1 cos 2θ + w2 cos 4θ ) ,

υ=

1 1 w1 sin 2θ + w2 sin 4θ , 2 2

where w1 and w2 are constants. Considering only the first quadrant, υ is seen to vanish at θ = 0 and θ = π /2, the value of w at these two points being –(w1 + w2 ) and (w1 —w2 ), respectively. The substitution for υ and w into (7.132) and (7.133) gives ⎫ y = cos θ a − [w1 − (2 − 3 cos 2θ ) w2 ] cos2 θ ,⎪ ⎬ κθ =

3 (w1 cos 2θ + 5w2 cos 4θ ) . a2

⎪ ⎭

(7.142)

The constant displacements w1 and w2 can be determined by inserting (7.142) into the variational equation (7.138). It is convenient at this stage to set λ = (S/E) / (1—η2 ) and denote the various integrals as

7.6

Torsional and Flexural Buckling of Tubes

π/2

A1 =

⎫ ⎪ ⎪ λ (2 − 3 cos 2θ ) cos6 θ dθ , ⎪ ⎪ ⎪ 0 ⎪ ⎪ π/2 ⎪ ⎪ ⎪ 6 ⎪ B2 = λ (2 − 3 cos 2θ ) cos θ dθ , ⎪ ⎬

λ cos4 θ dθ ,

π/2

B1 = 0

λ cos6 θ dθ ,

0

(7.143) ⎪ 3 15 π/2 ⎪ ⎪ λ cos2 2θ dθ , C2 = λ cos 2θ cos 4θ dθ , ⎪ ⎪ ⎪ 4 0 4 0 ⎪ ⎪ ⎪ π/2 π/2 ⎪ ⎪ 75 2 2 6 ⎭ C3 = λ cos 4θ dθ , B3 = λ (2 − 3 cos 2θ ) cos θ dθ ,⎪ 4 0 0

C1 =

π/2

A2 =

0

545

π/2

where λ is a known function of θ in view of (7.139) and (7.142). Using (7.138) and (7.142), and the fact that δw1 and δw2 are arbitrary variations of w1 and w2 , respectively, we thus obtain the pair of equations

κ 2 a4 w 2 w1 + C2 − B2 2 = A1 , a h a κ 2 a4 w1 κ 2 a4 w 2 C 2 − B2 2 + C3 + B3 2 = −A2 , h a h a C 1 + B1

κ 2 a4 h2

for the two dimensionless constants w1 /a and w2 /a. Setting ρ = κa2 /h, the solution can be written in the form ⎫ A1 C3 + B3 ρ 2 + A2 C2 − B2 ρ 2 w1 ⎪ = ⎪ 2 , ⎪ ⎪ ⎬ 2 2 2 a C1 + B1 ρ C3 + B3 ρ − C2 − B2 ρ ⎪ A1 C2 − B2 ρ 2 + A2 C1 + B1 ρ 2 w2 ⎪ ⎪ = − 2 .⎪ ⎭ 2 2 2 a C1 + B1 ρ C3 + B3 ρ − C2 − B2 ρ

(7.144)

When the ratios w1 /a and w2 /a have been computed from (7.143) and (7.144) for a selected value of ρ, the corresponding bending moment can be determined from (7.140). In view of the relations (7.142) and (7.143), the bending moment can be expressed in the dimensionless form 4hEρ M = M∗ π aσ ∗

0

π/2

λ cos2 θ dθ − 2A1

w 1

a

+ 2A2

w

w w 1 2 + B3 −2B2 a2

2

+ B1

a

w 2 2

a

w 2 1

a

(7.145)

,

where M∗ = π a2 hσ ∗ , with σ ∗ denoting the critical stress E (h/a) / 3 1 − v2 for the elastic buckling of the shell under uniform axial compression (Section 7.5). The critical bending moment Mc under pure bending is the maximum value of M predicted by (7.145) for varying values of ρ. Over the elastic range, S = E and

546

7

Plastic Buckling

η = v, giving the critical moment Mc ≈ 9.545M∗ as the maximum bending moment that corresponds to p ≈ 0.495 when v = 0.3. The solution is marginally improved if the nonlinearity of the strain– displacement relations is taken into consideration. This has been demonstrated by Gellin (1980), who computed the bending moment and curvature of the tube at the onset of buckling for different values of n, using a series of values of the parameter σ0 = σ∗

aσ0 3 1 − ν2 hE

and assuming ν = 0.3. His results are displayed in Fig. 7.19, which indicates that the critical curvature is relatively insensitive to the variation of σ 0 /σ ∗ for usual values of n. The theory has been found to be in reasonable agreement with available experimental data.

Fig. 7.19 Dimensionless bending moment and curvature at the onset of plastic buckling as functions of the ratio σ 0 /σ ∗ (after S. Gellin, 1980)

When the tube is not too long, the prebuckling ovalization of the cross section may be neglected, as has been shown by Seide and Weingarten (1961), Akserland (1965), and Reddy (1979) in their bifurcation analyses for buckling of the shell. The critical value of the greatest compressive stress in this case is found to be only slightly higher than the critical stress σ ∗ in pure axial compression. The same conclusion may be assumed to hold for the flexural buckling in the plastic range when the tube is relatively short.

7.7 Buckling of Spherical Shells 7.7.1 Analysis for a Complete Spherical Shell A complete spherical shell of uniform small thickness h is subjected to a uniform external pressure of intensity p per unit area of the middle surface. When the

7.7

Buckling of Spherical Shells

547

pressure is increased to a certain critical value, the spherical form of equilibrium is no longer guaranteed, and a nonuniform mode of deformation is possible as a result of buckling. It is assumed at the outset that the buckled shape of the middle surface is symmetrical with respect to a diameter of the sphere. The velocity field at the incipient buckling, referred to the spherical coordinates (r, φ, θ ), where φ is measured from one of the poles, may be written as υr = w,

υφ = u + zω,

υθ = 0,

where z is the radially outward distance from the middle surface, and ω is the rate of spin about the θ -axis. The thin shell approximation ε˙ rφ = 0 gives ω=

dw 1 u− , a dφ

where a denotes the radius of curvature of the middle surface. The nonzero components of the strain rate at a generic point of the shell material are ε˙ θθ = λ˙ θ − zκ˙ θ ,

ε˙ φφ = λ˙ φ − zκ˙ φ .

(7.146)

The quantities λ˙ θ ,λ˙ φ are the rates of extension of the middle surface, and κ˙ θ ,κ˙ φ are the rates of curvature of the middle surface. By (5.67), with the necessary change in sign for w, we have ⎫ 1 1 du ˙ ⎪ + w ,⎪ λφ = (u cot φ + w) , ⎬ a α dφ ⎪ 1 dw 1 d dw ⎭ κ˙ θ = 2 − u cot φ, κ˙ φ = 2 − u .⎪ a dφ a dφ dφ λ˙ θ =

(7.147)

The current state of stress prior to buckling is a balanced biaxial compression represented by σ θθ = σ φφ = –pa / 2h the remaining stress components being identically zero. The rate of change of the stress at the incipient buckling varies, however, along the meridian due to the nonuniformity of the mode of deformation. We consider a regular isotropic yield surface for the material which is stressed in the plastic range with a current uniaxial yield stress σ and tangent modulus T. The unit normal to the yield surface, considered in a nine-dimensional space, has the nonzero components nθθ = nφφ

1 = −√ , 6

nrr =

2 . 3

Considering a linearized elastic/plastic solid having the constitutive equation (7.100), the expressions for the circumferential and meridional rates of extension can be written as

548

7

Plastic Buckling

3T T 4T ε˙ θθ = 1 + τ˙θθ + 1 − (1 + 4ν) τ˙φφ , E E T 3T 4T ε˙ φφ = 1 + (1 + 4ν) τ˙θθ + 1 + τ˙φφ . E E Setting T = E in these equations, we recover the rate form of stress–strain relations for an isotropic elastic material. The above equations are easily solved for the stress rates to give τ˙θθ =

E α ε˙ θθ + β ε˙ φφ , 1+ν

τ˙φθ =

1 + 3T/E , 2 1 + (1 − 2ν) T/E

β=

E β ε˙ θθ + α ε˙ φφ , 1+ν

(7.148)

where α=

−1 + (1 + 4ν) T/E . 2 1 + (1 − 2ν) T/E

(7.149)

The remaining components of the stress rate are identically zero. It follows from (7.148) that τ˙ij ε˙ ij =

E 2 2 , α ε˙ θθ + 2β ε˙ θθ ε˙ φφ + α ε˙ φφ 1+ν

which gives the leading term in the uniqueness functional (7.104), the strain rates appearing in this expression being given by (7.146) and (7.147). The condition for uniqueness of the deformation mode is easily established by using the facts that the only nonzero components of the stress tensor are σ θθ and σ φφ , each being equal to —pa / 2h, and that the only nonzero components of the spin tensor are ωrφ = –ω and ωφr = ω. Since the shear components of the strain rate are identically zero, the uniqueness criterion (7.104) furnishes

h/2

−h/2 0

π

2z 2 2 sin φdφdz α ε˙ θθ + 2β ε˙ θθ ε˙ φφ + α ε˙ φφ 1+ a pa π 2 − (1 + ν) ω sin φdφ E 0 π p + (1 + ν) λ˙ θ + λ˙ φ w + uw sin φdφ > 0 E 0

to a close approximation. Substituting for the strain rates and the rate of spin, integrating through the thickness of the shell, and omitting the quantity d(uw sin φ) which does not contribute to the integral, the inequality may be expressed as

7.7

Buckling of Spherical Shells

549

du +w α (u cot φ + w) + 2β (u cot φ + w) dφ 0 % 2 du +α +w sin φdφ dφ 2 2 π$ dw dw du d w h2 2 α − − u cot φ + 2β −u cot φ + 12a2 0 dφ dφ dφ 2 dφ 2 2 % d w du sin φdφ +α − dφ dφ % π $ 2 pa dw 2 − (1 + ν) − 2w sin φdφ > 0 2Eh 0 dφ π

2

on neglecting terms of order (h/a)2 times the square of the rate of extension of the middle surface. The bifurcation would occur when the left-hand side of the above inequality is zero, the corresponding velocity field being that which minimizes the functional. Introducing the dimensionless quantities q = (1 + ν)

pa , 2Eh

k=

h2 , 12a2

the Euler–Lagrange differential equations satisfied by the two velocity components u and w can be easily written down following the standard technique of the calculus of variations. Taking due account of the factor sinφ that appears outside the square brackets in each integral, we obtain 2 du dw d u 2 + cot φ − β + α cot φ u + (α + β) (1 + k) α 2 dφ dφ dφ (7.150) 3 d2 w dw d w 2 + cot φ 2 − β + α cot φ −k α = 0, dφ 3 dφ dφ 3 d2 u d u du + 2 cot φ 2 + u cot φ + 2w + k −α (α + β) dφ dφ 3 dφ du

− cot φ α − β + αcosec2 φ u + β + αcosec2 φ dφ 4 (7.151) d2 w d3 w d w 2 + 2 cot φ 3 − β + αcosec φ +α dφ 4 dφ dφ 2 2

dw dw d w + cot φ α − β + αcosec2 φ + cot φ + 2w = 0. +q dφ dφ dφ 2 An immediate simplification of (7.150) is achieved by omitting k in the first factor, since it is small compared to unity. The above equations can be expressed in

550

7

Plastic Buckling

a more convenient form by setting u = dυ/dφ where υ is a new variable, and by introducing an operator H defined as H=

d d2 + cot φ + 2. 2 dφ dφ

In terms of the dependent variables υ and w, the differential equation (7.150) then becomes d [αH (υ) + (α + β) (w − υ) − kαH (w) + k (α + β) w] = 0 dφ The last term in the square brackets of this equation is seen to be negligible compared to the second. Integrating this equation, and setting the constant of integration to zero, we have αH (υ) + (α + β) (w − υ) − kαH (w) = 0.

(7.152)

Equation (7.151) can be similarly expressed in terms of the operator H and the new variable υ. The analysis is considerably simplified by noting the fact that the expressions in υ and w appearing in the square brackets of (7.151) are identical to one another except for the sign. The final result is easily shown to be (α + β) [H (υ) + 2 (w − υ)]+qH (w)+kαHH (w − υ)−(α + β) H (w − υ)] = 0. (7.153) on neglecting a term of order k compared to unity. The analysis for the bifurcation problem is therefore reduced to the solution of the pair of differential equations (7.152) and (7.153), the only restriction on the admissible velocity field being dυ/dφ = dw/dφ = 0 at φ = π and 0 = π in view of the symmetry of the field about the diameter passing through these points.

7.7.2 Solution for the Critical Pressure The nature of the differential equations (7.152) and (7.153) indicates that the solution may be expressed in terms of Legendre functions Pm of integer orders m. These functions are defined as m (m − 1) (2m)! Pm (cos φ) = (cos φ)m−2 (cos φ)m − 2 m 2. (2m − 1) 2 (m!) (7.154) m (m − 1) (m − 2) (m − 3) m−4 + − ··· . (cos φ) 2.4. (2m − 1) (2m − 3) The expression on the right-hand side of (7.154) is a polynomial of degree m in the variable cos φ. The Legendre functions of the first few orders are

7.7

Buckling of Spherical Shells P0 (cos φ) = 1, P3 (cos φ) =

551 1 P2 (cos φ) = 3 cos2 φ − 1 , 2 5 3 4 7 cos φ − 6 cos2 φ + . P4 (cos φ) = 8 5

P1 (cos φ) = cos φ,

1 cos φ 5 cos2 φ − 3 , 2

All these functions are found to satisfy a second-order linear differential equation, known as Legendre’s equation, which is d 2 Pm dPm + m (1 + m) Pm = 0. + cot φ 2 dφ dφ

(7.155)

Applying the operator H on the Legendre function Pm , and using (7.155), it is readily shown that H (Pm ) = −λm Pm ,

HH (Pm ) = λ2m Pm ,

λm = m (1 + m) − 2.

(7.156)

The usefulness of the operator H when dealing with Legendre functions now becomes obvious. To obtain a general solution to the eigenvalue problem, it is convenient to express the quantities v and w in the form of the infinite series υ=

∞ 1 m=0

Am Pm (cos φ) ,

w=

∞ 1

Bm Pm (cos φ) ,

m=0

where Am and Bm are arbitrary constants. Substitute these expressions into the differential equations (7.152) and (7.153), and using (7.156), we obtain the relations ∞ 1 m=0 ∞ 1 m=0

{[α + β + αλm ] Am − [α + β + kαλm ] Bm } Pm = 0, {[(α + β) (2 + λm ) + kλm (α + β + αλm )] Am 6 − 2 (α + β) − qλm + kλm (α + β + αλm ) Bm Pm = 0.

These equations will be satisfied only if the expressions appearing on the lefthand side for each value of m individually vanish. Hence, omitting the summation sign, we obtain the pair of equations ⎫ (α + β + αλm ) Am − (α + β + kαλm ) Bm = 0, ⎪ ⎬ [(α + β) (2 + λm ) + kλm (α + β + αλm )] Am ⎪ ⎭ − 2 (α + β) − qλm + kλm (α + β + αλm ) Bm = 0,

(7.157)

for the two typical constants Am and Bm . For the bifurcation to occur, (7.157) must admit nonzero values of these constants for some value of m, in which case the

552

7

Plastic Buckling

determinant of the coefficients of Am and Bm must vanish. Neglecting the small terms involving qk and k2 , the result may be expressed as

qλm (α + β + αλm ) − λm α 2 − β 2 − kαλm αλ2m − 2 (α + β) = 0 Since λm = 0 implies m = 1, representing a rigid-body motion, only λm = 0 will be relevant for bifurcation. Using the fact that α—β = 1, the dimensionless critical pressure is obtained as (α + β) + kα αλ2m − 2 (α + β) . q= (α + β) + αλm

(7.158)

To obtain the smallest value of q for which bifurcation may occur, it is convenient to regard the right-hand side of (7.158) as a continuous function of λm . Then setting dq/dλm = 0 to minimize q, the result is easily shown to be ⎫ ⎪ α+β ⎬ , αλm = − (α + β) + (7.159) k ⎪ ⎭ q = 2 k (α + β) − 2 k (α + β) , to a close approximation. Since λm is a large number, the critical pressure given by (7.159) cannot differ significantly from the smallest pressure based on successive integer values of m. For practical purposes, the second term on the right-hand side for q in (7.159) may be disregarded without any significant error. Using (7.149) for α and β we obtain the critical compressive stress for bifurcation in the form √ pa σ 2 (h/ a) , = =√ E 2Eh 3 (1 + ν) (1 − 2ν + E/ T)

(7.160)

which is the true tangent modulus formula for the plastic buckling of a spherical shell under external pressure. The same formula has been obtained earlier by Batterman (1969) using the rate equations of equilibrium instead of a variational principle. For any given material, the critical stress can be easily computed from (7.160) and (7.115) as a function of the ratio h/a, the results of the computation being presented graphically in Fig. 7.20. The deformation mode at the incipient buckling consists of a uniform radial contraction superposed on the eigenmode with m representing the nearest integer corresponding to the value of λm predicted by (7.159). Since Am / Bm ≈ k (α + β) in view of (7.157) and (7.159), the actual velocity field at the point of bifurcation may be written as w = −w0 (1 + cPm ) , (7.161) u = −cw0 k (α + β)Pm ,

7.7

Buckling of Spherical Shells

553

Fig. 7.20 Variation of critical stress with wall thickness for the plastic buckling of a complete spherical shell under uniform external pressure

where c is a constant and the prime denotes differentiation with respect to φ the quantity (1 + c)w0 being the radially inward velocity at φ = 0. The condition of continued loading of the radially compressed shell may be expressed as ε˙ θθ + ε˙ θθ < 0, which is equivalent to the restriction λ˙ θ + λ˙ φ − z κ˙ θ + κ˙ φ < 0 for all values of φ and z. Substituting from (7.147) and (7.161), and using the Legendre equation (7.155), this inequality is reduced to h cm (1 + m) μ + (1 − μ) Pm < 2 (1 + cPm ) , a √ where μ = k (α + β). This inequality can be satisfied for a range of values of c, giving the possible modes of deformation at bifurcation which occurs under increasing external pressure. It is interesting to note that the critical stress for buckling of an externally pressurized spherical shell is identical to that of an axially compressed short cylindrical

554

7

Plastic Buckling

shell, not only for an elastic material but also for an incompressible elastic/plastic material (v = 0.5). As in the case of cylindrical shells, slight geometrical imperfections in a spherical shell have a significant effect of lowering the critical pressure from the theoretical value (Hutchinson, 1972). An analysis for the plastic buckling of spherical shells based on the total strain theory has been discussed by Gerard (1962). The plastic buckling of shells of revolution has been investigated by Bushnell and Galletly (1974) and Bushnell (1982).

7.7.3 Snap-Through Buckling of Spherical Caps Consider a deep spherical cap of thickness h and radius a, which is subjected to an inward load P at the apex applied through a rigid circular boss of radius b, as shown in Fig. 7.21(a). As the load is gradually increased from zero, the material response is initially elastic, and the deflection at the apex increases monotonically with the intensity of loading. Over the practical range of values of h/a, plastic deformation inevitably begins before the failure occurs due to buckling. As the loading is continued in the plastic range, the load–deflection curve continues to rise with decreasing slope until the load attains a maximum, the corresponding deflection being about half the shell thickness. Subsequently, the load falls rapidly with increasing deflection and reaches a minimum when the deflection is about twice the shell thickness. Thereafter, the load increases again with deflection due to the strengthening effect of the membrane forces. If the shell is subjected to an incremental dead loading, a snap-through type of buckling would occur when the maximum value of the load is reached.

Fig. 7.21 Plastic collapse of a spherical cap centrally loaded through a rigid boss. (a) Geometry and loading, (b) conditions at incipient collapse

A complete elastic/plastic analysis of the problem, taking due account of the geometry changes, is actually required to obtain the load–deflection behavior of the shell, in order to predict the initiation of the snap-through action. For practical purposes, however, the maximum load can be approximately estimated by finding

7.7

Buckling of Spherical Shells

555

the point of intersection of the load–deflection curves obtained on the basis of purely elastic and rigid/plastic behaviors of the shell. The rigid/plastic curve begins with the theoretical collapse load, which is most conveniently determined by using the limited interaction yield condition. The analysis is similar to that used for a spherical cap under external pressure (Section 5.4) and involves the yield condition in the dimensionless form nφ = −1, −1 < nθ < 0 (φ0 ≤ φ ≤ α) , mθ = 1 (φ0 ≤ φ ≤ β) , mθ − mφ = 1 (β ≤ φ ≤ α) , where φ 0 = sin-1 (b/a). The angles α and β represent the extent of the deformation zone and the interface between the two plastic regimes, respectively. The shearing force s and the circumferential force nθ in the dimensionless form are easily shown to be ⎫

s = 1 − k¯qcosec2 φ tan φ,⎬ φ0 ≤ φ ≤ α, ⎭ n = − (1 − k¯q) sec2 φ, θ

where q¯ = P/2π M0 and k = h/4a. These relations follow from the equations of force equilibrium under zero surface loading. In view of the preceding relations, the equation of moment equilibrium becomes 1 d 1 − q¯ sec φ − − 1 cos φ, mφ sin φ = dφ k k dmφ 1 = − q¯ tan φ − (¯q − 1) cot φ, dφ k

⎫ ⎪ φ0 ≤ φ ≤ β,⎪ ⎬ ⎪ ⎭ β ≤ φ ≤ α. ⎪

(7.162)

The deformation mode at the incipient collapse involves hinge circles at φ = φ 0 and φ = α, requiring mφ = 1 and mφ, = –1 at these two sections, respectively. Using these as boundary conditions, (7.162) can be readily integrated to given the distribution of mφ. Since mφ must vanish at φ = β, we obtain

sec β + tan β (1 − k) sin β − sin φ0 = (1 − k¯q) ln sec φ0 + tan φ0 sin α cos β k (¯q − 1) ln − 1 = (1 − k¯q) ln . sin β cos α

⎫ ⎪ ,⎪ ⎬ ⎪ ⎪ ⎭

(7.163)

The angle α, defining the position of the plastic boundary, must be such that the bending moment mφ is a minimum at φ = α. The second equation of (7.162) therefore gives the relation q¯ = cos2 α +

1 2 sin α, k

(7.164)

from which the collapse load parameter q¯ can be determined for any assumed α. The corresponding values of φ 0 and β then follow from (7.163). For a given value

556

7

Plastic Buckling

of k, the collapse load increases as the boss size is increased, approaching a limiting value equal to 1/k as α tends to π/2. Since nθ < 1 for α < π /2 the preceding solution is statically admissible. The results of the calculation are shown in Fig. 7.21(b), which indicates that the region of plastic deformation is increasingly confined in the neighborhood of the loading boss as the radius of the boss is increased. The solution is also kinematically admissible, the associated velocity field being similar to that for a clamped spherical cap under uniform external pressure. Denoting the axial speed of the boss by δ˙ , the normal component of velocity of the material around it can be written as cos φ sec φ + tan φ , φ0 ≤ φ ≤ β, (7.165) w = δ˙ 1 − η ln sec φ0 + tan φ0 cos φ0 where sec β + tan β tan α −1 η = ln + sin β ln . sec φ0 + tan φ0 tan β In the outer plastic region (β ≤ φ ≤ α), the normal component of velocity is obtained by replacing the expression in the curly brackets of (7.165) with η sin β ln(tan α/tan φ). The velocity field implies the formation of hinge circles at φ = φ 0 and φ = α with appropriate discontinuities in the velocity gradient. The slope of the load–deflection curve at the incipient collapse has been determined by Leckie on the basis of a method due to Batterman (1964). A graphical plot of the large initial slope, included in Fig. 7.21(b), indicates that the carrying capacity would be considerably lowered due to the effect of the elastic deformation of the shell. The load–deflection curves shown in Fig. 7.22 have been obtained by Leckie and Penny (1968) using a series of carefully controlled tests. While these curves have the same general trend, the variation of load with√deflection is seen to be more pronounced for higher values of the parameter ρ = b/ ah. The theoretical collapse loads and the associated initial slopes of the load–deflection curves, predicted by the rigid/plastic analysis, are indicated by broken lines in the figure. We may consider a typical elastic load–deflection curve in which the plastic yielding is disregarded, the derivation of such a curve being that discussed by Ashwell (1959). The point of intersection of this curve with the corresponding rigid/plastic line is seen to provide a reasonable approximation for the maximum load at which the snap-through action is initiated. The snap-through buckling of a spherical √ cap results in an inversion of a central portion of the cap, having a radius of order hw0 , where w0 is the central deflection of the cap, and h the shell thickness The inverted cap has the same geometrical configuration as that obtained by removing a central portion of the cap, turning it over, and reuniting it with the parent cap around its edge. In actual practice, the transition between the inverted cap and the parent cap occurs through a narrow region √ in the form of a toroidal knuckle the width of which is of the order ah, where a is the mean radius of the shell. Over the initial stages of the post-buckling behavior, the variation of the load P with the central deflection w0 may be expressed by

Problems

557

Fig. 7.22 Experimental load–deflection curves for a spherical cap loaded inwardly through a rigid boss (after Leckie and Penny, 1968)

2.5 the formula Pa = 1.7λEw0.5 0 h , where E denotes Young’s modulus, while λ is a dimensionless parameter whose value is unity for purely elastic buckling (Calladine, 2001) Over the plastic range of buckling„ λ should depend on the ratio T/ E, where T is the tangent modulus at the incipient buckling of the shell.

Problems 7.1 A uniform straight column of slenderness ratio k is pin-supported at both ends, and is subjected to axial compression by equal and opposite forces P applied along the centroidal axis, The stress–strain curve of the material in the plastic range may be expressed by the equation

558

7

ε=

m αY σ σ + −1 , E E Y

Plastic Buckling

σ ≥ Y,

where α and m are dimensionless constants, the slope of the stress–strain curve being equal to E at the yield point. If the cross section of the column is rectangular, and the critical stress is denoted by σ according the tangent modulus theory, and by σ ∗ according to the reduced modulus theory, show that σ Y

1 + mα

σ Y

−1

m−1

π2 = 2 k

E , Y

σ∗ Y

1+

1 + mα

σ∗ Y

−1

m−1

=

2π k

E Y

7.2 A work-hardening elastic/plastic material yields according to the von Mises yield criterion, but obeys the non-associated flow rule furnished by the rate form of the Hencky stress–strain relation. Show that the constitutive equations for a balanced biaxial state of stress may be written as σ˙ xx = S α ε˙ xx + β ε˙ yy ,

σ˙ yy = S β ε˙ xx + α ε˙ yy ,

τ˙xy = S˙εxy / 1 + η ,

where S is the current secant modulus of the uniaxial stress–strain curve of the material, η is the modified contraction ratio obtained by replacing T with S in the expression for η, the parameters α and β being given by the expressions α=

1 ρ

3T 1+ , S

β =α−

1 , 1 + η

ρ = 4 (1 − η) 1 + η

7.3 Considering a state of plane stress, in which the current state is defined by σ x = –σ 1 , σ y = –σ 2 , and τ xy = 0, and assuming the rate form the Hencky stress–strain relation, derive the constitutive equations for an isotropic elastic/plastic material in the form σ˙ xx = S α ε˙ xx + β ε˙ yy ,

σ˙ yy = S β ε˙ xx + γ ε˙ yy ,

τ˙xy = S˙εxy / 1 + η

where S is the secant modulus of the uniaxial stress–strain curve, while α, β, and γ are dimensionless parameters expressed in terms of the ratios of the applied stresses to the effective stress σ¯ as T σ12 T σ1 σ 2 T ρα = 4 − 3 1 − , ρβ = 4η − 3 1 − , ργ = 4 − 3 1 − S σ¯ 2 S S σ¯ 2 S S T σ1 σ 2 , 2η = 1 − (1 − 2ν) ρ = 3 + (1 − 2ν) (1 + 2η) − 3 1 − E S E σ¯ 2

σ22 σ¯ 2

7.4 A rectangular plate, whose sides are of lengths a and b, respectively, where b ≤ a, is subjected to normal compressive stresses of magnitude σ along the shorter sides of the rectangle. Using the non-associated constitutive equations given in the preceding problem, show that the critical compressive stress for buckling is given by σ π 2 h2 = S 3ρb2

3T 2+ 1+ S

S a 2 mb 2 3 1 − 2ν T , + + − 2a 2 1+η E E mb

where m is an integer that minimizes the right-hand side of the above equation. Obtain the corresponding that holds for sufficiently large values of the aspect ratio a/b.

Problems

559

7.5 A rectangular plate of sides a and b is subjected to equibiaxial compressive stresses of magnitude σ normal to the edges of the plate. Using the non-associated constitutive equations of Prob.7.2, show that the critical stress for buckling is given by the expression

⎫ ⎧ ⎬ (1 + 3T/S) 1 + a2 /b2 π 2 h2 ⎨ σ = S 3a2 ⎩ 3 − (1 − 2ν) S/E 1 + (1 − 2ν) T/E ⎭ Using the Ramberg–Osgood equation with α = 3/7, E/σ 0 = 750 and c = 3, draws a graph of σ b2 /Eh2 against a/b when ν = 0.3, and compare it with that given by the Prandtl–Reuss theory. 7.6 A solid circular plate of radius a is subjected to a uniform radial compressive stress σ along its boundary. Using the constitutive equations given in Prob.7.2, obtain he critical stress for buckling of a clamped plate in the form σ k 2 h2 = S 12a2

1 + 3T/S , 4 (1 − η) 1 + η

2η = 1 − (1 − 2ν)

S E

where k is the smallest root of the equation J1 (k) = 0. Adopting the Ramberg–Osgood stress–strain curve of the preceding problem, and using ν = 0.3, obtain the variation of σ /E with h/a, and compare it with that given by the Prandtl–Reuss theory. 7.7 If the circular plate of the preceding problem is simply supported along its boundary .r = a, show that the critical stress for buckling under a uniform radial compression σ is given by 4 (1 − η) kJ1 (k) , = J0 (k) 1 + 3T/S

4a k= h

3σ S

1/2 (1 − η) 1 + η 1 + 3T/S

in view of the results given in Prob. 7.1. Assuming the Ramberg–Osgood equation of the preceding problem, compute the critical value of σ /E when ν = 0.3 and h/a = 0.05, and compare the result with that given by the Prandtl–Reuss theory.

Chapter 8

Dynamic Plasticity

In this chapter, we shall be concerned with the class of problems in which the plastic deformation is so rapid that the inertia effects cannot be disregarded. Problems of dynamic plasticity arise in the high-velocity forming of metals, penetration of highspeed projectiles into fixed targets, enlargement of cavities by underground explosion, and design of crash barriers related to collisions, to name only a few. The rate of loading and the size of the components are usually such that the deformation process can be described in terms of the propagation of elastic/plastic waves. However, simplified theories which disregard the wave propagation phenomenon are generally capable of providing useful information for practical purposes. In the case of structural members subjected to impact loading, the mode of plastic deformation can be most conveniently represented by the existence of discrete yield hinges that rapidly move away from the point of loading. The concept of moving yield hinges is a useful device for the dynamic analysis of structures.

8.1 Longitudinal Stress Waves in Bars 8.1.1 Wave Propagation Without Rate Effects The propagation of longitudinal elastic/plastic waves in thin rods or wires was first discussed independently by von Karman (1942), Taylor (1942), and Rakhmatulin (1945), although a theoretical treatment in a restricted sense was given earlier by Donnel (1930). Following von Karman and Duwez (1950), the theory will be developed here in terms of the nominal stress and strain, denoted here by σ and ε, respectively, and the initial coordinate x measured along the axis of the bar, which is assumed to have a uniform cross section. If the longitudinal velocity of the particle at any instant is denoted by v, and transverse plane sections are assumed to remain plane, the equation of motion is ∂υ ∂σ =ρ , ∂x ∂t where ρ is the initial density of the material, and σ is a known function of ε, tensile stress being taken as positive. Since υ = ∂u/∂t and ε = ∂u/∂x, where u is the J. Chakrabarty, Applied Plasticity, Second Edition, Mechanical Engineering Series, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77674-3_8,

561

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longitudinal displacement, the differential equation for a rate insensitive material becomes 2 ∂ 2u 2∂ u = c , ∂t2 ∂x2

c2 =

1 dσ , ρ dε

(8.1)

which is a quasi-linear wave equation governing the motion of the particle, with c = c(ε) denoting the speed of propagation of the wave. Equation (8.1) applies only during the loading process in which the stress continuously increases with time, and for common engineering materials, the wave speed decreases with increasing strain. If the loading is tensile, a critical stage would be reached when the total strain is equal to that at the onset of necking, and the plastic wave speed is then reduced to zero. The solution of the differential equation (8.1) is simplified by the fact that this equation is actually hyperbolic. Indeed, along any curve considered in the (x, t)plane, the variation of the velocity v is given by dυ =

∂ε ∂υ ∂υ ∂ε dx + dt = dx + c2 dt, ∂x ∂t ∂t ∂x

(8.2)

in view of the identity ∂υ/∂x = ∂ε/∂t, and the relation ∂υ/∂t = c2 (∂ε/∂x) which follows from (8.1). The variation of the strain ε along this curve is dε =

∂ε ∂ε dt + dx. ∂t ∂x

For given values of dυ and dε along the curve, the derivatives ∂ε/∂t and ∂ε/∂x can be uniquely determined from the last two equations unless the determinant of their coefficients vanishes. The considered curve will therefore be a characteristic if (dx)2 = c2 (dt)2 . Equation (8.1) is therefore hyperbolic, the characteristic directions and the differential relations holding along them being dx = ±c, dt

dυ = ±cdε.

(8.3)

where the second result follows on substitution for dx and dt from the first into the second expression of (8.2). Since c is a function of ε, the characteristic lines generally consist of two families of curves corresponding to the upper and lower signs in (8.3). When the characteristic is a straight line, c is a constant, which means that the stress, strain, and velocity remain constant along its length. Unloading begins in a given cross section as soon as the stress begins to decrease after reaching a certain maximum value σ ∗ , corresponding to a strain ε ∗ . The stress– strain relation for the unloading process may be written as σ − σ ∗ = E ε − ε∗ ,

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Longitudinal Stress Waves in Bars

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where E is Young’s modulus for the material. Substituting in the equation of motion, and using the fact that υ = ∂u/∂t, we obtain the differential equation 2 ∂ 2u d 2∂ u = c + 0 2 2 dx ∂t ∂x

σ∗ − c20 ε∗ , ρ

(8.4)

√ where c0 = E/ρ representing the speed of propagation of the elastic wave. Equation (8.4) is also hyperbolic with two families of straight characteristics in the (x, t)-plane, the characteristic relations being obtained in the same way as before with the result dx = ±c0, dt

ρc0 dυ = ±dσ .

(8.5)

Since σ ∗ and ε ∗ are functions of x, and are not known in advance, the shape of the loading/unloading boundary must be determined as a part of the solution. For a bar of finite length, waves are reflected from the ends of the bar, and the solution is strongly dependent on the nature of the boundary conditions. If a bar is subjected to impact loading at one end x = –l, the first wave to reach the other end x = 0 is always an elastic wave. The displacement of the particles during the propagation of the direct elastic wave is governed by (8.1) with c replaced by the elastic wave speed c0 . A general solution of this wave equation is u1 = f1 (c0 t − x) , where f1 is an arbitrary function that can be determined from the prescribed initial conditions. For the reflected wave, which propagates in the opposite direction, the displacement is u2 = f2 (c0 t + x) . The total displacement of a particle traversed by the incident and reflected waves is u = u1 +u2 , and the corresponding stress is σ =E

6 7 ∂ (u1 + u2 ) = E f2 (c0 t + x) − f1 (c0 t − x) , ∂x

where the prime denotes differentiation with respect to the argument c0 t– x. When the end x = 0 is free, the stress vanishes there for all t, and consequently f1 (c0 t) = f2 (c0 t), which gives f1 = f2 = f . The displacement and velocity at the free end therefore become u = 2f (c0 t) ,

υ = 2c0 f (c0 t) .

Thus, the displacement and velocity at the free end are doubled due to the reflection, a compression wave being reflected as a tension wave and vice versa. If the

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8 Dynamic Plasticity

end x = 0 is fixed, the total displacement u1 +u2 vanishes there for all t, and we have f2 = −f1 = f and the stress at the fixed end becomes σ = 2Ef (c0 t). The stress is therefore doubled in magnitude due to the reflection, and an elastic wave may therefore be reflected as a plastic wave. Across a wave front, the first derivatives of v and ε are necessarily discontinuous. In the case of weak waves, υ and ε are themselves continuous, and the wave front then coincides with a characteristic. In problems of plastic wave propagation, one frequently encounters shock waves in which the wave fronts are surfaces of discontinuity even for υ and ε. Shock waves are usually generated by a sudden change in velocity imposed at one end of the bar. If a shock wave front moves through a distance dx during a time interval dt, the displacement on either side of the wave front changes by the amount du = υ dt+ε dx. The condition of continuity of the displacement across the wave front therefore gives [υ] + cs [ε] = 0,

(8.6a)

where cs is the speed of propagation of the shock wave, and the square brackets represent the jump in the enclosed quantity when the front has passed a given cross section. The momentum equation for the element of length dx traversed by the wave front during the time dt yields ρcs [υ] + [σ ] = 0.

(8.6b)

The two jump conditions established in (8.6), together with the stress–strain equation, are sufficient to study the propagation of shock waves, provided the change in temperature due to impact is negligible. The elimination of [υ] between the two relations of (8.6) furnishes ρc2s = [σ ]/[ε]. The speed of propagation of the shock wave coincides with that of the weak wave only when the stress–strain curve is linear.

8.1.2 Simple Wave Solution with Application Consider the situation where ε and v are functions of the ratio x/t, but not of x and t individually. Then all the characteristics of positive slope are straight lines passing through the origin, the result being a centered fan of linear characteristics, having the equation x/t = c, known as simple waves. Since v and ε are functions of c only, we have ∂υ ∂ε dυ =t =t , dc ∂x ∂t the right-hand side of this expression being –c times the derivative of ε with respect to c. Thus

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Longitudinal Stress Waves in Bars

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dε 1 dσ dυ = −c =− . dc dc ρc dc Using the boundary condition υ = υ 0 along the characteristic that corresponds to σ = Y, the above equation is immediately integrated to give σ ρc0 (υ − υ0 ) = −

(c0 /c) dσ .

(8.7)

γ

Simple waves generally appear under instantaneous loading by a uniform stress or velocity at one end of the bar and exist in a region adjacent to one of constant stress or strain, which evidently satisfies (8.1) and the boundary condition. The initial loading of the bar is generally followed by an unloading process in which the stress progressively decreases in magnitude. To illustrate the basic principles, consider the normal impact of a cylindrical bar of length l, which is moving parallel to its longitudinal axis with a velocity –U against a stationary rigid target x = 0 (Lee, 1953). For mathematical convenience, we superimpose a constant velocity U to the whole system in the opposite direction and write the initial and boundary conditions as υ = 0, σ = 0 at t = 0, 0 < x ≤ t, υ = U, at x = 0, σ = 0 at x = l, t > 0. The stress and strain will be considered positive in compression. The characteristic field is shown in Fig. 8.1(a), where OA is the elastic wave front across which the stress rises instantaneously from zero to the yield stress Y. The fan OAD consists of plastic waves with OD representing the characteristic for υ = U. Within the triangle ODE, the material has a constant velocity U and is subjected to a constant stress that corresponds to the characteristic OD. At the free end x = l, the elastic loading wave is reflected as an unloading shock wave which propagates with a speed c0 and has the equation x = 2 l − c0 t. At A, the value of ρc0 υ is equal to Y before the reflection, in view of (8.6b) with cs = co and a change in sign for σ . The physical quantities along ABC after the reflection will be denoted by a prime. Since the velocity at the free end is doubled by the reflected wave, ρc0 υ is equal to 2Y at A, and the second relation of (8.5) applied to the boundary wave AC gives ρc0 υ − 2Y = σ as the stress vanishes at the free end after the reflection. At a generic point of AC, we may apply the jump condition (8.6b) with cs = –c0 and –σ written for σ , the result being − ρc0 υ − υ = σ − σ

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8 Dynamic Plasticity

Fig. 8.1 Longitudinal impact on a bar of finite length. (a) Characteristics in the plastic region and (b) distribution of plastic strain along its length

Eliminating υ between the above relations, the amount of stress discontinuity across the unloading wave may be expressed in terms of the stress and velocity before the reflection as σ − σ = Y −

1 (ρc0 υ − σ ) . 2

(8.8a)

The relationship between υ and σ is obtained from (8.7) by setting ρc0 υ 0 = Y and changing the sign of σ . Thus σ ρc0 υ = Y +

(c0 − c) dσ ,

(8.8b)

Y

where ρc2 = dσ/dε, given by the stress–strain curve. Since c ≤ co , the integral in (8.8b) is increasingly greater than σ – Y, and the expression in the parenthesis of (8.8a) steadily increases along AC from its minimum value zero at A. Thus the unloading shock wave is progressively absorbed during its propagation. If the impact velocity U is sufficiently high, the unloading wave will be completely

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Longitudinal Stress Waves in Bars

567

absorbed at a point C, where the stress discontinuity is reduced to zero. When the impact velocity is below a certain critical value, the unloading wave can continue through to the impact face at E, and the material is unloaded throughout the length of the bar. In the case of a supercritical impact velocity that terminates the unloading wave at C, the plastic region spreads from this point into an area above AE. The solution to this part of the impact problem involves discontinuities in stress and velocity derivatives, which must be admitted for the continuation of the loading–unloading boundary. For a subcritical impact velocity, which ensures that the unloading wave traverses the entire length of the bar, a new plastic region is initiated at a point

Fig. 8.2 Characteristic field in the propagation of longitudinal plastic waves involving regions of loading and unloading (after E. H. Lee, 1953)

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8 Dynamic Plasticity

where the unloaded bar has a stress equal to the maximum stress previously attained in the same section during loading. The complete characteristic field for this particular case obtained by Lee is shown in Fig. 8.2, where D is the point of initiation of the new plastic zone DEFGK. At point K, the loading–unloading boundary is intersected by the unloading wave reflected from the impact face. The permanent strain produced by the secondary plastic region is found to be small compared to that due to the primary plastic loading wave, as may be seen from Fig. 8.1(b), which displays the permanent strain distribution in the bar.

8.1.3 Solution for Linear Strain Hardening The problem of plastic wave propagation is greatly simplified by the assumption of a linear strain-hardening law, which makes the stress–strain curve consist of a pair of straight lines of slopes E and T in the elastic and plastic ranges. The elastic and plastic with constant speeds c0 and cp , respectively, √ waves then propagate √ where c0 = E/ρ and cρ = T/ρ. To illustrate the simplicity of this approach, we consider the same problem as that discussed before, a uniform bar of length l being assumed to strike a rigid wall x = 0 with a velocity U from right to left at time t = 0. For sufficiently large values of U, the elastic and plastic waves are simultaneously generated at x = 0 and are propagated along the length of the bar, the positions of the wave fronts at subsequent times being shown in Fig. 8.3. The solution, which is due to Lensky (1949), has been discussed by Cristescu (1967). During the time interval 0 ≤ t l/c0 , there are three distinct regions in the bar as indicated in Fig. 8.3(a). In the outer region, the material is undisturbed with υ1 = −U and σ1 = ε1 = 0, while in the central region that is traversed by the elastic waves only, we have σ2 = Y, ε2 = Y/E = ε0 (say), and υ2 = −U + c0 ε0 , in view of the jump conditions (8.6). In the region adjacent to the wall, where the material is brought to rest (υ 3 = 0) by the plastic shock wave, the compressive stress and strain are similarly obtained as σ3 = Y + ρcp (U − c0 ε0 ) ,

ε3 = ε0 + (U − c0 ε0 )/cρ .

It follows that the bar will be rendered plastic only if the velocity of impact satisfies the condition U > c0 ε0 . For t > 1/c0 , the elastic wave front moves backward after reflection from the free end of the bar, while the plastic wave front advances further to the right as indicated in Fig. 8.3(b). The reflected wave front completely unloads the region traversed by it, giving σ 4 = ε 4 = 0, and its velocity is reduced in magnitude to υ4 = 2c0 ε0 − U. At time t = ts the reflected elastic wave front meets the advancing plastic wave front at some section S, which is at a distance xs from the wall. Then ts = xs/cρ = 2(2 l − xs )/c0 , giving the relations xs = 2lcp / c0 + cp ,

ts = 2 l/ c0 + cp .

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Longitudinal Stress Waves in Bars

569

Fig. 8.3 Propagation of shock waves in an elastic/plastic bar of finite length striking a rigid target and the associated reflection and interaction of elastic and plastic waves

If the impact velocity U is not too large, the plastic wave will not propagate any further for t > ts , and reflected elastic waves will spread in both directions from S, as shown in Fig. 8.3(c). In the region between the two reflected waves, the particle velocities on both sides of section S must be the same, but the strains in the two portions will be different. By (6), the associated stresses and strains are given by σ5 = ρc0 (υ5 − υ4 ) , σ5 = σ3 − ρc0 υs , ε = (υ5 − υ4 )/c0 , ε5 = ε3 − ((υ5 − υ3 )/c0 ) . Substituting for σ3 , ε 3 , υ 3 , and υ 4 and noting the fact that σ5 = σ’5 to a close approximation for longitudinal equilibrium, we obtain cp 1 , υ5 = U − (U − c0 ε0 ) 3 − 2 c0

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8 Dynamic Plasticity

ε5 = (U − c0 ε0 )

c0 + cp 2c20

,

ε5

= (U − c0 ε0 )

c0 + cp 2c0 − cp 2c20 cp

.

(8.9) If the plastic strain stops at x = xs , the region to the right of section S must remain elastic, which requires ε5 < ε0 . The preceding solution is therefore valid for 1

(3c0 + cp )(c0 + cp ), the advancing plastic wave will be intercepted either by the elastic wave which propagates to the left and is subsequently reflected at x = 0 or by the elastic wave which propagates to the right and is subsequently reflected at x = l. The sequence of events leading to the first of these two possibilities is represented in the characteristic plane shown in Fig. 8.3(e). The reflected wave front propagating from x = 0 meets the plastic wave front at a section R when t = tr , the distance of the section from the rigid wall being x = xr . It is easily shown that xr = 2lcp / c0 − cp ,

te = 2 l/ c0 − cp .

In the case of the second possibility, the total distance traveled by the elastic wave front before it meets the plastic wave front is 2(2l – xs ) – xr , which leads to the relations 2 xr = 4lc0 cp / c0 + cp ,

2 tr = 4lc0 / c0 + cp .

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Longitudinal Stress Waves in Bars

571

In the critical situation, when both the elastic waves meet the plastic waves simultaneously, we have c20 − 4c0 cp − c2p = 0

or

c0 /cp = 2 +

√ 5 ≈ 4.24.

Thus, the leftward moving elastic wave from S meets the plastic wave on reflection when c0 /cp > 4.24, and the rightward moving elastic wave from S meets the plastic wave on reflection when c0 /cp > 4.24. During the time interval ts < t < 2l/c0 , the stress, strain, and velocity in regions 1–4 are identical to those previously given, while in region 5, these quantities are σ5 = Y, ε5 = ε0 , and υ5 = 3c0 ε0 − U. In the remaining regions 6 and 7, the stresses and strains are given in terms of the velocities as σ6 − Y = ρcp (υ6 − υ5 ) , σ7 − σ3 = −ρc0 υ7 , ε6 − ε0 = (υ6 − υ5 )/cp , ε7 − ε3 = −υ7 /c0 . Substituting for υ 5 , σ 3 , and ε3 and using the equilibrium and continuity conditions σ6 ≈ σ7 and υ6 = υ7 , respectively, we obtain the relations υ6 = υ7 =

2c0 cp ε0 , c0 + cp

ε7 =

ε0 c2 + c2p c0 U , ε6 = ε7 −2 − 0 − 1 ε0 . (8.11) cp cp c0 + cp cp

The solution can be continued in a similar manner for any given value of c0 /cp . The final configuration of the bar will contain two stationary discontinuities occurring at x = xs and x = xr , where there are abrupt changes in the cross section. Other examples of longitudinal wave propagation in bars have been discussed by White and Griffis (1947), Rakhmatulin and Shapiro (1948), De Juhasz (1949), Lebedev (1954), Ripperger (1960), Clifton and Bodner (1966), and Cristescu (1970), among others. The problem of combined longitudinal and torsional plastic waves in thin-walled tubes has been discussed by Clifton (1966), Goel and Malvern (1970), Ting (1972), and Wu and Lin (1974). The dynamic plastic behavior of extensible strings has been examined by Craggs (1954) and Cristescu (1964), and discussed at great length by Cristescu (1967).

8.1.4 Influence of Strain-Rate Sensitivity It is well known that the yield stress of engineering materials can be considerably higher under dynamic loading than under quasi-static loading. For example, the dynamic yield stress of annealed mild steel has been found to be more than double the quasi-static yield stress. The relevant experimental evidence has been provided by Duwez and Clark (1947), Campbell (1954), Goldsmith (1960), and Davies and Hunter (1963), along with a number of other investigators. The dynamic elastic modulus is essentially the same as the quasi-static elastic modulus, but the overall

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8 Dynamic Plasticity

dynamic stress–strain curve is generally much higher than the quasi-static curve as shown in Fig. 8.4, which is based on the experimental results of Kolsky and Douch (1962). The magnitude of the strain rate at which a given material begins to be rate sensitive varies from material to material, but rate dependence of the stress–strain curve is an important factor which must be included in the theoretical framework for a realistic prediction of the dynamic behavior (Campbell, 1972).

Fig. 8.4 Comparison of static and dynamic stress–strain curves for (a) copper and (b) aluminum (due to Kolsky and Douch, 1962)

For a work-hardening material having its uniaxial stress–strain curve given by σ = f(ε) under quasi-static conditions, the simplest constitutive equation relating the stress and strain to the rate of straining may be obtained on the assumption that the plastic part of the strain rate is a function of the overstress σ – f(ε), which is the difference between the dynamic and quasi-static yield stresses corresponding to a given strain. Since the elastic part of the strain rate is related to the stress rate by Hooke’s law, the constitutive equation becomes E˙ε = σ˙ + F σ − f (ε) .

(8.12)

The function F(z) must be positive for z > 0 and zero for z ≤ 0. Equation (8.12), which has been proposed by Malvern (1951), is a generalization of one due to Sokolovsky (1948b), who assumed the material to be ideally plastic in the quasi-static state. The expression F(z) = kzn , where k and n are positive constants,

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Longitudinal Stress Waves in Bars

573

has been found to fit experimental data reasonably well by Kukudjanov (1967). Lindholm (1964) has experimentally verified the constitutive relation σ = σ0 (ε) + σ1 (ε) 1n˙ε , where σ 0 (ε) defines the stress–strain curve under a unit strain rate. A different type of constitutive equation based on microstructural considerations has been examined by Steinberg and Lund (1989). In the high-velocity impact or explosive loading, of components, most of the heat generated by the plastic deformation remains in the specimen, causing a thermal softening of the material. Using a Hopkinson pressure bar recovery technique, the isothermal stress–strain curves for materials at high strain rates have been obtained by Nemat-Nasser et al. (1991), who also estimated the effect of the adiabatic rise in temperature on the dynamic response. Their experimental data over the plastic range of strains can be fitted by the constitutive equation m ε˙ T exp −λ −1 σ = σ0 ε 1 + ε˙ 0 T0 n

(8.13)

where σ 0 , ε0 , and T0 are reference values of the stress, strain rate, and temperature, respectively, while m, n, and λ are material parameters. The above equation is very similar to that suggested by Nemat-Nasser et al. (1994) and is found to be sufficiently accurate over the practical range. In general, both the elastic and plastic strain increments would be involved in a prescribed stress increment, whether or not the material is rate sensitive. Considering this fact, a constitutive equation has been proposed by Cristescu (1963) and Lubiner (1965), in which the instantaneous strain rate is expressed in terms of the stress rate in the generalized form E

∂σ ∂ε = φ (σ ,ε) + ψ (σ ,ε) . ∂t ∂t

(8.14)

When the material is not rate sensitive, ψ = 0 and φ is independent of t, while the derivatives in (8.14) may be considered with respect to any monotonically increasing parameter. For a highly rate-sensitive material, on the other hand, it is a good approximation to set φ = 1, and (8.14) then reduces to one of type (8.12). The problem of longitudinal wave propagation in an elastic/plastic bar with an arbitrary material response is therefore governed by (8.14), as well as the relations ∂υ ∂σ =ρ , ∂x ∂t

∂υ ∂ε = , ∂x ∂t

for the three unknowns σ , ε, and υ. The first equation above is the equation of motion, while the second is the equation of compatibility. This system of equations is hyperbolic, as may be shown by considering the variation of σ , ε, and v along any curve in the (x, t)-plane. The conditions under which the first derivatives of these

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8 Dynamic Plasticity

quantities may be discontinuous across the curve furnishes three families of real characteristics (Hopkins, 1968), which are given by dx = 0,

dx/dt = ±c,

c=

E/ρφ,

(8.15)

where c denotes the wave speed which is generally much higher than that given by (8.1). The differential relations holding along the characteristic curves dx/dt = ±c are easily shown to be dσ = ±ρcdυ − (ψ/φ) dt,

(8.16a)

where the upper and lower signs correspond with those in (8.15). The relationship that holds along the remaining family of characteristics dx = 0 is found to be Edε = φdσ + ψdt.

(8.16b)

Across a stationary wave front dx = 0, only ∂ε/∂x may be discontinuous, while all other first-order derivatives of σ , ε, and υ must be continuous. This is easily established from the condition of continuity of these quantities across the wave front, and the fact that dε = (∂ε/∂t) dt and so on along this line. The continuity of the time derivatives ∂ε/∂t and ∂υ/∂t implies the continuity of ∂υ/∂x and ∂σ/∂x, but no information is available for ∂ε/∂x which may therefore be discontinuous. Across a moving wave front defined by the second relation of (8.15), all the first-order derivatives of σ , ε, and υ may become discontinuous. The method of numerical integration of (8.16a) and (8.16b) along the characteristics has been expounded by Cristescu (1967). In the special case when ψ = 0 and φ is independent of t, the results are similar to those for the rate-independent material, the only difference being the existence of a third family of characteristics arising from the differential form of the constitutive relation. In the other extreme case√of φ = 1, the plastic wave speed becomes identical to the elastic wave speed c0 = E/ρ, while the characteristic lines (which are straight) and the differential relations along them are still given by (8.15) and (8.16) with c = c0 . It may be noted that the first term on the right-hand side of (8.13), when φ = 1, represents the elastic strain increment, and the second term represents the plastic strain increment. The stress and elastic strain increase as a result of the wave propagation, causing the plastic strain to increase along the stationary characteristics. More refined constitutive equations for the propagation of one-dimensional plastic waves with strain rate effects have been examined by Cristescu (1974). Generalized constitutive equations for viscoplastic solids, from the standpoint of continuum mechanics, have been presented by Perzyna (1963) and Mandel (1972). Other types of constitutive equations for the dynamic behavior of materials, based on the dislocation motion, have been examined by Nemat-Nasser and Guo (2003).

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8.1.5 Illustrative Examples and Experimental Evidence Some quantitative results for the plastic wave propagation in a rate-sensitive material have been given by Malvern (1951), who discussed an example in which the end x = 0 of a semi-infinite bar lying along the positive x-axis is instantaneously pulled with a longitudinal velocityv = −U at t = 0. The constitutive law adopted by Malvern corresponds to (8.14) with ψ = k σ − f (ε) ,

φ = 1,

f (ε) = Y (2 − Y/Eε)

(8.17)

where k is a material constant. The numerical computation is based on the values k = 106 /s, Y/E = 10–3 , E = 70 GPa, c = c0 = 5 km/s, and U = 15 m/s. The predicted dynamic yield stress for this material is about 10% higher than the quasi-static value for a constant strain rate of 200/s. Along the leading wave front x = c0 t, the stress and strain are not constant in the solution with strain rate effect, which allows the stress to rise above Y without exceeding the elastic limit. The stress attained in an element just after the passage of the leading wave front is given by dσ = −

k (σ − Y)2 dt 2σ

in view of (8.16a) and (8.17), together with the relations Eε = σ and ρc0 υ = −σ which hold along this shock wave. Integrating, and using the initial condition σ = ρc0 U at t = 0, which follows from (8.6b), we obtain the solution t=

2 k

(ρc0 U − Y) (ρc0 U − σ ) Y + 1n . (σ − Y) (ρc0 U − Y) (σ − Y)

(8.18)

It may be noted that ρc0 U = 3Y according to the assumed numerical data. Following the impulsive loading, the stress jumps instantaneously from 0 to 3Y, but rapidly falls to a value very nearly equal to Y as the wave propagates along the bar. If, on the other hand, the strain rate effect is disregarded, the stress would have a constant value Y all along the leading wave front. The solid curves in Fig. 8.5(a) represent the lines of constant strain in the (x, t)plane according to rate-dependent solution, while the broken straight lines passing through the origin correspond to the solution that neglects strain rate effects. The region of constant strain, that exists in the rate-independent solution above the wave front of the maximum strain ε = 0.0074, disappears in the rate-dependent solution, which predicts a gradual transition from plastic to elastic response across the chaindotted line. Along this unloading boundary, the quasi-static stress–strain relation σ = f(ε) holds, where σ and ε slowly decrease with increasing x and t. The stress–strain relations at various sections of the bar are displayed in Fig. 8.5(b), where the broken curve represents the quasi-static stress–strain relation. It is seen that the plastic deformation occurs with decreasing stress in sections closer to the impact end x = 0, and with increasing stress in sections farther from x = 0. This is a

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Fig. 8.5 Calculated results for the plastic wave propagation along a long bar made of rate-sensitive material. (a) Lines of constant strain and (b) stress–strain relations at different sections of the bar (after L.E. Malvern, 1951)

consequence of the high initial stress instantaneously attained at x = 0, and the fact that the strain rate at any section rapidly decreases with time. At sections sufficiently far away from the end x = 0, the stress–strain curves resemble the quasi-static curve. The strain distribution shown in Fig. 8.6 (a) indicates that the plateau of constant strain near the impact end disappears, which is contrary to that experimentally observed (Bell, 1968). The existence of the strain plateau can be theoretically predicted, however, by using a suitable modification of the constitutive equation. The plastic wave propagation in a bar of finite length, initially prestressed to the yield point with a longitudinal stress σ = Y, has been treated by Cristescu (1965), using the relations (8.17) with the same material constants as those employed in the preceding example. The bar is assumed to have an initial length l = 7.5 cm with one end fixed and the other end prescribed to move with a velocity υ = −Ut/t0 (0 < t ≤ t0 ) , υ = −U (t ≥ t0 ) , where U = 30 m/s and t0 = l2 μs. The solution has been obtained numerically by using the finite difference form of the differential relations (8.16) along the characteristics. Some of the computed results are displayed in Fig. 8.7(a) and (b), which exhibit the same general trends as those in the previous example. There is, of course, no strain plateau near the impact end of the bar.

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Longitudinal Stress Waves in Bars

577

Fig. 8.6 Strain rate effects on the longitudinal plastic wave propagation. (a) Strain distribution at t = 102.4 μs and (b) stress–time relation at a distance x = 16.2 cm from impact end

Fig. 8.7 Results for a bar of finite length subjected to a prescribed velocity at x = 0. (a) Time dependence of physical quantities at x = 3 cm and (b) dynamic stress–strain relations (after N. Cristescu, 1967)

578

8 Dynamic Plasticity

Several other problems involving the strain rate effect have been discussed by Cristescu (1972), Banerjee and Malvern (1975), and Cristescu and Suliciu (1982). The problem of plastic wave propagation due to the longitudinal impact of two identical bars, one of which is at rest and the other one fired from an air gun, has been considered by Cristescu and Bell (1970). The effects of strain rate history on the dynamic response have been examined by Klepaczko (1968) and Nicholas (1971). The application of the endochronic theory to the propagation of one-dimensional plastic waves has been discussed by Lin and Wu (1983). One of the earliest experimental investigations on the propagation of plastic waves, reported by von Karman and Duwez (1950), consisted in measuring the longitudinal strain in annealed copper wires subjected to impact loading by means of a drop hammer. Although strain rate effects were disregarded in their theoretical calculations, the overall agreement between theory and experiment was found to be reasonably good. Sternglass and Stuart (1953) applied small amplitude strain pulses to prestressed copper wires to study the propagation of plastic waves. The speed of propagation of the wave front was found to be that of elastic waves, and the wave speed of any part of the pulse was found to be much higher than the value given by the rate-independent theory. Further experimental supports for the rate-dependent theory have been provided by Alter and Curtis (1956), Malyshev (1961), Bianchi (1964), Dillons (1968), and others, and the results obtained by them were consistent with the differential constitutive law of type (8.12). Critical reviews of the published experimental work on the plastic wave propagation have been reported by Bell (1973), Clifton (1973), and Nicholas (1982). The development of plastic strains with time at various distances from the impact end of bars, made of mild steel and aluminum, has been experimentally determined by Bell (1960) and Bell and Stein (1962), using a diffraction grating method. The split Hopkinson pressure bar has been used by Lindholm (1964), following a method proposed by Kolsky (1949), to establish the dynamic properties and stress–strain curves for lead, aluminum, and copper under different values of the rate of strain. An electromagnetic method of measurement of the particle velocity at any section of the bar during the propagation of plastic waves along its length has been discussed by Malvern (1965) and Efron and Malvern (1969). Both these methods are quite accurate and can be used for obtaining the overall dynamic response from the moment of impact to the moment when the unloading begins. Further references to the theoretical and experimental investigations on the plastic wave propagation have been given by Nicholas (1982).

8.2 Plastic Waves in Continuous Media 8.2.1 Plastic Wave Speeds and Their Properties Consider an isolated wave front, not necessarily plane, which is advancing through an elastic/plastic material that work-hardens isotropically without exhibiting strain rate effects. We shall be concerned here with the propagation of weak waves across

8.2

Plastic Waves in Continuous Media

579

which discontinuities in the derivatives of stress, strain, and particle velocity may exist. Since the governing equations are quasi-linear, the weak waves are necessarily characteristics of the hyperbolic system. Any physical quantity in the immediate neighborhood of the wave front may be regarded as a function of the surface coordinates, the surface normal, and time t. Since the surface derivatives of a typical variable f must be continuous across the wave front, which has a velocity c in the direction of the normal n, the jump in the space and material derivatives of f across the wave front may be written as ∂f ∂f ∂f ni c, (8.19) = f˙ = − ∂xi ∂n ∂n where ni denotes the unit vector in the direction of n. The second expression follows from the fact that the rate of change of f relative to the motion of the wave front in the xi space is continuous, since f itself must be continuous. Each pair of square brackets in (8.19) will be taken to represent the value of the enclosed quantity behind the wave front in excess of that ahead of the front. In the analysis of the problem of plastic wave propagation, all rotational effects will be disregarded. Consequently, the stress rate entering into the constitutive equation will be taken as the material derivative of the true stress σ ij . If the material is prestressed in the plastic range, and obeys the von Mises yield criterion and the associated Prandtl–Reuss flow rule, the constitutive equation may be written in the form of equation (1.38), where nij = σ˙ ij = 2G ε˙ ij +

3 σ 2 sij/¯

The stress rate is therefore given by

ν 3α , ε˙ kl skl ≥ 0, ε ˙ s s ε˙ kk δij − kl kl ij 1 − 2ν 2σ¯ 2

(8.20)

where G is the shear modulus, ν is Poisson’s ratio, σ¯ is the equivalent stress, sij is the deviatoric stress tensor, and α = 3G/(H + 3G) with H denoting the current plastic modulus. In the case of unloading, indicated by ε˙ kl skl < 0, it is only necessary to set α = 0 in (8.20). Introducing the expression ∂υj 1 ∂υi ε˙ ij = + 2 ∂xj ∂xi for the true strain rate, where υ i denotes the particle velocity, (8.20) may be expressed as 2ν ∂υk ∂υi ∂υi 3α ∂υk σ˙ ij = G + + ∂ij − 2 skl sij . ∂xj ∂xi 1 − 2ν ∂xk σ¯ ∂xl Admitting possible discontinuities in the stress rate and the velocity gradient across the wave front, and using (8.19), we obtain the discontinuity relation ∂σij 2ν 3α λk nk δij − 2 λk nl skl sij , = G λi nj + λj ni + −c ∂n 1 − 2ν σ¯

580

8 Dynamic Plasticity

where λi , denotes the quantity ∂υi /∂n . Multiplying the preceding equation by nj following the summation convention, and using the relations σij nj = σi ,

sij nj = si ,

nj nj = 1,

where si and σ i are the deviatone and actual stress vectors acting across the wave front, we get −c

3α ∂σi = G λi + (1 − 2ν)−1 λj nj ni − 2 λk sk si . ∂n σ¯

(8.21)

The scalar parameter λk sk , which represents the discontinuity in the deviatoric work rate, must be positive for continued loading, in view of the sign convention for λi . When λk sk is negative, the work rate in the element is decreased by the passage of the wave front, and unloading would occur as a result. In order to complete the analysis, we must consider the equation of motion of the material element. Denoting the density of the material by ρ, which may be assumed constant since only elastic changes in volume are involved, the equation of motion may be written as ∂σij − ρ υ˙ i = 0. ∂xj

(8.22)

Using (8.19), the discontinuity relation corresponding to (8.22) is readily obtained, and the substitution σij nj = σi then results in

∂σi + ρcλi = 0. ∂n

The elimination of the stress gradient discontinuity ∂σi /∂n between this equation and (8.21) furnishes the connection equation (Jansen et al., 1972) as

c2 − c22 λi − c21 − c22 λj nj ni + c22 α/k2 λk sk si = 0,

(8.23)

√ where k = σ/ ¯ 3 is the current yield stress in shear, and c1 , c2 are the speeds of propagation of elastic dilatational and shear waves, given by c21

=

1−ν 1 − 2ν

2G , ρ

c22 =

G ρ

Equation (8.23) constitutes a set of three linear homogeneous equations in the unknown component of the discontinuity vector λi . The existence of nontrivial solutions for these unknowns requires the determinant of their coefficients in these equations to vanish. The result is a cubic equation for c2 having three distinct roots (Craggs, 1961; Mandel, 1962). One of these roots is obtained by a mere inspection of (8.23), the corresponding solution being

8.2

Plastic Waves in Continuous Media

c2 = c22 ,

581

λj nj = 0,

λk sk = 0.

The wave front therefore advances at the speed of the elastic shear wave. The associated discontinuity in the velocity gradient vector is tangential to the wave front and is perpendicular to the plane containing the deviatoric stress vector and the normal to the wave front. The other two roots of c2 are most conveniently obtained by forming the scalar products of (8.23) with the vectors ni and si , in turn. Setting λ j n j = λn ,

si ni = sn ,

λk sk = ω,

and denoting the magnitude of the deviatoric stress vector by s, the two scalar equations involving the unknown quantities λn and ω are found as ⎫

⎪ ⎬ c2 − c21 λn + c22 α/k2 sn ω = 0,

⎭ − c21 − c22 sn λn + c2 − c22 1 − αs2 /k2 ω = 0.⎪

(8.24)

If τ denotes the magnitude of the shear stress transmitted across the wave front, then s2 = s2n + τ 2 , and the two equations in (8.24) will be simultaneously satisfied for nonzero values of λn and ω if

c2 − c22

c2 − c21 + c22 c2 − c22 αs2n /k2 + c2 − c21 ατ 2 /k2 = 0 (8.25)

This equation indicates that there are two real roots for c2 , one of which lies between 0 and c22 , and the other between c22 and c21 . The former corresponds to slow waves and the latter to fast waves, their speeds of propagation being denoted by cs and c f, respectively. The roots of the above quadratic may be expressed as ⎫

√ 1 2 ⎬ c2f ,c2s = c1 + c22 − c22 αs2 /k2 ± n , ⎪ 2

⎪ η = c21 − c22 − c22 αs2 /k2 + 4c22 c21 − c22 ατ 2 /k2 .⎭

(8.25a)

In the special case of an elastic material (α = 0), the wave speeds cf and cs reduce to c1 and c2 , respectively, the discontinuities in the velocity gradient in the two cases being normal and tangential, respectively, to the wave front. The related problem of acceleration waves in solids has been studied by Thomas (1961) and Hill (1962).

8.2.2 A Geometrical Representation For an elastic/plastic material, a useful geometrical interpretation of the wave speed equation (8.25) is obtained by writing it in the alternative form

582

8 Dynamic Plasticity

c21 c21 − c2

αs2n + k2

c22 c22 − c2

ατ 2 =1 k2

(8.26)

√ √ This relation indicates that if αsn and ατ are plotted as rectangular coordinates, the locus of constant wave speed is an ellipse when c = cs and is a hyperbola

when c = cf . The semifocal distance in each case is equal to k c21/c22 − 1, and consequently, the family of ellipses is orthogonal to the family of hyperboles, as shown in Fig. 8.8 Moreover, it followsfrom (8.25a) that the sum of the squares of cs and cf is constant along any circle α s2n + τ 2 = constant, having its center at the origin of the stress plane (Ting, 1977).

Fig. 8.8 Elastic/plastic wave speeds in a continuous medium and their dependence on the deviatoric normal and shear stresses acting across the wave front

For a given deviatoric stress tensor sjj and the direction of propagation ni , the normal and tangential components of the deviatoric stress vector acting across the wave front can be found from the relations sn = sij ni nj ,

s2 = sij sjk ni nk , τ 2 = s2 − s2n

The plastic wave speeds then follow from (8.25a), and the associated directions of the velocity gradient discontinuity are given by (8.23). When the deviatoric stress vector is tangential to the wave front (sn = 0, s = τ ), (8.25a) furnishes

8.2

Plastic Waves in Continuous Media

c2f = c21 ,

583

c2s = c22 1 − ατ 2 /k2 .

Since ω vanishes for the fast wave and λn vanishes for the flow wave in view of (8.24), the discontinuities in the velocity gradient in the two cases are directed along the normal and tangent to the wave front, respectively. When the deviatoric stress vector is normal to the wave front (τ = 0), the wave propagates along a principal axis of the stress, and we have c2s = c22 ,

c2f = c21 − c22 αs2n /k2

αs2n /k2 ≤ c21 /c22 − 1.

It follows from (8.23) and the first equation of (8.24) that the directions of the velocity gradient discontinuity again coincide with the normal and tangent to the wave front for the fast and slow waves, respectively. When the above inequality is reversed, the values of cf and cs and the associated discontinuity directions are simply interchanged. When the rectangular axes of reference are taken along the principal axes of the stress, the deviatoric state of stress at any point can be represented by the appropriate Mohr circle. If the deviatoric principal stresses are denoted by s1 , s2 , and s3 , with s2 assumed to lie between s1 and s3 , we have the identity s1 + s2 + s3 = 0,

s1 > s2 > s3 .

Evidently, s1 > 0 and s3 < 0, while s2 can be either positive or negative. For given values of s1 , s2 , and s3 , the stress point (sn , τ ) lies within a region bounded by three circles, the largest of which has the equation sn −

s1 + s2 2

2 + τ2 =

s1 − s2 2

2 .

(8.27)

The two principal components s1 and s3 appearing in this equation cannot be chosen arbitrarily, since they are required to satisfy the yield criterion which may be expressed as 3 (s1 + s3 )2 + (s1 − s3 )2 = 4 k2 This restriction makes the family of Mohr’s circles (8.27) bounded by an envelope, which is easily shown to be an ellipse (Ting, 1977) having the equation 3 s n 2 τ 2 + = 1. 4 k k

(8.28)

The broken curve in Fig. 8.8 represents this ellipse for a typical value of α. The focal points A and B of the ellipse defined by (8.26) with c = cs lie inside or outside the defined by (8.28) depending on whether 4α/3 is greater or smaller than 2 ellipse c1/c22 − 1 . The situation cf = cs = c2 can arise only when the former condition is satisfied.

584

8 Dynamic Plasticity

The smallest value of c2f , which corresponds to τ = 0, is equal to c22 when the focal points are inside the envelope, and to c21 − 43 α22 when they are outside the envelope. The smallest value of c2s , on the other hand, corresponds to the point of tangency of the largest possible Mohr circle with the appropriate ellipse of constant cs . It can be obtained as a function of s2 by solving (8.26) and (8.27) simultaneously for sn and τ , after setting c = cs ,s1 + s3 = −s2 , (s1 − s3 )2 = 4 k2 − 3s22 , and establishing the condition for the quadratics to have equal roots. When s2 = 0, the smallest value of c2s is equal to (1 − α) c22 , corresponding to sn = 0.

8.2.3 Plane Waves in Elastic/Plastic Solids The preceding results are directly applicable to the propagation of plane waves in which the wave front is a plane surface advancing in a uniformly prestressed elastic/plastic medium. When the state of stress is a pure shear in which the normal stress vanishes across the plane (sn = 0,τ = k), the normal and tangential√components of the velocity gradient discontinuity propagate with speeds c1 and c2 1 − α respectively, in the direction of the normal to the plane. The strength of the former remains unchanged, while that of the latter steadily decreases as the plastic √ strain increases. In the case of a uniaxial prestress normal to the plane (sn = 2 k/ 3,τ = 0), the discontinuities in the tangential and normal velocity gradients propagate with speeds

c2 and c21 − 4αc22/3, respectively. The latter wave speed depends on the value of α, which increases with increasing plastic strain. Consider now the rectilinear propagation of a plane wave in an isotropic elastic/plastic body, which is in the form of a thick plate whose lateral dimensions are infinitely large. The wave travels through the thickness of the plate along the x-axis, the kinematical restrictions being ε˙ y = ε˙ z = 0,

ε˙ y e = −˙εy p =

1 p ε˙ x 2

which follow from the relation σ˙ y = σ˙ z holding throughout the body. The elastic stress–strain relations therefore give the elastic and plastic parts of ε˙ x E˙εx e = σ˙ x − 2ν σ˙ y ,

E˙εx p = 2 (1 − ν) σ˙ y − ν σ˙ x

Combining these two relations, the total rate of extension in the x-direction is expressed as E˙εx = (1 − 2ν) σ˙ x + 2σ˙ y

(8.29)

The material is assumed to be rate sensitive, with the plastic part of the strain rate satisfying the quasi-linear constitutive equation

8.2

Plastic Waves in Continuous Media

585

E˙εxp = φ1 σ˙ x + φ2 σ˙ y + ψ, where φ 1 and φ 2 are functions of stress and strain, and ψ is a function of the ρ dynamic overstress. In view of the expression for ε˙ x given above, the preceding relation becomes (2ν + φ1 ) σ˙ x − 2 (1 − ν) − φ2x σ˙ y + ψ = 0 The elimination of σ˙ y between (8.29) and the preceding equation leads to the relationship between σ˙ x and ε˙ x in the form 2 (1 − ν) − φ2x E˙εy = (1 − 2ν) {[2 (1 + ν) + 2φ1 − φ2 ] σ˙ x + 2ψ}

(8.30)

The characteristics of this hyperbolic system are given by dx/dt = ±c, where c is the speed of propagation of the wave. Along the characteristics, we have σ˙ x =

∂σx ∂σx ∂υx ∂υx = ±c = ±ρc = ±ρc2 = ρc2 ε˙ x . ∂t ∂x ∂t ∂x

It should be noted that ∂σx/∂t = σ˙ x when geometry changes are disregarded. Since ε˙ x and σ˙ x are not uniquely determined along the characteristics, the preceding two relations involving ε˙ x and σ˙ x furnish the result c2 =

E , ρ (1 − 2ν) (1 + 2λ)

λ=

2ν + φ1 2 (1 − ν) − φ2

(8.31)

The range of values of φ 1 and φ 2 for which the wave speed is real must satisfy the conditions φ2 = 2 (1 − ν) , φ2 − 2φ1 < 2 (1 + ν) as well as those obtained by reversing these inequalities. It is easy to see that ν/(1 − ν) < λ < 1,

K/ρ < c < c1

where K is the bulk modulus for the material. When φ 1 = φ 2 = 0, there is no instantaneous plastic strain in the material response, and (8.31) gives c = c1– . When ψ = 0, implying the absence of strain rate effects, the constitutive law gives φ1 = −φ2 = E/H, and the plastic wave speed given √ by (8.31) coincides with the value of cf given by (8.25a) with τ = 0 and s = 2k/ 3, where k is the yield stress in shear. In the (φ 1 , φ 2 )-plane, λ (and hence c) remains constant along straight lines passing through the point [−2v,2 (1 − v)], where the wave speed is indeterminate. It may be noted that the relation c = c1 holdsall along the straight line vφ2 +(1 − v) φ1 = 0, not merely at φ1 = φ2 = 0. In the σ˙ x ,σ˙ y -plane, (8.30) represents a straight line for given values of φ1 ,φ2 , and ψ. The slope of this straight line is equal to λ, and it intersects the axis σ˙ y = 0 at a point that depends on φ 1 . No plastic flow is possible

586

8 Dynamic Plasticity

along the straight line (1 − ν) σ˙ y − ν σ˙ x = 0, which therefore represents elastic unloading. The propagation of plane waves excluding the strain rate effects, but including the variation of the bulk modulus with the hydrostatic pressure, has been considered by Morland (1959), who also studied the interaction of loading and unloading waves when a pressure pulse is applied on the free surface. The propagation of cylindrical waves in an infinite medium has been discussed by Cristescu (1967) under torsional loading and by Jansen et al. (1972) under radially symmetric loading. The propagation of spherically symmetric waves in an unbounded medium has been investigated by Hunter (1957) and Hopkins (1960).

8.3 Crumpling of Flat-Ended Projectiles One of the simplest methods of studying the effect of high strain rates on the dynamic yield strength of metals consists in firing flat-ended cylindrical projectiles against rigid targets. The axial stress at the impact end immediately attains the yield limit, and a plastic wave moves away from the target plate rendering the projectile partially plastic. An elastic wave front initiated at the same time moves ahead of the plastic wave front, the region between the two wave fronts being stressed to the yield point. Due to the reflections of the elastic wave front from the free end of the bar and the advancing plastic wave front, the rear part of the projectile rapidly decelerates and comes to rest within a distance equal to the difference between the initial and final lengths of the cylinder. The extent of the deformed and undeformed portions of the projectile after impact depends on its kinetic energy before impact, as well as on the dynamic yield strength of the material.

8.3.1 Taylor’s Theoretical Model Let U denote the velocity of normal impact of a cylindrical projectile having an initial length L and an initial cross-sectional area A0 . At a generic stage of the dynamic process, the overall length of the projectile is reduced to l due to the piling up of material over a length h, leaving a nonplastic rear part of length x as shown in Fig. 8.9(b). For simplicity, the material is assumed to have a constant dynamic yield stress Y, and the radial inertia is disregarded so that the stress distribution may be considered as uniform over any given cross section (Taylor, 1948b). Within a small time interval dt, a nonplastic element of length –dx and area A0 passes through the advancing plastic boundary to come to rest as a plastic element of length dh and area A. Neglecting elastic strains, the continuity equation may be written as A dh =–A0 dx, which is equivalent to Aυ = A0 (u + υ) ,

(8.32)

8.3

Crumpling of Flat-Ended Projectiles

587

Fig. 8.9 Deformation of a flat-ended projectile fired at a speed U against a flat rigid target

where u is the current velocity of the rear part of the projectile, and υ is the velocity of the plastic boundary, the velocity of the nonplastic part relative to the plastic boundary being u + υ, Thus υ=

dh , dt

u+υ =−

dx . dt

(8.33)

The momentum of the elemental volume –A0 dx changes from −ρA0 udx to zero during the time dt, where ρ is the density of the material. Since the net force acting on the element is of magnitude (A–A0 )Y, the equation of motion becomes ρA0 (u + υ)u = (A − A0 ) Y.

(8.34)

Equations (8.32) and (8.34) are sufficient to express ρu2 /Y and υ/u in terms of a variable e = 1 − (A0/A), the result being e2 ρu2 = , Y 1−e

υ 1−e = . u e

(8.35)

To obtain the expressions for x and h at any instant, we consider the equation of motion of the rear portion of the projectile, which moves as a rigid body with a velocity u under an opposing force equal to A0 Y. Since the change in momentum of this portion is ρA0 xdu during the time interval dt, the equation of motion is ρx

du = −Y. dt

(8.36)

Combining this equation with the second equation of (8.33), and using (8.35), we obtain the differential equation ρux dx = du Ye

or

dx x (2 − e) . = de 2 (1 − e)2

Let e0 be the value of e at the moment of impact when x = L. Using this initial condition, the last equation is readily integrated to give

588

8 Dynamic Plasticity

x 2 L

=

1 − e0 1−e

e − e0 exp . (1 − e0 ) (1 − e)

(8.37)

The quantity e0 depends on the impact velocity U according to the first equation of (8.35) with e = e0 and u = U. When the projectile is brought to rest, e = 0 and x = x∗ (say), the relationship between x∗ /L and ρU2 /Y being obtained as

x∗ L

eo , = (1 − e0 ) exp − 1 − e0

e20 ρU 2 , = Y 1 − eo

(8.38)

in view of (8.37) and (8.35). It may be noted that x∗ /L decreases and e0 increases as the parameter ρU2 /Y is increased. The shape of the deformed part of the projectile at any instant can be determined by the integration of the equation dh υ =− = − (1 − e) , dx u+υ which is obtained from (8.33) and (8.35). In view of the initial conditions h = 0 and e = e0 at x = L, the solution may be written as h x = (1 − e0 ) − (1 − e) + L L

eo

x

e

L

de.

(8.39)

The integral is evaluated numerically for any given value of e0 and selected val√ ues of e, using (8.37) for x/L. Each value of e furnishes a radius a = a0/ 1 − e of the deformed part corresponding to a distance h from the target plate, where a0 is the initial radius of the cylinder. The shapes of the projectile after impact, predicted by the present theory for e0 = 0.5, 0.7 and e = 0.8, are shown in Fig. 8.9(c)–(e) for the case where the diameter was initially 0.3 of the height. The calculated value of h∗ /L and l∗ /L, where h∗ is the final length of the deformed part and l∗ is the final overall length, is plotted against ρU 2/Y in Fig. 8.10, which includes some experimental points obtained by Whiffen (1948). To determine how the various physical quantities vary with time t, measured from the beginning of the impact, it is necessary to integrate the differential equation for de/dt, which is most conveniently obtained from (8.36) and the first equation of (8.35). Indeed, the time derivative of the latter equation gives

2−e ρ du de = . 3/2 Y dt 2 (1 − e) dt

√ Eliminating du/dt by means of (8.36), and substituting for Y/ρ obtained by setting u = U and e = e0 in the first equation of (8.35), we obtain the differential equation de 2U =− dt x

√ 1 − e0 (1 − e)3/2 . eo 2−e

8.3

Crumpling of Flat-Ended Projectiles

589

Fig. 8.10 Results of calculation for the longitudinal 8impact of a flat-ended projectile. The measures values of l∗ /L and h∗ /L are indicated by • and . respectively

Since e = e0 at the moment of impact t = 0, the time interval t can be found numerically from the relation

Ut eo =√ L (1 − eo )

eo e

x (2 − e) de 2L (1 − e)3/2

,

(8.40)

where x/L is given by (8.37) as a function of e and e0. The parameter Ut/L, computed by numerical integration for the cases e0 = 0.5 and 0.7, is plotted against h/L in Fig. 8.11, which indicates how the plastic boundary moves away from the target plate with the time interval. The same problem has been analyzed by Lee and Tupper (1954) on the basis of the elastic and plastic waves in the projectile, taking into account the strain hardening of the material. The Taylor anvil test has been used to establish the constitutive modeling of materials by Johnson and Holmquist (1988) and Nemat-Nasser et al. (1994).

590

8 Dynamic Plasticity

Fig. 8.11 Advancement of the plastic boundary during the longitudinal impact of a flat-ended projectile

8.3.2 An Alternative Analysis The preceding theory, developed by Taylor (1948b), has been found to be in reason able agreement with experiment for relatively low-impact velocities ρU 2/Y ≤ 0.5 . The shapes of the slugs fired at greater speeds generally have a concave profile over the plastically strained part, instead of a convex profile predicted by Taylor’s theory. To explain this mushrooming effect associated with the high-speed impact of projectiles, Hawkyard (1969) proposed an alternative method in which the rate of plastic work done on the projectile is equated to the total external energy supplied to it. The resultant equations defining the final geometry of the projectile are fairly simple, and the predicted profiles are found to be in closer agreement with experiment. The amount of plastic work dW, which is done on an element of length –dx in changing its cross-sectional area from A0 to A before it comes to rest, is equal to −A0 Y ln (A/A0 ) dx. Since the time taken for this change is dt, the rate of plastic work done is ˙ = A0 Y (u + υ) 1n (A/A0 ) W in view of (8.33). During this time interval, the kinetic energy of the element is reduced from 12 ρu2 (−A0 dx) to zero, while the work done by the external force becomes equal to −A0 Y (dx + dh). Hence, the rate at which the total energy is

8.3

Crumpling of Flat-Ended Projectiles

591

supplied to the system is ρu2 . E˙ = A0 Y u + (u + υ) 2Y Neglecting the loss of kinetic energy due to impact, we may equate the rate of ˙ to the rate of supply of energy E˙ to give plastic work W

ρu2 A (u + υ) 1n − A0 2Y

= u.

(8.41)

This equation replaces (8.34) obtained on the basis momentum equilibrium. ( of Solving (8.32) and (8.41), and setting e = 1 − A0 A as before, we obtain the relations ρu2 1 = 1n − e, 2Y 1−e

1−e υ = . u e

(8.42)

The equation of motion (8.36) for the undeformed part of the projectile is now combined with the time derivative of the first equation of (8.42) to give ρu de = dt Y

1−e e

1−e u du =− . dt e x

(8.43)

In view of the second equation of (8.42), (8.43) can be combined with (8.33) to give dx x = , de 1−e

dh = −x. de

Integrating these two equations in succession, and using the initial conditions x = L and h = 0 when e = e0 , we obtain the solution x 1 − e0 = , L 1−e

1−e h . = (1 − e0 ) 1n L 1 − e0

(8.44)

The values of x and h at the end of the impact, denoted by x∗ and h∗ , respectively, are given by (8.44) on setting e = 0. The impact velocity U is related to e0 by the equation 1 ρU 2 − eo, = 1n 2Y 1 − e0 which follows from (8.42). The calculated shape of the projectile after impact, corresponding to e0 = 0.8 ρU 2 /Y = 1.62 according to (8.44), is shown in Fig. 8.9(f). The profile is seen to be concave in form, resembling what is experimentally observed. The calculated values of h∗ /L and l∗ /L predicted by the above analysis

592

8 Dynamic Plasticity

are compared with those given by Taylor’s theory in Fig. 8.10. The influence of strain hardening of the material has also been examined by Hawkyard (1969). The variation of the strain parameter e with time can be determined by the integration of (8.43) after substitution for u/U given by (8.42), the final differential equation being U (1 − e)2 de =− dt L e (1 − e0 )

'

e + 1n (1 − e) . eo + 1n (1 − e0 )

In view of the initial condition e = e0 at t = 0, the solution may be written as Ut = (1 − e) L

eo

'

e

eo + 1n (1 − eo ) ede . e + 1n (1 − e) (1 − e)2

(8.45)

The integral can be evaluated numerically for any given value of e0 . The results corresponding to e0 = 0.5 and e0 = 0.7 are compared in Fig. 8.11 with those predicted by Taylor’s theory. The two solutions do not seem to differ substantially from one another, though the shape of the projectile is predicted more ( realistically by the analysis based on the energy principle, particularly for ρU 2 Y ≥ 0.5.

8.3.3 Estimation of the Dynamic Yield Stress The measurement of the final overall length and the position of the plastic boundary, after the impact, provides a convenient means of estimating the dynamic yield stress of the material. For practical purposes, a simple formula for the yield point Y can be developed on the assumption that the velocity of the undeformed part of the projectile ( relative to the advancing plastic boundary has a constant magnitude c. Then dx dt = −c in view of (8.33), and the equation of motion (8.36) gives Y du = dx ρcx

or

u=U−

L Y ln , cρ x

(8.46)

in view of the condition u = U when x = L. If the distance traveled by the rear of the projectile at any instant is denoted by du/dt = u (du/ds), then (8.36) gives u

cρ du Y Y =− =− exp − (u − U) ds ρx ρL Y

in view of (8.46). Integrating, and using the conditions u = U when s = 0, and u = 0 when s = L − l∗ , we obtain the solution ρc2 Y

1−

l∗ L

=

cρU cρU − 1 + exp − . Y Y

8.4

Dynamic Expansion of Spherical Cavities

593

The parameter cρU/Y is equal to ln (L/x∗ ) in view of (8.46) with u = 0 and x = x∗ . The elimination of c from the above relation therefore gives Y ln (L/x∗ ) − (1 − x∗ /L) = 2 . ρU 2 (1 − l∗ /L) ln (L/x∗ )

(8.47)

If the deceleration of the rear of the projectile is assumed uniform, the formula for Y/ρU 2 would become that given by Taylor. Equation (8.47) should be sufficiently accurate for the estimation of the dynamic yield stress Y for a given impact velocity U, using the measured values of x∗ /L and l∗ /L. The experimental results of Whiffen (1948) and Hawkyard (1969) indicate that the values of Y computed from (8.47) for different sets of values of U, x∗ /L and l∗ /L are approximately the same for a given material. It is possible to define a mean strain rate as the ratio of the overall longitudinal plastic strain to the duration of the impact. If the rear portion of the projectile is assumed to move with a constant deceleration, the duration of impact is equal to 2(L — l∗ )/U by the simple kinematics of the rigid-body motion. Since the initial and final lengths of the plastically deformed part of the projectile are L – x∗ and l∗ – x∗ , respectively, the mean strain rate λ˙ may be written as λ˙ =

U L − x∗ ln . 2 (L − l∗ ) l∗ − x ∗

(8.48)

The value of λ˙ in each particular case may therefore be obtained from the direct measurement of the final undeformed length and the final overall length of the projectile. There seems to be very little variation of the duration of impact with the impact velocity, although there are fairly large variations in the other physical quantities. The buckling that would occur in a sufficiently long projectile due to the impact has been examined by Abrahamson and Goodier (1966) and Jones (1989). The related problem of dynamic plastic buckling of columns has been treated by Lee (1981).

8.4 Dynamic Expansion of Spherical Cavities The formation of spherically symmetric cavities in an infinitely extended medium under quasi-static conditions, originally discussed by Bishop et al. (1945), has been presented elsewhere (Chakrabarty, 2006). In this section, the problem of spherical cavity formation under dynamic conditions will be discussed for a material which obeys an arbitrary regular yield condition. The pressure applied at the cavity surface is supposed to be a given function of the current cavity radius. In the practical situation of cavity formation caused by high explosives, some kind of return motion following the expansion phase would be expected. Since the expansion process involves large plastic strains, it would be reasonable to begin by neglecting the elastic compressibility not only in the plastic region but also in the elastic region

594

8 Dynamic Plasticity

(Hopkins, 1960). The compressibility of the material will be allowed for in a subsequent treatment of the cavity expansion process.

8.4.1 Purely Elastic Deformation A spherical cavity of initial radius a0 is expanded into an infinitely extended medium which is assumed to be completely incompressible. The internal pressure p = p(t) is supposed prescribed at each instant of the expansion. Since the density ρ of the material remains constant by hypothesis, the equation for the conservation of mass may be written as ∂ 2 r υ = 0, ∂r where ν is the radial velocity of a typical particle currently situated at a radius r. This equation is immediately integrated to give

υ = a2 /r2 a˙ , a˙ = da/dt.

(8.49)

The particle velocity is therefore everywhere determined in terms of the velocity of the cavity surface. The associated components of the strain rate are 2 2a ∂υ ε˙ r = =− a˙ , ∂r r3

υ ε˙ θ = ε˙ φ = = r

a2 r2

a˙ .

(8.50)

For an isotropic material, the result ε˙ θ = ε˙ φ implies σθ = σφ throughout the medium, and the equation of motion in terms of the stresses and velocity becomes ∂σr 2 + (σr − σθ ) = ρ ∂r r

∂υ ∂υ +υ ∂t ∂r

.

(8.51)

The convective term represented by the second term in parenthesis will be retained even when the deformation is small, as it is not necessarily negligible under conditions of high-speed cavity formation. In the case of purely elastic deformation of the entire medium, the stress difference σθ − σr in (8.51) must be expressed as a function of r using the stress–strain relations before the integration can be carried out. Since the displacement is small in the elastic range, we may write υ ≈ ∂u/∂t, where u is the radial displacement, and (8.49) then integrates to

u = a3 − a30 /3r2 ≈ a2 /r2 ua

(8.52)

in view of the initial condition u = 0 when a = a0 , the quantity ua being the displacement a–a0 of the cavity surface. The strain components corresponding to (8.52) are

8.4

Dynamic Expansion of Spherical Cavities

εr = −

2a2 r3

595

ua ,

εθ = ε φ =

a2 r3

ua .

These results may be obtained by the direct integration of (8.50) using the same order of approximation. By the elastic stress–strain relations for an incompressible material (ν = 0.5), we have 2 2 a σθ − σr = E (εθ − εr ) = 2E 3 ua , 3 r

(8.53)

where E is Young’s modulus of the material. Substituting this into (8.51), and using (8.49), the equation of motion is reduced to 2 2 a ∂σr 2a a3 a = 4E 4 ua + ρ a ¨ + 1 − a2 . ∂r r r2 r2 r3 Integrating, and using the condition that the stress vanishes at infinity, the solution for the radial stress is obtained in the form 2 a a3 4 ρa a¨a + 2 − 3 a˙ 2 , σr = − E 3 u a − 3 r r 2r

(8.54)

which involves the velocity a˙ and acceleration ä of the cavity surface. The applied pressure p at the cavity surface is given by the boundary condition σr = −p at r = a, the result being p=

a0 3 4 E 1− + ρ a¨a + a˙ 2 . 3 a 2

(8.55)

If the initial pressure is denoted by p0 , the initial acceleration is equal to p0 /ρa0 , the initial velocity of the cavity being assumed to vanish. Multiplying both sides of (8.55) by a2 da, and using the identity 2a¨a + 3˙a2 a2 da = d a3 a2 , the resulting expressions can be integrated between the limits a0 and a to obtain

a

ao

pa2 da =

2 a0 2 1 2 3 + ρ a˙ a . E 1− 3 a 2

(8.56)

with a minor approximation. The evaluation of the integral is straightforward when p(a) is prescribed. A further quadrature of (8.56) after multiplying it by da furnishes the relationship between a˙ and a, permitting the cavity radius a(t) to be determined by integration. Plastic yielding begins when the yield condition σθ − σr = Y is first satisfied during the elastic expansion. Since σθ − σr has its greatest value at r = a in view of (8.53), yielding first occurs at the cavity surface when a = a1 and a˙ = a˙ 1 , such that

596

8 Dynamic Plasticity

a1 ≈ ao

Y , 1+ 2E

ρ a˙ 21

Y Y ≈ po − . E 3

The last result follows from (8.56) with the approximation p ≈ p0 during the elastic loading. It may be noted that the radial displacement at the cavity surface at the onset of yielding is quite independent of inertial effects. Although the displacement at this stage is quite small, the corresponding expansion velocity may be high.

8.4.2 Large Elastic/Plastic Expansion Subsequent to the commencement of yielding, an elastic/plastic boundary defined by r = c spreads outward from the cavity surface. The material outside this radius is elastic, while that within this radius is rendered plastic with the stresses satisfying the yield criterion σθ − σr = Y,

a ≤ r ≤ c.

The material is assumed to be nonhardening, so that the dynamic yield stress Y has a constant value throughout the plastic region. The velocity is still given by (8.49) everywhere in the medium, but the displacement in the elastic region is now given by ∂u = ∂t

c2 r2

υc

or

u=

c2 r2

r ≥ c,

uc ,

where uc is the displacement of the particles that are currently at the elastic/plastic boundary. The stress difference in the elastic region is σθ − σr = 2E

c2 r3

uc = Y

c3 r3

r ≥ c,

,

(8.57)

where the last expression follows from the fact that material at the elastic/plastic boundary is just at the point of yielding. Denoting by c0 the initial radius to the particle which is currently at radius c, the condition of incompressibility may be written as a3 − a30 = c3 − c30 ≈ 3c2 uc = (3Y/2E) c3 .

(8.58)

This equation relates the current cavity radius to the radius of the elastic/plastic boundary. The strains in the plastic region are given by

∂r0 εr = − ln ∂r

,

r εθ = εφ = ln r0

,

8.4

Dynamic Expansion of Spherical Cavities

597

where r0 is the initial radius to a typical particle, the relationship between r and r0 being obtained by integrating the equation dr/dt = ν along the path of the particle. The stress distribution in the elastic and plastic regions must be determined by integrating the equation of motion (8.51). Using (8.49) to eliminate ν, this equation is reduced to ∂σr 2 + (σr − σθ ) = ρ ∂r r

a2 r2

a¨ +

2a a3 1 − 3 a˙ 2 , r r

(8.59)

where σθ − σr is given by (8.53) in the elastic region (r ≥ c), and by (8.57) in the plastic region (r ≤ c). The integration in the elastic region is based on the condition that σ r is finite at r = ∞, while that in the plastic region involves the boundary condition σ r = –p at r = a. The result is easily shown to be a a3 −ρ r ≥ c, a¨a + 2 − 3 a˙ 2 , r 2r

r a a a3 3 2 2 +ρ 1− a¨a − 2 − 3 a˙ + a˙ , σr = −p + 2Y ln a r r 2 2r

2 σr = − Y 3

c3 r3

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ r ≤ c.⎪ ⎭ (8.60)

The fact that the stress in the plastic region contains a logarithmically divergent term indicates that the plastic region is always finite in extent. The condition of continuity of σ r across r = c furnishes

c 2 3 2 p = Y + 2Y ln + ρ a¨a + a˙ 3 a 2 in view of (8.60). Eliminating c/a from the above equation by means of (8.58), the applied pressure can be finally expressed as a30 2E 2 2 3 + Y ln 1 − 3 + ρ a¨a + a˙ 2 . p = Y 1 + ln 3 3Y 3 2 a

(8.61)

The first two terms on the right-hand side of (8.61) correspond to the quasi-static result for an ideally plastic incompressible material. Multiplying (8.61) by a2 da, integrating between the limits a1 and a, and using (8.56) corresponding to a = a1 , we get

a a0

2 2E 3Y 3 3 pa2 da = Y a0 + a − a30 1 + ln 9 4E 3Y a30 a3 1 3 3 + a ln 1 − 3 − a0 ln 3 − 1 + ρa3 a˙ 2 . 2 a a0

(8.62)

598

8 Dynamic Plasticity

This equation expresses the energy balance from the commencement of cavity expansion. The process comes to a stop when a = a2 and a = 0, the maximum cavity radius a2 being given by a a0

2 pa2 da = Y 9

2E 3Y 3 3 3 3 1 + ln + a2 ln 1 − a + a − a0 4E 0 3Y

a30

a3

− a30 ln

a32 a30

−1

(8.63) p

p

Since ε˙ r = −2˙εθ , and ε˙ θe − ε˙ re = 0 in the plastic region, the rate of plastic work done per unit volume is equal to (2Y/3) (˙εθ − ε˙ r ),which is nonnegative for a˙ ≥ 0 in view of (8.50). The preceding analysis is therefore valid at all stages of the expansion. When the expansion is so large that a0/a is small compared to unity, 3 the ratio c3/a is approximately equal to 2E/3Y, which is independent of the cavity radius. In this case, (8.63) reduces approximately to U=

8 2E π Ya32 1 + ln , 9 3Y

(8.64)

where U denotes the total internal energy of the explosion products utilized in the cavity formation. Equation (8.64) is a particularly simple result that furnishes the maximum cavity radius a2 from known values of U, Y, and E, when the cavity expansion is sufficiently large. The problem of fragmentation of a spherical shell due to an internal explosion has been discussed by Al-Hassani and Johnson (1969). It may be noted that (8.63) involves the total work done by the applied pressure and is quite independent of the rate of doing this work. Let the law of expansion of the volume of the explosion products be assumed isentropic, according to the relation p = p0 (a0 /a)3γ , where p0 is the initial pressure and γ > 1 is the index whose value is assumed constant over the considered range of expansion. The substitution of the above expression into the left-hand side of (8.63) results in

a2

a0

3(γ −1) p0 a30 a0 pa da = 1− . 3 (γ − 1) a2 2

(8.65)

Using (8.63) and (8.65), the ratio a2/a0 can be calculated for any given values of p0/Y, E/Y and γ . In actual practice, γ generally decreases during the expansion, and consequently a suitable mean value of γ should be used in (8.65). In the case of TNT, it is a good approximation to assume γ ≈ 3 for moderate expansions (Johnson, 1972). The variation of a2/a0 with p0/Y is shown graphically in Fig. 8.12 for E/Y = 450 and different values of γ . It is important to note that the final radius of the cavity achieved under dynamic conditions is the same as that under quasi-static conditions

8.4

Dynamic Expansion of Spherical Cavities

599

Fig. 8.12 Dependence of the maximum cavity radius on the intensity of initial pressure generated by an explosion

for the same expenditure of energy. An analysis for the cavity formation by deep underground explosion has been presented by Chadwick et al. (1964). Consider now the nature of the return motion following the end of the first expansion phase. Due to the isochronous nature of the process, the entire plastic region instantaneously unloads, and consequently, the initial part of the first contraction phase is purely elastic. This part of the return motion is terminated when plastic yielding again occurs at the cavity surface r = a, the yield condition then being σr − σθ = Y in the absence of the Bauschinger effect. Since σθ − σr = Y in the plastic region at the instant of unloading, and σθ − σr = 2Eu0 , at r = a due to the superposed elastic stresses during the unloading, plastic flow at the cavity surface requires ua = −Y/E, giving a cavity radius a3 = (1 − Y/E) a2 at the end of the elastic unloading. In the subsequent part of the first contraction phase, a new elastic/plastic boundary spreads outward creating a plastic zone where the yield condition is σr − σθ = Y. Since the return motion is very limited in extent, a detailed analysis of the contraction phase is of minor practical interest.

600

8 Dynamic Plasticity

The internal energy of the explosion products is mostly dissipated into plastic work, and the available energy at the end of the first expansion phase is relatively small. The details of the analysis for the subsequent motion, consisting of alternate contraction and expansion phases, becomes progressively more complicated. Eventually, a final shakedown state of purely elastic oscillating motion results. The initial motion is so highly damped by the plastic deformation produced in the first expansion phase that the subsequent motion is reduced to one of small oscillations with minor significance. It should be noted that in the hypothetical case of cavity formation from zero radius, only the first expansion phase is geometrically similar, while the subsequent phases are not.

8.4.3 Influence of Elastic Compressibility To simplify the analysis for the dynamic cavity expansion, when the compressibility of the elastic/plastic material is duly taken into account, we assume a constant speed U of the cavity surface which is expanded from zero radius. The plastic region at any instant is defined by a ≤ r ≤ c, where b = Ut, and the elastic region is defined by c ≤ r ≤ b, where b = Vt, with V denoting the speed of the elastic dilatational wave in the material. In terms of the radial displacement u, the stress–strain relations in the elastic region may be written as E

∂u = σr − 2νσθ, ∂r

E

u = (1 − ν) σθ − νσr, r

in view of the symmetry condition σθ = σφ , holding everywhere in the material. From the above relations, the elastic stress components are expressed as 3K ∂u u + 2ν , σr = (1 − ν) 1+ν ∂r r

3K σθ = 1+ν

∂u u + ν , ∂r r

(8.66)

where K denotes the elastic bulk modulus, equal to E/[3 (1 − 2ν)]. Substituting from (8.66) into the equation of motion (8.51), where υ is replaced by ∂u/∂t, and neglecting the convective term υ (∂u/∂r), we obtain the wave equation ∂ 2 u 2 ∂u 2u 1 ∂ 2u − + = , r ∂r ∂r2 r2 V 2 ∂t2 due to Forrestal and Luk (1988). The quantity V denotes the speed of propagation of the elastic wave front of radius b and is given by V = 2

1−ν 1 − 2ν

2G = ρ

1−ν 1+ν

3K ρ

8.4

Dynamic Expansion of Spherical Cavities

601

where ρ denotes the initial density of the material. Since the configuration maintains geometrical similarity during the expansion process, the dimensionless stresses σr/Y and σθ/Y, and the dimensionless displacement, u/c must be functions of the single variable r/c. Setting r ξ= , c

' 1+ν ρ c˙ u¯ = = c , V 1 − ν 3K

and using the fact that the first derivatives of the displacement are du¯ ∂u = , ∂r dξ

∂u du¯ = c˙ u¯ − ξ , ∂t dξ

(8.67)

where c˙ is a constant, the wave equation is transformed into the ordinary differential equation

d2 u¯ 2 du¯ 2¯u 1 − α2ξ 2 + − 2 = 0. dξ 2 ξ dξ ξ

(8.68)

The integration of (8.68) is facilitated by the substitution u¯ = ξ ψ, where ψ is a function of ξ , resulting in

d2 ψ 2 2 dψ + 2 2 − α ξ = 0. ξ 1 − α2ξ 2 dξ dξ 2 This is a first-order differential equation in dψ/dξ , which is easily integrated under the boundary condition dψ/dξ = 0 at the wave front r = b or ξ = 1/α, sin ce u¯ and du¯ /dξ both vanish at this boundary. A second integration then furnishes the quantity ψ in view of the boundary condition ψ = 0 at ξ = 1/α. Consequently, D 1 − α2ξ 2 dψ , =− dξ ξ4

ψ=

D (1 − αξ )2 (1 + 2αξ ) , 3ξ 3

where D is a constant of integration. These expressions yield the function u¯ and its derivative as 2D 1 − α 3 ξ 3 du¯ D (1 − αξ )2 (1 + 2αξ ) , . (8.69) =− u¯ = 3ξ 2 dξ 3ξ 3 The constant D is now obtained from the condition that the material at the elastic/plastic boundary r = c is at the point of yielding. Substituting from (8.69) into (8.66) with u = c¯u, and setting σθ − σr = Y at ξ = 1, we get

D = (Y/E) (1 + ν) / 1 − α 2

602

8 Dynamic Plasticity

The stress distribution in the elastic region (1 ≤ ξ ≤ 1/α) therefore becomes ⎫ 2 (1 − αξ ) (1 − 2ν) (1 + αξ ) + (1 + ν) α 2 ξ 2 ⎪ σr ⎪ ,⎪ =− ⎪ ⎬ 2 3 Y 3 (1 − 2ν) 1 + α ξ (1 − αξ ) (1 − 2ν) (1 + αξ ) − 2 (1 + ν) α 2 ξ 2 ⎪ σθ ⎪ ⎪ , ⎪ = ⎭ Y 3 (1 − 2ν) 1 + α 2 ξ 3

(8.70)

It may be noted that both the stress components vanish at ξ = 1/α as expected. The velocity distribution in the elastic region (1 ≤ ξ ≤ 1/α), by the second equation of (8.67), is given Y (1 + ν) 1 − α 2 ξ 2 dψ υ = −ξ 2 = c dξ E 1 − α2 ξ 2

(8.71)

The elastic/plastic interface velocity c˙ , which is c/a times the cavity velocity U, must be determined by considering the plastic region. In an incompressible material, the elastic wave front does not exist (α = 0), and the elastic region is then defined by ξ ≥ 1. The effect of compressibility on the dynamic expansion was first considered by Hunter and Crozier (1968). The dynamic expansion of an elastic/plastic spherical shell under blast loading has been examined by Baker (1960). Since the change in density is only of the elastic order, a constant value of ρ will be assumed in the equation of motion (8.51). The other equation necessary for the analysis in the plastic region is the compressibility equation 2υ ∂υ + = ∂c r

∂ ∂ +υ ∂t ∂r

σr + 2σθ 3K

.

(8.72)

The left-hand side of this equation is equal to −ρ/ρ, ˙ where the dot denotes the rate of change following the particle. In view of the yield criterion σθ − σr = Y, the two governing equations in the plastic region become ⎫ ∂υ ∂σr 2Y ∂υ ⎪ − =ρ +υ ,⎪ ∂r r ∂t ∂r ⎬ a≤r≤c ⎪ ∂υ 2υ 1 ∂σr ∂σr ⎪ ⎭ + = +υ . ∂R r K ∂t ∂r Using the similarity transformation as before with the help of the dimensionless quantities r ξ= , c

σr s= , Y

υ w= , c

ρ β=c , K

and noting the fact that c (∂υ/∂t) = −c2 ξ (dw/dξ ), the preceding equations can be expressed as

8.4

Dynamic Expansion of Spherical Cavities

dw ⎫ 2 K 2 2 ds ⎪ − = β β −ξ ,⎪ ⎬ dξ ξ Y dξ dw 2w Y ds ⎪ ⎭ + = (w − ξ ) , ⎪ dξ ξ K dξ

603

(8.73)

It has been shown by Forrestal and Luk (1988) that the velocity distribution can be estimated with insignificant error by omitting the convective terms on the righthand side of (8.73). Introducing this approximation, and eliminating s between the two equations of (8.73), we obtain the differential equation

dw 2w 2Y + =− . 1 − β2ξ 2 dξ ξ K

(8.74)

Integrating, and using the boundary condition w = (1 + ν) Y/E at ξ = 1, in view of (8.71), the solution for the dimensionless velocity is obtained as w=

Y 1 1 − β 2 ξ 2 1 + μβ 2 1 (1 + βξ ) (1 − β) , (8.75) − + + ln Kβ 2 ξ ξ2 1 + β2 2β (1 − βξ ) (1 + β)

where μ = (1 + ν)/[3 (1 − 2ν)]. The elimination of ξ (dw/dξ ) from the first equation of (8.73) with the help of (8.74) leads to the differential equation Kβ 2 ds = dξ Y

2w dw 2 . + w + 2 2 dξ 1−β ξ ξ 1 − β 2ξ 2

On substituting for w, in the second term of the parenthesis, and integrating under the boundary condition s = s0 and w = w0 at ξ = 1, the solution is found as Kβ 2 2 s = s0 + w − w20 , 2Y 1 − β2ξ 2 2 1 + μβ 2 (1 − ξ ) 1 (1 + βξ ) (1 − β) + ln ln . + − βξ (1 − βξ ) (1 + β) 1 − β2 ξ 1 − β2 ξ 2 (8.76) The quantities s0 and w0 , obtained by setting ξ = 1 in (8.70) and (8.76), respectively, may be written as 2 Y (1 + ν) μY (1 + ν) α 2 , w0 = s0 = − = . 1+ 3 (1 − 2ν) (1 + α) 3 K (1 − 2ν) K Since the boundary condition ν = U at the cavity surface r = a is equivalent to w = a/c at ξ = a/c in view of the relation c˙ = (c/a) U, the ratio c/a is obtained as the solution of the transcendental equation c a

a2 + μκ 2 c2 a2 − λ2 c2 /a2

+

1 + ρU 2 /Y 1 (a − λc) (1 + λ) . ln = 2λ 1 − λ2 (a + λc) (1 − λ)

(8.77)

604

8 Dynamic Plasticity

√ where λ = U ρ/K. This equation can be solved for λc/a with any given value of λ, ν and Y/K, and the ratio c/a then follows as the end result. The cavity pressure p is finally obtained by setting s = −p/Y and ξ = a/c in (8.76), the result being 1 ρU 2 2 λ2 c2 p (1 + ν) μλ2 c2 − = − + ν) 1+ (1 Y 3 2 Y (1 − ν) (1 + α) α 2 3a2

c a2 + μλ2 c2 1 − λ2 c2 (a − λc) (1 + λ) . +2 + ln + ln −1 a a2 − λ2 c2 a2 − λ2 c2 (a + λc) (1 − λ) (8.78) Although this expression is more complicated than the corresponding formula for an incompressible material, the evaluation of the cavity pressure from (8.78) is straightforward, once c/a has been computed from (8.77) by a trial-and-error procedure. The dynamic expansion of a cylindrical cavity in a compressible elastic/plastic medium has been treated by Luk and Amos (1991). The results for the cavity expansion from zero radius with a constant velocity U in an incompressible elastic/plastic material may be directly obtained by setting a0 = 0, a˙ = U, and a¨ = 0 in the more general solution given earlier, the cavity pressure in this case being expressible in the form 3 c 2 9ρU 2 c3 p = 1 + ln 3 + , Y 3 4E a a3

c3 2E . = 3Y a3

(8.79)

In view of (8.61) and (8.58), the relationship between the interface velocity c˙ = (c/a) U and the cavity velocity U for a compressible elastic/plastic material is compared with that for the incompressible material in Fig. 8.13(a), assuming ν = 13 and Y/K = 0.00435. The cavity pressures in the two cases are compared with one another in Fig. 8.13(b) for identical material properties. The cavity pressure obtained by replacing in (8.79) the ratio c3 /a3 by its quasi-static value E/[3 (1 − ν) Y] for a compressible material, almost coincides with the exact numerical solution represented by the broken curve. A numerical solution for a compressible work-hardening material has been given by Luk et al. (1991) assuming a power law of hardening.

8.5 Mechanics of Projectile Penetration The problem considered here is that of penetration into sufficiently thick targets caused by the motion of rigid projectiles striking at normal incidence with a velocity v0 . The depth of penetration that occurs before the projectile comes to rest depends on the mechanical properties of the target material as well as on the geometry of the projectile. The projectile produces a cylindrical tunnel in the target, about the size of the shank diameter, and a thin layer of material between the target and the nose of the projectile is melted during the penetration. Consequently, the resistance to penetration due to friction is usually small compared to that due the normal pressure

8.5

Mechanics of Projectile Penetration

605

Fig. 8.13 Expansion of a spherical cavity in an elastic/plastic medium. (a) Interface velocity against cavity velocity and (b) cavity pressure against cavity velocity

exerted on the surface of the projectile. The problem has been treated by Backman and Goldsmith (1978) for an incompressible target material using the results for the expansion of spherical cavities in an elastic/plastic medium. The elastic compressibility of the material has been taken into account by Forrestal et al. (1988), who developed empirical formulas for the penetration mechanics based on spherical and cylindrical cavity expansion processes, and performed terminal-ballistic experiments to support their theory.

8.5.1 A Simple Theoretical Model We begin by considering an ideally plastic target material, and an ogival shape of the projectile nose defined by a pair of circular arcs meeting the shank tangentially as shown in Fig. 8.14(b). The radius of each circular arc is denoted by na, where a is the radius of the shank and n is a constant factor. A generic point on the nose surface is subjected to a normal pressure p and a tangential traction μp, where μ is the coefficient of friction between the nose and the surrounding material. The resultant axial thrust acting on the ogival nose in the positive z-direction is given by the expression P = 2π an

π/2 θ0

(cos θ + μ sin θ ) prdθ ,

606

8 Dynamic Plasticity

where θ denotes the angle of inclination of the normal to the surface with the axis of symmetry, and θ = θ 0 corresponds to the apex O. From simple geometry, the coordinates of a generic point on the nose are r = a [n sin θ − (n − 1)] ,

z = 1 − na cos θ .

Fig. 8.14 Various shapes of projectile. (a) Spherical nose projectile, (b) ogival nose projectile, and (c) conical nose projectile

The first of these relations immediately gives sinθ0 = (n − 1)/n, and the preceding expression for the axial force becomes P = 2π a n

2 2

π/2

θ0

(cos θ + sin θ ) (sin θ − sin θ0 ) pdθ ,

(8.80)

where p is an unknown function of θ having its greatest value at the apex θ = θ 0 . The height of the ogival nose is √ l = na cos θ0 = a 2n − 1. The hemispherical nose shown in Fig. 8.14(a) is the limiting case of an ogival nose, and corresponds to n = 1 giving θ 0 = 0. The conical nose shown in Fig. 8.14(c) cannot be obtained by a limiting process, since the nose in this case meets the shank with a discontinuous slope. To obtain an approximate solution to the penetration problem, it is assumed that the normal pressure p acting on the nose is equal to the internal pressure necessary to expand a spherical cavity from zero radius in the same medium with a radial

8.5

Mechanics of Projectile Penetration

607

velocity equal to the local particle velocity in the direction normal to the nose surface. The normal pressure distribution on the nose penetrating into a compressible elastic/plastic target is therefore obtained from (8.79) on setting U = υ cos θ , where ν is the current projectile velocity, and replacing the quantity c3 /a3 by its quasi-static value, the result being expressed in the dimensionless form p E 2 3ρυ 2 cos2 θ = , 1 + ln + Y 3 3 (1 − υ) Y 4 (1 − ν) Y

(8.81)

where ρ is the density of the target material. Equation (8.81) implies that the pressure acting at the base θ = π/2 is equal to the quasi-static expansion pressure. Substituting (8.81) into (8.80) and integrating, the penetration force is obtained as 2 ρυ P = π a2 Y A + B , Y

(8.82)

where A and B are dimensionless constants, given in terms of the mechanical properties of the target material, the geometry of the projectile nose and the friction coefficient as ⎫ l E 2 ⎪ 2 π − θ0 − (n − 1) 1 + ln 1+μ n ,⎪ A= ⎪ ⎬ 3 3 (1 − ν) Y 2 a .

⎪ l 3 q 4n − 1 μ 3 2 π l2 ⎪ ⎪ − − l) + B= n − θ + (n ⎭ 0 1−ν 8n2 8 2 2 a 2 n2 a2 (8.82a) The parameters θ0 and l/a in (8.82a) are defined by the value of n. When the frictional effect is disregarded (μ = 0), the constant A reduces to the quasi-static value of p/Y for the spherical cavity expansion process. The variation of the projectile velocity ν with time t and the depth of penetration h can be determined by considering the equation of rigid-body motion of the projectile. Since the rate of change of momentum is m(dυ/dt), where m is the mass of the projectile, the equation of motion is P = −m

dυ dυ = −mυ . dt dh

Substituting P from (8.82), and integrating under the initial condition υ = υ 0 when t = 0 and h = 0, we obtain % $ % ⎫ ⎪ ⎪ Bρ Bρ ⎪ t= tan−1 υ0 − tan−1 υ ,⎪ ⎪ ⎬ AY aY ⎪ ⎪ A + B ρυ02 /Y πρa2 P0 1 1 ⎪ ⎪ ln ln h= , = ⎪ ⎭ m 2B 2B P A + B ρυ 2 /Y

π a2 Y m

'

Y Bρ

$

(8.83)

608

8 Dynamic Plasticity

where P0 denotes the initial value of the force. It follows that the force decreases exponentially with the increase in depth of penetration. The projectile comes to rest when υ = 0, the corresponding depth of penetration h∗ being given by

πρa2 m

2 Bρυ 1 0 ln 1 + h∗ = , 2B AY

(8.84)

where A and B are obtained from (8.82a) for any given n. When the nose is hemispherical (n = 1), these expressions are simplified to A=

2 E μπ 1+ , 1 + ln 3 2 3 (1 − ν) Y

B=

3 (1 + μπ/4) . 8 (1 − ν)

The total time of penetration t∗ is obtained from the first relation of (8.83) by setting υ = 0, and using the appropriate value of n to compute the constants A and B. When the nose of the projectile is a circular cone of vertex angle 2α, the normal component of velocity at any instant has a constant value equal to νsin α. The normal pressure p, which is also a constant, is given by (8.81) with θ = π/2 − α, and the axial force acting on the conical surface is P = π a2 p (1 + μ cot α) for a coefficient of friction μ. The substitution of (8.81) with θ = π/2 − α into the above relation results in (8.82), where the constants are now given by ⎫ 2 E ⎪ A = (1 + μ cot α) 1 + ln ,⎪ ⎬ 3 3 (1 − ν) Y . ⎪ 3 E ⎭ B = (1 + μ cot α) ln tan2 α. ⎪ 4 3 (1 − ν) Y

(8.85)

The variations of the projectile velocity and depth of penetration with time are defined by (8.83) and (8.85). The final penetration is still given by (8.84) as a function of the striking velocity v0 , provided A and B are determined from (8.85). In Fig. 8.15, the theoretical predictions for the ogival nose (n = 6) and conical nose (α = 18.4◦ ) projectiles are compared with some available experimental results (Forrestal et al., 1988). The mechanical properties of the target material, which is prestrained aluminum, consist of v = 13 , Y = 400 Mpa, E = 68.9 Gpa, and ρ = 2.71 103 kg/m3 , the projectiles being made of managing steel with a = 15.2 mm, l = 3a, and m = 0.024 kg. The theoretical calculations are based on the assumption that the coefficient of friction μ would lie between 0.02 and 0.1 in the dynamic penetration. The agreement between theory and experiment is seen to be reasonably good, considering the uncertainty that exists in selecting the appropriate value of μ.

8.5

Mechanics of Projectile Penetration

609

Fig. 8.15 Depth of penetration of rigid projectiles into thick metallic targets as a function of the striking velocity. (a) Ogival nose projectile (n = 6) and (b) conical nose projectile (α = 18.4◦ )

The simplest way of taking into account the work-hardening property of the target material is to assume that the internal pressure in the dynamic cavity expansion is increased by the same amount as that for the quasi-static expansion in an incompressible elastic/plastic medium. If the stress–strain curve in the plastic range is approximated by a straight line of slope T, then (8.81), for the normal pressure on the projectile nose, is modified to E 2 π 2T 3ρυ 2 cos2 θ p = 1 + ln + + Y 3 3 (1 − ν) Y 9Y 4 (1 − ν) Y

(8.86)

in view of a known result for the quasi-static expansion of spherical cavities (Chakrabarty, 2006). A similar expression can be written for a nonlinear hardening. Empirical formulas based on a numerical solution to the spherical cavity expansion process, using a power law of hardening, have been given by Forrestal et al. (1991), who obtained good agreement of their theoretical predictions with experimental results. The high-speed impact and penetration of long rods has been investigated by Tate (1969) and Rosenberg and Dekel (1994), while the associated ricochet problem has been examined by Johnson et al. (1982). Explicit formulas for the penetration dynamics of rigid projectiles into thick plates have been presented by Chen and Li (2003).

610

8 Dynamic Plasticity

8.5.2 The Influence of Cavitation At sufficiently high velocities of penetration, the target material sometimes flows away due to its own inertia to produce a hole which is larger in diameter than that of the projectile, Fig. 8.16. This phenomenon, known as cavitation, has the effect of enhancing the resistance to penetration, since a part of the kinetic energy is absorbed by the plastic deformation involved in the enlargement of the hole. The problem of cavitation in the penetration process has been discussed by Hill (1980), who assumed the normal pressure distribution over the projectile nose, defined by a convex function r = r(z), in the form p = q + kρυ 2

d dr r , dz dz

(8.87)

where q is the pressure corresponding to the quasi-static process, and k is a dimensionless constant that depends on the shape of the nose. Cavitation would occur over the region where the value of p given by (8.87) is found to be negative. The resistive force acting on the projectile in the absence of cavitation is P = 2π

a

prdr = 2π

0

0

l

dr dz = π qa2 . p r dz

provided r dr/dz vanishes at both z = 0 and z = l. The resistance to penetration in this case is therefore independent of the dynamic factor. Since for a conventional nose, p decreases monotonically from z = 0 to z = l, cavitation would begin at the base of the nose when the velocity attains a critical value υ c . If the radius of curvature of the profile at this point is denoted by na, then d 2 r/dz2 = −1/na at z = 1, where p = 0 at the incipient cavitation. When d 2 r/dz2 = 0 at z = 1, the critical velocity is υc =

nq/kp.

Fig. 8.16 Projectile penetration with cavitation. (a) Steady-state cavity formation and (b) geometry of conventional projectiles

8.5

Mechanics of Projectile Penetration

611

For υ > υ c the target loses contact with the nose at a point which progressively moves toward the tip, and the resistive force becomes P=

π r02

q + kρυ

2

dr dz

2 ,

υ ≥ υc ,

(8.88a)

r=r0

where r = r0 corresponds to the point on the nose at which p = 0 according to (8.87). Consequently, d dr = 0, r q + kρυ 2 dz dz r=r0

υ ≥ υc ,

(8.88b)

Let the local radius of the hole due to cavitation be denoted by λa. Since the enlarged cavity may be imagined to have been produced by a projectile of radius λa without cavitation, we may write p = π λ2 a2 q. Inserting this expression in (8.88a), and using the relation nq = kpυ c , we get

dr n dz

2 =

λ 2 a2 υc 2 −1 , 2 ro υ

n

d2 r λ2 a 2 υ c 2 = , dz2 ro3 υ

(8.89)

at the point where the material breaks contact with the nose. The second equation of (8.89) follows from the first on combining it with (8.88b). Equations (8.89) are sufficient to determine r0 /a and λ in terms of the velocity ratio v/vc for a given shape of the nose. In the case of an ogival nose, the profile of the nose has a constant radius of curvature na, the equation of the circular profile being [r + (n − 1) a]2 + (z − l)2 = n2 a2 , √ where l is the height of the nose, equal to a 2n − 1. In view of the above relation, equation (8.89) yields ⎫ % 1 λυc 2/3 ⎪ ⎪ ⎪ = n− ⎪ ⎬ υ n−1 4/3 2 $ 2 % # 2/3 ⎪ ⎪ λυc a λυc λυc ⎪ ⎪ n− = n− ⎭ υ r0 υ υ

λυc υ

2/3

a r0

$

(8.90)

The elimination of a/r0 between these two equations furnishes the relationship between λ and υ/υ c as

612

8 Dynamic Plasticity

λυc υ

2/3

= n − (n − 1)2/3 n −

υc2 υ2

1/3 , υ ≥ υc

(8.91)

As υ tends to infinity, the parameter λυ c /υ tends to a limiting value that depends on n, the limiting position r = r0∗ of the point where cavitation is initiated being given by

r0∗ = na

n−1 n

1/3 n − 1 2/3 1− n

in view of (8.90). Consequently, the contact is always maintained over a finite part of the nose. The pressure distribution on the nose over the region of contact according to (8.87) is given by (n − 1) n2 a3 p−q = − 1, kρυ 2 [r + (n − 1) a]3

0 ≤ r ≤ r0 .

(8.92)

The pressure decreases monotonically from its greatest value at the tip, vanishing at r = r0 ≤ a, when υ ≥ υc . The cavitation velocity υ c can be determined by an optimal fit of (8.92) with the experimentally measured value λ for a given material and any particular value of n defining the ogival nose. When n = l, the nose is hemispherical, and the projectile could be a spherical ball. Considering this as a limiting case of an ogival nose, the pressure distribution is uniform and can be written as p = q − kρυ 2 , 0 ≤ r ≤ a, √ so long as there is no cavitation (υ < υc ). When v exceeds vc = q/kρ the pressure is zero everywhere except at the pole, where there is a concentrated force P = π λ2 a2 q with λ = υ/υc . The nose therefore makes contact with the target only at the pole over the cavitation range (υ > υ c ). The results for a conical nose cannot be obtained from those for the ogival nose by a limiting process, because of the sharp corner existing at the base r = a. The pressure distribution is, however, uniform according to (8.87) and is given by p = q + kρυ 2 tan α, where α is the semiangle at the vertex. The cavitation in this case is only possible at the base, where there is a singularity in pressure and may occur at any velocity of the projectile. Since the resistive force is pA = qλ2 A where A = π a2 is the area of the base, we have p λ = =1+ q 2

kp υ 2 tan2 α. q

8.5

Mechanics of Projectile Penetration

613

Assuming q/Y to be given by the first term on the right-hand side of (8.81), the constant k can be determined from the experimentally measured values of the cavity radius λa. It remains to establish how the velocity υ of the projectile varies with the depth of penetration h, the striking velocity υ 0 being given. For a mass m of the projectile, the equation of motion is mυ

dυ = −λ2 qA. dh

For a conventional nose, λ = 1 when υ ≤ υ c but is a function of υ when υ > υ c . A direct integration of the above equation therefore results in m h −h= qA ∗

υ 0

ξ dξ mυc2 1+ , υdυ = 2 2qA 1 λ

υ ≥ υc,

(8.93)

where ξ = (υ0 /υc )2 and h∗ are the final penetration when the projectile comes to rest. Since υ = υ 0 when h = 0, we have h∗ =

β mn dξ , 1+ 2 2 kρA 1 λ

υ0 ≥ υc ,

where β = (υ0 /υc )2 . When the incidence velocity υ 0 is less than υ c , the quantity in the curly brackets must be replaced by β. Indeed, in the absence of friction and cavitation, qah∗ = mυ02 /2 for a conventional nose. In the particular case of an ogival nose, the integral in (8.93) is evaluated by using (8.91) for λ.(ξ ), and the results are displayed graphically in Fig. 8.17. For a hemispherical nose, on the other hand, we have λ2 = ξ , and the integral is then equal to In β. For a conical nose, an analysis similar to above gives h∗ =

mn cρ 2 ln 1 + υ0 , 2 kρA q

where c = k tan2 α. The results for a composite nose, in which an ogival frustum is surmounted by a circular cone, can be similarly obtained. Some experimental evidence in support of the theoretical prediction involving cavitation has been reported by Hill (1980), who used copper targets and steel bullets with various head shapes. In the case of hypervelocity impact, for which the speed of the projectile exceeds the elastic wave speed in the target material, the pressure generated on the cavity surface is large compared to the yield stress of the material. Since the rise in temperature caused by the impact is extremely high, the material can be treated as a fluid for the analysis of the penetration problem. When a meteor traveling in a highly eccentric orbit strikes the surface of a planet, there is complete volatilization of some material during the formation of the crater, and the situation is effectively similar to that encountered in a subsurface explosion (Johnson, 1972).

614

8 Dynamic Plasticity

Fig. 8.17 Relationship between depth of penetration and squared striking velocity for projectiles with ogival heads

8.5.3 Perforation of a Thin Plate A projectile having a conical nose or a sharp ogival nose strikes a target in the form of a thin plate of uniform thickness h0 with a sufficiently high velocity v0 , the axis of the projectile being normal to the plane of the plate, Fig. 8.18(a). It is assumed that the projectile penetrates the plate without shattering it and produces a lip of height b as it leaves behind a circular hole of radius equal to the shank radius a. In the simplified model, each element of the lip is assumed to reach its final position through rotation under a uniaxial tension in the circumferential direction. If the initial radius to a typical element is denoted by s, and the final distance of the element from the outer edge of the lip is denoted by x, then the condition of plastic incompressibility requires h0 s ds = ha dx

or

dx/ds = h0 s/ha,

where h is the local thickness of the lip. In view of the assumed uniaxial state of stress, we have the additional relation

h ln h0

1 a = − ln 2 s

or

h = h0

s . a

8.5

Mechanics of Projectile Penetration

615

Fig. 8.18 Perforation of a flat plate by a projectile with pointed nose. (a) Geometry of lip formation and (b) fractional change in velocity against square of the striking velocity

Eliminating h/h0 between the preceding relations, the resulting differential equation for x can be integrated to give x 2 s 3/2 , = a 3 a

h = h0

3x 2a

1/3 ,

(8.94)

in view of the boundary condition x = 0 when s = 0. The remaining boundary condition x = b at s = a indicates that the total height of the lip is b = 2a/3. The radial pressure exerted by the projectile when the lip is fully formed is given by p=

h0 x 1/3 h , Y= Y a a b

when the material is nonhardening with a uniaxial yield stress equal to Y. The pressure varies, therefore, from zero at x = 0 to h0 Y/a at x = b. It may be noted that the strain at x = 0 is infinitely large. Suppose that the material work-hardens according to the power law σ = σ 0 εk , where σ 0 and k are constants. Then the plastic work done per unit volume in a typical element is σ ε/(1 + k), where σ is the hoop stress corresponding to the final hoop strain ε = ln(a/s). Hence, the total plastic work expended during the formation of the lip is

616

8 Dynamic Plasticity

W1 =

2π h0 σ0 1+k

1

ε1+k sds =

0

2π a2 h0 σ0 1+k

∞

e−2e ε1+k dε.

0

The integral on the right-hand side is equal to (2 + k)/22+k , where (ξ) is the well-known gamma function of a positive variable |. The expression for the plastic work therefore becomes Γ (1 + k) . (8.95a) W1 = π a2 h0 σ0 21+k To obtain the work done by the inertia forces, let r denote the radius of the hole at any instant of time t. Since the mass of material displaced at time t is equal to π ρh0 r2 , where ρ is the density of the material, the work done by the distribution of the radial accelerating force F is W2 =

a

Fdr = πρh0

z

0

a

d 2 dr r dr. dt dt

The work expended in overcoming the frictional resistance will be disregarded, and W2 will be evaluated on the basis of a constant projectile velocity equal to υ 0 (Thomson, 1955). Considering an ogival nose defined by r = na (sin θ − sin θ0 ) ,

sin θ0 = (n − 1) /n,

n > 1,

and setting dr/dt = υ 0 cot θ and dθ /dt = (υ 0 /na) cosec θ , we obtain the expression W2 = πρho υ02

π/2

θ

2nar cot θ cos θ − r2 cos ec2 θ cot θ dθ .

Inserting the expression for r(θ ), and carrying out the integration, the result is found to be 1 n − (1 + 2n) . (8.95b) W2 = πρh0 υ02 n2 ln n−1 2 Let the speed of the projectile decrease from υ 0 to υ f during the perforation. Equating the loss of kinetic energy of the projectile to the total work done on the material, we have

m υ02 − υf2 = 2 (W1 + W2 ) , where m0 is the mass of the projectile. The substitution of W1 and W2 from (8.95) into the preceding relation furnishes the square of the velocity ratio as

υf υ0

2 =1−

m0 m

n (1 + k) 2 ln + 2n − + 2n) , (1 n−1 2k η

(8.96)

8.6

Impact Loading of Prismatic Beams

617

where η = ρυ02/σ0 and m0 = π ρh0 a2 is the mass of the displaced material. When the projectile has a conical nose, dr/dt = υ 0 a/l, and an independent analysis leads to the formula

υf υ0

2

m0 =1− m

(1 + k) 2a2 + . 2k η l2

The residual velocity υ f for a standard ogival nose is somewhat higher than that for a conical nose having the same a/1 ratio. Figure 8.18(b) shows how the ratio (υ 0 – υ f )/υ 0 varies with the parameter η in the case of an ogival nose projectile for different values of m0 /m. Due mainly to the neglect of plastic bending of the plate beyond the radius r = a, the theoretical prediction is found to underestimate the velocity change (Goldsmith and Finnegan, 1971) An experimental investigation on the perforation of target plates by the normal and oblique impact of projectiles has been reported by Piekutowski et al (1996). In the case of perforation of a thin plate by the normal impact of a flat-ended cylinder, a plate plug is generally formed by shearing and is ejected from the target as the projectile passes through the plate. The diameter of the plug is approximately equal to that of the projectile, and the velocity of its ejection differs only marginally from the residual velocity of the projectile (Recht and Ipson, 1963). The analysis of the perforation problem involving truncated projectiles has been discussed by Zaid and Paul (1958). Useful experimental results for impact on finite plates have been reported by Calder and Goldsmith (1971). For very high velocities of impact, the effect of the rate of straining on the resistance to shear becomes significant, as has been shown by Chou (1961). The problem of ricochet of the deforming projectile after impact with plates has been investigated by Zukas and Gaskill (1996).

8.6 Impact Loading of Prismatic Beams 8.6.1 Cantilever Beam Struck at Its Tip Consider a uniform cantilever of length l which is struck transversely at the end by an object of mass m0 moving with velocity U, the mass being assumed to be attached to the beam during the plastic deformation that follows. The material is assumed to be rigid/plastic with constant uniaxial yield stress Y. The kinetic energy of the moving object is absorbed by a plastic hinge which is initially formed at the tip of the cantilever. As the inertia effect progressively decreases, the plastic hinge moves along the beam toward the built-in end, causing a permanent change in curvature over the distance traversed by the hinge. This problem has been investigated theoretically and experimentally by Parkes (1955) based on the rigid/plastic model and by Symonds and Fleming (1984) on an elastic/plastic model. At any instant of time t measured from the moment of impact, let the plastic hinge H be situated at a distance ξ l from the loaded end of the cantilever, Fig. 8.19(a). The bending moment at the hinge is equal to –M0 , where M0 is the yield moment that

618

8 Dynamic Plasticity

Fig. 8.19 Uniform cantilever struck at its tip. (a) Deformed configuration at any instant and (b) position of plastic hinge as a function of time

depends on the yield stress, the area of cross section, and the shape factor. Since the shearing force is zero at the hinge, where the bending moment is a relative maximum, we may analyze the portion of the beam between the hinge and the tip as a rigid body. Since the inertia force per unit length acting at a generic point of the beam is (m/l)(∂ 2 w/∂t2 ) acting in the upward sense, where m is the total mass of the beam and w the downward deflection, the application of D’Alembert’s principle for the dynamic equilibrium of forces and moments gives ⎫ m ξ t d2 w d 2 w0 ⎪ ⎪ dx = 0,⎪ m0 2 + ⎬ 2 l 0 dt dt ξt ⎪ x d2 w d 2 w0 ⎪ M 0 + m0 ξ l 2 + m ξ− dx = 0, ⎪ ⎭ 2 dt l dt 0

(8.97)

where w0(t) denotes the deflection at x = 0 and is assumed to be sufficiently small. Since the deformed part of the beam between the tip and the hinge at any instant rotates about x = ξ l as a rigid body, the velocity of a generic point is given by x dw0 ∂w = 1− . ∂t ξ l dt

(8.98)

Substituting in (8.97), and introducing a constant deflection δ together with a set of dimensionless quantities ρ, z, and τ , which are defined as δ=

m0 lU 2 , 2M0

ρ=

m x , z= , 2m0 l

τ=

tU 2δ

we obtain a pair of ordinary differential equations for w0 in a mathematically convenient form. The result may be expressed as

8.6

Impact Loading of Prismatic Beams

619

⎫ w ¨ 0 + ρ ξw ¨ 0 + ξ˙ w ˙ 0 = 0, ⎬ ρ 2 2δ + ξ w ¨0 + ¨ 0 + ξ ξ˙ w˙ 0 = 0,⎭ 2ξ w 3

(8.99)

where the dot denotes differentiation with respect to the dimensionless time τ . Since the expression in parenthesis of the first equation of (8.99) forms an exact differential, this equation is immediately integrated once to give (1 + ρξ ) w˙ 0 = 2δ in view of the initial conditions ξ = 0 and dw0 /dt = U (or w˙ 0 = 2δ ) when τ = 0. A suitable combination of the two equations in (8.99) gives

¨ 0 + 2ξ ξ˙ w˙ 0 = 6δ ρ ξ 2w The expression in parenthesis of the above equation is an exact differential, and integration gives ρξ 2 w˙ 0 = 6δτ , on using the initial condition ξ = 0 when τ = 0. The relationship between the velocity of the tip of the cantilever and the time interval over the range 0 ≤ ξ ≤ 1 is therefore given parametrically as w˙ 0 = 2δ/ (1 + ρξ ) , 3τ = ρξ 2 / (1 + ρξ ) ,

0 ≤ τ ≤ τ0 = ρ/ (3 + 3ρ) ,

(8.100)

where τ 0 is the dimensionless time when the plastic hinge reaches the built-in end of the cantilever. Figure 8.19(b) indicates how the hinge position varies with time during the initial phase of bending. For τ > τ 0 , the plastic hinge remains fixed at the built-in end, and the hinge moment is no longer a relative maximum. The nonzero shear force that exists at the hinge can be determined from the equation of vertical motion. Considering the angular motion, we set ξ = 1 and ξ˙ = 0 in the second equation of (8.99) to have 2 ¨ 0 + 2δ = 0. 1+ ρ w 3 Integrating, and using the condition that w0 is continuous at τ = τ 0 , we obtain the solution w˙ 0 =

6δ (1 − τ ) , τ0 ≤ τ ≤ 1. 3 + 2ρ

(8.101)

Since w ˙ 0 = 0 when τ = 1, the motion stops at this value of τ . Hence the duration of the motion is t∗ = 2δ/U = m0 lU/M0 , which is independent of the mass of the beam.

620

8 Dynamic Plasticity

The shape of the deformed beam at any instant can be determined by solving the appropriate differential equation for w. It follows from (8.98), (8.100), and (8.101) that ⎫ 2δ (ξ − z) ∂w ⎪ ⎪ = , 0 ≤ τ ≤ τ0 , ⎬ ∂τ ξ (1 + ρξ ) ∂w 6δ (1 − τ ) (1 − z) ⎪ ⎭ = , τ0 ≤ τ ≤ 1,⎪ ∂τ 3 + 2ρξ

(8.102)

the relationship between τ and ξ in the interval 0 ≤ τ ≤ τ 0 being given by the second equation of (8.100). The differentiation of this equation with respect to ξ gives ρξ (2 + ρξ ) dτ , = dξ 3 (1 + ρξ )2

0 ≤ τ ≤ τ0 .

The above relation can be used to change the independent variable from τ to ξ in the first equation of (8.102), the result being ∂w 2δρ (ξ − z) (2 + ρξ ) , = ∂ξ 3 (1 + ρξ )3

0 ≤ z ≤ ξ.

Integrating, and using the boundary condition w = 0 at the plastic hinge z = ξ , we obtain the solution w 1 + ρξ 2 1 ξ −z ρ (ξ − z) , 0 ≤ τ ≤ τ0 , = ln − 2− δ 3ρ 1 + ρz 3 1 + ρξ (1 + ρξ ) (1 + ρz) (8.103) which holds over the length 0 ≤ z ≤ ξ. The remainder of the beam is undeformed, giving w = 0 for ξ ≤ z ≤ 1. When τ > τ0 , the expression for w is readily obtained by integrating the second equation of (8.102), and using the condition of continuity of w at τ = τ 0 (ξ = 1). It is easily shown that w 1+ρ 1−z 2 = 3τ (2 − τ ) ln + δ 3 + 2ρ 3ρ 1 + ρz 1 1−z ρz 3ρ − 2+ + , 3 1+ρ (1 + ρz) (3 + 2ρ)

(8.104) τ0 ≤ τ ≤ 1,

Equations (8.103) and (8.104) furnish the vertical displacement of the bent beam throughout the motion following the impact. The slope of the deflection curve at z = 1 is nonzero for τ 0 ≤ τ ≤ 1, implying a discontinuity which occurs in the same sense as that permitted by the plastic hinge. The final shape of the deformed cantilever and the deflection of the tip of the beam as a function of time are of special practical interest. Setting τ = 1 in (8.104), and denoting the limiting deflection by w∗ , we obtain the relation

8.6

Impact Loading of Prismatic Beams

621

1+ρ 2 1−z w∗ = ln + δ 3ρ 1 + ρz 3 (1 + ρ) (1 + ρz)

(8.105)

giving the shape of the cantilever when it has come to rest. The deflection at the tip of the cantilever during its motion is obtained by setting z = 0 in (8.103) and (8.104), the result being ⎫ 3 ξ (2 + ρξ ) w0 ⎪ ⎪ = ln (1 + ρξ ) − , 0 ≤ τ ≤ τ , 0 ⎬ δ 2ρ 3 (1 + ρξ )2 ⎪ 3τ (2 − τ ) 2 2 + 7ρ/3 w0 ⎭ , τ0 ≤ τ ≤ 1⎪ = + ln (1 + ρ) − δ 3 + 2ρ 3ρ (1 + ρ) (3 + 2ρ) (8.106) where ξ is given by (8.100) as a function of τ < τ0 . If the mass of the beam is small compared to that of the striking object, so that ρ tends to zero, the plastic hinge moves almost instantaneously to the built-in end, and (8.104) reduces to w/δ = τ (2 − τ ) (1 − z) ,

0 ≤ τ ≤ 1.

It follows that δ is identical to the final deflection of the tip of the cantilever when the mass of the beam is negligible in comparison with the striking mass. The beam rotates in this case as a rigid body with the plastic hinge at the built-in end, the time taken by the striking object to come to rest being twice the time required by it to travel the same distance with a constant velocity U. The influence of the value of ρ on the final shape of the beam and the time dependence of the tip deflection are shown in Figs. 8.20 and 8.21, respectively. The problem of central impact of a simply supported beam can be similarly treated (Ezra, 1958). The transverse impact of a beam built-in at both ends has been considered by Parkes (1958) and Jones (1989). The influence of elastic deformation of the beam has been investigated by Symonds and Fleming (1984).

8.6.2 Rate Sensitivity and Simplified Model When the ratio of the kinetic energy input to the greatest possible elastic energy in the beam is sufficiently large, the strain-rate dependence of the yield stress must be included in the analysis for a realistic prediction of the dynamic behavior. Since the strain-rate influence generally changes the mode of deformation, a simple correction factor for the yield stress could not be applied to the rate-independent theory without discrimination. To illustrate the procedure, we consider the cantilever beam of Fig. 8.19(a) and assume the mass m0 to be attached to the tip instead of being dropped on the beam. Following Bodner and Symonds (1962), the relationship between the uniaxial stress σ and the plastic strain rate ε˙ will be taken in the form of (8.13), specialized by setting n = λ = 0, and by replacing m with 1/n. The power law,

622

8 Dynamic Plasticity

Fig. 8.20 Final shape of the cantilever beam due to impact loading with different ratios of the attached mass to the mass of the beam

Fig. 8.21 Deflection of the cantilever tip as a function of time for various mass ratios

8.6

Impact Loading of Prismatic Beams

623

σ =1+ Y

1/n ε˙ , α

(8.107)

where α and n are empirical constants, has been found to fit with Manjoine’s experimental data (1944) reasonably well by taking a ≈ 400/s, n ≈ 5 for mild steel, and a ≈ 6500/s, n ≈ 4 for aluminum. In the case of high strain rates with mild steel specimens, different values of these constants have been suggested by Hashmi (1980). In view of (8.107), and the fact that ε˙ varies linearly with the vertical distance y measured from the axis of the beam, the bending moment for a beam of rectangular section of width b and depth h is given by

h/2

M = 2b

σ dy = M0

0

2 1+ 2 ε˙ 0

ε0 0

1/n ε˙ ε˙ dε˙ , α

where M0 is the quasi-static value of the fully plastic moment, equal to bh2 Y/4, and ˙ with κ˙ denoting ε˙ 0 is the maximum strain rate in the cross section, equal to κh/2, the curvature rate of the bent axis. The relationship between the bending moment M and the curvature rate κ˙ may therefore be written as M =1+ M0

1/n κ˙ , β

β=

2α 1 n . 1+ h 2n

(8.108)

The nature of the (M, κ) ˙ curve is therefore identical to that of the (σ, ε˙ ) curve and is completely defined by the empirical constants α and n for a given depth of the beam. Due to the effect of strain hardening of the material, the actual bending moment will depend not only on the curvature rate but also on the curvature of the beam, as has been shown experimentally by Apsden and Campbell (1966). For the cantilever beam subjected to a tip impulse, as we have seen, a plastic hinge starts at the tip and moves toward the built-in end, when the material is ideally plastic. A rate-sensitive material, on the other hand, requires the plastic region to initially extend over the whole length of the beam without the formation of a localized plastic hinge. This is a consequence of the fact that the bending moment is a continuous function of the curvature rate, which vanishes at the tip of the cantilever and has its greatest value at the built-in end. The plastic zone continually shrinks during the motion and becomes zero when the beam comes to rest. The shape of the ratesensitive cantilever at an intermediate stage of motion is compared with that of the ideally plastic cantilever during its first phase of motion in Fig. 8.22. For simplicity, the analysis will be carried out on the assumption that the outer portion of the cantilever is one of constant slope ψ that varies with time t as the motion continues. It would be instructive at first to derive the rate-independent solution corresponding to the above simplified model (Mentel, 1958). Since the total momentum of the beam together with the attached mass has a constant value equal to the applied tip impulse I, while the resultant angular momentum about the built-in end is equal to the angular impulse Il – M0 t, we have

624

8 Dynamic Plasticity

Fig. 8.22 Simplified deformation mode for a cantilever subjected to tip impulse. (a) Perfectly plastic material and (b) rate-sensitive material

⎫ 1 ˙ 2 ⎪ ˙ l = I, ⎪ mψξ l + m0 ψξ ⎬ 2 1 ⎪ ˙ 2 l − ξ l2 + m0 ψξ ˙ l2 = Il − M0 t,⎪ mψξ ⎭ 2 3

(8.109)

where m is the mass of the beam, m0 is the attached mass, and the dot denotes the time derivative. These equations are immediately solved for ψ˙ and t as functions of ξ, the result being ψ˙ =

1 , m0 ξ (1 + ρξ ) l

Il t= 3M0

ρξ 2 1 + ρξ

,

where ρ denotes the ratio m/2m0. The time taken by the hinge to reach the built-in end (ξ = 1) and the corresponding angle of rotation are found to be t0 =

ρ 1+ρ

Il ρ (4 + 3ρ) I 2 , ψ0 = . 3M0 6 (1 + ρ)2 M0 m0

(8.110)

It is interesting to note that the duration of the first phase is identical to that given by (8.100), if we set I = m0 U. Since the second equation of (8.109) continues to hold (with ξ = 1) in the second phase, during which the whole beam rotates about the fixed end, the angular velocity is ψ˙ =

3 (Il − M0t ) , (3 + 2ρ) m0 l2

t ≥ t0 .

In view of the initial condition ψ = ψ 0 when t = t0 , this equation is readily integrated to give

8.6

Impact Loading of Prismatic Beams

625

3 (t − t0 ) M0 (t + t0 ) ψ − ψ0 = 1− , 2l (3 + 2ρ) m0 l

t ≥ t0

(8.111)

The beam comes to rest when ψ˙ = 0, the total time of impact t∗ and the final angle of rotation ψ∗ being found as t∗ =

Il , M0

ψ∗ =

I2 . 2M0 m0

Surprisingly, both t∗ and ψ∗ are independent of the mass ratio ρ according to this simplified analysis, the result for t∗ being in agreement with that obtained in the previous solution. The plastic work done during the second phase is M0 (ψ ∗ — ψ 0 )> which is used up in absorbing a part of the kinetic energy input equal to I2 /2m0 . For sufficiently small values of ρ, most of the kinetic energy is absorbed in the second phase.

8.6.3 Solution for a Rate-Sensitive Cantilever The inclusion of rate dependence of the yield moment completely changes the kinematics of the dynamic response of the beam, Fig. 8.22(b). The outer portion CA of constant slope ψ, at any instant, has unloaded from the plastic state, while the inner segment OC has its bending moment increasing from M0 at C to a magnitude M0 at 0. Let M denote the magnitude of the bending moment at a typical section in the plastic region at a distance sl from the free end. If the inertia effects in the plastic region are disregarded, the shearing force has a constant value R, and the bending moment distribution may be written as M = M0 + R (s − ξ ) l = M0 − R (1 − s) l. Eliminating R and M between these relations in turn, and using (8.108) to express the ratios M/M0 and M0 /M0 in terms of the corresponding curvature rates, we have κ˙ = κ˙ 0

s−ξ 1−ξ

n ,

Rl (κ˙ 0 /β)1/n = . M0 1−ξ

(8.112)

The curvature rate increases from zero at s = ξ to attain its greatest value κ 0 at the fixed end s = 1. Since ∂θ/∂s = −lκ, where θ is the local slope of the plastic segment, the local angular velocity is θ˙ = l

1 s

κ˙ ds = lκ˙ 0

1−ξ 1+n

1−

s−ξ 1−ξ

1+n (8.113)

in view of the boundary condition θ˙ = 0 at s = 1. Setting s = ξ in (8.113) furnishes the angular velocity ψ˙ of the rigid outer segment as

626

8 Dynamic Plasticity

1−ξ . 1+n Similarly, integrating the equation ∂ w/∂s ˙ = −lθ˙ , where w ˙ is the particle velocity, and using the boundary condition w˙ = 0 at s = 1, we obtain the velocity distribution in the plastic region as ψ˙ = lκ˙ 0

w˙ = l κ˙ 0 2

1−ξ 1+n

$ % 1−ξ s − ξ 2+n 1− . (1 − s) − 2+n 1−ξ

(8.114)

In particular, the velocity w ˙ c at the rigid/plastic interface s = ξ and the free-end velocity w˙ a which exceeds w˙ c by the amount ξ ψ˙ are given by w ˙ c = l2 κ˙ 0

(1 − ξ )2 , 2+n

w˙ a = l2 κ˙ 0

1−ξ 2+n

1+

ξ . 1+n

(8.115)

In all the preceding relations, ξ is a function of t to be determined. Once this is known, along with κ˙ 0 , the physical quantities κ, θ , and w can be found by time integration, the initial conditions being κ = θ = w = 0 at t = 0. The two other equations necessary for the mathematical formulation of the problem are furnished by the principle of impulse and momentum. The equation of linear momentum in the vertical direction and the equation of angular momentum about the built-in end are easily shown to be ⎫ 1 ⎪ ⎪ Rdt = m0 w˙ a + mξ (w˙ c + w˙ a ) cos ψ + G, ⎬ 2 0 , t ⎪ ξ 1 ⎭ ˙ a + 2w˙ c ) + m0 w˙ a l + H ⎪ Il − Rdt = mlξ (w˙ c + w ˙ a ) − (w 2 3 0

I−

t

(8.116)

where G and H represent the linear and angular momentum, respectively, of the plastic region OC and are given by G≈m

1 ξ

w ˙ ds,

H ≈ ml

ξ

1

w˙ (1 − s) ds.

It turns out that G and H make only minor contributions to the total momentum during the motion. For practical purposes, it is therefore a good approximation to take G≈

ml2 κ˙ 0 (1 − ξ )3 , 3 (2 + n)

H≈

ml3 κ˙ 0 (1 − ξ )4 . 3 (2 + n)

(8.117)

The first expression of (8.117) is obtained by assuming w˙ to be given by the righthand side of the first equation of (8.115) with j written for ξ, so that the fixed-end conditions w˙ = ∂ w/∂s ˙ = 0 at s = 1 are identically satisfied. The expression for H is also obtained in the same way, but the numerical factor in the denominator is

8.6

Impact Loading of Prismatic Beams

627

adjusted in such a way that both equations in (8.116) furnish identical initial values of κ˙ 0 . Indeed, setting t = ξ = 0 in (8.116) results in (κ˙ 0 )t=0 =

3 (2 + n) l = cnβ (3 + 2) m0 l2

(say) ,

(8.118)

in view of (8.115) and (8.117). The integrals appearing on the left-hand side of (8.116) may be evaluated in an approximate manner, without introducing significant errors, by using the initial value of κ˙ 0 for expressing the integrands, which are then obtained from (8.112) as c Rl ≈ , M0 1−ξ

M ≈ 1 + c. M0

Introducing this approximation, and using (8.115) for the velocities, the momentum equations (8.116) are easily reduced to λ−c

τ

dτ =φ 1 −ξ 0

1−ξ 2+n

λ − (1 + c) τ = φ

2ρ ρ n−1 1 2ρ 1+ ξ − ξ 2 cos ψ + + 3 3 1+n 3 n+1

1−ξ 2+n

1+

2ρ ξ ρ + + 3 1+n 3

(8.119)

3−ξ ξ2 , 1+n

the nondimensional quantities λ, τ , and φ introduced here being defined as I t λ= √ , τ= l M0 m

M0 m , φ = l2 κ˙ 0 . m M0

By eliminating the common factor φ(1 — ξ)/(2 + n) between the two equations (8.119), and solving the resulting equation numerically under the initial condition ξ = 0 when τ = 0, we obtain τ as a function of ξ. The second equation of (8.119) then gives φ as a function of ξ, and the shape of the deformed cantilever is finally obtained from the distribution of θ determined by the integration of (8.113). The free-end deflection of the cantilever is given by wa = l

t 0

w˙ a l

t

dt = 0

φ

1−ξ 2+n

ξ 1+ dτ , 1+n

(8.120)

˙ of the rigid portion is similarly computed which follows from (8.115). The slope ψ as a function of time. The total time t∗ required by the beam to come to rest corresponds to φ = 0, or τ = λ/(l + c) in view of the second equation of (8.119). Hence

Il t∗ = M0

1/n −1 3 (2 + n) lh 2n . 1+ 1 + 2n 2α (3 + 2ρ) m0 l2

(8.121)

628

8 Dynamic Plasticity

Some of the computed results, obtained by Ting (1964) using λ = 293, ρ = √ 0.305, n = 5.0, α = 1036/s, and η = l m/M0 = 0.039s, are displayed graphically in Fig. 8.23, the duration of impact in this case being t∗ = 0.064 s (ψ∗ = 1.033 radians). The broken curves are based on the approximation cos ψ ≈ 1, which neglects the change in geometry. The theory has been found to be in good overall agreement with experiment by Bodner and Symonds (1962). The agreement is not so good, however, when the rate-independent theory is used. ˙ of the rigid segment The preceding theory indicates that the angular velocity ψ decreases almost linearly with time. Using the linear relationship between ψ and t as an approximation (Ting, 1964), and assuming an initial value of ψ equal to 3l/(3 + 2ρ)m0 l, it is easily shown that

Fig. 8.23 Variation of curvature rate and slope at the built-in end of an impulsively loaded ratesensitive cantilever which is progressively rendered plastic (after Bodner and Symonds, 1962)

8.6

Impact Loading of Prismatic Beams

ψ≈

3It t 2− ∗ , 2 (3 + 2ρ) m0 l t

629

ψ∗ =

3I 2 , 3 (3 + 2ρ) (1 + c) M0 m0

(8.122)

in view of the terminal condition ψ˙ = 0 at t = t∗ and the initial condition ψ = 0 at t = 0. The terminal slope ψ∗ predicted by (8.122) is somewhat smaller than that given by (8.111). Further solutions on the dynamics of beams including strain-rate effects, using simplified models, have been discussed by Perrone (1965) and Lee and Martin (1970), among others. The influence of strain hardening of the material has been considered by Jones (1967) and Perrone (1970). As a consequence of the elastic response of the material, an elastic flexural wave develops at the point of impact and propagates along the length of the cantilever before it is reflected from the fixed end. The reflected wave moves back toward the tip of the cantilever and meets the traveling plastic hinge midway along the beam. The interaction between the primary bending wave and the reflected wave significantly modifies the deformation mode from that predicted by the rigid/plastic theory. The subsequent deformation of the cantilever depends to a large extent on the ratio of the tip mass to the mass of the beam material. At the base of the cantilever, the bending moment oscillates in magnitude and sense, producing some reversed bending in this region. The final deflection of the tip of the cantilever is found to be about the same as that predicted by the rigid/plastic theory (Stronge and Yu, 1993).

8.6.4 Transverse Impact of a Free-Ended Beam A uniform beam of mass 2m and length 2l, initially at rest, is subjected to a concentrated impact load at its midpoint, such that the central section instantaneously attains a velocity U which is subsequently maintained constant. For simplicity, we propose to analyze the equivalent problem of a beam moving with a uniform normal velocity U, the central section of the beam being suddenly brought to rest by mean of a rigid stop, Fig. 8.24(a). Over a sufficiently small time interval after the beam strikes the stop, plastic hinges occur not only at the central section but also at two other sections each at a distance ξ l from the center. The problem has been treated by Symonds and Leth (1954) for a beam of finite length, the corresponding problem for infinitely extended beams having been discussed by Lee and Symonds (1952), Conroy (1952, 1956), and Shapiro (1959). Because of symmetry, it is only necessary to consider one-half of the beam with segments OH and HA subjected to end moments of magnitude M0 , which is assumed to have a constant value. The shear force vanishes at the moving hinge H, where the bending moment has a relative maximum, but a shear force of magnitude R/2 exists at O, where R denotes the reaction exerted by the stop at any instant t. The segment OH rotates about O with an angular velocity dφ/dt while HA rotates with an angular velocity dψ/dt and translates with zero acceleration of its mass center. OH acquires a permanent deformation from the traveling plastic hinge H, but the material to the right of this hinge at any instant is undeformed. Since the moments of inertia of

630

8 Dynamic Plasticity

Fig. 8.24 Beam of infinite length subjected to transverse impact. (a) Deformation mode following impact and (b) graphical presentation of results

OH and HA about the respective mass centers are mξ 3 l2 /12 and m(1 – ξ )3 l2 /12, respectively, the equations of angular rigid-body motion of these two segments may be written as 1 2 2 d2 φ mξ l = −2M0 , 3 dt2

d2 ψ 1 m (1 − ξ )3 l2 2 = M0 . 12 dt

(8.123)

Since the deformed segment OH is actually curved, φ must be interpreted as the angle made by the chord joining O and H. The velocity of the mid-section of the outer segment is always a constant, and its value must be equal to the impact velocity U. It follows from the kinematics of the motion that ξ

1 U dψ dφ + (1 − ξ ) = . dt 2 dt l

(8.124)

It may be noted that the linear acceleration is discontinuous across the moving hinge H. Equations (8.123) and (8.124) form the basis for finding the three unknown quantities φ, ψ, and ξ . Introducing the dimensionless quantities ω=

l dψ M0 t l dφ , λ= , τ= , U dt U dt mlU

and using a dot to denote differentiation with respect to τ , the preceding equations (8.123) and (8.124) can be expressed in the more convenient form ⎫ λ˙ = 12/ (1 − ξ )3 ,⎬ ω˙ = −6/ξ 3 , 1 ⎭ ξ ω + (1 − ξ ) λ = 1 2

(8.125)

8.6

Impact Loading of Prismatic Beams

631

At the moment of impact, H coincides with O, and the angular velocity of HA vanishes. Hence the initial conditions are ξ = 0,

λ = 0,

φ=ψ =0

when τ = 0.

The last equation of (8.125) therefore indicates that ω tends to infinity as ξ tends to zero, such that ξω = 1 in the initial state, implying a singularity in the angular velocity at the point of impact. To obtain the solution of (8.125), we begin by differentiating the last of these ˙ equations with respect to τ , and using the other two equations to eliminate ω ˙ and λ, the results being easily shown to be 6 (1 − 2ξ ) 1 . ω − λ ξ˙ = 2 ξ 2 (1 − ξ )2

(8.126)

Differentiating this equation with respect to τ , and substituting again for ω ˙ and λ˙ using (8.125), we obtain the differential equation for ξ as

ξ (1 − ξ ) (1 − 2ξ ) ξ¨ + 1 − 3ξ + 3ξ 2 ξ 2 = 0. Since the independent variable τ does not appear explicitly, a first integration can be easily carried out by setting ξ˙ = η, so that ξ¨ = η(dη/dξ). The preceding equation then becomes 1 − 3ξ + 3ξ 2 η dn + = 0. dξ ξ (1 − ξ ) (1 − 2ξ ) Since we are concerned only with the situation ξ < above equation results in η=

1 2,

the integration of the

√ dξ 1 − 2ξ = , dτ Cξ (1 − ξ )

(8.127)

where C is a constant to be determined later. Integrating again, and using the initial condition ξ = 0 when τ = 0, we obtain the solution τ=

C 1 − 2ξ . 1 − 1 + ξ − ξ2 5

(8.128)

Substituting (8.127) for ξ˙ into (8.126), and combining the resulting expression with the last equation of (8.125), the dimensionless angular velocities are found as √ ω =1+

1 − 2ξ , ξ

√ 1 − 2ξ λ=2 1− , 1−ξ

(8.129)

632

8 Dynamic Plasticity

on setting C = 16 to satisfy the condition λ = 0 when τ = 0. The angles φ and ψ at any instant during the motion of the outer hinge are ⎫ ξ (2 − ξ ) ⎪ mU 2 ⎪ τ+ ωdτ = ,⎪ ⎬ M 12 0 0 0 t ⎪ ξ2 mU 2 τ mU 2 U ⎪ ⎪ 2τ − λdt = λdτ = ψ= , ⎭ l 0 M0 0 M0 16

U φ= l

t

mU 2 ωdt = M0

τ

(8.130)

in view of (8.127) and (8.129), the dimensionless timeτ being given by (8.128) with C = 16 . The reaction R at the stop is most conveniently obtained from the condition of moment equilibrium of the segment OH about its mass center. Thus 1 3 2 d2 φ = Rξ l − 8M0 . mξ l 3 dt2 The elimination of d2 φ/dt2 between this equation and the first equation of (8.123) immediately furnishes R = 6M0 /ξ l. For sufficiently small values of ξ , the beam behaves as being infinitely long. In 2 this case, (8.127) and (8.128) √ give ξ ≈ 6/ξ and τ ≈ ξ /12, so that the parameter Rl/M0 becomes equal to 3/τ approximately. An elastic/plastic analysis for an infinitely long beam with an arbitrary moment–curvature relation has been given by Duwez et al. (1950). The preceding analysis remains valid until the angular velocities of the inner and outer segments of the beam become equal to one another. Setting ω = λ, and using (8.129) and (8.128), the corresponding values of ξ and τ are found to be ξ0 =

√ 1/2 5−2 ≈ 0.486,

τ0 ≈ 0.0265.

During the subsequent motion (τ > τ 0 ), there is a single plastic hinge occurring at the central section of the beam, and the two halves rotate as rigid bodies with an angular velocity dφ/dt, the relevant equations in dimensionless form being ω˙ = −3,

φ˙ = mU 2 /M0 ω,

τ ≥ τ0 .

The first equation defines the angular motion, while the second equation follows from the definition of ω. The integration of the above equations gives ω = A − 3τ ,

φ=

mU 2 M0

3 B + Aτ − τ 2 , 2

τ ≥ τ0 ,

(8.131)

where A and B are constants. Since ω = ω0 ≈ 1.3456 and φ = φ 0 ≈ 0.0877mU2 /Mo when τ = τ 0 , in view of (8.129) and (8.130), we have A ≈ 1.425 and B ≈ 0.051.

8.7

Dynamic Loading of Circular Plates

633

The motion stops when ω = 0, and this corresponds to τ ≈ s 0.495, giving φ ≈ 0.389mU2 /M0 as the limiting angle of rotation. The reaction R at the support for τ > τ 0 discontinuously changes to a constant value equal to 3M0 /l. Figure 8.24(b) shows the variation of φ, ψ, and R with time over the range 0 ≤ τ ≤ 0.05. The neglect of geometry changes, which is implicit in the analysis, would be justified if the value of φ 0 is sufficiently small, preferably less than about 0.15 radians (say). The assumption of a rigid/plastic material, on the other hand, requires that the kinetic energy absorbed in the plastic deformation greatly exceeds the elastic energy stored in the beam. The range of validity of the solution may therefore be approximately defined as kM0 l 5 mU 2 < , < El M0 3 where EI is the flexural rigidity of the beam, and k is a numerical factor (presumably of the order of 10) that may be found from experiments. Outside this range, the elastic deformation of the beam must be considered for a realistic prediction of the dynamic behavior. A variety of related problems on the dynamic plastic behavior of beams have been considered by Lee and Symonds (1952), Symonds (1953), and Seiler and Symonds (1954) using force pulses; by Symonds (1954), Johnson (1972), and Nonaka (1977) for blast loading; by Cotter and Symonds (1955), Martin and Symonds (1966), and Yu and Jones (1989) for impulsive loading. The dynamics of elastic/plastic beams has been treated by Bleich and Salvadori (1953), Seiler et al. (1956), Martin and Lee (1968), Symonds and Fleming (1984), and Reid and Gui (1987). The influence of axial restraints has been examined by Symonds and Mentel (1958), and that of shear has been considered by Karunes and Onat (1960). The dynamic load characteristics of curved bars have been discussed by Owens and Symonds (1955), Perrone (1970), and Stronge et al. (1990). The dynamic plastic response of frames has been considered by Rawlings (1964), Symonds (1980), and Raphanel and Symonds (1984). The dynamic plastic behavior of a circular beam subjected to impact loading has been discussed by Yu et al. (1985), and that of a right-angled bent cantilever loaded at its tip has been examined by Reid et al. (1995). Lower and upper bound principles in the dynamic loading of structures have been discussed by several authors including Kalisky (1970) and Stronge and Yu (1993). A great deal of published work on the dynamic failure of structural members caused by severe plastic deformations has been reported by Wierzbicki and Jones (1989).

8.7 Dynamic Loading of Circular Plates 8.7.1 Formulation of the Problem The theory of plastic bending of circular plates, developed in Chapter 4, can be extended to include inertia effects which arise under dynamic loading conditions. In accordance with the usual assumption for thin plates, shearing stresses normal to the

634

8 Dynamic Plasticity

plate surface are neglected in comparison with the bending stresses parallel to the surface. The resultant shearing force must be included, however, in the equation of dynamic equilibrium. For a rotationally symmetric state of stress, the nonzero stress resultants Mr , Mθ , and Q, denoting the radial and circumferential bending moments and the transverse shearing force, respectively, are functions of the radial coordinate r and time t. If the downward deflection of the plate is denoted by w, which is also a function of r and t, the inertia force per unit area acting in the upward sense is −μ(∂ 2 w/∂t2 ), where μ is the surface density of the plate. Then the equations of dynamic equilibrium may be written as ∂ (rMr ) − Mθ − rQ = 0, ∂r

∂ ∂ 2w (rQ) + rp = μr 2 , ∂r ∂t

(8.132)

where ρ denotes the local intensity of the normal pressure acting on the plate. Eliminating Q from the first equation of (8.132) by means of the integrated form of the second equation, we obtain the governing differential equation ∂ (rMr ) − Mθ = − ∂r

r 0

∂ 2w p − μ 2 r dr, ∂t

(8.133)

which is independent of the mechanical properties of the plate material. For a plastically deforming plate made of an ideally plastic isotropic material, these properties are specified by the yield condition involving Mr , Mθ , and a fully plastic moment M0 , and the associated flow rule defining the ratio of the rates of change of the principal curvatures κ r and κ θ , which are given by κr = −

∂ 2w , ∂r2

κθ = −

1 ∂w . r ∂r

(8.134)

The rate of change of all physical quantities is specified by the partial derivative with respect to t, the effect of geometry changes being disregarded. If the material yields according to the maximum shear stress criterion of Tresca, the yield locus in the moment plane is the hexagon ABCDEF shown in Fig. 8.25(a). The generalized strain rate having components κ˙ r = ∂ κ˙ r /∂t and κ˙ θ = ∂ κ˙ θ /∂t, corresponding to each side of the hexagon, is represented by a vector directed along the exterior normal to the side. If the stress state is represented by a vertex of the hexagon, the flow vector may have any direction between those defined by the two limiting normals. In most physical problems, the plate may be divided into a central circular region together with surrounding annular regions, each of which corresponds to a different plastic regime. Across a circle separating two plastic regimes, the radial moment Mr and the shear force Q must be continuous but the circumferential moment Mg may be discontinuous. The deflection w and the velocity ∂w/∂t must also be continuous across , but all the second derivatives of w may become discontinuous as will be shown later. When the velocity slope ∂ 2 w/∂r ∂t and hence the circumferential curvature rate κ˙ θ are discontinuous across , it is called a hinge circle, which may be either stationary or moving with respect to the plate. Since the discontinuity

8.7

Dynamic Loading of Circular Plates

635

Fig. 8.25 Dynamic of pulse-loaded circular plates. (a) Yield condition and (b) growth of central deflection with time (p ≤ 2 p0 )

must be considered as the limit of a narrow annulus of rapid change in slope, the ratio κ˙ r /˙κθ becomes infinite at the hinge circle. It follows that a hinge circle can be associated only with the side CD and FA, as well as the corners A, C, D, and F of the yield hexagon (Hopkins and Prager, 1954). To establish the relations between the possible discontinuities across a hinge circle , we represent this circle by the equation r = ρ(t). Since w and ∂w/∂t are continuous across , the derivative of these quantities along this curve is also continuous. Consequently,

∂w dp ∂w + = 0, ∂t dt ∂t

dp ∂ 2 w ∂ 2w + = 0, ∂t2 dt ∂r∂t

(8.135)

where the square brackets denote the jump in the enclosed quantities. Since ∂w/∂t is continuous, the first equation of (8.135) indicates that the slope ∂w/∂r can be discontinuous only across a stationary hinge circle (dρ/dt = 0). The second equation of (8.135), on the other hand, shows that the acceleration ∂ 2 w/∂t2 must be continuous across a stationary hinge circle, but discontinuous across a moving hinge circle. Since ∂w/∂t is continuous across a moving hinge circle, its tangential derivative gives

dp ∂ 2 w ∂ 2w = 0, + ∂r∂t dt ∂ 2 r

dp

= 0. dt

The radial curvature ∂ 2 w/∂t2 is therefore discontinuous across a moving hinge circle. Due to the continuity of the radial bending moment, the space and time derivatives of Mr are both discontinuous across a moving hinge circle.

636

8 Dynamic Plasticity

8.7.2 Simply Supported Plate Under Pressure Pulse A circular plate which is simply supported round its edge r = a is subjected to a uniformly distributed normal pressure of intensity ρ that is brought on suddenly at t = 0. The pressure is maintained constant during a small time interval 0 < t < t0 , and then suddenly removed at t = t0 . The dynamic plastic action will occur only if ρ exceeds the value ρ 0 necessary for quasi-static plastic collapse (Section 4.1). We are therefore concerned here with p > p0 = 6M0 /a2 . For ρ = ρ 0 , the inertia forces do not arise because the deformation occurs indefinitely slowly. The dynamic behavior of the plate for ρ > ρ 0 is found to depend on whether or not ρ is greater or less than 2ρ 0 . Following Hopkins and Prager (1954), the former will be referred to as high load and the latter as medium load. We begin by considering the medium load, characterized by ρ 0 ≤ ρ ≤ 2ρ 0 The first phase of the dynamic behavior of the plate corresponds to the time interval 0 ≤ t ≤ t0 , during which the applied pressure is held constant throughout the plate. The equilibrium equation (8.133) therefore simplifies to ∂ (rMr ) = Mθ − M0 ∂r

3pr2 − λ , p0 a2

(8.136)

where λ (r,t) =

μ M0

r 0

∂ 2w rdr ∂t2

(8.137)

Here, use has been made of the relation p0 a2 = 6M0 . Equations (8.136) and (8.137) hold for all values of the ratio p/p0 . Under medium loads, the middle surface of the entire plate may be assumed to deform into a conical surface, as in the case of plastic collapse under the pressure p = p0 . We therefore take w (r,t) =

r p0 1− f (t) , μ a

(8.138)

where f(t) is a function of t to be determined. This expression for w satisfies the boundary condition w = 0 at r = a for all t ≥ 0. The rates of curvature associated with (8.138) are κ˙ r = −

∂ 3w = 0, ∂r2 ∂t

κ˙ θ = −

1 ∂ 2w p0 = f (t) . r ∂r∂t μar

For f (t) > 0, these relations correspond to the flow rule associated with side AB of the yield hexagon, requiring Mθ = M0 and 0 ≤ Mr ≤ M0 . It follows from (8.137) and (8.138) that

8.7

Dynamic Loading of Circular Plates

λ (r,t) =

637

r2 a2

2r 3− f (t) . a

Substituting this into (8.133), and using the yield condition Mθ = M0 , the resulting equation can be readily integrated. Since Mr = M0 at r = 0, we get r pr2 r2 Mr 1 − =1− + f (t) . M0 p0 a2 a2 2a The boundary condition Mr = 0 at r = a will be satisfied for all t if f”(t) is a constant equal to 2(p/p0 – 1). Since w = ∂w/∂t = 0 at t = 0, this gives f (t) =

p − 1 t2 . p0

(8.139)

The deflection of the plate as a function of r and t is completely defined by (8.138) and (8.139). The bending moment distribution is, however, independent of t, the radial moment being finally given by 2 Mr r r r p 1− =1− 2 2− − . M0 a a p0 a

(8.140)

The bending moment Mr according to (8.140) monotonically decreases from M0 at r = 0 when p0 < p < 2p0 , and the yield condition is therefore nowhere violated. This completes the solution for the first plastic phase of the motion. At t = t0 , the load is suddenly removed, but the motion continues until the kinetic energy acquired during the application of the load is dissipated by plastic work. The second plastic phase therefore corresponds to p = 0 and t > t0 . All conditions of the problem in the second phase can be satisfied by assuming w to be given by (8.138) together with f”(t) = –2. In view of the yield condition Mθ = M0 , and the conditions of continuity of w and ∂w/∂t at t = t0 , we have ⎫ pt0 (2t − t0 ) − t2 ,⎪ ⎪ ⎬ p0 t ≥ t0 r ⎪ r2 Mr ⎪ ⎭ =1− 2 2− . M0 a a

f (t) =

(8.141)

Since Mr lies between 0 and M0 , the yield condition is nowhere violated, the only restriction on the solution being pt0 f (t) = 2 − t ≥ 0, p0

or

t ≤ t∗ =

pt0 . p0

At the instant t = t∗ , the entire plate comes to rest, since f’(t) = 0 implies that the velocity w is identically zero. The final deflection of the plate is found from (8.138) and (8.141) as

638

8 Dynamic Plasticity

p w = μ ∗

p r − 1 t02 1 − , p0 a

(8.142)

which indicates that the deflection of the plate in the final stage at any radius r is p/p0 times that at t = t0. Figure 8.25(b) shows the manner in which the ratio w/w∗ varies with t/t∗ for given values of p/p0 in the medium load range. The dynamic bending problem for circular and annular plates under a linearly distributed pressure pulse has been considered by Jones (1968a,b).

8.7.3 Dynamic Behavior Under High Loads The bending moment distribution (8.140) is not admissible for p > 2p0 , since the predicted value of Mr then exceeds M0 in the neighborhood of the plate center. Indeed, ∂ 2 Mr/ ∂r2 is positiveat r = 0 according to (8.140) when p > 2p0 , indicating that Mr is a relative minimum at r = 0, when Mr = M0 . We may therefore expect Mr = Mθ = M0 in a central region of the plate within some radius p0 , which corresponds to the corner A of the yield locus. It follows from (8.136) and (8.137) that this region moves down with a constant acceleration equal to p/μ. Thus ∂ 2 w/∂t2 = p/μ,

0 ≤ r ≤ ρ0 .

In the remainder of the plate, the plastic regime AB should apply, and a conical mode of deflection may be assumed as before. In view of the initial conditions w = ∂w/∂t = 0 at t = 0, and the condition of continuity of w across r = ρ 0 , the expression for the deflection during the initial phase of the motion may be written as ⎧ 2 pt ⎪ ⎪ 0 ≤ r ≤ ρ0 , ⎨ , w (r,t) = 2μ2 pt ⎪ a−r ⎪ ⎩ , ρ0 ≤ r ≤ a, 2μ a−ρ0

(8.143)

where ρ 0 is independent of t but depends on the applied pressure p. The boundary condition w = 0 at r = a is identically satisfied, and the curvature rates associated with the velocity field (8.143) are κ˙ r = κ˙ θ = 0, κ˙ r = 0,

0 ≤ r ≤ ρ0 , pt κ˙ θ = , μr (a − ρ0 )

ρ0 ≤ r ≤ a.

Since κ θ is discontinuous across r = ρ 0 , this radius coincides with a stationary hinge circle. The above relations are seen to be compatible with the assumed plastic regimes inside and outside the circle of radius ρ 0 . It may be noted that the slope ∂w/∂r is discontinuous across the hinge circle. To obtain the bending moment distribution in the region ρ 0 ≤ r ≤ a, we observe that the acceleration ∂ 2 w/∂t2 is equal to p/μ inside the circle r = ρ 0 , and equal to

8.7

Dynamic Loading of Circular Plates

639

(p/μ)(a – r)/(a – p0 ) outside the circle r = ρ 0 . The substitution in (8.137) therefore results in p 3r2 − (r − ρ0 )2 (2r + ρ0 ) , ρ0 ≤ r ≤ a. λ (r,t) = p0 a2 (r − ρ0 ) Equation (8.136) is now readily integrated to obtain an expression for Mr outside the circle r = ρ 0 . Since Mr = M0 at r = ρ 0 in view of the continuity of the bending moment, and Mr = 0 at r = a in view of the boundary condition, we get 2a3 p = , 0 ≤ t ≤ t0 , p0 (a − ρ0 )2 (a + ρ0 ) Mr a (r − ρ0 )3 (r + ρ0 ) , ρ0 ≤ r ≤ a. =1− M0 r (a − ρ0 )3 (a + ρ0 )

(8.144) (8.145)

The last equation indicates that Mr decreases monotonically from M0 at r = p0 to 0 at r = a, and the yield condition is nowhere violated. The central deflection of the plate at t = t0 is δ0 = pt02 /2μ. It may be noted that ρ 0 /a tends to unity as p/p0 tends to infinity. A second plastic phase begins with the sudden removal of the load at t = t0 . During this phase, the radius of the hinge circle decreases from p0 to some radius p = p(t) at any instant, so that the kinetic energy of the plate decreases due to dissipation by plastic work. The velocity field in the second phase may therefore be taken in the form ⎧ pt ⎪ 0 , 0 ≤ r ≤ ρ, ∂w ⎨ μ = pt a−r ⎪ ∂t ⎩ 0 , ρ ≤ r ≤ a. μ a−ρ

(8.146)

Since ρ(t0 ) = ρ 0 , the velocity is automatically continuous at t = t0 . The acceleration ∂ 2 w/∂t2 vanishes inside the circle r =ρ and assumes the value (pt0 /μ)ρ (a – r)/(a – p)2 outside this circle. It follows from (8.137) that 3a r2 − ρ 2 − 2 r3 − ρ 3 p , ρ ≤ r ≤ a. λ = t0 ρ p0 a2 (a − ρ)2 Setting Mθ = M0 and p = 0 in (8.136), substituting the above expression for λ, and using the boundary conditions Mr = M0 and ∂Mr /∂r = 0 at r = ρ and Mr = 0 at r = a, the integration of the resulting equation furnishes (a − ρ) (a + 3ρ)

dρ 2p0 a3 , =− dt pt0

t0 ≤ t ≤ t1 ,

(8.147)

where t = t1 represents the instant when the hinge circle reaches the center of the plate. The function ρ(t) is obtained by integrating (8.147) under the initial condition ρ = ρ 0 at t = t0 , the result being

640

8 Dynamic Plasticity

M a =1− M0 r

r−ρ a−ρ

2

ρ (4a − 3ρ) + 2 (a − ρ) r − r2 , ρ ≤ r ≤ a, (8.148) (a − ρ) (a + 3ρ)

in view of (8.144). Evidently, t1 = pt0 /2p0 , corresponding to ρ = 0. It may be noted that (8.149) establishes the same relationship between ρ/a and t/t1 as (8.144) does between ρ 0 /a and p/2p0. Since 0 < Mr < M0 according to (8.148), the yield condition is nowhere violated. Consider now the deflection of the plate during the time interval t0 < t < t1 . Since the region inside the circle r = ρ has been associated with the single plastic regime A, the deflection in this region is found by a straightforward integration of the first equation of (8.146), using the initial condition w = pt02 /2μ at t = t0 . Thus

2p0 t t ρ ρ2 = , 1− 1− 2 = a a pt0 t1

t0 ≤ t ≤ t1,

(8.149)

in view of (8.149). For the region outside the circle of radius ρ, it is convenient to change the independent variable from t to ρ in the second equation of (8.146) using

w=

pt2 pt0 (2t − t0 ) = 0 2μ 2μ

ρ p ρ2 1− 1− 2 −1 , p0 a a

0 ≤ r ≤ ρ, (8.150)

(8.147), the result being p2 t02 3ρ ∂w r =− , 1+ a 1− ∂ρ 2μp0 a a

ρ ≤ r ≤ a.

For the annular region ρ 0 ≤ r ≤ a, which has been associated only with the plastic regime AB, the preceding equation is readily integrated under the initial condition w = (pt02 /2μ)(a − r)/(a − ρ0 ) at ρ = ρ 0 to obtain the solution pt2 a r 3 ρ0 + ρ p ρ0 − ρ 1+ , ρ0 ≤ r ≤ a. w= 0 1− + 2μ a a − ρ0 p0 a 2 a (8.151) Let τ denote the time when the contracting hinge circle coincides with a given radius r. Setting t = τ and ρ = r in (8.149), we get r pt0 r2 1− τ= 1− 2 . 2p0 a a The material at radius r belongs to plastic regime A for t ≤ τ and to plastic regime AB for t > τ , the deflection at t = τ being pt0 w (r,τ ) = 2μ

r p r2 1− 1− 2 −1 , p0 a a

8.7

Dynamic Loading of Circular Plates

641

which is obtained by setting ρ = r in (8.150). The integration of the deflection equation under the condition w = w(r, τ ) when ρ = r then furnishes pt2 w= 0 2μ

p r+ρ p2 r−ρ r 1+ −1 , 1− 2 + 1− p0 a a 2a a

ρ ≤ r ≤ ρ0 . (8.152)

Equations (8.150), (8.151), and (8.152) provide the complete solution for the deflection of the plate during the second plastic phase. The deflection at the end of this phase within the circle of radius ρ 0 is pt0 w (r,t1 ) = 2μ

r p r2 r 1− 1+ + 2 −1 , p0 a a 2a

0 ≤ r ≤ ρ0 .

(8.153)

The slope ∂w/∂r is continuous across r = ρ but discontinuous across r = ρ 0 , the discontinuity being of amount pt02 /2μ(a − ρ0 ). The jump condition (8.135) is found to be satisfied across r = ρ at each stage of the interval t0 ≤ t ≤ t1 . The central deflection of the plate is δ1 = (pt02 /2μ)(p/p0 − 1) when t = t1 , obtained by setting r = 0 in the preceding equation. The shape of the deflected circular plate at the instant t = t1 for different values of p/p0 is displayed in Fig. 8.26.

Fig. 8.26 Deformed shape of a pulse-loaded simply supported circular plate at the end of the second dynamic phase for different values of p/p0 ≥2

For t > t1 , there is a third plastic phase during which the entire plate is in the regime AB, and the velocity field may then be written as ∂w p0 r = 1− f (t) , ∂t μ a

t1 ≤ t ≤ t ∗ ,

(8.154)

which is identical in form to that given by (8.146), the instant when the plate comes to rest being denoted by t∗ . The integration of the above equation results in w (r,t) = w (r,t1 ) +

p0 r 1− f (t) , μ a

642

8 Dynamic Plasticity

where f (t) is defined in such a way that f ( t1 ) = 0. The equation of dynamic equilibrium (8.136), where p = 0 and Mθ =M0 , can be satisfied along with the boundary conditions by taking f”(t) = –2. Integrating, and using the initial conditions f(t1 ) = pt0 /p0 = 2ti and f(t1 )=0, which ensure the continuity of the velocity and deflection at t = t1 , we get f (t) = (t − t1 ) (3t1 − t) ,

f (t) = 2 (2t1 − t) ,

t∗ = 2t1

The bending moment distribution during this phase is the same as that given by (8.148) with ρ = 0, while the deflection of the plate is expressed as w (r,t) = w (r,t1 ) +

r p0 1− (t − t1 ) (3t1 − t) , μ a

t1 ≤ t ≤ t∗ .

(8.155)

The final deflection w∗ (which corresponds to t = t∗ ), considered over the region inside the circle r = ρ 0 , is obtained from (8.153) and (8.155) as w∗ =

pt02 2μ

r p r2 2r 1− + 2 −1 3+ 2p0 a a a

0 ≤ r ≤ ρ0 .

(8.156)

Outside this circle, the shape of the deformed plate is conical, and the deflection at any radius r is (a – r)/(a — p0 ) times that at r = ρ 0 , which is directly obtained from (8.156). The central deflection of the plate in the final stage is δ∗ = (pt02 /2μ)(3p/2p0 − 1). Figure 8.27 shows the variation of w∗ /δ∗ with r/a for several values of the ratio 2p0 /p and indicates how the central curved part of the plate increases with increasing pressure. The corresponding problem for a built-in circular plate has been analyzed by Perzyna (1958) and Florence (1966). The dynamic behavior of circular plates under a central circular loading has been investigated by Conroy (1969) and Liu and Strange (1996).

Fig. 8.27 Final shape of the deformed plate in relation to the central deflection for p/p0 ≥ 2. The solid circles indicate the positions of hinge circles

8.7

Dynamic Loading of Circular Plates

643

8.7.4 Solution for Impulsive Loading A simply supported circular plate is subjected to a blast-type loading which instantaneously imparts a uniform transverse velocity U to the entire plate except at r = a. The impulsive action is immediately withdrawn so that the plate is free from transverse loads thereafter. During the first phase of the dynamic plastic deformation, a central part of the plate of steadily decreasing radius ρ continues to move with velocity U, while the surrounding annulus involves a conical flow field with w = U at r = ρ. The velocity field may therefore be written as ∂w U, 0 ≤ r ≤ ρ = ∂t U (a − r) / (a − ρ) , ρ ≤ r ≤ a.

(8.157)

There is a plastic moving hinge at r = ρ across which the velocity slope is discontinuous. Both the principal curvature rates vanish in the circular region 0 ≤ r ≤ ρ, which corresponds to the plastic regime A, giving Mr = Mθ = M0. Since kr = 0 and κ˙ θ > 0 in the annular region ρ ≤ r ≤ a, it corresponds to the plastic regime AB for which Mθ = M0 , and the differential equation (8.133) for the bending moment becomes μρU ∂ (rMr ) = M0 + ∂r (a − ρ)2

r

ρ

(a − r) rdr,

ρ ≤ r ≤ a.

The bending moment must satisfy the boundary conditions Mr = 0 at r = a and Mr = M0 at r = p, while ∂Mr /∂r must vanish at r = ρ for the shearing force to be continuous. The integration of the above equation under these conditions furnishes Mr , which is the same as (8.148), while the differential equation for ρ is found to be

(a − ρ) (a + 3ρ)

2M0 a dp =− . dt μU

(8.158)

Integrating, and using the initial condition ρ = a when t = 0, the solution is obtained as

12M0 t ρ ρ2 t 1− 1− 2 = = , a t1 a μa2 U

(8.159)

where t1 denotes the time corresponding to ρ = 0. The defection of the plate during the time interval 0 ≤ t ≤ t1 is obtained by the integration of (8.157). Using (8.158), to change the independent variable from t to ρ, the second equation of (8.157) may be rewritten as ∂w Ut1 r 3ρ =− 1− , 1+ ∂ρ a a a

ρ ≤ r ≤ a.

644

8 Dynamic Plasticity

If t = τ denotes the time at which the moving plastic hinge coincides with a given circle of radius r, then

r 3ρ 1+ τ = t1 1 − a a in view of (8.159). Since w = Uτ when t = τ or ρ = r, the integration of the above differential equation for w furnishes

ρ2 r−ρ r+ρ r , ρ ≤ r ≤ a. 1− 2 + w = Ut1 1 − a a 2a a

(8.160)

Within the circle 0 ≤ r ≤ ρ, the deflection at any instant has a constant value equal to Ut. The slope ∂w/∂r is continuous across r = p, although the velocity gradient is not. The deflection at the end of this phase is obtained by setting p = 0 in (8.160). The rest of the analysis for impulsive loading is essentially the same as that for the high-pressure pulse considered before. The deflection during the second phase may therefore be written as

1 t r t r2 r , w = Ut1 1 − −1 3− 1+ + 2 + a a 2a 2 t1 t1

t1 ≤ t ≤ t∗ (8.161)

where t1 is given by (8.159). The acceleration at r = 0 has a constant value equal to –U/t1 , and the bending moment distribution satisfying the differential equation and the boundary conditions is given by (8.148) with ρ = 0. The motion stops when t = t∗ , where t∗ = 2t1 =

μa2 U . 6M0

Setting t = 2t1 in (8.161), the shape of the deformed plate after it has finally come to rest is obtained as (Wang, 1955) w∗ =

r 1 r2 2r Ut1 1 − + 2 .β 3+ 2 a a a

(8.162)

It may be noted that the slope of the deflected plate is discontinuous at r = 0 over the range t1 ≤ t ≤ t∗ . The central deflection of the plate finally attains the value δ∗ = 3Ut1 /2. The ratio w/δ∗ is plotted as a function of r/a in Fig. 8.28 for t/t1 = 0.5, 1.0, and 2.0, the last two values defining the ends of the two plastic phases. The corresponding solution for a clamped circular plate has been given by Wang and Hopkins (1954). The solution for impact loading of an annular plate clamped at the inner radius has been discussed by Shapiro (1959), Florence (1965), and Johnson (1972). An analysis for a clamped circular plate impulsively loaded over a central circular area has been presented by Weirzbicki and Nurick (1996).

8.8

Dynamic Loading of Cylindrical Shells

645

Fig. 8.28 Deformed shape of a simply supported circular plate under blast loading at different instants of time (t1 = μa2 U/12M0 )

The influence of the shape of the pressure pulse on the dynamic behavior has been examined by Youngdahl (1971) and Krajcinovic (1972). The dynamic behavior of circular plates under a central pulse loading has been investigated by Florence (1977). The effect of transverse shear on the dynamic plastic response has been studied by Kumar and Reddy (1986). An analysis for the dynamic bending problem of square plates has been presented by Cox and Morland (1959), and that of rectangular plates by Jones (1970). The use of mode approximation in predicting the dynamic plastic response of plates has been discussed by Chon and Symonds (1977). The influence of rate sensitivity on the dynamic behavior has been considered by Perrone (1967) and Perrone and Bhadra (1984). The effect of membrane forces on shape changes in dynamically loaded plates has been investigated by Jones (1971) and Symonds and Wierzbicki (1979). The dynamic buckling of rectangular plates in the plastic range has been treated by Goodier (1968). A variety of other problems on the dynamic plastic behavior of plates have been considered by Nurick et al. (1987), Jones (1989), Yu and Chen (1992), and Zhu (1996).

8.8 Dynamic Loading of Cylindrical Shells 8.8.1 Defining Equations and Yield Condition Consider a circular cylindrical shell of radius a and thickness h, subjected to a uniform radial pressure p, whose initial value is greater than the quasi-static collapse pressure under identical boundary conditions. Such a loading will produce accelerated plastic flow that requires the inclusion of inertia effects in the theoretical framework. The state of stress in the shell is characterized by the axial bending moment Mx and the circumferential force Nθ acting per unit length of the circumference. The state of strain rate, on the other hand, is defined by the radially

646

8 Dynamic Plasticity

inward velocity w. As in the case of static analysis (Section 5.1), Mx will be taken as positive if it corresponds to tensile stresses on the inner surface, while Nθ will be reckoned positive when it is tensile in nature. The duration of the applied pressure is assumed small enough to justify the neglect of geometry changes in the analysis. If the surface density of the material of the shell is denoted by μ, the inertia force per unit area of the middle surface is μ(∂ 2 w/∂t2 ) acting in the radially outward sense, and the equations of dynamic equilibrium are ∂Mx − Q = 0, ∂x

∂Q Nθ ∂ 2w + +p=μ 2 , ∂x a ∂t

where Q is the shearing force per unit circumference and x is the distance measured along the length of the shell. The elimination of Q between the above equations gives the differential equation Nθ ∂ 2w ∂Mx + +p=μ 2 ∂x a ∂t

(8.163)

for simplicity, the material is assumed to have a constant uniaxial yield stress Y, the fully plastic values of the resultant force and moment being N0 = Yh and M0 = Yh2 /4, respectively. It is convenient at this stage to introduce the dimensionless qualities Mx mx = , M0

Nθ nθ = , N0

pa q= , Yh

ξ =x

2 , ah

τ=

t . t0

Denoting the differentiation with respect to τ by a superimposed dot, (8.163) can be expressed in the dimensionless form w ¨ ∂ 2 mx + 2 (nθ + q) = , 2 δ ∂ξ

(8.164)

where δ = Yht02 /2μa denotes a representative constant deflection of the middle surface. This equation can be integrated with the help of an appropriate yield condition and a suitable choice of the deflection function w(ξ , τ ). The analysis of the dynamic problem is greatly simplified by the use of the square yield condition shown in Fig. 8.29(a). It is an approximation not only to the Tresca yield condition (shown broken) but also to the von Mises yield condition (not shown). Referred to the dimensionless variables ξ and τ , the generalized strain rates may be defined as w˙ λ˙ θ = − , a

κ˙ x = −

2 ∂ 2w ˙ . ah ∂ξ 2

(8.165)

8.8

Dynamic Loading of Cylindrical Shells

647

Fig. 8.29 Dynamic loading of a cylindrical shell. (a) Yield condition and (b) clamped shell with a pressure pulse

When the stress point lies on one of the sides of the yield locus, the vector (N0 λ˙ θ ,M0 κ˙ x ) is directed along the outward normal to this side. At a corner of the yield locus, the vector must lie between the two extreme normals defined there. In any particular problem, the stress profile generally includes two or more plastic regimes, and the flow rule in each case can be easily established (Hodge, 1955). Considerations of equilibrium require the bending moment and the shearing force to be continuous, while cohesion of the material demands that the deflection and ˙ and w are all continuous across the velocity are continuous. Thus, mx ,∂mx /∂ξ,w, any boundary, moving or stationary. The velocity slope ∂ w/∂ξ ˙ may, however, be discontinuous across a hinge circle, which corresponds to a finite value of w˙ and an 2 . Such conditions can only hold at the corners A infinitely large value of ∂ 2 w/∂ξ ˙ and B of the field locus when nθ is compressive.

8.8.2 Clamped Shell Loaded by a Pressure Pulse A cylindrical shell of length 2 l is rigidly clamped at both ends, Fig. 8.29(b), and is instantaneously loaded at t = 0 by a uniform radial pressure p which is held constant for a sufficiently small time interval t0 . The static collapse pressure p0 , which must be exceeded for dynamic actions, can be determined in the same way as that using the hexagonal yield condition (Section 5.1). Considering one-half of the shell defined by 0 ≤ x = l, the incipient velocity at collapse is taken as w˙ = ξ, which gives κ˙ x = 0 and λ˙ < 0. The stress profile is therefore entirely on side AB, with x = 0 corresponding to point A and x = l to point B. Setting nθ = – 1, q = q0 , and w = 0 in (8.164), and integrating it under the boundary conditions mx = 1 at ξ = 0, and mx = 1 at ξ = ω, we obtain the solution

648

8 Dynamic Plasticity

mx = (q0 − 1) ξ (ω − ξ ) + 2 (ξ/ω) − 1. Since the bending moment is a relative maximum at x = l, the derivative ∂mx /∂ξ must vanish at ξ = ω, giving the dimensionless collapse pressure 2 q0 = 1 + 2 , ω

ω=l

2 . ah

For a range of values of q > q0 , it is reasonable to suppose that the same plastic regime AB is applicable for dynamic loading. Since the velocity at each instant is then proportional to ξ , the expressions for w˙ and w may be written as w˙ = w˙ 0 (ξ/ω) ,

w = w0 (ξ/ω) ,

(8.166)

where w0 (t) is the deflection at the central section ξ = ω. Substituting into the equilibrium equation (8.164), and setting nθ = – 1, we obtain the differential equation for mx as d 2 mx wξ ¨ . = −2 (q − 1) + 2 δω dξ

(8.167)

The assumed mode of deformation implies the formation of hinge circles at ξ = 0 and ξ = ω. The preceding equation may therefore be integrated under the boundary conditions ∂mx /∂ξ = 0 at ξ = ω and mx = –1 at ξ = 0, resulting in w ¨0 ξ2 ξ ω− . mx = −1 + (q − 1) ξ (2ω − ξ ) − 2δ 3ω The remaining boundary condition mx = 1 at ξ = ω furnishes the central acceleration w ¨ 0 = 3 (q − q0 ) δ,

0 ≤ τ ≤ 1,

(8.168)

in view of the expression for q0 , and the bending moment distribution in the first plastic phase becomes 1 ξ2 mx = −1 + (q − 1) ξ (2ω − ξ ) − (q − q0 ) ξ 3ω − , 2 ω

0 ≤ τ ≤ 1 (8.169)

Evidently, mx = mx (ξ ) in this phase, being independent of time. Integrating (8.168), and using the initial conditions w˙ 0 = w0 = 0 at τ = 0, we get w˙ = 3 (q − q0 ) τ , δ

3 w0 = (q − q0 ) τ 2 , δ 2

0 ≤ τ ≤ 1.

(8.170)

The preceding solution will be acceptable if the maximum bending moment does occur at ξ = ω. Since mx (ω)has the value q – (3q0 – 2), which is negative for q < 3q0 – 2, the applied pressure must satisfy the inequalities

8.8

Dynamic Loading of Cylindrical Shells

q0 = 1 +

649

2 6 < q < 1 + 2 = qc ω2 ω

(say).

(8.171)

When q exceeds the upper limit qc over this range, considered as the medium load range, mx is a relative minimum at ξ = ω, and its value exceeds unity in the neighborhood of this section, thereby violating the yield condition. Suppose that the applied pressure is instantaneously removed at t = t0 , which marks the beginning of a second plastic phase. For a sufficiently short shell subjected to medium load, the stress profile should continue to be on side AB. The central acceleration and the bending moment distribution are therefore given by (8.168) and (8.169), respectively, with q = 0, the latter quantity being

mx = − 1 + 2ωξ − ξ

2

1 ξ2 . + q0 ξ 3ω − 2 ω

(8.172)

The yield condition will not be violated in the neighborhood of ξ = 0 so long as m x ≤ 0. By (8.172), this condition is equivalent to q0 ≥

4 3

or

ω≤

√ 6.

Such shells will be regarded as short shells, as opposed to long shells for which the above inequalities are reversed. Since m x (ω) < 0 during the second plastic phase, the bending moment distribution (8.172) is acceptable for short shells. Integrating (8.168) after setting q = 0, and using the conditions of continuity of w˙ 0 and w0 at τ = 1, we get w˙ 0 = 3 (q − q0 τ ) δ, 3 w˙ 0 = q (2τ − 1) − q0 τ 2 δ, 2

⎫ ⎬ 1 ≤ τ ≤ q/q0 ,⎭

(8.173)

in view of (8.170). The shell comes to rest (w˙ = 0) when τ = τ ∗ = q/q0. The final deflection of the shell is w∗ = (ξ/ω)w∗0 , and it follows from (8.173) that w∗ 3q = δ 2

ξ q −1 , q0 < q < qc , q0 ω

ω2 ≤ 6.

(8.174)

The final value of the central deflection is therefore equal to 1.5δ when q and ω √ have their limiting values of 2 and 6, respectively. For long shells (ω2 > 6) under medium loads (q < qc ), the preceding solution still holds for τ < 1, but that for τ > 1 the solution is modified since the plastic regime does not apply throughout the shells. It is natural to expect that a region 0 < ξ < ρ near the built-in end would correspond to the plastic regime AD, for which mx = –1 and w˙ = 0. This portion of the shell therefore becomes rigid after being previously deformed. The section ξ = ρ defines the instantaneous position of the hinge circle which requires mx = –1 there. The remaining portion ρ ≤ ξ ≤ ω of the half-shell corresponds to the plastic regime AB and

650

8 Dynamic Plasticity

involves continuation of the plastic deformation. The velocity in this region may be written as ξ −ρ , ρ ≤ ξ ≤ ω, τ ≥ 1, (8.175) w˙ = w˙ 0 ω−ρ so that κ˙ x = 0, and w˙ is automatically made continuous across ξ = ρ. The distribution of mx in the deforming region can be determined by integrating (8.167) with q = 0, nθ = –1, and the expression w ¨ =w ¨0

ξ −ρ ω−ρ

,−

ρ˙ w ¨0 ω−ρ

ω−ξ ω−ρ

,

ρ ≤ ξ ≤ ω.

Using the boundary conditions ∂mx /∂ξ = 0 and ξ = ρ at the central section £ = co, the solution to the differential equation (8.167) in the region ρ < ξ < ω is obtained in the form 1 w ¨0 ρ˙0 w˙ 0 ω − ξ − w ¨0 + . (8.176) mx = 1 + (ω − ξ )2 1+ 2δ 6δ ω−ρ ω−ρ Since the bending moment and its derivative must be continuous across the hinge circle, the conditions mx = –1 and ∂mx /∂ξ = 0 at ξ = ρ must also be satisfied. Hence ρ˙ w˙ 0 2 ρ˙ w˙ 0 . −w ˙ 0 = 4δ, − 2w˙ 0 = 6δ 1 + ω−ρ ω−ρ (ω − ρ)2 These two relations may be combined together to express w˙ 0 and ρ˙ w ˙ 0 in terms of ρ, the result being w ¨ 0 = −2δ 1 +

6 (ω − ρ)2

,

6 ρ˙ w˙ 0 = 2δ 1 − . ω−ρ (ω − ρ)2

(8.177)

The substitution from (8.177) into (8.176) finally gives the bending moment distribution in the form

ω−ξ mx = 1 − 2 3 − 2 ω−ρ

ω−ξ ω−ρ

2 ,

ρ ≤ ξ ≤ ω.

(8.178)

It follows from the boundary conditions on mx that –1 ≤ mx ≤ 1 for ρ < ξ < ω. Since w ¨ 0 = ρ˙ (dw/dρ), ˙ the elimination of ρ˙ between the two relations of (8.177) leads to the differential equation d w˙ 0 + dρ

(ω − ρ)2 + 6 (ω − ρ)2 − 6

w˙ 0 = 0. ω−ρ

8.8

Dynamic Loading of Cylindrical Shells

651

Since ρ = 0 at τ = 1, when w˙ 0 = 3(q−q0 )δ, the integration of the above equation results in 3ω (q − q0 ) (ω − ρ)2 − 6 w˙ 0 , = δ ω2 − 6 (ω − ρ)

0≤ρ ≤ω−

√ 6.

(8.179)

To determine the variation of ρ with time, we substitute (8.179) into the second equation of (8.177) and obtain the differential equation 2 dρ = dτ 3ω

ω2 − 6 q − q0

2ω = 3

4 − 3q0 q − q0

.

Thus ρ˙ is a constant, which means that ρ varies linearly with the time, the result of integration of the above equation being ρ 2 = ω 3

4 − 3q0 q − q0

(τ − 1) ,

1 ≤ τ ≤ τ ∗.

(8.180)

√ The velocity everywhere vanishes when ρ = ω − 6, and the motion is terminated, the duration of the motion τ ∗ being obtainable from (8.180). The central deflection of the shell for τ ≥ 1 can be found by the integration of (8.179), using the fact that w˙ = ρ˙ (dw0 /dρ), and substituting for ρ. The result is easily shown to be $ %

6 ρ w0 3 3 (q − q0 ) ρ (2ω − ρ) + ln 1 − , = (q − q0 ) 1 + δ 2 4 − 3q0 ω 2 ω2 − 6 ω2 − 6

ω2 > 6

(8.181) The deflection at a generic section can be determined by integrating a similar equation obtained from (8.175) and (8.179), together with the change √ of variable to ρ. The final shape of the shell obviously corresponds to ρ = ω − 6. The ratio w0 /δ is plotted against t/t∗ in Fig. 8.30 for ω2 = 3 and 6, and for three different values of P/P0 The derivation of the uppermost broken curve is based on the analysis for high loads which is given below.

8.8.3 Dynamic Analysis for High Loads Consider the range of loads for which q > qc , applied to sufficiently short shells characterized by ω2 ≤ 6. In this case, a central portion of the shell is in regime B, while the remainder of the shell is in regime AB, the two portions being separated by a hinge circle. During the first plastic phase (0 ≤ τ ≤ 1), the hinge circle is fixed at a section ξ = α 0 , the velocity and deflection of the outer portion of the shell being given by w˙ = w˙ 0 ξ/α0 ,

w = w0 ξ/α0,

0 ≤ ξ ≤ α0 ,

(8.182)

652

8 Dynamic Plasticity

Fig. 8.30 Central deflection of a clamped cylindrical shell as a function of time under a uniform pressure pulse

where w0 represents a uniform deflection of the central portion α 0 ≤ ξ ≤ ω. Since mx = 1 and nθ = –1 over the length α 0 ≤ ξ ≤ ω, it follows from (8.164) that w ¨ 0 = 2 (q − 1) δ, which gives on integration w˙ 0 = 2 (q − 1) τ , w

w0 = (q − 1) τ 2 , δ

0 ≤ τ ≤ 1.

(8.183)

The bending moment distribution in the region 0 ≤ ξ ≤ α 0 and the quantity α 0 are determined by the integration of (8.167), where α 0 is written for ω, using the boundary conditions mx = –1 at ξ = 0 and mx = 1, ∂mx /∂ξ = 0 at ξ =α 0. Thus mx = 1 − 2 (1 − ξ/α0 )3 , mx = 1,

nθ = −1,

nθ = −1,

(q − 1) α02 = 6,

or

0 ≤ ξ ≤ α0 ,

α0 ≤ ξ ≤ ω,

α0 =

(8.184)

6/ (q − 1).

For τ > 1, the load is absent, and the hinge circle separating the two regions progressively moves toward the central section, its position at any instant being denoted by ξ = α. The velocity (but not the deflection) is still given by (8.182) with a written for α 0 , the central acceleration being given by w0 = −2δ. Integrating, and using the initial conditions w0 = 2(q – 1)δ at τ = 1, we get

8.8

Dynamic Loading of Cylindrical Shells

653

w˙ 0 = 2 (q − τ ) δ,

1 ≤ τ ≤ τ1 ,

where τ 1 is the value of r when the hinge circle reaches ξ = ω. The velocity and displacement in the central region during the second plastic phase are given by w˙ = 2 (q − τ ) , δ

w = (2τ − 1) q − τ 2 , α ≤ ξ ≤ w, δ

(8.185)

The generalized stresses in this region are mx , = 1 and nθ = – 1. In the region 0 ≤ ξ ≤ α, the bending moment distribution can be directly written from the conditions mx (0) = −1,mx (α) = 1,mx (α) = 0, and mx (0) = 2, the result being α2 ξ α2 ξ 3 2 , +ξ − 1+ mx = −1 + 3 − 2 α 2 α3

0 ≤ ξ ≤ α.

(8.186)

Inserting this expression into the differential equation (8.164), setting nθ = – 1 and q = 0, and using the fact that w˙ ξ w˙ α˙ ξ = 2 (q − τ ) , = −2 1 + (q − τ ) , δ α δ α α

0 ≤ ξ ≤ α,

(8.164) is found to be satisfied if α is given by the differential equation (q − τ )

6 + α2 dα = , dτ 2α

which is readily integrated under the initial condition α = α 0 when τ = 1 to give q−1 6 + α2 = 2 q −τ 6 + α0

' or

α=

6τ , q−τ

(8.187)

in view of the last equation of (8.184). Since α = ω when τ = τ 1 , (8.187) furnishes

τ1 = qω2 / 6 + ω2 = q/qc in view of (8.171). The instant τ = τ 1 marks the end of the second plastic phase, since the hinge circle reaches the central section and can go no further. In order to complete the solution for the second plastic phase (1 ≤ τ < τ 1 ), it is necessary to find the deflection in the region 0 ≤ ξ ≤ α by the integration of the differential equation ∂w 2 q ξ = 2δ (q − τ ) = δ ξ (q − τ ) − 1, ∂τ α 3 τ

0 ≤ ξ ≤ α.

654

8 Dynamic Plasticity

The solution is straightforward for the region 0 ≤ ξ ≤ α 0 , which has always been in the plastic regime AB, the initial condition for this region being w0 /δ = (q – 1) ξ /α at t = 1, in view of (8.182) and (8.183). The integration of the above equation therefore gives τ 3 2 w ξ −1 −1 1 =√ q sin − sin √ δ q q 6 2 3 5 τ (q − τ ) − q q − 1 , 0 ≤ ξ ≤ α0 . q−τ + 2 2

(8.188)

The elements in the region α 0 ≤ ξ ≤ α have passed from regime B to regime AB at different instants as they have been traversed by the moving hinge circle. Let τ be the value of τ for which a typical section of the shell coincides with the hinge circle. Setting α = ξ and τ = τ in (8.187), we have

τ = qξ 2 / 6 + ξ 2 . The deflection of the element at this instant is given by (8.185) with τ = τ . Using this as the initial condition, the solution for the deflection in the region α 0 ≤ ξ ≤ α is easily shown to be τ 3 2 5 −1 −1 ξ τ (q − τ ) q sin − sin √ q−τ + 2 q 2 6 0.5ξ 2 −q 1+ , α0 ≤ ξ ≤ α. 6 + ξ2

ξ w =√ δ 6

(8.189)

It is readily verified that the deflection is continuous at ξ = α 0 . The continuity of the deflection at ξ = α is also ensured by the fact that the right-hand side of (8.189) at ξ = a coincides with that given by (8.185). For τ > τ 1 , the entire stress profile is in plastic regime AB, and the velocity distribution throughout the shell is given by the first equation of (8.182) with w to written for α 0 . The bending moment distribution over the entire shell becomes ω2 ξ ω2 ξ 2 , 0 ≤ ξ ≤ ω, + ξ2 − 1 + mx = −1 + 3 3 − 2 ω 2 ω3

(8.190)

obtained by simply replacing α by w in (8.186). The substitution in the differential equation (8.167) with q = 0 then gives the central acceleration as w ¨ 2 = − 1 + 2 = −3q0 . δ ω In view of the continuity of the velocity w ˙ at τ = τ 1 , the integration of the above equation results in the velocity field

8.8

Dynamic Loading of Cylindrical Shells

w˙ ξ = 3 (q − q0 τ ) , δ ω

655

τ1 ≤ τ ≤ τ ∗ , 0 ≤ ξ ≤ ω,

where τ ∗ = q/q0 , representing the instant when the motion is terminated. A straightforward integration furnishes the deflection at any point as 3 w ξ = τ (2q − q0 τ ) + λ (ξ ) , δ 2 ω

q q ≤τ ≤ , qc q0

(8.191)

where λ(ξ ) must be determined from the condition of continuity of w at τ = τ = q/qc . Using (8.188) and (8.189), it is easily shown that ⎫ ⎪ 3 ⎪ −1 1 −1 1 ⎪ − sin √ q sin − q − 1 , 0 ≤ ξ ≤ α0 , ⎪ ⎬ 2 qc q ⎪ 3 ω ξ qξ qξ ⎪ ⎪ ⎪ − q, α − ≤ ξ ≤ ω. λ (ξ ) = q tan−1 √ − tan−1 √ 0 ⎭ 2 2 2 6 + ξ 6 6 (8.192) qξ λ (ξ ) = 2

Setting τ = q/q0 in (8.191), the final shape of the deformed middle surface of the shell is obtained as w∗ 3q2 ξ = + λ (ξ ) , δ 2q0 ω

q ≥ qc ,

ω2 ≤ 6.

(8.193)

When q=qc , we have α 0 –w and λ(ξ ) = –3qξ/2ω, which reduces (8.193) to (8.174) as expected. Setting ξ = ω in (8.193) and using (8.192), the central deflection in the final phase is found to be given by q 3 w0 = q 3τ − − 1 − q0 τ 2 , δ 2qc 2

q q ≤τ ≤ , qc q0

(8.194)

where q0 and qc depend only on ω and are given by (8.171). Figure 8.31 displays the final deformation pattern of the clamped shell when ω2 = 3, each curve being based on a definite value q. For longer shells (ω2 > 6), the stress and velocity distributions are identical to those for short shells during the period of application of the load, but the subsequent part of the solution, following the load removal, is modified, due to the presence of a second hinge circle which begins at ξ – 0 and moves along the length of the shell. The effects of blast loading under different end conditions on the dynamic plastic behavior have been discussed by Hodge (1956b, 1959). The dynamic plastic response of cylindrical shells under a band of pressure has been discussed by Eason and Shield (1956), Kuzin and Shapiro (1966), Youngdahl (1972), and Li and Jones (2005). The influence of membrane forces has been examined by Jones (1970) and Galiev and Nechitailo (1985). The dynamic plastic response of spherical caps under pulse and impact, loadings has been investigated by Sankaranarayanan (1963, 1966). The dynamics of an impulsively loaded

656

8 Dynamic Plasticity

Fig. 8.31 Final shape of the deformed meridian of a clamped cylindrical shell subjected to a highpressure pulse

cylindrical shell based on a generalized yield condition has been investigated by Lellep and Torn (2004). The progressive crumpling of cylindrical tubes under axial compression has been discussed by Abramowicz and Jones (1984) and Jones (1989). The dynamic plastic buckling of cylindrical shells has been considered by Vaughan and Florence (1970) and Jones and Okawa (1976) and that of a complete spherical shell by Jones and Ahn (1974). A simplified method of analysis for the plastic buckling, based on energy considerations, has been discussed by Gu et al. (1996).

8.9 Dynamic Forming of Metals 8.9.1 High-Speed Compression of a Disc Consider the rapid compression of a short circular cylinder between a pair of parallel platens, or dies, the speed of compression being such that the inertia effects are significant. The lower die z = 0 is stationary, while the upper die z = h is assumed to move down with a constant speed U, the coefficient of friction μ between the dies and the cylindrical block being assumed to be constant (Haddow, 1965). The elastic and plastic stress waves initiated at the upper die travel up and down the block several times during the compression process. The load acting on the upper platen, which is significantly in excess of the quasi-static value in the early part of the process, appreciably decreases during the final stages. Only the incipient com-

8.9

Dynamic Forming of Metals

657

pression of a thin block of uniform diameter will be considered in what follows, ignoring the effect of wave propagation on the dynamic process. Since the distribution of stresses and strains is symmetrical about the axis of the block, which is assumed to coincide with the z-axis as shown in Fig. 8.32(a), the equation of radial motion of a typical element in cylindrical coordinates (r, θ , z) may be written as σr − σθ ∂τrz ∂σr + + =ρ ∂r r ∂z

∂u ∂u ∂u +u +ω , ∂t ∂r ∂z

where u and w are the radial and axial components of the velocity, and ρ is the density of the material. If the influence of barreling is disregarded, σ r , σ θ , σ z , and u are independent of z, but the presence of die friction requires τ rz to vary with z so that τ rz = μp at z = 0 and τ rz = –μp at z = h, where ρ is the die pressure. The multiplication of the preceding equation by dz and integration between the limits 0 and h therefore result in ∂u σr − σθ 2μp ∂u ∂σr + − =ρ +u . (8.195) ∂r r h ∂t ∂r

Fig. 8.32 High-speed compression of short cylinders. (a) Condition of loading and (b) mean die pressure against kinetic energy of impact

In the absence of barreling, the velocity field corresponds to a uniform compression of the block and is consequently given by

658

8 Dynamic Plasticity

u=

Ur Uz , w=− , 2h h

(8.196)

satisfying the condition of plastic incompressibility. Since ε˙ r = ε˙ θ according to (8.196), we have σ r = σ θ for an isotropic material, and the yield criteria of both Tresca and von Mises reduce to σr − σz = σθ − σz = Y, where σ z = –p, and the material is considered as ideally plastic with a uniaxial yield stress Y. Substituting from (8.196) into (8.195), setting σ e = σ r , and using the yield criterion, we obtain the differential equation dp 2μp 3ρU 2 r . + =− dr h 4 h2 Since σ r must vanish along the cylindrical surface, the boundary condition is p = Y at r = a. The integration of the above equation therefore furnishes p 2μ (a − r) 3ρU 2 1 a r 3ρU 2 1 = 1− − − exp + . (8.197) Y 8Yμ 2μ h h 8Yμ 2μ h This equation predicts the die pressure distribution as a function of r. The inertia effect represented by the parameter ρU2 /Y is therefore to increase the die pressure at any given radius. The mean die pressure p¯ corresponding to (8.197) is easily shown to be given by p¯ 2μa h h a 3ρU 2 1 h = − exp 1− − 1+ Y qμ 8Yμ 2μ h 2μa h 2μa (8.198) 2 3 a ρU − . + 4Yμ 4μ h When the ratio ρU2 /Y is vanishingly small, this formula reduces to (3.57), obtained for the quasi-static compression. Expanding exp(2μa/h) in ascending powers of μa/h, and neglecting terms containing powers of μ higher than the fourth, (8.198) can be reduced to 2μa 3ρU 2 a 2 p¯ , =1+ + Y 3h 16Y h which is sufficiently accurate for small values of μ. Figure 8.32(b) shows the variation of the mean die pressure with ρU2 /Y for different values of a/h in the special case of μ = 0.1. In the case of steel, for instance, the inertia effect is significant when the speed of compression exceeds about 50 m s–1 . The situation where the speed of compression varies with time has been considered by Lippmann (1966) and Dean (1970).

8.9

Dynamic Forming of Metals

659

The inclusion of the strain rate sensitivity of the material, based on the homogeneous deformation mode (8.196), is quite straightforward. Since the effective strain rate ε˙ has a constant value equal to U/h, it is only necessary to replace Y in the preceding analysis by the modified yield stress n n U ε˙ =Y 1+ , Y 1+ α αh where α and η are material constants to be determined by experiment. In particular, the simplified formula for small μ is easily shown to be modified to n U 2μa p¯ 3ρU 2 a 2 . = 1+ 1+ + Y 3h αh 16Y h

(8.199)

This is a complete generalization of the well-known Siebel formula for the plastic compression of short cylinders. It involves the estimation of the coefficient of die friction, the speed of compression, and the empirical constants characterizing the dynamic response of the material. A upper bound analysis for the dynamics of a closed die forging process has been discussed by Scrutton and Marasco (1995).

8.9.2 Dynamic Response of a Thin Diaphragm A thin circular diaphragm of initial thickness h0 is rigidly held along its periphery r = a and is subjected to a uniform velocity U normal to its plane (Hudson, 1951). An elastic wave front immediately sweeps inward from the edge, producing a radially outward motion of the material particles. At any later instant, a plastic bending wave generated at the edge travels some distance toward the center, producing a bulged shape of the diaphragm as shown in Fig. 8.33(a). The annular region swept over by the bending wave forms a surface of revolution, which is assumed to have come to rest, while the flat central region yet unaffected by the wave retains its normal velocity U. Since we are dealing with large plastic strains, all elastic effects other than those of the initial stress wave will be neglected, the material being effectively considered as rigid/plastic in the dynamic analysis of the process. At any time t after the beginning of the process, let b denote the radius of the central flat portion of the diaphragm that has been uniformly deformed to a thickness h under the action of a constant normal velocity U and a variable radial velocity υ induced at r = b. During a time internal dt, an elemental ring of width ds just ahead of the bending wave impulsively rotates to form a part of the bulge after being swept over by the wave. Since the radial velocity of the ring relative to that of the wave is equal to υ–b, we have ds = (υ –b) dt. The radial and transverse components of the displacement of the inner edge of the ring are –b dt and U dt, respectively, giving √ ds = U 2 + b2 dt. Consequently,

660

8 Dynamic Plasticity

Fig. 8.33 Impact loading of a circular diaphragm. (a) Geometry of deformation and (b) deformed shape for different initial velocities

ds = υ − b˙ = U 2 + b˙ 2 dt

or

υ 2 − U2 b˙ = . 2υ

(8.200)

Since b˙ is negative, it follows from (8.200) that v < U throughout the deformation. The assumed uniformity of thickness in the flat central region requires a state of balanced biaxial tension σ r = σ θ to exist in this region. The radial and circumferential strain rates are therefore equal to one another, and the rate equation of incompressibility is h˙ + 2˙εθ = 0 h

or

h˙ 2υ + = 0. h b

(8.201)

when considered at the interface r = b. If r0 denotes the initial radius to a particle that is currently at a radius r in the interior of the central region, then the integrated form of the flow rule gives εθ = ε r = 12 ln (h0 /h). Hence ∂r r = = r0 ∂r0

h0 . h

The assumed uniformity of the stress and strain in the central region is incompatible with the existence of inertia forces, the effect of which is disregarded in the yielding and flow of the material. Consequently, σ r is equal to the current yield stress σ , which is a function of the total compressive thickness strain equal to ln(h0 /h). Let q denote the resultant radial stress, including the inertia effect, in the material just ahead of the bending wave at r = b. As the wave sweeps inward over an annular element of width ds, the work done by the stress during the time interval dt is equal to –qhbdθ (υ dt), where dθ is the angle subtended by the element at the center of the disc. Since the corresponding change in kinetic energy of the element is –(ρ/2)h ds dθ (υ 2 + U2 ), we have

8.9

Dynamic Forming of Metals

q=p

661

υ 2 + U2 2υ

ds = ρ U 2 + b˙ 2 dt

(8.202)

in view of (8.200). If the radially outward accelerating force acting on a typical element of mass phr dr dθ in the central region at any instant is denoted by dF dθ = (∂F/∂r), dr dθ , then the equation of motion for this element may be written as ' ∂F h d2 h0 ∂ 2r 2 = ρhr 2 = ρhr . 2 ∂r ∂t h0 dt h It is reasonable to suppose that the rate of work done by the distribution of this force over the central region is equal to that produced by the stress difference q – σ r occurring at r = b. Then

υ ∂r dF = ∂t b

(q − σr ) hbυ =

b

r 0

∂F dr. ∂r

Substituting for ∂F/∂r and integrating, we obtain the result ' 1 q = σ + ρb2 4

h d2 h0 dt2

h0 h

.

(8.203)

The right-hand side of this equation depends only on the thickness ratio h/h0 . The solution to the system of equations (8.201) to (8.203) requires the specification of an initial value of v at the bending wave r = b. Assuming a linear distribution of velocity in the central region, Hudson (1951) obtained the initial condition ' υ = 2Y

1−ν ρE

at t = 0,

where E is Young’s modulus, and ν is Poisson’s ratio for the material of the diaphragm. The motion begins at t = 0 when the stress at the clamped edge rises suddenly from zero to the initial yield stress Y. We begin with the situation where the material is nonhardening, so that σ – Y throughout the motion. It is convenient to introduce the dimensionless parameters Ut z b ξ= = , α= , a a a

υ β= , η= U

h0 , h

λ=

4Y , ρU 2

where z is the current height of the central flat part above the initial plane of the diaphragm. Equations (8.200) and (8.201) immediately become 2β

dα = − 1 − β2 , dξ

α

dn = ξ η, dξ

(8.204)

662

8

Dynamic Plasticity

while the elimination of q between (8.202) and (8.203) leads to the differential equation α2 d2 η dα 2 = (λ − 4) + , 4 dξ η dξ which can be combined with the second equation of (8.204) to eliminate η. After some algebraic manipulation using the first equation of (8.204), the resulting equation is reduced to 1 + β2 2 + β2 dβ . (8.205) = −λ + α dξ 2β 2 Equations (8.204) and (8.205) form a set of three basic differential equations for the three unknowns α, η, and β as functions of ξ , the initial conditions being α = η = 1,

β = β0 =

λ (1 − ν) Y/E

when ξ = 0.

To carry out the integration, we combine (8.205) with the two equations of (8.204) in turn to obtain the results 1 − β 2d β dα , =− α 2 − (2λ − 3) β 2 + β 4

dη 2β 3 d β . = η 2 − (2λ − 3) β 2 + β 4

Although these equations can be integrated exactly, it is more convenient for practical purposes to introduce a minor approximation (since β 4 is usually a small fraction) to express the solution in the form (m−1)/4m2 ⎫ ⎪ β02 − β 2 ⎪ mβ 2 − 1 ⎪ exp α= ,⎪ ⎪ ⎪ 2 ⎬ 4m mβ0 − 1 ⎫ 2 1/2m2 ⎪ mβ0 − 1 β 2 − β02 ⎬ ⎪ ⎪ ⎪ ⎪ , η= exp ⎭ ⎭ ⎪ 2m mβ 2 − 1

(8.206)

where m = λ − 32 . These relations furnish η, as a function of α parametrically through β. Finally, ξ can be found as a function of β by the numerical integration of (8.205). When λ is sufficiently large, so that m ≈ λ, the thickness h and the height z of the central flat part are closely approximated by the relations h ≈ h0

4/λ b , a

ρ z b ≈U 1− . a Y a

(8.207)

The diaphragm is therefore deformed into a conical shape, the central deflection being proportional to the initial velocity √ U. The radial velocity v of the flat part rapidly increases to its terminal value U/ λ. When the diaphragm is completely formed

8.9

Dynamic Forming of Metals

663

(α = 0), thickness vanishes at the center, no doubt as a result of the mathematical idealization. The total time √ for the deflection, which is known as the swing time, is shows the shape of the profiles of the approximately equal to a ρ/Y. Figure 8.33(b)√ completely deformed diaphragm for β 0 = 0.04 λ with different values of λ. The simplest way of taking into account the effect of work-hardening of the material is to consider a mean yield stress based on the given stress–strain curve. It is therefore only necessary to replace Y in the expression for λ by a quantity that depends on ε = ln(h0 /h). The general effect of work-hardening is to decrease the central deflection for a given initial velocity U. The predicted shape of the profile is found to be in complete qualitative agreement with observation. Figure 8.34, which has been obtained experimentally by Keil (1960), indicates the difference between the static and dynamic behaviors of a clamped circular diaphragm. An energy method of analysis for a circular membrane subjected to impact loading has been discussed by Boyd (1966). The propagation of plastic waves in an impulsively loaded circular membrane has been discussed by Munday and Newitt (1963) and Cristescu (1967).

Fig. 8.34 Comparison of static and dynamic responses of clamped circular diaphragms (after A.H. Keil, 1960)

664

8 Dynamic Plasticity

8.9.3 High-Speed Forming of Sheet Metal A variety of techniques have been developed in recent years for the forming of sheet metal using loading rates that are large enough to have a dominating effect on the deformation process. Chemical explosives are commonly used to generate the energy which is transmitted to the workpiece through an intervening medium such as water. The forming die, together with the workpiece, is immersed in water contained in a forming tank, and the air in the die cavity is often evacuated with the help of a vacuum pump. The explosive charge is located at a suitable stand-off distance from the workpiece, and the kinetic energy released by the detonation of the charge is utilized in forming the component. Detailed accounts of the high-rate forming of metals have been presented by Rinehart and Pearson (1965). One of the common methods of explosive forming of sheet metals is indicated in Fig. 8.35(a). An air-backed circular blank is clamped around the periphery and is subjected to a shock wave generated by the detonation of an explosive at a stand-off distance approximately equal to the blank radius. The shock wave provides the necessary kinetic energy which is dissipated in doing the plastic work. The mean initial velocity necessary for the material to attain a given polar strain in the deformed state can be approximately estimated from an assumed distribution of the thickness strain in the bulge (Johnson, 1972). Experiments tend to indicate that the final hoop strain is approximately equal to the radial strain, which means that the equivalent strain is approximately equal to the compressive thickness strain whose greatest value occurs at the pole. In the explosive-forming process, the final shape of the blank depends on such factors as the hydrostatic head, the stand-off distance, the type of explosive, and the

Fig. 8.35 Explosive forming of circular blanks. (a) Experimental setup and (b) experimental results for stand-off distances of 3, 6, and 9 inches (after Travis and Johnson, 1962)

Problems

665

weight of the charge. Figure 8.35(b), which is due to Travis and Johnson (1962), indicates how the hoop strain and thickness strain distributions in the deformed blank are affected by the ratio of the stand-off distance to the blank radius. When the size of the blank is relatively small, a closed system in which the water is contained in the die that is closed by a plate is sometimes used for maintaining the pressure for a longer interval of time. The related problem of deep drawing of cylindrical cups using an explosive device has been investigated by Johnson et al. (1965). An interesting situation arises in the high-speed blanking of sheet metal, using a typical setup of die and punch. The width of the zone of shearing, which is confined near the area of clearance between the die and punch, decreases as the speed of the operation increases. The energy required for the separation of the blank from the stock generally increases with an increase in the speed of blanking, and this phenomenon may be attributed to the strain rate sensitivity of the material. The nonuniformity of the sheared edge, which is an essential feature of the process, constitutes a major challenge for its improvement. Useful experimental results on high-speed blanking have been reported by Johnson and Travis (1966), Balendra and Travis (1970), and Dowling et al. (1970). An electromagnetic method of highspeed bulging of tubes has been investigated by Aizawa et al. (1990).

Problems 8.1 A vertical wire of length l and cross-sectional area A is made of a rigid-perfectly plastic material with a rate-sensitive yield stress. σ . The wire is suspended from a ceiling and has an attached mass m at the lower end which is given an initial axial velocity u0 . The axial displacement and velocity of the mass at any instant t are denoted by x and u, respectively. Neglecting the mass of the wire, form the equation of motion, and using (8.107) show that the final displacement δ when the motion comes to a stop is given by

δ =μ l

ξ0 0

ξ dξ , 1 + ξ 1/n

ξ=

u , αl

μ=

mlα 2 YA

where ξ 0 denotes the initial value of ξ . Assuming n = 5 and ξ 0 = 32, determine the numerical value of δ/l in terms of μ Obtain also the result based on the approximation ξ ≈ ξ0 in the denominator of the integrand and verify that it differs only marginally from the exact value. 8.2 A thick-walled spherical shell made of a nonhardening rigid/plastic material with a uniaxial yield stress Y, and having an internal radius a0 and external radius b0 , is subjected to an explosive internal pressure that produces large plastic expansion of the shell. Let p denote the magnitude of the pressure at any instant when the radii of the shell have increased to a and b respectively. Using the equation of motion and the yield criterion, together with the incompressibility condition, show that

dλ a2 a (p/Y) − 2 ln (b/a) λ= 1+ 2 + 4− 1+ , dξ b 1 − a/b b

λ=

ρυa 2 , 2Y

ξ = ln

a a0

666

8 Dynamic Plasticity

where ρ is the density of the material and υ a = da/dt is the radial velocity at r = a. Using the relation p = pe (a0 /a)3γ , this equation can be integrated numerically with the initial condition λ = 0 when ξ = 0 for any given values of b0 /a0 , p0 /Y, and γ . Verify that the hoop stress is tensile throughout the shell, vanishing at r = a when p = Y, which may be regarded as the criterion for the dynamic rupture. 8.3 A thin spherical shell having an initial thickness h0 , mean radius a0 , and density ρ is subjected to an internal explosion by detonating a concentric spherical charge. The explosion pressure is large enough to be approximated by the relation p = ρhυ, ˙ where υ is the radial velocity and h the instantaneous thickness when the shell radius is a. Using the relation p = p0 (a0 /a)3γ , where p0 is the effective detonation pressure and γ is a constant isentropic expansion index, show that

ν 2 U

'

a 3(γ −1) 2p0 , = 1− 0 3 (γ − 1) Y a

U=

Ya0 ρh0

Note that the velocity υ rapidly approaches a constant value obtained by omitting the second term in the curly brackets. Based on this constant velocity, show the time taken by the shell to rupture (defined by p = Y) and the corresponding shell radius are given by ' U tf = a0

(γ − 1)

3Y 2p0

af a0

−1 ,

p0 = Y

af 3γ a0

8.4 A lateral load P is suddenly applied at the tip of a cantilever of length l and mass m at t = 0, and held constant for a short period. For moderate values of P exceeding the static collapse load P0 = M0 /l, the beam rotates as a rigid body about a plastic hinge formed at the built-in end. Considering the equations of motion of a segment of length x measured from the tip, and using th boundary conditions, obtain the distribution of shearing force Q and bending moment M in terms of p = P/P0 as x 3x Q 2− , = −p + (p − 1) P0 2l l

M x x x 3− , 1≤p≤3 = − p − (p − 1) M0 l 2l l

Verify that the reaction at the built-in end vanishes when p = 3, and must continue to do so when p > 3, for which the plastic hinge is located at a distance b0 < l from the tip of the cantilever. Considering the equations of motion of the segment of length b0 x, and using the boundary conditions, show that Q x 2 = −p 1 − , P0 b0

M x =− M0 l

x2 3x + 2 3− b0 b0

,

b0 3 = , l p

p≥3

over the length 0 ≤ x ≤ b0 , the remainder of the cantilever being at rest. Verify that the tip acceleration is constant during this phase, giving a tip deflection of w0 = p2 t2 (P0 /3m) when p ≥ 3. 8.5 Considering the range of loading p ≥ 3, applied over a short period 0 ≤ t ≤ t0 , suppose that the load is suddenly removed. The plastic hinge then moves away from x = b0 toward the built-in end to assume a position x = b at any instant t, the part of the cantilever traversed by the plastic hinge being curved. Form the equations of linear and angular momentum to show that b = 3t/pt0 during this phase. Setting δ = P0 t0 2 /m, prove that the tip deflection and the shape of the curved portion are given by

Problems

667

p2 w0 = δ 3

b 1 + 2 ln , b0

2p2 w = δ 3

ln

b x + −1 , x b

(b0 ≤ x ≤ b)

On reaching the built-in end x = l, the plastic hinge becomes anchored at the root, while the cantilever rotates as a rigid whole with a constant angular acceleration until the motion ceases. Show that the tip velocity during this phase and the associated tip deflection are given by w ˙0 2 = δ t0

3t 2p − , t0

w0 = δ

p2 t 3t p + −p p− 1 + 2 ln , t0 t0 3 3

2p t p ≤ ≤ 3 t0 3

8.6 A free-ended beam of length 2l, made of a rigid/plastic material with a fully plastic moment M0 , is subjected to a central impact load P which increases with time. Show that a plastic hinge begins to form at the middle of the beam when q = Pl/4 M0 is equal to unity. For q > 1, each half of the beam rotates as a rigid body with an angular acceleration α, while the central hinge moves on with a linear acceleration a. Considering the equations of motion of one-half of the beam, show that μl2 a = 2 (4q − 3) , M0

μl3 α = 12 (q − 1) M0

where μ is the mass per unit length of the beam. Derive an expression of the bending moment at a distance x from the middle of the beam, and show that the maximum value of the moment has the magnitude M0 when 3x/l = q/(q – 1), the value of q at this stage being 5.725 given by the cubic 4

q 3 3

− 10.5

q 2 3

+6

q 3

−1=0

8.7 An annular plate, made of a rigid/plastic Tresca material, is clamped along its inner edge r = a and is given a constant normal velocity U along its free outer edge r = b. A plastic hinge initiated along the outer edge at t = 0 travels inward to some radius ρ = λb at a generic instant. Assuming a velocity field w ˙ = U (r − ρ)/(b − ρ) for ρ ≤ r ≤ b, using the equations of dynamic equilibrium with Mθ = - M0 , and setting t0 = μ b2 U/12 M0 , where μ denotes the surface density of the material, show that t0 λ˙ = −

2 (1 + 2λ) 1 , q=− (1 − λ) (1 + 3λ) (1 − λ) (1 + 3λ)

where q is the value of Q/bM0 at r = b. Show also that the time interval t and the deflection w at any radius r can be expressed in terms of the parameter λ as

1 t = t0 (1 − λ) 1 − λ2 , w = Ut0 (ξ − λ)2 (1 + ξ + 2λ) , 2

λ≤ξ ≤1

where ξ = r/b. Derive also the radial bending moment distribution in this part of the plate in the form Mr = −1 + M0

⎫ ⎧ 2 2 2 ξ − λ ⎨ (2 − λ − ξ ) λ + ξ + 2λξ − 2 (3 − 2λ) λ ⎬ , ⎭ 1−λ ⎩ (1 − λ)3 (1 + 3λ)

λ≤ξ ≤1

668

8 Dynamic Plasticity

8.8 In the preceding problem, let the imposed velocity U be suddenly removed at the instant t = t1 , when the plastic hinge reaches the clamped edge r = a. The plate then rotates about this edge with a deceleration, the deflection of the plate at any subsequent instant being expressed in the form w (ξ ,t) = w1 + U

ξ −α 1−α

t ≥ t1 = t0 (1 − α)2 (1 + α) ,

f (t) ,

where α = a/b, and w1 is given by the preceding expression of w considered at λ = α. The function f (t) must satisfy the initial conditions f (t1 ) = 1 and f (t1 ) = 0. Show that the equation of dynamic equilibrium for the bending moment, along with the boundary conditions, can be satisfied if

1 f (t) = (t − t1 ) 1 − 2

t − t1 t22 − t1

,

t2 − t1 = t0 (1 − α)2 (3 + α)

where t2 denotes the instant when the motion stops. Verify that the final deflection of the plate exceeds w1 by the amount (Ut0 /2) (ξ – α) (1 – α) (3 + α), and hence obtain the final shape of the plate. 8.9 A rigid/plastic circular cylindrical shell of length 2l, mean radius a, and thickness h, is clamped at both ends and subjected to an explosive blast loading, in which the external pressure p rises instantaneously to a peak value and then decays exponentially with time according to the law q=

pa = λq0 e−τ , Yh

q0 = 1 +

2 , ω2

ω=l

2 , ah

τ=

t t0

where λ > 1 is a constant, and t0 is a constant time. Let x = ξ l denote the axial distance measured from the left-hand edge of the shell. Using the square yield condition, show that the axial bending moment, radial velocity, and deflection of the shell for moderate values of λ are given by λq e−τ − 1

2 0 mx = −1 + ξ 3 − ξ 2 + ξ (1 − ξ )2 , 1 ≤ λ ≤ 3 − q0 − 1 q0 3 3 w˙ τ w − λ 1 − e−τ ξ = q0 λ 1 − e−τ − τ ξ , = q0 τ λ − δ 2 δ 2 2 where δ = Yht0 2 /2aμ with μ denoting the surface density, while the dot denotes the derivative with respect to τ . Determine the central deflection of he shell when the motion comes to a stop. 8.10 Consider the range λ ≥ 3 − 2/q0 , for which there is a moving hinge circle located at ξ = ρ. The stress point remains on the same side of the locus over the region 0 ≤ ξ ≤ ρ, while the remainder of the shell corresponds to the corner of the yield locus. Integrating the equation of motion, obtain the radial acceleration for the two portions of the shell, and hence show that 3ξ w ˙ = δ 2

τ

0

q0 − 1 dτ λq0 e−τ − 1 − ρ ρ2

w ˙ = λq0 1 − e−τ − τ δ

(ρ ≤ ξ ≤ 1)

(0 ≤ ξ ≤ ρ) ,

Problems

669

Invoking the continuity of the radial velocity at ξ = ρ, dividing the resulting equation by ρ, and then differentiating with respect to τ , show that the parameter ρ, the time τ 1 for which ρ = 1, and the final deflection of the central section are given by 2 3 (q0 − 1) τ 1 − e−τ1 (0 ≤ τ ≤ τ =3− ), λ 1 τ1 q0 λq0 1 − e−τ − τ

3 3 w0 τ τ τ + = q0 λ τ2 − 1 − τ2 1 + 2 q0 − 1 τ1 1 + 1 δ 2 3 2 2 2

ρ2 =

where τ2 > τ1 denotes the instant when the motion stops. Assuming q0 = 1.4, obtain a graphical plot for the variation of w0 /δ with λ, covering the range 1 ≤ λ ≤ 10.

Chapter 9

The Finite Element Method

In the numerical solution of engineering problems, it is often convenient to assume the physical domain to consist of an assemblage of a finite number of subdomains, called finite elements, which are connected with one another along their interfaces. The distribution of a governing physical parameter within each element is approximated by a suitable continuous function, which is uniquely defined in terms of its values at a specified number of nodal points that are usually located along the boundary of the element. The solution to the original boundary value problem is often reduced to that of a variational problem involving the nodal point values of the unknown parameter. In this chapter, we shall be concerned with a rigid/plastic formulation of the finite element method, a complete elastic/plastic formulation of the problem being available elsewhere (Chakrabarty, 2006).

9.1 Fundamental Principles The technological forming of metals generally involves plastic strains which dominate over the elastic strains, and the rigid/plastic approximation of material response is therefore appropriate in most cases. At any stage of the deformation process, the current shape of the workpiece and the associated strain distribution are supposed to be known from the previous computation. It is further assumed that the influence of geometry changes during an incremental loading of the workpiece may be disregarded in a variational formulation for the estimation of the strain increment suffered by each material element. The geometry of the workpiece can be subsequently updated on the basis of the computed strain increment, and the solution can be continued in a stepwise manner. The simplest rigid/plastic formulation of the finite element method, widely used for the analysis of metal-forming processes (Kobayashi et al., 989), will be described in what follows.

9.1.1 The Variational Formulation In a typical boundary value problem, the surface traction Fj is prescribed over a part of the boundary, and the velocity vj is prescribed over the remainder. Let ε˙ ij denote J. Chakrabarty, Applied Plasticity, Second Edition, Mechanical Engineering Series, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77674-3_9,

671

672

9 The Finite Element Method

the true strain rate corresponding to any kinematically admissible velocity field, and let σij denote the associated true stress that is not necessarily in equilibrium. Among a sufficiently wide class of admissible velocity fields, the actual field corresponds to a stationary value of the functional U=

σij ε˙ ij dV −

Fj vj dS.

The variational principle is analogous to the conventional upper bound technique for the estimation of the yield point load. The finite element approach, because of its discretization, allows the consideration of a much wider class of velocity fields than that possible in the usual upper bound analysis. The restriction imposed on the admissible velocity field by the incompressibility of the material can be removed by introducing a large positive constant , known as the penalty constant, which allows the variational functional to be written in the modified form 1 2 dV − Fj vj dS. (9.1) U = σij ε˙ ij dV + ε˙ kk 2 ˙ and setting the first variation δU to Denoting the volumetric strain rate ε˙ kk by λ, zero, we get

σ¯ δ ε˙¯ dV +

λ˙ δ λ˙ dV −

Fj δvj dS = 0,

(9.2)

where σ¯ and ε¯˙ are the effective stress and strain rate, respectively. The parameter , which is similar to the elastic bulk modulus, must be carefully chosen, since too large a value of would cause difficulties in the convergence, while too small a value of would result in unusually large changes in volume. An appropriate choice of seems to be that for which λ˙ is restricted to the order of 10–3 times the mean effective strain rate in the material (Kobayashi et al., 1989). Rigid zones, which generally coexist with plastically deforming zones in most metal-forming processes, may be identified by the occurrence of effective strain rates that are smaller than a certain limiting value ε˙ 0 . These regions can be approximately included in the analysis by setting σ¯ = hε˙¯ in the first integral of (9.2), where h is a constant. Such a modification of the variational equation is necessary only over those regions which are considered as nearly rigid. Realistic estimates of the extent of the deforming zone can be achieved, without adversely affecting the convergence of the numerical analysis, by taking ε¯ to have an assigned limiting value of 10–2 approximately.

9.1.2 Velocity and Strain Rate Vectors The finite element analysis of the boundary value problem begins with the specification of a velocity distribution within each element. The velocity must have continuous first derivatives within the element and must satisfy the condition of

9.1

Fundamental Principles

673

continuity across its interfaces with the adjacent elements. The components of the velocity at any point within an element are completely defined by those at a sufficient number of nodal points, which are generally located along the boundary of the element. Considering a three-dimensional velocity field with rectangular components u, v, and w, it is possible to express them in the form u=

1

N α uα ,

v=

1

Nα vα ,

w=

1

Nα wα

(9.3)

where uα, vα , and wα are the velocity components at the αth node, Nα is the associated shape function, and the summation extends over all the nodal points of the element. The explicit forms of the shape function in specific cases will be discussed later. Equation (9.3) can be conveniently written in the matrix form v = Nq

(9.4)

where N is the shape function matrix, v is the velocity vector for a generic particle, and q is the nodal velocity vector. These vectors are defined as vT = {u, v, w} ,

qT = {u1 , v1 , w1 , u2 , v2 , w2 , . . .} ,

where the superscript T denotes the transpose. The shape function matrix assumes the form ⎤ N1 0 0 N2 0 0 N3 0 0 . . . N = ⎣ 0 N1 0 0 N2 0 0 N3 0 . . . ⎦ 0 0 N1 0 0 N3 0 0 N3 . . . ⎡

(9.5)

The total number of columns in the N-matrix is equal to the nodal degrees of freedom, defined by the number of nodal velocity components. The components of the true strain rate within the element can be expressed in terms of the nodal velocities and the derivatives of the shape functions. If a typical component of the velocity at the αth node is denoted by qi (α) then vi = Nα qi (α) in view of (9.4), and the expression for the true strain rate tensor becomes ε˙ ij =

∂vj 1 ∂vi 1 ∂Nα ∂Nα + = qi (α) + qj (α), 2 ∂xj ∂xi 2 ∂xi ∂xi

Denoting the rates of extension in the coordinate directions by ε˙ x , ε˙ y , and ε˙ z , and the associated rates of engineering shear by γ˙xy , γ˙yz , and γ˙yz , the strain rate vector ε˙ may be defined by its transpose 7 6 ε˙ T = ε˙ x , ε˙ y , ε˙ z , ε˙ xy , ε˙ yz , ε˙ zx .

674

9 The Finite Element Method

In view of the preceding expression for the strain rate, the components of the vector are given by 1 1 Pα uα , ε˙ y = Qα uα, ε˙ z = Rα w α , 1 1 γ˙xy = (Qα uα + Pα vα ) , γ˙yz = (Rα vα + Qα wα ) ,

ε˙ x =

1

(9.6)

together with a similar expression for γ˙zx , where we have introduced the notation Pα =

∂Nα , ∂x

Qα =

∂Nα , ∂y

Rα =

∂Nα . ∂z

(9.7)

It may be noted that the parameters Pα , Qα , and Rα are generally functions of the space variables. Equation (9.6) may be written in the matrix form ε˙ = Bq

(9.8)

where B denotes the strain rate matrix, which can be written explicitly as ⎡

p1 ⎢ 0 ⎢ ⎢ 0 B=⎢ ⎢ Q1 ⎢ ⎣ 0 R1

0 Q1 0 P1 R1 0

0 0 R1 0 Q1 P1

P2 0 0 Q2 0 R2

0 Q2 0 P2 R2 0

0 0 R2 0 Q2 P2

P3 0 0 Q3 0 R3

0 Q3 0 P3 R3 0

0 0 R3 0 Q3 P3

⎤ ... ...⎥ ⎥ ...⎥ ⎥. ...⎥ ⎥ ...⎦ ...

The number of columns in the B-matrix is evidently identical to the nodal degrees of freedom of the considered element. The equivalent or effective strain rate ε˙¯ and the volumetric strain rate λ, which occur in the finite element analysis, need to be expressed in matrix forms involving the nodal velocity vector q. Since ε˙¯ =

1/2 2 ε˙ ij ε˙ ij 3

(9.9)

when the material is isotropic, it follows from (9.8) that ε¯˙ 2 = ε˙ T D˙ε = qT Sq., S = BT DB

(9.10)

For a three-dimensional deformation mode, D is a diagonal matrix whose first three diagonal elements are equal to 23 and the last three diagonal elements are equal to 13 . In the special case of plane stress, as we shall see later, the matrix D is not diagonal. In view of the first three relations of (9.6), the volumetric strain rate is λ˙ = CT q

9.1

Fundamental Principles

675

where CT is a row vector obtained by adding the vectors represented by the first three rows of B. Thus CT = {P1 , Q1 , R1 , P2 , Q2 , R2 , . . .} .

(9.11)

It is interesting to note that this vector is also obtained by premultiplying the matrix B by a row vector whose first three elements are unity and the remaining three elements are zero.

9.1.3 Elemental Stiffness Equations The global integral appearing in the variational equation (9.2) is actually an assembly of integrals taken over the individual elements in the deforming body. The derivation of the stiffness equation in the matrix form at the elemental level is therefore essential for the establishment of the global stiffness equation. Denoting typical elements of the matrices N, S, and C by the quantities Nij , Sij , and Cj , respectively, and using (9.4), (9.10), and (9.11), we have σ¯ σ¯ 1 σ¯ 1 ˙2 1 ˙ σ¯ δ ε¯ = δ ε¯ = δ Sij qi qj = Sij qi δqj , 2 2 ε˙¯ ε˙¯ ε˙¯ 2 λ˙ δ λ˙ = Cj qj δ (Ci qi ) = Ci Cj qj δqi , Fj δvj = Fj δ Nji qi = Nji Fj δqi . The expressions on the left-hand side of the above relations are actually the successive integrands in the variational equation (9.2) in relation to a typical element. Since δqj is arbitrary, the result becomes ∂U/∂qi =

( σ¯ ε˙¯ Sij qj dV +

Ci Cj qj dV −

Nji Fj dS = 0,

In matrix notation, the equation for the minimization of the functional U therefore takes the form (9.12) (σ¯ /ε˙¯ )Sq dV + CCT q dV − NT F dS = 0 where F is a column vector representing the applied force (Fx , Fy , Fz ) on the considered element, the last integral being the equivalent nodal point force. The preceding result represents a set of nonlinear simultaneous equations for the nodal point velocities. By assembling equations of type (9.12) over all the elements in the deforming body, we obtain the global equation for the boundary value problem. The solution to the stiffness equation (9.12) is generally obtained by an iterative procedure based on a linearization with the help of Taylor’s expansion of the functional U in the neighborhood of an assumed solution point q = q0 . The condition δU = 0 may therefore be written as

676

9 The Finite Element Method

∂ 2U ∂U + qj = −fi + kij qj = 0 ∂qi ∂qi ∂qj where the derivatives are considered at q = q0 , while q denotes the first-order correction to the nodal velocity q0 . The matrix form of the preceding equation is kq = f

(9.13)

where k denotes the elemental stiffness matrix, and f is the residual of the nodal point force vector obtained by setting q = q0 on the left-hand side of (9.12). Since the first derivative ∂U/∂qi is represented by (9.12), it is easily shown that σ¯ 1 ∂ σ¯ ∂ 2U Sij dV + Sik qk Sjm qm dV + ci cj dV = ∂qi ∂qj ε˙ ∂ ε˙¯ ε˙¯ ε˙¯ ( in view of (9.10). Setting ∂ σ¯ /ε˙¯ ε˙¯ = η, the elemental stiffness matrix can be expressed as k=

(σ¯ /ε˙¯ )S dV +

(η/ε˙¯ )SqqT ST dV +

CCT dV

(9.14)

The stiffness equations are most conveniently solved by an iterative method in which σ¯ /ε˙¯ is assumed constant (η = 0 ) during each iteration (Oh, 1982). The computation may begin by assuming that that the effective strain rate ε˙¯ in each element is the same as that in the previous step. Since σ¯ then follows from the computed value of ε¯ , the ratio σ¯ /ε˙¯ is easily evaluated in each element, leading to the stiffness matrix k. The elemental stiffness equations are then assembled to form the global stiffness equations (Section 9.4), which are solved under the prescribed boundary conditions. The solution for the velocity correction furnishes an updated nodal velocity field, and a modified strain rate in each element, which can be used to test the convergence of the solution. When the effective strain rate ε˙¯ in a given element is found to be less than a preassigned value, it may be considered as nonplastic with an effective stress proportional to ε˙¯ . The stiffness equation for such an element should be modified by replacing σ¯ /ε˙¯ in the leading integral of (9.12) with a constant h. The stiffness matrix k is similarly obtained by setting σ¯ /ε˙¯ = h and η= 0 in (9.14). The penalty function therefore enables us to separate the deforming region from the undeforming one.

9.2 Element Geometry and Shape Function 9.2.1 Triangular Element It is evident from the preceding discussion that the shape function Nα for any given geometry of the element is a fundamental quantity in the finite element analysis. Equation (9.3) indicates that if xβ ,yβ denote the rectangular coordinates of the βth node, then

9.2

Element Geometry and Shape Function

677

Nα xβ ,yβ = δαβ , where δαβ is the familiar Kronecker delta. It should be noted that any scalar function f(x, y) can be expressed in the same way as the velocity components are, using the same shape functions. In the case of two-dimensional problems, the simplest finite element is a triangle whose vertices are defined by the coordinates (x1 , y1 ), (x2 , y2 ), and (x3 , y3 ). The coordinates of any point P within the triangle can be expressed in terms of those of its vertices using the transformation x = L1 x1 + L2 x2 + L3 x3 , y = L1 y1 + L2 y2 + L3 y3 ,

(9.15)

where L1 , L2 , and L3 are the ratios of the areas of the three triangles, formed by joining the generic point P to the vertices 1, 2, and 3, to the total area A of the triangular element shown in Fig. 9.1. It follows from this definition that L1 = 0 along the side 2–3, L2 = 0 along the side 3–1, and L3 = 0 along the side 1–2. In view of the identity L1 +L2 + L3 = 1, (9.15) can be solved for the area coordinates L1 , L2 and L3 to give Lα = (aα + bα x + cα y) /2A, 2A = (x1 − x2 ) (y2 − y3 ) − (x2 − x3 ) (y1 − y2 ) ,

(9.16)

where aα , bα , and cα depend on the coordinates of the vertices of the triangle 1–2–3 and are given by ⎫ a1 = x2 y3 − x3 y2 , b1 = y2 − y3 , c1 = x3 − x2 , ⎬ a2 = x3 y1 − x1 y3 , b2 = y3 − y1 , c2 = x1 − x3 , ⎭ a3 = x1 y2 − x2 y1 , b3 = y1 − y2 , c3 = x2 − x1 .

Fig. 9.1 Triangular elements. (a) Linear element and (b) quadratic element

(9.17)

678

9 The Finite Element Method

It is important to note that Lα xβ ,yβ = δαβ , which represents a fundamental property of the area coordinates, similar to that for the shape functions. A linear triangular element consists of three nodes located at its vertices, as shown in Fig. 9.1 (a), the velocity components at any point within the triangle being assumed to vary linearly with x and y. The continuity of the velocity at the nodal points therefore ensures its continuity along the sides of the triangle. The velocity distribution within the triangle may be written as u = N1 u1 + N2 u2 + N3 u3 , v = N1 v1 + N2 v2 + N3 v3 , where the shape functions N1 , N2 , and N3 are linear functions of x and y. It follows from (9.15) and the linearity of the area coordinates that these functions are identical to L1 , L2 , and L3 respectively. Thus N1 = (a1 + b1 x + c1 y) / 2A,

N2 = (a2 + b2 x + c2 y) / 2A,

N3 = (a3 + b3 x + c3 y) / 2A. (9.18)

Elements which involve shape functions that are identical to the functions defining the coordinate transformation of type (9.15) are known as isoparametric elements. The linear triangular element is therefore isoparametric. The relevant components of the strain rate matrix B for the linear triangular element are y2 − y3 y3 − y1 , P2 = , 2A 2A x3 − x2 x1 − x3 Q1 = , Q2 = , 2A 2A

P1 =

P3 = − (P1 + P2 ) , (9.19) Q3 = − (Q1 + Q2 ) ,

in view of (9.7) and (9.18). It may be noted that the components of the strain rate corresponding to the linear triangular element are constant over each element. A useful integral involving exponents of the area coordinates taken over the area of the triangle is

p

L1m L2n L3 dxdy =

m! n! p!(2A) (m + n + p + 2)!

(9.20)

where m, n, and p are integers. The result follows from the fact that the Jacobian of the transformation of coordinates is J=

∂x ∂y ∂x ∂y − = 2A ∂L1 ∂L2 ∂L2 ∂L1

The linear triangular element can be used for treating finite deformation problems by using a linear interpolation of the Lagrangian strains from the midpoints of the sides of the triangle, as has been shown by Flores (2006). In a quadratic triangular element, there are three primary nodes located at the vertices of the triangle, and three secondary nodes at the midpoints of the sides of the triangle, as shown in Fig. 9.1(b). The velocity components u and v, which are

9.2

Element Geometry and Shape Function

679

assumed to be quadratic functions of x and y, are again continuous across the sides of the triangle and are expressed in terms of the nodal values as u = N1 u1 + N2 u2 + N3 u3 + N4 u4 + N5 u5 + N6 u6 v = N1 v1 + N2 v2 + N3 v3 + N4 v4 + N5 v5 + N6 v6 where (uα ,vα ) denote the velocity vector at a typical nodal point α. The shape functions associated with the six nodal velocities can be expressed in terms of the area coordinates L1 , L2 , and L3 , the result being easily shown to be N1 = L1 (2L1 − 1) , N4 = 4L1 L2 ,

N2 = L2 (2L2 − 1) , N5 = 4L2 L3 ,

N3 = L3 (2L3 − 1) , N6 = 4L3 L1 .

(9.20)

The elements of the strain rate matrix can be determined in the same way as that for the linear triangle. It can be seen that the quadratic element with straight sides is not isoparametric. It is possible, however, to construct curvilinear triangles in this case to form isoparametric elements (Zienkiewicz, 1977).

9.2.2 Quadrilateral Element In the solution of special problems, it is often convenient to use quadrilateral elements with nodal points located at the corners. The element is generally defined parametrically in terms of auxiliary coordinates (ξ , η), known as natural coordinates, so that the quadrilateral is transformed into a square defined by ξ ± 1 and η = ± 1, as shown in Fig. 9.2. The shape functions are bilinear in ξ and η according to the relations

Fig. 9.2 Linear quadratic element. (a) Natural coordinates and (b) physical coordinates

680

9 The Finite Element Method

N1 = N3 =

1 4 1 4

(1 − ξ ) (1 − η) ,

N2 =

(1 + ξ ) (1 + η) ,

N4 =

1 4 1 4

(1 + ξ ) (1 − η) , (1 − ξ ) (1 + η) .

(9.21)

It is easy to see that Nα (ξα ,ηα ) = δαβ , where (ξα ,ηα ) are the natural coordinates of a typical node. The transformation of the natural coordinates (ξ ,η) into the physical coordinates (x, y) is given by x = N1 x1 + N2 x2 + N3 x3 + N4 x4 , y = N1 y1 + N2 y2 + N3 y3 + N4 y4 .

(9.22)

The condition of constancy of the slope dy/dx of each side of the quadrilateral is therefore identically satisfied. Since the coordinate transformation involves the same shape functions as those in the velocity relations u=

1

Nα uα ,

v=

1

Nα vα ,

where α varies from 1 to 4, the linear quadrilateral element is isoparametric. If we consider an 8-node isoparametric element in which the secondary nodes are located at the midpoints of the sides of the square in the (ξ , η)-plane, the shape of the element in the physical plane is a curvilinear quadrilateral defined by shape functions that are obtained on multiplying the right-hand side of (9.23) by suitable linear functions of ξ and η. The elements of the strain rate matrix B involve the derivatives of the shape functions with respect to x and y. Since the corresponding derivatives with respect to ξ and η are given by

∂Nα /∂ξ ∂Nα /∂η

∂x/∂ξ ∂y/∂ξ = ∂x/∂η ∂y/∂η

∂Nα /∂ξ ∂Nα /∂y

,

where the square matrix is the well-known Jacobian matrix having a determinant J, which is the Jacobian of the transformation, and is given by J=

∂x ∂y ∂x ∂y − ∂ξ ∂η ∂η ∂ξ

(9.23)

A straightforward process of inversion of the preceding matrix equation furnishes 1 ∂y/∂η −∂y/∂ξ ∂Nα /∂x ∂Nα /∂ξ = . ∂Nα /∂y ∂Nα /∂η J −∂x/∂η ∂x/∂ξ In the case of a liner quadrilateral element, the partial derivatives appearing in the Jacobian matrix and its inverse are readily obtained from (9.22) and (9.23), the result being 8 J = (x13 y24 − x24 y13 ) + (x34 y12 − x12 y34 ) ξ + (x23 y14 − x14 y23 ) η, (9.24)

9.2

Element Geometry and Shape Function

681

where xij = xi − xj and yij = yi − yj . Since ∂Nα /∂x and ∂Nα /∂y are equal to Pα and Qα , respectively, according to (9.7), we have ⎧ ⎫ ⎪ ⎪ P1 ⎪ ⎪ ⎨ ⎬ P2 = ⎪ ⎪ P3 ⎪ ⎪ ⎩ ⎭ P4 ⎧ ⎫ Q1 ⎪ ⎪ ⎪ ⎨ ⎪ ⎬ Q2 = Q3 ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ Q4

1 8J

1 8J

⎧ y24 ⎪ ⎪ ⎨ −y13 −y24 ⎪ ⎪ ⎩ y13 ⎧ −x24 ⎪ ⎪ ⎨ x13 x24 ⎪ ⎪ ⎩ −x13

−y34 ξ +y34 ξ +y12 ξ −y12 ξ +x34 ξ −x34 ξ −x12 ξ +x12 ξ

⎫ −y23 η ⎪ ⎪ ⎬ +y14 η , −y14 η ⎪ ⎪ ⎭ +y23 η ⎫ +x23 η ⎪ ⎪ ⎬ −x14 η . +x14 η ⎪ ⎪ ⎭ −x23 η

(9.25)

The results for the 8-node isoparametric quadrilateral element are evidently more complex. It is more convenient in this case to evaluate Pα and Qα numerically for selected values of ξ and η, using the method employed for the linear element.

9.2.3 Hexahedral Brick Element For treating three-dimensional problems, the quadrilateral element must be replaced by a brick element with eight corners. The simplest isoparametric element involves a node at each of the eight corners of the element, which assumes the form of a cube defined by ξ = ± 1, η = ± 1, and ζ = ± 1, in the associated natural coordinate system (ξ , η, ζ). The shape functions are defined as Nα =

1 (1 + ξα ξ ) (1 + ηα η) (1 + ζα ζ ) , 8

(9.26)

where (ξ α , ηα , ζα ) are the natural coordinates of a typical node α. Since ξα2 = ηα2 = ζα2 − 1, the above expression indicates that Nα = 1 at the αth node, while Nα = 0 at all other nodes due to the vanishing of at least one of its factors. The velocity distribution is given by (9.3), while the coordinate transformation is x=

1

Nα xα ,

y=

1

Nα yα ,

z=

1

N α zα ,

(9.27)

where (xα , yα , zα ) are the rectangular coordinates of the αth node. Figure 9.3 shows the brick element defined in both the natural coordinates and the physical coordinates. The velocity field for the brick element can be expressed as u=

1

Nα uα ,

v=

1

Nα vα ,

w=

1

Nα wα

where (uα , vα , wα ) are the rectangular components of the velocity vector at the αth node. The 8-node brick element is therefore isoparametric. The Jacobian matrix

682

9 The Finite Element Method 8

8

7

7

5 4

ζ

4 3

z

3 y

η ξ

6

5

6

x

1

1

2

2

(a)

(b)

Fig. 9.3 Three–dimensional brick element depicted in (a) natural coordinate system, and (b) rectangular coordinate system

⎡

∂x/∂ξ J = ⎣ ∂x/∂η ∂x/∂ζ

∂y/∂ξ ∂y/∂η ∂y/∂ζ

⎤ ∂z/∂ξ ∂z/∂η ⎦ ∂z/∂ζ

(9.28)

for the transformation of coordinates is easily formed for any selected values of (ξ , η,ζ), using (9.26) and (9.27). This matrix and can be inverted numerically to determine the quantities Pα , Qα , and Rα , using the expression ⎫ ⎫ ⎧ ⎫ ⎧ ⎧ ⎨ Pα ⎬ ⎨ ∂Nα /∂x ⎬ ⎨ ∂Nα /∂ξ ⎬ Qα = ∂Nα /∂y = j−1 ∂Nα /∂η ⎭ ⎭ ⎩ ⎭ ⎩ ⎩ Rα ∂Nα /∂z ∂Nα /∂ζ

(9.29)

Since the column vector on the right-hand side of (9.29) is easily obtained using (9.26), the column vector on the left-hand side can be evaluated for any selected node α. The determinant of the Jacobian matrix (9.28) furnishes the value of J in each particular case.

9.3 Matrix Forms in Special Cases 9.3.1 Plane Strain Problems We begin with the situation where the resultant velocity of each particle is parallel to a given plane, which is considered as the (x, y)-plane, the rectangular components of the velocity of a typical particle being denoted by (u, v). Adopting the 4-node isoparametric quadrilateral element shown in Fig. 9.2, the nodal velocity vector q and the shape function matrix N may be written as

9.3

Matrix Forms in Special Cases

683

qT = {u1 v1 u2 v2 u3 v3 u4 v4 } , N1 0 N2 0 N3 0 N4 0 , N= 0 N 1 0 N2 0 N3 0 N4

(9.30)

where N1, N2 , N3 , and N4 are given by (9.21) in terms of the natural coordinates (ξ , η). The vectors representing the particle velocity and the strain rate are u v= = Nq, v

⎫ ⎧ ⎨ ε˙ x ⎬ = Bq, ε˙ = ε˙ y ⎭ ⎩ γ˙xy

where B is the strain rate matrix having the form ⎡

⎤ P1 0 P2 0 P3 0 P4 0 B = ⎣ 0 Q1 0 Q2 0 Q3 0 Q4 ⎦ , Q1 P 1 Q2 P 2 Q3 P 3 Q4 P 4

(9.31)

where Pε and Q4 are given by (9.25) as functions of (ξ , η) in terms of the global coordinates of the nodal points. The equivalent strain rate ε˙¯ and the volumetric strain rate λ˙ can be evaluated from (9.10) and (9.11), respectively, the relevant matrix and vector being of the form ⎡ ⎤ 100 2⎣ 0 1 0 ⎦, D= 3 0 0 12

⎧ ⎫ ⎨1⎬ C = BT 1 . ⎩ ⎭ 0

(9.32)

The set of nonlinear equations for the unknown nodal velocities are finally obtained by considering (9.12) for each individual element. The solution is most conveniently obtained by using linearized stiffness equations of type (9.13) as explained before in general terms.

9.3.2 Axially Symmetrical Problems In problems of axial symmetry, the finite element is taken in the form of a ring whose cross section is identical to the two-dimensional element. For an isoparametric ring element, the global coordinates (r, z) of a generic point inside the element are related to the natural coordinates (ξ , n) by the transformation r=

1

Nα r α ,

z=

1

N α zα ,

(9.33)

where (rα , zα ) are the coordinates of a typical node, and Nα is the corresponding shape function that depends on (ξ , η). For the 4-node quadrilateral element shown in Fig. 9.4, the associated shape functions are given by (9.30). The radial and circumferential components of the velocity of a typical particle are given by

684

9 The Finite Element Method

Fig. 9.4 Axisymmetric ring element having the cross section of a 4-node quadrilateral in the meridian plane

u=

1

w=

Nα u α ,

1

Nα wα ,

where (uα wα ) are the nodal velocity components, the shape function matrix N being identical to that in (9.27). The vector representing the strain rate then becomes ⎧ ⎫ ⎡ ∂/∂r ε˙ r ⎪ ⎪ ⎪ ⎬ ⎢ ⎨ ⎪ ε˙ z 0 =⎢ ε˙ = ⎣ 1/r ε ˙ ⎪ ⎪ θ ⎪ ⎭ ⎩ ⎪ γ˙rz ∂/∂z

⎤ 0 ∂/∂z ⎥ ⎥ u = Bq. 0 ⎦ w ∂/∂r

(9.34)

The nodal velocity vector q is given by (9.30) with vα replaced by wα , and the strain rate matrix B has the modified form ⎡

P1 ⎢ 0 B=⎢ ⎣ B1 Q1

0 Q1 0 P1

P2 0 B2 Q2

0 Q2 0 P2

P3 0 B3 Q3

0 Q3 0 P3

P4 0 B4 Q4

⎤ 0 Q4 ⎥ ⎥, 0 ⎦ P4

(9.35)

where Pα and Qα are given by (9.26) with xij and yij replaced rij by and zij , respectively, the expression for J in (9.24) being similarly modified, while the third row of (9.35) is given by Bα = Nα /r = Nα /(N1 r1 + N2 r2 + N3 r3 + N4 r4 ) . The evaluation of ε˙¯ and λ˙ is identical to that for the plane strain case, the matrix D and the vector C being similar to those given by (9.32). The final stiffness equations can be handled in the same way as those for plane strain.

9.4

Sheet Metal Forming

685

9.3.3 Three-Dimensional Problems In the three-dimensional finite element analysis, it is generally convenient to use the 8-node isoparametric brick element shown in Fig. 9.3. In the natural coordinate system (ξ , η, ζ ), this element is a cube, and the deformation mode is specified by the nodal velocity vector q, where qT = [u1

v1

w1

u2

...

v2

u8

v8

w8 ]

The transformation between the natural and global coordinates defined by (9.27) in terms of the shape functions are given by (9.26), the shape function matrix being ⎡

N1 N = ⎣0 0

0 N1 0

0 0 N1

N2 0 0

0 N2 0

0 0 N2

. . . . N8 .... 0 .... 0

0 N8 0

⎤ 0 0 ⎦ N8 ..

(9.36)

The velocity field within the element and the associated strain rate vector, having the six rectangular components given by (9.6), may be written as ⎧ ⎫ ⎨u ⎬ u = v = Nq, ε˙ = Bq ⎩ ⎭ w where B is the strain rate matrix for the 8-node brick element, in which the components of train rate are given by (9.26). It is easily shown that ⎡

P1 ⎢ 0 ⎢ ⎢ 0 B=⎢ ⎢ Q1 ⎢ ⎣ 0 R1

0 Q1 0 P1 R1 0

0 0 R1 0 Q1 P1

P2 0 0 Q2 0 R2

0 Q2 0 P2 R2 0

0 0 R2 0 Q2 P2

. .. .. .. .. ..

P8 0 0 Q8 0 R8

0 Q8 0 P8 R8 0

⎤ 0 0 ⎥ ⎥ R8 ⎥ ⎥ 0 ⎥ ⎥ Q8 ⎦ P8

(9.37)

The Jacobian of the transformation of coordinates, given by the determinant of the matrix (9.28), is easily evaluated numerically in each particular case. The square matrix S, which defines the stiffness matrix, is given by (9.10), where D is a diagonal matrix in which each of the six diagonal element is 2/3.

9.4 Sheet Metal Forming 9.4.1 Basic Equations for Sheet Metals In the plastic forming of sheet metal, the stress component in the thickness direction is generally disregarded. A state of plane stress therefore exists in each element of the sheet, which is assumed to be orthotropic with the anisotropic axes coinciding

686

9 The Finite Element Method

with the rolling, transverse, and thickness directions. Considering a biaxial loading of the sheet, we choose a set of rectangular axes in which the x- and y-axes are directed along the rolling and transverse directions, respectively, the z-axis being taken along the normal to the sheet. The effective stress in a material element is given by the yield criterion, and may be defined in such a way that it reduces to the current uniaxial yield stress in the rolling direction. Using the quadratic yield criterion for simplicity (Section 6.2), we write 1/2 2H F+H 2N σx σy + σy2 + τxy 2 σ¯ = σx 2 − G+H G+H G+H

(9.38)

where F, G, H, and N are parameters defining the state of planar anisotropy of the sheet metal. When the hypothesis of strain equivalence is adopted for the hardening of the material, the effective strain rate in a deforming element may be written as ε˙¯ = (G + H)

2 /4 ε˙ x2 + ε˙ x ε˙ y + ε˙ y2 + γ˙xy

1/ 2 ,

G2 + GH + H 2

so that ε˙¯ becomes identical to the longitudinal strain rate ε˙ x in the case of a simple tension applied in the rolling direction. This is easily verified by setting σ = σ as the only nonzero stress in the associated flow rule, which may be written as ε˙ y γ˙xy ε˙ x = = = λ˙ + H) σ + H) σ (G (F N x − Hσy y − Hσx

(9.39)

where λ˙ is a positive scalar. The thickness strain rate follows from the incompressibility condition ε˙ z = −(˙εx + ε˙ y ) Since the rate of plastic work per unit volume is equal to σ¯ λ˙ in view of (9.39) and (9.38), the effective strain rate according to the hypothesis of work equivalence is equal to λ˙ , giving ε˙¯ =

√

G+H

(F + H) ε˙ x2 + 2H ε˙ x ε˙ y + (G + H) ε˙ y2 FG + GH + HF

+

2 γ˙xy

2N

1/2 ,

(9.40)

which is obtained by expressing the stresses in terms of the strain rates using the flow rule (9.39), and substituting them into the field criterion (9.38). The ratios of the anisotropic parameters F, G, H, and N can be determined from the measured R-values of the sheet in the rolling, transverse, and at 45◦ to the rolling direction. The strain rate vector in the plane stress formulation is identical to those for plane strain. It is convenient to express the effective strain rate in the matrix form ε˙ =

ε˙ T D˙ε

as before, but the forms of the square matrix D for the anisotropic material depends on the hardening hypothesis and is expressed by

9.4

Sheet Metal Forming

⎡

2 1 (G + H) ⎣ 1 2 D= 2S 0 0 2

687

⎤ 0 0 ⎦, 1 2

⎡

⎤ F+H H 0 G+H ⎣ H G + H 0 ⎦ , (9.41) D= T 0 0 T/2 N

where S = G2 +GH+H2 and T = FG+GH+HF. The first expression in (9.41) corresponds to the hypothesis of strain equivalence and the second expression to that of work equivalence. When the material exhibits normal anisotropy with a uniform R-value, it is only necessary to set F = G, H = RG, N = (1 +2R)G, and T = NG in the preceding relations. The shape function matrix and the strain rate matrix for a given shape of the element are identical to those for plane strain. The application of the preceding theory to the flange drawing and bore-expanding processes has been reported by Lee and Kobayashi (1975). A finite element formulation of the problem based on the biquadratic yield criterion (Section 6.2) has been discussed by Gotoh (1978, 1980).

9.4.2 Axisymmetric Sheet Forming In the case of out-of-plane deformations of the sheet metal, such as in the hydraulic bulging and punch stretching, the deformed sheet at each stage may be regarded as a membrane with a state of plane stress existing in each element. Additional equations are obviously necessary to determine the deformed shape of the sheet metal, and the distribution of stress and strain in the workpiece. When the deformed sheet forms a surface of revolution, it may be approximated by a succession of conical frustums, each frustum being treated as a finite element, Fig. 9.5. Consider a line element along the meridian, extending between the nodal points 1 and 2 with coordinates (r1 , z1 ) and (r2 , z2 ) respectively. If the radial and axial coordinates of a generic particle are denoted by r and z, and the corresponding components of the velocity are denoted by u and w, respectively, then

z

2

Fig. 9.5 Finite element approximation of a surface of revolution developed in the axisymmetric forming of a sheet metal

2 (ξ = 1) 1 (ξ = – 1)

1

r

688

u=

9 The Finite Element Method

1 1 1 1 (u1 + u2 ) + (u2 − u1 ) ξ , r = (r1 + r2 ) + (r2 − r1 ) ξ , 2 2 2 2

−1 ≤ ξ ≤ 1

where u1 and u2 are the radial velocities at the nodal points 1 and 2, respectively. Similar relations may be written down for the components w and z. If φ is the angle made by the surface normal with the axis of symmetry, which coincides with the z-axis, then the circumferential and meridional components of the strain rate are given by (u2 + u1 ) + (u2 − u1 ) ξ u = r (r2 + r1 ) + (r2 − r1 ) ξ 1/2 z2 − z1 2 ∂u u2 − u1 + α, ˙ α = 1+ + φ˙ tan φ = ε˙ φ = ∂r r2 − r1 r2 − r1 ε˙ =

(9.42)

The effective stress and strain rate for a uniform R-value material with normal anisotropy, according to the hypothesis of work equivalence, may be written as 1/2 2R 2 2 σ¯ = σθ − σθ σφ + σφ 1+R 1/2 1+R 2R ε˙ θ2 + ε˙¯ = √ ε˙ θ ε˙ φ + ε˙ φ2 1+R 1 + 2R in view of (9.38) and (9.40). The effective strain can be expressed in a matrix form as before in terms of a square D, the result being ˙ ε¯ = ε˙ T Dε,

ε˙ =

ε˙ θ ε˙ φ

,

l+R l+R D= l + 2R R

R l+R

(9.43)

The finite element procedure is based on a variational principle similar to (9.1). Since the condition of incompressibility need not be dealt with, the penalty function may be omitted for the variational formulation. It is convenient in this case to multiply the integrands in ( ) by a small time increment Δt and write the functional in the modified form U = σ (ε) dV − Fj (uj ) dS, ˙ and uj = vj t, not to be confused with the radial velocity. where ¯ε = εt, Considering the variation of U, and using the fact that δ σ¯ = Hδ (¯ε ,), where H is the plastic modulus, we have

δU =

h

σ + H (¯ε) δ (¯ε) dA − Fj Nij δ(qj ) dS ε

(9.44)

where h is the local sheet thickness, A is the surface area of the element, Nji is a typical element of the shape function transpose matrix of which the nonzero components

9.4

Sheet Metal Forming

689

are N11 = N22 = (1 – ξ )/2 and N31 = N42 = (1 + ξ )/2, and Δqj is a typical component of the nodal displacement increment vector, similar to the nodal velocity vector. The effective strain increment ¯ε can be expressed in terms of the components of the strain increment by replacing the strain rates appearing on the right-hand side of (9.39) by the corresponding strain increments. In matrix notation, the effective strain increment becomes 1/2 , ¯ε = (εT )D(ε)

εT = εθ εφ ,

in view of (9.43). The strain increments are evidently given by the strain rates in (9.43) multiplied by Δt. It follows from (9.43) that (Toh and Kobayashi, 1985) ∂U = ∂qi,

h

σ¯ + H ai dA − Fj Nji dS, ¯ε

ai =

∂ T ε D (ε) (9.45) ∂qi

The second derivative of the functional U with respect to a nodal velocity component follows from (9.45) and is easily shown to be ∂ 2U = ∂qi, ∂qj bij =

h

σ¯ hσ¯ ai aj dA, +H bij + cij dA − 2 ¯ε (¯ε)

∂ 2 T ∂ T ∂ (ε) ε ε D(ε), cij = D ∂qi ∂qj ∂qi ∂qi

(9.46)

The element stiffness equations for the sheet metal-forming process now given by kij qj = fi where kij and –fi are given by the first equations of (9.46) and (9.45), respectively, while Δqj represents the displacement correction vector. The formation of the global stiffness equations and their solution can be carried out in the same way as described before. A finite element formulation of the problem based on a nonlinear membrane shell theory has been considered by Wang and Budiansky (1978). An implementation of the nonquadratic yield criterion (Section 6.2) into the finite element formulation has been presented by Wang (1984).

9.4.3 Sheet Forming of Arbitrary Shapes In general, the out-of-plane deformation of a sheet metal is complicated by the fact that the principal axes of the stress and strain increments are not known in advance. It is therefore necessary to extend the analysis given above to deal with the general sheet forming process. Assuming a state of normal anisotropy of the sheet metal as

690

9 The Finite Element Method

before, the effective stress and strain increments are obtained from (9.38) and (9.40) in the form 1/2 2R 1 + 2R 2 σx σy + σy2 + 2 τxy σ¯ = σx2 − 1+R 1+R ¯ε =

2 2 1 2 1/2 1+R γxy (1 + R) (εx )2 + εy + 2R (εx ) εy + εy + 1 + 2R 2 (9.47)

according to the hypothesis of work equivalence. Introducing a square matrix D, the effective strain increment can be expressed in the matrix form 61/2 7 ¯ε = εT D(ε) ,

⎡ l+R R l+R ⎣ R l+R D= l + 2R 0 0

⎤ 0 0 ⎦ 1/2

(9.48)

It is convenient to discretize the sheet metal into an assemblage of linear triangular elements and consider a set of rectangular axes in which the x- and y-axes are taken along the plane of the sheet, and the z-axis along the normal. The components of the increment of displacement at any point of the element during a time increment Δt may be expressed in terms of the nodal values in the matrix form ⎧ ⎫ ⎤ ⎡ N1 0 0 N2 0 0 N3 0 0 ⎨ u ⎬ (9.49) u = v = Nq, N = ⎣ 0 N1 0 0 N2 0 0 N3 0 ⎦ ⎩ ⎭ 0 0 N1 0 0 N2 0 0 N3 w where N1 , N2 , and N3 are the shape functions, which are identical to the area coordinates and are given by (9.19), while q is the nodal displacement increment vector given by qT = [u1

v1

w1

u2

v2

w2

u3

v3

w3 ]

Each normal component of the true strain in the coordinate directions is the logarithm of the ratio of the final and initial material line elements originally coinciding with each coordinate axis. These are the logarithmic normal components of the Lagrangian strain tensor, and their sufficiently small increments in the surface, if the deforming sheet may be written with sufficient accuracy as εx =

2 ∂ 1 ∂ (u) + (w) , ∂x 2 ∂x

εy =

2 ∂ 1 ∂ (v) + (w) ∂y 2 ∂y

(9.50a)

where {u, v, w} are the components of the displacement of a generic particle. The increment of the surface shear strain, to the same order of approximation, may be written as γxy =

∂ ∂ ∂ ∂ (v) + (u) + (w) (w) ∂x ∂y ∂x ∂y

(9.50b)

9.5

Numerical Implementation

691

In view of (9.49), the various derivatives appearing in the above equations may be expressed as 1 ∂ (u) = Pα uα , ∂x

1 ∂ (v) = Pα vα , ∂x

1 ∂ (w) = Pα wα ∂x

1 ∂ (u) = Qα uα , ∂y

1 ∂ (v) = Qα vα , ∂y

1 ∂ (w) = Qα wα ∂y

where Pα and Qα are given by (9.19), where α varies from 1 to 3. Substituting from the above into equation (9.49), the vector representing the strain increment may be written in the matrix form ⎧ ⎫ ⎫ ⎧ ⎨ εx ⎬ ⎨ εx ⎬ , ε = εy = Bq + εy ⎩ ⎭ ⎩ ⎭ γxy γxy

⎡

⎤ P1 0 0 P2 0 0 P3 0 0 B = ⎣ 0 Q1 0 0 Q2 0 0 Q3 0 ⎦ Q 1 P1 0 Q 2 P2 0 Q 3 P3 0 (9.51) where B is the strain increment matrix. The second column vector in (9.50) arises from the change of the deforming sheet metal and is given by εx =

2 1 1 Pα w α , 2

εy =

2 1 1 Qα wα , 2

= γxy

1

P α wα

1

Qα wα

Using the same variational principle as that for the axisymmetric forming process, we arrive at the element stiffness equation, which is still governed by (9.45) and (9.46), but the strain increment vector ε and the associated matrix appearing in these equations now correspond to (9.51) and (9.48), respectively. It may be noted that the first derivatives appearing in (9.44 ) can be expressed in terms of Pα , Qα , and the associated components of the nodal displacement increment. A finite element formulation for sheet metal forming, including planar anisotropy of the sheet, has been presented by Yang and Kim (1987). A simplified method of finite element analysis based on the total strain theory of plasticity has been discussed by Majlessi and Lee (1988). The influence of bending of the sheet, which is locally important in a variety of sheet-forming processes, has been incorporated in the finite element formulation by Huh et al. (1994).

9.5 Numerical Implementation 9.5.1 Numerical Integration The elemental stiffness equation involves volume and surface integrals which generally require some kind of numerical integration in which the integrand is evaluated at a finite number of points, called integration points, within the limits of integration. We begin with the one-dimensional situation in which a scalar function f (x) is to be integrated over the range a ≤ x ≤ b. Introducing the natural coordinate ξ , such

692

9 The Finite Element Method

that 2x = (b + a) + (b – a)ξ , the formula for the numerical integration can generally be expressed as

b a

1 f (x)dx = (b − a) 2

1

−1

1 1 (b − a) wi F(ξi ) 2 n

F(ξ )dξ =

(9.52)

i=1

where wi is a weight factor associated with the integration point ξ = ξ i and n is the number of integration points. Simpson’s one-third rule of integration is a special case of (9.52), where n = 3, and w1 = w3 = 1/3, w2 = 4/3, the integration points being ξ 1 = –1, ξ 2 = 0, ξ 3 = 1. In the finite element analysis, it is customary to employ the Gaussian quadrature, as it requires the minimum number of integration points for the same degree of accuracy, The Gaussian quadrature formula for n integration points gives the exact result when F (ξ ) is a polynomial of degree less than or equal to 2n – 1. Setting F (ξ ) = ξ s in (9.52), and integrating, we have n 1

wi ξis = 0 (s = 1, 3, 5,..., 2n−1),

i=1

n 1 i=1

wi ξis =

2 s+1

(s = 0, 2, 4,...,2n−2)

(9.53) These relations enable us to determine the integration points and weight factors for any selected value of n. Considering n = 2, and setting s = 0,. . .,3, we have w1 + w2 = 2, w1 ξ 1 + w2 ξ 2 = w1 ξ 1 3 + w2 ξ 2 3 = 0, w1 ξ 1 2 + w2 ξ 2 2 = 2/3, giving the solution w1 = w2 = 1,

1 − ξ1 = ξ2 = √ (n = 2) 3

Similarly, considering n = 3, and setting s = 0,. . .,5 in (9.53), we obtain a set of six equations which are easily solved to give 5 w1 = w3 = , 9

8 w 2 = , − ξ1 = ξ 3 = 9

3 , 5

ξ2 = 0

From the geometrical point of view, the Gaussian integration formulas corresponding to n = 2 and n = 3 are equivalent to linear and quadratic approximations, respectively, of the given function F (ξ ). The integration points in the Gaussian quadrature for any given value of n are in fact the roots of the equation Pn ( ξ ) = 0, where Pn ( ξ ) denotes the Legendre polynomial of degree n. Consider now a scalar function f (x, y), which is defined over a two-dimensional isoparametric element with natural coordinates (ξ , η). If the number of integration points in the ζ and η directions be taken as m and n respectively, the integral of f (x, y) over the area of the element may be written as

9.5

Numerical Implementation

I=

f (x, y)dx dy =

1

693 1

−1 −1

F(ξ , η) J (ξ , η) dξ dη =

m 1 n 1

wi wj F(ξi , ηj ) J(ξi , ηj )

i=1 j=1

(9.54) where J (ξ , η) is the Jacobian of the transformation given by (9.23), while wi and wj are the weight factors corresponding to the integration points ξ i and ηj , respectively. In the case of axial symmetry, involving the volume integration of a function f (r, z) defined over a ring element shown n Fig. 9.4, the integration formula becomes I = 2π

f (r, z) r dr dz = 2π

m 1 n 1

wi wj F(ξi , ηj ) r(ξ , η) J(ξi , η)

(9.55)

i=l j=l

where J (ξ , η) is given by (9.23) with x and y replaced by r and z, respectively, while r = N1 r1 + N2 r2 + N3 r3 + N4 r4 . The numerical integration formula for the general three-dimensional situation can be written down as a straightforward extension of (9.55). In the Gaussian quadrature for two- or three-dimensional cases, the integration points and the weight factors in each coordinate direction for a given number of integration points are the same as those in the one-dimensional case. Setting φ(ξ , η) = F(ξ , η)J(ξ , η), in the double integral (9.55), and assuming m = n =2, we have four Gaussian integration points, each having a weight factor of unity, and the integration formula then becomes I = φ(− α, − α) + φ(− α, α) + φ(α, − α) + φ(α, α),

√ α = 1/ 3

(9.56a)

The assumption m = n = 3, on the other hand, gives us nine Gauss points, one of which is located at the center of the element with a weight factor of 64/81, four of which are located along the diagonals η = ±ξ with a weight factor of 25/81 for each one, the remaining four being along the axes ξ = 0 and η = 0 each with a weight factor of 40/81. The Gaussian integration formula then becomes I=

64 40 [φ(− α, 0) + φ(0, − α) + φ(α, 0) + φ(0. α)] φ(0, 0) + 81 81 25 [φ(− α, − α) + φ(− α, α) + φ(α, − α) + φ(α, α)] , + 81

α=

√ 0.6

(9.56b)

In problems of axial symmetry using quadrilateral ring elements, the Gaussian integration formula for the volume integral can be expressed exactly in the same forms as (9.56), provided we set φ(ξ ,η) = 2π F(ξ ,η) r(ξ ,η) J(ξ ,η),

694

9 The Finite Element Method

In the case of a linear triangular element, for which J = 2A, it is customary in the finite element analysis to consider a single integration point located at the centroid of the triangle, and the integral becomes I = 2A F0 , where F0 is the value of F at the centroid, which corresponds to L1 = L2 = L3 = 1/3. When the same element is used as an axisymmetric ring element, a similar approximation to the volume integral gives I = 4π Ar0 F0 , where r0 is the radius to the centroid of the triangle In a linear quadrilateral element, the condition of constancy of volume cannot be satisfied at all points except for a uniform mode of deformation. In the finite element formulation, this difficulty is usually overcome by using a single-point integration scheme for dealing with the volumetric strain rate term. If, on the other hand, four linear triangular elements are arranged to form a quadrilateral, then the plastic incompressibility condition can be satisfied over the entire quadrilateral (Nagtegaal et al., 1974). In the solution of metal-forming problems, the reduced integration scheme is frequently used, particularly for the evaluation of the stresses

9.5.2 Global Stiffness Equations The finite element analysis of a physical problem is based on dividing the body into a large number of finite elements which are joined together at their nodal points. It is customary to assign the global node numbers and the element numbers sequentially as shown in Fig. 9.6(a). The physical constraints require the velocity vector at any nodal point to be identical to that of the individual elements sharing the same nodal point. The force vector at a given nodal point, on the other hand, is the sum of the forces associated with the elements having this nodal point in common (Desai and Abel, 1972). 1

5

9

13

17 o

1 2 2 3

7

4 6 5 7

12 18

14

10 8 11

13

K= 19

15

o 3 4

6 8

14

9 12

(a)

16

20

(b)

Fig. 9.6 Finite element mesh and associated global stiffness matrix. (a) Element node numbering, (b) typical banded matrix

It is customary to have the elemental nodes numbered in the same sequence for each element in the assemblage. Adopting the elemental node numbering to be

9.5

Numerical Implementation

695

indicated in Fig. 9.2, the elemental stiffness equation (9.13) may be expressed in terms of 2 × 2 submatrices in the form ⎫ ⎧ e ⎫ ⎡ e e e e ⎤ ⎧ e ⎪ {q1 e } ⎪ ⎪ ⎪ ⎪ ⎪ {f1 } ⎪ k11 e k12 e k13 e k14 e ⎪ ⎨ ⎥ {q2 } ⎬ ⎨ {f2 e } ⎬ ⎢ k21 e e T k k k 22 23 24 ⎢ e e e e ⎥ = , kji = kij e e ⎦ ⎣ k31 {q3 } ⎪ {f3 } ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ e k32 e k33 e k34 e ⎪ ⎩ ⎭ ⎭ ⎩ {q4 e } {f4 e } k41 k42 k43 k44 (9.57) where the last relation follows from the symmetry of the stiffness matrix. The submatrices introduced in (9.57) are defined as e e k 2i−1, 2j−1 kij = ke 2i, 2j−1

ke 2i−1, 2j , ke 2i, 2j

7 e6 qj =

uj e , vj e

7 e6 fj =

fj e gj e

(9.58)

The global stiffness equation is formed by a suitable combination of the elemental stiffness equations, taking into account the connectivity of the elements, and may be written as KV = T

(9.59)

where K is the global stiffness matrix, T is the global load vector, and V the global velocity change vector, which differs from the elemental vector q only in the node numbering. Referring to Fig. 9.6(a), and considering, for example, element 2, whose connectivity with the neighboring elements is defined by the nodal points (2, 3, 7, 6), the submatrices of K associated with these nodes are found as [K22 ] = [k11 1 ] + [k44 2 ],

[K23 ] = [k23 2 ],

[K26 ] = [k12 1 ] + [k43 2 ]

[K 33 ] = [k11 2 ] + [k44 3 ],

[K36 ] = [k36 2 ],

[K37 ] = [k12 2 ] + [k43 3 ]

[K66 ] = [k22 1 ] + [k33 2 ] + [k11 4 ] + [k44 5 ],

[K67 ] = [k32 2 ] + [k41 5 ]

[K77 ] = [k22 2 ] + [k33 3 ] + [k11 5 ] + [k44 6 ],

[K27 ] = [k27 2 ]

The remaining submatrices of the global stiffness matrix can be similarly established by considering the connectivity of the other elements. When any two nodal points do not belong to the same element, the corresponding submatrix becomes a null matrix. Due to the limited influence of the element connectivity, the global stiffness matrix is a sparse matrix, which can be arranged in a banded form, as indicated in Fig. 9.6(b). With the help of an appropriate node numbering, the band width can be kept down to a minimum. The global stiffness equations are most conveniently solved by the Gaussian elimination technique using a linear equation solver. In a skyline solver, the matrix coefficients are stored column-wise, starting from the first diagonal element and ending with the last nonzero element. The computational time required to solve the matrix equation is found to be proportional to the square of the semi-bandwidth of the matrix. It is therefore necessary to number the nodes in such a way that the band width is a minimum.

696

9 The Finite Element Method

9.5.3 Boundary Conditions The solution of the global stiffness equations requires due consideration of the boundary conditions. In general, the boundary surface of the workpiece consists of a part on which the traction is prescribed, a part Sv on which the velocity is prescribed, and a part ST which is the tool–workpiece interface. The imposition of the traction boundary condition on SF in the form of nodal point forces is straightforward. Consider, for example, the three-dimensional brick element shown in Fig. 9.3, and suppose that the lower surface 1–2–3–4 is subjected to normal and tangential tractions specified by F. Since ζ = 1 over this surface, the nonzero shape functions are given by (9.21), and the shape function matrix reduces to

N1 N= 0

0 N1

N2 0

0 N2

N3 0

0 N3

N4 0

0 N4

The associated nodal point force vector is easily determined from the expression (NT F) dx dy = (NT F) J dξ dη (9.60) f0 = where J is the Jacobian of the transformation, given by (9.28), and the integral extends over the area of the entire surface SF For a nodal point on Sv over which the velocity is prescribed, the velocity correction is zero, and the corresponding stiffness equation needs to be omitted. In the finite element solution, the simplest way to impose the velocity boundary condition ΔVm = 0 at a nodal point m is to set the diagonal element of the mth row of the stiffness matrix to unity and replace the remaining elements in the corresponding row and column by zeros, as indicated below. ⎫ ⎧ ⎫ ⎤⎧ ⎡ T1 ⎪ K11 K12 . . 0 . . K1n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ K21 K22 . . 0 . . K2n ⎥ ⎪ V T2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎥⎨ ⎪ ⎢ ⎨ ⎬ ⎬ ⎥ ⎢ . . . . . . . . . . ⎥ ⎢ = ⎢ 0 0 ; . 1 . . 0 ⎥ Vm ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ . . . . . . . . . ⎦⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ⎪ ⎭ ⎩ Vn Kn1 Kn2 . . 0 , . Knn Tn On the remainder of the surface, ST , representing the interface between the die and the workpiece, the boundary condition is of the mixed type, as the velocity is prescribed along the normal to the interface of contact, while the tangential traction is prescribed in the direction of relative sliding between the die and the workpiece. The tangential stress is usually specified in terms of a constant coefficient of friction, or as a constant frictional stress mk, where 0 ≤ m ≤ 1. When the element surface does not conform with the die surface, an additional approximation is necessary to obtain the associated distribution of nodal forces. Once the velocity solution is obtained for the entire workpiece, the geometry of the workpiece must be updated by changing the coordinates of the various nodes.

9.6

Illustrative Examples

697

For a two-dimensional problem, the rectangular coordinates of a typical nodal point (xj ,yj ) are changed by the amounts xj = uj t, yj = vj t where t denotes the increment of time scale. The nodal point strains are similarly updated from the available values of the strain rate. In metal-forming analysis, the time increment may be taken as that for which the next free node of the workpiece comes in contact with the die surface. Since the deformation that occurs in metal-forming processes is generally large, the size and shape of the element soon become unacceptable as the deformation continues, and the imposition of the boundary conditions also becomes increasingly difficult. In order to overcome theses difficulties, it is necessary to modify the mesh system periodically, so that the mesh size remains sufficiently small, and also to transfer the information from the old mesh system to the new one through interpolation. The difficulty can be largely overcome by using a spatially fixed meshing scheme, as has been discussed by Derbalian et al. 1978) and by Mori et al. (1983). An area-weighted averaging method of evaluating a parameter at a node internal to an original linear quadrilateral element, which is found to be sufficiently accurate in metal-forming analysis, has been discussed by Kobayashi et al. (1989). The complete elastic/plastic formulation for large strain finite element analysis has been discussed by McMeeking and Rice (1975) and Nagtegaal and DeJong (1981). A useful discussion of the elastic/plastic formulations in relation to metal-forming problems has been made by Rebelo and Wertheimer (1986).

9.6 Illustrative Examples Numerous solutions to the metal-forming problems, based on the various types of finite element formulation presented in Section 9.1, have been given in detail by Kobayashi et al. (1989). A number of these solutions are based on the variational method, and a few of these will be briefly discussed in what follows in order to illustrate the application of the preceding theory.

9.6.1 Compression of a Cylindrical Block In the axial compression of a cylindrical block between a pair of flat dies, the plastic deformation is inhomogeneous due to the presence of friction at the interfaces, and the mean compressive stress exceeds the uniaxial yield stress of the material (Section 3.4). The deformation of the block is characterized by a barreling of the free surface, a part of which comes in contact with the die during the compression. When the ratio of the height of the block to its diameter is sufficiently small, the barreling consists of a single bulge in which the maximum diameter occurs at the central cross section of the

698

9 The Finite Element Method

block. For fairly large values of the height/diameter ratio, a double bulge is sometimes observed. The analysis may be based on a constant frictional stress equal to mk along the interface between the die and the workpiece. The finite element analysis carried out by Lee and Kobayashi (1971) for a cylindrical block with an initial height/diameter ratio of 2.5 reveals the formation of a double bulge, as depicted in Fig. 9.7. The double bulge gives way to a single bulge as the height/diameter ratio progressively decreases to sufficiently small values during the continued compression.

Fig. 9.7 Grid distortion patterns in the axial compression of a cylinder with an initial hight/diameter ratio of 2.5 (after Lee and Kobayashi, 1971)

The analysis may also be used to predict the limit of workability of ductile materials due to the formation of surface cracks during the compression. This has been investigated experimentally by Kudo and Aoi (1967), who measured the equatorial surface stains in upsetting solid cylindrical specimens under various frictional conditions until the surface cracks were observed. The computed strain paths of the critical element, obtained from the finite element analysis for a block of a unit initial height/diameter ratio under various frictional conditions, are plotted in Fig. 9.8, which also includes the experimental results referred to above. The limit set by the occurrence of surface cracks, based on the experimental data, may be approximated by the criterion 2ε θ + εz = 0.8 to a close approximation. Finite element solutions to the axial compression of hollow cylinders have been discussed by Chen and Kobayashi (1978) and also by Hartley et al. (1979). A finite. element analysis of the upsetting process based on the total strain theory of plasticity has been reported by Vertin and Majlessi (1993).

9.6.2 Bar Extrusion Through a Conical Die Consider the axisymmetric extrusion of a cylindrical billet through a conical die, along which the frictional stress has a constant value equal to mk, the container wall being assumed to be perfectly smooth. The material in the container approaches the die with a uniform unit speed, and it leaves the die with a uniform speed equal to

9.6

Illustrative Examples

699

1.0 Theory Experiment

lubricated

Fracture

=0 .2

.1

m

m=

Hoop Strain

grooved dies

0.6

0.4 5 m= 0.3

unlubricated

0.8

m

=0

m=

0

0.4

0.2

0

–0.4

–0.2

–0.6 Axial Strain

–0.8

–1.0

Fig. 9.8 Strain paths of an equatorial element during axial compression of a cylinder with an initial height/diameter ration of unity

b2 /a2 , where a and b denote the final and initial radii of he billet. The assumed finite element mesh for the extrusion problem is shown in Fig. 9.9, where the corners of the die have been slightly modified by straight lines joining he nodal points closest to the corners, in order to avoid singularities of velocity components near the edge of the die. The origin of the coordinate (r, z) is taken on the axis of symmetry at O with the r-axis coinciding with the exit plane of the die, as shown in the figure. Along the die face AB, the boundary conditions are τ = mk,

w = u cot α,

along

z = (r − a) cot α,

a≤r≤b

where α denotes the semiangle of the die. The extruded part of the billet, which moves as a rigid body, is entirely free of surface tractions. The remaining boundary conditions may be written as u = 0, w = −1,

Fz = 0,

along

Fr = 0 on

r=0 z = c;

and

r=b

w = −(b2 /a2 )

on

z=0

700

9 The Finite Element Method

Fig. 9.9 Geometry and finite element grid pattern for the axisymmetric extrusion of a cylindrical billet through a conical die

r

B α τ

C

A

1 b

b 2/a 2 a

z O

The problem can be treated as one of steady state in which the geometrical configuration does not change with time. A complete elastic/plastic analysis for extrusion through a sigmoidal die until the attainment of the steady state has been discussed by Lee et al. (977). The results presented here have been obtained by the rigid/plastic method by Chen and Kobayashi (1978) and Chen et al. (1979). In the finite element analysis for the extrusion of a work-hardening material, the components of the strain rate at the center of each element are initially assumed to be same as those in a nonhardening material. Starting from a selected point on the plane of entry, where the effective strain is zero, the rate of change of the effective strain is determined from the known values at the surrounding element centers. Since the velocity of the selected point is found from the element interpolation formula, the effective strain and the new position of the particle are then easily obtained from the increment of time. This procedure is sequentially repeated, following the path of the particle, until the exit plane is reached. The flow lines emanating from different points on the entry plane determined this way furnish the shape of the distorted grid, and also the distribution of the effective strain throughout the deforming region. Since the distribution of the effective stress follows from the given stress– strain curve of the material, the new distribution of nodal point velocities can be computed in order to carry out the next iteration. When the velocity solution converges after a few iterations, the mean extrusion pressure and the distribution of the die pressure can be determined for the given frictional condition, die angle, and fractional reduction in area. ◦ The steady-state grid distortion pattern for frictionless extrusion through a 90 conical die, obtained by the finite element solution, is displayed in Fig. 9.10, where the upper half holds for a nonhardening material and the lower half for a workhardening material (SAE 1112 steel). The difference between the two patterns is due to the restriction of metal flow that occurs in a work-hardening material. The ◦ distribution of radial and axial velocities within the die for α = 45 and b/a = 2,

9.6

Illustrative Examples

701

Fig. 9.10 The distortion of an initial square grid in a cylindrical billet extruded through a 90◦ conical die (after Chen et al., 1978)

furnished by the finite element solution, is shown in Fig. 9.11. The material adopted in this solution is the same as that used in an experimental investigation by Shabaik and Thomsen (1968), who obtained remarkably similar results for the velocity distribution. The computation also reveals that the hydrostatic part of the stress in a region near the center of the deforming region becomes tensile at sufficiently large reductions, leading to the possibility of a central crack which is frequently observed.

9.6.3 Analysis of Spread in Sheet Rolling In the rolling of sheets and slabs, in which the width of the workpiece is less than about five times the length of the arc of contact, the usual assumption of plane strain is not justifiable. The amount of lateral spread that occurs in such cases is quite appreciable, and must be taken into consideration in the analysis of the rolling process. A finite element analysis of the problem using the three-dimensional brick element has been carried out by Li and Kobayashi (1982), who adopted the nonsteadystate approach for the solution. Figure 9.12 shows a narrow strip of the workpiece with the arrangement of an element in the upper half of the material within the arc of contact. The deformation of the material entering the roll gap with a bite is considered in a step-by-step manner based on a constant frictional stress, while updating the material properties and the coordinates of the nodal points at the end of each step. A steady state is assumed to be reached when the associated roll torque has attained a steady value, and the spread contour has become stationary. The computation has been carried out with a friction stress equal to 0.5 k, and using R/h0 = 160, two different values of w0 /h0 , and several values of the reduction in thickness, where R denotes the roll radius, 2 h0 is the initial slab thickness, and 2w0 is the initial slab width, the material used in the analysis being annealed

702

9 The Finite Element Method 0

0.5

z / b = 1.0

0

z / b = 1.0 –1.0

0.8 0.8 0.6

0.6

–0.5

0.4

w

u

0 –2.0

–1.0

0.3

0.4

0.2

–3.0

–1.5

0.1

0.3 0.2

0.1

0

–2.0

–4.0 –4.5 0

0.5

1.0 r/ b

–2.5

0

1.0

0.5 r/ b

Fig. 9.11 Distribution of radial and axial velocities of particles moving through a 90◦ conical die with b / a = 2 (after Chen and Kobayashi, 1982)

AISI 1018 steel. The final values of the thickness and width of the rolled stock are denote by 2hf and 2wf , respectively. The computed value of the mean lateral spread is plotted against the final reduction in height in Fig. 9.13a, which shows excellent agreement with some experimental results reported by Kobayashi et al. (1989). During the rolling process, not only the thickness but also the cross-sectional area of the rolled stock progressively decreases due to the effect of the lateral spread. The solid curves in Fig. 9.13(b) show the variation of the reduction in cross section of the rolled stock with the reduction in height for w0 /h0 =1 and 3, while the broken straight line indicates the plane strain situation in which the reduction in cross-sectional area is equal to the reduction in height. The spread in rolling has also been investigated approximately by Lahoti and Kobayashi (1974), and, by a finite element analysis, by Kanazawa and Marcal (1982). The finite element solution for the compression of a rectangular block has been discussed by Park and Kobayashi (1984), and that of a ring of square cross section has been considered by Park and Oh (1987). The shape rolling of bars of various cross sections, using the finite element method, has been investigated by Park and Oh (1990).

9.6

Illustrative Examples

703 y

x

2w f

ne it pla

ex ne

y pla

entr

O 2hf

2w o

z 2ho

Fig. 9.12 A schematic view of sheet rolling with lateral spread indicating the location of a typical finite brick element

25

10

[(w f –w 0) / w 0] × 102

8

w 0 / h 0 =1.0

6 4 3.0 2 0

0

20 10 [(h 0 –h f) / h 0] × 102

(a)

Reduction in area, percent

Theory Experiment

3.0

20

w0 / h0 15 1.0 10 5 0

0

10 20 Reduction in height, percent

(b)

Fig. 9.13 Results for sheet rolling with lateral spread. (a) Variation of overall spread with reduction in height, (b) variation of change in cross section with change in height

704

9 The Finite Element Method

9.6.4 Deep Drawing of Square Cups As a final example, consider the deep drawing of a square cup using a flat punch, the base of the punch being a square of sides 2a The schematic view of the process is similar to that shown in Fig. 2.28(a). The cup is drawn from a square blank whose sides have an initial length equal to 2b0 , the initial blank thickness being denoted by h0 . In the finite element formulation, the continuous blank holding force is replaced by a set of concentrated forces acting at the nodal points along the periphery of the blank. The frictional condition at the interfaces between the tools and the sheet metal is assumed to be governed by Coulomb’s law with a constant coefficient of friction. Denoting the die and punch profile radii by rd and rp respectively, and the punch corner radius by rc , the geometry of the process is defined as b0 = 2.75, a

h0 = 0.043, a

rp rd = = 0.25, a a

rc = 0.16. a

The material is aluminum killed steel having a uniform R-value equal to 1.6, the planar stress–strain curve being given by the power law σ = Cε n , where n = 0.228 and C = 739 MPa. The blank-holding force is taken as 4.9 kN, the friction coefficient being 0.2 over the punch and 0.04 over the die.

y

x

0 Under Punch

Under Blankholder

Fig. 9.14 Finite element mesh in a square blank to be drawn into a cup with square base

9.6

Illustrative Examples

705

The finite element mesh used in the analysis of the square cup drawing is shown in Fig. 9.14, which indicates a choice of finer mesh over the region where the thickness is expected to vary rapidly. The thinning of the sheet is found to have maximum values over the punch and die profile radii, particularly along the diagonal of the square. In Fig. 9.15, the computed distribution of the thickness strain across the diagonal of the formed cup, based on a = 20 mm, is compared with that obtained experimentally by Thomson (1975) for a given punch load. The predicted strain distribution has the same trend as that of the experimental one, though there is an appreciable difference in the magnitude of the strain. The discrepancy is due to the fact that the observed punch penetration is 1.50 a, which is significantly higher than the value 1.01 a predicted by the finite element solution and may be attributed to the difference in the frictional conditions existing in the experiment. Similar results based on a simplified finite element analysis have been reported by Majlessi and Lee (1993). Rigid/plastic finite element solutions for the hydrostatic bulging of circular diaphragms have been given by Lee and Kobayashi 1975) and Kim and Yang (1985a), and of elliptical diaphragms by Chung et al. (1988). The bulging of rectangular diaphragms has been investigated experimentally by Duncan and Johnson (1968) and numerically by Yang and Kim (1987). The large strain elastic/plastic

–80

y

0

x

Theory (d = a) Experiment (d = 1.5 a)

Thickness strain, percent

–60

–40

–20

Under punch

Under blankholder

0

Fig. 9.15 Comparison of theoretical and experimental distributions of thickness strain along the diagonal of a square blank (after Toh and Kobayashi, 1985)

+20

0

40 60 20 Initial distance from blank center, mm

706

9 The Finite Element Method

finite element formulation has been applied to the axisymmetric punch stretching problem by Kim et al. (1978) and to the plane strain bending of sheets by Oh and Kobayashi (1980). A rigid/plastic finite element solution to the punch stretching problem, based on the nonquadratic yield criterion, has been presented by Wang (1984). An elastic/plastic finite element analysis for the deep drawing of anisotropic cups has bee reported by Saran et al. (1990), and the associated problem of flange wrinkling has been investigated by Kim et al. (2000) and Correia et al. (2003).

Appendix: Orthogonal Curvilinear Coordinates

Cylindrical Coordinates The position of a typical particle is defined by the coordinates (r, θ , z) taken in the radial, circumferential, and axial directions, respectively. If the associated components of the velocity are denoted by (u, v, w), respectively, then the components of the true strain rate are ε˙ r = ε˙ θ =

∂u ∂r

1 ∂v u+ , r ∂θ ε˙ z =

∂w , ∂z

∂v v 1 ∂u − + , ∂r r r ∂θ 1 ∂v 1 ∂w + γ˙ θz = , 2 ∂z r ∂θ 1 ∂u ∂w γ˙ rz = + . 2 ∂z ∂r

γ˙ rθ =

1 2

If σr , σθ , and σz denote the normal stresses and τrθ , τθz , and τrz the shear stresses, then the equations of equilibrium in the absence of body forces are 1 ∂τrθ ∂τrz σr − σθ ∂σr + + + = 0, ∂r r ∂θ ∂z r 1 ∂σθ ∂τθz 2τrθ ∂τrθ + + + = 0, ∂r r ∂θ ∂z r ∂τrz 1 ∂τθz ∂σz τrz + + + = 0. ∂z r ∂θ ∂z r

Spherical Coordinates The coordinate system is defined by (r, φ, θ ), where r is the length of the radius vector, φ is the angle made by the radius vector with a fixed axis, and θ is the angle measured round this axis. If the velocity components in the coordinate directions are denoted by (u, v, w), then the components of the true strain rate are J. Chakrabarty, Applied Plasticity, Second Edition, Mechanical Engineering Series, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77674-3,

707

708

Appendix: Orthogonal Curvilinear Coordinates

∂u 1 ∂v v 1 ∂u ε˙ r = , γ˙rφ = − + , ∂r 2 ∂r r r ∂φ 1 1 1 ∂w w 1 ∂v ∂v ε˙ φ = − cot φ + u+ , γ˙rφ = , r ∂φ 2 r ∂φ r r sin φ ∂θ 1 1 ∂w w 1 ∂u ∂w ε˙ θ = − + u + v cot φ+cosec φ , γ˙rθ = . r ∂θ 2 ∂r r r sin φ ∂θ Denoting the normal stresses by σr , σφ , and σθ and the shear stresses by τrφ , τφθ , and τrθ , the equations of equilibrium in the absence of body forces can be written as ∂σr 1 ∂τrφ 1 ∂τrθ 1 + + 2σr − σφ − σθ + τrφ cot φ = 0, ∂r r ∂φ r sin φ ∂θ r 6 ∂τrφ 1 ∂σφ 1 ∂τφθ 1 7 + + σφ − σθ cot φ + 3τrφ = 0, ∂r r ∂φ r sin φ ∂θ r 1 ∂τφθ 1 ∂σθ 1 ∂τrθ + + + 3τrθ + 2τφθ cot φ = 0. ∂r r ∂φ r sin φ ∂θ r When the deformation is infinitesimal, the preceding expressions for the components of the strain rate may be regarded as those for the strain itself, provided the components of the velocity are interpreted as those of the displacement.

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Name Index

A Abrahamson, G.R., 593 Abramowicz, W., 533, 656 Ades, C.S., 541 Adie, J.F., 198 Ahn, C.S., 656 Aizawa, T., 665 Akserland, E.L., 546 Alexander, J.M., 94, 97, 103, 112, 184, 186, 198, 445, 530 Alter, B.E.K., 578 Amos, D.E., 604 Aoki, I., 447 Apsden, R.J., 623 Ariaratnam, S.T., 528 Asaro, R.J., 1 Ashwell, D.G., 556 Astuta, T., 487, 499 Atkins, A.G., 203, 205 Avitzur, B., 169, 177, 198 Azrin, M., 448 B Backman, M.E., 605 Backofen, W.A., 10, 103, 448 Bailey, J.A., 430 Baker, J.F., 487 Baker, W.E., 602 Balendra, R., 665 Baltov, A., 28 Banerjee, A.K., 578 Barlat, F., 421 Batdorf, S.R., 518 Batterman, S., 524, 528, 552, 556 Bell, J.F., 576, 578 Belytschko, T., 268, 270 Bhadra, P., 645 Bianchi, G., 578 Bijlaard, P.P., 518, 528

Biron, A., 323, 379 Bishop, J.F.W., 2, 15, 69, 167, 593 Bland, D.R., 24 Blazynski, T.Z.V., 181 Bleich, H.H., 633 Bodner, S.R., 571, 621, 628 Bourne, L., 465 Boyce, W.E., 233 Boyd, D.E., 663 Bramley, A.N., 412, 416 Brazier, L.G., 541 Bridgman, P.W., 11, 162, 163 Brooks, G.M., 323, 327, 358 Budiansky, B., 2, 26, 103, 455, 689 Bushnell, D., 554 C Caddell, R.M., 203, 205, 419 Calder, C.A., 617 Calladine, C.R., 198, 255 Campbell, J.D., 571, 572, 623 Casey, J., 26 Chaboche, J.L., 38 Chadwick, P., 599 Chakrabarty, J., 6, 12, 39, 43, 44, 45, 72, 76, 95, 97, 103, 109, 123, 126, 166, 185, 189, 192, 202, 206, 257, 275, 276, 278, 418, 420, 428, 439, 444, 446, 461, 466, 489, 526, 532, 533, 593, 609, 671 Chan, K.C., 61, 416, 451 Chan, K.S., 451 Chater, E., 451 Chawalla, E., 487 Chen, C.C., 198, 205, 698, 700, 701 Chen, F.K., 461 Chen, F.L., 645 Chen, W.F., 18, 487, 496, 499 Cheng, S.Y., 166 Chenot, J.L., 188, 200

743

744

Name Index

Chern, J., 94 Chiang, D.C., 122, 463 Chitkara, N.R., 443 Chon, C.T., 645 Chou, P.C., 617 Chu, C.C., 131, 451 Chung, S.Y., 112, 184 Chung, W.J., 705 Cinquini, C., 237, 302 Clark, D.S., 571 Clifton, R.J., 571 Cockroft, M.G., 177 Cole, I.M., 181 Collins, I.F., 169, 181, 203, 270, 275, 443, 444 Conroy, M.F., 629, 642 Coon, M.D., 323 Corona, A., 541 Cotter, B.A., 633 Cox, A.D., 645 Craggs, J.W., 571, 580 Cristescu, N., 568, 571, 573, 574, 576, 577, 578, 586, 663 Crossland, B., 11 Crozier, R.J.M., 602 Curtis, C.W., 578

Dugdale, D.S., 157 Duncan, J.L., 451, 705 Durban, D., 148, 198, 445, 514, 522 Duszek, M., 237, 243 Duwez, P.E., 561, 571, 578, 632

D Dafalias, Y.F., 29, 33 Danyluk, H.T., 147, 157 Davidenkov, N.N., 162 Davies, C.J., 11 Davies, R.M., 571 De Juhasz, K.J., 571 Dean, T.H., 658 Dekel, E., 609 Demir, H.H., 327 Denton, A.A., 184 DePierre, V., 177 DeRuntz, J.A., 379 Desdo, D., 151 Dillamore, I.L., 414 Dillons, O.W., 578 Ding, J.L., 422 Dinno, K.S., 385 Dodd, B., 419 Donnell, L.H., 541 Dorn, J.F., 407 Douch, L.S., 572 Dowling, A.R., 665 Drucker, D.C., 16, 19, 23, 26, 38, 209, 211, 294, 315, 381 Dubey, R.N., 528 Duffill, A.W., 185

G Galiev, S.U., 655 Galletly, G.D., 554 Gaskill, B., 617 Gaydon, P.A., 85 Gellin, S., 530, 546 Gerard, G., 518, 541, 554 Gere, J.M., 496, 505, 515, 517, 526, 530, 538 Ghosh, A.K., 451 Gill, S.S., 323, 385 Gjelsvik, A., 518 Goel, R.P., 571 Goldsmith, W., 571, 605, 617 Goodier, J.N., 593, 645 Gotoh, M., 424, 465, 687 Graf, A., 451 Green, A.P., 17, 62, 73, 77, 80 Griffis, Le Van, 571 Gu, W., 656 Gui, X.G., 633 Gunasekera, J.S., 220 Gurson, A.L., 2

E Eason, G., 155, 248, 297, 324, 326, 655 Efron, L., 578 Eisenberg, M.A., 38 El-Ghazaly, H.A., 518 El-Sebaie, M.S., 122, 463 Engesser, F., 486 Ewing, D.J.F., 72, 73 Ezra, A.A., 621 F Finnegan, S.A., 617 Fleming, W.T., 617, 621, 633 Fliigge, W., 343, 348, 370 Florence, A.L., 642, 644, 645, 656 Fogg, B., 128 Ford, H., 15, 18, 62, 94, 95 Forrestal, M.J., 600, 603, 605, 608, 609 Freiberger, W., 297, 393 Fukui, S., 128

H Haar, A., 148 Haddow, J.B., 144, 147, 157, 169, 211, 243, 656

Name Index Hailing, J., 198 Hamada, H., 522 Han, C.H., 198 Han, D.J., 18 Hart, E.W., 10 Hashmi, M.S.J., 623 Hassani, H.A., 131, 445 Hawkyard, J.B., 172, 176, 220, 590, 592, 593 Haydi, H.A., 367 Haythornthwaite, R.M., 233, 244, 255 Hecker, S.S., 447, 448 Hector, L.G., 111 Hencky, H., 26 Hetnarski, R.B., 29, 265 Hill, R., 1, 2, 15, 17, 19, 20, 21, 25, 42, 43, 45, 52, 59, 62, 69, 91, 97, 102, 108, 126, 140, 144, 151, 160, 161, 169, 170, 177, 205, 212, 214, 406, 407, 409, 418, 421, 425, 435, 438, 465, 480, 484, 581, 610, 613 Hillier, M.J., 65 Hodge, P.G., 21, 28, 85, 89, 97, 237, 243, 260, 264, 265, 268, 269, 270, 284, 320, 321, 323, 326, 328, 334, 339, 340, 341, 350, 351, 355, 356, 358, 361, 372, 376, 379, 647, 655 Hoffman, G.A., 402 Hong, H.K., 33 Hopkins, H.G., 229, 232, 240, 260, 297, 574, 586, 594, 635, 636, 644 Horne, M.R., 494 Horner, M.R., 323 Hosford, W.F., 419, 451 Hu, L.W., 263, 472 Hu, T.C., 391 Huang, S., 2 Hudson, G.E., 659, 661 Huh, H., 691 Hundy, B.B., 17, 62 Hunter, S.C., 571, 586, 602 Hutchinson, J.W., 2, 166, 412, 451, 484, 487, 554 I Ilahi, M.F., 97, 445 Ilyushin, A.A., 26 Inoue, T., 518 Ipson, T.W., 617 Ishlinsky, A., 26, 157 Issler, W., 402 Ivanov, G.V., 349 J Jackson, L.R., 407, 409 Jaeger, T., 238, 265, 302

745 Jansen, D.M., 580, 586 Jaumann, J.J., 25 Jiang, W., 28, 34 Johansen, K.W., 265, 281, 293 Johnson, R.W., 203, 220, 705 Johnson, W., 76, 112, 131, 169, 172, 176, 184, 193, 203, 211, 220, 265, 270, 279, 288, 440, 533, 589, 598, 609, 613, 633, 644, 664, 665, 705 Jones, L.L., 281, 629 Jones, N., 593, 621, 629, 633, 645, 655, 656 Juneja, B.L., 220 K Kachanov, A., 26 Kachi, Y., 33 Kaftanoglu, B., 103, 445 Kalisky, S., 633 Kamalvand, H., 499 Kamiya, N., 258 Karunes, B., 633 Kasuga, Y., 122 Kato, B., 518 Kawashima, I., 258 Keck, P., 456 Keeler, S.P., 103, 447 Keil, A.H., 663 Ketter, R.L., 496 Khan, A.S., 2 Kim, J.H., 103, 220, 445, 705, 706 Kim, M.U., 220 Klepaczko, J., 578 Klinger, L.G., 412 Kliushnikov, V.D., 26 Kobayashi, S., 103, 112, 122, 172, 177, 181, 188, 196, 208, 214, 217, 220, 419, 463, 671, 672, 687, 689, 705, 706 Koide, M., 445 Koiter, W.T., 24 Kojic, H., 410 Kolsky, H., 572, 578 Kondo, K., 255 K¨onig, J.A., 302 Koopman, D.C.A., 270 Kozlowski, W., 294 Krajcinovic, D., 237, 645 Kudo, H., 169, 177, 196, 698 Kuech, R.W., 379 Kukudjanov, V.N., 573 Kumar, A., 645 Kummerling, R., 214 Kuzin, P.A., 655

746 Kwasczynska, K., 169 Kyriakides, A., 536, 541 L Lahoti, G.D., 177, 181, 217, 702 Lakshmikantham, C., 361, 372 Lamba, H.S., 33 Lambert, E.R., 188 Lance, R.H., 233, 270, 347, 375 Landgraf, R.W., 30 Lange, K., 168 Lebedev, N.E., 571 Leckie, F.A., 385, 556, 557 Lee, C.C., 33 Lee, C.H., 26, 172, 220, 698, 706 Lee, C.W., 37 Lee, D., 451, 691, 705 Lee, E.H., 26, 28, 220, 362, 366, 367, 375, 379, 565, 568, 589, 629 Lee, L.C., 366 Lee, L.H.N., 131, 528, 593, 700 Lee, L.S.S., 629 Lee, S.H., 687, 705 Lee, S.L., 379 Lee, W.B., 2, 61, 416, 451 Lengyel, B., 186 Lensky, V.S., 568 Leth, C.F., 629 Leung, C.P., 358 L´evy, M., 16 Li, S., 530, 609, 655, 701 Lian, J., 421, 451 Lianis, G., 15, 18, 62 Lin, G.S., 518 Lin, H.C., 571, 578 Lin, S.B., 422 Lin, T.H., 2 Lindholm, U.S., 573, 578 Ling, F.F., 198 Liou, J.H., 33 Lippmann, H., 151, 214, 658 Liu, D., 642 Liu, J.H., 388 Liu, T., 642 Liu, Y.H., 384 Lockett, F.J., 156, 157 Lode, W., 13 Logan, R.W., 419 Lu, L.W., 499 Lubiner, J., 26, 573 Ludwik, P., 7 Luk, V.K., 600, 603, 604 Lund, O., 573

Name Index M Macdonald, A., 168 Maclellan, G.D.S., 203 Male, A.T., 177 Malvern, L.E., 571, 572, 575, 576, 578 Malyshev, V.M., 578 Mamalis, A.G., 220, 533 Mandel, J., 26, 574, 580 Manjoine, H.J., 623 Manolakos, D.E., 265 Mansfield, E.H., 270 Marcal, A.V., 297, 702 Marciniak, Z., 448, 451 Markin, A.A., 444 Markowitz, J., 472 Marshall, E.R., 163 Martin, J.B., 629, 633 Massonnet, C.E., 270, 281, 286, 297, 402, 475 Mazumdar, J., 248 McCrum, A.W., 85 McDowell, D.L., 33 Megarefs, C.J., 393 Meguid, S.A., 444 Mellor, P.B., 97, 112, 122, 123, 184, 185, 265, 412, 416, 419, 445, 448, 450, 451, 463 Mentel, T.J., 623, 633 Meyer, C.E., 158 Miles, J.P., 43, 166 Mitchell, L.A., 198 Miyauchi, K., 453 Montague, P., 323 Moore, G.G., 181, 184, 414 Mori, K., 697 Morland, L.W., 586, 645 Morrison, A.L., 205 Mr¨oz, Z., 28, 29, 33, 169 Munday, G., 663 Murakami, S., 253, 258 Myszkowsky, S., 258 N Nadai, A., 31 Naghdi, P.M., 24, 26, 28, 97, 258 Nagpal, V., 220 Nakamura, T., 348, 370 Nardo, S.V.N., 323 Naruse, K., 422 Naziri, H., 460 Neal, B.G., 80 Neale, K.W., 131, 445, 451 Nechitailo, N.V., 655 Needleman, A., 166, 414 Nemat-Nasser, S., 94, 97, 573, 574, 589

Name Index Nemirovsky, U.V., 237 Newitt, D.M., 663 Newmark, N.M., 97 Nicholas, T., 578 Nieh, T.G., 11 Nimi, Y., 198 Nine, H.D., 126 Nonaka, T., 633 Norbury, A.L., 159 Nordgren, R., 97 Novozhilov, V.V., 28 Nurick, G.N., 645 O Oblak, M., 258 Oh, S.I., 214, 676, 702, 706 Ohashi, Y., 253, 258, 323 Ohno, N., 33 Okawa, D.M., 656 Okouchi, T., 323 Onat, E.T., 233, 255, 297, 299, 331, 344, 347, 366, 367, 374, 393, 633 Osakada, K., 198 Osgood, W., 9 Owens, R.H., 633 P Padmanabhan, K.A., 11 Palgen, L., 38 Palusamy, S., 358 Park, J.J., 220, 702 Parkes, E.W., 617, 621 Parmar, A., 419, 445, 450 Paul, B., 340, 341, 617 Payne, D.J., 385 Pearce, R., 418, 460 Pearson, C.E., 502, 513 Pell, W.H., 238 Penny, R.K., 556, 557 Perrone, N., 89, 629, 633, 645 Perzyna, P., 574, 642 Phillips, A., 28, 37 Pian, T.H.H., 255 Popov, E.P., 33, 248 Prager, W., 25, 26, 28, 35, 97, 228, 229, 233, 238, 297, 302, 303, 344, 391, 393, 635, 636 Prandtl, L., 4, 16 Presnyakov, A.A., 11 Pugh, H., 186 Pugsley, A., 530 R Raghavan, K.S., 448 Rakhmatulin, H.A., 561, 571

747 Ramberg, W., 9 Ranshi, A.S., 76, 81 Raphanel, J.L., 633 Rawlings, B., 633 Recht, R.F., 617 Reddy, B.D., 198, 530 Reddy, V.V.K., 645 Rees, D.W.A., 29, 541 Reid, S.R., 530, 633 Reiss, R., 393, 402 Reuss, E., 16 Rice, J.R., 1, 66, 413, 434, 697 Richards, C.E., 73 Richmond, O., 205, 402 Rinehart, J.S., 664 Ripperger, E.A., 571 Robinson, M., 349, 358 Rogers, T.G., 97 Rosenberg, Z., 609 Ross, E.W., 97 Rowe, G.W., 203 Rychlewsky, R., 302 S Sachs, G., 412 Sagar, R., 220 Salvadori, N.G., 633 Samanta, S.K., 169, 208 Samuel, T., 159 Sanchez, L.R., 126 Sankaranarayanan, S., 97, 243, 339 Save, M.A., 281, 297, 402 Sawczuk, A., 28, 237, 238, 243, 265, 270, 284, 286, 289, 292, 302, 323, 328, 466 Sayir, M., 339 Schumann, W., 260 Seide, P., 546 Seiler, J.A., 633 Senior, B.W., 129 Sewell, M.J., 480, 484, 486, 513, 516 Shammamy, M.R., 97, 102, 445 Shanley, F.S., 482 Shapiro, G.S., 349, 571, 629, 644, 655 Shaw, M.C., 163 Sherbourne, A.N., 258, 367, 518 Sheu, C.Y., 302 Shield, R.T., 26, 145, 148, 152, 155, 209, 211, 233, 294, 297, 305, 324, 326, 381, 390, 391, 393, 655 Shrivastava, H.P., 514, 530 Shull, H.F., 263 Sidebottom, O.M., 33 Siebel, E., 167, 199, 659

748 Sinclair, G.B., 161 Skrzypek, J.J., 29, 265 Sobotka, Z., 238, 281 Sokolovsky, V.V., 59, 572 Sortais, H.C., 208 Southwell, R.V., 533 Sowerby, R., 451 Spencer, A.J.C., 193 Spurr, C.E., 72 Srivastava, A., 258 Stein, A., 578 Steinberg, D., 573 Sternglass, E.J., 578 Storakers, B., 102 Storen, S., 66 Stout, M.G., 415, 447 Stronge, W.J., 258, 629, 633 Stuart, D.A., 578 Suliciu, I., 578 Swift, H., 8, 65, 112, 177, 184, 430 Symonds, P.S., 617, 621, 629, 633, 645

T Tabor, D., 158 Tadros, A.K., 448 Tan, Z., 444 Tanaka, M., 258 Tate, A., 609 Taylor, G.I., 1, 92, 561, 586, 589, 590, 593 Tekinalp, B., 297 Thomas, H.K., 581 Thomsen, E.G., 196, 701 Thomson, T.R., 705 Thomson, W.T., 616 Timoshenko, S., 496, 505, 515, 517, 526, 530, 538 Ting, T.C.T., 571, 582, 583, 628 Tirosh, J., 122, 161, 198 Toh, C.H., 689, 705 Toth, L.C., 430 Travis, F.W., 664, 665 Tresca, H., 14 Triantafyllidis, N., 126, 414 Tseng, N.T., 33 Tsuta, T., 2 Tsutsumi, S., 112, 122 Tugcu, P., 518 Tupper, S.J., 589 Turvey, G.J., 258

Name Index U Unksov, L.P., 168 V Valanis, K.C., 38 Van Rooyen, G.T., 172 Vaughan, H., 656 Venter, R., 440 Vial, C., 419 Voce, E.B., 8 von Karman, Th., 148, 486, 487, 561, 578 von Mises, R., 1, 13, 16 W Wagoner, R.H., 415 Wallace, J.F., 122, 184, 414 Wang, A.J., 240, 644 Wang, N.M., 97, 445, 689, 706 Wang, X., 131 Wasti, S.T., 532 Weil, N.A., 97 Weingarten, V.l., 546 Weiss, H.J., 89 Weng, G.J., 28 Wertheimer, T.B., 697 Whiffen, A.C., 588, 593 White, M.P., 571 Whiteley, R.L., 460 Wierzbicki, T., 533, 633, 645 Wifi, S.A., 103, 112 Williams, B.K., 181, 203 Wilson, D.V., 451, 460 Wilson, W.R.D., 111 Wistreich, J.G., 203 Woo, D.M., 97, 103, 112 Wood, R.H., 281 Woodthorpe, J., 418 Wu, H.C., 2, 32, 39, 412, 422, 431, 578 X Xu, B.Y., 379 Y Yakovlev, S., 444 Yamada, Y., 445, 447 Yamaguchi, K., 451 Yang, D.Y., 97, 198, 220, 691, 705 Yeom, D.J., 358 Yin, Y., 2 Yoshida, K., 38 Yossifon, S., 122 Youngdahl, C.K., 645, 655 Yu, T.X., 131, 258, 541, 629, 633, 645

Name Index Z Zaid, M., 617 Zanon, P., 237 Zaoui, A., 2 Zaverl, R., 451

749 Zhang, L.C., 258, 541 Zhao, I., 451 Zhu, L., 645 Ziegler, H., 26, 28, 402 Zukas, J.A., 617

Subject Index

A Admissible fields, 42, 213, 481 Angular velocity hodograph, 288, 310 Anisotropic hardening, 33 Anisotropic material flow rule, 407, 413 work-hardening, 409, 413 yield criterion, 405–407 Anisotropic parameters, 408–410 Anisotropy effects on plane plastic flow, 405–407 on plastic torsion, 424 on plates and shells, 466 on sheet metal forming, 444, 445 on slipline fields, 438–440 on stress-strain curves, 414–416 Annular plates, 372 Associated flow rule, 15 Axisymmetric problems compression of blocks, 169, 175 conical flow field, 147 extrusion of billets, 184 indentation, 155 necking in tension, 161–163 tube sinking, 177 wire drawing, 198–199 yield point in tubes, 144 B Back pull factor, 204 Bar drawing, 214–218 Bauschinger effect, 6, 26 Beam columns, 500 Bending of plates and shells, 227, 394 of prismatic beams, 73–76 Bending moments, 76, 228, 303 Bifurcation, 43, 480–484 Blanking, 664

Blast loading, 633 Bounding surface, 33–36 Brinell hardness, 158 Buckling of circular plates, 519–522 of cylindrical shells, 522–524, 656 of eccentric columns, 489 of narrow beams, 500 of rectangular plates, 511–516 of spherical shells, 546–550 of straight columns, 482 C Cantilevers, 73–76, 506, 625–629 Cavitation, 610–614 Cavity expansion compressible material, 604 incompressible material, 602 Characteristics axial symmetry, 164 plane plastic strain, 413 plane stress, 49–52 propagation of waves, 564, 569 torsion of bars, 502 Circular plates limit analysis, 240, 466 optimum design, 294 Collapse load, 19 circular plates, 227–229, 472 conical shells, 379 cylindrical shells, 321 hollow square plates, 87 noncircular plates, 258–261 spherical shells, 355–358, 471 Column buckling, 479–480, 487 Combined hardening, 28–30 Compression of hollow cylinders, 175, 698 of rectangular blocks, 212

751

752 Compression (cont.) of solid cylinders, 166–170 Compression test, 6 Conical shells, 367–369 Constitutive equations, 39 Constraint factor, 68–71 Contraction ratio, 4–5 Converging flow, 145 Crumpling of projectiles, 586–593 Curvature rates, 228, 260, 295 Curvilinear coordinates, 707 Cyclic plasticity, 35 Cylindrical coordinates, 707 Cylindrical shells limit analysis, 323 plastic buckling, 522–524 Cylindrical tubes combined loading, 167 flexural buckling, 537–541 torsional buckling, 537–541 D Deep drawing of anisotropic blanks, 460 of isotropic blanks, 452 Deflection of beams, 618, 621 of circular plates, 243–253, 634 of cylindrical shells, 379, 647 Deformation theory, 253, 476 Deviatoric plane, 12 Deviatoric stress, 11 Diffuse necking, 65 Direct forces, 332 Discontinuity in stress, 63–64 in velocity, 59–61 Dissipation function, 260, 261, 267 Double modulus, 571 Drawbeads, 125 Drawing processes bar drawing, 214 cup drawing, 704 tube drawing, 179 wire drawing, 198–208 Drawing stress, 179 Drucker’s postulates, 26 Dynamic analysis of cavity formation, 593, 600 of circular diaphragms, 659 of projectile penetration, 604 of structural members, 487, 633 Dynamic expansion of cavities, 593–604

Subject Index Dynamic forming of metals, 656–665 Dynamic loading of cantilevers, 617, 622 of circular plates, 633 of cylindrical shells, 645 of free-ended beams, 629 E Earing of deep-drawn cups, 464 Effective strain, 129 Effective stress, 21 Elastic/plastic analysis of circular plates, 243 of cylindrical shells, 554 Elastic/plastic material, 9 Elemental stiffness equation, 675 Elliptic plates, 308 Empirical stress-strain equations, 7 Endochronic theory, 38 Engineering strain, 3 Equation of motion, 563, 602, 613 Equilibrium equations axial symmetry, 139 circular plates, 229 cylindrical shells, 315 noncircular plates, 260 plane stress, 51 shells of revolution, 342 Equivalent strain, 60, 408, 422 Equivalent stress, 21, 408, 422 Explosive forming, 664 Extrusion of metals through conical dies, 186, 196 through square dies, 189, 193 Extrusion pressure, 187 Extrusion ratio, 188 F Fatigue failure, 30 Finite element method, 265, 671 element shape functions, 673 elemental stiffness equations, 675 global stiffness equations, 695 Finite elements isoparametric elements, 678 quadrilateral elements, 679 three-dimensional brick element, 679 triangular elements, 676 Finite element solutions axisymmetric extrusion, 698 compression of a cylinder, 697 lateral spread in rolling, 701 plastic collapse of plates, 265 square cup drawing, 704

Subject Index Finite expansion of a hole in infinite plate, 91 of a spherical cavity, 593 Flange drawing, 130, 687 Flange wrinkling, 126 Flexural buckling, 537 Flow rules L´evy-Mises, 55 Prandtl-Reuss, 17 Tresca’s associated, 17 Flush nozzles, 385 Forging of metals high speed forming, 664 quasi-static forgoing, 479, 571 Forming limit diagram, 447 Fractional reduction, 188 Friction coefficient, 188 Friction factor, 200 G Generalized strain rates, 228, 259, 349 Generalized stresses, 228, 259, 349 Geometry changes, 20 Global stiffness equation, 675 Grooved sheet in tension, 61 H Hardening rules combined or mixed, 28 isotropic, 21 kinematic, 26 Hardness of metals, 157 Head shape, 613 Hencky theory, 26, 66, 258 Hinge circle, 229, 321, 338 Hinge rotation, 277 Hodograph, 272 Hole expansion, 89, 97 Homogeneous work, 188 Hooke’s law, 16 Hydrostatic stress, 11 I Ideal die profile, 205 Ideally plastic material, 20 Impulsive loading, 633, 643, 644 Incompressibility, 15, 20 Indentation of anisotropic medium, 436 by circular flat punch, 152 by conical indenter, 155 by rectangular flat punch, 208 by spherical indenter, 157 Instability in tension

753 of circular diaphragms, 134 of cylindrical bars, 31, 161 of plane sheets, 97, 413, 445 Interaction curves, 143, 241, 320 Invariants, 11 Ironing of cups, 126 I-section beams, 81 Isotropic hardening, 21 Isotropic material, 17 J Jacobian matrix, 682 Jacobian of transformation, 678 Jaumann stress rate, 24 Jump conditions, 564, 641 K Kinematically admissible, 20 Kinematic hardening, 26 Kinetic energy, 586 Kronecker delta, 169, 677 L Lateral buckling, 499 Lateral spread, 212 Length changes in torsion, 430 L´evy-Mises equations, 55 Limit analysis circular plates, 227 conical shells, 367 cylindrical shells, 313 extrusion of billets, 184 hollow square plates, 81 rectangular plates, 261, 270 spherical shells, 353 triangular plates, 279 Limited interaction, 350 Limiting drawing ratio, 460 Limit strains, 451 Limit theorems, 18 Linearized yield condition, 319, 456 Linear programming, 270 Load–deflection relations, 248 Loading surface, 33 Localized necking, 60, 64, 412 Lode variables, 17, 62 Logarithmic strain, 690 Longitudinal stress waves, 561 Lower bound loads, 82, 317, 351 Lower bound theorem, 20 L¨uders bands, 7 Ludwik equation, 9 M Material derivative, 93

754 Maximum work principle, 69 Membrane forces, 253, 346 Metal forming problems bar drawing, 214 deep drawing, 111, 452 extrusion, 184 forging, 212, 214, 659 steckel rolling, 217 tube sinking, 178 wire drawing, 198 Micromechanical model, 29 Minimum weight design of circular plates, 284 of cylindrical shells, 382 of elliptic plates, 308 Mixed hardening, 28 Moment-curvature relation, 489 N Necking in tension, 163 Neutral loading, 23 Nodal forces, 696 Nodal velocities, 679 Nominal stress, 3 Nominal stress rate, 39 Nonquadratic yield function, 421, 445 Normality rule, 16 Normal stress, 138 Notched strip in tension, 67 Numerical integration, 691 O Objective stress rate, 25 Ogival nose, 611 Optimum design, 296, 393 Optimum die angle, 128 Orthotropic material, 410 P Penalty constant, 672 Perfectly plastic material, 624 Perforation of a plate, 617 Plane plastic strain, 272 Plane strain analogy, 270 Plane strain compression, 434 Plane stress anisotropic material, 392, 420 isotropic material, 47 Plastic anisotropy, 405 Plastic buckling, 484 Plastic collapse, 73 Plastic instability, 65, 132, 445, 450 Plastic modulus, 4, 29 Plastic wave propagation, 568, 579

Subject Index Plate bending, 265 Poisson’s ratio, 4 Polycrystalline aggregate, 2 Prandtl–Reuss relations, 17 Preferred orientation, 405 Pressure vessels, 382 Principal curvatures, 272 Principal line theory, 151 Principal stresses, 11 Projectile crumpling, 586 Projectile penetration, 610 Proportional limit, 3 Pulse loading of circular plates, 635 of cylindrical shells, 645 Punch load, 106, 122, 462 Punch stretching, 107 Punch travel, 122, 125, 459 Pure bending, 487, 545 Pure shear, 13 R Ramberg-Osgood equation, 9 Rate of extension, 46, 413 Rate of rotation, 25 Rate sensitive material, 576, 624 Recrystallization, 10 Rectangular plates, 261, 274 Reduced modulus, 486 Redundant work, 202 Residual stress, 94, 186 Reversed loading, 29 Rigid body rotation, 25 Rigid/plastic material, 16 Rotational symmetry, 227, 230 R-value of sheet metal, 416 S Sandwich approximation, 319, 320 Shape functions, 681 Shear force, 306 Shear modulus, 16 Sheet metal forming deep drawing, 111, 452 explosive forming, 664 stretch forming, 97 Shell buckling, 522 Shells of revolution, 342 Shock waves, 568 Simple shear, 73, 425 Simple waves, 564 Slipline fields, 150, 440 Slip planes, 2 Snap through buckling, 554

Subject Index Spherical cavity, 594 Spherical coordinates, 707 Spherical shell, 353, 552 Spin tensor, 45 Stability criterion, 43 Statically admissible, 19 Stiffness matrix, 685 Strain equivalence, 414 Strain hardening isotropic hardening, 21 kinematic hardening, 26 mixed hardening, 28 Strain hardening exponent, 454 Strain rate, 10 Strain rate effect, 577, 578 Strain rate sensitivity, 10 Stress discontinuity, 64, 185 Stress profile, 249, 357 Stress rate, 24 Stress resultant, 272, 315, 393 Stress space, 12, 334 Stress-strain curves, 3, 30, 416, 559 Stress–strain relations, 26 Stretch forming by hydrostatic pressure, 144, 272 by rigid punch head, 447 Strong discontinuity, 579 Strong support, 78 Superplasticity, 10 Swaging process, 140 Swift equation, 8 T Tangent modulus, 4 Tensile test, 3 Three-dimensional problems, 208 Torque–twist relation, 31 Torsional buckling, 586 Torsion test, 431 Tresca criterion, 14 Tresca’s associated flow rule, 17 True strain, 3 True strain rate, 24 True stress, 3 True stress rate, 39 Tube buckling, 542 Tube extrusion, 193 Tube sinking, 178 Two-surface theory, 33 U Uniqueness criterion, 41 Uniqueness of stress, 39

755 Unloading process, 32, 95, 565 Unloading waves, 568 Upper bound loads, 219, 292 Upper bound theorem, 20 V Variational principle, 215, 524, 552, 672, 688 Velocity discontinuity, 64 Velocity field, 172, 196, 213, 267 Virtual velocity, 20, 170 Virtual work principle, 19 Voce equation, 8 Volume constancy, 6, 118 Von Mises criterion, 14 W Warping function, 426 Wave front, 574 Wave propagation longitudinal waves, 571 planes waves, 563 three-dimensional waves, 681 Weak support, 77 Weak waves, 564 Wire drawing conical die profile, 199 ideal die profile, 205 Work equivalence, 415 Work-hardening, 21 Y Yield condition bending of plates, 229, 390 cylindrical shells, 316, 319 shells of revolution, 342 Yield criterion anisotropic, 407 regular, 13 singular, 13 Tresca, 14 von Mises, 13 Yield function, 11, 407 Yield hinge, 261, 324 Yield line solutions distributed loading, 286 elliptical plates, 289 rectangular plates, 264 semi-circular plates, 293 Yield locus, 13, 58, 228, 320 Yield point, 3, 19 Yield surface, 12, 33, 333, 336 Young’s modulus, 4

Mechanical Engineering Series (continued from page ii) D. Gross and T. Seelig, Fracture Mechanics with Introduction to Micro-mechanics K.C. Gupta, Mechanics and Control of Robots R. A. Howland, Intermediate Dynamics: A Linear Algebraic Approach D. G. Hull, Optimal Control Theory for Applications J. Ida and J.P.A. Bastos, Electromagnetics and Calculations of Fields M. Kaviany, Principles of Convective Heat Transfer, 2nd ed. M. Kaviany, Principles of Heat Transfer in Porous Media, 2nd ed. E.N. Kuznetsov, Underconstrained Structural Systems P. Ladevèze, Nonlinear Computational Structural Mechanics: New Approaches and Non-Incremental Methods of Calculation P. Ladevèze and J.-P. Pelle, Mastering Calculations in Linear and Nonlinear Mechanics A. Lawrence, Modern Inertial Technology: Navigation, Guidance, and Control, 2nd ed. R.A. Layton, Principles of Analytical System Dynamics F.F. Ling, W.M. Lai, D.A. Lucca, Fundamentals of Surface Mechanics: With Applications, 2nd ed. C.V. Madhusudana, Thermal Contact Conductance D.P. Miannay, Fracture Mechanics D.P. Miannay, Time-Dependent Fracture Mechanics D.K. Miu, Mechatronics: Electromechanics and Contromechanics D. Post, B. Han, and P. Ifju, High Sensitivity and Moiré: Experimental Analysis for Mechanics and Materials R. Rajamani, Vehicle Dynamics and Control F.P. Rimrott, Introductory Attitude Dynamics S.S. Sadhal, P.S. Ayyaswamy, and J.N. Chung, Transport Phenomena with Drops and Bubbles A.A. Shabana, Theory of Vibration: An Introduction, 2nd ed. A.A. Shabana, Theory of Vibration: Discrete and Continuous Systems, 2nd ed. Y. Tseytlin, Structural Synthesis in Precision Elasticity

Mechanical Engineering Series J. Chakrabarty, Applied Plasticity, Second Edition G. Genta, Vibration Dynamics and Control R. Firoozian, Servo Motors and Industrial Control Theory G. Genta and L. Morello, The Automotive Chassis, Volumes 1 & 2 F. A. Leckie and D. J. Dal Bello, Strength and Stiffness of Engineering Systems Wodek Gawronski, Modeling and Control of Antennas and Telescopes M. Ohsaki and KiyohiroIkeda, Stability and Optimization of Structures: Generalized Sensitivity Analysis A.C. Fischer-Cripps, Introduction to Contact Mechanics, 2nd ed. W. Cheng and I. Finnie, Residual Stress Measurement and the Slitting Method J. Angeles, Fundamentals of Robotic Mechanical Systems: Theory, Methods and Algorithms, 3rd ed. J. Angeles, Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms, 2nd ed. P. Basu, C. Kefa, and L. Jestin, Boilers and Burners: Design and Theory J.M. Berthelot, Composite Materials: Mechanical Behavior and Structural Analysis I.J. Busch-Vishniac, Electromechanical Sensors and Actuators J. Chakrabarty, Applied Plasticity K.K. Choi and N.H. Kim, Structural Sensitivity Analysis and Optimization 1: Linear Systems K.K. Choi and N.H. Kim, Structural Sensitivity Analysis and Optimization 2: Nonlinear Systems and Applications G. Chryssolouris, Laser Machining: Theory and Practice V.N. Constantinescu, Laminar Viscous Flow G.A. Costello, Theory of Wire Rope, 2nd ed. K. Czolczynski, Rotordynamics of Gas-Lubricated Journal Bearing Systems M.S. Darlow, Balancing of High-Speed Machinery W. R. DeVries, Analysis of Material Removal Processes J.F. Doyle, Nonlinear Analysis of Thin-Walled Structures: Statics, Dynamics, and Stability J.F. Doyle, Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms, 2nd Edition P.A. Engel, Structural Analysis of Printed Circuit Board Systems A.C. Fischer-Cripps, Introduction to Contact Mechanics A.C. Fischer-Cripps, Nanoindentation, 2nd ed. J. García de Jalón and E. Bayo, Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge W.K. Gawronski, Advanced Structural Dynamics and Active Control of Structures W.K. Gawronski, Dynamics and Control of Structures: A Modal Approach G. Genta, Dynamics of Rotating Systems (continued after index)

J. Chakrabarty

Applied Plasticity, Second Edition

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J. Chakrabarty Visiting Professor Department of Mechanical Engineering Florida State University Tallahassee FL 32303 USA [email protected]

ISSN 0941-5122 ISBN 978-0-387-77673-6 e-ISBN 978-0-387-77674-3 DOI 10.1007/978-0-387-77674-3 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009934696 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Mechanical Engineering Series Frederick F. Ling Editor-in-Chief

The Mechanical Engineering Series features graduate texts and research monographs to address the need for information in contemporary mechanical engineering, including areas of concentration of applied mechanics, biomechanics, computational mechanics, dynamical systems and control, energetics, mechanics of materials, processing, production systems, thermal science, and tribology.

Advisory Board/Series Editors Applied Mechanics

F.A. Leckie University of California, Santa Barbara D. Gross Technical University of Darmstadt

Biomechanics

V.C. Mow Columbia University

Computational Mechanics

H.T. Yang University of California, Santa Barbara

Dynamic Systems and Control/ Mechatronics

D. Bryant University of Texas at Austin

Energetics

J.R. Welty University of Oregon, Eugene

Mechanics of Materials

I. Finnie University of California, Berkeley

Processing

K.K. Wang Cornell University

Production Systems

G.-A. Klutke Texas A&M University

Thermal Science

A.E. Bergles Rensselaer Polytechnic Institute

Tribology

W.O. Winer Georgia Institute of Technology

Series Preface

Mechanical engineering, an engineering discipline forged and shaped by the needs of the industrial revolution, is once again asked to do its substantial share in the call for industrial renewal. The general call is urgent as we face profound issues of productivity and competitiveness that require engineering solutions, among others. The Mechanical Engineering Series features graduate texts and research monographs intended to address the need for information in contemporary areas of mechanical engineering. The series is conceived as a comprehensive one that covers a broad range of concentrations important to mechanical engineering graduate education and research. We are fortunate to have a distinguished roster of consulting editors on the advisory board, each an expert in one of the areas of concentration. The names of the consulting editors are listed on the facing page of this volume. The areas of concentration are applied mechanics, biomechanics, computational mechanics, dynamic systems and control, energetics, mechanics of materials, processing, production systems, thermal science, and tribology. Austin, Texas

Frederick F. Ling

This book is humbly dedicated to the loving memory of MA INDIRA who continues to be the source of real inspiration to me.

Preface

The past few years have witnessed a growing interest in the application of the mechanics of plastic deformation of metals to a variety of engineering problems associated with structural design and technological forming of metals. Written several years ago to serve as a companion volume to the author’s earlier work under the title Theory of Plasticity, which comprehensively expounds the fundamentals of plasticity of metals, the present work seems to have stood the test of time and has established itself as a comprehensive reference work that is equally useful for classroom purposes. While the earlier work is mainly concerned with the application of the theory to the solution of elastic/plastic problems, limit analysis of framed structures, and problems in plane plastic strain involving slipline fields, several important areas of plasticity related to the analysis of multidimensional structures and various metal-forming processes had to be left out for obvious reasons. The present text is intended to fill this gap and to make available to the reader in a single volume a detailed account of a wide range of useful results that are scattered in numerous periodicals and other sources. The fundamentals of the mathematical theory of plasticity are discussed in Chapter 1 with sufficient details, in order to eliminate the need for frequent references to the author’s earlier volume. The theory of plane plastic stress and its applications to structural analysis and sheet metal forming are presented in Chapter 2. The axially symmetrical plastic state, as well as a few three-dimensional problems of plasticity, is treated in Chapter 3. The plastic behavior of plates and shells, mainly from the point of view of limit analysis, is discussed with several examples in Chapters 4 and 5. The plasticity of metals with fully developed orthotropic anisotropy and its application to the plastic behavior of anisotropic sheets are presented in Chapter 6. The generalized tangent modulus theory of buckling in the plastic range for columns, plates, and shells is treated in Chapter 7 from the point of view of the bifurcation phenomenon. Chapter 8 deals with a wide range of topics in dynamic plasticity, including the wave propagation, armor penetration, and structural impact in the plastic range. The fundamentals of the rigid/plastic finite element method, with special reference to its application to metal-forming processes, are presented in Chapter 9, where several examples are included for illustration. The publication of the revised second edition of Applied Plasticity is deemed necessary not only for the obvious need for updating the book but also for the purpose ix

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of making it more suitable for the teaching of appropriate courses on plasticity at the graduate level. During the preparation of the second edition, several parts of the text have been extensively revised in the light of the recent developments of the subject, and new references to the published literature have been made in appropriate places. The discussion of the finite element method in plasticity, previously relegated to an appendix in the first edition, has now been expanded into a new chapter to permit a more complete treatment of the subject. A new section has been added in Chapter 4 to discuss the yield line theory for plate bending, not only for the derivation of complete solutions but also for the estimation of upper bounds on the limit load. A set of homework problems has been included at the end of each chapter for the benefit of both the student and the instructor, many of these problems having been designed to supplement the text. The references to the published literature have now been collected together and placed at the very end of the book for the sake of the expected convenience of the reader. The book in its present form would be suitable for teaching advanced graduate level courses on plasticity and metal forming to students of mechanical and manufacturing engineering, as well as on structural plasticity to students of civil and structural engineering. The book will also be found useful for teaching courses on dynamic plasticity to both the mechanical and civil engineering students. Though intended primarily for research workers in the field of plasticity, senior undergraduate students and practicing engineers are also likely to benefit from this book to a large extent. I take this opportunity to express my gratitude to the late Professor J. M. Alexander, formerly of Imperial College, London, who not only stimulated my interest in plasticity but also encouraged me to undertake the task of writing this book. I am also grateful to Dr. Frederick F. Ling, the Editor-in-Chief of this Series, for his encouragement and support for the publication of the second edition of Applied Plasticity. It is a pleasure to offer my sincere thanks to Ms. Jennifer Mirski, the Assistant Engineering Editor of Springer for her helpful cooperation and support during the preparation of the manuscript. Finally, I am deeply indebted to my wife Swati, who gracefully accepted the hardship of many lonely hours to enable me to complete this work in a satisfactory manner. J. Chakrabarty

Contents

1 Fundamental Principles . . . . . . . . . . . . 1.1 The Material Response . . . . . . . . . . 1.1.1 Introduction . . . . . . . . . . . . 1.1.2 The True Stress–Strain Curve . . . 1.1.3 Empirical Stress–Strain Equations . 1.2 Basic Laws of Plasticity . . . . . . . . . . 1.2.1 Yield Criteria of Metals . . . . . . 1.2.2 Plastic Flow Rules . . . . . . . . . 1.2.3 Limit Theorems . . . . . . . . . . 1.3 Strain-Hardening Plasticity . . . . . . . . 1.3.1 Isotropic Hardening . . . . . . . . 1.3.2 Plastic Flow with Hardening . . . . 1.3.3 Kinematic Hardening . . . . . . . 1.3.4 Combined or Mixed Hardening . . 1.4 Cyclic Loading of Structures . . . . . . . 1.4.1 Cyclic Stress–Strain Curves . . . . 1.4.2 A Bounding Surface Theory . . . . 1.4.3 The Two Surfaces in Contact . . . 1.5 Uniqueness and Stability . . . . . . . . . . 1.5.1 Fundamental Relations . . . . . . . 1.5.2 Uniqueness Criterion . . . . . . . 1.5.3 Stability Criterion . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . .

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2 Problems in Plane Stress . . . . . . . . . . . . . 2.1 Formulation of the Problem . . . . . . . . . 2.1.1 Characteristics in Plane Stress . . . . 2.1.2 Relations Along the Characteristics . 2.1.3 The Velocity Equations . . . . . . . 2.1.4 Basic Relations for a Tresca Material 2.2 Discontinuities and Necking . . . . . . . . . 2.2.1 Velocity Discontinuities . . . . . . .

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2.2.2 Tension of a Grooved Sheet . . . . . 2.2.3 Stress Discontinuities . . . . . . . . 2.2.4 Diffuse and Localized Necking . . . 2.3 Yielding of Notched Strips . . . . . . . . . . 2.3.1 V-Notched Strips in Tension . . . . . 2.3.2 Solution for Circular Notches . . . . 2.3.3 Solution for Shallow Notches . . . . 2.4 Bending of Prismatic Beams . . . . . . . . . 2.4.1 Strongly Supported Cantilever . . . . 2.4.2 Weakly Supported Cantilever . . . . 2.4.3 Bending of I-Section Beams . . . . . 2.5 Limit Analysis of a Hollow Plate . . . . . . 2.5.1 Equal Biaxial Tension . . . . . . . . 2.5.2 Uniaxial Tension: Lower Bounds . . 2.5.3 Uniaxial Tension: Upper Bounds . . 2.5.4 Arbitrary Biaxial Tension . . . . . . 2.6 Hole Expansion in Infinite Plates . . . . . . 2.6.1 Initial Stages of the Process . . . . . 2.6.2 Finite Expansion Without Hardening 2.6.3 Work-Hardening von Mises Material 2.6.4 Work-Hardening Tresca Material . . 2.7 Stretch Forming of Sheet Metals . . . . . . 2.7.1 Hydrostatic Bulging of a Diaphragm 2.7.2 Stretch Forming Over a Rigid Punch 2.7.3 Solutions for a Special Material . . . 2.8 Deep Drawing of Cylindrical Cups . . . . . 2.8.1 Introduction . . . . . . . . . . . . . 2.8.2 Solution for Nonhardening Materials 2.8.3 Influence of Work-Hardening . . . . 2.8.4 Punch Load and Punch Travel . . . . 2.9 Ironing and Flange Wrinkling . . . . . . . . 2.9.1 Ironing of Cylindrical Cups . . . . . 2.9.2 Flange Wrinkling in Deep Drawing . Problems . . . . . . . . . . . . . . . . . . . . . .

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3 Axisymmetric and Related Problems . . . . . . . . 3.1 Basic Theory and Exact Solutions . . . . . . . . 3.1.1 Fundamental Relations . . . . . . . . . . 3.1.2 Swaging in a Contracting Cylinder . . . 3.1.3 Fully Plastic State in a Cylindrical Tube 3.1.4 Plastic Flow Through a Conical Channel 3.2 Slipline Fields and Indentations . . . . . . . . . 3.2.1 Relations Along the Sliplines . . . . . . 3.2.2 Indentation by a Flat Punch . . . . . . . 3.2.3 Indentation by a Rigid Cone . . . . . . .

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3.2.4 The Hardness of Metals . . . . . . . . . . . 3.3 Necking of a Cylindrical Bar . . . . . . . . . . . . 3.3.1 Stress Distribution in the Neck . . . . . . . 3.3.2 Initiation of Necking . . . . . . . . . . . . . 3.4 Compression of Short Cylinders . . . . . . . . . . . 3.4.1 Compression of Solid Cylinders . . . . . . . 3.4.2 Estimation of Incipient Barreling . . . . . . 3.4.3 Compression of a Hollow Cylinder . . . . . 3.5 Sinking of Thin-Walled Tubes . . . . . . . . . . . . 3.5.1 Solution Without Strain Hardening . . . . . 3.5.2 Influence of Strain Hardening . . . . . . . . 3.6 Extrusion of Cylindrical Billets . . . . . . . . . . . 3.6.1 The Basis for an Approximation . . . . . . . 3.6.2 Extrusion Through Conical Dies . . . . . . 3.6.3 Extrusion Through Square Dies . . . . . . . 3.6.4 Upper Bound Solution for Square Dies . . . 3.6.5 Upper Bound Solution for Conical Dies . . . 3.7 Mechanics of Wire Drawing . . . . . . . . . . . . . 3.7.1 Solution for a Nonhardening Material . . . . 3.7.2 Influence of Back Pull and Work-Hardening 3.7.3 Ideal Wire-Drawing Dies . . . . . . . . . . 3.8 Some Three-Dimensional Problems . . . . . . . . . 3.8.1 Indentation by a Rectangular Punch . . . . . 3.8.2 Flat Tool Forging of a Bar . . . . . . . . . . 3.8.3 Bar Drawing Through Curved Dies . . . . . 3.8.4 Compression of Noncircular Blocks . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Plastic Bending of Plates . . . . . . . . . . . . . . . . 4.1 Plastic Collapse of Circular Plates . . . . . . . . . 4.1.1 The Basic Theory . . . . . . . . . . . . . 4.1.2 Circular Plates Carrying Distributed Loads 4.1.3 Other Types of Loading of Circular Plates 4.1.4 Solutions Based on the von Mises Criterion 4.1.5 Combined Bending and Tension . . . . . . 4.2 Deflection of Circular Plates . . . . . . . . . . . . 4.2.1 Basic Equations . . . . . . . . . . . . . . 4.2.2 Deflection of a Simply Supported Plate . . 4.2.3 Deflection of a Built-In Plate . . . . . . . 4.3 Influence of Membrane Forces . . . . . . . . . . 4.3.1 Simply Supported Circular Plates . . . . . 4.3.2 Built-In Circular Plates . . . . . . . . . . 4.4 Plastic Collapse of Noncircular Plates . . . . . . . 4.4.1 General Considerations . . . . . . . . . . 4.4.2 Uniformly Loaded Rectangular Plates . . .

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4.4.3 Finite Element Analysis for Plate Bending 4.5 Plane Strain Analogy for Plate Bending . . . . . . 4.5.1 The Use of Square Yield Condition . . . . 4.5.2 Application to Rectangular Plates . . . . . 4.5.3 Collapse Load for Triangular Plates . . . . 4.6 Yield Line Theory for Plates . . . . . . . . . . . . 4.6.1 Basic Yield Line Theory . . . . . . . . . . 4.6.2 Elliptical Plate Loaded at the Center . . . 4.6.3 A Plate Under Distributed Loading . . . . 4.6.4 Yield Line Upper Bounds . . . . . . . . . 4.6.5 Examples of Upper Bounds . . . . . . . . 4.7 Minimum Weight Design of Plates . . . . . . . . 4.7.1 Basic Principles . . . . . . . . . . . . . . 4.7.2 Circular Sandwich Plates . . . . . . . . . 4.7.3 Solid Circular Plates . . . . . . . . . . . . 4.7.4 Elliptical Sandwich Plates . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Plastic Analysis of Shells . . . . . . . . . . . . . . . . . . 5.1 Cylindrical Shells Without End Load . . . . . . . . . 5.1.1 Basic Equations . . . . . . . . . . . . . . . . 5.1.2 Yield Condition and Flow Rule . . . . . . . . 5.1.3 Shell Under Uniform Radial Pressure . . . . . 5.1.4 Shell Under a Band of Pressure . . . . . . . . 5.1.5 Solution for a von Mises Material . . . . . . . 5.2 Cylindrical Shells with End Load . . . . . . . . . . . 5.2.1 Yield Condition and Flow Rule . . . . . . . . 5.2.2 Shell Under Radial Pressure and Axial Thrust 5.2.3 Influence of Elastic Deformation . . . . . . . 5.3 Yield Point States in Shells of Revolution . . . . . . . 5.3.1 Generalized Stresses and Strain Rates . . . . . 5.3.2 Yield Condition for a Tresca Material . . . . . 5.3.3 Approximations for a von Mises Material . . . 5.3.4 Linearization and Limited Interaction . . . . . 5.4 Limit Analysis of Spherical Shells . . . . . . . . . . 5.4.1 Basic Equations . . . . . . . . . . . . . . . . 5.4.2 Plastic Collapse of a Spherical Cap . . . . . . 5.4.3 Spherical Cap with a Covered Cutout . . . . . 5.4.4 Solution for a Tresca Sandwich Shell . . . . . 5.4.5 Extended Analysis for Deeper Shells . . . . . 5.5 Limit Analysis of Conical Shells . . . . . . . . . . . 5.5.1 Basic Equations . . . . . . . . . . . . . . . . 5.5.2 Truncated Shallow Shell Under Line Load . . 5.5.3 Shallow Shell Loaded Through Rigid Boss . . 5.5.4 Centrally Loaded Shell of Finite Angle a . . .

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5.6 Limit Analysis of Pressure Vessels . . . . . . . . . 5.6.1 Plastic Collapse of a Toroidal Knuckle . . . 5.6.2 Collapse of a Complete Pressure Vessel . . . 5.6.3 Cylindrical Nozzle in a Spherical Vessel . . 5.7 Minimum Weight Design of Shells . . . . . . . . . 5.7.1 Principles for Optimum Design . . . . . . . 5.7.2 Basic Theory for Cylindrical Shells . . . . . 5.7.3 Simply Supported Shell Without End Load . 5.7.4 Cylindrical Shell with Built-In Supports . . 5.7.5 Closed-Ended Shell Under Internal Pressure Problems . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Plastic Anisotropy . . . . . . . . . . . . . . . . . . . . 6.1 Plastic Flow of Anisotropic Metals . . . . . . . . . 6.1.1 The Yield Criterion . . . . . . . . . . . . . 6.1.2 Stress–Strain Relations . . . . . . . . . . . 6.1.3 Variation of Anisotropic Parameters . . . . . 6.2 Anisotropy of Rolled Sheets . . . . . . . . . . . . . 6.2.1 Variation of Yield Stress and Strain Ratio . . 6.2.2 Localized and Diffuse Necking . . . . . . . 6.2.3 Correlation of Stress–Strain Curves . . . . . 6.2.4 Normal Anisotropy in Sheet Metal . . . . . 6.2.5 A Generalized Theory for Planar Anisotropy 6.3 Torsion of Anisotropic Bars . . . . . . . . . . . . . 6.3.1 Bars of Arbitrary Cross Section . . . . . . . 6.3.2 Some Particular Cases . . . . . . . . . . . . 6.3.3 Length Changes in Twisted Tubes . . . . . . 6.3.4 Torsion of a Free-Ended Tube . . . . . . . . 6.4 Plane Strain in Anisotropic Metals . . . . . . . . . 6.4.1 Basic Equations in Plane Strain . . . . . . . 6.4.2 Relations Along the Sliplines . . . . . . . . 6.4.3 Indentation by a Flat Punch . . . . . . . . . 6.4.4 Indentation of a Finite Medium . . . . . . . 6.4.5 Compression Between Parallel Platens . . . 6.5 Anisotropy in Stretch Forming . . . . . . . . . . . 6.5.1 Basic Equations for Biaxial Stretching . . . 6.5.2 Plastic Instability in Tension . . . . . . . . . 6.5.3 Forming Limit Diagram . . . . . . . . . . . 6.6 Anisotropy in Deep Drawing . . . . . . . . . . . . 6.6.1 The Radial Drawing Process . . . . . . . . . 6.6.2 Use of the Linearized Yield Condition . . . 6.6.3 The Limiting Drawing Ratio . . . . . . . . . 6.6.4 Earing of Deep-Drawn Cups . . . . . . . . . 6.7 Anisotropy in Plates and Shells . . . . . . . . . . .

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6.7.1 6.7.2 6.7.3 Problems .

Bending of Circular Plates . . . . . Plastic Collapse of a Spherical Cap Reinforced Circular Plates . . . . . . . . . . . . . . . . . . . . . . . . .

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466 469 472 475

7 Plastic Buckling . . . . . . . . . . . . . . . . . . . . . 7.1 Buckling of Axially Loaded Columns . . . . . . . 7.1.1 Analysis for Bifurcation . . . . . . . . . . 7.1.2 Analysis for Instability . . . . . . . . . . . 7.2 Behavior of Eccentrically Loaded Columns . . . . 7.2.1 Moment-Curvature Relations . . . . . . . 7.2.2 Analysis for a Pin-Ended Column . . . . . 7.2.3 Solution for an Inelastic Beam Column . . 7.3 Lateral Buckling of Beams . . . . . . . . . . . . 7.3.1 Pure Bending of Narrow Beams . . . . . . 7.3.2 Buckling of Transversely Loaded Beams . 7.4 Buckling of Plates Under Edge Thrust . . . . . . 7.4.1 Basic Equations for Thin Plates . . . . . . 7.4.2 Buckling of Rectangular Plates . . . . . . 7.4.3 Rectangular Plates Under Biaxial Thrust . 7.4.4 Buckling of Circular Plates . . . . . . . . 7.5 Buckling of Cylindrical Shells . . . . . . . . . . . 7.5.1 Formulation of the Rate Problem . . . . . 7.5.2 Bifurcation Under Combined Loading . . 7.5.3 Buckling Under Axial Compression . . . . 7.5.4 Influence of Frictional Restraints . . . . . 7.5.5 Buckling Under External Fluid Pressure . 7.6 Torsional and Flexural Buckling of Tubes . . . . . 7.6.1 Bifurcation Under Pure Torsion . . . . . . 7.6.2 Buckling Under Pure Bending . . . . . . . 7.7 Buckling of Spherical Shells . . . . . . . . . . . . 7.7.1 Analysis for a Complete Spherical Shell . 7.7.2 Solution for the Critical Pressure . . . . . 7.7.3 Snap-Through Buckling of Spherical Caps Problems . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Dynamic Plasticity . . . . . . . . . . . . . . . . . . . . . . . 8.1 Longitudinal Stress Waves in Bars . . . . . . . . . . . . 8.1.1 Wave Propagation Without Rate Effects . . . . . . 8.1.2 Simple Wave Solution with Application . . . . . . 8.1.3 Solution for Linear Strain Hardening . . . . . . . 8.1.4 Influence of Strain-Rate Sensitivity . . . . . . . . 8.1.5 Illustrative Examples and Experimental Evidence 8.2 Plastic Waves in Continuous Media . . . . . . . . . . . . 8.2.1 Plastic Wave Speeds and Their Properties . . . . .

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8.2.2 A Geometrical Representation . . . . . . . . . 8.2.3 Plane Waves in Elastic/Plastic Solids . . . . . 8.3 Crumpling of Flat-Ended Projectiles . . . . . . . . . 8.3.1 Taylor’s Theoretical Model . . . . . . . . . . 8.3.2 An Alternative Analysis . . . . . . . . . . . . 8.3.3 Estimation of the Dynamic Yield Stress . . . . 8.4 Dynamic Expansion of Spherical Cavities . . . . . . 8.4.1 Purely Elastic Deformation . . . . . . . . . . 8.4.2 Large Elastic/Plastic Expansion . . . . . . . . 8.4.3 Influence of Elastic Compressibility . . . . . . 8.5 Mechanics of Projectile Penetration . . . . . . . . . . 8.5.1 A Simple Theoretical Model . . . . . . . . . . 8.5.2 The Influence of Cavitation . . . . . . . . . . 8.5.3 Perforation of a Thin Plate . . . . . . . . . . . 8.6 Impact Loading of Prismatic Beams . . . . . . . . . . 8.6.1 Cantilever Beam Struck at Its Tip . . . . . . . 8.6.2 Rate Sensitivity and Simplified Model . . . . 8.6.3 Solution for a Rate-Sensitive Cantilever . . . . 8.6.4 Transverse Impact of a Free-Ended Beam . . . 8.7 Dynamic Loading of Circular Plates . . . . . . . . . . 8.7.1 Formulation of the Problem . . . . . . . . . . 8.7.2 Simply Supported Plate Under Pressure Pulse 8.7.3 Dynamic Behavior Under High Loads . . . . 8.7.4 Solution for Impulsive Loading . . . . . . . . 8.8 Dynamic Loading of Cylindrical Shells . . . . . . . . 8.8.1 Defining Equations and Yield Condition . . . 8.8.2 Clamped Shell Loaded by a Pressure Pulse . . 8.8.3 Dynamic Analysis for High Loads . . . . . . 8.9 Dynamic Forming of Metals . . . . . . . . . . . . . . 8.9.1 High-Speed Compression of a Disc . . . . . . 8.9.2 Dynamic Response of a Thin Diaphragm . . . 8.9.3 High-Speed Forming of Sheet Metal . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 The Finite Element Method . . . . . . . . . 9.1 Fundamental Principles . . . . . . . . . 9.1.1 The Variational Formulation . . . 9.1.2 Velocity and Strain Rate Vectors . 9.1.3 Elemental Stiffness Equations . . 9.2 Element Geometry and Shape Function . 9.2.1 Triangular Element . . . . . . . 9.2.2 Quadrilateral Element . . . . . . 9.2.3 Hexahedral Brick Element . . . . 9.3 Matrix Forms in Special Cases . . . . . 9.3.1 Plane Strain Problems . . . . . .

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671 671 671 672 675 676 676 679 681 682 682

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Contents

9.3.2 Axially Symmetrical Problems . . . 9.3.3 Three-Dimensional Problems . . . . 9.4 Sheet Metal Forming . . . . . . . . . . . . . 9.4.1 Basic Equations for Sheet Metals . . 9.4.2 Axisymmetric Sheet Forming . . . . 9.4.3 Sheet Forming of Arbitrary Shapes . 9.5 Numerical Implementation . . . . . . . . . 9.5.1 Numerical Integration . . . . . . . . 9.5.2 Global Stiffness Equations . . . . . . 9.5.3 Boundary Conditions . . . . . . . . 9.6 Illustrative Examples . . . . . . . . . . . . . 9.6.1 Compression of a Cylindrical Block . 9.6.2 Bar Extrusion Through a Conical Die 9.6.3 Analysis of Spread in Sheet Rolling . 9.6.4 Deep Drawing of Square Cups . . .

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683 685 685 685 687 689 691 691 694 696 697 697 698 701 704

Appendix: Orthogonal Curvilinear Coordinates . . . . . . . . . . . . .

707

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

709

Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

743

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751

Chapter 1

Fundamental Principles

1.1 The Material Response 1.1.1 Introduction In a single crystal of many metals, the main mechanism of plastic deformation is simple shear parallel to preferred planes and directions, which at ordinary temperatures coincides with those of the highest atomic density. Slip is initiated along a particular plane and in a given direction when the associated component of the shear stress attains a critical value under increasing external load. The amount of plastic deformation in a single crystal is specified by the glide strain, which is the relative displacement of two parallel slip planes at a unit distance apart. When there are several possible slip directions in a crystal lattice, the displacement of any point in the crystal due to simultaneous shears in the appropriate directions can be found from simple geometry. The mechanism of slip-induced plasticity in single crystals, governed by the glide motion of dislocations along corresponding slip planes, has been the subject of numerous investigations in the past. The change in shape of a single crystal requires, in general, the operation of five independent slip systems (von Mises, 1928). This is due to the fact that an arbitrary state of strain is specified by the six independent components of the symmetric strain tensor, while the sum of the normal strain components vanishes by the condition of constancy of volume of the plastic material. The existence of five independent slip systems in a single crystal is necessary for the material to be ductile in the polycrystalline form. Face-centered cubic metals, having 12 potential slip systems in each crystal grain, satisfy this requirement and are known to have high degrees of ductility, while hexagonal close-packed metals having relatively low symmetry are noted for limited ductility at room temperatures. The ductility of a polycrystalline metal also requires slip flexibility which enables the five independent slip systems to operate simultaneously within a small volume of the aggregate. Mathematical theories of slip-induced plasticity in single crystals have been developed by Hill (1966), Hill and Rice (1972), and Asaro (1983). Some attempts have been made in the past to relate the tensile yield stress of polycrystalline metals in terms of the shear yield stress of the corresponding single crystals. Assuming each crystal grain to undergo the same uniform strain as the polycrystalline metal, and by minimizing the sum of the magnitudes of a set of geometrically possible shears, Taylor (1938) determined the uniaxial stress–strain

J. Chakrabarty, Applied Plasticity, Second Edition, Mechanical Engineering Series, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77674-3_1,

1

2

1 Fundamental Principles

curve for an aluminum aggregate, in good agreement with the experimental curve. The selection of active slip planes from a very wide range of possible combinations, on the basis of the minimum principle, is not necessarily unique. Taylor’s approach has been generalized by Bishop and Hill (1951), who developed a method of deriving upper and lower bounds on the yield function for a polycrystalline metal under any set of combined stresses. The plastic behavior of polycrystalline aggregates in relation to that of single crystals, including the effect of elastic deformation, has been similarly examined by Lin (1957, 1971). A useful review of the recent developments of the micromechanics of polycrystal plasticity has been presented by Khan and Huang (1995). Taylor’s theoretical model ensures the compatibility of strains across the grain boundaries, but fails to satisfy the conditions of equilibrium across these boundaries. In order to satisfy both the conditions of compatibility and equilibrium, a self-consistent model has been proposed by Kröner (1961) and by Budiansky and Wu (1962). These authors approximated each individual crystal grain by a spherical inclusion embedded in an infinitely extended homogeneous elastic matrix. The relationship between the stresses and the strains in the individual grains and those applying to the aggregate has been obtained by an averaging process based on an elastic/plastic analysis for the inclusion problem. Berveiller and Zaoui (1979) modified the theoretical model by introducing a plastic accommodation factor based on the assumption of isotropy of the elastic and plastic responses of both the crystal grain and the aggregate. A self-consistent model in which the anisotropy of the material response is allowed for has been developed by Hill (1965) and Hutchinson (1970), who considered the individual crystal grain as an ellipsoidal inclusion embedded in an elastic/plastic matrix. Another extension of the self-consistent model based on equivalent body forces has been put forward by Lin (1984). The macroscopic theory of plasticity, which the present volume is concerned with, is based on certain experimental observations on the behavior of ductile metals beyond the elastic limit under relatively simple states of combined stress. The theory is capable of predicting the distribution of stresses and strains in polycrystalline metals, not only in situations where the elastic and plastic strains are of comparable magnitudes but also in situations where the plastic strains are large enough for the elastic strains to be disregarded. The mathematical formulation of the plasticity problem is essentially incremental in nature, requiring due consideration of the complete stress and strain history of the elements that have been deformed in the plastic range. An interesting theoretical model, based on the nucleation of voids in the deforming material including interaction of neighboring voids, has been developed by Gurson (1977) and Tvergaard (1982). A theory of plastic yielding and flow of porous materials has been advanced by Tsuta and Yin (1998). A practical method of predicting the macroscopic behavior of polycrystalline aggregates from microstructural data has been discussed by Lee (1993).

1.1

The Material Response

3

1.1.2 The True Stress–Strain Curve In order to deal with large plastic deformation of metals, it is necessary to introduce the concept of true stress and true strain occurring in a test specimen. The true stress σ defined as the applied load divided by the current area of cross section of the specimen, can be significantly different from the nominal stress s, which is the load divided by the original area of cross section. If the initial and current lengths of a tensile specimen are denoted l0 and l, respectively, then the engineering strain e is equal to the ratio (l – l0 )/l0 , while the true strain ε is defined in such a way that its increment dε is equal to the ratio dl/l, where dl is the corresponding increase in length. It follows that the total true strain produced by a change in length from l0 to l during the tensile test is ε = ln

l = ln (1 + e). l0

(1.1)

Similarly, in the case of simple compression of a specimen, whose height is reduced from h0 to h during the test, the engineering strain is of magnitude e = (h0 – h)/h0 , while the magnitude of the true strain is ε = ln

h0 h

= − ln (1 − e).

As the deformation is continued in the plastic range, the true stress becomes increasingly higher than the nominal stress in the case of simple tension and lower than the nominal stress in the case of simple compression. The true strain, on the other hand, is progressively smaller than the engineering strain in simple tension and higher than the engineering strain in simple compression, as the deformation proceeds. There is sufficient experimental evidence to suggest that the macroscopic stress– strain curve of a polycrystalline metal in simple compression coincides with that in simple tension when the true stress is plotted against the true strain. Figure 1.1(a) shows the true stress–strain curve of a typical engineering material in simple tension. The longitudinal true stress σ existing in a test specimen under an axial load is a monotonically increasing function of the longitudinal true strain ε. The straight line OA represents the linear elastic response with A denoting the proportional limit. The elastic range generally extends slightly beyond this to the yield point B, which marks the beginning of plastic deformation. The strain-hardening property of the material requires the stress to increase with strain in the plastic range, but the slope of the stress–strain curve progressively decreases as the strain is increased until fracture occurs at G. Since plastic deformation is irreversible, unloading from some point C on the loading curve would make the stress–strain diagram follow the path CE, where E lies on the ε-axis for complete unloading and represents the amount of permanent or plastic strain corresponding to C. On reloading the specimen, the stress–strain

4

1 Fundamental Principles

Fig. 1.1 True stress–strain curve in simple tension: (a) loading and unloading with reloading and (b) idealized stress–strain behavior

curve follows the path EFG forming a hysteresis loop of narrow width, where F is a new yield point and FG is virtually a continuation of BC. Following Prandtl (1928), the stress–strain curve may be idealized by neglecting the width of the hysteresis loop and assuming the unloading path to be a straight line parallel to OA. The idealized curve, shown in Fig. 1.1(b), implies that the reloading proceeds along the path ECG, passing through the point C where the previous unloading started. Furthermore, the proportional limit A is assumed to coincide with the initial yield point, the corresponding stress being denoted by Y. It follows from the idealized stress–strain curve that the recoverable elastic strain at any point of the curve is equal to σ /E, where E is Young’s modulus and σ the current stress. Any stress increment dσ is associated with an elastic strain increment of amount dσ /E and a plastic strain increment of amount dσ /H, where H is the plastic modulus representing the current slope of the curve for the stress against plastic strain. Since the total strain increment is dσ /T, where T is the tangent modulus denoting the slope of the (σ , ε)-curve, we obtain the relationship 1 1 1 = + . T E H

(1.2)

The difference between T and H decreases rapidly with increasing strain, the elastic strain increment being increasingly small compared to the plastic strain increment. The material is said to be nonhardening when H = T = 0, which is approximately satisfied by a material that is heavily prestrained. Each increment of longitudinal strain dε in a tensile specimen is accompanied by a lateral compressive strain increment of magnitude dε’, the ratio dε’/dε being known as the contraction ratio denoted by η. Over the elastic range of strains, the contraction ratio has a constant value equal to Poisson’s ratio v, but once the yield

1.1

The Material Response

5

point is exceeded, the contraction ratio becomes a function of the magnitude of the strain. For an isotropic material, the elastic and plastic parts of the lateral strain increment have the magnitudes vdεe and dε p /2, respectively, where dε p = dε – dσ /E denotes the plastic component of the longitudinal strain increment. Consequently, the total lateral strain increment is given by 2dε = dε – (1 – 2v) dσ /E, and the contraction ratio is therefore expressed as T 1 1 − (1 − 2v) . η= 2 E

(1.3)

Thus, η depends on the current value of T. As the loading is continued in the plastic range, the tangent modulus progressively decreases from its elastic value E and the contraction ratio rapidly increases from v to approach the fully plastic value of 0.5. It follows that the elastic compressibility of the material becomes negligible when the tangent modulus is reduced to the order of the current yield stress σ . For an incompressible material, the relationship between the nominal stress s and the true stress σ is easily shown to be s = σ exp ( ∓ ε), where the upper sign corresponds to simple tension and the lower sign to simple compression of the test specimen. The distinction between the behaviors in tension and compression, in relation to the engineering strain, is illustrated in Fig. 1.2.

Fig. 1.2 The stress–strain behavior of metals with respect to nominal stress, true stress, and engineering strain: (a) simple tension and (b) simple compression

The tensile test is unsuitable for obtaining the stress–strain curve up to large values of the strain, since the specimen begins to neck when the rate of work-hardening

6

1 Fundamental Principles

decreases to a critical value. At this stage, the applied load attains a maximum, and the specimen subsequently extends under decreasing load. Setting the differential of the axial load P = σ A to zero, where A is the current cross-sectional area of the specimen, we get dσ /σ = –dA/A. On the other hand, the condition of zero incremental volume change, which holds very closely at the onset necking, gives –dA/A = dl/l = dε. The condition for plastic instability in simple tension therefore becomes dσ = σ. de

(1.4)

As the extension continues beyond the point of necking, the plastic deformation remains confined in the neck, which grows rapidly under decreasing load leading to fracture across the minimum section. The true stress–strain curve cannot be continued beyond the point of necking without introducing a suitable correction factor (Section 3.3). A neck is also formed in a cylindrical specimen subjected to a uniform fluid pressure on the lateral surface, and the amount of uniform strain at the point of necking is exactly the same as that in uniaxial tension (Chakrabarty, 1972). The strain-hardening characteristic of materials covering a fairly wide range of strains is most conveniently obtained by the simple compression test, in which a solid cylindrical block is axially compressed between a pair of parallel platens. However, due to the presence of friction between the specimen and the platens, the deformation of material near the regions of contact is constrained, resulting in a barreling of the specimen as the compression proceeds. Since the compression of the cylinder then becomes nonuniform, the true stress–strain curve cannot be derived from the compression test without introducing a suitable correction factor (Section 3.4). Several methods have been proposed in the past to eliminate the effect of friction on the stress–strain curve, but none of them seems to be entirely satisfactory. On the other hand, a state of homogeneous compression is very nearly achieved by inserting thin sheets of ptfe (polytetrafluoroethylene) between the specimen and the compression platens. The compressed ptfe sheets not only act as an effective lubricant but also help to inhibit the barreling tendency by exerting a radially outward pressure on the material near the periphery. It is necessary to apply the axial load, on an incremental basis, and to replace the deformed ptfe sheets with new ones before each application of the load. The true stress and the true strain are obtained at each stage from the measurement of the applied load and the current specimen height, together with the use of the constancy of volume. Consider an annealed specimen which is loaded in simple tension past the yield point and is subsequently unloaded to zero stress so that there is a certain amount of residual strain left in the specimen. If an axial compressive load is now applied, the specimen will begin to yield under a stress that is somewhat lower than the original yield stress in tension or compression. While the yield stress in tension at the time of unloading is much greater than Y owing to strain hardening of the material, the yield stress in compression is usually found to be lower than Y. A similar lowering of the yield stress is observed if the specimen is loaded plastically in compression and then pulled in tension. This phenomenon is known as the Bauschinger effect,

1.1

The Material Response

7

which occurs whenever there is a reversal of stress in a plastically deformed element. The phenomenon is generally attributed to residual stresses in the individual crystal grains due to the presence of grain boundaries in a polycrystalline metal. In some metals, such as annealed mild steel, the load at the elastic limit suddenly drops from an upper yield point to a lower yield point, followed by an elongation of a few percent under approximately constant stress. At the upper yield point, a discrete band of deformed metal, known as Lüders band, appears at approximately 45◦ to the tensile axis at a local stress concentration. During the yield point elongation, several bands usually form at several points of the specimen and propagate to cover the entire length. At this stage, the load begins to rise with further strain and the stress– strain curve then continues in the usual manner as a result of strain hardening. The upper yield point depends on such factors as the rate of straining, eccentricity of the loading, and the rigidity of the testing machine, but its value is usually 10–20% higher than the lower yield point.

1.1.3 Empirical Stress–Strain Equations In the theoretical treatment of plasticity problems, it is generally convenient to represent the true stress–strain curve of the material by a suitable empirical equation that involves constants to be determined by curve fitting with the experimental curve. For sufficiently large strains, the simplest empirical equation frequently used in the literature is the simple power law proposed by Ludwik (1909), which is σ = Cεn ,

(1.5)

where C is a constant stress and n is n dimensionless constant, known as the strainhardening exponent, whose value is generally less than 0.5. Although (1.5) corresponds to an infinite initial slope, it does provide a reasonably good fit with the actual stress–strain curve over a fairly wide range of strains. Since dσ /dε is equal to nσ /ε according to (1.5), the true strain at the onset of necking in simple tension is ε = n in view of (1.4). A nonhardening material corresponds to n = 0 with C representing the constant yield stress of the material. When the material is assumed to be rigid/plastic having a distinct initial yield stress Y, the simple power law (1.5) needs to be suitably modified. One such modification, sometimes used in the solution of special problems, is σ = Y + Kεn , where K has the dimension of stress and n is an exponent. Although this equation predicts a nonzero initial yield stress, it does not provide a better fit with the actual stress–strain curve over the relevant range of strains. The preceding equation includes, as a special case, the linear strain-hardening law (n = 1), with K denoting the constant plastic modulus. A more successful empirical equation involving a definite yield point is the modified power law

8

1 Fundamental Principles

σ = C(m + ε)n ,

(1.6)

where C, m, and n are constants. The stress–strain curve defined by (1.6), which is due to Swift (1952), is essentially the Ludwik curve (1.5) with the σ -axis moved through a distance m in the direction of the ε-axis. The parameter m therefore represents the amount of initial prestrain with respect to the annealed state. If the same stress–strain curve is fitted by both (1.5) and (1.6), the value of n in the two cases will of course be different. It follows from (1.4) and (1.6) that the magnitude of the true strain at the point of tensile necking is equal to n – m for m ≤ n and zero for n ≥ m. Figure 1.3(a) shows the Swift curves for several values of n based on a typical value of m.

Fig. 1.3 Nature of empirical stress–strain equations. (a) Swift equation and (b) Voce equation

For certain applications, it is sometimes more convenient to employ a different type of empirical equation proposed by Voce (1948). The Voce equation, which also involves an initial yield stress of the material, may be expressed as σ = C(1 − me−nε ),

(1.7)

where C, m, and n are material constants, and e is the exponential constant. The stress–strain curve defined by (1.7) exhibits an initial yield stress equal to (1 – m)C and tends to become asymptotic to the saturation stress σ = C. The slope of the stress–strain curve varies linearly with the stress according to the relation dσ /dε – n(C – σ ). Thus, the initial state of hardening of the material is represented by m, while the rapidity of approach to the saturation stress is represented by n. The stress–strain curves defined by (1.7) for a given value of m and several values of n are displayed in Fig. 1.3(b).

1.1

The Material Response

9

The preceding stress–strain equations can be used for elastic/plastic materials, with comparable elastic and plastic strains, provided ε is replaced by the plastic component εp . Since the elastic component of the strain εe is equal to σ /E, the simple power law of type (1.5) relating the stress to the plastic strain furnishes the total strain in the form m−1 σ σ , (1.8) ε= 1+α E σ0 where m = 1/n, σ 0 is a nominal yield stress, and α is a dimensionless constant. The stress–strain curve defined by (1.8), which is due to Ramberg and Osgood (1943), bends over with an initial slope equal to E, the plastic modulus H associated with any stress a being given by E/H = m(σ /σ 0 )m-1 . It may be noted that the secant modulus of the stress–strain curve is equal to E/(l + α) at the nominal yield point σ = σ 0 according to (1.8). Figure 1.4(a) shows several curves of this type for constant values of σ 0 /E and α.

Fig. 1.4 True stress–strain curves for elastic/plastic materials: (a) Ramberg–Osgood equation and (b) modified Ludwik equation

In some cases, the elastic/plastic analysis can be considerably simplified by using a stress–strain relation that corresponds to the Ludwik curve with its initial part replaced by a chord of slope equal to E. The stress–strain law is then given by the pair of equations σ =

Eε, ε ≤ Y/E, Y(Eε/Y)n , ε ≥ Y/E.

(1.9)

10

1 Fundamental Principles

The material has a sharp yield point at σ = Y, and the slope of the stress–strain curve changes discontinuously from E to nE at the yield point, Fig. 1.4(b). The value of the tangent modulus T at any stress σ ≥ Y is equal to nE(Y/σ )m–l , where m = l/n. The discontinuity in slope at σ = Y can be eliminated, however, by modifying the stress–strain equation in the plastic range as σ =Y

1−n Eε − nY n

n , ε ≥ Y/E.

The tangent modulus according to this equation is easily shown to be T = E(Y/σ )m–l , where m = l/n as before. Since T = σ at the incipient necking of a bar under simple tension, the magnitude of the uniform true strain exceeds n by the amount (1 – n)Y/E, which is negligible for most metals. The rate of straining has a profound influence on the yield strength of metals, particularly at elevated temperatures. The mechanical response of materials to high strain rates and temperatures is generally established on the basis of uniaxial stress– strain curves obtained under constant values of the strain rate and temperature. For a given temperature, the combined effects of strain ε and strain rate ε˙ can be expressed by the empirical equation σ = Cεn ε˙ m ,

(1.10)

where C, m, and n are constants, the parameters m being known as the strain rate sensitivity which is generally less than 0.2 for most metals and alloys. The higher the value of m, the greater is the strain at the onset of tensile necking (Hart, 1967). Since most of the heat generated during a high-speed test remains in the specimen, leading to an adiabatic rise in temperature, the results for a given test must be adjusted appropriately so that they correspond to a constant temperature. In the hot working of metals, the working temperature is high enough for recovery and recrystallization to occur without significant grain growth. Since the rate of work-hardening is then exactly balanced by the rate of thermal softening, the yield stress is practically independent of the strain except for sufficiently small values of the strain. There are certain metals and alloys known as superplastic materials, which exhibit very large neck-free tensile elongations prior to failure (Backofen et al., 1964). These materials are characterized by high values of the strain rate sensitivity m, which is generally greater than 0.4. The tensile fracture in superplastic materials is caused by the evolution of cavities at grain boundaries rather than by the development of diffuse necks. Conventional superplasticity is observed at relatively low strain rates, usually ranging from 10–4 /s to 10–3 /s, although recent studies have revealed the existence of superplasticity at considerably higher strain rates in certain alloys and composite materials. The relationship between the flow stress σ and the strain rate ε˙ in superplastic materials is often expressed by the equation σ = K ε˙ m ,

1.2

Basic Laws of Plasticity

11

where m is the strain rate sensitivity and K is generally independent of the strain. The elongation to failure in a tensile specimen of superplastic material increases with increasing values of m. In fine-grained superplastic materials, an m value of about 0.5 is fairly common, and the deformation takes place mainly by grain boundary sliding. Materials which are rendered superplastic by the generation of internal stresses through thermal or pressure cycling can have a strain rate sensitivity as high as unity. The superplasticity of metals and alloys has been discussed at length in the books by Presnyakov (1976), Padmanabhan and Davies (1980), and Nieh et al. (1997).

1.2 Basic Laws of Plasticity 1.2.1 Yield Criteria of Metals The macroscopic theory of plasticity is based on certain experimental observations regarding the behavior of ductile metals. The theory rests on the assumption that the material is homogeneous and is valid only at temperatures for which thermal phenomena may be neglected. For the present purpose, it is also assumed that the material is isotropic and has identical yield stresses in tension and compression. The plasticity of metals with fully developed states of anisotropy will be discussed in Chapter 6. A further simplific`ation results from the experimental fact that the yielding of metals is unaffected by a moderate hydrostatic pressure (Bridgman, 1945; Crossland, 1954). A law governing the limit of elastic behavior, consistent with the basic assumptions, defines a possible criterion of yielding under any combination of the applied stresses. The state of stress in any material element may be represented by a point in a nine-dimensional stress space. Around the origin of the stress space, there exists a domain of elastic range representing the totality of elastic states of stress. The external boundary of the elastic domain defines a surface, known as the initial yield surface, which may be expressed in terms of the components of the true stress σ ij , as f (σij ) = constant. Since the material is initially isotropic, plastic yielding depends only on the magnitudes of the three principal stresses, and not on their directions. This amounts to the fact that the yield criterion is expressible as a function of the three basic invariants of the stress tensor. The yield function is therefore a symmetric function of the principal stresses and is also independent of the hydrostatic stress, which is defined as the mean of the three principal stresses. Plastic yielding therefore depends on the principal components of the deviatoric stress tensor, which is defined as sij = σij − σ δij ,

(1.11)

12

1 Fundamental Principles

where σ denotes the hydrostatic stress, equal to (σ 1 + σ 2 + σ 3 )/3. Since the sum of the principal deviatoric stresses is sij = 0, the principal components cannot all be independent. It follows that the yield criterion may be expressed as a function of the invariants J2 and J3 of the deviatone stress tensor, which are given by (Chakrabarty, 2006) ⎫ 1 2 1 (s1 + s22 + s23 ) = sij sij ,⎪ ⎬ 2 2 1 1 ⎪ ⎭ J3 = s1 s2 s3 = (s31 + s32 + s33 ) = sij sjk ski . 3 3

J2 = − (s1 s2 + s2 s3 + s2 s1 ) =

(1.12).

The absence of Bauschinger effect in the initial state implies that yielding is unaffected by the reversal of the sign of the stress components. Since J3 changes sign with the stresses, an even function of this invariant should appear in the yield criterion. In a three-dimensional principal stress space, the yield surface is represented by a right cylinder whose axis is equally inclined to the three axes of reference, Fig. 1.5. The generator of the cylinder is therefore perpendicular to the plane σ 1 + σ 2 + σ 3 = 0, known as the deviatoric plane. Since σ 1 = σ 2 = σ 3 , along the geometrical axis of the cylinder, it represents purely hydrostatic states of stress. Points on the generator therefore represent stress states with varying hydrostatic part, which does not have any influence on the yielding. The yield surface is intersected by the deviatoric plane in a closed curve, known as the yield locus, which is assumed to be necessarily convex. Fig. 1.5 Geometrical representation of yield criteria in the principal stress space

Due to the assumed isotropy and the absence of the Bauschinger effect, the yield locus must possess a six-fold symmetry with respect to the projected stress axes and the lines perpendicular to them, as indicated in Fig. 1.6(b). In an experimental

1.2

Basic Laws of Plasticity

13

Fig. 1.6 Deviatoric yield locus. (a) Tresca hexagon and von Mises circle and (b) general shape of the locus

determination of the initial yield locus, it is therefore only necessary to apply stress systems covering a typical 30◦ segment of the yield locus. This may be achieved by introducing the Lode (1926) parameter μ, which is defined as μ=

√ 2σ2 − σ3 − σ1 = − 3 tan θ , σ3 − σ1

σ1 > σ2 > σ3 ,

(1.13)

where θ is the counterclockwise angle made by the deviatoric stress vector with the direction representing pure shear. To obtain the shape of the yield locus, it is only necessary to apply stress systems varying between pure shear (μ = ±, θ = 0) and uniaxial tension or compression (μ = ±1, θ = ±π/6). The yield locus is called regular when it has a unique tangent at each point and singular when it has sharp corners. The simplest yield criterion expressed in terms of the invariants of the deviatoric stress tensor is J2 = k2 , suggested by von Mises (1913), where k is a constant. The yield function does not therefore involve J3 at all. In terms of the stress component referred to an arbitrary set of rectangular axes, the von Mises yield criterion may be written as ⎫ 2 2 2 ⎪ sij sij = s2x + s2y + s2z + 2(τxy + τyz + τzx ) = 2 k2 ⎪ ⎬ or (1.14) ⎪ ⎭ 2 2 2 2 2 2 2⎪ (σ − σ ) + (σ − σ ) + (σ − σ ) + 6(τ + τ + τ ) = 6 k . x

y

y

z

z

x

xy

yz

zx

The second expression in (1.14) follows from the first on subtracting the identically zero term (sx + sy + sz )2 /3 and noting the fact that sx – sy = σ x – σ y , etc. The constant k is actually the yield stress in simple or pure shear, as may be seen

14

1 Fundamental Principles

by setting σ x = σ and σ y = –σ as the only nonzero √ stress components. According to (1.14), the uniaxial yield stress Y is equal to 3 k, which is obtained by considering σ x = Y as the only nonzero stress. The von Mises yield surface is evidently a right circular cylinder having its geometrical axis perpendicular to the deviatoric plane. The principal deviatoric stresses according to the von Mises criterion may be expressed in terms of the deviatoric angle θ as s1 =

π

π 2 2 2 Y cos + θ , s2 = Y sin θ , s3 = − Y cos −θ . 3 6 3 3 6

(1.15)

In the case of plane stress, the actual principal stresses σ 1 and σ 2 may be expressed in terms of θ using the fact that the sum of these stresses is equal to –3s3 which ensures that σ 3 is identically zero. On the basis of a series of experiments involving the extrusion of metals through dies of various shapes, Tresca (1864) concluded that yielding occurred when the magnitude of the greatest shear stress attained a certain critical value. In terms of the principal stresses, the Tresca criterion may be written as σ1 − σ3 = 2 k,

σ1 ≥ σ2 ≥ σ3 ,

(1.16)

where k is the yield stress in pure shear, the uniaxial yield stress being Y = 2k according to this criterion. All possible values of the principal stresses are taken into account when the Tresca criterion is expressed by a single equation in terms of the invariants J2 and J3 , but the result is too complicated to have any practical usefulness. For a given uniaxial yield stress Y, the Tresca yield surface is a regular hexagonal cylinder inscribed within the von Mises cylinder. The Tresca yield surface is not strictly convex, but each face of the surface may be regarded as the limit of a convex surface of vanishingly small curvature. The deviatoric yield loci for the Tresca and von Mises criteria are shown in Fig. 1.6(a). The maximum difference between the two criteria occurs in √ pure shear, for which the von Mises criterion predicts a yield stress which is 2/ 3 times that given by the Tresca criterion. Experiments have shown that for most metals the test points fall closer to the von Mises yield locus than to the Tresca locus, as indicated in Fig. 1.7. If the latter is adopted for simplicity, the overall accuracy can be improved by replacing 2k in (1.16) by mY, where m is an empirical constant lying between 1 √ and 2/ 3.

1.2.2 Plastic Flow Rules The following discussion is restricted to an ideally plastic material having a definite yield point and a constant yield stress. The influence of strain hardening will be discussed in the next section. The yield locus for the idealized material retains its size and shape so that the material remains isotropic and free from the Bauschinger effect. Each increment of strain in the plastic range is the sum of an elastic part

1.2

Basic Laws of Plasticity

15

Fig. 1.7 Experimental verification of the yield criterion for commercially pure aluminum (due to Lianis and Ford, 1957)

which may be recovered on unloading, and a plastic part that remains unchanged on unloading. The elastic part of the strain increment is given by the generalized Hooke’s law, while the plastic part is governed by what is known as the flow rule. The consideration of the plastic deformation of a polycrystalline metal in relation to that of the individual crystals leads to the existence of a plastic potential that is identical to the yield function (Bishop and Hill, 1951). The plastic strain increment, regarded as a vector in a nine-dimensional space, is therefore directed along the outward normal to the yield surface at the considered stress point. Denoting the unit vector along the exterior normal by nij , the associated flow rule for a nonhardening material with a regular yield surface may be written as p

dεij = nij dλ, nij dσij = 0,

(1.17)

where dλ is a positive scalar representing the magnitude of the plastic strain increment vector. The condition nij dσ ij = 0 implies that the stress point must remain on the yield surface during an increment of plastic strain. When nij dσ ij < 0, indicating unloading from the plastic state, the plastic strain increment is identically zero. Since the components of nij are proportional to ∂f/∂σ ij , where f defines the yield function, of J2 and J3 , the principal axes of nij coincide with those of σ ij . The flow rule therefore implies that the principal axes of stress and plastic strain increment p coincide for an isotropic solid. The plastic incompressibility condition dεii = 0 is identically satisfied, while the symmetric tensor nij satisfies the relations nij nij = 1,

nii = 0.

For a singular yield criterion, (1.17) holds for all regular points of the yield surface. At a singular point of the yield surface, the normal is not uniquely defined, and the plastic strain increment vector may lie anywhere between the normals to the

16

1 Fundamental Principles

faces meeting at the considered edge. When nij dσ ij < 0, the element unloads from the plastic state, and the plastic strain increment vanishes identically. The relation p

dσij dεij = 0, which holds for both loading and unloading for a regular yield surface, may be assumed to hold even for a singular point of a yield surface when the material is nonhardening (Drucker, 1951). When the yield criterion is that of von Mises, f (σij ) = 12 sij sij , which gives √ ∂f /∂σij = sij . The deviatoric stress vector √ is then of magnitude 2k, and √ the unit normal to the yield surface is nij = sij / 2 k. Replacing the quantity dλ/ 2 k by dλ in (1.17), the associated flow rule may be expressed as p

dεij = sij dλ, sij dsij = 0.

(1.18)

The ratios of the components of the plastic strain increment are therefore identical to those of the deviatoric stress. This relationship was proposed independently by Levy (1870) and von Mises (1913), both of whom used the total strain increment instead of the plastic strain increment alone. It therefore applies to a hypothetical rigid/plastic material whose elastic modulus is infinitely large. The extension of the flow rule to allow for the elastic part of the strain is due to Prandtl (1924) in the case of plane strain and to Reuss (1930) in the case of complete generality. The increment of plastic work per unit volume according to (1.18) and (1.14) is p

dW p = σij dεij = sij sij dλ = 2 k2 dλ in view of (1.11). Since plastic work must be positive, dλ is seen to be necessarily positive for plastic flow. Using the generalized form of Hooke’s law, the elastic strain increment may be written as dεije =

dsij 1 − 2v + dσkk δij , 2G 3E

(1.19)

where G is the shear modulus and v is Poisson’s ratio. When a stress increment satisfying sij dsij = 0 is prescribed, the elastic strain increment is known, but the plastic strain increment cannot be found from the flow rule alone. The flow rule associated with the Tresca criterion furnishes ratios of the components of the plastic strain increment depending on the particular side or corner of the deviatoric yield hexagon. If we consider the side AB of the hexagon, Fig. 1.6(a), the yield criterion is given by (1.16) and the normality rule furnishes p

p

dε1 = −dε3 > 0,

p

dε2 = 0.

1.2

Basic Laws of Plasticity

17

When the stress point is at the corner B of the yield hexagon, defining the equal biaxial state σ 1 = σ 2 , the plastic strain increment vector can lie between the normals for the sides meeting at B, giving p

dε1 > 0,

p

dε2 > 0,

p

p

p

dε3 = −(dε1 + dε2 ).

Similar relations hold for the other sides and corners of the yield hexagon. In each case, the rate of plastic work per unit volume is 2k times the magnitude of the numerically largest principal plastic strain rate. An interesting feature of Tresca’s associated flow rule is that it can be written down in the integrated form whenever the stress point remains on a side, remains at a corner, or moves from a side to a corner, but not when it moves from a corner back to a side. Let ψ denote the counterclockwise angle made by the plastic strain increment vector with the direction representing pure shear in the deviatoric plane. Evidently, ψ depends on the nature of the plastic potential, which is a closed curve similar to the yield locus. For an experimental verification of the flow rule, it is convenient to introduce the Lode parameter v (not to be confused with Poisson’s ratio), which is defined as p

ν=

p

√ = − 3 tan ψ,

p

2dε2 − dε3 − dε1 p dε3

p − dε1

p

p

p

dε1 > dε2 > dε3 .

(1.20)

For a regular yield function and plastic potential, ν = 0 when μ = 0, and ν = – 1 when μ = – 1. In the case of the von Mises yield criterion and the associated Prandtl–Reuss flow rule, μ = ν or θ = ψ for all plastic states. Tresca’s yield criterion and its associated flow rule, on the other hand, correspond to ν = 0 for 0 ≥ μ ≥ –1, and 0 ≥ ν ≥ – 1 for μ = –1. The (μ, ν) relations corresponding to the Tresca and von Mises theories are shown in Fig. 1.8. The experimental results of Hundy and Green (1954), included in the figure, clearly support the Prandtl–Reuss rule for the plastic flow of isotropic materials. p Suppose that a plastic strain increment dεij is associated with σ ij stress satisfying the yield criterion, while σij∗ is any other plastic state of stress, so that f (σij∗ ) = p f (σij ) = constant. The work done by σij∗ on the given plastic strain increment dεij has a stationary value for varying σ ij , when ∂ ∗ p σij dεij − f (σij∗ )dλ = 0, ∗ ∂σij where dλ is the Lagrange multiplier (Hill, 1950a). Carrying out the partial differentiation, we have p

dεij = p

∂ ∗ f (σij ) dλ. ∂σij∗

Since dεij is associated with σ ij according to the normality rule, the above equation is satisfied when σij∗ equals σ ij apart from a hydrostatic stress. The rate of plastic

18

1 Fundamental Principles

Fig. 1.8 Experimental verification of the (μ, v) relation due to Hundy and Green (+) and due to Lianis and Ford (o)

work is then a maximum in view of the convexity of the yield function. The maximum work theorem, which is due to von Mises (1928), may therefore be stated as

p σij − σij∗ dεij ≥ 0.

(1.21)

From the geometrical point of view, the result is evident from the fact that the vector representing the stress difference forms a chord of the yield surface and consequently makes an acute angle with the exterior normal defined at the actual stress point. The identity of the yield function and the plastic potential has a special significance in the mathematical theory of plasticity. Further details regarding the plastic stress–strain relations with and without strain-hardening of the material have been presented by Chen and Han (1988).

1.2.3 Limit Theorems In an elastic/plastic body subjected to a set of external forces, yielding begins in the most critically stressed element when the load attains a critical value. Under increasing load, a plastic zone continues to spread while the deformation is restricted to the elastic order of magnitude due to the constraint of the nonplastic material. When the plastic region expands to a sufficient extent, the constraint becomes locally ineffective and large plastic strains become possible. For a material whose rate of hardening

1.2

Basic Laws of Plasticity

19

is of the order of the yield stress, only a slight increase in load can produce an overall distortion of appreciable magnitude. If the material is nonhardening, and the change in geometry is disregarded, the load approaches an asymptotic value which is generally known as the yield point load. The basic theorems for the approximate estimation of this load have been obtained by Hill (1951) using a rigid/plastic model and by Drucker et al. (1952) using an elastic/plastic model. The elastic/plastic asymptotic load, frequently referred to as the collapse load in the context of structural analysis, is very closely attained while the elastic and plastic strains are still comparable in magnitudes. During the collapse, the deformation may therefore be assumed to occur under a constant load while changes in geometry are still negligible. Let σ˙ ij and vj denote the stress rate and particle velocity, respectively, under a distribution of boundary traction rate T˙ j at the incipient collapse. If the elastic and plastic components of the associated strain rate are denoted by ε˙ ije and p ε˙ ij , respectively, then by the rate form of the principle of virtual work we have

T˙ j vj dS =

σ˙ ij ε˙ ij dV =

p σ˙ ij ε˙ ije + ε˙ ij dV,

(1.22)

where the integrals are taken over the entire surface enclosing a volume V of the considered body. The integral on the left-hand side vanishes at the instant of colp lapse, since T˙ j = 0, while the scalar product σ˙ ij ε˙ ij vanishes in an ideally plastic material in view of (1.17). It follows therefore from (1.22) that σ˙ ij ε˙ ije = 0 at the incipient collapse, which indicates that σ˙ ij and ε˙ ije individually vanish, in view of the elastic stress–strain relation. A stress field is regarded as statically admissible if it satisfies the equilibrium equations and the stress boundary conditions without violating the yield criterion. Let σij and ε˙ ij denote the actual stress and strain rate in the considered body, and σij∗ any other statically admissible state of stress. If Tj and Tj∗ denote the surface tractions corresponding to σij and σij∗ , respectively, then by the virtual work principle, ∗ ∗ k − τ ∗ [u] dSD ≥ 0, σij − σij ε˙ ij dV + Tj − Tj vj dS = where τ ∗ ≤ k is the magnitude of the shear component of σ ij along a surface SD that actually involves a tangential velocity discontinuity of magnitude [u]. The inequality follows from (1.21) and the fact that the strain rate is purely plastic during the collapse. The assumed stress field may involve stress discontinuities across certain internal surfaces, which are limits of thin elastic regions of rapid but continuous variations of the stress. Since Tj = Tj∗ over the part SF of the surface where the traction is prescribed, the above inequality becomes

Tj vj dSv ≥

Tj∗ vj dSv ,

(1.23)

20

1 Fundamental Principles

where Sv denotes the part of the surface over which the velocity is prescribed. The above inequality (1.23) constitutes the lower bound theorem, which states that the rate of work done by the actual surface tractions on Sv is greater than or equal to that done by the surface tractions associated with any statically admissible stress field. The theorem provides a lower bound on the load itself at the incipient collapse when the prescribed velocity is uniform over this part of the boundary. A velocity field is considered as kinematically admissible if it satisfies the plastic p incompressibility condition ε˙ ii = 0 and the velocity boundary conditions. Let σ ij and vj denote the actual stress and velocity, respectively, in a deforming body, and v∗j any other kinematically admissible velocity producing a strain rate ε˙ ij∗ . Since the rate of deformation is purely plastic during the collapse, the associated work rate σij∗ εij∗ is uniquely defined by the flow rule. By the virtual work principle, we have

Tj v∗j dS =

σij ε˙ ij∗ dV +

∗ τ u∗ dSD ,

where τ is the magnitude of the actual shear stress and [u∗] is the magnitude of ∗ . Since the tangential discontinuity in the virtual velocity along a certain surface SD τ ≤ k, and σij ε˙ ij∗ ≤ σij ε˙ ij∗ in view of (1.21), the elastic strain rate being zero, the preceding expression furnishes

Tj vj dSv ≤

σij ε˙ ij dV +

k u∗ dSD −

Tj v∗j dSF ,

where use has been made of the fact that v∗j = vj on Sv . This result constitutes the upper bound theorem of limit analysis. When the last term of (1.24) is zero, the theorem states that the rate of work done by the actual surface tractions on Sv is less than or equal to the rate of dissipation of internal energy in any kinematically admissible velocity field. When the prescribed velocity is uniform on Su , the theorem provides an upper bound on the load itself at the instant of collapse. In a rigid/plastic body, no deformation can occur before the load reaches the yield point value. Over the range of load varying between the elastic limit and the yield point, the body remains entirely rigid, even though partially plastic. Under given surface tractions over a part SF of the boundary, and given velocities over the remainder Sv , the state of stress at the yield point is uniquely defined in the region where deformation is assumed to occur. On the other hand, the mode of deformation at the incipient collapse is not necessarily unique for an ideally plastic material. When positional changes are disregarded, the physically possible mode compatible with the rate of hardening can be singled out by specifying the traction rate on SF as an additional requirement (Hill, 1956). If geometry changes are duly taken into account, the deformation mode is found to be unique so long as the rate of workhardening exceeds a certain critical value (Hill, 1957). The nature of nonuniqueness

1.3

Strain-Hardening Plasticity

21

associated with an ideally plastic body has been illustrated with an example by Hodge et al. (1986).

1.3 Strain-Hardening Plasticity 1.3.1 Isotropic Hardening The most widely used hypothesis for strain hardening assumes the yield locus to increase in size during continued plastic deformation without change in shape. The yield locus is therefore uniquely defined by the final plastic state of stress regardless of the actual strain path (Hill, 1950a). According to this postulate, the material remains isotropic throughout the deformation, and the Bauschinger effect continues to be absent. The state of hardening at any stage is therefore specified by the current uniaxial yield stress denoted by σ¯ . When √ the yield criterion is that of von Mises, the current radius of the yield surface is 2/3 times σ¯ , and we have 1/2 3 σ¯ = sij sij 2 1/2

2 2 1 2 2 2 =√ + τyz + τzx . σx − σy + σy − σz + (σz − σx )2 + 6 τxy 2 (1.25) The quantity σ¯ is known as the equivalent stress or effective stress, which increases with increasing plastic strain. To complete the hardening rule, it is necessary to relate σ¯ to an appropriate measure of the plastic deformation. As a first hypothesis, it is natural to suppose that σ¯ is a function of the total plastic work per unit volume expended in a given element. The work-hardening hypothesis may be therefore stated mathematically as σ¯ = φ

p σij dεij ,

(1.26)

where the integral is taken along the strain path. Thus, no hardening is produced by the hydrostatic part of the stress which causes only an elastic change in volume. The function φ can be determined from the true stress–strain curve in uniaxial tension, where σ¯ is exactly equal to the applied tensile stress σ , and the incremental plastic work per unit volume is σ times the longitudinal plastic strain increment equal to dε – dσ /E. The argument of the function φ is simply the area under the curve for σ plotted against the quantity ln(l/l0 ) – σ /E, up to the ordinate σ , where l/l0 is the length ratio at any stage of the extension. A second hypothesis, frequently used in the literature, assumes σ¯ to be a function of a suitable measure of the total plastic strain during the deformation. In analogy ρ to the expression for σ¯ , we introduce a positive scalar parameter dε , known as the equivalent or effective plastic strain increment, defined as

22

1 Fundamental Principles

2 p p 1/2 dε = dεij dεij 3 2 2

2 1/2 2 p 2 p 2 p 2 p = + 2 dγyzp + 2 dγzxp . dεx + dεy + dεz + 2 dγxy 3 (1.27) p

p

The above definition implies that in the case of a uniaxial tension, dε is equal to the longitudinal plastic strain increment, provided the yield function is regular. The strain-hardening hypothesis may now be stated mathematically as σ¯ = F

dε

p

,

(1.28)

where the integral is taken along the strain path of a given element. Thus, the amount of hardening depends on the sum total of all the incremental plastic strains and not merely on the difference between the initial and the final shapes of the element. Both (1.26) and (1.28) imply that the longitudinal tensile stress is the same function of ln(l/l0 ) in uniaxial tension as the compressive stress is of ln(h0 /h) in simple compression, where h0 /h is the associated height ratio of the specimen. For a work-hardening Prandtl–Reuss material, the quantity dλ appearing in the flow rule (1.18) can be directly related to the equivalent stress and plastic strain p increment. Since sij = 0, it follows from (1.18), (1.27), and (1.25) that dε = (2σ¯ /3) dλ, and consequently, p

σij dεij = sij sij dλ =

2 2 p σ¯ dλ = σ¯ dε , 3

(1.29)

indicating that in this case the two hypotheses (1.26) and (1.28) are completely equivalent. Inserting the value of dλ from (1.29), the Prandtl–Reuss flow rule may be written as p

p

dεij =

3dε 3dσ¯ sij = sij, 2σ¯ 2H σ¯

(1.30)

p

where H = dσ¯ /dε , representing the current rate of work hardening of the material. Another important result for a Prandtl–Reuss material, which follows from (1.18) and (1.25), is p

σij dεij = sij dsij dλ =

2 p σ¯ dσ¯ dλ = dσ¯ dε , 3

(1.31)

where dσ¯ must be positive for plastic flow. The side of (1.31) is also right-hand p p equal to H(dε )2 , where H is a given function of dε . Adding the elastic and plastic strain increments given by (1.19) and (1.30), we obtain the complete Prandtl–Reuss strain–strain relation in the incremental form.

1.3

Strain-Hardening Plasticity

23

When yielding occurs according to the Tresca criterion (1.16), it is necessary to replace 2k by the current uniaxial yield stress σ¯ . According to Tresca’s associated flow rule, the increment of plastic work per unit volume is σ¯ times the magnitude of the numerically largest principal plastic strain increment denoted by dερ . If the work-hardening hypothesis is adopted, it follows that σ¯ = F

p dε ,

where the integral is taken along the strain path. When the stress point remains on a side, remains at a corner, or moves from a side to a corner, and the principal axes of stress and strain increments do not rotate with respect to the element, the argument of the function F is equal to the magnitude of the numerically greatest principal plastic strain in the element.

1.3.2 Plastic Flow with Hardening For continued plastic flow of a work-hardening material, the stress increment vector must lie outside the current yield locus, so that dσ¯ > 0. When the yield locus is regular, having a unique normal at each point, the plastic strain increment may be written as dεij = h−1 (nkl dσkl ) nij , nkl dσkl ≥ 0, p

(1.32)

where nij is the unit normal to the yield surface in a nine-dimensional stress space, and h (equal to 2H/3) is a parameter representing the rate of hardening. The equality in (1.32) represents neutral loading since it implies that the stress point remains on the same yield locus. When nkl dσ kl < 0, the element unloads and no incremental plastic strain is involved. The scalar products of (1.32) with dσ ij and dεij in turn furnish the result 2 p p p dσij dεij = h−1 dσij nij = hdεij dεij ≥ 0.

(1.33)

The equality holds not only for neutral loading but also for unloading from the plastic state. Since (1.21) holds whether the material work-hardens or not, the basic inequalities for a work-hardening element that is currently in a plastic state may be stated as

p p (1.34) σij − σij∗ dεij ≥ 0, dσij dεij ≥ 0. In both cases, the equality holds for neutral loading and unloading. If inequalities (1.34) are taken as the basic postulates for the plastic flow of work-hardening materials, the normality rule and the convexity of the yield surface can be easily deduced, as has been shown by Drucker (1951).

24

1 Fundamental Principles

Let P1 and P2 be two arbitrary stress points located on the two sides of a singular point P, as shown in Fig. 1.9(a). According to the first inequality of (1.34), each of the vectors P1 P and P2 P must make an acute angle with the plastic strain increment vector at P. This condition is evidently satisfied if the direction of the plastic strain increment lies between the normals PN1 and PN2 corresponding to the meeting surfaces. A further restriction is imposed by the second inequality of (1.34), which states that the plastic strain increment vector must make an acute angle with the p stress increment vector. In Fig. 1.9(b), the vector dεij may therefore lie anywhere between the normals PN1 and PN2 so long as the vector dσ ij lies within the angle T1 PT2 formed by the tangents at P. If the loading condition is such that dσ ij lies p outside this angle, the direction of dεij coincides with the normal that makes an acute angle with dσ ij . The flow rule at a singular point has been discussed by Koiter (1953), Bland (1957), and Naghdi (1960).

Fig. 1.9 Geometrical representation of the plastic stress–strain relation at a singular point of the yield surface

In the stress–strain relations considered so far, the strain increment dε ij must be interpreted as ε˙ ij dt, where ε˙ ij is the true strain rate and dt is an increment of time scale. The stress increment dσ ij is similarly given by a suitable measure of the stress rate, which must be defined in such a way that it vanishes in the event of a rigidbody rotation of the considered element. The most appropriate stress rate in the ◦ theory of plasticity is the Jaumann stress rate σ ij , which is related to the material rate of change σ˙ ij of the true stress by the equation ◦

σ = σ˙ ij − σik ωjk − σjk ωik ,

(1.35)

1.3

Strain-Hardening Plasticity

25

where ωij is the rate of rotation of the considered element. The tensors ε˙ ij and ωij are the symmetric and antisymmetric parts, respectively, of the velocity gradient tensor ∂vi /∂xj and are given by ε˙ ij =

1 2

∂vj ∂vi + ∂xj ∂xi

, ωij =

1 2

∂vj ∂vi − ∂xj ∂xi

.

(1.36)

Equation (1.35), which is originally due to Jaumann (1911), has been rederived by several investigators including Hill (1958) and Prager (1961a). The Jaumann ◦ stress rate σ i j is the rate of change of the true stress σ ij referred to a set of axes ◦ which participate in the instantaneous rotation of the element. Both σ ij; and σ i j have the same scalar product with any tensor whose principal axes coincide with those of σ ij . For an isotropic material, therefore, the material rate of change of the yield function f(σ ij ) is ∂f ◦ ∂f σ ij . σ˙ ij = f˙ = ∂σij ∂σij Since ∂f /∂σij is in the direction of the unit normal nij , the Jaumann stress rate satisfies the condition that the yield function has a stationary value during the neutral loading of a plastic element. No other definition of the objective stress rate, vanishing in the event of a rigid-body rotation of the element, satisfies this essential requirement. The constitutive equation for an elastic/plastic solid relates the strain rate to the stress rate, considered in the Jaumann sense. Combining the elastic and the plastic parts of the strain rate, the incremental constitutive equation for an isotropic workhardening material may be written as dεij =

3 1 v δij dσkk + nij nkl dσkl dσij − 2G 1+v 2H

(1.37)

for nkl dσ kl ≥ 0 in an element currently stressed to the yield point, the yield surface being considered as regular. The scalar product of (1.37) with nij furnishes the result

nij dσij =

2GH nij dεij . 3G + H

It follows that nij dσ ij ≥ 0 for nij dσ ij ≥ 0 when H > 0. Equation (1.37) therefore has the unique inverse

v 3G δij dεkk − nij nkl dεkl dσij = 2G dεij + 1 − 2v 3G + H

(1.38)

whenever nkl dσ kl ≥ 0. Equation (1.38) holds equally well for a nonhardening material (H = 0), but the magnitude of the last term of (1.37) becomes indeterminate

26

1 Fundamental Principles

when H = 0. When nkl dσ kl < 0, or nkl dεkl < 0, implying unloading of the element from the plastic state, the last terms of (1.37) and (1.38) must be omitted. Some computational aspects of the work-hardening Prandtl–Reuss theory of plasticity have been examined by Mukherjee and Liu (2003). An interesting strain space formulation of the constitutive relations for elastic/plastic solids has been developed by Casey and Naghdi (1981). A generalized constitutive theory for finite elastic/plastic deformation of solids has been developed by Lee (1969) and Mandel (1972) and further discussed by Lubiner (1990). A critical review of the subject of finite plasticity has been made by Naghdi (1990). A simplified stress–strain relation, proposed by Hencky (1924), assumes each component of the total plastic strain in any element to be proportional to the corresponding deviatoric stress. Although physically unrealistic, the Hencky theory does provide useful approximations when the loading is continuous and the stress path does not deviate appreciably from a radial path. For a work-hardening material, when the yield surface develops a corner at the loading point, the Hencky theory satisfies Drucker’s postulates (1.34) over a certain range of nonproportional loading paths, as has been shown by Budiansky (1959) and Kliushnikov (1959). When the material is rigid/plastic, and strain hardens according to the Ludwik power law (1.5), the Hencky theory coincides with the von Mises theory even for nonproportional loading during an infinitesimal deformation of the element, as has been shown by Ilyushin (1946) and Kachanov (1971).

1.3.3 Kinematic Hardening The simplest hardening rule that predicts the development of anisotropy and the Bauschinger effect, exhibited by real metals, is the kinematic hardening rule proposed by Prager (1956b) and Ishlinsky (1954). It is postulated that the hardening is produced by a pure translation of the yield surface in the stress space without any change in size or shape. If the initial yield surface is represented by f(σ ij ) = k2 in a nine-dimensional space, where k is a constant, the subsequent yield surfaces may be represented by the equation f σij − αij = k2 ,

(1.39)

where α ij is a tensor specifying the total translation of the center of the yield surface at a generic stage, as indicated in Fig. 1.10(b). To complete the hardening rule, it is further assumed that during an increment of plastic strain, the yield surface moves in the direction of the exterior normal to the yield surface at the considered stress point. Following Shield and Ziegler (1958), we therefore write p

dαij = cdεij ,

(1.40)

where c is a scalar parameter equal to two-thirds of the current slope of the uniaxial stress–plastic strain curve of the material. When c is a constant, (1.40) reduces to the

1.3

Strain-Hardening Plasticity

27

Fig. 1.10 Geometrical representation of the hardening rule considered in the stress space. (a) Isotropic hardening and (b) kinematic hardening

p

integrated form αij = cεij , the deformation being assumed small. In general, c may be regarded as a function of the equivalent plastic strain, the increment of which is defined by (1.27) in terms of the components of the plastic strain increment tensor. Since the material becomes anisotropic during the hardening process, the principal axes of stress and plastic strain increment do not coincide, unless the principal axes remain fixed in the element as it deforms. The loading condition df = 0, which ensures that the stress point remains on the yield surface, furnishes

∂f p p = 0 = dσij − cdεij dεij dσij − dαij ∂σij

(1.41)

in view of (1.39) and (1.40) and the normality rule for the plastic strain increment vector. If the initial yield surface is that of von Mises, f σij = 12 sij sj , and the yield criterion at any stage of the deformation becomes sij − αij sij − αij = 2 k2 ,

(1.42)

where k is the initial yield stress in pure shear. The associated flow rule furnishes the plastic strain increment as p

dεij =

∂f dλ = sij − αij dλ, ∂σij

(1.43)

where dλ is a positive scalar. Combining (1.43) with (1.41), and using (1.42), it is easily shown that dλ =

1 (skl − αkl ) dσkl . 2ck2

(1.44)

28

1 Fundamental Principles

The plastic strain increment for the kinematic hardening is completely defined by (1.43) and (1.44). A modified form of Prager’s hardening rule has been proposed by Ziegler (1959), while other types of kinematic hardening have been examined by Baltov and Sawczuk (1965), Phillips and Weng (1975), and Jiang (1993). When the deformation is large, the stress increment entering into the constitutive equation must be carefully defined, and this question has been examined by Lee et al. (1983) and Naghdi (1990).

1.3.4 Combined or Mixed Hardening The concept of kinematic hardening has been extended to include an expansion of the yield surface along with a translation by Hodge (1957), Kadashevich and Novozhilov (1959), and Mröz et al. (1976). Equation (1.39) is then modified by replacing its right-hand side with a function of the total equivalent plastic strain, whose increment is given by (1.27). Assuming the von Mises yield criterion for the initial state, the combined hardening rule may be stated as 2 sij − αij sij − αij = σ¯ 2 , 3

(1.45)

where dα ij is still given by (1.40). The right-hand side of (1.45) is the square of the current radius of the displaced yield cylinder. The associated plastic strain increment p p dεij is given by (1.43) with dλ = 3dε /2σ¯ , as may be seen by substituting (1.43) into (1.27) and using (1.45). The plastic strain increment therefore becomes p

3dε dσ¯ p = sij − αij , dεij = sij − αij 2σ¯ hσ¯

(1.46)

where h is a measure of the isotropic part of the rate of hardening, the anisotropic part being represented by the parameter c. The differentiation of the yield criterion (1.45) gives 2 p σ¯ dσ¯ = (skl − αkl ) dσkl − cdεkl 3 in view of (1.40) and the fact that dskk = 0. Substituting from (1.46) and using (1.45), we obtain the relation

c 3 1+ dσ¯ = (skl − αkl ) dσkl . h 2σ¯

(1.47)

The flow rule corresponding to the combined hardening process is completely defined by (1.46) and (1.47), the loading condition being specified by dσ¯ > 0 for an element that is currently plastic. Following the early experimental work due to Naghdi et al. (1958), the distortion of the yield surface under continued plastic deformation has been subsequently examined by several investigators.

1.3

Strain-Hardening Plasticity

29

The resultant strain increment in any element deforming under the combined hardening rule from an initially isotropic state may be written down on the assumption that the material continues to remain elastically isotropic. Then the elastic strain increment, which is given by the generalized Hooke’s law, may be written as d εije

1 1 1 − 2v = dsij + dσkk δij , 2G 3 1+v

(1.48)

where G is the shear modulus and v is Poisson’s ratio. Taking the scalar product of the above equation with the tensor sij – α ij and using (1.47), we have sij − αij d εije =

c+h p σ¯ d ε . 2G

On the other hand, the scalar product of (1.46) with the same tensor sij – αij gives p p sij − αij d εij = σ¯ d ε in view of (1.45). The last two equations are added together to obtain the relation H p sij − αij d εij = 1 + σ¯ d ε , 3G

(1.49)

where H denotes the plastic modulus corresponding to the current state of stress and is defined as H=

3 (c + h) . 2

Thus, H is the slope of the uniaxial stress–plastic strain curve corresponding to a longitudinal plastic strain equal to the total equivalent plastic strain suffered by the given element. Further results related to the mixed hardening rule have been given by Mröz et al. (1976), Rees (1981), and Skrzypek and Hetnarski (1993). A micromechanical model for the development of texture with plastic deformation in polycrystalline metals has been considered by Dafalias (1993). Consider the special case of proportional loading in which the stress path is a radial line in the deviatoric plane. Let the state of stress at the initial yielding be denoted by s0ij , satisfying the yield criterion s0ij s0ij = 2Y 2 /3. Since the plastic strain increment tensor in this case may be written as

p d εij

3d ε =± 2Y

p

s0ij ,

where the upper sign corresponds to continued loading and the lower sign to any subsequent reversed loading in the plastic range, the deviatoric stress increment is

30

1 Fundamental Principles

dsij =

p cd εij

±

dσ¯ Y

s0ij

3H =± 2Y

s0ij d ε

p

(1.50)

by the simple geometry of the loading path and the assumption of simultaneous translation and expansion of the yield surface in the stress space. During the unloading of an element from the plastic state, followed by a reversal of the load, the components of the deviatoric stress steadily decrease in magnitude. Plastic yielding would occur under the reversed loading when the vector representing the deviatoric stress changes by a magnitude equal to the current diameter of the yield surface.

1.4 Cyclic Loading of Structures 1.4.1 Cyclic Stress–Strain Curves The investigations of low-cycle fatigue in mechanical and structural components have resulted in the development of considerable interest in the study of plastic behavior of materials under cyclic loading. In uniaxial states of stress involving symmetric cycles of stress or strain, an annealed material usually undergoes cyclic hardening, and the hysteresis loop approaches a stable limit as shown in Fig. 1.11(a). If the material is sufficiently cold-worked in the initial state, cyclic softening would occur and the hysteresis loop would again stabilize to a limiting state. Based on a family of stable hysteresis loops, obtained by the cyclic loading of a material with different constant values of the strain amplitude, we can derive a cyclic strainhardening curve, such as that shown in Fig. 1.11(b), which may be compared with the standard strain-hardening curve for the same material (Landgraf, 1970). If the cyclic loading is continued in the plastic range, the stable hysteresis loops are repeated and failure eventually occurs due to low-cycle fatigue. Under certain stress cycles with materials exhibiting cyclic softening, the plastic strain may continue to grow in a unidirectional sense, causing failure by the phenomenon of ratcheting. Let us suppose that a specimen that is first loaded in tension to a stress equal to σ is subsequently unloaded from the plastic state and then reloaded in compression. It follows from above that yielding would again occur when the magnitude of the applied compressive stress becomes 2σ¯ − σ , where σ¯ depends on the magnitude of the previous plastic strain. If we assume the relations h=

2 βH, 3

c=

2 (1 − β) H, 3

where β is a constant less than unity, then dσ¯ = βdσ , which gives σ¯ − Y = β (σ − Y). The initial yield stress σ ’ in compression during the reversed loading is therefore given by σ − Y = (2β − 1) (σ − Y) .

1.4

Cyclic Loading of Structures

31

Fig. 1.11 Cyclic loading curves in the plastic: (a) constant strain cycles and (b) cyclic stress–strain curves

It follows that σ ≷ Y according as β ≷ 12 , irrespective of the rate of hardening. If the specimen is subjected to a complete cycle of loading and unloading with the longitudinal plastic strain varying between the limits – ε∗ to ε∗ , then the magnitude of the final stress under a constant plastic modulus H exceeds σ by the amount 4βHε∗ , which vanishes only when the hardening is purely kinematic. The shear stress–strain curve of a material under cyclic loading can be derived from the experimental torque–twist curve for a solid cylindrical bar subjected to cyclic torsion in the plastic range. Let T denote the applied torque at any stage of the loading, and let θ be the corresponding angle of twist per unit length. Since the engineering shear strain at any radius r is γ = rθ , we have T = 2π 0

a

τ r2 dr =

2π θ3

aθ

τ γ 2 dγ ,

0

where a denotes the external radius of the bar, and τ = τ (γ ) is the shear stress at any radius. Multiplying both sides of the above equation by θ 3 , and differentiating it with respect to θ , it is easy to show that θ

dT + 3T = 2π a3 τ (aθ ) , dθ

(1.51)

where τ (aθ ) is the shear stress at the boundary r = a, corresponding to a shear strain equal to aθ . The preceding relation provides a means of obtaining the (τ , γ )-curve from an experimental (T, θ )-curve during the loading process (Nadai, 1950). Consider now the unloading and reversed loading of a bar that has been previously twisted in the plastic range by a torque T0 producing a specific angle of twist

32

1 Fundamental Principles

θ 0. For a given value of θ 0 , the shear stress acting at any radius r, when the specific angle of twist has decreased to θ may be expressed as τ (rθ ) = τ0 (rθ0 ) + f [r (θ − θ0 )] ,

(1.52)

where τ 0 denotes the local shear stress at the moment of unloading. The function f represents the change in shear stress caused by the unloading or reversed loading. The torque acting at any stage is T = T0 + 2π

a

r2 f [r (θ − θ0 )]dr.

0

Setting ξ = r(θ – θ 0 ), which gives dξ = (θ – θ 0 ) dr, the preceding relation can be expressed as T − T0 =

2π (θ − θ0 )3

a(θ−θ0 )

ξ f (ξ ) dξ .

0

Multiplying both sides of this equation by (θ – θ 0 )3 and differentiating the resulting expression partially with respect to θ , we have (θ − θ0 )

∂T + 3 (T − T0 ) = 2π a3 f [a (θ − θ0 )] , ∂θ

since T0 is a function of θ 0 only. Substituting for f[a(θ –θ 0 )] from the above equation into (1.52), the shear stress at r = a is finally obtained as τ (aθ ) = τ0 (aθ0 ) +

1 ∂T . + − θ 3 − T ) (θ ) (T 0 0 ∂θ 2π a3

(1.53a)

Since ∂T/∂θ is positive, both terms in the curly brackets of (1.53a) are negative during unloading and reversed loading. The residual shear stress at r = a at the end of the unloading process corresponds to T = 0, the corresponding residual shear strain being found directly from the given torque–twist curve. Suppose that the reversed loading in torsion is terminated when T = T1 and θ = θ 1 , the corresponding shear stress at r = a being denoted by τ 1 (aθ 1 ). If the bar is again unloaded, and then reloaded in the same sense as that in the original loading, an analysis similar to the above gives the shear stress at the external radius in the form 1 ∂T − θ τ (aθ ) = τ1 (aθ1 ) + + − T . (1.53b) (θ (T ) ) 1 1 ∂θ 2π a3 Equations (1.51) and (1.53) completely define the cyclic shear stress–strain curve based on an experimentally determined cyclic torque–twist curve Wu et al. (1996). The derivative ∂T/∂θ is piecewise continuous, involving a jump at each reversal of the applied torque, the correspondence between the various points in the two

1.4

Cyclic Loading of Structures

33

Fig. 1.12 The cyclic shear stress–strain curve derived from the cyclic torque–twist curve for a solid cylindrical bar

cyclic curves being indicated in Fig. 1.12. It may be noted that (1.53) can be directly obtained from (1.51) if we simply replace T, θ , and τ in this equation by the appropriate differences of the physical quantities.

1.4.2 A Bounding Surface Theory The anisotropic hardening rule described in the preceding section cannot be applied without modifications to predict the plastic behavior of materials under cyclic loading with relatively complex states of stress (Dafalias and Popov, 1975; Lamba and Sidebottom, 1978). Following an earlier work by Mröz (1967a), various types of theoretical model involving two separate surfaces in the stress space have been widely discussed in the literature, notably by Tseng and Lee (1983), McDowell (1985), Ohno and Kachi (1986), and Hong and Liou (1993), among others. The two-surface model assumes the existence of a bounding surface that encloses the current yield surface throughout the loading history. Both the yield surface and the bounding surface can expand and translate, and possibly also deform in the stress space, as the loading and unloading are continued in the plastic range. The general features of the two-surface theory are illustrated in Fig. 1.13(a), where the yield surface or the loading surface S and the bounding surface S’ are represented by circles with centers √ C and C’, respectively. The current radii of the surfaces S and S’ are denoted by 2/3 times σ¯ and τ¯ , respectively, while the position vectors of the centers C and C’ are denoted by α ij and α’ij , respectively. The equation for the loading surface is given by (1.35), while that of the bounding surface is expressed as

sij − αij

2 sij − αij = τ¯ 2 . 3

34

1 Fundamental Principles

Fig. 1.13 Yield surface S and bounding surface S’ in the deviatoric stress space. (a) Both surfaces are in translation and (b) only the yield surface is in translation

For each point P on the yield surface S, there is a corresponding image point P’ on the bounding surface, the distance between the two points P and P’, which are defined by the vectors sij and s’ij , respectively, is an important parameter that enters into the theoretical framework. There are several possible ways of relating the two stress points, the one suggested by Mröz being sij − αij = (τ¯ /σ¯ ) sij − αij . The variation of α ij and α’ij with continued loading must be defined by appropriate hardening rules, for which there are several possibilities. Since the two-surface theory is not without its limitations in predicting the material response under cyclic loading (Jiang, 1993, 1994), we shall describe in what follows the simplest theoretical model that is consistent with the basic purpose of the theory. It is assumed, for simplicity, that the bounding surface at each stage is a circular cylinder concentric with the initial yield surface, which is taken as the von Mises cylinder. The radius of the bounding surface S’ therefore increases with the amount of plastic deformation following the isotropic hardening rule. The yield surface S, on the other hand, undergoes simultaneous expansion and rotation according to the mixed hardening process, Fig. 1.13(b). At a generic stage of the loading, the deviatoric stresses sij and s’ij associated with the surfaces S and S’, respectively, satisfy the relations 2 sij − αij sij − αij = σ¯ 2 , 3

sij sij =

2 2 τ¯ , 3

(1.54)

where α ij is the back stress defining the center of the current yield surface. So long as the two surfaces are separated from one another, the translation of the yield surface

1.4

Cyclic Loading of Structures

35

is assumed to be governed by Prager’s kinematic hardening rule, which requires the yield surface to move in the direction of the plastic strain increment. Thus p

dαij = cd εij , αij αij ≤

2 (τ¯ − σ¯ )2 , 3

(1.55)

where c represents the kinematic part of the rate of hardening of the material. The inequality in (1.55) ensures that the yield surface is not in contact with the bounding p surface. The plastic strain increment d εij is given by the flow rule (1.46), which applies to the mixed hardening process, the equivalent plastic strain increment d ερ corresponding to a given strain increment dεij being found from (1.49). The plastic modulus H at any stage is given by the relation H=

3 (h + c) , 2

where h represents the isotropic part of the rate of hardening and is two-thirds of the current slope of the curve obtained by plotting σ¯ against ε¯ p . In the case of cyclic loading, the parameter c depends not only on the accumulated plastic strain ε¯ p but also on the distance between the loading point P and its image point P’ on the bounding surface. For simplicity, the image point is considered here as the point of intersection of the outward normal to the yield surface at P with the bounding surface, Fig. 1.14(a). If ψ denotes the included angle between the vectors representing the deviatoric stress sjj and the reduced stress sij – α jj , then by the geometry of the triangle OPP’, we have s¯ 2 + δ 2 + 2¯sδ cos ψ = τ¯ 2 ,

Fig. 1.14 Simplified two-surface model for cyclic plasticity. (a) Separate loading and bounding surfaces and (b) the two surfaces are in contact

36

1 Fundamental Principles

√ where s¯ and δ are 3/2 times the lengths of the vectors OP and PP’, respectively. The above equation immediately gives δ = −¯s cos ψ +

τ¯ 2 − s¯ 2 sin2 ψ.

(1.56)

The quantities ψ and s¯ appearing in (1.56) can be determined from the relations 1/2 3sij sij − αij 3 , s¯ = sij sij . cos ψ = 2sσ 2

(1.57)

The plastic modulus H evidently depends on both ε¯ p and δ. For practical purposes, H can be estimated by using the empirical relation H=

dτ¯ dε

p

γ δ exp β , τ¯

(1.58)

where β and γ are dimensionless constants to be determined from experimental data p on uniaxial stress cycles. Since δ/τ¯ and dτ¯ /d ε monotonically decrease during the process, (1.58) implies a fairly rapid decrease in the value of H. When δ = 0, the plastic modulus becomes identical to the slope of the curve for τ¯ against ε¯ p . The quantities σ¯ and τ¯ are functions of ε¯ p alone and can be expressed by the empirical equations σ¯ = σ0 1 − m exp (− n¯εp ) , τ¯ = τ0 1 − m exp (− n ε¯ p ) ,

(1.59)

where σ 0 and τ 0 are saturation stresses, while m, n, m’, and n’ are appropriate dimensionless constants. The hardening rate parameters h and c at any given stage follow from (1.58) and (1.59). Suppose that all the physical quantities have been found for a generic stage of the cyclic loading. During an additional strain increment dεij satisfying the inequalp ity (sij – αij ) dεij > 0, the equivalent plastic strain increment d ε is computed from p (1.49), and the associated plastic strain increment tensor d εij then follows from p (1.46). Since the elastic strain increment d εije is equal to dεij – d εij , the deviatoric stress increment dsij is obtained from (1.48), where the second term in the curly brackets is equal to 2G dεkk δij . The incremental displacement dαij of the yield surp face is determined from (1.55) and the fact that c = 23 (H − dσ¯ /d ε ). The new stress tensors sij and α ij , together with the updated values of σ¯ and τ¯ obtained from (1.59), enable us to compute the new values of δ/τ¯ and H using (1.58) and (1.59), thereby completing the solution to the incremental problem.

1.4

Cyclic Loading of Structures

37

1.4.3 The Two Surfaces in Contact When the yield surface comes in contact with the bounding surface, the position of the stress point P will generally require the two surfaces to remain in contact during the subsequent loading process. Consider first the situation where P coincides with the point of contact T between the two surfaces, and the coincidence is then maintained following the stress path. Since the back stress αij in this case is in the direction of the deviatoric stress sij , it follows from simple geometry that σ¯ sij , αij = 1 − τ¯

sij sij =

2 2 τ¯ . 3

(1.60)

The yield surface is now assumed to expand at the same rate as the bounding surface so that dσ¯ = dτ¯ during this loading phase. The plastic modulus H is therefore continuous when the contact begins, in view of (1.58), and the incremental translation of the center of the yield surface is given by the relation σ¯ dτ¯ sij . dsij − dαij = 1 − τ¯ τ¯

(1.61)

This expression implies that the motion of the yield locus consists of a radial expansion of the circle together with a rigid body sliding along the bounding surface. Suppose now that the stress point P on the yield surface lies between the points T and D, where CD is parallel to the common tangent to the two surfaces in contact, Fig. 1.14(b). This condition, together with the condition of contact can be stated mathematically as αij αij =

2 (τ¯ − σ¯ )2 , 3

2 2 2 τ¯ ≥ sij sij ≥ (τ¯ − σ¯ )2 + σ¯ 2 . 3 3

Assuming dσ¯ and dτ¯ to be equal to one another as before, thus allowing a slight discontinuity in the plastic modulus as the contact is established, the translation of the center of the yield surface may be written as dαij = λdsij − sij dμ,

(1.62)

where λ and dμ are scalar parameters. The modified hardening rule expressed by (1.62) is consistent with the experimental observation of Phillips and Lee (1979). It implies that the radial expansion of the yield surface is accompanied by its sliding and rolling over the bounding surface. The unknown parameters in (1.62) can be determined from the condition that dα ij is orthogonal to α ij , and the fact that the stress point remains on the yield surface. Thus αij dαij = 0,

2 sij − αij dsij − dαij = σ¯ dσ . 3

(1.63)

38

1 Fundamental Principles

Taking the scalar product of (1.62) with α ij , and using the first relation of (1.63), we get dμ/λ = αij dsij /(αkl skl ) .

(1.64)

Using the contact condition, the second condition of (1.63) is easily reduced to 2 sij dαij = sij − αij dsij − σ¯ dτ¯ . 3 The scalar product of (1.62) with sjj and the substitution from above lead to the expression p sij − αij dsij − 23 H σ¯ d ε λ= , dskk − (dμ/λ) skl skl p

where H is the plastic modulus equal to dτ¯ /d ε during this phase. Equations (1.64) p and (1.65) define the hardening parameters λ and dμ when d ε is known for a given dsij . In the special case when (1.60) are applicable, we get λ = 1 − σ¯ /τ¯ , and dμ = λ (dτ¯ /τ¯ ), and the hardening rule then reduces to (1.61). Assuming the strain increment dεij to be prescribed, the corresponding value p of d ε can be approximately estimated from (1.49), which is not strictly valid for the situation considered here. The associated deviatoric stress increment dsij is then obtained as before, and the parameters λ and dμ are determined from (1.64) and p (1.65). An improved value of d ε subsequently follows from the relation 1 H p 1+ sij dαij , σ¯ d ε = sij − αij d εij − 3G 2G

(1.66)

where dα ij is given by (1.61). Equation (1.66) is obtained by taking the scalar products of (1.46) and (1.48) with the tensor sjj – αij , and using (1.63). The computation may be repeated until the difference between successive values of the effective plastic strain increment becomes negligible. The quantity τ¯ is still given by (1.59), but the value of σ¯ over this range is obtained from the relation σ¯ = τ¯ + (σ¯ ∗ − τ¯ ∗ ), where the asterisk refers to the instant when the two surfaces first come in contact with one another during the loading. The theoretical treatment of cyclic plasticity based on a single-surface model has been examined by Eisenberg (1976) and Drucker and Palgen (1981). The constitutive modeling of large strain cyclic plasticity has been discussed by Chaboche (1986), Lemaitre and Chaboche (1989), and Yoshida and Uemori (2003), among other investigators. The plastic response of materials under cyclic loading has also been discussed in recent years on the basis of an interesting theory of plasticity that does not require the specification of a yield surface. The theory, which is essentially due to Valanis (1975, 1980), who called it the endochronic theory of plasticity, is based on the concept of an intrinsic time that depends on the deformation history, the relationship

1.5

Uniqueness and Stability

39

between the two quantities being regarded as a material property. The theory also introduces an intrinsic time scale which is a function of the intrinsic time, the rate of change of the various physical quantities being considered with respect to this time scale. For further details of the endochronic theory of plasticity, together with some physical applications, the reader is referred to Wu et al. (1995).

1.5 Uniqueness and Stability 1.5.1 Fundamental Relations Consider the quasi-static deformation of a conventional elastic/plastic body whose plastic potential is identical to the yield function, which is supposed to be regular and convex. The current shape of the body and the internal distribution of stress are assumed to be known. We propose to establish the condition under which the boundary value problem has a unique solution and examine the related problem of stability. When positional changes are taken into account, it is convenient to formulate the boundary condition in terms of the rate of change of the nominal traction, which is based on the configuration at the instant under consideration. When body forces are absent, the equilibrium equation and the stress boundary condition for the rate problem may be written in terms of the nominal stress rate ˙tij and the nominal traction rate F˙ j as ∂ ˙tij = 0, ∂xi

F˙ j = li ˙tij ,

(1.67)

where xi denotes the current position of a typical particle and li the unit exterior normal to a typical surface element. The relationship between the unsymmetric nominal stress rate ˙tij and the symmetric true stress rate σ˙ ij , referred to a fixed set of rectangular axes (Chakrabarty, 2006), may be written as ˙tij = σ˙ ij − σjk

∂vi ∂vk + σij , ∂xk ∂xk

(1.68)

where vi denotes the velocity of the particle. The constitutive equations, on the other ◦ hand, must involve the Jaumann stress rate σ ij , which is the material rate of change of the true stress σij with respect to a set of rotating axes, and is given by (1.35). The elimination of σ˙ ij between (1.35) and (1.68) gives ˙tij = σ˙ ij + σij ε˙ kk + σik ωjk − σjk ε˙ ik

(1.69)

This equation relates the nominal stress rate ˙tij directly to the Jaumann stress rate σ˙ ij . Using the interchangeability of dummy suffixes, the scalar product of (1.69) with ∂vj /∂xi can be expressed as

40

1 Fundamental Principles

∂v ˙tij j = ∂xi

◦

[−4pt] [−6pt]σ ij

+ ε˙ kk σij

∂vi ∂vk ε˙ ij − σij 2˙εik ωjk + ∂xk ∂xj

(1.70)

in view of the symmetry of the tensors σ ij and ε˙ ij . This relation is derived here for later use in the analysis for uniqueness and stability. The constitutive law for the conventional elastic/plastic solid is such that the strain rate is related to the stress rate by two separate linear equations defining the loading and unloading responses. For an isotropic solid, when an element is currently plastic, the constitutive equation for loading may be written down by using the rate form of (1.38). The stress rate is therefore given by v 3G ◦ σ ij = 2G ε˙ ij + ε˙ kk δij − ε˙ kl nkl nkl , ε˙ kl nkl ≥ 0, 1 − 2v 3G + H v ◦ σ ij = 2G ε˙ ij + ε˙ kk δij , ε˙ kl nkl ≥ 0. 1 − 2v

(1.71)

Consider now a fictitious solid whose constitutive law is given by the first equation of (1.71), whenever an element is currently plastic, regardless of the sign of ε˙ kl nkl . Such a solid may be regarded as a linearized elastic/plastic solid, in which ◦ the stress rate corresponding to a strain rate ε˙ ij is denoted by τ ij . The scalar product of (1.71) with ε˙ ij then furnishes ◦ ◦ σ ij ≥ σ ij ε˙ ij = 2G ε˙ ij ε˙ ij +

2 v 3G 2 , − ε˙ kk ε˙ ij nij 1 − 2v 3G + H

(1.72)

where the equality holds only in the loading part of the current plastic region. In contrast to the bilinear elastic/plastic solid, the linearized solid has identical loading and unloading responses for any plastic element. ◦ ◦ Let (σ ij , ε˙ ij ) and (σ ∗ij , ε˙ ij∗ ) denote two distinct combinations of stress and strain rates in an element of the actual elastic/plastic solid corresponding to a given state ◦ ◦ of stress. The stress rates for the linearized solid in the two states are τ ij and τ ∗ij , respectively. If the element is currently plastic, and the two strain rates do not both call for instantaneous unloading, the scalar product of ε˙ ij∗ with the appropriate equation of (1.71) shows that ◦ ◦ σ ij ε˙ ij∗ ≤ τ ij ε˙ ij∗ = 2G ε˙ ij ε˙ ij∗ +

v 3G ∗ ∗ − nij nkl . ε˙ ij ε˙ kk ε˙ ij ε˙ kl 1 − 2v 3G + H

(1.73)

The equality holds when εij∗ calls for further loading, whatever the nature of ε˙ ij∗ . ◦ ◦ The inequalities satisfied by σ ∗ij ε˙ ij∗ and σ ∗ij ε˙ ij∗ are similar to (1.72) and (1.73), respectively. Consequently,

σ ij − σ ∗ij ◦

◦

◦ ◦ ε˙ ij − ε˙ ij∗ ≥ τ ij − σ ∗ij ε˙ ij − ε˙ ij∗ .

(1.74)

1.5

Uniqueness and Stability

41

If the difference between the unstarred and the starred quantities is denoted by the prefix , then it follows from above that ◦

◦

σij ε˙ ij ≥ τij ε˙ ij = 2G ˙εij ˆεij +

v (˙εkk )2 1 − 2v

(1.75)

with equality holding for instantaneous loading produced by both ε˙ ij and ε˙ ij∗ . When ◦ both the states call for instantaneous unloading, the relationship between σ ij and ◦ ˙εij is given by the second equation of (1.71), while that between τ ij and ˙εij is ◦ ◦ given by the first equation of (1.71), leading to the inequality σ ij ˙εij ≥ τ ij ˙εij . For an element that is currently elastic, there is the immediate identity

v 2 σ ij ˙εij = τ ij ˙εij = 2G ˙εij ˙εij + (˙εkk ) . 1 − 2v ◦

◦

(1.76)

◦

◦

It follows, therefore, that the inequality σ ij ˙εij ≥ τ ij ˙εij holds throughout the elastic/plastic body and under all possible conditions of loading and unloading. This result will now be used for the derivation of the uniqueness criterion.

1.5.2 Uniqueness Criterion Consider the typical boundary value problem in which the nominal traction rate F˙ j is specified on a part SF of the current surface of the body, and the velocity vj on the remainder Sv . Suppose that there could be two distinct solutions to the problem, involving the field equations (1.67), (1.69), and (1.71), together with the prescribed boundary conditions. If the difference between the two possible solutions is denoted by the prefix , then in the absence of body forces, we have ∂ ˙tij = 0, ∂xi

F˙ j = li ˙tij ,

in view of (1.67). The application of Green’s theorem to integrals involving surface S and volume V gives

F˙ j vj dS =

li ˙tij vj dS =

˙tij

∂ ∂xi

∂ ˙tij vj dV. ∂xi

The integral on the left-hand side vanishes identically, since F˙ j = 0 on SF and vj = 0 on Sv by virtue of the given boundary conditions. The condition for having two possible solutions therefore becomes

˙tij

∂ vj dV = 0. ∂xi

42

1 Fundamental Principles

The left-hand side of the above equation must be positive for uniqueness (Hill, 1958). Using (1.70), expressed in terms of the difference of the two possible states, a sufficient condition for uniqueness may be written as ∂ ∂ ◦ σ ij ˙εij + σij ˙εkk ˙εij − 2˙εik ω˙ jk − (vi ) (vk ) dV > 0. ∂xk ∂xj (1.77) for the difference vj of every possible pair of continuous velocity fields taking prescribed values on Sv . For applications to physical problems, it is preferable to replace the above condition by a slightly over-sufficient criterion, using the fact that ◦ ◦ σ ij ˙εij ≥ τ ij ˙εij throughout the body. Uniqueness is therefore assured when ∂ ∂ ◦ τ ij ˙εij − σij 2˙εik ωjk + (vi ) (vk ) dV > 0 ∂xk ∂xj

(1.78)

for all continuous difference fields vj vanishing on Sv . The term in εkk has been neglected, since the contribution made by it is small compared to that arising from ◦ a similar term in the quantity τ ij ˙εij , which is given by (1.75) and (1.76) in the plastic and elastic regions, respectively. It follows from (1.77) and (1.78) that the condition for uniqueness for the linearized elastic/plastic solid also ensures uniqueness for the actual elastic/plastic solid (Hill, 1959). If the constraints are rigid, so that vj = 0 on Sv , every difference field vj is a member of the admissible field vj , and the uniqueness criterion reduces to

∂vi ∂vk τ ij ε˙ ij − σij 2 ε˙ ik ωjk + dV > 0 ∂xk ∂xj ◦

(1.79)

for all continuous differentiable fields vj vanishing at the constraints. Splitting the tensors ∂vi /∂xk and ∂vk /∂xj into their symmetric and antisymmetric parts, it is easily shown that ∂vi ∂vk = σij ε˙ ik ε˙ jk − ωik ωjk . σij ∂xk ∂xj The remaining two triple products cancel one another by the symmetry and antisymmetry properties of their factors. In view of the above identity, the uniqueness criterion (1.79) becomes ◦ τ ij ε˙ ij − σij 2 ε˙ ik ωjk + ε˙ ik ε˙ jk − ωik ωjk dV > 0. (1.80) The leading term in square brackets is given by (1.72) for the current plastic region and by the same equation with the last term omitted for the elastic region. In the treatment of problems involving curvilinear coordinates, it is only necessary to regard the components of the tensors appearing in (1.80) as representing the curvilinear components.

1.5

Uniqueness and Stability

43

In a number of important physical problems, a part Sf of the boundary is submitted to a uniform fluid pressure p, which is made to vary in a prescribed manner. In this case, the change in the load vector on a given surface element, whose future orientation is not known in advance, cannot be specified. It can be shown that the nominal traction rate for the pressure-type loading is ∂v ∂v ˙Fj = p˙ lj + p lk k − lj k , ∂xj ∂xk where p˙ is the instantaneous rate of change of the applied fluid pressure. When the boundary value problem has two distinct solutions under a given pressure rate p˙ so that ˙p = 0, then the preceding relation gives ∂ ∂ ˙ Fj = p lk (vk ) − lj (vk ) on Sf . ∂xj ∂xk

(1.81)

It is assumed that the remaining surface area of the body is partly under a prescribed nominal traction rate F˙ j and partly under a prescribed velocity vj . Since F˙ j = 0 on Sf , the bifurcation condition (1.77) must be modified by replacing the right-hand side of this equation with the surface integral ∂ ∂ (vk ) − lj (vk ) vj dSf . p lk ∂xj ∂xk The uniqueness criterion (1.78) is therefore modified by subtracting the same quantity from the left-hand side of the inequality. In particular, (1.80) is modified to (Chakrabarty, 1969b).

◦ τ ij ε˙ ij − σij 2˙εik ωjk + εˆ ik ε¨ jk − ωik ωjk dV − p

lk ε˙ kj + ωkj − lj ε˙ kk vj dSf > 0. (1.82)

If the functional in (1.82) vanishes for some nonzero field vj , bifurcation in the linearized solid may occur for any value of the traction rate on Sf and pressure rate on Sf . In the actual elastic/plastic solid, on the other hand, bifurcation will occur only for those traction rates which produce no unloading of the current plastic region. When the material is rigid/plastic, the admissible velocity field is incompressible ◦ (˙εkk = 0), and the scalar product τ ij ε˙ ij becomes equal to 23 H ε˙ ij ε˙ ij , while the triple produce σij ε˙ ik ωjk vanishes due to the coaxiality of the principal axes of stress and strain rate, leading to a considerable simplification of the problem. The condition for uniqueness in rigid/plastic solids has been discussed by Hill (1957), Chakrabarty (1969a), and Miles (1969).

1.5.3 Stability Criterion Consider an elastic/plastic body which is rigidly constrained over a part Sv of its external surface, while constant nominal tractions are maintained over the remain-

44

1 Fundamental Principles

der SF . The deformation of the body will be stable if the internal energy dissipated in any geometrically possible small displacement from the position of equilibrium exceeds the work done by the external forces. Since these two quantities are equal to one another when evaluated to the first order, it is necessary to consider secondorder quantities for the investigation of stability. The stress and velocity distributions throughout the body are supposed to be given in the current state, which is taken as the initial reference state for the stability analysis. At any instant during a small virtual displacement of a typical particle, its velocity is denoted by ωj and the associated true stress by sij . Then the instantaneous rate of dissipation of internal energy per unit mass of material in the neighborhood of the particle is (sij /ρ)(∂ωi /∂zj ), where zj is the instantaneous position and ρ the current density. The rate of change of this quantity following the particle is sij ∂wi ρ˙ ∂wi ∂wk ∂ ∂ ∂wi ∂w ˙i 1 + sij − + wk s˙ij − sij = , ∂t ∂zk ρ ∂zj ρ ρ ∂zj ∂zj ∂zk ∂zj where w˙ j is the instantaneous acceleration of the considered particle (Chakrabarty, 1969a). The operator appearing in the first parenthesis represents the material rate of change and may be denoted by D/Dt for compactness. If the initial true stress is σ ij , the initial velocity vj , and the initial stress rate σ˙ ij , the above expression considered in the initial state furnishes ρ0

D Dt

sij ∂wi ρ ∂zj

= σ˙ ij t=0

∂vi + σij ∂xj

∂ v˙ i ∂vk ∂vi ∂vi ∂vk + − ∂xj ∂xk ∂xj ∂xk ∂xj

(1.83)

in view of the compressibility condition ρ˙ = −ρ (∂vk /∂xk ) in the initial state. Since the rate of dissipation of internal energy per unit volume in the initial state is σij (∂vi /∂xj ), the internal energy dissipated per unit initial volume during an interval of time δt required by the additional displacement δuj may be written as δU = σij

∂vi D sij ∂wi 1 δt + ρ0 (δt)2 , ∂xj 2 Dt ρ ∂zj t=0

(1.84)

which is correct to the second order irrespective of the strain path. If the nominal traction and its rate of change in the initial state are denoted by Fj , and F˙ j , respectively, the work done by the surface forces during the virtual displacement is δW =

1 Fj + F˙ j δt δuj dS = Fj δuj dS 2

in view of the boundary conditions F˙ j = 0 on SF and vj = 0 on Sv . Expressing δuj in terms of the initial velocity vj and the initial acceleration v˙ j , we have δW = δt

1 Fj vj dS + (δt)2 2

Fj v˙ j dS

(1.85)

1.5

Uniqueness and Stability

45

to second order. Since the total internal energy dissipated during the additional displacement must exceed the work done by the external forces, a sufficient condition for stability is δUdV − δW > 0. Substituting from (1.83), (1.84), and (1.85), the left-hand side of this inequality can be written entirely as a volume integral, since

Ff vj dS =

σij

∂vi dV, ∂xj

Ff v˙ j dS

σij

∂ v˙ i dV, ∂xj

by the principle of virtual work, the nominal traction being the same as the actual traction in the initial state. We therefore have σ˙ ij

∂vi + σij ∂xj

∂vk ∂vi ∂vi ∂vk − ∂xk ∂xj ∂xk ∂xj

dV > 0

as the required condition for stability. Substituting for σ˙ ij , using (1.68), and introducing the true strain rate ε˙ ij and the spin tensor ωij in the initial state, the stability criterion is finally obtained as

∂vi ∂vk σ ij ε˙ ij + σij ε˙ kk ε˙ ij − 2˙εik ωjk − dV > 0 ∂xk ∂xj ◦

(1.86)

for all continuous differentiable velocity fields vj vanishing at the Since constraints. the expression in the curly brackets of (1.86) is equal to ˙tij ∂vj /∂xi in view of (1.70), the surface integral of the scalar product F˙ j vj must be positive for the stability of the elastic/plastic solid (Hill, 1958). In the special case of rigid/plastic solids, an analysis similar to that presented above has been given by Chakrabarty (1969a). The stability functional (1.86) may be compared with the uniqueness functional (1.77), which involves the difference field vj instead of the velocity field vj. Due to the nonlinearity of the material response, the difference between two possible solutions is not necessarily a solution itself, and consequently (1.86) is always satisfied when (1.77) is. It follows that a partially plastic state in which the boundary value problem has a unique solution is certainly stable. When the solution is no longer unique, the partially plastic state may still be stable, and a point of bifurcation is therefore possible before an actual loss of stability. At such a stable bifurcation, the load must continue to increase with further deformation in the elastic/plastic range.

46

1 Fundamental Principles

Problems 1.1 For a certain application involving an elastic/plastic material, the stress–strain curve in the plastic range needs to be replaced by a straight line defined by σ = Y + T ε. The actual strain-hardening curve can be represented by σ = Y (Eε/Y) .n. If the linear strain-hardening law predicts the same area under the stress–strain curve as that given by the power law curve, over the range 0 ≤ ε ≤ ε 0 , when both the hardening laws are extended backward to ε = 0, show that Eε0 = Y

1 + n 1/n , 1−n

2n T = E i−n

1 − n 1/n . 1+n

1.2 For an element of work-hardening material yielding according to the von Mises yield criterion under biaxial compression, show that the principal stresses can be expressed in terms of the polar angle θ of the deviatoric stress vector as 2σ¯ σx = − √ cos θ, 3

π 2σ¯ σy = − √ sin −θ , 6 3

where σ¯ is the equivalent stress. Show also that the components of the plastic strain increment, according to the Prandtl–Reuss flow rule, can be expressed in terms of the angle θ and the current plastic modulus H as p d εx = − cos

π 6

+θ

dσ¯ H

,

d ε y p = sin θ

dσ¯ H

.

1.3 For an element of von Miss material deforming under a plane a strain tension in the x-direction and a stress-free state in the y-direction, show that√the applied stress and √ the deviatoric angle at the initial yielding are given by σ e = Y/ c and 2cos θ e = 3/c, where c = 1 – ν + ν 2 . If a prismatic beam made of such a material having a depth 2 h is bent to an elastic/plastic curvature, so that the depth of the elastic core becomes 2c, prove that the bending couple M is given by √ a2 M = 2 + 2 3c (cos θ ) ξ dξ , Me h 1

ξ=

y , h

c/h

where M√ e is the bending moment at the elastic limit. Assuming a mean value of cos θ, equal to cos θe , obtain the moment–curvature relationship in the dimensionless form

κ 2 M e = m − (m − 1) , Me κ

m=

3 (3c)1/4 . 2

1.4 An ideally plastic bar of circular cross section is rendered completely plastic by the combined action of an axial force N and a twisting moment T. If the ratio of the rate of extension to the rate of twist at the yield point is denoted by aα/3, show that the normal and shear stress distributions over the cross section of the bar are given by σ α = , Y α 2 + 3r2 /a2

r/a τ = . Y α 2 + 3r2 /a2

Problems

47

Denoting the fully plastic values of the axial force and twisting moment by N0 and T0 , respectively, and setting λ = 3 + α 2 , obtain the interaction relationship in the parametric form N 2 = α (λ − α) , N0 3

√ T 2 3 = λ − α 2 (λ − α) . T0 3

1.5 A block of isotropic material is compressed in the x-direction by a pair of rigid smooth dies, while the deformation in the y-direction is completely suppressed by constraints. If the material strain hardens linearly with a constant tangent modulus T, show that the polar equation of the stress path in the deviatoric plane is given by σ¯ = Y

√ √

3 sin θ − (1 − 2ν) (T/E) cos θe

αT/E

3 sin θe − (1 − 2ν) (T/E) cos θ

βT , exp (θe − θ) E

where σ¯ is the equivalent stress, and θ is the deviatoric angle having a value θ e at the initial yielding, while α and β are dimensionless parameters defined as α=

3 + (1 − 2ν)2 (T/E)2 3 + (1 − 2ν)2 (T/E)

√ β=

3 (1 − 2ν) (1 − T/E)

3 + (1 − 2ν)2 (T/E)

.

1.6 The plastic modulus of a certain kinematically hardening material varies with the equivalent plastic strain according to the relation H = Kn exp −n¯ε p , where K and n are material constants. A specimen of this material is first pulled in tension until the longitudinal stress is equal to σ 0 and is then subjected to a complete loading cycle which involves a plastic strain amplitude of amount ε∗. Show that the longitudinal stress at the end of the loading cycle exceeds σ 0 by the amount σ = −K 1 − exp (−4nε∗ ) exp −nε ∗ . 1.7 For a material that hardens according to the combined hardening rule, the isotropic and kinematic parts of the plastic modulus H are assumed to be in the ration β/(1 – β), where β is a constant. Assumie the plastic modulus to be given by H = Kn exp −n¯ε p Considering a complete loading cycle of a specimen involving a constant strain amplitude of amount ε ∗ , following a stress equal to σ 0 applied by simple tension, show that the tensile stress at the end of the cycle exceeds σ 0 by the amount σ = K 1 + exp ( − 2nε ∗ ) (2β − 1) + exp −2nε ∗ exp −nε ∗ . 1.8 The plastic yielding and flow of a certain isotropic material can be predicted with sufficient accuracy by modifying the von Mises yield criterion in the form J2 1 −

9J32 4J23

1/3 = k2

48

1 Fundamental Principles

where k is the yield stress in pure shear. Show that the uniaxial yield stress according to this criterion is Y = 1.853 k. Considering a state of plane stress defined by σ 3 = 0 and denoting σ 2 /σ 1 by α, prove that the ratio of the two in-plane plastic strain increments according to the associated flow rule is given by p

d ε2

p =

d ε1

⎧ 2

⎫ 2 ⎪ ⎬ + (1 + α) (2 − α) 2α 2 − 2α − 1 ⎪ 2α − 1 ⎨ 6 1 − α + α 2−α ⎪ ⎭ ⎩ 6 1 − α + α 2 2 + (1 + α) (2α − 1) 2 − 2α − α 2 ⎪

Chapter 2

Problems in Plane Stress

In many problems of practical interest, it is a reasonable approximation to disregard the elastic component of strain in the theoretical analysis, even when the body is only partially plastic. In effect, we are then dealing with a hypothetical material which is rigid when stressed below the elastic limit, the modulus of elasticity being considered as infinitely large. If the plastically stressed material has the freedom to flow in some direction, the distribution of stress in the deforming zone of the assumed rigid/plastic body would approximate that in an elastic/plastic body, except in a transition region near the elastic/plastic interface where the deformation is restricted to elastic order of magnitude. The assumption of rigid/plastic material is generally adequate not only for the analysis of technological forming processes, where the plastic part of the strain dominates over the elastic part, but also for the estimation of the yield point load when the rate of work-hardening is sufficiently small (Section 1.2). In this chapter, we shall be concerned with problems in plane stress involving rigid/plastic bodies which are loaded beyond the range of contained plastic deformation.

2.1 Formulation of the Problem A plate of small uniform thickness is loaded along its boundary by forces acting parallel to the plane of the plate and distributed uniformly through the thickness. The stress components σ z , τ xz , and τ vz are zero throughout the plate, where the z-axis is considered perpendicular to the plane. The state of stress is therefore specified by the three remaining components σ x , σ y , and τ xy , which are functions of the rectangular coordinates x and y only. During the plastic deformation, the thickness does not generally remain constant, but the stress state may still be regarded as approximately plane provided the thickness gradient remains small compared to unity.

2.1.1 Characteristics in Plane Stress For greater generality, we consider a nonuniform plate of small thickness gradient, subjected to a state of generalized plane stress in which σ x, σ y , and τ xy denote the J. Chakrabarty, Applied Plasticity, Second Edition, Mechanical Engineering Series, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77674-3_2,

49

50

2

Problems in Plane Stress

stress components averaged through the thickness of the plate. In the absence of body forces, the equations of equilibrium are ∂ ∂ hτxy = 0, (hσx ) + ∂x ∂y

∂ ∂ hτxy + hσy = 0, ∂x ∂y

(2.1)

where h is the local thickness of the plate. If yielding occurs according to the von Mises criterion, the stresses must also satisfy the equation 2 σx2 − σx σy + σy2 + 3τxy = σ12 − σ1 σ2 + σ22 = 3k2 ,

(2.2)

principal stresses in the xy-plane, and k is the yield stress where σ 1 and σ 2 are the√ in pure shear equal to 1/ 3 times the uniaxial yield stress Y. In the (σ 1, σ 2 )-plane, (2.2) represents an ellipse having its major and minor axes bisecting the axes of reference. Suppose that the stresses are given along some curve C in the plastically deforming region. The thickness h is regarded as a known function of x and y at the instant under consideration. Through a generic point P on C, consider the rectangular axes ( X, y) along the normal and tangent, respectively, to C. If we rule out the possibility of a stress discontinuity across C, the tangential derivatives ∂σx /∂y,∂σy /∂y, and ∂τxy /∂y must be continuous. Since ∂h/∂x and ∂h/∂y are generally continuous, the equilibrium equation (2.1) indicates that ∂σx /∂x and ∂τxy /∂xare also continuous across c. For given values of these derivatives, the remaining normal derivative ∂σy /∂x can be uniquely determined from the relation ∂σx ∂σy ∂τxy + 2σy − σx + 6τxy =0 2σx − σy ∂x ∂x ∂x

(2.3)

obtained by differentiating the yield criterion (2.2), unless the coefficient 2σ y – σ x is zero. When σ x = 2σ y , the above equation gives no information about ∂σy /∂x, which may therefore be discontinuous across C. It follows that S is a characteristic when the stress components are given by σx = σ ,

σy =

1 σ, 2

τxy = τ ,

where σ is the normal stress, and τ the shear stress transmitted across C. In view of the yield criterion (2.2), the relationship between σ and τ is σ 2 + 4τ 2 = 4k2

(2.4)

which defines an ellipse E in the (σ , τ )-plane, as shown in Fig. 2.1. If ψ denotes the angle of inclination of the tangent to E with the σ -axis, reckoned positive when τ decreases in magnitude with increasing σ , then tan ψ = ∓

σx − σy σ dτ =± =± . dσ 4τ 2τxy

(2.5)

2.1

Formulation of the Problem

51

Fig. 2.1 Yield envelope for an element in a state of plane plastic stress including the associated Mohr circle for stress

The expression in the parenthesis is the cotangent of twice the counterclockwise angle made by either of the two principal stresses with the x-axis. It follows that there are two characteristic directions at each point, inclined at an angle π /4 + ψ/2 on either side of the algebraically greater principal stress direction. The two characteristics are identified as α- and β-lines following the convention indicated in Fig. 2.2, when σ 1 > σ 2 . The two characteristics coincide when ψ = ±π/2. The normal and shear stresses acting across the characteristics are defined by the points of contact of the envelope E with Mohr’s circle for the considered state of stress. The locus of the highest and lowest points of the circle is an ellipse representing the yield criterion (2.2), which may be written as

Fig. 2.2 Characteristics directions in plane stress (σ 1 >σ 2 >0), designated by α- and β-lines

52

2

Problems in Plane Stress

F (σ0 ,τ0 ) = σ02 + 3τ02 = 3k2 , where

(2.6) 1 3 σ0 = (σ1 + σ2 ) = σ , 2 4

1 τ0 = (σ1 − σ2 ) . 2

Thus, τ 0 is numerically equal to the maximum shear stress and σ 0 the mean normal stress in the plane. Since σ cannot exceed 2k in magnitude, the material being ideally plastic, σ 0 be numerically less than 3k/2 and τ 0 numerically greater than k/2 for the characteristics to be real. Since dτ /dσ 0 is equal to −rsin ψ in view of (2.6) and Fig. 2.1, the acute angle which the tangent to the yield locus makes with the σ -axis must be less than π /4 for the stress equations to be hyperbolic. When τ 0 is numerically less than k/2, there is no real contact between the Mohr circle and the envelope E, and the stress equations then become elliptic. In the limiting case of |τ 0 | = k/2, the characteristics are coincident with the axis of the numerically lesser principal stress, the values of the principal stresses being ±(2k, k) in the limiting state.

2.1.2 Relations Along the Characteristics When the equations are hyperbolic, it is convenient to establish equations giving the variation of the stresses along the characteristics. Following Hill (1950a), we take the axes of reference along the normal and tangent to a typical characteristic C at a generic point P as before. Setting 2σ y = σ x tan ψ in (2.3), we get ∂τxy ∂σx ± tan ψ = 0, ∂x ∂x where the upper sign corresponds to the β-line and the lower sign to the α-line, respectively. The substitution of ∂σ x /∂x from above into the first equation of (2.1) gives ∂τxy ∂τxy ∂h ∂h ± tan ψ = ± σx + τxy tan ψ. h ∂x ∂y ∂x ∂y Regarding the curve C as a β-line, we observe that the space derivatives along the σ and β-lines at P are ∂ ∂ ∂ − sin ψ , = cos ψ ∂sα ∂x ∂y

∂ ∂ = , ∂sβ ∂y

and the preceding equation therefore reduces to h

∂τxy ∂h ∂h = tan ψ + (σ tan ψ + τ ) sin ψ ∂sα ∂sα ∂sβ

(2.7)

2.1

Formulation of the Problem

53

since σ x = σ and τxy = τ at the considered point. To obtain the derivative ∂τ xy /∂sα at P, let φ β denote the counterclockwise angle made by the tangent to C with an arbitrary fixed direction. If the x-axis is now taken in this direction, then 1 τxy = − σ sin 2φβ − τ cos 2φβ . 4 Differentiating this expression partially with respect to sα and then setting φ β = π /2. we obtain ∂τxy ∂φβ 1 ∂φβ ∂τ ∂σ = σ − = tan ψ 2τ − ∂sα 2 ∂sα ∂sα ∂sα ∂sα in view of (2.5). Inserting in (2.7) and rearranging, the result can be expressed in the form d (hσ ) − 2 hτ dφβ = − (σ sin ψ + τ cos ψ)

∂h dsα ∂sβ

(2.8)

along an α-line. Similarly, considering the curve C to be an α-line, it can be shown that d (hσ ) + 2 hτ dφα = − (σ sin ψ + τ cos ψ)

∂h dsβ ∂sα

(2.9)

along a β-line, where φ α is the counterclockwise orientation of the α-line with respect to the same fixed direction. Evidently, dφ β - dφ α =dψ. If the thickness distribution is given, (2.8) and (2.9) in conjunction with (2.4) would enable us to determine the stress distribution and the characteristic directions in the hyperbolic part of the plastic region. When the thickness is uniform, (2.8) and (2.9) reduce to the relations dσ − 2τ dφβ = 0 along an α − line, dσ + 2τ dφα = 0 along an β − line,

(2.10)

which are analogous to the well-known Hencky equations in plane strain. In the solution of physical problems, it is usually convenient to express the yield criterion parametrically through the angle ψ, or a related angle θ such that tan θ =

√ 3 sin ψ,

−

π π ≤ψ ≤ , 2 2

−

π π ≤θ ≤ . 3 3

(2.11)

The angle θ denotes the orientation of the stress vector in the deviatoric plane with respect to the direction representing pure shear in the plane of the plate. Indeed, it follows from (2.4), (2.5), and (2.11) that

54

2

Problems in Plane Stress

⎫ 4 sin ψ 4 σ ⎪ ⎪ = = √ sin θ , ⎪ ⎪ k 3 2 ⎪ ⎬ 1 + 3 sin ψ ⎪ 4 cos ψ τ ⎪ = = 1 − sin2 θ .⎪ ⎪ ⎪ ⎭ k 3 2 1 + 3 sin ψ

(2.12)

√ Since 2(σ 1 +σ 2 ) = 3σ and 2(σ1 −σ2 ) = σ 2 + 16τ 2 , we immediately get σ1 +σ2 = √ 3k sin θ and σ 1 – σ 2 = 2k cos θ , and the principal stresses become

π , σ1 = 2k sin θ + 6

π σ2 = 2k sin θ − 6

(2.13)

It is also convenient at this stage to introduce a parameter λ which is defined in the incremental form dλ =

1 2

dσ − dψ τ

=2

dσ 1 tan ψ − dψ σ 2

which is assumed to vanish with ψ or θ . Substituting from (2.12) and integrating, we get 1 λ = tan−1 (2 tan ψ) − ψ 2 1 2 1 = sin−1 √ sin θ − sin−1 √ tan θ . 2 3 3

(2.14)

Evidently, –π /4 ≤ λ ≤ π /4 over the relevant range. If ω denotes the counterclockwise angle made by a principal stress axis with respect to a fixed axis, then

1 dω = d φα + ψ 2

1 = d φβ + ψ . 2

Dividing (2.8) and (2.9) by hσ throughout, and substituting for dσ /σ , σ /τ , dφ α , and dφ β , we finally obtain the relations ⎫ 1 + 3 sin2 ψ ∂h dh ⎪ ⎪ =− d (λ − ω) + 2 tan ψ dsα ,⎪ ⎪ ⎬ h 2h cos ψ ∂sβ ⎪ ⎪ 1 + 3 sin2 ψ ∂h dh ⎪ dsβ .⎪ =− d (λ − ω) + 2 tan ψ ⎭ h 2h cos ψ ∂sα

(2.15)

along the α- and β-lines, respectively. Similar equations may be written in terms of θ using (2.11). Numerical values of λ are given in Table 2.1 for the whole range of values of ψ and θ . When h is a constant, (2.15) reduces to

2.1

Formulation of the Problem

55

Table 2.1 Parameters for plane strees characteristics Ψ degrees

Ψ Radians

λ radians

θ degrees

Ψ degrees

λ radians

0 10 20 30 40 50 60 70 80 90

0 0.17453 0.34907 0.52360 0.89813 0.87266 1.04720 1.22173 1.39626 1.57080

0 0.25177 0.4570 0.59527 0.68435 0.73722 0.76616 0.77992 0.78473 0.78540

0 10 20 30 35 40 45 50 55 60

0 5.843 12.130 19.471 23.845 28.977 35.264 55.542 55.542 90.000

0 0.15089 0.30013 0.44556 0.51581 0.58352 0.64757 0.75558 0.75558 0.78540

λ − ω = constant along an α-line, λ − ω = constant along a β-line,.

(2.16)

In analogy with Hencky’s first theorem, we can state that the difference in values of both λ and ω between a pair of points, where two given characteristics of one family are cut by a characteristic of the other family, is the same for all intersecting characteristics. It follows that if a segment of one characteristic is straight, then so are the corresponding segments of the other members of the same family, the values of λ and ω being constant along each straight segment. If both families of characteristics are straight in a certain portion of the plastic region, the state of stress is uniform throughout this region.

2.1.3 The Velocity Equations Let (vx , vy ) denote the rectangular components of the velocity averaged through the thickness of the plate. The material is assumed as rigid/plastic, obeying the von Mises yield criterion and the Lévy–Mises flow rule. In terms of the stresses σ x , σ y , and τ xy , the flow rule may be written as ∂vy /∂x ∂vx /∂y + ∂vy /∂x ∂vx /∂x = = . 2σx − σy 2σy − σx 6τxy

(2.17)

Consider a curve C along which the stress and velocity components are given, and let the x- and y-axes be taken along the normal and tangent, respectively, to C at a typical point P. Assuming the velocity to be continuous across C, the tangential derivatives dvx /dy and dvy /dy are immediately seen to be continuous. From (2.17), the normal derivatives dvx /dx and dvy /dx can be uniquely determined unless 2σ y – σ x = 0, which corresponds to dvy /dy = 0. Thus, C is a characteristic for the velocity field if it coincides with a direction of zero rate of extension. There are two such directions at each point and they are identical to those of the stress characteristics.

56

2

Problems in Plane Stress

Since the characteristics are inclined at an angle π /4+ψ/2 to the direction of the algebraically greater principal strain rate ε˙ 1 , the condition ε˙ = 0 along a characteristic gives 1 ε˙ 1 + ε˙ 2 = sin ψ = ε˙ 1 − ε˙ 2 3

σ1 + σ2 σ1 − σ2

1 = √ tan θ . 3

(2.18)

The range of plastic states for which the characteristics are real corresponds to |σ 0 |≤3|τ 0 | and are represented by the arcs AB and CD of the von Mises ellipse shown in Fig. 2.3. The numerically lesser principal strain rate vanishes in the limiting states, represented by the extremities of these arcs, where the tangents make an acute angle of π /4 with the σ 0 -axis. Fig. 2.3 Plane stress yield loci according to Tresca and von Mises for a material with a uniaxial yield stress Y

The velocity of a typical particle is the resultant of its rectangular components vx and vy . The resolved components of the velocity vector along the α and β-lines, denoted by u and v, respectively, are related to the rectangular components as u = vx cos φα + vy sin (ψ + φα ) , v = −vx sin φα + vy cos (ψ + φα ) , where φ α denotes the counterclockwise angle made by the α-line with the x-axis. The preceding relations are easily inverted to give vx = [u cos (ψ + φα ) − v sin φα ] sec ψ, vy = [u sin (ψ + φα ) − v cos φα ] sec ψ.

(2.19)

Differentiating vx partially with respect to x, and using the fact that dvx /dx = 0 when φ α = −ψ, we obtain ∂φα ∂v − (u tan ψ + v sec ψ) = 0. ∂sα ∂sα

2.1

Formulation of the Problem

57

Similarly, equating dvy /dy to zero after setting φ α = −ψ in the expression for the partial derivative of vy with respect to y, we get ∂φβ ∂v + (u sec ψ + v tan ψ) =0 ∂sβ ∂sβ in view of the relation dφ β – dφ α = dψ The velocity relations along the characteristics therefore become du − (u tan ψ + v sec ψ) dφα = 0 along an α-line, dv + (u sec ψ + v tan ψ) dφβ = 0 along an β-line,

(2.20)

When the characteristic directions are known at each point of the field, the velocity distribution can be determined from (2.20). For ψ = 0, these equations reduce to the well-known Geiringer equations in plane plastic strain. Since the thickness strain rate has the same sign as that of –(σ +σ 2 ), a thinning of the sheet corresponds to σ 0 > 0 and a thickening to σ 0 < 0. If the rate of change of thickness following the element is denoted by h, it follows from the associated flow rule that σn + σ1 ∂ω 1 ∂h ∂h h˙ = +ω =− , h h ∂t ∂s 2σ1 − σn ∂s

(2.21)

where w is the speed of a typical particle, s is the arc length along the momentary flow line, and (σ n , σ t ) are the normal stress components along the normal and tangent, respectively, to the flow line. The change in thickness during a small interval can be computed from (2.21). Evidently, whenever there is a discontinuity in the velocity gradient, there is also a corresponding jump in ∂h/∂t, leading to a discontinuity in the surface slope of an initially uniform sheet.

2.1.4 Basic Relations for a Tresca Material If the material yields according to Tresca’s yield criterion with a given uniaxial yield stress Y = 2k, the yield locus is a hexagon inscribed in the von Mises ellipse. When the principal stresses σ 1 and σ 2 have opposite signs, the greatest shear stress occurs in the plane of the sheet, and the yield criterion becomes 2 2 = 4k2 , σx + σy ≤ 2k. (σ1 − σ2 )2 = σx − σy + 4τxy

(2.22)

As in the case of plane strain, the stress equations are hyperbolic, and the characteristics are sliplines bisecting the angles between the principal stress axes. Since the shear stresses acting across the characteristics are of magnitude k, the envelope

58

2

Problems in Plane Stress

of the Mohr’s circles then coincides with the yield locus. The variation of the normal stress σ along the characteristics is obtained from (2.8) and (2.9) by setting τ = k, ψ = 0, and dφ α = dφ β = dφ, the expression in each parenthesis being then equal to k. For a uniform sheet, these relations reduce to the well-known Hencky equations in plane strain. Since the thickness strain rate vanishes by the associated flow rule, the velocity equations are also hyperbolic and the characteristics are again the sliplines, the velocity relations along the characteristics being given by the familiar Geiringer equations in plane strain. When the principal stresses σ 1 and σ 2 have the same sign, the greatest shear stress occurs out of the plane of the applied stresses, and the numerically greater principal stress must be of magnitude 2k for yielding to occur. On the (σ 0 , τ 0 )plane, the yield criterion is defined by the straight lines σ 0 ±τ 0 = ±2k. In terms of the principal stresses, the yield criterion may be written as (σ1 + 2k) (σ2 ± 2k) = 0,

σ1 σ2 ≥ 0.

This equation can be expressed in terms of the (x, y) components of the stress, using the fact that σ1 +σ 2 = σ x + σ y and σ 1 σ 2 = σ x σ y − τ2 xy the result being 2 − σx σy + 2k σx + σy = 4k2 , 2k ≤ σx + σy ≤ 4k. τxy

(2.23)

The partial differentiation of (2.23) with respect to x reveals that the stress derivatives are uniquely determined unless σ x = ±2k. Hence, there is a single characteristic across which the normal stress is of magnitude 2k. In other words, the stress equations are parabolic with the characteristic coinciding with the direction of the numerically lesser principal stress, whose magnitude is denoted by σ . When the xand y-axes are taken along the normal and tangent, respectively, to the characteristic, we have

σ sin 2ω. (2.24) σx = ± 2k and σy = ±σ , τxy = ∓ k − 2 where ω denotes the counterclockwise angle made by the characteristic with a fixed direction which is temporarily considered as the x-axis. Inserting (2.24) into the equilibrium equation (2.1), and setting ω = π /2 after the differentiation, we get

σ ∂ω 1 ∂h ∂ hσ σ ∂ω 1− =− , =− 1− h , 2k ∂s h ∂n ∂s 2k 2k ∂n

(2.25)

where ds and dn are the line elements along the characteristic and its orthogonal trajectory, forming a right-handed pair of curvilinear axes. When the thickness is uniform, ω is constant along each characteristic, which is therefore a straight line defined by y = x tan ω+f (ω), where f(ω) is a function of ω to be determined from the stress boundary condition. The curvature of the numerically greater principal stress trajectory is

2.2

Discontinuities and Necking

59

∂ω ∂ω cos ω ∂ω = sin ω − cos ω = − . ∂n ∂x ∂y x + f (ω) cos2 ω

(2.26)

Inserting from (2.26) into the second equation of (2.25), and using the fact that ds = sec ω dx along a characteristic, the above equation is integrated to give (Sokolovsky, 1969), 1−

σ g(ω) , = 2k x + f (ω) cos2 ω

(2.27)

where g(ω) is another function to be determined from the boundary condition. Since the numerically lesser principal strain rate vanishes according to the associated flow rule, it follows that the velocity equations are also parabolic, and the characteristic direction coincides with the direction of zero rate of extension. The tangential velocity v remains constant along the characteristic, and the normal velocity u follows from the condition of zero ate of shear associated with these two orthogonal directions.

2.2 Discontinuities and Necking 2.2.1 Velocity Discontinuities In a nonhardening rigid/plastic solid, the velocity may be tangentially discontinuous across certain surfaces where the shear stress attains its greatest magnitude k. For a thin sheet of metal, it is also necessary to admit a necking type of discontinuity involving both the tangential and normal components of velocity. To be consistent with the theory of generalized plane stress, the strain rate is considered uniform through the thickness of the neck, whose width b is comparable to the sheet thickness h. Since plastic deformation is confined in the neck, the rate of extension vanishes along its length, and the neck therefore coincides with a characteristic. The relative velocity vector across the neck must be perpendicular to the other characteristic in order that the velocity becomes continuous across it. Localized necking cannot occur, however, when the stress state is elliptic. Let v denote the magnitude of the relative velocity vector which is inclined at an angle ψ to the direction of the neck, Fig. 2.4a. The rate of extension in the direction perpendicular to the neck is (v/b) sin ψ, and the rate of shear across the neck is (v/2b) cos ψ. The condition of the zero rate of extension along the neck therefore gives the principal strain rates as (Hill, 1952). ε˙ 1 =

v v v (1 + sin ψ) , ε˙ 2 = − (1 − sin ψ) , ε˙ 3 = − sin ψ, 2b 2b b

(2.28)

irrespective of the flow rule. These relations imply that the axis of ε˙ 1 is inclined at an angle π/4 + ψ/2 to the neck. For a von Mises material, ψ is related to the stress according to (2.18), while for a Tresca material, ψ = 0 when the stress state

60

2

Problems in Plane Stress

Fig. 2.4 Velocity and stress discontinuities in plane stress including the associated principal directions

is hyperbolic. In the case of a uniaxial tension, ε˙ 1 = −2˙ε2 for any regular yield inclination of the neck to function, giving ψ = sin–1 1/3 ≈ 19.47◦ , the angle of √ the direction of tension being β = π/4 + ψ/2 = tan−1 2 ≈ 54.74◦ . For a Tresca material, on the other hand, ψ can have any value between 0 and π/2 under a uniaxial state of stress. When the material work-hardens, a localized neck is able to develop only if the rate of hardening is small enough to allow an incremental deformation to remain confined in the incipient neck. For a critical value of the rate of hardening, the deformation is just able to continue in the neck, while the stresses elsewhere remain momentarily unchanged. Since the force transmitted across the neck remains momentarily constant, we have −

dσ dY H dh = = = d ε, h σ Y Y

where σ is the normal stress across the neck, Y the current yield stress, H the rate of hardening, and dε the equivalent strain increment. If the Lévy–Mises flow rule is adopted, then dh − = h

σ1 + σ2 2Y

dε =

3σ d ε. 4Y

The last two equations reveal that the critical rate of hardening is equal to 3σ /4 or (σ 1 +σ 2 )/2. Using (2.12), the condition for localized necking to occur may therefore be expressed as

2.2

Discontinuities and Necking

√ H 3 sin ψ σ1 + σ2 ≤ = √ . Y 2Y 1 + 3 sin2 ψ

61

(2.29)

As It is also necessary to have –1≤σ 1 /σ 2 ≤2 for the equations to be hyperbolic. √ ψ increases from 0 to π/2, the critical value of H/Y increases from 0 to 3/2. In the case of a uniaxial tension (3 sin ψ = 1), localized necking can occur only if H/Y ≤ 0.5. Thus, for a sheet of metal with a rounded stress–strain curve, a gradually increasing uniaxial tensile stress a produces a diffuse neck when H = σ , and eventually a localized neck when H = σ /2. A microstructural model for the shear band type of strain localization has been examined by Lee and Chan (1991).

2.2.2 Tension of a Grooved Sheet Consider a uniform rectangular sheet of metal whose thickness is locally reduced by cutting a pair of opposed grooves in an oblique direction across the width, Fig. 2.5a. The grooves are deep enough to ensure that plastic deformation is localized there when the sheet is pulled longitudinally in tension. The width of the sheet is large compared to the groove width b, which is slightly greater than the local sheet thickness h so that a uniform state of plane stress exists in the grooves. The material in the grooves is prevented from extending along its length by the constraint of the adjacent nonplastic material. The principal strain rates in the grooves are therefore given by (2.28) in terms of the angle of inclination of the relative velocity with which the sides of the grooves move apart. If the material is isotropic, the directions of the principal stresses σ 1 and σ 2 are inclined at angles π /4 + ψ/2 and π /4 – ψ/2, respectively, to the direction of the grooves, where σ 1 > σ 2, the principal axes of stress and strain rate being coincident.

Fig. 2.5 Necking of grooved and notched metal strips under longitudinal tension

Let φ denote the counterclockwise angle made by the prepared groove with the direction of the applied tensile force P. Then the normal and shear stresses acting across the grove are P sinφ/lh and P cos φ/lh, respectively, where l is the length of the groove. By the transformation relations for the stress, we have (σ1 + σ2 ) + (σ1 − σ2 ) sin ψ = (2P/lh) sin φ, (σ1 − σ2 ) cos ψ = (2P/lh) cos φ.

62

2

Problems in Plane Stress

These equations can be solved for the principal stresses in terms of the angles φ and ψ, giving σ1 =

P [sin (φ − ψ) + cos φ] P [sin (φ − ψ) − cos φ] , σ2 = . hl cos ψ hl cos ψ

(2.30)

From (2.28) and (2.30), Lode’s well-known stress and strain parameters are obtained as ⎫ 3 cos φ − sin (φ − ψ) ⎪ 2σ2 − σ1 − σ3 =− ,⎪ μ= σ1 − σ3 sin (φ − ψ) + cos φ ⎬ 3 (1 − sin ψ) 2˙ε2 − ε˙ 1 − ε˙ 3 ⎪ ⎪ ⎭ =− ν= . ε1 − ε3 (1 + 3 sin ψ)

(2.31)

Equation (2.31) form the basis for establishing the (μ, v) relationship for a material using the measured values of φ and ψ. The ends of the sheet must be supported in such a way that they are free to rotate in their plane to accommodate the relative movement necessary to permit the localized deformation to occur. The shape of the deviatoric yield locus may be derived from the fact that the length of the deviatoric stress vector is 1/2 2 2 P = σ1 − σ1 σ2 + σ22 s= 3 hl

2 sin2 (φ − ψ) + 3 cos2 ψ , √ 3 cos ψ

(2.32)

√ and the angle made by the stress vector is θ = tan−1 ( − μ/ 3) with the direction representing pure shear. If the yield locus and plastic potential have a sixfold symmetry required by the isotropy and the absence of the Bauschinger effect, it is only necessary to cover a 30◦ segment defined by the direction of pure shear (μ = v = 0) and that of uniaxial √ tension (μ = v = –1). This is accomplished by varying φ between 90◦ and tan−1 2 ≈ 54.7◦ , and measuring ψ for each selected value of φ. It may be noted that according to the Lévy–Mises flow rule (μ = v), the relationship tan φ = 4 tan ψ always holds. The preceding analysis, due to Hill (1953), is equally applicable to the localized necking caused by the tension of a sheet provided with a pair of asymmetrical notches as shown in Fig. 2.5b. If the notches are deep and sharp, and the rate of work-hardening is sufficiently low, plastic deformation is localized in a narrow neck joining the notch roots. This method may be used for the determination of the yield criterion and the plastic potential for materials with sufficient degrees of pre-strain, provided it is reasonably isotropic. The method has been tried with careful experiments by Hundy and Green (1954), and by Lianis and Ford (1957), using specimens which can be effectively tested to ensure that they are actually isotropic. These investigations have confirmed the validity of the von Mises yield criterion and the associated plastic potential for several engineering materials, as indicated in Fig. 1.8.

2.2

Discontinuities and Necking

63

2.2.3 Stress Discontinuities We begin by considering the normal and shear stresses acting over a surface which coincides with a characteristic. When the state of stress is hyperbolic, there is only one stress circle that can be drawn through the given point on the Mohr envelope without violating the yield criterion. Since the stress states on both sides of the characteristic are represented by the same circle, all components of the stress are continuous. When the stress state is parabolic, and the yield criterion is that of Tresca, the principal stress acting along the tangent to the characteristic can have any value between 0 and ±2 k, permitting a discontinuity in the numerically lesser principal stress. When the considered surface is not a characteristic, a stress discontinuity is always possible with two distinct plastic states separated by a line of stress discontinuity. Let σ 1 , σ 2 be the principal stresses on one side of the discontinuity (σ 1 ≥ σ 2 ), and σ 1 , σ 2 , those on the other side (σ 1 ≥ σ 2 ) . The angles of inclination of σ 1 and σ 2 with the line of discontinuity are denoted by θ and θ , respectively, reckoned positive as shown in Fig. 2.4b. Since the normal and shear stresses across the line of discontinuity must be continuous for equilibrium, we have (σ1 + σ2 ) − (σ1 − σ2 ) cos 2θ = σ1 + σ2 + σ1 − σ2 cos 2θ , (σ1 − σ2 ) sin 2θ = σ1 − σ2 sin 2θ .

(2.33)

If the von Mises yield criterion is adopted, the stresses on each side of the discontinuity must satisfy (2.2). Considering the stress components along the normal and tangent to the discontinuity, specified by n and t, respectively, and using the continuity conditions σ n = σ n and τ nt = r nl , it is easily shown from (2.2) that σ n – σ t = σ t giving 2 cot 2θ = cot 2θ +

σ1 + σ2 σ1 − σ2

cosec2θ .

(2.34)

The elimination of σ 1 – σ 2 between the two equations of (2.33), and the substitution from (2.34), lead to the relation 2 σ1 − σ2 = (σ1 + σ2 ) − 3 (σ1 − σ2 ) cos 2θ .

(2.35)

Since σ 1 – σ 2 is then given by the yield criterion, the principal stresses and their directions are known on one side of the discontinuity when the corresponding quantities on the other side are given. Considering the Tresca criterion for yielding, suppose that (σ 1 , σ 2 ) represents a hyperbolic state, so that σ 1 –σ 2 = 2k and |σ 1 +σ 2 | ≤ 2k. If the (σ 1 , σ 2 ) state is also hyperbolic, then σ 1 –σ 2 = 2k by the yield criterion, and the continuity conditions (2.33) furnish θ = θ , and

64

2

σ1 = σ1 − 2k cos 2θ ≥ 0,

Problems in Plane Stress

σ2 = σ2 − 2k cos 2θ ≤ 0.

If the (σ 1 , σ 2 ) state is parabolic, the yield criterion is either σ 1 = 2k(σ 2 ≥ 0) or σ 2 = –2k(σ 1 ≤ 0). In the first case, (2.33) gives

⎫ σ2 ⎪ tan θ = 1 + cos 2θ − co sec 2θ , ⎬σ k 2 ≤ cos 2θ . #

σ2 σ2 σ2 σ2 ⎪ 2k =− − cos 2θ 1 + cos 2θ − ,⎭ 2k k 2k k

(2.36)

The corresponding results for the second case are obtained from (2.36) by replacing σ 2 and σ 2 with –σ 1 and σ 1 , respectively, tan θ with cos θ , and reversing the sign of cos 2θ . Exceptionally, when the stress normal to the discontinuity is ±2k, the other principal stress on either side can have any value between 0 and ±2k. Such a discontinuity may be considered as the limit of a narrow zone of a continuous sequence of plastic states. When yielding occurs according to the von Mises criterion, all the stress components must be continuous across a line of velocity discontinuity. This is evident for a necking type of discontinuity (since the neck must coincide with a characteristic), across which the stress is necessarily continuous. For a shearing discontinuity, the shear stress across the line of discontinuity is of magnitude k, and since the normal stress is continuous for equilibrium, the remaining stress must also be continuous in view of the yield criterion. As a consequence of this restriction, the velocity must be continuous across a line of stress discontinuity. The rate of extension along a line of stress discontinuity, which is the derivative of the tangential velocity, is evidently continuous. Since the flow rule predicts opposite signs for this component of the strain rate on the two sides of the stress discontinuity, the rate of extension must vanish along its length. The discontinuity may therefore be regarded as the limit of a narrow zone of elastic material through which the stress varies in a continuous manner.

2.2.4 Diffuse and Localized Necking It is well known that the deformation of a bar subjected to a longitudinal tension ceases to be homogeneous when the rate of work-hardening of the material is less than the applied tensile stress. At the critical rate of hardening, the load attains its maximum, and the subsequent extension of the bar takes place under a steadily decreasing load. Plastic instabilities of this sort, leading to diffuse local necking, also occur when a flat sheet is subjected to biaxial tension in its plane. The strainhardening characteristic of the material is defined by the equivalent stress σ , and the equivalent total strain ε, related to one another by the true stress–strain curve in uniaxial tension. Using the Lévy–Mises flow rule in the form

2.2

Discontinuities and Necking

65

d ε1 dε d ε2 d ε3 = =− = , 2σ1 − σ2 2σ2 − σ1 σ1 + σ2 2σ

(2.37)

and the differential form of the von Mises yield criterion (2.2) where 3k2 is replaced by σ 2 , it is easily shown that the stress and strain increments in any element must satisfy the relation d σ1 d ε1 + d σ2 d ε2 = d σ d ε. Consider a rectangular sheet whose current dimensions are b1 and b2 along the directions of σ 1 and σ 2 , respectively. If the applied loads hb2 σ 1 and hb2 σ 2 attain stationary values at the onset of instability, where h denotes the current thickness, then d σ1 db1 = = d ε1 , σ1 b1

d σ2 db2 = = d ε2 , σ2 b2

in view of the constancy of the volume hb1 b2 of the sheet material. Combining the preceding two relations, we have dσ dε

= σ1

d ε1 dε

2

+ σ2

d ε2

2

dε

.

Substituting from (2.37), and using the expression for σ 2 , and setting σ 2 /σ 1 = p, the condition for plastic instability is obtained as (Swift, 1952; Hillier, 1966) (1 + ρ) 4 − 7ρ + 4ρ 2 H = 3/2 , σ 4 1 − ρ + ρ2

(2.38)

where H = dσ /dε denotes the critical rate of hardening. The quantity on the lefthand side of (2.38) is the reciprocal of the subtangent to the generalized stress–strain curve. The variation of the critical subtangent with stress ratio ρ is shown in Fig. 2.6 for both localized and diffuse necking. If the stress ratio is maintained constant throughout the loading, the total equivalent strain at instability is obtained directly from (2.38), if we adopt the simple power law σ = C εn , which gives H/σ = n/ε. In the case of variable stress ratio, the instability strain will evidently depend on the prescribed loading path. In the biaxial tension of sheet metal, failure usually occurs by strain localization in a narrow neck following the onset of instability. The phenomenon can be explained by considering the development of a pointed vertex on the yield locus, which allows the necessary freedom of flow of the plastic material in the neck. When both the principal strains are positive, experiments seem to indicate that the neck coincides with the direction of the minimum principal strain ε2 in the plane of the sheet. The incremental form of the Hencky stress–strain relations may be assumed to hold at the incipient neck, where the subsequent deformation remains

66

2

Problems in Plane Stress

Fig. 2.6 Critical subtangent to the effective stress–strain curve as a function of the stress ratio

confined (Stören and Rice, 1975). The principal surface strains in the neck may therefore be written as 2σ1 − σ2 2σ2 − σ1 ε1 = ε , ε2 = ε , 2σ 2σ where σ and ε are the equivalent stress and total strain, respectively. The elimination of σ 2 and σ 1 in turn between these two relations gives the principal stresses σ1 = (2ε1 + ε2 )

2σ 2σ , σ2 = (2ε2 + ε1 ) . 3ε 3ε

Assuming the power law σ = C εn for the generalized stress–strain curve, the incremental form of the first equation above at the inception of the neck is found as d σ1 2d ε1 + d ε2 dε , = − (1 − n) σ1 ε (2 + α) ε1

(2.39)

where α ≥ 0 denotes the constant strain ratio ε2 /ε 1 , prior to the onset of neck2 ing. 2The quantity2 de is the increment of the equivalent total strain ε. Since ε = 4 3 ε1 + ε1 ε2 + ε2 according to the Hencky theory, we get dε (2 + α) d ε1 + (1 + 2α) d ε2 . = ε 2 1 + α + α 2 ε1 The neck is characterized by a discontinuity in dε1 , but dε2 must be regarded as continuous across the neck. Since the material outside the neck undergoes neutral loading at its inception, we set dε 2 = 0. Combining the last two equations, and using

2.3

Yielding of Notched Strips

67

the fact that dσ 1 /σ 1 = dε1 for the load across the neck to be stationary, the condition for localized necking is obtained as

2 (1 − n) (2 + α) − ε1 + 2 2 + α 2 1+α+α

d ε1 = 0. ε1

Since dε1 = 0 for the development of the neck, the expression in the curly brackets must vanish, giving the limit strain over the range 0 < α < 1 in the form (2 − ρ) (2ρ − 1)2 + 3n 3α 2 + n (2 + α)2 = , ε1 = 2 (2 + α) 1 + α + α 2 6 1 − ρ + ρ2

(2.40)

where ρ is the stress ratio σ 2 /σ 1 , equal to (1 + 2α)/(2 + α). The value of ε 1 given by (2.40) may be compared with that predicted by (2.38) for a material with a given value of n. The limit strain is seen to be higher than the instability strain except when α = 0, for which both the conditions predict ε 1 = n. For α < 0,√the neck forms along the line of zero extension, which is inclined at angle tan−1 −α to the direction of the minimum principal strain in the plane of the sheet. The strain ratio then remains constant during the incremental deformation, and the onset of necking is given by (2.29), with Y = a, the limit strain being easily shown to be 2−ρ n ε1 = n , − 12 ≤ α ≤ 0. = 1+ρ 1+α The curve obtained by plotting ε 1 against ε 2 corresponding to localized necking in a given material is called the forming limit diagram, which represents the failure curve in sheet stretching. This will be discussed in Section 6.5 on the basis of a different physical model including anisotropy of the sheet metal.

2.3 Yielding of Notched Strips 2.3.1 V-Notched Strips in Tension Consider the longitudinal extension of a rectangular strip having a pair of symmetrical V-notches of included angle 2α in the plane of the strip. The material is assumed to be uniformly hardened, obeying the von Mises yield criterion and the associated Lévy–Mises flow rule. When the load attains the yield point value, the region of incipient plastic flow extends over the characteristic √ field shown in Fig. 2.7. The triangular region OAB is under a uniaxial tension 3 k parallel to the √ notch face, and the characteristics are straight lines inclined at an angle β = tan−1 2 to the notch face. Within the fan OBC, one family of characteristics are straight lines passing through the notch root, and the state of stress expressed in polar coordinates (r, φ) is given by

68

2

Problems in Plane Stress

Fig. 2.7 Characteristic field in a sharply notched metal strip under longitudinal tension (α ≥ 70.53)

σr = k cos φ, σφ = 2k cos φ, τrφ = k sin φ,

(2.41)

so that the equilibrium equations and the yield criterion are identically satisfied. It follows from (2.5) that cos ψ = 2tan √ φ within the fan. Along the line OB, ψ = sin−1 31 , giving φ = β = tan−1 2. The baseline from which φ is measured therefore makes an angle 2β with the notch face. The curved characteristics in OBC are given by r (dφ/dr) = − cos ψ = −2 tan φ, or

r2 sin φ = constant.

The curved characteristics approach the baseline asymptotically, if continued, and are inflected where φ = π/2 – β. The region OCO is uniformly stressed, and the principal stress axes coincide with the axes of symmetry. From geometry, angle COD is equal to φ 0 – (2β + α – π ), where φ 0 is the value of φ along OC, the corresponding value of ψ being denoted by ψ 0 . Since the algebraically lesser principal stress direction in OCO is parallel to 00, angle COD is also equal to π /4 – ψ 0 /2. The relation cot ψ0 = 2tan φ 0 therefore gives 2 tan φ0 + tan 2 (α + 2β − φ0 ) = 0,

(2.42)

which can be solved for φ 0 when α lies between π − 2β and π /2, the limiting values of φ 0 being 0 and β, respectively. Since σ 1 + σ 2 = 3k cos φ 0 and σ1 − σ2 = k 1 + 3 sin2 φ0 within OCO in view of (2.41), the constraint factor is

2.3

Yielding of Notched Strips

σ1 f =√ = 3k

3 cos φ0 +

69

1 + 3 sin2 φ0 , π − 2β ≤ α ≤ π/2. √ 2 3

(2.43)

The field of Fig. 2.7, which is due to Hill (1952), has been extended by Bishop (1953) in a statically admissible manner to show that the solution is in fact complete. As α decreases from √ π/2, the constraint factor increases from unity to reach its highest value of 2/ 3 when α = π – 2β ≈ 70.53◦ . The field in this case shrinks to a coincident pair of characteristics along the transverse axis of symmetry. In general, the constraint factor is closely approximated by the empirical formula

π − α , π − 2β ≤ α ≤ π/2. f = 1 + 0.155 sin 4.62 2

(2.44)

For all sharper notches, the characteristic field and the constraint factor are identical to those for α = π – 2β. Indeed, by the maximum work principle, the constraint factor cannot decrease when material√is added to reduce the notch angle, while the value of f certainly cannot exceed 2/ 3 since no stress component can exceed 2k in magnitude. The yield point load can be associated with a deformation mode consisting of localized necking along both characteristics through the center D of the minimum section. If the ends of the strip are moved longitudinally with a unit speed relative to D, the particles on the transverse axis of symmetry must move inward with a speed equal to tan(π /4 – ψ/2). The vector representing the relative velocity of particles across the neck is then perpendicular to the other characteristic as required. Strictly speaking, there is no opportunity for the deformation to occur outside the localized necks.

2.3.2 Solution for Circular Notches Consider, now, the longitudinal tension of a strip with symmetrical circular notches of radius c, the roots of the notch being at a distance 2α apart. For sufficiently small values of the ratio a/c, the characteristic field is radially symmetric as shown in Fig. 2.8, and the stress distribution is defined by (2.13) where σ 2 and σ 1 represent the radial and circumferential stresses denoted by σ r and σ φ, respectively. The substitution into the equation of radial equilibrium gives σφ − σr 2k cos θ dθ 2 d σr , = = , or r =√ dr r r dr 3 + tan θ where r is the radius of a generic point in the field. The boundary condition σ r = 0 at the notch surface is equivalent to θ = π/6 at r = c, and the integration of the above equation results in

70

2

Problems in Plane Stress

Fig. 2.8 Characteristic field in a circularly notched strip under longitudinal tension (a/c ≤ 1.071)

√ √ r2 3 π π π = 3 0 − sec θ exp ≤θ ≤ . , 2 2 6 6 3 c

(2.45)

The characteristics coincide when θ = π /3, and this corresponds to r/c ≈ 2.071. The angular span of the circular root covered by the field in this limiting case is 4β – π ≈ 38.96◦ , which is obtained from (2.14) as twice the difference between the values of λ corresponding to θ = π /3 and θ = π /6. The characteristic field is easily constructed using (2.45) and (2.14), and the fact that the polar angle φ measured from the transverse axis is, by (2.16), equal to the decrease in the value of λ from that on the transverse axis. The resultant longitudinal force per unit thickness across the minimum section is P=2 c

a+c

σφ dr = 2

c

a+c

π d , (rσr ) dr = 4k (a + c) sin θ0 − dr 6

where θ 0 is the value of θ at the center of the minimum section where r = a + c, and is directly given by (2.45). The constraint factor is f =

P c 2 π a = √ 1+ sin θ0 − , 0 ≤ ≤ 1.071. √ a 6 c 2 3ka 3

(2.46)

The value of f computed from (2.46) exceeds unity by the amount 0.226a/(a + c) to a close approximation. For higher values of a/c, the characteristics coincide along a central part of the transverse axis, and the longitudinal force per unit thickness is 4k(a – 1.071c) over the central part, and 2k(2.071c) over the remainder of the minimum section, giving the constraint factor

2.3

Yielding of Notched Strips

71

1 c f = √ 2 − 0.071 , a 3

a ≥ 1.071. c √ As a/c increases, f approaches its asymptotic value of 2/ 3, which is the ratio of the maximum shear stresses in pure shear and simple tension. Localized necking would occur at the yield point along the characteristics through C if the rate of work-hardening of the material is not greater than the uniaxial yield stress of the material. Consider, now, the solution for Tresca’s yield criterion and its associated flow rule. Since no stress can exceed 2k, which is now equal to the uniaxial yield stress, the constraint factor f cannot exceed unity. On the other hand, f is unity for a strip of width 2a. Hence, the actual constraint factor is f = 1, whatever the shape of the notch. A localized neck forms directly across the minimum section when the yield point is attained, provided the rate of work-hardening is not greater than 2k.

2.3.3 Solution for Shallow Notches The preceding solutions hold only when the notches are sufficiently deep. In the case of shallow notches, the deformation originating at the notch roots spreads across to the longitudinal free edges. Considering a sharply notched bar with an included angle 2α, a good approximation to the critical notch depth may be obtained by extending the characteristic field further into the specimen. With reference to Fig. 2.9, which shows one-quarter of the construction, the solution involves the extension ABCEFG of the basic field OABC. The extended field is bounded by a stress-free boundary AG generated from a point on the notch face, the material lying beyond AG being assumed unstressed. The construction begins with the consideration of the curvilinear triangle CBE, which is defined by the β-line CB and the conditions of symmetry along CE. Since cotφ = 2 tan ψ along CB, the boundary conditions are

Fig. 2.9 Critical width of a V-notched strip subjected to longitudinal tension

72

2

λ=

π −φ− 2

1 2

tan−1

1 2

tan φ , ω = −γ + φ +

1 2

Problems in Plane Stress

tan−1

1 2

tan φ ,

along CB in view of (2.14), where ω denotes the counterclockwise angle made by the algebraically greater principal stress with the longitudinal axis of symmetry, while γ = σ + 2β – 3 π /4. Since ω = 0 along CE by virtue of symmetry, the values of λ and ω are easily obtained throughout the field CBE using the characteristic relations (2.16). Starting with the known coordinates of the nodal points along CB, and using the fact that φα = ω −

π ψ + 4 2

,

φβ = ω +

π ψ + 4 2

,

(2.47)

where ψ is obtained from (2.14), the coordinates of each point of the field can be determined numerically by the mean slope approximation for small arcs considered along the characteristics, Since AB is a straight characteristic, all the β-lines in the field ABEF are also straight, though not of equal lengths. The angles ψ, ω, and λ. along AF are therefore identical to those along BE. The known values of φ β and φ α along BE and AF furnish the coordinates of the nodal points of AF by simple geometry and the tangent approximation. Since all characteristics meet the stress-free boundary OAG at a constant angle β = 54.74◦ , we have the boundary conditions ψ = 19.47◦ and λ = 25.53◦ along AG. Starting from point A, where ω = π /2 – α, and using (2.16), the values of λ and ω throughout the field AFG are easily determined. The angles φ α and φ β at the nodal points of the field are then computed from (2.47), and the rectangular coordinates are finally obtained by the mean slope approximation. The stress-free boundary AG generated as a part of the construction has a maximum height ω∗ , which is the critical semiwidth of the strip. The numerical computation carried out by Ewing and Spurr (1974) suggests the empirical formula

π a = 1 − 0.286 sin 4.62 −α , ∗ ω 2

(2.48)

which is correct to within 0.5% over the range 70.5◦ ≤ α ≤ 90◦ . This formula actually provides an upper bound on the critical semiwidth, since all specimens wide enough to contain the extended field are definitely not overstressed, as may be shown by arguments similar to those used for the corresponding plane strain problem (Chakrabarty, 2006). When the semiwidth w of the notched bar is less than w∗ , we can find an angle α ∗ > α such that w∗ (α ∗ ) = w. Then, the corresponding constraint factor f(α ∗ ) calculated from (2.46) would provide a lower bound on the yield point load. Indeed, the yield point load cannot be lowered by the addition of material required to reduce the notch angle from 2α ∗ to the actual value 2α. The constraint factor for subcritical widths is closely approximated by the lower bound value, which may be expressed empirically as

2.4

Bending of Prismatic Beams

73

a , a ≤ w ≤ w∗ . f = 1 + 0.54 1 − w In the case of an unnotched bar, the above formula reduces to f = 1. The deformation mode then consists of a localized neck inclined at an angle β = 54.74◦ to the tension axis. Such a neck is also produced in a tensile strip with either a single notch or a symmetric central hole (Fig. 2.10). The tension of single-notched strips has been investigated by Ewing and Richards (1973), who also produced some experimental evidence in support of their theoretical prediction.

Fig. 2.10 Initiation of a localized neck in a flat sheet inclined at an angle β to the direction of tension

2.4 Bending of Prismatic Beams 2.4.1 Strongly Supported Cantilever A uniform cantilever of narrow rectangular cross section carries a load kF per unit thickness at the free end, just sufficient to cause plastic collapse, the weight of the cantilever itself being negligible. The material is assumed to obey the von Mises yield criterion and the Lévy–Mises flow rule. Consider first the situation where the cantilever is rigidly held at the built-in end. If the ratio of the length l of the beam to its depth d is not too large, the characteristic field in the yield point state will be that shown in Fig. 2.11. The deformation mode at the incipient collapse consists of rotation of the rigid material about a center C on the longitudinal axis of symmetry. The solution involves localized necking along EN and localized bulging along NF, together with a simple shear occurring at N. The vector representing the relative velocity of the material is inclined to EF at an angle ψ which varies along the discontinuity. Although a local bulging can only occur in a strain-softening material, the solution may be accepted as a satisfactory upper bound on the collapse load (Green, 1954a) for ideally plastic materials. In the triangular regions ABD and GHK, the state of stress √ is a uniform longitu3 k, the characteristics dinal tension and compression, respectively, of magnitude √ being straight lines inclined at an angle β = tan−1 2 to the free edge. The region ADE is an extension of the constant stress field round the singularity A, the corresponding stresses being given by (2.41), where the polar angle φ is measured from a baseline that is inclined at an angle 2β to the free edge AB. The curved characteristics in ADE have the equation r2 sin φ = constant, while the relation cot ψ = 2 tan φ holds for the characteristic angle ψ. The normal stresses vanish at N, and the stress components along the curve ENF in plane polar coordinates are

74

2

Problems in Plane Stress

Fig. 2.11 Characteristic fields for an end-loaded cantilever with strong support (i/d ≤ 5.65)

σr = k sin θ , σθ = 2k sin θ , τrθ = −k cos θ ,

(2.49)

so that the equilibrium equations and the yield criterion are identically satisfied, the polar angled being measured counterclockwise with respect to the longitudinal axis. The relation cot ψ = 2 cot θ immediately follows from (2.5) and (2.49). The polar equation to the curve ENF, referred to C taken as the origin, is given by r (dθ/dr) = cot ψ = 2 cot θ ,

r 2 cos θ = constant

or

Since the characteristic directions are everywhere continuous, the value of ψ at E may be written as −1

ψ0 = tan

1 cot φ0 2

−1

= tan

1 tan α , 2

2.4

Bending of Prismatic Beams

75

where φ 0 and α are the values of φ and θ , respectively, at E. It follows that φ 0 = π /2 – α. From geometry, angle AEC is 2β – φ 0 + α, which must be equal to π /2 + ψ 0, and substituting for φ 0 and ψ 0 in terms of α, we obtain 2α − tan−1

1 2

tan α = π − 2β.

(2.50)

The solution to this transcendental equation is α ≈ 51.20◦ , which gives ψ 0 ≈ 31.88◦ . The fan angle EAD is δ = α + β– π /2 = ψ 0 /2. It is interesting to note that the state of stress in the deforming region in plane stress bending varies from pure tension at the upper edge to pure shear at the center, whereas in plane strain bending the stress state is pure shear throughout the region of deformation. The geometry of the field is completely defined by the dimensions b and R, representing the lengths AE and CE, respectively. The ratios b/d and R/d depend on the given ratio l/d, and are determined in terms of F/d from the conditions: (a) the sum of the vertical projections of AE and CE is equal to d/2; and (b) the resultant vertical force transmitted across AENFK per unit thickness is equal to kF. The resultant force acting across ENF is most conveniently obtained by regarding CENF as a fully plastic region, the normal and shear stresses across CE and CF being of magnitudes σ and τ directed as shown. The pair of conditions (a) and (b) furnishes the relations R sin α + b cos λ = d/2, R (σ cos α − τ sin α) + b (τ sin λ − σ cos λ) = kF/2,

(2.51)

where σ = 2k sin α and τ = k cos α in view of (2.49), while λ = α + 2β – π /2 ≈ 70.68◦ . Substituting for σ and τ , and inserting the values of σ and λ, equations of (2.51) are easily solved for b/d and R/d as F R F b = 0.6075 − 0.9696 , = −0.0941 + 1.1741 . d d d d

(2.52)

The ratio F/d at the yield point is finally determined from the condition that the resultant moment of the forces acting on AENFK about the center of rotation C is equal to the moment of the applied force about the same point. Thus

KF (l + R cos α − b cos λ) = σ R2 + b2 + 2Rb sin ψ0 + 2τ Rb cos ψ0 . Substituting for σ and τ , using the values of σ , λ, and ψ 0 inserting the expressions for b/d and R/d from (2.52), the above equation may be rearranged into the quadratic

76

2 Problems in Plane Stress

0.4342 −

1 F F2 − 0.2600 − 0.5288 2 = 0, d d d

(2.53)

when F/d has been calculated from (2.53) for a given value of l/d, the ratios b/d and R/d follow from (2.52). Since R = 0 when F/d ≈ 0.080, the proposed field applies only for l/d ≤ 5.65. For higher l/d ratios, the characteristic field is modified in the same way as that in the corresponding plane strain problem, the collapse load being closely approximated by the empirical formula 1 1 d = 0.20 + 2.18 , ≥ 5.65. F d d The values of F/d, b/d, and R/d corresponding to a set of values of l/d < 5.65 are given in Table 2.2. The plane stress values of F/d are found to be about 14% lower than the corresponding plane strain values (Chakrabarty, 2006) over the Table 2.2 Results for an end-loaded cantilever Strong support

Work support

l/d

F/d

b/d

R/d

l/d

F/d

b/d

δ

1.33 1.62 2.00 2.55 3.36 4.72 5.65

0.346 0.287 0.233 0.182 0.137 0.086 0.80

0.272 0.329 0.382 0.431 0.475 0.514 0.530

0.313 0.243 0.180 0.120 0.067 0.019 0

0.328 0.275 0.255 0.177 0.134 0.095 0.079

0.492 0.482 0.477 0.477 0.482 0.492 0.498

0.536 0.531 0.527 0.523 0.518 0.515 0.512

54.74 49.06 43.26 37.53 31.82 26.07 23.49

whole range of values of l/d. The yield point load for a tapered cantilever has been discussed by Ranshi et al. (1974), while the influence of an axial force has been examined by Johnson et al. (1974). Let M denote the bending moment at the built-in end under the collapse load ktF, where t is the thickness of the cantilever. Since the fully plastic moment under pure √ √ bending is M0 = 3 ktd2 , the ratio M/M0 is equal to 4Fl/ 3 d2 , and (2.53) may be written in the form F F M ≈ 1 + 1.23 0.49 − M0 d d

(2.54)

which is correct to within 0.3% for F/d ≤ 0.62. The elementary theory of bending assumes M/M0 ≈1 irrespective of the shearing force. The results for the end-loaded cantilever are directly applicable to a uniformly loaded cantilever if we neglect the effect of surface pressure on the region of deformation. Since the resultant vertical load now acts halfway along the beam, the collapse load for a uniformly loaded cantilever of length 2l is identical to that for an end-loaded cantilever of length l.

2.4

Bending of Prismatic Beams

77

2.4.2 Weakly Supported Cantilever Consider a uniform cantilever of depth d, which fits into a horizontal-slot in a rigid vertical support. The top edge of the beam is clear of the support, so that the adjacent plastic region is able to spread into the slot under the action of a load kF per unit thickness at the free end. The length of the cantilever is l, measured from the point where the bottom edge is strongly held. The characteristic field, due to Green (1954b), is shown in Fig. 2.12. It consists of a pair of triangles ABN and CEF, under uniaxial tension and compression, respectively, and a singular field CEN where one family of characteristics is straight lines passing through C. The√characteristics in the uniformly stressed triangles are inclined at an angle β − tan−1 2 to the respective free edges. The curved characteristics in CEN are given by the polar equation r2 sin φ = constant, where φ is measured from a datum making an angle 2β to the bottom edge CF. The stresses in this region are given by (2.41) with an overall reversal of sign.

Fig. 2.12 Characteristic field for an end-loaded cantilever with weak support (l/d ≥ 1.33)

For a given depth of the cantilever, the field is defined by the angle δ at the stress singularity, and the length b of the characteristic CN. The height of the triangle ABN is a = d − b sin (β + δ) .

(2.55)

The stress is discontinuous across the neutral point N, about which the cantilever rotates as a rigid body at the incipient collapse. The normal and shear stresses over CN are

78

2

Problems in Plane Stress

σ = −2 k cos (β − δ) , τ = sin (β − δ) . √ Since the tensile stress across the vertical plane DN is 3 k parallel to AB, the condition of zero resultant horizontal force across CND and the fact that the resultant vertical force per unit thickness across this boundary is equal to kF furnish the relations √ b [τ cos (β + δ) − σ sin (β + δ)] = 3 ka, b [τ sin (β + δ) + σ cos (β + δ)] = kF. Substituting for σ and τ , and using the value of β, these relations can be simplified to F = sin2 δ, d

√ √ a 1 F 2 2 3 = + − tan (β + δ) b 3 3 b

(2.56)

When F/b and a/b have been calculated from (2.56) for a selected value of δ, the ratio d/b follows from (2.55), while l/b is obtained from the condition of overall moment equilibrium. Taking moment about N of the applied shearing force kF and also of the tractions acting over CND, we get $ √ 2% 3a l b . = cos (β + δ) + cos (β + δ) + b F 2b2

(2.57)

Numerical values of l/d, F/d, b/d, and a/d for various values of δ are given in Table 2.2. As l/d decreases, δ increases to approach the limiting value equal to β. The angle between the two characteristics CN and EN decreases with decreasing l/d, becoming zero in the limit when the triangle CEF shrinks to nothing. The ratios F/d and l/d attain the values 0.328 and 1.332, respectively, in the limiting state. √ The ratio M/M0 at the built-in section (through C), which is equal to 4Fl/ 3d2 , can be calculated from the tabulated values of l/d and F/d. The results can be expressed by the empirical formula F M F ≈ 1 + 1.45 0.34 − , M0 d d

(2.58)

which is correct to within 0.5% for F/d < 0.33. Equations (2.54) and (2.58) are represented by solid curves in Fig. 2.13, the corresponding relations for plane strain bending being shown by broken curves. Evidently, M exceeds M0 over the whole practical range, indicating that the constraining effect of the built-in condition outweighs the weakening effect of the shear except for very short cantilevers. In the case of a strong support, the maximum value of M/M0 is about 1.121 in plane strain and 1.074 in plane stress, corresponding to F/d equal to about 0.28 and 0.25, respectively.

2.4

Bending of Prismatic Beams

79

Fig. 2.13 Influence of transverse shear on the yield moment in the plane stress and plane strain bending of beams

2.4.3 Bending of I-Section Beams Consider an I-beam, shown in Fig. 2.14a, whose transverse section has an area Aω for the web and Af for each flange including the fillets. The depth of the web is denoted by d, and the distance between the centroids of the flanges is denoted by h. It is assumed that the flanges yield in simple tension or compression, while the

Fig. 2.14 Yield point states for I-section beams subjected to combined bending and shear

80

2

Problems in Plane Stress

web yields under√combined bending and shear. The bending moment carried by the flanges is Mf = 3 khAf , since √ one flange is in tension and the other in compression with a stress of magnitude 3 k. For a uniform cantilever of length l, carrying a load ktF at the free end, where t is the web thickness, the bending moment existing at the fixed end is M = ktFl. Hence the bending moment shared by the web is

√ Mω = M − Mf = k tFl − 3hAf giving 4Fl 4h Mω = √ − 2 M0 d 3d

Af Aw

,

where M 0 is the √ fully plastic moment of the web under pure bending, equal to √ 2 3 kd t/4 = 3 kAωd/4. When the cantilever is strongly supported at the builtin end, Mω/M 0 is given by the right-hand side of (2.54), and the collapse load is given by 1.23

l F2 F 4h Af − 1 = 0. + 2.31 − 0.60 − d2 d d d Aw

(2.59a)

For a weak end support, Mω/M 0 is given by the right-hand side of (2.58), and the equation for the collapse load becomes 1.45

l F2 F 4h Af − 1 = 0. + 2.31 − 0.49 − d2 d d d Aw

(2.59b)

Equation (2.59), due to Green (1954b), is certainly valid over the practical range of values of l/d for standard I-beams. Since the flanges do not carry any shearing load, Mf = M0 – M 0 , where M0 is the fully plastic moment of the I-beam under pure bending. The relation Mf = M – Mw therefore gives M =1+ M0

M0 Mw − 1 , M0 M0

M0 4h =1+ M0 d

Af Aw

.

The relationship between the bending moment and the shearing force at the yield point state of an I-beam is now obtained on substitution from (2.54) and (2.58), where M/M0 is replaced by Mw /M 0 . Considering the strong support, for instance, we have & F F 4h Af M = 1 + 1.23 0.49 − 1+ . (2.60) M0 d d d Aω The variation of M/M0 with F/d is displayed in Fig. 2.14b. A satisfactory lower bound solution for M/M0 at the yield point of an I-beam has been derived by Neal

2.5

Limit Analysis of a Hollow Plate

81

(1961). The influence of axial force on the interaction relation, based on the characteristic field in plane stress, has been investigated by Ranshi et al. (1976).

2.5 Limit Analysis of a Hollow Plate 2.5.1 Equal Biaxial Tension A uniform square plate, whose sides are of length 2α, has a circular hole of radius c at its center. The plate is subjected to a uniform normal stress σ along the edges in the plane of the plate. As the loading is continued into the plastic range, plastic zones spread symmetrically outward from the edge of the hole and eventually meet the outer edges of the plate when the yield point is reached. We begin by considering the von Mises yield criterion and its associated Lévy–Mises flow rule. For a certain range of values of c/a, the characteristic field would be that shown in Fig. 2.15a. The stress distribution within the field is radially symmetrical with the radial and circumferential stresses given by (2.13), where σ 1 = σ φ and σ 2 = σ r , the spatial distribution of the angle θ being given by (2.45). Assuming θ = α at the external boundary r = a, where ar = a, we obtain

√ 2 σ π c2 2 π = √ sin α − , 2 = √ cos α exp − 3 α − . Y 6 6 a 3 3

(2.61)

The deformation mode at the incipient collapse consists of localized necking along the characteristics through A, permitting the rigid corners to move diagonally

Fig. 2.15 Equal biaxial tension of a square plate with a central circular hole. (a) 0.483 ≤ c/a ≤ and (b)0.143 ≤ c/a ≤ 0.483

82

2

Problems in Plane Stress

outward. Since the characteristics of stress and velocity exist only over the range π /6 ≤ α ≤ π /3, the solution is strictly valid for 0.483 ≤ c/a ≤ 1. When α = π /3, the two characteristics coincide at A, and σ attains the value 0.577 Y. For c/a ≤ 0.483, (2.61) provides a lower bound on the yield point load, since the associated stress distribution is statically admissible in the annular region between the hole and the broken circle of radius a, shown in Fig. 2.15b. The remainder of the plate is assumed to be stressed below the yield limit under balanced biaxial stresses of magnitude σ , a discontinuity in the circumferential stress being allowed across the broken circle. On the other hand, an upper bound solution is obtained by extending the localized necks as straight lines from r = c∗ = 2.071c to r = a, permitting the same mode of collapse as in (a). Since σ φ =2σ r = 2k along the straight part of the neck, the longitudinal force per unit thickness across BAE is

a

aσ = c

σφ dr =

c∗ c

d c∗ 2Y , (rσr )dr + 2k a − c∗ = √ a − dr 2 3

where the second step follows from the equation of stress equilibrium. The upper bound therefore becomes 2Y c σ = √ 1 − 1.035 , a 3

0.143 ≤

c ≤ 0.483. a

(2.62a)

For c/a ≤ 0.143, a better upper bound is provided by the assumption of a homogeneous deformation mode in which the rate of plastic work per unit volume is 2U/a, where U is the normal velocity of each side of the square. Since the rate of external work per unit plate thickness is 8aσ U, we obtain the upper bound π c2 σ =Y 1− 2 , 4a

0≤

c ≤ 0.143. a

(2.62b)

The difference between the lower and upper bounds, given by (2.61) and (2.62), respectively, is found to be less than 3% over the whole range of values of c/a. When the material yields according to the Tresca criterion, a lower bound solution is obtained from the stress distribution σ r = Y(1– c/r), σ φ = Y in the annulus c ≤ r ≤ a, and σ r = σ φ = σ in the region r = a. The continuity of the radial stress across r = a gives the lower bound a = Y(1 – c/a). To obtain an upper bound, we assume localized necking along the axes of symmetry normal to the sides of the square, involving a diagonally outward motion of the four rigid corners with a relative velocity v perpendicular to the necks. For a unit plate thickness, the rate of internal work in the necks is 4(a – c)vY, while the rate of external work is 4aσ v, giving the upper bound σ = Y(1 – c/a). Since the upper and lower bounds coincide, it is in fact the exact solution for the yield point stress.

2.5

Limit Analysis of a Hollow Plate

83

2.5.2 Uniaxial Tension: Lower Bounds Suppose that the plate is brought to the yield point by the application of a uniform normal stress σ over a pair of opposite sides of the square. The effect of the circular cutout is to weaken the plate so that the yield point value of σ is lower than the uniaxial yield stress Y. A lower bound solution for the yield point stress may be obtained from the stress discontinuity pattern of Fig. 2.16a, consisting of four uniformly stressed regions separated by straight lines, across which the tangential stress is discontinuous. The material between the circular hole of radius ρa the inner square of side ρa is assumed stress free. The conditions of continuity of the normal and shear stresses across each discontinuity are given by (2.33). If the principal stresses are denoted by the symbols s and t where s ≤ t, the stress boundary conditions require t1 = σ , s3 = 0, t4 = 0, where the subscripts correspond to the numbers used for identifying the regions of uniform stress. Let α denote the counterclockwise angle which the direction of the algebraically lesser principal stress in region 2 makes with the vertical. Then the clockwise angles made by this principal axis with the discontinuities bordering regions 1, 3, and 4 are θ1 = α − θ1 , θ3 =

π π + α − θ3 , θ4 = − −α . 2 4

Fig. 2.16 Uniaxial tension of a square plate with a circular hole. (a) Stress discontinuities and (b) graphical representation of yields equalities

84

2

Problems in Plane Stress

From geometry, the counterclockwise angles made by the algebraically greater principal stress axis with the line of discontinuity in regions 1, 3, and 4 are given by π tan θ1 = 1 − ξ , tan θ3 = ρ/ (ξ − ρ) , θ4 = − . 4 Considering the discontinuity between regions 4 and 2, and using (2.33), the quantities (s2 + t2 )/s4 and (s2 – t2 )/s4 can be expressed as functions of α. The consideration of the discontinuity between regions 3 and 2 then furnishes α and the ratio t3 /s4 . Finally, the continuity conditions across the boundary between regions 1 and 2 furnish s4 and s1 in terms of the applied stress σ . The results may be summarized in the form ⎫ 2ρ ρσ ⎪ ⎪ , tan 2α = , s1 = ⎪ ⎪ ⎪ (1 − ρ) (1 − ξ ) ξ ⎪ ⎪ ⎬ 2 2 σ (ξ − 2ρ) σ ξ + 4ρ (2.63) , s2 − t2 = , s2 + t 2 = ⎪ ξ (1 − ρ) ξ (1 − ρ) ⎪ ⎪ ⎪ ⎪ ρσ σ ⎪ ⎪ , s4 = . −t3 = ⎭ (1 − ρ) (ξ − ρ) 1−ρ Since regions 3 and 4 are in uniaxial states of stress, the magnitudes of t3 and s4 must not exceed the yield stress Y. For the von Mises criterion, the required inequalities in regions 1 and 2 follow from (2.2) and (2.63). When ρ is sufficiently small, the greatest admissible value of a is that for which region 4 is at the yield limit, giving the lower bound σ = Y (1 − ρ) ,

0 ≤ ρ ≤ 0.204.

For higher values of ρ, region 4 is not critical, and we need to examine the following yield inequalities for the estimation of the lower bound: σ (1 − ρ) (1 − ξ ) ≤ 1 2 , Y (1 − ρ)2 (1 − ξ )2 − ρ (1 − ρ) (1 − ξ ) + ρ 2 / ξ (1 − ρ) σ ≤ 1 2 , Y ξ 2 − ρξ + 4ρ 2 /

(1 − ρ) (ξ − ρ) σ ≤ , Y ρ

(2.64a)

(2.64b,c)

The parameter ξ must be chosen in the interval 0 ≤ ξ ≤ 1 such that the inequalities (2.64) admit the greatest value of σ . If the right-hand sides of these inequalities are plotted as functions of ξ for a given ρ, the greatest admissible value of σ /Y is the largest ordinate of the region below all the curves, as indicated in Fig. 2.16b. Thus, a/Y is given by the point of intersection of the curves (a) and (b), if this point is below the line (c), and by the point of intersection of (a) and (c), if curve (b) passes above this point. It turns out that the former arises for 0.204 ≤ ρ ≤ 0.412, and the latter for 0.412 ≤ p ≤ 1.

2.5

Limit Analysis of a Hollow Plate

85

When the Tresca criterion is adopted, the inequalities corresponding to regions 1 and 2 only are modified, the stress distribution being statically admissible if s1 ≤ Y, s2 − t2 ≤ Y, − t3 ≤ Y, s4 ≤ Y. Since s2 – t2 is greater than s4 over the whole range in view of (2.63), it is only necessary to consider the first three of the above inequalities. To obtain the best lower bound, the first two conditions should be taken as equalities for relatively small values of ρ, while the first and third conditions should be taken as equalities for relatively large values of ρ. Using (2.63), the results may be expressed as ρ=

ξ (1 − ξ ) σ (1 − ξ ) (1 − ρ) = , √ , ρ (3ξ − 2) 2 − ξ Y

1+ρ , ξ= 2

σ (1 − ρ)2 = , Y 2ρ

0.44 ≤ ρ ≤ 1.

⎫ 0 ≤ ρ ≤ 0.44, ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(2.65)

The lower and upper bound solutions given in this section are essentially due to Gaydon and McCrum (1954), Gaydon (1954), and Hodge (1981). All the bounds discussed here apply equally well to the uniaxial tension of a square plate of side 2a, containing a central square hole with side 2c, where c = ρa.

2.5.3 Uniaxial Tension: Upper Bounds An upper bound on the yield point stress is derived from the velocity field involving straight localized necks which run from the edge of the hole to the stress-free edges of the plate, as shown in Fig. 2.17a. The rigid triangles between the two pairs of neck move away vertically toward each other, while the rigid halves of the remainder of the plate move horizontally at the incipient collapse. Let v denote the velocity of one side of the neck relative to the other, and ψ the angle of inclination of the neck to the relative velocity vector.For a von Mises material, the rate of plastic work per unit volume in the neck is kν 1 + 3 sin2 ψ/b in view of (2.28), where b is the width

Fig. 2.17 Velocity discontinuity patterns for the plastic collapse of a uniaxially loaded square plate with a circular hole

86

2

Problems in Plane Stress

of the neck. Since the length of each neck is a(1 – ρ) cosec β, where β is the angle of inclination of the neck to the free edge of the plate, the rate of internal work in the necks per unit plate thickness is W = 4kva (1 − ρ) cos ecβ 1 + 3 sin2 ψ.

(2.66)

Equating this to the rate of external work, which is equal to 4σ aU = 4σ avcos (β – ψ) per unit thickness, we obtain the upper bound (1 − ρ) cos ecβ 1 + 3 sin2 ψ σ . = √ Y 3 cos (β − ψ) Minimizing σ with respect to β and ψ, it is found that the best upper bound corresponds to β = π /4 + ψ/2 and ψ = sin−1 31 , giving σ = Y (1 − ρ) .

(2.67)

Since the upper bound coincides with the lower bound in the range 0 ≤ ρ ≤ 0.204, the exact yield point stress is Y(l – ρ) for a von Mises material over this range. For a Tresca material, the rate of plastic work per unit volume in a neck is kv(l + sin ψ)/b in view of (2.28), whatever the state of stress in the neck. The upper bound solution is therefore modified to σ (1 − ρ) cos ecβ (1 + 3 sin ψ) = . Y 2 cos (β − ψ) This has a minimum value for β = ψ = π /2, and the best upper bound is precisely that given by (2.67). The necks coincide with the vertical axis of symmetry, and the two halves of the plate move apart as rigid bodies at the incipient collapse. For relatively large values of ρ, a better upper bound is obtained by assuming that each quarter of the plate rotates as a rigid body with an angular velocity ω about a point defined by the distances ξ a and ηa as shown in Fig. 2.17b. The deformation mode involves localized necking and bulging, with normal stresses of magnitude 2k acting along the horizontal and vertical axes of symmetry. Since the normal component of the relative velocity vector is of magnitude ω|x – ξ a| along the x-axis, and to ω|y – ηa| along the y-axis, the rate of internal work per unit thickness of the quarter plate is W = 2 kω

a ρa

|x − ξ a |dx+ |

a

ρa

|y − ηa| dy .

Carrying out the integration, the result may be expressed as W = 2 kωa2

1 + ρ2 − (1 + ρ) (ξ + η) + ξ 2 + η2 .

(2.68)

2.5

Limit Analysis of a Hollow Plate

87

The rate of external work on the quarter plate is σ aU, where U = aω(η − 12 ) is the normal component of the velocity of the center of the loaded side. Equating the rates of external and internal work done, and setting to zero the partial derivatives of σ with respect to ξ and η, we get 2ξ = 1 + ρ,

2η = (1 + ρ) +

σ , 2k

for the upper bound to be a minimum. The best upper bound on a is therefore given by

σ 2 2k

+ 2ρ

σ 2k

− 2 (1 − ρ)2 = 0.

It is easily verified that the conditions for a minimum σ are the same as those required for equilibrium of the quarter plate under the tractions acting along the neck and the external boundaries. The solution to the above quadratic is σ = −ρ + 2k

2 − 4ρ + 3ρ 2 ,

(2.69)

√ where k = Y/ 3 for a von Mises material, and k = Y/2 for a Tresca material. Evidently, (2.67) should be used in the range 0 ≤ ρ ≤ 0.42, and (2.69) in the range of 0.42 ≤ ρ ≤ 1 for a von Mises material. The ranges of applicability of (2.67) and (2.69) for a Tresca material are modified to 0 ≤ ρ ≤ 13 and 13 ≤ ρ ≤ 1, respectively. The upper and lower bound solutions are compared with one another in Fig. 2.18a and b, which correspond to the von Mises and Tresca materials, respectively.

Fig. 2.18 Bounds on the collapse load for a square plate with a circular hole. (a) von Mises material and (b) Tresca material

88

2

Problems in Plane Stress

2.5.4 Arbitrary Biaxial Tension Suppose that a uniform normal stress λσ (0 ≤ λ ≤ 1) is applied to the horizontal edges of the plate, in addition to the stress σ acting on the vertical edges. For a given value of λ, let σ be increased uniformly to its yield point value. A graphical plot of λσ against σ at the yield point state defines a closed interaction curve such that points inside the curve represent safe states of loading. Since such a curve must be convex, a simple lower bound may be constructed by drawing a straight line joining the points representing the lower bounds corresponding to uniaxial and equal biaxial tensions acting on the plate. Upper bounds on the yield point stress for an arbitrary λ can be derived on the basis of the velocity discontinuity patterns of Fig. 2.17. Considering mode (a), the rate of external work done by the stress λσ is obtained from the fact that the speed of the rigid triangle is equal to –v sin(β – ψ), the length of its base being 2a (1 – ρ)cot β. Equating the total external work rate to the internal work rate given by (2.66), we get (1 − ρ) 1 + sin2 ψ

σ = Y sin β cos (β − ψ) − λ (1 − ρ) cos β sin (β − ψ) for a von Mises material. The upper bound has a minimum value when β = π /4 + ψ/2 and 3 sin ψ = (1 + z)/(l – z), where z = λ (l – ρ), the best upper bound for the considered deformation mode being σ 1−ρ , λ (1 − ρ) ≤ 0.5. = Y (1 − λ) (1 − ρ) + λ2 (1 − ρ)2

(2.70)

When λ(l–ρ) ≥ 0.5, the best configuration requires β = π /2, giving σ = 2Y(l − √ ρ)/ 3 as the upper bound. For a Tresca material, the quantity 1 + 3 sin2 ψ in (2.66) must be replaced by (1 + sin ψ), and the upper bound is then found to be σ = Y(1– ρ), on setting β = ψ = π /2 for all values of λ. If the rotational mode (b) is considered for collapse, the rate of external work per unit thickness is σ a2 ω(η − 12 ) due to the horizontal stress, and −λσ a2 ω(ξ − 12 ) due to the vertical stress. Equating the total external work rate to the internal work rate given by (2.68), and minimizing σ /2k with respect of ξ and η, we get 2ξ = (1 + ρ) −

λσ , 2k

2η = (1 + ρ) +

σ , 2k

and the best upper bound on the yield point stress is then given by σ 2

σ

+ 2ρ (1 − λ) − 2 (1 − ρ)2 = 0 1 + λ2 2k 2k

(2.71)

for both the von Mises and Tresca materials with the appropriate value of k. When λ = 0, E (2.70) and (2.71) coincide with (2.67) and (2.69), respectively. When

2.6

Hole Expansion in Infinite Plates

89

λ = 1, these bounds do not differ appreciably from those obtained earlier for the von Mises material, while coinciding with the exact solution for the Tresca material. For sufficiently small values of ρ, a better bound for the von Mises √ material is obtained by dividing the right-hand side of (2.62b) by the quantity 1 − λ + λ2 , which is appropriate for an arbitrary λ. The bounds on the collapse load for a square plate with reinforced cutouts have been discussed by Weiss et al. (1952), Hodge and Perrone (1957), and Hodge (1981).

2.6 Hole Expansion in Infinite Plates 2.6.1 Initial Stages of the Process An infinite plate of uniform thickness contains a circular hole of radius a, and a gradually increasing radial pressure ρ is applied round the edge of the hole, Fig. 2.19. While the plate is completely elastic, the radial and circumferential stresses are the same as those in a hollow circular plate of infinite external radius and are given by σr = −

ρa2 , r2

σθ =

ρa2 . r2

Fig. 2.19 Elastic and plastic regions around a finitely expanded circular hole in an infinite plate

Each element is therefore in a state of pure shear, the magnitude of which is the greatest at r = a. Yielding therefore begins at the edge of the hole when ρ is equal to the shear yield stress k. If the pressure is increased further, the plate is rendered plastic within some radius c, the stresses in the nonplastic region being σr = −

kc2 kc2 , σ = , θ r2 r2

r ≥ c.

90

2

Problems in Plane Stress

Since the stresses have opposite signs, the velocity equations must be hyperbolic in a plastic region near the boundary r = c, with the characteristics inclined at an angle π /4 to the plastic boundary. Over a range of values of c/a, the plastic material would be entirely rigid. The equation of equilibrium is σθ − σr ∂σr = . ∂r r If the von Mises criterion is adopted, the stresses may be expressed in terms of the deviatoric angle φ as σr = −2k sin

π

+φ ,

6

σθ = 2k sin

π 6

−φ ,

(2.72)

where φ = 0 at r = c. Inserting the relation σ θ – σ r > = 2k cos φ into the equilibrium equation, we have cos

π

+φ

6

∂φ ∂r

=−

cos φ . r

Using the boundary condition at r = c, this equation is readily integrated to give the radial distribution of φ as √ c2 = e 3φ cos φ. 2 r

The relationship between the applied pressure ρ and the plastic boundary radius c is given parametrically in the form ρ = 2k sin

π 6

+α ,

√ c2 3α = e cos α, a2

(2.73)

where α is the value of φ at r = a. As α increases from zero, ρ increases from k. The pressure attains its greatest value 2k when α = π /3, giving c = a

1 π/√3 e 2

1/2

≈ 1.751.

The characteristics at this stage envelop the edge of the hole, which coincides with the direction of the numerically lesser principal stress, equal to -k. If the hole is further expanded, the plastic boundary continues to move outward, and the inner radius p of the rigid part of the plastic material is such that c/ρ = 1.751 throughout the expansion. Since an ideally plastic material cannot sustain a stress greater than 2k in magnitude, the plate must thicken to support the load which must increase for continued expansion. If the material yields according to the Tresca criterion, σ θ – σ r = 2k in a plastic annulus adjacent to the boundary r = c. Substituting in the equilibrium equation and integrating, we obtain the stress distribution

2.6

Hole Expansion in Infinite Plates

91

c σr = −k 1 + 2In , r

c σθ = k 1 − 2In . r

(2.74)

in view of the continuity of the√ stresses across r = c. The applied pressure attains its greatest value 2k when c/a = e ≈ 1.649, and the corresponding circumferential stress vanishes at the edge of the hole. A further expansion of the hole must involve thickening of the plate, while a rigid annulus of plastic material exists over the region ρ ≤ r≤ c, where c/p = 1.649 at all stages of the continued expansion.

2.6.2 Finite Expansion Without Hardening A solution for the finite expansion of the hole will now be carried out on the basis of the Tresca criterion, neglecting work-hardening. The circumferential stress then changes discontinuously across r = ρ from zero to – k, the value required by the condition of the zero circumferential strain rate in the presence of thickening. The angle of inclination of the velocity characteristics to the circumferential direction √ changes discontinuously from cot−1 2 to zero across r = ρ. In the plastic region defined by a≤ r≤ p, the equations defining the stress equilibrium and the Lévy– Mises flow rule are ⎫ ∂ 2σr − σθ v ⎪ h ∂v ⎪ = ,⎪ (hσr ) = (σθ − σr ) , ∂r r ∂r 2σθ − σr r ⎬ (2.75) σr + σθ v ∂h 1 ∂h ⎪ +v =− , ⎪ ⎪ ⎭ h ∂ρ ∂r 2σθ − σr r where h is the local thickness and v the radial velocity with ρ taken as the time scale. These equations must be supplemented by the yield criterion which becomes σ r = –2k in the region r ≤ p. The set of equations (2.75) is hyperbolic with characteristics dρ = 0 and dr – vdρ = 0 in the (r, ρ)-plane. The solution for a hole expanded from a finite radius may be obtained from that expanded from zero radius by discarding the part of the solution which is not required. Indeed, it is immaterial whether the pressure at any radius is applied by an external agency or by the displacement of an inner annulus (Hill, 1949). Since the plate is infinite, the stress and velocity at any point must be functions of a single parameter ξ = r/p, so that ρ

d ∂ = , ∂r dξ

ρ

∂ d = −ξ . ∂ρ dξ

Setting σ r = –2k and σ θ = –2ks in (2.75), where s is a dimensionless stress variable; they are reduced to the set of ordinary differential equations dh h = (s − 1) , dξ ξ

2−s v dv =− , dξ 1 − 2s ξ

1+s h ξ dh 1− = . v dξ 1 − 2s ξ (2.76)

The elimination of dh/dξ between the first and third equations of (2.76) leads to

92

2

v (1 − s) (1 − 2s) , = ξ 2 1 − s + s2

Problems in Plane Stress

dv (1 − s) (2 − s) . =− dξ 2 1 − s + s2

Eliminating v between the above pair of equations, we have ξ

d dξ

2s − s2 1 − s + s2

=

3 (1 − s)2 . 1 − s + s2

(2.77)

In view of the boundary condition s = 12 when ξ = 1, the integration of (2.77) results in 1 1 3 (1 − s)2 1 1 − 2s 1 −1 1 − 2s + tan . (2.78) In =− + In √ √ ξ 3 1−s 2 1 − s + s2 3 3 The elimination of ξ between (2.77) and the first equation of (2.76) gives d h dh

2s − s2 1 − s + s2

=−

3 (1 − s) . 1 − s + s2

If the initial thickness of the plate is denoted by h0 , then h = h0 when s = 12 , and the above equation is integrated to h 1 − 2s 2 . = [2 (1 − s)]−1/3 exp √ tan1 √ h0 3 3

(2.79)

If r0 denotes the initial radius to a typical particle, the incompressibility of the plastic material requires h0 r0 dr0 = hr dr at any given stage of the expansion. Since r0 = ρ when r = p, we obtain the relation r2 1− 2 =2 ρ

1

ξ

h ξ dξ , h0

(2.80)

where the integration is carried out numerically using the relations (2.78) and (2.79). As ξ decreases from unity, s decreases from 0.5 and becomes zero at r = ρ ∗ , which is given by In

π ρ 1 1 = − + (In 3) + √ ≈ 0.518, ∗ ρ 3 2 6 3

c ρast

≈ 2.768,

in view of (2.78). The thickness ratio at r = p∗ is h∗ /h0 ≈ 1.453 in view of (2.79). The solution is therefore complete for c/a ≤ 2.768, where a is the current radius of the hole. For larger expansions of the hole, σ θ becomes positive when r/c ≤ 0.361, and the yield criterion reverts to σ θ – σ r = 2k. Equation (2.76) is then modified in such a way that an analytical solution is no longer possible. A numerical solution furnishes a/c ≈ 0.280 when the hole is expanded from zero radius, the value of h/h0 at the edge of the hole being 3.84 approximately. This is in close agreement with the value obtained experimentally by Taylor (1948a).

2.6

Hole Expansion in Infinite Plates

93

2.6.3 Work-Hardening von Mises Material When the material work-hardens and obeys the von Mises yield criterion with the Lévy–Mises flow rule, (2.75) must be supplemented by the yield criterion which is written parametrically through an auxiliary angle φ as

π 2σ σr = − √ sin +φ , 6 3

π 2σ σθ = − √ sin −φ , 6 3

(2.81)

where σ is the current yield stress in uniaxial tension or compression. We suppose that the uniaxial stress–strain curve is represented by the equation σ = σ0 1 − me−ne ,

(2.82)

where σ 0 and n are the empirical constants. The initial yield stress is Y = (1− m)σ 0 , and the current slope of the stress–strain curve is H = n(σ 0 – σ ), which decreases linearly with increasing stress. The material work-hardens only in the region r ≤ ρ, where ρ = 0.571c, since the plastic material beyond this radius remains rigid. It is convenient to define the dimensionless quantities ξ=

r , ρ

η=

h , h0

s=

σ . σ0

We consider the expansion of a hole from zero radius, so that the stresses and strains in any element depend only on ξ . On substitution from (2.81), the last two equations of (2.75) become √ dv 3 + tan φ v =− √ , dξ 3 − tan φ ξ

ξ −v η

2 tan φ dη v =− √ . dξ ξ 3 − tan φ

(2.83)

To obtain the differential equation for s, we write the expression for the circumferential strain rate in terms of the material derivative by using the Lévy–Mises flow rule. Thus v 2σθ − σr ∂σ ∂σ = +v . r 2Hσ ∂ρ ∂r Substituting for σ r , σ θ , and H, and introducing the parameter ξ = r/ρ, this equation is reduced to ξ − v ds 2n sec φ v =− √ . (2.84) 1 − s dξ ξ 3 − tan φ Inserting the expressions for σ r and σ θ from (2.81) into the first equation of (2.75) gives

94

2

Problems in Plane Stress

π dφ 2 1 dη 1 ds + +φ +ξ = −√ tan . η dξ s dη 6 dξ 3 − tan φ

Eliminating dη/dξ and ds/dξ from the above equation by means of (2.83) and (2.84), we finally obtain the differential equation for φ in the form dφ

√ 3 − tan φ dξ

π √ v √ 1−s = 1 + 3 tan φ sec + φ − 2. 3 sec φ + n ξ s 6

(ξ − v)

(2.85)

Equations (2.84), (2.85), and the first equation of (2.83) must be solved simultaneously for the three unknowns v, s, and φ, using the boundary conditions v = 0, s = 1 – m, and φ = π /3 when ξ = 1. The second equation of (2.83) can be subsequently solved for η under the boundary condition η = 1 when ξ = 1. The expressions for all the derivatives given by (2.83) to (2.85) become indeterminate at ξ = 1, but the application of L’Hospital’s rule furnishes √ 3 dφ dv dη 1 ds λ = , = =− , =− , ξ = 1, dξ 2 dξ dξ 2 (1 + λ) dξ 2 (1 + λ) √ where λ = (2/ 3)mn/(1−m). The first derivatives of all the physical quantities are therefore discontinuous across ξ = 1. For a nonhardening material, it is easy to see that the stress gradients ∂σ r /∂r and ∂σ θ /∂ r have the values zero and −3k/2ρ, respectively, just inside the radius r = ρ. Once the thickness distribution has been found, the initial radius ratio r0 /ρ to a typical element can be calculated from (2.80) for any assumed value of ξ . The current radius σ for a hole expanded from an initial radius σ 0 is obtained from the relation 1 a20 1 ηξ dξ = 1− 2 . 2 ρ a/ρ The stress distribution is shown graphically in Fig. 2.20 for a material with m = 0.60 and n = 9.0. The thickness distribution is displayed in Fig. 2.21 for both work-hardening and nonhardening materials. The selected material is similar to that used by Alexander and Ford (1954), who analyzed the corresponding elastic/plastic problem using the Prandtl–Reuss theory. A rigid/plastic analysis for a plate of variable thickness has been given by Chern and Nemat-Nasser (1969). From the known variations of r0 /ρ, s, and φ with ξ , the internal pressure necessary to expand a circular hole from an internal radius a0 to a final radius a is found by using the first equation of (2.81). On releasing an amount q of the expanding pressure, the plate is left with a certain distribution of residual stresses. This is obtained by adding the quantities q(a2 /r2 ) and −q(a2 /r2 ) to the values of σ r , and σ θ , respectively, at the end of the expansion, so long as the unloading is elastic. For sufficiently large values of q, secondary yielding would occur on unloading, and the

2.6

Hole Expansion in Infinite Plates

95

Fig. 2.20 Stress distribution in the finite expansion of a circular hole in an infinite plate of workhardening material

Fig. 2.21 Thickness variation in an infinite plate containing a finitely expanded circular hole

analysis becomes quite involved. A complete analysis of the unloading process for c/a ≤ 1.751, taking secondary yielding into account, has been given by Alexander and Ford (1954) and by Chakrabarty (2006).

96

2

Problems in Plane Stress

2.6.4 Work-Hardening Tresca Material Suppose, now, that the material obeys Tresca’s yield criterion and its associated flow rule, the stress–strain curve of the material is given by (2.82). For ρ ≤ r ≤ c, where p = 0.607c, the stress distribution is still given by (2.74), where k = Y/2, since there is no strain hardening in this region. Indeed, the flow rule corresponding to the yield condition σ θ – σ r – Y implies the thickness strain to be zero, and the velocity vanishes identically in view of the incompressibility condition and the boundary condition. For r ≤ ρ, the stresses over a certain finite region would be σ r = – σ and σ θ = 0, corresponding to a corner of the yield hexagon. The associated flow rule gives ε˙ r < 0,

ε˙ θ > 0,

ε˙ z > 0.

The sum of the three strain rates must vanish by the condition of plastic incompressibility. The rate of plastic work per unit volume is σ ε˙ , where ε = – εr , indicating that the relationship between σ and ε is the same as that in uniaxial tension or compression. It follows from the plastic incompressibility equation written in the integrated form that hr = eε = h0 r 0

1−s m

−1/n (2.86)

in view of (2.82), with s = σ /σ 0 . Since σ θ = 0, the first equation of (2.75) reveals that hrσ = h0 ρY in view of the boundary conditions at r = ρ. Equation (2.86) therefore gives Y r0 = e−ε = ρ σ

1−m s

1−s m

1/n .

(2.87)

Differentiating (2.87) partially with respect to r, and noting the fact that ε = ln(∂r0 /∂r), we have −

1−m s

1 1 + s n (1 − s)

1−s m

2/n

1 ∂s = . ∂r ρ

Integration of this equation under the boundary condition s = 1− m at r = ρ, and using the integration by parts, furnishes the result ξ=

1−m s

1−s m

2/n

1−m + mn

s

1−s m

2/n−1

ds s

(2.88)

1−m

where ξ = r/p. The integral can be evaluated exactly for n = 2 and n = 4, but the numerical integration for an arbitrary value of n is straightforward. The thickness change can be calculated from the relation

2.7

Stretch Forming of Sheet Metals

97

ρY 1−m h = = . h0 rσ ξs

(2.89)

This solution will be valid so long as the thickness strain rate is positive. Since the strain is a function of ξ only, this condition is equivalent to dh/dξ < 0, which gives -ds/dξ < s/ξ in view of (2.89). Using the expression for ∂ s /∂r, the condition for the validity of the solution may be written as ξ≤

1−m s

s 1+ n (1 − s)

1−s m

2/n .

This condition will be satisfied for most engineering materials √ for all values of r0 /ρ ≥ 0. For a nonhardening material, h0 /h = ξ and r0 /ρ = 2ξ − 1, giving a/c = 0.303 for a hole expanded from zero radius. For a work-hardening material with m = 0.6 and n = 9.0, it is found that h/h0 ≈ 1.69 and r/c ≈ 0.143 at the edge of the hole when its initial radius is zero. The computed results for the Tresca theory are plotted as broken curves in Figs. 2.20 and 2.21, which provide a visual comparison with the results corresponding to the von Mises theory. The Tresca theory for a hypothetical material with an exponentially rising stress– strain curve has been discussed by Prager (1953), Hodge and Sankaranarayanan (1958), and Nemat-Nasser (1968). A rigid/plastic analysis for the hole expansion under combined radial pressure and twisting moment has been given by Nordgren and Naghdi (1963). An elastic/plastic analysis for the finite expansion of a hole in a nonhardening plate of variable thickness has been presented by Rogers (1967). An elastic/plastic small strain analysis for a linearly work-hardening Tresca material has been given by Chakrabarty (1971).

2.7 Stretch Forming of Sheet Metals 2.7.1 Hydrostatic Bulging of a Diaphragm A uniform plane sheet is placed over a die with a circular aperture and is firmly clamped around the periphery. A gradually increasing fluid pressure is applied on one side of the blank to make it bulge through the aperture. If the material is isotropic in the plane of the sheet, the bulge forms a surface of revolution, and the radius of curvature at the pole can be estimated at any stage from the measurement of the length of the chord to a neighboring point and its corresponding sagitta. The polar hoop strain can be estimated from the radial expansion of a circle drawn from the center of the original blank. The stress–strain curve of the material under balanced biaxial tension obtained in this way is capable of being continued up to fairly large strains before instability. The process has been investigated by Hill (1950b), Mellor (1954), Ross and Prager (1954), Weil and Newmark (1955), Woo (1964), Storakers (1966), Wang and Shammamy (1969), Chakrabarty and Alexander (1970), Ilahi et al. (1981), and Kim and Yang (1985a), among others.

98

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Let r denote the current radius to a typical particle, and r0 the initial radius, with respect to the vertical axis of symmetry. The local thickness of the bulged sheet is denoted by t, and the inclination of the local surface normal to the vertical by φ, as shown in Fig. 2.22. The ratio of the initial blank thickness t0 to the blank radius a is assumed small enough for the bending stress to be disregarded. The circumferential and meridional stresses, denoted by σ θ and σ φ , respectively, must satisfy the equations of equilibrium which may be written in the form ∂ rtσφ = tσθ , ∂r

σφ sin φ =

pr , 2t

(2.90)

where ρ is the applied fluid pressure. If the meridional and circumferential radii of curvature are denoted by ρ φ and ρ θ , respectively, the equations of normal and tangential equilibrium may be expressed as σφ p σθ + = , ρθ ρφ t

σφ P = , ρθ 2t

(2.90a)

where ρθ = r cos ecφ,

ρφ =

∂r sec φ. ∂φ

The elimination of p/t between the two equations of (2.90a) immediately furnishes

Fig. 2.22 Bulging of a circular diaphragm by the application of a uniform fluid pressure

2.7

Stretch Forming of Sheet Metals

99

σθ ρθ =2− . σφ ρφ This equation indicates that σθ ≶ σ φ for ρ θ ≷ ρ θ . The principal surface strains εθ ,εφ and the thickness strain εt at any stage are

r εθ = ln r0

∂r t , εφ = ln . sec φ , εt = ln ∂r0 t0

(2.91)

The condition for incompressibility requires ε t = – (εθ + ε φ ). If the radial velocity is denoted by v the components of the strain rates may be expressed as ˙t v ∂v + φ˙ tan φ, ε˙ t = , ε˙ θ = , ε˙ φ = r ∂r t where the dot denotes rate of change following the particle. Eliminating v between the first two of the above relations, we obtain the equation of strain rate compatibility ∂ (r˙εθ ) = ε˙ φ − φ˙ tan φ. ∂r

(2.92)

It will be convenient to take the initial radius r0 as the independent space variable, and the polar compressive thickness strain ε 0 as the time scale, to carry out the analysis. Introducing an auxiliary angle ψ, representing the angle made by the deviatone stress vector with the direction representing pure shear, the von Mises yield criterion and the associated Lévy–Mises flow rule can be simultaneously satisfied by writing σθ = √2 σ sin π6 + ψ , σφ = √2 σ cos ψ, 3 3 ε˙ θ = ε˙ sin ψ, ε˙ φ = ε˙ cos π6 + ψ ,

(2.93)

where σ and ε˙ are the equivalent stress and strain rate, respectively. Introducing dimensionless variables ξ=

σ pa r0 , s= , q= , a c t0 C

where C is a constant stress, and using the fact that ∂r/∂r0 = cosφ exp(εφ ), r = rφ exp(εφ ), and t = t0 exp(ε1 ) in view of (2.91), (2.90) and (2.92) can be combined with (2.93) to obtain the set of differential equations

π ∂ + ψ exp (−εθ ) , ξ s cos ψ exp −εφ = s cos φ sin ∂ξ 6 √ 3 s sin φ = qξ sec ψ exp 2εθ + εφ , 4

(2.95)

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Problems in Plane Stress

π ∂ + ψ − φ˙ sin φ exp −εφ , ξ ε˙ sin ψ exp (εθ ) = ε˙ cos φ cos ∂ξ 6

(2.96)

These equations must be supplemented by the strain-hardening law σ = Cf(ε), where ε is the equivalent total strain. Since σ θ = σ φ = σ at the pole (ξ = 0), while ε˙ = 0 at the clamped edge (ξ = 1), the boundary conditions may be written as ψ=

π and ε˙ = 1 at ξ = 0; 6

ψ = 0 at ξ = 1.

When the distributions of the relevant physical parameters have been found for any given polar strain ε0 , the shape of the bulge can be determined by the integration of the equation ∂z ∂r = − tan φ = − sin φ exp εφ . ∂r0 ∂r0 Using (2.95), this equation may be written in the more convenient form √ 3 q ∂ z =− ξ sec ψ exp 2εθ + 2εφ , ∂ξ a 4 s

(2.97)

which must be solved numerically under the boundary condition z = 0 at ξ = 1. The polar height h is finally obtained as the value of z at ξ = 0. The polar radius of curvature ρ is given by ρ = a

2σ0 t0 pa

exp (−ε0 ) =

2s0 q

exp (−ε0 ) ,

where σ 0 is the value of σ θ or σ φ the pole ξ = 0. Plastic instability occurs when the pressure attains a maximum, and this corresponds to dρ/ρ being equal to dσ 0 /σ 0 – dε 0 during an incremental deformation of the bulge. This condition can be used to establish the point of tensile instability in the bulging process. Suppose that the values of s, ψ, and φ are known at each point for the mth stage of the bulge. In order to continue the solution, we must find the corresponding distribution of ε˙ . Using the boundary condition ε˙ = 1 at ξ = 0, (2.96) is therefore solved numerically for ε˙ , the quantity φ being found from the values of φ in the previous and current stages of the bulge. The values of ε θ and ε φ for the (m + l)th stage are then obtained from their increments, using (2.93) and an assigned change in ε0 . It is convenient to adopt the power law of hardening σ = Cεn . Since s is a known function of ε, (2.94) can be solved for ψ, assuming a value of q for the (m + l)th stage, and using (2.95). The correct value of q is obtained when the boundary conditions ψ = 0 at ξ = 1 and ψ = π /6 at ξ = 0 are both satisfied.

2.7

Stretch Forming of Sheet Metals

101

If, for a certain stage of the bulge, ε˙ is found to vanish at ξ = 1, indicating neutral loading of the clamped edge, the condition ε˙ = 0 at ξ = 1 must be satisfied at all subsequent stages. The material rate of change of (2.95) gives φ˙ =

π n q˙ √ 3 sin + +ψ − ε˙ + ψ˙ tan ψ tan φ q 6 ε

in view of (2.93). The ratio q˙ /q in this case should be found by substituting the above expression for φ into (2.96), and integrating it under the boundary conditions ε˙ = 1 at ξ = 0, and ε˙ = 0 at ξ = 1, the quantity ψ˙ being given by the previous and current values of ψ. √ Initially, however, it is reasonable to assume 3 tanψ ≈ 1 – nξ 2 as a first approximation, which is appropriate over the whole bulge except at ξ = 1 (when n = 0). Since changes in geometry are negligible, (2.94) reduces in this case to √ 1 ∂ 1 − 3 tan ψ (s cos ψ) = 0. (s cos ψ) + ∂ξ 2ξ This equation is readily integrated under the boundary condition s = s0 at ξ = 0, resulting in √

n

n 3 qξ s0 exp − ξ 2 , φ = s cos ψ = exp ξ 2 , 2 4 2s0 4

(2.98)

in view of (2.95). The power law of hardening permits the Lévy–Mises flow rule to be replaced by the Hencky relations, so that the strain rates in (2.93) are replaced by the strains themselves. Substituting for the strain ratio εφ /εθ into the equation of strain compatibility, obtained by eliminating r between the first two relations of (2.91), we get √ √ 3 ∂εθ φ2 − cot ψ − 3 εθ = − ∂ξ 2ξ 2ξ

(2.99)

to a sufficient accuracy. Inserting the expressions for cot ψ and φ, this equation can be integrated under the boundary condition εθ = 0 at ξ = 1. Since εθ = εθ /2 at ξ = 0, we obtain −1/2

nx 1 ε0 q 3/4 = 2 2ε0 ≈ [8] . dx (1 − nx) exp s0 2 8−n 0 Using the above expression for q, the hoop strain εθ = ε sin ψ can be expressed as a function of εθ and ξ . It is sufficiently accurate to put the result in the form

−3/4 nξ 2 . 1 − nξ 2 ε sin ψ ≈ 12 ε0 1 − ξ 2 1 − 8−n

(2.100)

102

2

Problems in Plane Stress

Since s/s0 = (ε/ε0 )n , the distributions of s, ε, ψ, and φ can be determined from (2.98) and (2.100) for any small value of ε0. The shape of the bulge and the polar height are obtained from the integration of the equation ∂z/∂ξ = -a φ, the result being n q z 1 − ξ2 1 + ≈ 1 + ξ2 , a 4s0 32

h 1 ≈ a 2

1/2 16 + n . ε0 8−n

The initial shape of the bulge is therefore approximately parabolic for all values of n. Since φ changes at the rate φ = φ/2ε0 in view of (2.98), the solution for the von Mises material can proceed by integrating (2.96) as explained earlier. Figure 2.23 shows the surface strain distribution for various values of ε0 in a material with n = 0.2, the variation of ε0 with the polar height being displayed in Fig. 2.24. The theoretical predictions are found to agree reasonably well with available experimental results on hydrostatic bulging. A detailed analysis for the initial deformation of the diaphragm has been given by Hill and Storakers (1980).

Fig. 2.23 Distribution of circumferential and meridional true strains in the hydrostatic bulging process when n = 0.2 (after Wang and Shammamy, 1969)

2.7.2 Stretch Forming Over a Rigid Punch A flat sheet of metal of uniform thickness t0 is clamped over a die with a circular aperture of radius a, and the material is deformed by forcing a rigid punch with a hemispherical head. The axis of the punch passes through the center of the aperture and is normal to the plane of the sheet. The deformed sheet forms a surface of revolution with its axis coinciding with that of the punch. Due to the presence of friction between the sheet and the punch, the greatest thinning does not occur at the

2.7

Stretch Forming of Sheet Metals

103

Fig. 2.24 Variation of the polar thickness strain with polar height during the bulging of a circular diaphragm

pole but at some distance away from it, and fracture eventually occurs at this site. In a typical cupping test, known as the Erichsen test, a hardened steel ball is used as the punch head, and the height of the cup when the specimen splits is regarded as the Erichsen number, which is an indication of the formability of the sheet metal. The process has been investigated experimentally by Keeler and Backofen (1963) and theoretically by Woo (1968), Chakrabarty (1970a), Kaftanoglu and Alexander (1970), and Wang (1970), among others. Finite element methods for the analysis of the stretch-forming process have been discussed by Wifi (1976), Kim and Kobayashi (1978), and Wang and Budiansky (1978). In the theoretical analysis of the forming process, the coefficient of friction μ will be taken as constant over the entire surface of contact. The radius of the punch head, denoted by R, is somewhat smaller than the radius of the die aperture, Fig. 2.25. Let t denote the local thickness of an element currently at a radius r and at an angular distance φ from the pole, the initial radius to the element being denoted by r0 . Over the region of contact, the equations of tangential and normal equilibrium are ∂ rtσφ = tσθ + μpR tan φ, t σθ + σφ = pR, ∂r

(2.101)

where p is the normal pressure exerted by the punch, and r = R sin φ. The elimination of pR between the above equations gives ∂ rtσφ = tσθ (1 + μ tan φ) + μtσφ tan φ. ∂r If the material obeys the von Mises yield criterion, the stresses are given by (2.93). In terms of the dimensionless variables ξ = r0 /a and s = σ /C, where C is a constant stress, the above equation becomes

104

2

Problems in Plane Stress

Fig. 2.25 Stretch forming of a circular blank of sheet metal over a hemispherical-headed punch

∂ ξ s cos ψ exp −εφ ∂ξ

π = s (cos φ + μ sin φ) sin + ψ + μ sin φ cos ψ exp (−εθ ) , 6 where sin φ = (a/R)ξ exp (εθ ). In view of (2.93) and the second equation of (2.101), the pressure distribution over the punch head is given by

√ pR = s sin ψ + 3 cos ψ exp −εθ − εφ . t0 C

(2.103)

The geometrical relation r = R sin φ furnishes φ˙ = ε˙ θ tan φ over the contact region. Substituting into the compatibility equation (2.92), and using (2.93) for the components of the strain rate, we get π ∂ + ψ − tan2 φ sin ψ exp εφ . ξ ε˙ sin ψ exp (εθ ) = ε˙ cos φ cos ∂ξ 6 When the stresses and strains are known at the mth stage of the process, (2.104) can be solved using the condition ε˙ = 1 at ξ = 0, the polar compressive thickness strain ε0 being taken as the time scale. The computed distribution of ε˙ and an assigned increment of ε0 furnish the quantities εθ , εφ , and φ, while s follows from the given stress–strain curve. Equation (2.102) is then solved for ψ at the (m + l)th stage under the boundary condition ψ = π/6 at ξ = 0, and the stresses are finally obtained from (2.93).

2.7

Stretch Forming of Sheet Metals

105

Over the unsupported surface of unknown geometry, the equation of meridional equilibrium is given by (2.101) with p = 0. Since the circumferential and meridional curvatures at any point are sin φ/r and (∂φ/∂r) cos φ, respectively, the equation of normal equilibrium is ∂φ σθ sin φ + σφ cos φ = 0. r ∂r Using the relations (2.91) and (2.93), this may be rewritten as r0 cos ec φ

√ ∂φ 1 = − 1 + 3 tan ψ exp εφ − εθ . ∂r0 2

If the angle of contact is denoted by β, then ξ = ξ ∗ = (R/a) sin β exp(−εθ∗ ) at φ = β, where the asterisk refers to the contact boundary. The integration of the above equation results in ln

tan (φ/2) tan (β/2)

=−

1 2

ξ ξ∗

√ dξ 1 + 3 tan ψ exp εφ − εθ . ξ

(2.105)

The remaining equilibrium equation and the compatibility equation in the dimensionless form are identical to (2.94) and (2.96), respectively. To continue the solution from a known value of β at the mth stage, and the corresponding distributions of ψ and s, (2.96) is solved numerically for ε using the condition of continuity across ξ = ξ ∗ , the distribution of φ˙ being obtained from the previous values of φ. The distribution of φ for the (m + l)th stage is then obtained from (2.105), assuming a value of β and the previous distribution of ψ. The correct value of β is that for which the continuity condition for ψ across ξ = ξ ∗ is satisfied, when (2.94) is solved for ψ with the boundary condition ψ = 0 at ξ = 1. The total penetration h of the punch at any stage can be computed from the formula h R = (1 − cos β) + a a

1 ξ∗

sin φ exp εφ dξ .

(2.106)

in view of the relation ∂z/∂ξ = −a tan φ, where φ is given by (2.105) over the unsupported region. The resultant punch load is P = 2l Rt∗ σφ∗ σ £ sin2 β, and the substitution for t∗ and σφ∗ , furnishes

π P = s∗ sin2 β sin + ψ ∗ exp − εθ∗ + εφ∗ . 2π Rt0 C 6

(2.107)

Equations (2.106) and (2.107) define the load–penetration relation parametrically through β. When the load attains a critical value, a local neck is formed at the thinnest section (leading to fracture) due to some kind of instability of the biaxial stretching.

106

2

Problems in Plane Stress

If the strain hardening is expressed by the power law s = ε n√ , the Hencky theory may be used for the solution of the initial problem. Assuming 3tanψ ≈ 1 − n ξ 2 as a first approximation, and omitting the negligible friction terms in (2.102), it is found that s is given by the first equation of (2.98) throughout the deformed sheet. Furthermore, φ = a ξ /R for 0 ≤ ξ ≤ ξ ∗ , and n aξ ∗2 exp ξ 2 − ξ ∗2 , ξ ∗ ≤ ξ ≤ 1, ξR 4

φ=

in view of (2.105). The strain compatibility equation (2.99), which is the same as that for the stretch-forming process, gives 3/4 3/4 φ2 ∂ sin ψ = − . ε 1 − nξ 2 1 − nξ 2 ∂ξ 2ξ

(2.108)

Substituting for φ, and using the conditions of continuity of ε and ψ across ξ = ξ ∗ , we obtain the expression for the polar thickness strain as β2 ε0 = 2

1+

1 ξ ∗ 2 x∗

x

(1 − nx)

3/4

n ∗ x − x dx , exp 2

(2.109)

where x = ξ 2 and x∗ = ξ ∗2 = (RB/a)2 . The distribution of ψ is now obtained by the integration of (2.108). By (2.106), the polar height is h=

2 1 2 Rβ

1+

1 x∗

x − x∗ dx exp n , 4 x

(2.110)

while the punch load is easily found from (2.107). Starting with a sufficiently small value of ε 0 , for which a complete solution has just been derived, the analysis can be continued by considering (2.104) and (2.96) as explained before. The distribution of thickness strain and the load–penetration relationship are shown graphically in Figs. 2.26 and 2.27 for a material with n = 0.2, assuming R = a and μ = 0.2. The theory seems to be well supported by available experimental results. It is found that the punch load required for a given depth of penetration is affected only slightly by the coefficient of friction.

2.7.3 Solutions for a Special Material The solution to the stretch-forming problem becomes remarkably simple when the material is assumed to have a special strain-hardening characteristic. From the practical point of view, such a solution is extremely useful in understanding the physical behavior of the forming process and in predicting certain physical quantities with reasonable accuracy. The stress–strain curve for the special material is represented by

2.7

Stretch Forming of Sheet Metals

107

Fig. 2.26 Distribution of thickness strain in the stretch-forming process using a hemispherical punch with R = a (after N.M. Wang, 1970)

Fig. 2.27 Dimensionless load–penetration behavior in the punch stretching process using R = a (n = 0.2, μ = 0.2)

108

2

Problems in Plane Stress

σ = Y exp (ε) , where Y is the initial yield stress. The stress–strain curve is unlike that of any real metal, but the solution based on it should provide a good approximation for sufficiently prestrained metals (Hill, 1950b). Considering the hydrostatic bulging process, it is easy to see that the assumed strain-hardening law requires the bulge to be a spherical cap having a radius of curvature ρ, which is given by simple geometry as ρ=

h2 + a2 = a cos ec α, 2h

(2.111)

where α is the semiangle of the cap. Indeed, it follows from (2.90a) that σθ = σφ = σ = pρ/2t when ρθ = ρφ = ρ, indicating that tσ is a constant at each stage. Since t = t0 exp(–ε), we recover the assumed stress–strain curve. Substituting from (2.91) into the relation εθ = εφ = ε/2 given by the flow rule, and using the fact that r = ρ sin φ, we get r0

∂φ = sin φ ∂r0

or

r0 tan (φ/2) = a tan (α/2)

in view of the condition r0 = a at φ = α. The strain distribution over the bulge is therefore given by ε = 2 ln

r 1 + cos φ hz = 2 ln = 2 ln 1 + 2 . r0 1 + cos α a

(2.112)

It follows from (2.112) that ε = 0 at φ = α, indicating that there is no straining at the clamped edge, which merely rotates to allow the increase in bulge height. The magnitude of the polar thickness strain is ε0 = 4 ln sec

α h2 = 2 ln 1 + 2 . 2 a

(2.113)

This relation is displayed by a broken curve in Fig. 2.24 for comparison. To obtain the velocity distribution, we consider the rate of change of (2.112) following the particle, as well as that of the geometrical relation r = a (sin φ/ sin α) , taking α as the time scale. The resulting pair of equations for v and φ˙ may be solved to give v cos φ − cos α z = = , r sin α a

φ˙ =

sin φ r = . sin α a

The rate of change of the above expression for z and the substitution for φ˙ furnish the result z˙ = v cot φ, which shows that the resultant velocity of each particle is along the outward normal to the momentary profile of the bulge.

2.7

Stretch Forming of Sheet Metals

109

The relationship between the polar strain and the polar radius of curvature obtained for the special material provides a good approximation for a wide variety of metals. From (2.111) and (2.113), it is easily shown that

a ≈ 2ε0 exp − 38 ε0 . ρ to a close approximation. When the pressure attains a maximum, the parameter tσ /ρ has a stationary value at the pole, giving 1 1 dσ0 1 dρ 11 − =1+ ≈ σ0 dε0 ρ dε0 8 2ε0 in view of the preceding expression for ρ as a function of εθ . For the simple power law σ0 = Cε0n , the polar strain at the onset of instability therefore becomes ε0 =

4 11

(+2n) .

4 The instability strain is thus equal to 11 for a nonhardening material (n = 0). This explains the usefulness of the bulge test as a means of obtaining the stress– strain curve of metals for large plastic strains. The special material is also useful in deriving an analytical solution for stretch forming over a hemispherical punch head, provided friction is neglected (Chakrabarty, 1970). The stress–strain curve is then consistent with the assumption of a balanced biaxial state of stress throughout the deforming surface. We begin with the unsupported region, for which the equilibrium in the normal direction requires

ρφ σφ 1 ∂r tan φ =− =− σθ ρθ r ∂φ in view of the first equation of (2.90a) with p = 0. The assumption σ θ = σ φ therefore implies ρ θ = –ρφ over the unsupported region, giving ∂r = −r cot φ ∂φ

or

r sin α = , a sin φ

(2.114)

in view of the boundary condition φ = α at r = a. Since r = R sin β at φ = β, the angles β and α are related to one another by the equation sin α =

R 2 sin β a

(2.115)

It may be noted that the meridional radius of curvature changes discontinuously from –R to R across φ = β. The shape of the unsupported surface is given by the differential equation

110

2

Problems in Plane Stress

sin α ∂r ∂z = − tan φ =a , ∂φ ∂φ sin φ which is integrated under the boundary condition z = 0 at φ = α to obtain z tan (φ/2) = sin α ln . a tan (α/2)

(2.116)

The unsupported surface is actually a minimal surface since the mean curvature vanishes at each point. It is known from the geometry of surfaces that the only minimal surface of revolution is the catenoid. Indeed, the elimination of φ between (2.114) and (2.116) leads to the geometrical relation

z α r = sin α cosh cos ecα + ln tan a a 2 which is the equation of a catenoid. The flow rule requires εθ = εφ = ε/2, and the substitution from (2.91) and (2.114) gives r0

∂φ = − sin φ ∂r0

or

r0 tan (α/2) = , a tan (φ/2)

in view of the boundary condition r0 = a at φ = α. The expressions for r and r0 furnish the compressive thickness strain as 1 + cos α r ε = 2 ln = 2 ln , R sin β ≤ r ≤ a. r0 1 + cos φ

(2.117)

Since tσ is constant at each stage due to the assumed strain-hardening law, and the fact that ε = ln (t0 /t) , the first equation of (2.90) is identically satisfied. Over the region of contact, ρθ = ρφ = R, and equilibrium requires σθ = σφ = σ in the absence of friction, giving p = 2σ t/R. In view of the relations εθ = ε φ and r = R sinφ, the initial radius r0 to a typical particle is given by r0

∂φ = sin φ ∂r

or

r0 tan (α/2) tan (φ/2) , = a tan2 (β/2)

in view of the condition of continuity across φ = β. The compressive thickness strain therefore becomes r (1 + cos α) (1 + cos φ) , 0 ≤ r ≤ R sin β, (2.118) = 2 ln ε = 2 ln r0 (1 + cos β)2 The continuity of the strains evidently ensures the continuity of the stresses across the contact boundary. The thickness has a minimum value at the pole when there is no friction between the material and the punch head. The total penetration of the punch at any stage is obtained from (2.116) and the fact that the height of the pole above the contact boundary is equal to R(1–cos β).

2.8

Deep Drawing of Cylindrical Cups

111

Hence tan (β/2) h = (1 − cos β) + sin2 β ln R tan (α/2)

(2.119)

in view of (2.115). Available experimental results indicate that the relationship between h/R and β is practically independent of the material properties when the punch is well lubricated. It is therefore a good approximation to assume that the relationship between h/R and P/2Rt0 C is independent of the strain-hardening characteristic. Then, for the simple power law σ = Cε n , the load–penetration relationship is given parametrically through β by (2.119) and the formula P = 2π Rt0 C

1 + cos β 1 + cos α

2

1 + cos α n sin2 β 2 ln . 1 + cos β

(2.120)

This expression is obtained from the fact that t∗ σ ∗ is equal to t0 C(ε ∗ )n exp(–ε∗ ), where ε ∗ is given by (2.117) with φ = β. It is easily shown that the punch load given by (2.120) does not have a stationary value in the interval 0 < β 2.76. = 4 tan + cot β + 2M0 α α α β2 + 1 (4.115) For 1 ≤ α ≤ 2.76, Equation (4.113) gives a lower value of the collapse load for a simply supported plate. As α tend to infinity, the ratio β/α tends to unity, and the right-hand side of (4.115) tends to the asymptotic value of 2 + π . A graphical plot of the complete upper bound solution, furnished by (4.113) and (4.115), is included as broken lines in Fig. 4.25 for a visual comparison with the exact solution. The collapse load for a simply supported elliptical plate which carries a uniformly distributed load of intensity p may be obtained in a similar manner. In the case of complete collapse of the plate, the rate of internal energy dissipation is the same as that for the centrally loaded plate, but the rate of external work done is now becomes (pw0 /3)(π a b). The work equation therefore furnishes the upper bound solution as 2 b2 α +1 pb2 = 3 1 + . (4.116) =3 M0 α2 a2 The upper bound solution defined by (4.116) does not differ significantly from the exact solution (Sawczuk, 1989), which is known only over the range 1 ≤ α ≤ 3.52. As indicated earlier, the upper bound load would be exactly doubled if the plate were fully clamped. The yield line upper bound for a uniformly loaded rectangular plate has been given in Section 4.5 (ii). When a part of the boundary of the plate is unsupported, a realistic collapse mode requires the center of the yield line fan to be located outside the plate. It is necessary in this case to introduce the factor (1 − ρ1 /ρ) in the last integral of (4.109), where ρ 1 denotes the length of the radius vector to the point of intersection of a generic

4.7

Minimum Weight Design of Plates

293

yield line with the free edge of the plate. Consider, for example, the plastic collapse of a semi-circular plate which is simply supported along the curved edge and is loaded by a uniform line load q per unit length along the straight edge AB, which is unsupported. Assuming the yield line pattern shown in Fig. 4.28b, in which the center O of the fan is taken at a distance a from the free edge, we have π/4 D = M0 w0 −π/4

ρ1 1− ρ

π/4 2ρ 2 ρ 4 1+ 2 − dφ = M0 w0 2 − sec2 φ sec2 φ dφ = M0 w0 . ρ 3 ρ −π/4

(4.117)

in view of the relations ρ1 = a sec φ, ρ = 2a cos φ. If the distance of a generic point of the free edge from its center is denoted by x, then the rate of external work done is a W=

qwdx = qw0 −a

a a 2 ρ1 1 dx = qaw0 1− 1 − sec2 φ sec2 φ dφ = qaw0 . ρ 2 3

−a

−π/4

The work equation W = D finally gives the collapse load q = 2M0 /a, which is essentially due to Johansen (1943). Since the acute angle between the normal to the semi-circular edge and the radius vector nowhere exceeds π/4, the solution is also statically admissible, and the yield line load is therefore the actual collapse load for the considered plate.

4.7 Minimum Weight Design of Plates 4.7.1 Basic Principles Consider the problem of design of a flat plate which is just at the point of collapse under given conditions of loading and support, the thickness of the plate being allowed to vary in such a way that the total volume of the plate is a minimum. The material is assumed to be homogeneous so that the design for minimum volume is identical to that for minimum weight. Let M1 , M2 be the principal bending moments at any point in the plate that collapses under a distribution of transverse pressure p. If w denotes the rate of deflection of the plate whose middle surface is of area A, then pwdA = (M1 κ˙ 1 + M2 κ˙ 2 ) dA = M0 κ˙ 0 dA, where κ˙ 1 ,κ˙ 2 are the principal curvature rates of the middle surface, while κ˙ 0 is an effective curvature rate that depends on the yield function and is given by √ 1/2 2/ 3 κ˙ 12 + κ˙ 1 κ˙ 2 + κ˙ 22 Mises . κ˙ 0 = (1/2) [|κ˙ 1 | + |κ˙ 2 | + |κ˙ 1 + κ˙ 2 |] , Tresca

(4.118)

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4 Plastic Bending of Plates

The plastic moment M0 depends on the local plate thickness, which is denoted by h in the optimum design. If M1∗ , M2∗ denote the principal moments in any other design which is capable of supporting the given distribution of pressure p, then by the principle of virtual work,

pwdA =

∗ M1 κ˙ 1 + M2∗ κ˙ 2 dA ≤

M0∗ κ˙ 0 dA,

where M0∗ is the plastic moment in the second design, characterized by a thickness ∗ distribution h . The inequality arises from the fact that (κ˙ 1 , κ˙ 2 ) need not be associated with M1∗ , M2∗ . It follows from the preceding relations that

M0 − M0∗ κ˙ 0 dA ≤ 0.

(4.119)

Consider first a sandwich plate which has a light-weight core of constant thickness H between two identical face sheets of variable thickness h made of the given material (h 0, the velocity field is kinematically admissible. For the analysis of the spherical vessel, it is instructive to consider the differential equation of equilibrium involving the shearing force Q. Setting r1 = R, r = R sin φ and pr = –p in the second equation of (5.47) and normalizing the forces by the yield force YH, we have ds + s cot φ + nθ + nφ = 2q. dφ Eliminating s cot φ by means of (5.72), where –q must be written for q, leads to the simple differential equation ds + nθ = q. dφ Introducing the yield condition nθ = 1, and using the boundary condition s = 0 at φ = α, where mφ is a maximum, the above equation is readily integrated to give s = (1 − q) (α − φ) , nθ = 1, α ≤ φ ≤ β. nφ = q − (1 − q) (α − φ) cot φ,

(5.135)

For simplicity, the hexagonal yield condition for the bending moment will be replaced by the square yield condition defined by mθ = ±1 and mφ = ±1. Since κ˙ θ is expected to be positive in view of the assumed mode of collapse, we set mθ = 1 in the second equation of (5.66) to obtain the differential equation d s mφ sin φ = cos φ + sin φ. dφ k Substituting from (5.135), this equation is integrated under the boundary condition mφ = 1 at φ = α to give mφ = 1 −

1−q sin α + (α − φ) cot φ , β ≤ φ ≤ α. 1− k sin φ

(5.136)

The continuity condition mφ = –h2 /H2 at junction φ = β, where sin β = a/R, furnishes the relationship between q and a as ⎫ ⎧ ' ⎨ 2 aH h2 a a⎬ = 1+ 2 . (5.137) (1 − q) (α − β) − 1 − 2 − sin α + ⎩ R ⎭ 4R2 R H Since q < 1 due to the weakening effect of the nozzle, the stress distribution predicted by (5.135) and (5.136) is statically admissible. The velocity distribution

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5 Plastic Analysis of Shells

can be found from the associated flow rule, which gives λ˙ φ = κ˙ φ = 0, the corresponding velocity equations being du − w = 0, dφ

dw + u = c, dφ

in view of (5.67), where c is a constant velocity. Using the boundary condition u = w = 0 at φ = α, the solution is obtained as w = −c sin (α − φ) ,

u = c [1 − cos (α − φ)] .

(5.138)

It follows from (5.67) and (5.138) that λ˙ θ > 0 and κ˙ θ > 0 when c is positive, indicating that the velocity field is kinematically admissible. The continuity of the velocity vector across φ = β requires w0 = c (cos β − cos α) ,

u0 = c (sin α − sin β) .

The velocity distribution throughout the region of deformation is therefore completely determined in terms of a single constant c, which is of arbitrary magnitude at the incipient collapse. The collapse pressure p must be determined from the condition that the resultant of the meridional and shearing forces is continuous across the interface φ = β (Fig. 5.26). Resolving horizontally and vertically, we get Nβ cos β − Qβ sin β = Q0 ,

Nβ sin β + Qβ cos β = pa/2,

where Q0 is the value of Q in the cylinder at x = 0 and the subscript β refers to quantities at φ = β. The second equation of the above pair is equivalent to (5.72), considered at φ = β, with the necessary sign change of q. Since Nβ cos β − Qβ sin β = YH q cos β − (1 − q) (α − β) cosecβ in view of (5.135), while Q0 is the value of Yhs at ξ = 0 given by (5.134), the collapse pressure is given by ⎧ ⎫ ⎨ ' ⎬ 2 a R h a 1 − ηq, q 1 − 2 − (1 − q) (α − β) = ⎭ H h⎩ a R

(5.139)

where β = sin–1 (a/R). For given values of the ratios h/a, H/R, and a/R, (5.137) and (5.139) can be solved simultaneously for q and a. In Fig. 5.27, the parameters q and α – β at the incipient collapse are plotted against a/R for different values of H/R in the special case of η = 1. The collapse pressure is actual for the assumed yield condition, but provides only an upper bound for a non-hardening Tresca material.

5.7

Minimum Weight Design of Shells

389

Fig. 5.27 Collapse pressure and the extent of the associated plastic region in a spherical pressure vessel with a projecting cylindrical nozzle

5.7 Minimum Weight Design of Shells 5.7.1 Principles for Optimum Design The problem of optimum design considered here involves the requirement of minimizing the weight of the material of the shell that is capable of supporting given loads. For a homogeneous material, this is equivalent to finding the minimum volume of shell, which will provide us with a basis for comparison with any actual design. Since the shell carries both direct forces and bending moments, it is convenient to discuss the design criterion in terms of the rate of plastic energy dissipation D per unit area of the middle surface. The dissipation rate is a monotonically increasing function of the local shell thickness h and depends also on the velocity vj of the middle surface. For a shell of variable thickness h which is just at the point of collapse under a boundary traction Tj and a velocity field vj , we have

D vj dA =

Tj vj dA

(5.140)

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5 Plastic Analysis of Shells

in the absence of body forces, the integrals being extended over the entire area of the middle surface. Consider now a neighboring thickness distribution h∗ = h + δh for the same geometry of the middle surface, where δh is an infinitesimally small variation in thickness. The dissipation rate for the shell of thickness h∗ in the deformation mode vj may be written as ∂ D∗ vj = D vj + D vj δh ∂h on neglecting higher-order terms, provided D(vj ) is a continuously differentiable function of h. Since the corresponding distribution of the generalized stresses in the shell of thickness h∗ is not necessarily in equilibrium, it follows from the upper bound theorem of limit analysis that

D∗ vj dA ≥

Tj vj dA.

Substituting for D∗ (vj ), and using (5.140), the preceding inequality is reduced to

∂ D vj δhdA ≥ 0. ∂h

(5.141)

An immediate consequence of inequality (5.141) is that if δD/δh is a positive constant for the shell of thickness distribution h, then

δhdA = δ

hdA ≥ 0.

This inequality implies that the volume of the shell of thickness h, which is designed to collapse under the given loads, is a relative minimum when δD/δh has a constant value over the middle surface. The minimum volume criterion established here, which is due to Shield (1960b), is an extension of that given earlier for the bending of plates (Section 4.5). A stronger result can be obtained for the ideal sandwich shell which consists of a core of thickness H between two thin identical face sheets each of thickness h/2. The membrane forces and bending moments across any section are carried by the face sheets, while the core carries only the shearing force. The strain rate in each of the face sheets may be considered as constant, so that the dissipation rate D is proportional to h. The core thickness H is assumed to be prescribed, and the face sheet thickness is to be determined for minimum volume so that the shell is just at the point of collapse under the given loads Tj . To obtain the condition for minimum volume of the sandwich shell, consider any sheet thickness distribution h∗ for which the shell is at or below collapse under the loads Tj and in the velocity pattern vj . Since the dissipation rate for this shell is D∗ (vj ) = (h∗ /h)D(vj ), we have

5.7

Minimum Weight Design of Shells

391

∗ h /h D vj dA ≥

Tj vj dA

by the upper bound theorem of limit analysis. The elimination of the right-hand side of the above inequality by means of (5.140) results in

∗ h − h D vj /h dA ≥ 0.

(5.142)

If the shell of thickness h is such that D(vj )/h is a positive constant, then (5.142) reduces to

∗

h dA ≥

hdA.

Thus, a sandwich shell designed to collapse in a mode that makes the ratio D/h a constant over the middle surface provides an absolute minimum for its volume under the prescribed loads (Shield, 1960). The condition D/h = constant for the optimum design of a sandwich shell is independent of the design thickness due to the linear dependence of D on h. For a solid shell, on the other hand, the condition ∂D/∂h = constant for the optimum design involves the thickness h, which renders the problem more complicated. The question of uniqueness of the optimum design has been examined by Hu and Shield (1961), and some criteria for minimum cost design have been discussed by Prager and Shield (1967).

5.7.2 Basic Theory for Cylindrical Shells Consider the particular case of a circular cylindrical shell of sandwich construction under axially symmetrical loading. When the applied load is an internal pressure p, which may vary with the axial coordinate x, the axial force Nx is a constant, and equilibrium also requires dMx = Q, dx

dQ Nθ =p− . dx a

These equations follow from (5.3) with a change in sign for p. The shear force Q is easily eliminated between the above equations to give ∂ 2 Mx Nθ = p. + 2 a ∂x

(5.143)

The circumferential bending moment Mθ is a passive or induced moment that arises from the fact that the corresponding curvature rate κ˙θ vanishes because of symmetry. The rate of dissipation of internal energy per unit area of the middle surface is

392

5 Plastic Analysis of Shells

D = Mx κ˙ x + Nθ λ˙ θ + Nx λ˙ x ,

(5.144)

where κ˙ x is the axial curvature rate, and λ˙ θ ,λ˙ x are the circumferential and axial extension rates of the middle surface. They are given by κ˙ x =

d2 w , dx2

λ˙ θ =

w , a

λ˙ x =

du , dx

(5.145)

where u and w are the axial and radially outward velocities of the middle surface. Suppose that the material of the face sheets is ideally plastic obeying Tresca’s yield criterion and the associated flow rule. The corresponding yield conditions in terms of the generalized stresses have been discussed in Sections 5.1 and 5.2. When the axial force is absent (Nx = 0), the condition D/h = constant for minimum volume restricts the stress point to be at one of the corners of the yield hexagon shown in Fig. 5.28a. Indeed, D vanishes along the sides BC and EF of the hexagon and is a function of x along the remaining sides of the hexagon. For the stress point lying at the corners, the condition D/h = constant may be written as λ˙ = constant, λ˙ ≥ H κ, ˙ corners A and D, λ˙ + H κ˙ = constant, λ˙ ≤ H κ, ˙ points B, C, E, F,

(5.146)

where λ˙ and κ˙ denote the absolute values of λ˙ θ and κ˙ x , respectively. The above relations are directly obtained by setting the values Nθ and Mx in (5.144) for the considered stress point and using the fact that N0 = Yh and M0 = YHh/2 for the sandwich shell having a uniaxial yield stress Y.

Fig. 5.28 Yield condition for a cylindrical sandwich shell. (a) Yield locus for no axial force and (b) a part of the yield surface for nonzero axial force

When the axial force is present (Nx = 0), the part of the yield surface for which the membrane forces N0 and Nx are positive is shown in Fig. 5.28b. It can be shown that only the stress states represented by points on the edges AG, AH, BG, FH, GK, HK, and KL, and also those on the plane AGKH can be associated with a rate of

5.7

Minimum Weight Design of Shells

393

deformation for which the condition D/h = constant is satisfied. Considering, for instance, stress points on the edges GK and HK, the condition D/h = constant is found to imply 2λ˙ + H κ˙ = constant,

λ˙ θ ≥ 0,

2λ˙ x = H κ. ˙

(5.147)

Similar relations can be established for the other plastic regimes relevant to the optimum design. The preceding results, as well as their applications to be discussed later in this section, are essentially due to Shield (1960). When the material yields according to the von Mises criterion, the stress point has the freedom to move along the yield locus, which for zero axial load is an ellipse circumscribing the Tresca hexagon. By the flow rule (5.9) associated with the yield condition (5.8), the dimensionless stress resultants can be expressed in terms of the generalized strain rates. Thus nθ = N0 λ˙ θ /D,

mx = 4M0 κ˙ x /3D,

3 n2θ + m2x = 1. 4

Since D/h is a constant for the optimum design, we write D = chY, where c is an arbitrary positive constant that can be associated with the mode of collapse. The above relations then give nθ = λ˙ θ /c,

mx = 2H κ˙ x /3c,

λ˙ 2θ + (H/3) λ˙ 2x = c2 .

(5.148)

In view of (5.145), the quantities nθ and mx are obtained from (5.148) as functions of the velocity w, the spatial distribution of which is determined by the numerical integration of the last equation of (5.148). The bending moment distribution can be found by the integration of (5.143) after eliminating Nθ by means of the relation Nθ /Mx = N0 nθ /M0 mx = 3λ˙ θ /H 2 κ˙ x . The design thickness h at each section is finally obtained from the ratio of Mx to mx . Applications of the basic theory to the von Mises material have been discussed by Reiss and Megarefs (1969). The optimum design of cylindrical shells based on the calculus of variation and the Tresca theory has been considered by Freiberger (1956), following an earlier work by Onat and Prager (1955).

5.7.3 Simply Supported Shell Without End Load As a first example illustrating the preceding theory, consider an open-ended cylindrical shell of length 2l loaded by a uniform internal pressure p. If the shell is simply supported at both ends, which correspond to x = 0 and x = 2l, the bending moment Mx and the radial velocity w must vanish at these sections. For relatively short shells, the mode of collapse should involve bending of the entire shell, and the stress point F in Fig. 5.28a will therefore apply throughout the shell. Using the second equation

394

5 Plastic Analysis of Shells

of (5.146), where λ˙ = λ˙ θ and κ˙ = κ˙ x , the differential equation for w in the optimum design is obtained as aH

d2 w − w = −w0 dx2

(5.149)

in view of (5.145), the quantity w0 being a positive constant velocity. Integrating, and using the boundary conditions w = 0 at x =0 and x = 2l, the solution is found as cosh (α − ξ ) w = w0 1 − , cosh α √ √ where ξ = x/ aH and α = l/ aH. The velocity field will be associated with the stress state at corner F if the first inequality of (5.146) is also satisfied. This gives w ≤ −d2 w/dξ 2

or

cosh α ≤ 2 cosh (α − ξ ) .

This condition will be satisfied throughout the shell if it holds at the central section ξ = α. Hence α ≤ cosh−1 2 or

√ l ≤ 1.317 aH.

The thickness distribution in the optimum design, when the shell is sufficiently short, is obtained by setting Mx = –YHh/2 and Nθ = Yh/2 in the equilibrium equation (5.143), the resulting differential equation for h being aH

2pa d2 h . =− 2 Y dx

(5.150)

Since the bending moment must vanish at the ends of the shell, the boundary conditions are h = 0 at x = 0 and x = 2l, and the solution is hY cosh (α − ξ ) =2 1− , α ≤ 1.317. pa cosh α

(151)

It is interesting to note that the variation of hY/pa with ξ in the case of short shells is identical to that of w/w0 with ξ , except for a scale factor. For longer shells (α ≥ 1.317), a central portion of the shell is stressed by Nθ alone, corresponding to corner A of the yield locus, Fig. 5.28a. The stress state √ in the remainder of the shell corresponds to point F and covers a length d = δ aH on either side of the central portion, where δ is a constant. Over the central portion, the condition λ˙ θ = constant for minimum volume requires w to be constant in the region. Considering the end portion 0 ≤ x ≤ d, the velocity distribution is easily determined from the differential equation (5.149) and the boundary conditions w = 0 at x = 0 and dw/dx = 0 (and hence continuous) at x = d, the result being

5.7

Minimum Weight Design of Shells

395

cosh (δ − ξ ) w = w0 1 − , 0 ≤ ξ ≤ δ. cosh δ

(5.152)

The constant δ is obtained from the requirement λ˙ = H κ, ˙ or w = –d2 w/dξ 2 at ξ = δ, giving δ = cosh−1 2 = 1.317. By the condition of continuity at ξ = δ, the velocity in the central portion of the shell is w = w0 /2. The inequality w ≤ –d2 w/dξ 2 in the end portion and the inequality w ≥ –d2 w/dξ 2 in the central portion are identically satisfied. The thickness distribution in each portion of the shell for minimum volume is determined from the equilibrium equation and the appropriate yield condition. In the end portion 0 ≤ x ≤ δ, the resulting differential equation is (5.150), which must be solved under the boundary conditions h = 0 at x = 0 and x = d, the bending moment at these sections being zero. The solution is easily shown to be hY 1 = 2 1 − √ [sinh ξ + sinh (δ − ξ )] , pa 3

0 ≤ ξ ≤ δ.

(5.153a)

In the central portion of the shell, the stress point corresponds to Mx = 0 and N0 = Yh, and (5.143) immediately furnishes hY/pa = 1,

δ ≤ ξ ≤ α,

(5.153b)

only one-half of the shell being considered because of symmetry. The thickness changes discontinuously from 0 to pa/Y at x = d. Since the thickness gradient dh/dx is zero in the central portion, but is nonzero in the end portion, the shearing force Q = dMx /dx is also discontinuous at x = d. The discontinuity in Q is removed, however, by adding a flange of vanishingly small width but of finite area of cross section at x = d. The total volume V of the face sheets in the optimum design is now determined by integration over the area of the middle surface, using the expression

l

V = 4π a 0

√ hdx = 4π a aH

a

hdξ .

(5.154)

0

Substituting from (5.151) and (5.153) for short and long shells, respectively, we obtain ⎧ ⎨ 8π a2 l (p/Y) (1 − tanh α/α) α ≤ 1.317,

√ (5.155) V= ⎩ 8π a2 l (p/Y) 1 − 3 − δ /α , α ≥ 1.317, √ A term equal to 1/ 3α has been included in the square brackets of (5.155) to take account of the flanges so that volume is continuous at α = δ = 1.317.

396

5 Plastic Analysis of Shells

It is instructive to compare the volume for the optimum design with that for the constant thickness design for the sandwich shell. To this end, we express the equilibrium equation (5.143) in the dimensionless form d 2 mx + 2nθ = 2q, dξ 2 where q = pa/Yh0 , with h0 /2 denoting the thickness of each face sheet. For a sufficiently short shell, the state of stress involves mx < 0 and nθ > 0, so that side AF of the yield hexagon applies throughout the shell. Using the yield condition 2nθ – mx = 2 to eliminate nθ from the equilibrium equation, we get d 2 mx + mx = 2 (q − 1) . dξ 2 In view of the boundary conditions mx = 0 at ξ = a and dmx /dξ = 0 at ξ = a, the solution becomes cos (α − ξ ) π (5.156) , α≤ . mx = 2 (q − 1) 1 − cos α 2 The velocity equation associated with the yield condition gives w = w0 sin ξ (ξ ≤ a) in view of the boundary condition w = 0 at ξ =0. The velocity slope dw/dξ is discontinuous at ξ = α < π /2, giving rise to a hinge circle, where mx = – 1. By (5.156), this condition furnishes q=

π 2 − cos α , α≤ . 2 − 2 cos α 2

(5.157)

For longer shells (α ≥ π/2), the plastic state of stress is represented by corner A of the yield hexagon. Then mx = 0 and nθ = 1, giving q = 1 for plastic collapse of the simply supported shell, the associated velocity field being w = w0 sin ξ (ξ ≤ π/2) ,

w = w0 (ξ ≥ π/2) .

Since the total volume of the face sheet material in the constant thickness design is V0 = 4π alh0 , the ratio of the face sheet volumes for the optimum and constant thickness designs may be written from (5.157) as 2q (1 − tanh α/α) V , α ≤ 1.317, √ = V0 3 − δ /α α ≥ 1.317. q 1−

(5.158)

The parameter q is given by (5.157) when α ≤ π /2 and is equal to unity when α ≥ π /2. The ratio V/V0 is plotted as a function of α in Fig. 5.29 on the basis of (5.158) for the simply supported shell. The discontinuities in slope of the curves at α = 1.317 and α = 1.571 are due to the change in character of the solution for the

5.7

Minimum Weight Design of Shells

397

Fig. 5.29 Ratio of face sheet volume in the optimum design to that in the constant thickness design of a cylindrical shell under uniform internal pressure

optimum design and constant thickness design, respectively. The saving of material effected by the minimum volume design is quite appreciable for short shells.

5.7.4 Cylindrical Shell with Built-In Supports √ A circular cylindrical shell of semilength l = α ah is provided with built-in support at its open ends and is subjected to a uniform internal pressure p. The condition D/h = constant for the minimum volume design requires the velocity slope to vanish at the clamped edges √ when the shell is at the point of collapse. A portion of the shell of length b = β ah exists at each end where the state of stress is represented by point B in Fig. 5.28a. If the shell is sufficiently short, the remainder of the shell would correspond to point F of the yield hexagon. By the second equation of (5.146), the velocity equation for the optimum design is aH

d2 w ± w = ±w0 , dx2

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5 Plastic Analysis of Shells

where the upper sign applies to the end portions and the lower sign to the central portion of the shell. Considering only one-half of the shell and using the boundary conditions w = dw/dx = 0 at x = 0,

dw/dx = 0 at x = l,

and the condition of continuity of w at x = b, the solution for the velocity at the incipient collapse is obtained as 0 ≤ ξ ≤ β, w = w0 (1 − cos ξ ) , w = w0 1 − cos β cosh (α − ξ )/cosh (α − β) ,

β ≤ ξ ≤ α.

(5.159)

Since dw/dx must be continuous at ξ = β for the minimum volume design, the relationship between β and α is tan β = tanh (α − β) .

(5.160)

The compatibility of the velocity field with the stress field associated with corners B and F requires w≤

d2 w (0 ≤ ξ ≤ β) , dξ 2

w≤−

d2 w (β ≤ ξ ≤ α) . dξ 2

In view of (5.160), the above inequalities will be satisfied throughout the respective portions of the shell if 2 cos β ≥ 1,

cosh (α − β) ≤ 2 cos β.

It turns out that the first inequality is satisfied when the second inequality is. Combining the second inequality with (5.160), it is easily shown that α−β ≤

1 cosh−1 4, 2

β ≤ 0.659,

α ≤ 1.691.

(5.161)

The thickness distribution in the shell for α ≤ 1.691 is given by the solution of the differential equation aH

d2 h 2pa ±h=± , 2 dx Y

(5.162)

which is obtained by setting mx = ±YHh/2 and Nθ = Yh in (5.143). The upper sign applies to 0 ≤ x ≤ b and the lower sign to b ≤ x ≤ a. Since the bending moment must be continuous at x = b, it must vanish at this section. Further, symmetry requires the shearing force to vanish at x = l. Hence the boundary conditions are

5.7

Minimum Weight Design of Shells

h = 0 at ξ = β,

399

dh/dξ = 0 at ξ = α.

These conditions, together with the fact that dh/dξ changes sign at ξ = β while retaining its numerical values for the shearing force to be continuous, furnish the solution hY/pa = 2 1 − (cos ξ/ cos β) + 2 tan β sin (β − ξ ) , hY/pa = 2 1 − cosh (α − ξ ) / cosh (α − β) ,

0 ≤ ξ ≤ β, β ≤ ξ ≤ α.

(5.163)

For longer shells with α ≥ 1.691, the minimum volume condition D/h = constant can only be satisfied by introducing a central region of constant radial velocity associated with corner A in Fig. 5.28a. The stress points B and F of the yield hexagon then correspond to the regions 0 ≤ ξ ≤ γ and γ ≤ ξ ≤ δ, respectively, where γ and δ are dimensionless constants. The velocity distribution in these two regions can be written as w = w0 (1 − cos ξ ) , 0 ≤ ξ ≤ γ, w = w0 1 − cos γ cosh (δ − ξ ) / cosh (δ − γ ) ,

(5.164)

γ ≤ ξ ≤ δ.

The velocity at ξ = γ and the velocity slope at ξ = δ are automatically made continuous. The condition of continuity of dw/dξ at ξ = γ, and the requirement w = –d2 w/dξ 2 at ξ = δ furnish tan γ = tanh (δ − γ ) , 2 cos γ = cosh (δ − γ ) , in view of (5.164). These two equations are easily solved for γ and δ to give δ−γ =

1 cosh−1 4, 2

γ = 0.659,

δ = 1.691.

The velocity in the region δ ≤ ξ ≤ a is evidently w = w0 /2 by the condition of continuity. The thickness distribution over the region 0 ≤ ξ ≤ δ is determined by solving (5.162) under the conditions h = 0 at ξ = γ and ξ = δ, so that the bending moment is continuous, and the fact that dh/dξ merely changes its sign at ξ = γ. The result is hY = 2 1 − 1.265 cos ξ + 1.249 sin (γ − ξ ) , pa $ % hY 2 {sinh (ξ − γ ) + sinh (δ − ξ )} , =2 1− pa 3

⎫ ⎪ 0 ≤ ξ ≤ γ ,⎪ ⎪ ⎬ ⎪ ⎪ γ ≤ ξ ≤ δ, ⎪ ⎭

. (5.165a)

Since 2 sin γ = sinh (δ − γ ) = 32 . In the remainder of the considered half of the shell, the state of stress is given by Mx = 0, Nθ = Yh and (5.143) furnishes

400

5 Plastic Analysis of Shells

hY/pa = 1,

δ ≤ ξ ≤ α.

(5.165b)

Since dh/dx is discontinuous at ξ = δ, a flange of vanishingly small width but of finite area of cross section must be added at this section to maintain continuity of the shearing force Q. To obtain the total volume V of the face sheets of the shell designed for minimum volume, it is only necessary to insert in (5.154) the expressions for h in (5.163) and (5.165) for short and long shells, respectively, and to evaluate the integral. Including the volume of the flanges appropriately, so that the results match when α = 1.691, the minimum volume can be written in the dimensionless form 2q (1 − 2 sin β/α) V , √ = V0 6 − δ /α, q 1−

α ≤ 1.619, α ≥ 1.619,

(5.166)

where V0 = 4πalh0 represents the volume of a shell of constant face sheet thickness h0 , and q denotes the quantity pa/h0 Y. The relationship between q and a for the sandwich shell is directly obtained from (5.12) and (5.16) with w replaced by α. When a is very small, q ≈ 2/α 2 , while α ≈ 2 tan β in view of (5.160), so that V/V0 = 12 in the limit α = 0. The ratio V/V0 is plotted as a function of a in Fig. 5.29, which indicates that the saving of material effected by the optimum design is higher for the clamped shell than for the simply supported shell.

5.7.5 Closed-Ended Shell Under Internal Pressure Suppose now that a circular cylindrical shell of length 2l is closed at the ends by rigid plates and is subjected to a uniform internal pressure p. The rigid plates not only produce an axial force Nx = pa/2 per unit circumference but also give rise to clamped edge conditions at the ends of the shell. As in the case of a shell of constant thickness, the yield condition requires Nθ = N0 throughout the shell, only stress states represented by the edges GK and HK of Fig. 5.28b being involved in the minimum volume design. Hence the relationship between the axial force and the bending moment is pa Nx YH h− , =± Mx = ±M0 1 − N0 2 2Y

(5.167)

where the upper sign holds for the line GK and the lower sign for the line HK. The former applies to two identical outer portions of the shell, each having a √ length b =√β aH, and the latter applies to a central portion of length 2 (l − b) = 2 (α − β) aH. The condition D/h = constant, which is equivalent to (5.147) with λ˙ and κ˙ denoting λ˙ θ and ±κ˙ x , respectively, leads to the differential equation

5.7

Minimum Weight Design of Shells

aH

401

d2 w ± 2w = ±2w0 , dx2

where w0 is a positive constant. These two equations cover the entire shell for all values of l and are subject to the boundary conditions w = dw/dx = 0 at x = 0,

dw/dx = 0 at x = l.

Further, w and dw/dx must be continuous at the interface x = b. The solution for √ the velocity field is given by (5.159) with a factor of 2 for each of the quantities ξ , β, and α, the relationship between β and α being tan

√

√ 2β = tanh 2 (α − β) .

(5.168)

Inserting (5.167) into the equilibrium equation (5.143), and setting Nθ = Yh, the differential equation for the thickness h is obtained as aH

2pa d2 h . ± 2h = ± Y dx2

Since Mx must vanish at x = b by the condition of continuity, and the shearing force Q must vanish at x = l because of symmetry, we have h=

pa at ξ = β, 2Y

dh = 0 at ξ = α. dξ

In addition, dh/dξ must change sign at ξ = β without changing its absolute value for the shearing force to be continuous. The solution to the above equation therefore becomes ⎫

√ √ ⎬ hY/pa = 1 − 12 (cos ξ/cos β) + tan 2β sin 2 (β − ξ ) , 0 ≤ ξ ≤ β,⎪ √ √ ⎭ β ≤ ξ ≤ α. ⎪ hY/pa = 1 − 12 cosh 2 (α − ξ ) / cosh 2 (α − β) , (5.169) In Fig. 5.30, the thickness distribution for a closed-ended shell is compared with √ that for an open-ended shell when α = 2. The ratio of the volume of the face sheets for the optimum design to that for the constant thickness design is obtained from (5.154) and (5.169) as ⎧ ⎨

V =q 1− ⎩ V0

sin

√

⎫ 2β ⎬

√ 2α

⎭

=

⎧

2 + α2 ⎨ 1 + α2

⎩

1−

sin

√

⎫ 2β ⎬

√ 2α

⎭

,

(5.170)

where the last step follows from (5.33a) with ω replaced by α. When α tends to zero, β/a tends to 12 and V/V0 tends to unity. The variation of V/V0 with α for the closed-ended shell, computed from (5.168) and (5.170), is included in Fig. 5.29.

402

5 Plastic Analysis of Shells

Fig. 5.30 Thickness distribution in a clamped cylindrical sandwich shell designed for minimum weight under uniform internal pressure

The presence of the axial force appreciably reduces the saving of material caused by the optimum design. The minimum weight design of cylindrical shells based on the von Mises yield criterion has been presented by Reiss and Magarefs (969). The optimum design of closed pressure vessel heads has been discussed by Hoffman (1962) and by Save and Massonnet (1972). The minimum weight design of conical shells has been considered by Reiss (1974, 1979). The plastic design of shells of revolution for constant strength has been examined by Ziegler (1958), Issler (1964), and Dokmeci (1966). The minimum weight design of membrane shells of revolution subjected to uniform external pressure and vertical load has been investigated by Richmond and Azarkhin (2000).

Problems 5.1 A short cylindrical shell of mean radius a and wall thickness h is under a uniform internal pressure p, and is simply supported at both ends, the length of the shell being denoted by 2l. Adopting the linearized yield condition for a Tresca sandwich shell, show that the dimensionless collapse pressure and the associated velocity field are given by. pa 2 − cos ω = , Yh 2 (1 − cos ω)

sin (ω − ξ ) w = w0 sin ω

√ where ξ = x 2/ah, with x denoting the axial distance measured from the central section, ω is the value of ξ at x = l, and w0 is the deflection rate at the center.... 5.2 A cylindrical shell of length l, thickness h, and mean radius a is clamped at one end and is free at the other. The shell is subjected to a uniform radial pressure p to reach the point of plastic collapse. Using the yield condition for a Tresca sandwich shell, show that the intensity of the pressure and the velocity distribution are given by

Problems

403

pa 2 cosh ω − 1 = , N0 2 (cosh ω − 1)

sinh (ω − ξ ) w = w0 sinh ω

where ξ and ω denote the same quantities as those in the preceding problem, and w0 is the deflection rate at the free end of the shell. 5.3 A cylindrical shell of wall thickness h and mean radius a is free at both ends and is subjected to a radially outward √ ring load P per unit circumference applied at x = 0. Introducing the parameter ξ = x 2/ah, the ends of the shell are defined by ξ = – α and ξ = β. Assuming α and β to be sufficiently small, so that the shell can collapse without the formation of a hinge circle, and using the square yield condition defined by mx = ±1 and nθ = ±1, show that the hoop force changes sign at ξ = λ, where 2λ2 = α 2 + β 2 , and that the collapse load is given by P = N0

h β −α , α2 + β 2 − √ a 2

α ≤ 1,

β≤

√

2+

1 + α2

5.4 For higher values of β in the preceding problem, show that a hinge circle must form at ξ = 1 + α 2 , the outer portion of the shell remaining rigid, the associated collapse load being given by the modified expression P = N0

h α + 1 + α2 , 2a

α < 1,

β>

√ 2 + 1 + α2

√ Over the range α > 1 and β > 2 2, prove that the collapse mode involves a hinge circle at ξ = 0, and the corresponding collapse load is given by P = N0

√ h α+β , 1 G when X > Y, together with two similar inequalities for the first three parameters. Evidently, (6.1) reduces to the von Mises criterion when L = M = N = 3F = 3G = 3H = 3/2Y2 , where Y is the uniaxial yield stress of the isotropic material. For an arbitrary orthotropic material, the first three relations of (6.2) give 2F = Y −2 + Z −2 − X −2 , 2G = Z −2 + X −2 − Y −2 , 2H = X −2 + Y −2 − Z −2 . The state of anisotropy in an element is, therefore, specified by the directions of the three principal axes of anisotropy, and the values of the six independent yield stresses X, Y, Z, R, S, and T, which depend on the degree of previous cold work as well as on the subsequent heat treatment (Hill, 1948). The yield criterion is expressed in the form (6.1) only when the principal axes of anisotropy are taken as the axes of reference. For an arbitrary set of rectangular axes, the form of the yield criterion must be changed by the appropriate transformation of the stress components. When the state of anisotropy is rotationally symmetric about the z-axis, the form of the yield criterion must be independent of the choice of x-and y-axes. Since X = Y and R = S for such a symmetry, it is obviously necessary to set F = G and L = M. To obtain additional conditions for the rotational symmetry, we rewrite (6.1) in the form

2 2 (F + H) σx + σy + 2Fσz σz − σx − σy − 2 (F + 2H) σx σy − τxy

2 2 2 + 2 (N − F − 2H) τxy + τzx = 1. + 2L τyz Since the first four terms of the above expression are invariants for a fixed z-axis, the coefficient of the last term must be identically zero for the yield criterion

6.1

Plastic Flow of Anisotropic Metals

407

to be unaffected by any rotation of the x- and y-axes. The necessary and sufficient conditions for the anisotropy to be rotationally symmetric about the z-axis therefore becomes N = F + 2H = G + 2H, L = M. The number of independent parameters defining a rotationally symmetrical state of anisotropy is therefore reduced to three, which may be taken as the uniaxial yield stresses along and perpendicular to the axis of symmetry and the shear yield stress with respect to these two directions.

6.1.2 Stress–Strain Relations To derive the relations between the stress and the strain increments for an anisotropic material, we adopt the usual normality rule of plastic flow, assuming the plastic potential to be identical to the yield function. Referred to the principal axes of anisotropy, when the elastic strains are disregarded, the strain increment tensor for an orthotropic material may be written as dεij =

∂f dλ, ∂σij

where 2f(σ ij; ) denotes the expression on the left-hand side of (6.1), and dλ is necessarily positive for plastic flow in an element that is stressed to the yield point. Setting 2τ 2 yz = τ 2 yz +r2 zy , etc., in the field function (6.1), and treating all nine components of the stress tensor as independent, we obtain the strain increment relations ⎫ dεx = H σx − σy + G (σx − σz ) dλ, dγxy = Nτxy dλ, ⎪ ⎬ dεy = F σy − σz + H σy − σx dλ, dγyz = Lτyz dλ, , ⎪ ⎭ dεz = G (σz − σx ) + F σz − σy dλ, dγzx = Mτzx dλ,

(6.3)

which constitute a generalization of the Lévy–Mises flow rule. The sum of the three normal strain components is seen to be zero, satisfying the condition of plastic incompressibility. When the principal axes of stress coincide with those of anisotropy, the principal axes of the strain increment also occur in the same directions. In general, however, the principal axes of stress and strain increment do not coincide for an anisotropic material. The preceding results are due to Hill (1948), although similar relations have been given by Jackson et al. (1948) and Dorn (1949). The ratios of the anisotropic parameters can be determined by carrying out tensile tests on specimens cut at suitable orientations with respect to the principal axes of anisotropy. It is, of course, necessary for this purpose that the anisotropy is uniformly distributed through a volume of sufficient extent in order to allow the preparation of the specimens. For a tensile specimen cut parallel to the x-axis of anisotropy, the ratios of the principal strain increments are

408

6 Plastic Anisotropy

dεx :dεy :dεz = G + H: − H: − G.

(6.4)

A longitudinal extension is therefore accompanied by a contraction in each transverse direction, unless the yield stresses differ so much that one of the parameters G and H is negative. The magnitude of the incremental transverse strain is greater in the direction of the lesser yield stress. Tensile tests carried out on specimens cut parallel to the y- and z-axes of anisotropy similarly furnish the ratios F/H and GIF, respectively, providing an immediate test on the theory in view of the identity (HIG)(G/F)(FIH) = 1. Where the theory is applicable, the measurement of strain ratios in the appropriate tensile specimens provide an indirect method of finding the ratios of the yield stresses along the three principal axes of anisotropy.

6.1.3 Variation of Anisotropic Parameters It is assumed at the outset that the material has a very pronounced state of anisotropy, and that further changes in anisotropy during cold work are negligible over the considered range of strains. The yield stresses of the material in the different directions then increase in strict proportion as the material deforms, the factor of proportionality being denoted by a parameter h which increases monotonically from unity to represent the amount of hardening. Thus X = hXo, Y = hYo, etc., where the subscript zero denotes the initial value, giving F = Fo/h2 , G= Go/h2 , etc. The anisotropic parameters therefore decrease in strict proportion, and their ratios remain constant during the deformation. The scalar parameter h is a dimensionless form of the equivalent stress σ¯ which may be defined as '

3 3 =h , σ¯ = h 2 (F0 + G0 + H0 ) 2c

c = F0 + G6 + H0 ,

so that h is equal to σ¯ /Y for an isotropic material with an initial yield stress Y. The substitution for F = F0 /h2 , etc., in (6.1) furnishes h2 in terms of the initial values of the anisotropic parameters, and the expression for the equivalent stress becomes 2 2 3 F0 σy − σz + G0 (σz − σx )2 + H0 σx − σy σ¯ = 2c (6.5) 1/2 2 2 2 +2L0 τyz + 2M0 τzx + 2N0 τxy . As in the case of isotropic solids, σ¯ may be regarded as a function of an equivalent strain whose increment may be defined according to the hypothesis of strain equivalence as dε¯ =

2 3

2 dεx2 + dεy2 + dεz2 + 2dγxy + 2dγxz2

1/2

.

(6.6)

6.1

Plastic Flow of Anisotropic Metals

409

When a uniaxial tension X is applied in the x-direction, the ratios of the nonzero components of the strain increment are given by (6.4), and the equivalent stress and strain increments become ' 3 G0 + H0 X, σ¯ = 2 F0 + G0 + H0

2 G20 + G0 H0 + H02 dε = √ dεx . 3 (G0 + H0 )

(6.7)

Similar expressions for σ¯ and dε are obtained for uniaxial tensions Y and Z applied in the y- and z-directions, respectively. A comparison of the stress–stress curves along the three principal axes of anisotropy provides a direct means of testing the hypothesis. Consider the alternative hypothesis in which σ¯ is assumed to be a function of the total plastic work per unit volume of the element. This has been proposed by Jackson et al. (1948) and was later followed by Hill (1950a). The increment of plastic work per unit volume is dW = σij dεij = σij

∂f dλ = 2fdλ = dλ ∂σij

by Euler’s theorem of homogeneous functions and by the yield criterion expressed as 2 f = 1. From (6.3), we have Gdεy − Hdεz = (FG + GH + HF) σy − σz dλ, together with two similar relations obtained by cyclic permutation. The substitution for the normal stress differences into the yield criterion (6.1) then gives 1

Gdεy − Hdεz 2 2dγyz2 = (dλ)2 . + F FG + GH + HF L

Since dW = σ¯ d¯ε , where d¯ε is the equivalent strain increment according to the hypothesis of work equivalence, we have 1/2 2 dλ dλ = . (F dε¯ = 0 + G0 + H0 ) σ¯ 3 h Substituting for dλ, and using the relations F = F0 /h2 , etc., the equivalent strain increment according to the work-hardening hypothesis is finally obtained as

dε¯ =

1/2 2 (F0 + G0 + H0 ) 3 1/2 2 2γyz2 G0 dεy + H0 dεz × F0 ··· + + ··· . F 0 G0 + G 0 H0 + H 0 F 0 L0

(6.8)

410

6 Plastic Anisotropy

For a uniaxial tension X parallel to the x-axis of anisotropy, the expressions for σ¯ and d¯ε are ' 3 G0 + H0 X, σ¯ = 2 F0 + G 0 + H0

' 2 F0 + G0 + H0 dε¯ = dεx . 3 G0 + H 0

For an isotropic material, the equivalent strain increments defined by (6.6) and (6.8) are identical, and the two hypotheses for the hardening process are therefore equivalent. For an anisotropic material, the two hypothesis are distinct, and the predicted stress–strain curves along one axis of anisotropy derived from another will generally be different in the two cases. The choice of the appropriate expression for the equivalent strain increment for a particular material must be decided by experiment. The more general case of hardening of an orthotropic material, involving both expansion and translation of the yield surface in the stress space, has recently been discussed by Kojic et al. (1996).

6.2 Anisotropy of Rolled Sheets 6.2.1 Variation of Yield Stress and Strain Ratio In a rolled sheet of metal, the principal axes of anisotropy are along the rolling, transverse, and through-thickness directions at each point of the sheet. Let the axes of reference be so chosen that the x-axis coincides with the direction of rolling, the y-axis with the transverse direction in the plane, and the z-axis with the normal to the plane. If the sheet is subjected to forces in its plane, the only nonzero stress components are σ x , σ y ,and τ xy , and the yield criterion (6.1) reduces to 2 = 1. (G + H) σx2 − 2Hσx σy + (H + F) σY2 + 2 Nτxy

(6.9)

Let σ denote the uniaxial yield stress of the sheet metal in a direction making a counterclockwise angle α with the rolling direction. The stress components corresponding to a uniaxial tension σ applied in the α-direction are σx = σ cos2 α,

σy = σ sin2 α,

τxy = σ sin α cos α.

(6.9a)

The substitution of (6.9a) into the yield criterion (6.9) furnishes σ as a function of α for any given state of anisotropy, the result being −1/2 . (6.10) σ = F sin2 α + G cos2 α + H + (2 N − F − G − 4H) sin2 α cos2 α The uniaxial yield stress σ can be shown to have maximum and minimum values along the anisotropic axes, and also in the directions α = ±α0 , where

6.2

Anisotropy of Rolled Sheets

411

−1

α0 = tan

N − G − 2H . N − F − 2H

(6.11)

When N is greater than both F + 2H and G +2H, the yield stress has maximum unequal values in the x- and y-directions, and minimum equal values in the α0 -directions. If N is less than both F + 2H and G+2H, the yield stress has minimum unequal values in the x- and y-directions, and maximum equal values in the α0 -directions. When N lies between F + 2H and G + 2H, there is no real α0 , and σ is a maximum in the x-direction and a minimum in the y-direction if F > G, and vice versa if F where α0 is given by (6.11) and the two possible necks are then equally inclined to the direction of the applied tension. A localized neck is able to develop only if the rate of work-hardening of the material is lower than a certain critical value, for which it is exactly balanced by the rate of reduction of thickness in the neck. Since the normal stresses transmitted across the neck are proportional to the applied tension a, we have dε dσ dσ σ = −dεz = , or = , σ 1+R dε 1+R

(6.15)

where dε is the longitudinal strain increment, and R is given by (6.13). The critical subtangent to the appropriate stress–strain curve for localized necking is therefore (1+ R) times that for diffuse necking. Hence, a localized neck can be expected to form on a superimposed diffuse neck as in the case of an isotropic sheet metal. In the case of a plane sheet subjected to biaxial stresses σ1 and σ2 in the rolling and transverse directions, respectively, the condition of the zero rate of extension along the localized neck, which makes an angle ß with the direction of σ1, furnishes

tan2 β = −

dε1 (G + H) σ1 − Hσ2 = , dε2 Hσ1 − (F + H) σ2

in view of the flow rule (6.3). Assuming σ 1 > σ 2 , the range of stress ratios for which the necking can occur is given by σ 2 /σ 2 ≤ H/(F + H), which ensures dε2 ≤ 0. An analysis for localized necking based on the total strain theory along with the assumption of a yield vertex has been presented by Storen and Rice (1975),

414

6 Plastic Anisotropy

the analysis being similar to that given in Section 2.1. The development of localized necks as a result of void growth has been considered by Needleman and Triantafyllidis (1978). Consider now the initiation of diffuse necking in a thin sheet under biaxial tensile stresses σ 1 and σ 2 along the rolling and transverse directions, respectively. As in the case of isotropic sheets, the uniform deformation mode becomes unstable when the rate of hardening becomes critical. In terms of the initial values of the anisotropic parameters, the yield criterion (6.9) may be written as (G0 + H0 ) σ12 − 2H0 σ1 σ2 + (F0 + H0 ) σ22 = h2 .

(6.16)

It follows from the stress–strain relations and the differentiated form of (6.16) that dσ1 dε1 + dσ2 dε2 = (dh/h) dλ = (dσ¯ /σ¯ ) dλ. If the applied loads simultaneously attain their maximum at the onset of instability, then dσ 1 /σ 1 = dε1 and dσ 2 /σ 2 = dε 2 , and the preceding relation gives dε1 2 dε2 2 1 dσ¯ + σ2 = σ1 σ¯ dλ dλ dλ = (G + H)2 σ13 − H [(H + 2G) σ1 + (H + 2F) σ2 ] σ1 σ2 + (F + G)2 σ23 . The hypothesis of √ work equivalence will be adopted here for simplicity. Using the relation dλ = h 3/2c dε on the left-hand side, substituting for G, H, and F on the right-hand side, and denoting the stress ratio σ 2 /σ1 by ρ, the instability condition is finally obtained in the form

3 (G0 + H0 )2 − H0 (H0 + 2G0 ) ρ − H0 (H0 + 2F0 ) ρ 2 + (F0 + H0 )2 ρ 3 3/2 2 (F0 + G0 + H0 )1/2 (G0 + H0 ) − 2H0 ρ + (F0 + H0 ) ρ 2 (6.17) in view of (6.16). The expression on the right-hand side, which is due to Moore and Wallace (1964), can be evaluated for any given p using the measured r-values in the rolling and transverse directions. An instability condition similar to (6.17) follows for the hypothesis of strain equivalence. The physical significance of the instability condition (6.17), which reduces to (2.38) when the material is isotropic, has been discussed by Dillamore et al. (1972). 1 dσ¯ = σ¯ dε

6.2.3 Correlation of Stress–Strain Curves Consider a uniaxial tension σ applied to a specimen cut of an angle a to the direction of rolling. When the hypothesis of strain equivalence is adopted, the equivalent strain increment is most conveniently obtained by using the property of its invariance. Thus, by taking the x-axis temporarily along the axis of the specimen, we get

6.2

Anisotropy of Rolled Sheets

dεx = dε,

415

dεy = −

R dε, 1+R

dεz = −

1 dε, 1+R

where r is given by (6.13). The substitution from above into (6.6) then gives the equivalent strain increment d¯ε , while the equivalent stress σ¯ is directly obtained by inserting (6.9a) in (6.5), the results being ' σ¯ =

3 (1 + R) F0 ξ σ, 2 (F0 + G0 + H0 )

√ 2 1 + R + R2 dε¯ = dε, √ 3 (1 + R)

where ξ = sin2 α + (G0 /F0 ) cos2 α. Let the uniaxial stress–strain curve in the rolling direction be defined by the equation σ = f(ε). In view of (6.7), the equivalent stress– strain curve is given by ⎧ ⎫ ' ⎨ √ 3 3 (G0 + H0 ) ε¯ ⎬ G0 + H0 f σ¯ = . (6.18) 2 F 0 + G 0 + H 0 ⎩ 2 G2 + G H + H 2 ⎭ 0 0 0 0 Substituting for σ and ε into (6.18), and introducing the R-values in the rolling and transverse directions, respectively, the equation for the stress–strain curve in the direction α is obtained as ' ' R y 1 + Rx 1 + Rx 1 + R + R2 f ε . (6.19) σ = ξ Rx 1 + R 1 + R 1 + Rx + R2x The stress–strain curve transverse to the rolling direction is obtained by setting ξ = 1 and R = Ry in (6.19). When the hypothesis of work equivalence is adopted, the equivalent strain increment corresponding to a uniaxial tension σ is equal to (σ/σ¯ )dε, and (6.19) is then replaced by ' σ =

Ry ξ Rx

' Ry 1 + Rx 1 + Rx f ε . 1+R ξ Rx 1 + R

It is evident that the stress–strain curve predicted by this relation will be generally different from that predicted by (6.19), except when the sheet is isotropic in its plane. The effective stress–strain behavior of anisotropic sheet metals has been investigated by Wagoner (1980) and Stout et al. (1983). Consider now a state of balanced biaxial tension σ in the plane of the sheet, which is equivalent to a uniaxial compression σ normal to the plane. If the increment of the compressive thickness strain is denoted by dε, then dεx =

G0 F0 + G 0

dε,

dεy =

F0 F0 + G 0

dε,

dεz = −dε,

on setting σ x = σ y = σ in the stress–strain relations. The expressions for the equivalent stress σ¯ and the equivalent strain increment dε therefore become

416

6 Plastic Anisotropy

' 3 F0 + G 0 σ, σ¯ = 2 F0 + G0 + H0

G0 + H0 dε = F0 + G0

'

F02 + F0 G0 + G20 G20 + G0 H0 + H02

dε,

in view of (6.5) and (6.6). The substitution into (6.18) shows that the stress–strain curve in the through-thickness direction according to the hypothesis of strain equivalence is given by ' σ =

Ry

⎫ ⎧ ' ⎨ 2 + R R + R2 ⎬ R 1 + Rx 1 + Rx x y x y ε . f Rx + R y ⎩ Rx + R y 1 + Rx + R2x ⎭

(6.20)

The hypothesis of work equivalence, on the other hand, leads to the equation for the through-thickness stress–strain curve as ' σ =

Ry

' 1 + Rx 1 + Rx Ry f ε . Rx + Ry Rx + Ry

Experimentally, such a curve is most conveniently obtained by the bulge test, in which a thin circular blank of sheet metal is clamped round the periphery and deformed by uniform fluid pressure applied on one side. Due to the symmetry of the loading, a state of balanced biaxial tension exists at the pole of the bulge, where the compressive thickness strain e is equal to the sum of the two orthogonal surface strains. In Fig. 6.2, the stress–strain curves in the thickness direction given by the above equations are compared with the bulge test curve obtained by Bramley and Mellor (1966). The derived curves are based on their measured R-values and the experimental stress–strain curve in the rolling direction. The hypothesis of strain equivalence is evidently in better agreement with experiment, at least for the materials used in this investigation, as has been shown by Chakrabarty (1970b). Further experimental results available in the literature tend to suggest that the strain-hardening hypothesis is preferable to the work-hardening hypothesis for most engineering materials. A crystallographic method of predicting the anisotropic behavior of sheet metals has been developed by Chan and Lee (1990).

6.2.4 Normal Anisotropy in Sheet Metal In many applications, the anisotropy in the plane of the sheet is small and can be disregarded by considering a state of planar isotropy with a uniform mean R-value. This provides a radical simplification to the problem, since the yield criterion and the flow rule then become independent of the choice of coordinate axes in the plane of the sheet. Since H/F = H/G = R and N/F =1+2R, when the anisotropy is rotationally symmetric about the z-axis, the yield criterion and the flow rule become

6.2

Anisotropy of Rolled Sheets

417

Fig. 6.2 Equibiaxial stress–strain curves for anisotropic sheets, (a) Experimental curve, (b) theoretical curve based on the uniaxial curve in the rolling direction and the hypothesis of strain equivalence, (c) theoretical curve derived from the uniaxial curve on the basis of work equivalence

σx2 −

2R 1 + 2R 2 = Y 2, σx σy + σy2 + 2 τxy 1+R 1+R

dεy sdγxy dλ dεx = = = , (1 + R) σx − Rσy (1 + R) σy − Rσx (1 + 2R) τxy (1 + R) Y

(6.21) (6.22)

where Y is the uniaxial yield stress of the material in the plane of the sheet. For the present purpose, it is convenient to take the equivalent stress σ¯ as identical to the current value of the planar yield stress and redefine the equivalent strain increment dε¯ according to the hypothesis of strain equivalence as 1+R dε = 2

1/2 2 2 2 3 dεx + dεy + dεx − dεy + 4dγxy 1 + R + R2

,

(6.23a)

so that it reduces to the longitudinal strain increment for a uniaxial tension applied in the plane of the sheet. The corresponding expression for dε according to the hypothesis of work equivalence takes the form dε =

1/2 1+R 2 . (1 + R) dεx2 + dεy2 + 2Rdεx dεy + 2dγxy 2

(6.23b)

418

6 Plastic Anisotropy

Expressed in terms of the principal stresses (σ 1 , σ 2 ) in the plane of the sheet, (6.21) represents an ellipse whose major and minor axes coincide with those of the von Mises ellipse. For R > 1 , the effect of anisotropy is to elongate the ellipse along the major axis, and slightly contract it along the minor axis for a given planar yield stress μY, Fig. 6.3 (a). The extremities of the√major axis represent states of balanced biaxial stress of magnitude μY, where μ = (1 + R)/2, as may be seen on setting σ x = σ y = σ and τ xy = 0 in (6.21). A piecewise linear approximation to the nonlinear yield criterion is achieved by replacing the ellipse with a hexagon obtained by elongating the inclined sides of the Tresca hexagon (Chakrabarty, 1974). The new yield locus is defined by σ1 = ±μY, σ2 = ±μY, σ1 − σ2 = ±Y.

Fig. 6.3 Biaxial yield loci for sheet metals with normal anisotropy. (a) Quadratic yield function and its linearization for R 1, (b) quadratic and nonquadratic yield functions for different R-values

It may be noted that the hexagon meets the ellipse not only at points representing uniaxial and biaxial states of stress, but also at points where the principal stress ratio is μ2 (μ2 − 1), which is equal to (R + l)/(R − l). The flow rule associated with the hexagonal yield locus requires the strain increment vector to be directed along the exterior normal when the stress point is on one of the sides of the hexagon. At a corner of the hexagon, the strain increment vector may assume any position between the normals to the two sides meeting in the corner. The increment of plastic work per unit volume is uniquely defined for all points of the hexagon. The preceding theory, though generally adequate for R > 1, does not account for an anomalous behavior observed in materials with R < 1. Indeed, Woodthorpe and Pearce (1970) have found that the yield stress in balanced biaxial tension for rolled aluminum, having an R-value lying between 0.5 and 0.6, is significantly higher than the uniaxial yield stress in the plane of the sheet. A modification of the theory is therefore necessary for dealing with such materials. Following Hill (1979),

6.2

Anisotropy of Rolled Sheets

419

the modified yield criterion in terms of the principal stresses σ 1 and σ 2 may be taken as |σ1 + σ2 |1+m + (1 + 2R) |σ1 − σ2 |1+m = 2 (1 + R) Y 1+m ,

(6.24)

where m is an additional parameter that depends on the state of normal anisotropy. It is easy to show that (6.24) reduces to (6.21) when m = 1. The parameter R in (6.24) is indeed the customary strain ratio in simple tension, as may be seen from the plastic flow rule which may be expressed in the form |dε1 − dε2 | |dε1 + dε2 | dλ = = . m m |σ1 + σ2 | (1 + 2R) |σ1 − σ2 | (1 + R) Y m

(6.25)

The signs of dε1 +dε2 and dε1 −dε2 are the same of those of σ1 +σ2 and σ1 −σ2 , respectively. The parameter dλ in (6.25) is a positive scalar equal to the equivalent strain increment dε defined by the hypothesis of work equivalence. Indeed, it follows from (6.24) that 2dW = (σ1 + σ2 ) (dε1 + dε2 ) + (σ1 − σ2 ) (dε1 − dε2 ) = 2Ydλ. Eliminating σ1 + σ2 and σ1 − σ2 between (6.24) and (6.25), the corresponding expression for the equivalent strain increment is obtained as

dε

1+n

=

1 (1 + R)n |dε1 + dε2 |1+n + (1 + 2R)−n |dε1 − dε2 |1+n , 2

where n = 1/m. For m = 1, this expression reduces to (6.23b) when the principal components alone are considered. The equivalent strain increment defined by the hypothesis of strain equivalence is obtained directly from (6.23a). For a workhardening material, it is only necessary to replace the planar yield stress Y by the equivalent stress σ¯ , which is a given function of the total equivalent strain. In the case of an equal biaxial tension σ , the yield criterion (6.24) gives (σ /Y)1+m = (1 + R)/2m , which indicates that a can be greater than Y for R < 1 with suitable values of m. Parmar and Mellor (1978b) derived M-values of 0.5 for rim steel (R ≈ 0.44), 0.7 for soft aluminum (R ≈ 0.63), and 0.8 for software brass (R ≈ 0.86), on the basis of their experimental results using the bulge test and the hole expansion test. Similar experimental results for different materials have been examined by Kobayashi et al. (1985). Since the m-value depends on the R-value, these two parameters cannot be chosen independently for any given material. Other types of nonquadratic yield function for normal anisotropy of sheet metal have been considered by Logan and Hosford (1980), Vial et al. (1983), and Dodd and Caddell (1984).

420

6 Plastic Anisotropy

6.2.5 A Generalized Theory for Planar Anisotropy The inadequacy of the quadratic yield function in predicting the ratio of the equibiaxial and uniaxial yield stresses for materials with R-values less than unity has already been noted. In the case of planar anisotropy, similar discrepancies are expected when the average R-value of the sheet is less than unity. The difficulty can be overcome by considering a suitable extension of the nonquadratic yield criterion in (6.24) to include the effect of planar anisotropy. The simplest yield criterion of this type, when the x- and y-axes are taken along the rolling and transverse directions, respectively, may be taken in the form σx + σy v + 2a |σx |v − σy v + b 2τxy v v/2 2 2 + c σx + σy + 4τxy = (2Z)v ,

(6.26)

where a, b, c, and v > 1 are the dimensionless anisotropic parameters, while Z is the through-thickness yield stress of the material (Chakrabarty, 1993a). A state of planar isotropy requires a = b = 0, and (6.26) then reduces to (6.24) with v = 1 + m and c = 1 + 2R. When v = 2, the above yield criterion becomes identical to (6.9) if we set G−F N − 2H 4H a= , b=2 − 1, c = 1 + , G+F F+G F+G and invoke the relation f + g = 1/Z2 . The anisotropic parameters appearing in (6.26) can be determined from the measurement of the R-values in the rolling, transverse, and 45◦ directions, as well as by the estimation of the ratio of the equibiaxial yield stress to the uniaxial yield stress in one of these directions. Using (6.26) the anisotropic coefficients can be expressed as 4a =

v v ⎫ 2Z v 2Z 2Z ⎪ − , 2 (1 + c) = + ,⎪ ⎬ Y X Y v v v ⎪ 2Z 2Z 2Z ⎪ ⎭ 2b = 2 − − , S X Y

2Z X

v

(6.27)

where X, Y, and S denote the uniaxial yield stresses in the rolling, transverse, and 45◦ directions, respectively, in the plane of the sheet. The uniaxial yield stress σ in any direction making an angle α with the rolling direction is obtained by inserting (6.9a) into (6.26), the result being

2Z σ

v = 1 + c + b (sin 2α)v + 2a

cos2 α

v

v . − sin2 α

(6.28)

Setting dσ /dα = 0, the stationary values of the yield stress are found to occur along the anisotropic axes, as well as in the directions α = ±α0 , where α0 is given by the expression

6.2

Anisotropy of Rolled Sheets

421

(tan α0 )2−v + (tan α0 )v = 2v−2 1 − tan2 α0

b . a

Thus, α0 depends only on the parameters v and a/b. When a = 0, we have α0 = π/4 irrespective of the value of v, a result which is in accord with experiment. Other types of nonquadratic yield functions for planar anisotropy have been discussed by Barlat and Lian (1989), and Hill (1990). Taking the plastic potential to be identical to the yield function (6.26), the components of the strain increment in the plane of the sheet can be written down from the normality rule. It is convenient to express the stress–strain relations as ⎫ v−2 v−2 + (σx + σy ) σx + σy dμ,⎪ dεx + dεy = a σx |σx |v−2 − σy σy ⎪ ⎪ ⎪ ⎬ v−2 v−2 v−2 + c(σx − σy )(2τ ) dμ, + σy σy dεx + dεy = a σx |σx | ⎪ ⎪ ⎪ v−2 ⎪ v−2 ⎭ dμ, + cτxy (2τ ) dγxy= bτwy 2τxy (6.29) where τ denotes the magnitude of the maximum shear stress in the plane of the sheet, and dμ denotes a positive scalar factor of proportionality. For a uniaxial tension σ acting at an angle α to the rolling direction, the transverse strain increment is given by the expression 1 1 dεx + dεy − dεx − dεy cos 2α − dγxy sin 2α 2 2 v−1 1 v−1 c − 1 + b (sin 2α)v − 2α cos2 α sin2 α =− σ 2

v−1 − sin2 α cos2 α dμ,

dεw =

which is obtained on substitution from (6.9a) and (6.29). The ratio of the transverse strain increment dεω to the thickness strain increment dεz = −(dεx + dεy ) is the ft-value in the considered direction and is the R-value in the considered direction and is found as v−1 2 v−1 sin α − sin2 α cos2 α (c − 1) + b (sin 2α)v − 2a cos2 α R= . (6.30) v−1 2 v−1 2 + 2a cos2 α − sin α It is interesting to note that the R-value is independent of v when α = 0, π/4, and π/2. Denoting these three R-values by Rx , Rs , and Ry , respectively, and using (6.30), we obtain the relations a=

Ry − Rx , Ry + Rx

2Rx Ry b = 2 Rs − , Rx + R y

c=1+

4Rx Ry , Rx + Ry

(6.31)

422

6 Plastic Anisotropy

which are the same as those for the quadratic yield function. The remaining parameter v is given by the three independent relations

2Z S

ι

= 2 (1 + Rs ) ,

2Z X

ν

4Ry (+Rx ) = , Rx + R y

2Z Y

ι

4Rx 1 + Ry = , Rx + R y (6.31a)

which are easily obtained by suitable combinations of (6.27) and (6.31). The first expression in (6.31a) is generally the most appropriate one. It follows from (6.26) and (6.29) that the plastic work increment per unit volume is given by 2dW = σx + σy dεx + dεy + σx − σy dεx − dεy + 4τxy dγxy = (2Z)v dμ, which immediately indicates that dμ is a measure of the equivalent strain increment based on the hypothesis of work equivalence. If the equivalent stress σ¯ is defined in such a way that it reduces to the applied stress when a uniaxial tension acts in the rolling direction, then in view of (6.28) and (6.31), we can express it in the form (2Z/σ¯ )v = 1 + 2a + c = 4Ry (1 + Rx ) / Rx + Ry . For a uniaxial tension σ applied in any direction α in the plane of the sheet, the ratio σ¯ /σ is obtained by combining (6.33) with (6.28). The equivalent strain increment dε, based on the hypothesis of strain equivalence, is given by (6.23a) with R replaced by Rx . The results for the rolling, transverse, and 45◦ directions are v ⎫ Rx + Ry Ry − Rx σ¯ 1+R ⎪ ⎪ +η ,⎪ = ⎪ ⎬ σ 1 + Rx 2Ry 2Ry ' ⎪ dε 1 + R x 1 + R + R2 ⎪ ⎪ ⎪ , = ⎭ dε 1 + R 1 + Rx + R2x

(6.32)

where R is the strain ratio associated with the given direction; dε is the longitudinal strain increment; and η is a parameter whose value is 1 for α = 0, is –1 for α = π /2, and is 0 for α = –π /4. In the case of an equibiaxial tension σ , the equivalent stress directly follows from (6.33) with Z = σ . The equivalent strain increment is obtained from (6.23a), using the fact that dεx + dεy = dε, dε x – dεy = β a dε and dγxy = 0, where dε is the compressive thickness strain increment at any stage of the loading, and β = 2 ν-2 . In Figs. 6.4 and 6.5, some comparison has been made of the theoretical predictions with the experimental data reported by Naruse et al. (1993) and Lin and Ding (1995). An interesting explanation of the observed anomaly in biaxial tension has been advanced by Wu et al. (1997) with the help of kinematic hardening along with the quadratic yield function.

6.2

Anisotropy of Rolled Sheets

423

Fig. 6.4 Theoretical and experimental variations of the R-value and yield stress ratio in the plane of the sheet for annealed aluminum

Fig. 6.5 Theoretical and experimental yield loci for cold rolled aluminum sheet exhibiting planar anisotropy

424

6 Plastic Anisotropy

Fig. 6.6 Plastic torsion of an anisotropic bar. (a) Direction of shear stress in relation to that of the characteristic, components of shear stress on a transverse section

Better agreement with experimental data can be achieved by considering a higher order yield function, such as the biquadratic yield function proposed by Gotoh (1977). Taking the axes of reference along the principal axes of anisotropy, the yield. criterion may be expressed as 2 4 +Kτxy = 1. (6.33) Aσx4 +Bσx3 σy +Cσx2 σy2 +Dσx σy3 +Eσy4 +(Fσx2 +Gσx σy +Hσy2 )τxy

The nine anisotropic coefficients appearing in (6.33) can be uniquely determined from the measurement of the uniaxial yield stresses and R-values in directions mak◦ ◦ ◦ ing angles of 0◦ , 22.5◦ (or 67.5 ), 45 , and 90 with the direction of rolling, in addition to that of the through-thickness yield stress. The R-value of the sheet metal corresponding to a uniaxial tension in any given direction follows from the flow rule associated with the yield criterion. Evidently, the distribution of the yield stresses and R-values, according to (6.33), is defined by curves which pass through all the experimental points included in Figs. 6.6 and 6.7. An isotropic material corresponds to B = D = – 2A, C = 3A, E = A, F = –G = H = 6A, K = 9A, and the yield function (6.33) then becomes a perfect square of the von Mises yield function.

6.3 Torsion of Anisotropic Bars 6.3.1 Bars of Arbitrary Cross Section A prismatic bar of arbitrary cross section is twisted by terminal couples about an axis parallel to the generator. The material is considered as rigid/plastic, and the applied torque is assumed sufficient to render the bar fully plastic. The state of anisotropy is orthotropic with the z-axis of anisotropy parallel to the generator. Choosing the x- and y-axes in an end section of the bar, the components of the velocity may be taken as

6.3

Torsion of Anisotropic Bars

425

u = −yz,

v = xz,

w = w (x, y) ,

the rate of twist per unit length being taken as unity. The nonzero components of the strain rate corresponding to the above velocity field are given by 2γ˙xy =

∂w − y, ∂x

2γ˙yz =

∂w + x. ∂y

The nonzero stress components are τxz and τyx which must satisfy the equilibrium equation and the yield criterion, these equations being ∂τyz ∂τxz + = 0, ∂x ∂y

2 τxz

k12

+

2 τyz

k22

= 1,

(6.34)

where k1 = (2M)–/1/2 and k2 = (2L)–/1/2 are the yield stresses in simple shear in the x- and y-directions, respectively. Differentiating the second equation of (6.34) partially with respect to x and y, and using the first equation, we get τyz ∂τxz τxz ∂τxz − 2 = 0, 2 k2 ∂x k1 ∂y

τyz ∂τyz τxz ∂τyz − 2 = 0. 2 k2 ∂x k1 ∂y

These equations are seen to be hyperbolic, and the characteristic through a generic point P in the (x, y)-plane is in the direction 2 k2 τxz dy = tan ψ = − , dx k1 τyz

(6.35)

where ψ is the counterclockwise angle made by the tangent to the characteristic with the positive x-axis. Using the preceding relations, it is easy to show that dτxz = dτyz = 0 along a characteristic (Hill, 1954). It follows that the characteristics are straight, and the resultant shear stress r along each one of them is constant in direction and magnitude. Since the lateral surface of the bar is stress free, the resultant shear stress must be directed along the tangent to the boundary S at the point where it is cut by the characteristic. The magnitude of the shear stress along any characteristic can be calculated from (6.35) and the second equation of (6.34). The characteristics meet the boundary at varying angles when the bar is anisotropic. The resultant shear strain rate at a point of the cross section is orthogonal to the characteristic, as may be seen from the flow rule associated with the yield function.

426

6 Plastic Anisotropy

Indeed, the components of the rate of shear are in the ratio γyz = γyz

k1 k2

2

τyz = − cot ψ τxz

in view of (6.35), establishing the condition of orthogonality of the strain rate. The component of the rate of shear in the characteristic direction is therefore zero. The substitution for γ˙xz and γ˙yz above relation gives

∂w ∂w − y cos ψ + − x sin ψ = 0. ∂x ∂x

This equation is hyperbolic with characteristics identical to (6.35). The variation of the rate of warping along a characteristic is given by ∂w ∂w ∂w = cos ψ + sin ψ = y cos ψ − x sin ψ, ∂s ∂x ∂y where ds is a line element along the characteristic. If w0 denotes the value of w at any given point on the characteristic, the integration of the above equation furnishes ω = ω0 +

(ydx − xdy) = ω0 +

pds,

(6.36)

where p is the perpendicular distance from the origin to the considered element, reckoned positive when the vector ds has a clockwise moment about the z-axis, Fig. 6.6(a). The integration begins from the point of intersection of the characteristic with a stress discontinuity G, formed by the intersection of characteristics. Since the shear strain rate vanishes along G, (6.36) also holds along G. Any projecting corner of S is the source of one of the branches of G, while a reentrant corner is a point of singularity that generates a fan of characteristics. For the evaluation of the torque T in the fully plastic state it is convenient to introduce a stress function φ such that the shear stress components are given by τxz =

∂φ , ∂y

τyz = −

∂φ , ∂x

(6.37)

satisfying the equilibrium equation identically. The lines of constant φ are the shearing stress trajectories in the (x, y)-plane. Theexternal boundary C is also a contour line of φ, and we may choose φ = 0 along this boundary. Referring to Fig. 6.6(b), the applied torque is found to be

6.3

Torsion of Anisotropic Bars

T=

427

xτyz − yτxz dxdy = 2

φdxdy.

(6.38)

The second expression of (6.38) follows on substituting from (6.37), integrating by parts, and using the boundary condition φ = 0. If we now introduce the transformation k2 x = kξ , k1 y = kη, where k is an arbitrary constant having the dimension of stress, the yield criterion expressed by the second equation of (6.34) becomes

∂φ ∂ξ

2 +

∂φ ∂η

2 = k2 .

In the (ξ , η)-plane, the stress function therefore satisfies the same equation as that for an isotropic bar with a shear yield stress k. The characteristics in the (ξ , η)-plane are normal to the transformed contours of φ, and correspond to the characteristics in the (x, y)-plane. Equation (6.38) now becomes T=

2 k2 k1 k2

φdξ dη =

k2 k1 k2

T∗,

(6.39)

where T∗ denotes the fully plastic torque for an isotropic bar having the transformed cross section. The stress function for the anisotropic bar is obtained directly from 1/2 the fact that ∂φ/∂s along a characteristic is of magnitude k12 sin2 ψ + k22 cos2 ψ which is easily obtained from (6.35), (6.37), and (6.34).

6.3.2 Some Particular Cases As a simple application of the theory, let us consider a bar of elliptical cross section whose semiaxes are in the ratio k1 :k2 , the equation of the ellipse being y2 x2 + = 1, a2 b2

a k1 = . b k2

The corresponding contour in the (ξ , η)-plane becomes the circle ξ 2 + η2 = c2 , where kc = k1 b = k2 a. Since T ∗ = 23 π kc3 for the isotropic bar, (6.39) furnishes the fully plastic torque for the anisotropic bar as 2 T= π 3

k 3 c3 k1 k2

=

2 2 π k1 ab2 = π k2 a2 b. 3 3

428

6 Plastic Anisotropy

Since the lines of shearing stress in the (ξ, η)-plane are concentric circles, the stress trajectories in the (x, y)-plane are concentric ellipses. The characteristics are radial lines in both planes, and the warping is absent. When the cross section of the anisotropic bar is a circle of radius c, the transformed contour is an ellipse with semiaxes a = ck2 /k and b = ck1 /k. When k1 >k2 , a stress discontinuity extends along the η-axis over a length 2b(1–a2 /b2 ), which is the distance between the centers of curvatures at the extremities of the major axis. The corresponding discontinuity along the y-axis covers a length 2c(1 − k22 /k12 ). If the degree of anisotropy is sufficiently small, it is a good approximation to take the yield point torque for the transformed section as the mean of those for circular sections of radii a and b. Thus 3 3 3 π k π 3 3 3 k1 + k 2 . a +b = c T≈ 3 k1 k2 3 k1 k2 It follows from (6.36), and the nature of the characteristic field, that the warping is positive in the first and third quadrants, and negative in the others. As a final example, consider a bar whose cross section is a rectangle having sides of lengths a and b parallel to the x- and y-axes, respectively. The transformed contour is also a rectangle with sides α = ak2 /k and β = bk1 /k, parallel to the ξ and η-axes, respectively. Since T∗ = kβ 2 (3α–β)/6 for α ≥ β, the actual yield point torque is obtained from (6.39) as 1 T= 6

k3 β 2 k1 k2

(3α − β) =

1 3 k1 b 6

3a k1 − b k2

,

k1 a ≥ b k2

(6.40)

The solution for a/b ≤ k1 /k2 is obtainedby merely interchanging a, b and k1 , k2 in (6.40). The stress discontinuities emanating from the corners of the rectangle meet on the x- or y-axes accordingly as a/b is greater or less than k1 /k2 , the axial discontinuity being of length |ak2 —bk1 |/k. The warping of the cross section can be calculated from (6.36) in the same way as that employed for an isotropic bar (Chakrabarty, 2006).

6.3.3 Length Changes in Twisted Tubes A thin-walled cylindrical tube is twisted in the plastic range, the ends of the tube being supported in such a way that it is free to extend or contract in the axial direction. We begin with the situation where the tube may be assumed to remain isotropic during the deformation. Consider a small element of the tube formed by cross sections at a unit distance apart, so that the displacement of a particle on one edge due to its rotation relative to the other is equal to the engineering shear strain γ . Thus, a typical material line element OP, inclined at an angle ψ to the direction of

6.3

Torsion of Anisotropic Bars

429

Fig. 6.7 Torsion of an anisotropic tube. (a) Geometry of finite shear deformation, (b) stress acting on an element of the tube wall

shear, rotates to a new position OQ such that PQ = γ . If the lengths OP and OQ are denoted by l0 and l, respectively, Fig. 6.7(a), then from simple geometry,

l2 = l02 + 2l0 γ cos ψ + γ 2 = l02 1 + 2γ sin ψ cos ψ + γ 2 sin2 ψ in view of the relation l0 sin ψ = 1. Differentiating this expression with respect to ψ, it is found that the ratio l/l0 has a maximum value when − 2 cot 2ψ = γ , l = l0 tan ψ, l cos ψ = 1.

It follows that the angles of inclination of OP and OQ to the direction of shear are complementary when i/iq is a maximum. The same conclusion holds for the line elements OP’ and OQ’ in the initial and final states, respectively, for which the length ratio is a minimum. The directions OR and OR’, which correspond to zero resultant change in length, must be each inclined at an angle 2ψ to the direction of shear, since R’N = RN = γ /2, where ON is parallel to the tube axis. Suppose, now, that the tube becomes progressively anisotropic during the torsion, and that the anisotropic axes coincide at each stage with the directions of the greatest relative extension and contraction. Since the axial strain ε is small compared to the shear strain y, these directions may be assumed identical to those for an isotropic tube. Let the x- and y-axes of anisotropy, which correspond to OQ’ and OQ, respectively, at any given stage, make an angle φ with the direction of twist and the axis of the tube, respectively, Fig. 6.7(b). The nonzero components of the stress are σx = −σy = −τ sin 2φ,

τxy = τ cos 2φ,

where τ denotes the applied shear stress. The substitution into the yield criterion (6.9) then furnishes

430

6 Plastic Anisotropy

−1 τ 2 = 2 N + (F + G + 4H − 2 N) sin2 2φ . The ratios of the components of the strain increment associated with the yield criterion are obtained from (6.3) as dεy dγxy dεx dεz = = = tan 2φ. G + 2H F + 2H G−F N The axial strain increment dε and the engineering shear strain increment dγ produced by the torsion are dε = dεx sin2 φ + dεy cos2 φ − 2dγxy sin φ cos φ, dγ = dεy − dεx sin 2φ + 2dγxy cos 2φ. Substituting for the strain increment components appearing on the right-hand side of these relations, we obtain the ratio of dε to dγ in the form (N − G − 2H) sin2 φ − (N − F − 2H) cos2 φ sin 2φ dε = . dγ 2 N + (F + G + 4H − 2 N) sin2 φ

(6.41)

The value of dε/dγ is initially zero, when the tube is isotropic (φ = π /4). For small angles of twist, φ is slightly greater than π /4, and dε has the same sign as that of F – G, or equivalently of X–Y. With increasing angle of twist, φ eventually approaches π /2, and dε is finally positive if N > G + 2H. For many metals, the anisotropic parameters vary in such a way that the tube lengthens continuously during the torsion (Swift, 1947; Bailey et al. 1972). In exceptional cases, however, the length of the tube may progressively decrease with increasing angle of twist (Toth et al., 1992). As the tube becomes increasingly anisotropic during the torsion, the ratios F/H, G/H, and N/3H vary continuously from their common initial value of unity to approach some limiting values in an asymptotic manner. These limiting ratios can be found by direct measurements after subjecting the tube to a sufficiently large angle of twist. Each ratio of the anisotropic parameters at a generic stage of the torsion depends on its asymptotic value and the magnitude of the shear strain, according to a mathematical function which may be assumed to be the same for all three ratios. A suitable choice of such a function enables us to determine the length change at any stage by the integration of (6.41). Conversely, an actual measurement of dε and dγ for each increment of torque provides a means for determining the variation of the ratios of the anisotropic parameters under increasing strain.

6.3.4 Torsion of a Free-Ended Tube The change in length that takes place during the free-end twisting of a thin-walled tube should be taken into consideration for the derivation of the shear stress–strain

6.3

Torsion of Anisotropic Bars

431

curve from the torsion test. Let a0 and t0 denote the initial mean radius and wall thickness, respectively, of a thin-walled tube which is twisted in the plastic range with a torque T and an angle of twist θ per unit length of the tube. If the current internal and external radii of the tube are denoted by a1 and a2 , respectively, then the applied torque is T = 2π

a2

τ r2 dr =

a1

2π θ3

a2 θ

a1 θ

τ γ 2 dγ .

Multiplying both sides of this equation by θ 3 differentiating with respect to θ , and considering a mean shear stress τ¯ through the thickness of the tube, we get 3T + θ

d d dT = 2π τ¯ a22 (a2 θ) − a21 (a1 θ ) . dθ dθ dθ

Since the wall thickness is small compared to the mean radius a and remains practically unchanged during the torsion, it is reasonable to introduce the approximations da1 /a1 ≈ da2 /a2 ≈ −dε, a32 − a31 ≈ 3t0 a2 ≈ 3t0 a20 e−2ε dT t0 dτ¯ 1 θ τ2 − τ1 ≈ θ ≈ , τ1 + τ2 ≈ 2τ¯ a dθ 2π a3 dθ where ε denotes the axial strain, which is approximately equal to the magnitude of the hoop strain, the elastic deformation being disregarded. The expression for the mean shear stress is therefore closely approximated by the formula τ¯ ≈

T 2π t0 a20

dε 1 + 2ε + θ dθ

(6.42)

to a sufficient accuracy. The evaluation of the mean shear stress τ¯ requires the measurement of T,θ , and ε simultaneously at each stage of the loading. The corresponding mean shear strain γ¯ is approximately equal to α0 θ , the change in radius being small. A more elaborate expression for the shear stress in the free-end torsion of a thin-walled tube has been given by Wu et al. (1997). The shear stress–strain curve obtained from the torque-twist curve in the free-end torsion of a thin-walled tube, using (6.42), is found to coincide with that derived from the fixed-end torsion of the tube. The fixed-end torsion naturally gives rise to an axial compressive stress that suppresses the axial strain, while producing small amounts of circumferential and thickness strains in the twisted tube. Figure 6.8 shows the observed variation of the axial and hoop strains with the shear strain during the free-end torsion of an extruded aluminum tube, obtained experimentally by Wu (1996). The accumulation of axial strain in the finite torsion of tubular specimens with both free and fixed ends has been investigated by Wu et al. (1998).

432

6 Plastic Anisotropy

Fig. 6.8 Development of axial and hoop strains in finitely twisted thin-walled tubes in the plastic range (after H.C. Wu, 1996)

6.4 Plane Strain in Anisotropic Metals 6.4.1 Basic Equations in Plane Strain Consider the class of problems in which the plastic flow is restricted in the plane perpendicular to the z-axis of anisotropy, which coincides with a principal axis of the strain rate. The condition γ˙xz = γ˙yz = 0 requires τ xz = τ yz = 0 by the flow rule, indicating that σz is a principal stress. Using the plane strain condition dε z = 0 in the third equation of (6.3), we get σz =

Gσx + Fσy . G+F

In the special case when σy = 0, the above relation indicates that σz is greater or smaller than σx /2 according as G is greater or smaller than F. Substituting for σz in (6.1), and setting τ xz = τ yz = 0, the yield criterion is reduced to 2 FG 2 = 1. σx − σy + 2 Nτxy H+ F+G

6.4

Plane Strain in Anisotropic Metals

433

It is convenient at this stage to introduce a dimensionless parameter c defined as c=1−

N (F + G) (−∞ < c < 1) . 2 (FG + GH + HF)

(6.43)

Thus, c is positive when N is less than both F + 2H and G + 2H, and negative when N is greater than both F + 2H and G + 2H. If the anisotropy is rotationally symmetrical about the z-axis, c = 0. In view of (6.43), the yield criterion for anisotropic materials is expressed as 2 σx − σy 2 = k2 , + τxy 4 (1 − c)

(6.44)

where k = (2 N) –1/2 is the yield stress in pure shear associated with the x- and yaxes. The yield stress σ in plane strain tension in the direction making an angle α with the x-axis is obtained by inserting (6.9a) into (6.44), the result being 1/2 1−c σ = 2k . 1 − c sin2 2α Since σ has equal values in the directions ±α and ±(π /2–α), the angular variation of σ is symmetrical about the x- and y-axes, √ as well as about their bisectors. When c is positive, σ has a minimum value of 2 k 1 − c along the anisotropic axes, and a ◦ maximum value √ of 2 k in the 45 directions. When c is negative, σ has a maximum value of 2 k 1 − c along the anisotropic axes, and a minimum value of 2 k in the 45◦ . A more general theory for plane strain has been discussed by Rice (1973). Let vx and vy denote the components of velocity of a typical particle with respect to the anisotropic axes. The components of the strain rate in the plane of plastic flow referred to the anisotropic axes are ∂vy ∂vx ∂vv 1 ∂vx , ε˙ y = , γ˙xy = + . ε˙ x = ∂x ∂y 2 ∂y ∂x It follows from the flow rule associated with the yield criterion (6.44) that 2τxy 2γ˙xy . = (1 − c) ε˙ x − ε˙ y σx − σy

(6.45)

If ψ and ψ denote the angles of inclination of a principal stress direction and the corresponding principal strain rate direction, respectively, with respect to the x-axis, then 2ψ = (1 − c) tan 2ψ in view of (6.45). When c =0, it follows that ψ = ψ only for ψ = 0, π/4, and π/2, as expected. In terms of the velocity gradients, (6.45) can be expressed as ∂vx ∂vy ∂vy ∂vx + − = 2 (1 − c) τxy . σx − σy ∂y ∂x ∂x ∂y

434

6 Plastic Anisotropy

The last equation, together with the incompressibility condition, the equilibrium equations, and the yield criterion, constitutes a set of five equations for the three stress components and the two velocity components. The parameters k and c in the theoretical framework can be experimentally determined by carrying out the plane strain compression test at 0◦ and 45◦ to one of the axes of anisotropy. As in the case of isotropic solids, the governing equations are hyperbolic with characteristics in the directions of maximum rate of shear at each point of the deforming zone. The characteristics for the stress are identical to those for the velocity, but these curves do not generally coincide with the trajectories of the maximum shear stress in the plane of plastic flow. Due to the incompressibility of the material, the rate of extension vanishes along the characteristics, which are known as the sliplines.

6.4.2 Relations Along the Sliplines The two orthogonal families of sliplines will be designated by α and β following the convention that the acute angle made by the algebraically greater principal stress in the considered plane is measured counterclockwise with respect to the α-direction. If φ denotes the counterclockwise angle made by a typical α-line with the x-axis, as shown in Fig. 6.9(a), then dy/dx = tan φ along an α-line and dy/dx = –cot φ along a β-line. Since the left-hand side of (6.45) is equal to –cot 2φ, we have σx − σy cos 2φ + 2 (1 − c) τxy sin 2φ = 0.

(6.46)

Fig. 6.9 Orientation of sliplines, and the stresses acting across them in a curvilinear element

It follows from (6.44) and (6.46) that the outward drawn normal to the yield locus, obtained by plotting τxy against (σ x –σ y )/2, makes a counterclockwise angle of 2φ with respect to the τ xy -axis. This result actually holds for any convex yield function, under condition of plane strain, as has been shown by Rice (1973).

6.4

Plane Strain in Anisotropic Metals

435

Equation (6.46) can be solved simultaneously with the yield criterion (6.44) to give τxy cos 2φ = , k 1 − c sin2 2φ

σx − σy 2 (1 − c) sin 2φ = − . k 1 − c sin2 2φ

(6.47)

Let σα , σ β , and τ αβ denote the stress components referred to the local sliplines taken as a pair of curvilinear axes in the plane, as shown in Fig. 6.9(b). Then σα − σβ = σx − σy cos 2φ + 2τxy sin 2φ, ταβ = σx − σy sin 2φ − 2τxy cos 2φ. Substituting from (6.47), and setting p = –(σα + σβ )/2, which is the mean compressive stress in the plane of plastic flow, the (α, β) components of the stress can be expressed as ⎫ ds ds ⎪ σβ = −p + k , ταβ = 2ks, ⎪ σα = −p − k , ⎬ dφ dφ (6.48) ⎪ ⎪ ⎭ 2 2s = 1 − c sin 2φ. If the radii of curvature of the α- and β-lines at a generic point are denoted by Rα and Rβ , respectively, the equilibrium equations in the curvilinear coordinates (α, β) may be written as σα − σβ ∂ταβ 2ταβ ∂σα + + − = 0, ∂sα Rβ ∂sβ Rα σα − σβ ∂ταβ 2ταβ ∂σβ + + + = 0, ∂sβ Rα ∂sα Rβ where sα and sβ are the arc lengths along the respective sliplines. Since the curvatures 1/Rα and 1/Rβ are equal to ∂φ/∂Sα and ∂φ/∂Sβ , respectively, while the normal stress difference σ α –σ β is equal to = –dταβ /dφ in view of (6.48), the second and third terms of each of the above equations cancel one another, while the remaining terms give ∂ ds ∂φ p+k = 0, + 4ks ∂sα dφ ∂sα ∂φ ∂ ds + 4ks p−k = 0, ∂sβ dφ ∂sβ in view of (6.48). These equations can be expressed in the integrated form (Hill, 1949) to obtain the characteristic relations

436

6 Plastic Anisotropy

p + 2 kω = constant along an α line, p − 2 kω = constant along a β line,

(6.49)

The parameter ω depends on the angle φ and the anisotropic parameter c according to the relation 2ω =

ds +4 dφ

φ 0

√ c sin 2φ cos 2φ + E 2φ, c , sdφ = − 1 − c sin2 2φ

(6.50)

where E denotes the standard elliptic function of the second kind and is defined as

α

E (α, m) =

1 − m2 sin2 θ dθ .

0

For an isotropic material, ω = φ, and (6.49) reduces to the well-known Hencky equations for plane strain. If the components of the velocity along the α- and β-lines are denoted by u and v, respectively, the condition of zero rate of extension along the sliplines can be easily reduced to du − vdφ = 0 along an α−line, (6.51) dv + udφ = 0 along a β−line, These relations are the same as the Geiringer equations for isotropic solids. In analogy to Hencky’s first theorem, it is easily shown that the difference in the values of ω or p between a pair of points, where two given sliplines of one family are intersected by a slipline of the other family, remains constant along their lengths. It follows that if one segment of a slipline is straight, the corresponding segments of sliplines of the same family are also straight, and the straight segments are consequently of equal lengths.

Fig. 6.10 Indentation of the plane surface of a semi-infinite anisotropic medium by a rigid flat punch

6.4

Plane Strain in Anisotropic Metals

437

6.4.3 Indentation by a Flat Punch Consider, as an example, the indentation of the plane surface of a large block of metal by a rigid flat punch. At the incipient plastic flow, the sliplines emanating from the corners A and A of the punch reach the free surface of the block. The slipline field shown in Fig. 6.10 is an extension of the Prandtl field for isotropic solids. In the triangular regions ABD and A B D , the state of stress is a uniform compression 2p0 parallel to the surface, where p0 is the value of p in these regions. If ψ denotes the angle of inclination of the plane surface with the x-axis of anisotropy, it follows from (6.44) that p0 = k

1/2

1−c

.

1 − c sin2 2ψ

(6.52)

If λ denotes the acute angle made by the α-lines with the plane surface, then φ = λ – ψ along AB and A B . Since the direction of the algebraically greater principal stress is normal to the plane, making an angle π /2 – ψ with the x-axis, (6.46) gives cot 2 (λ − ψ) + (1 − c) tan (π − 2ψ) = 0, or

(6.53) λ=ψ+

1 2

−1

cot

[(1 − c) tan 2ψ] .

√ √ It is easily shown that λ lies between cot−1 1 − c and π/2 cot−1 1 − c whatever the value of ψ. Since the state of anisotropy cannot change appreciably during the deformation preceding the yield point, we are justified in treating c as a constant throughout the plastic region. It is assumed that the material beneath the punch is uniformly stressed with the α sliplines inclined at an angle λ to the punch face, regardless of the frictional condition. This is compatible with a rigid-body motion of the plastic triangle ACA which is attached to the punch. The slipline field is completed by introducing the 90◦ centered fans ACD and A CD , where the sliplines are radial lines and circular arcs. From (6.49), the value of p in ACD increases from p0 on AD to p0 + 2kE on AC, where

π √ π/2 , c = E=E 1 − c sin2 2θ dθ . 2 0 The principal compressive stresses in the region ACA , directed normal and parallel to the punch face, are 2(p0 + kE) and 2kE, respectively. The normal pressure q on the punch face is therefore given by q p0 = +E = 2k k

1−c 1 − c sin 2ψ 2

1/2 +E

π √ , c 2

(6.54)

438

6 Plastic Anisotropy

in view of (6.52). When the degree of anisotropy is small, it is convenient to expand P0 /k and E in ascending powers of c, and neglect all terms of order c2 and above. Then the result becomes

q π c π ≈ 1+ − + cos2 2ψ . 2k 2 2 4 For an isotropic solid (c = 0), the punch pressure reduces to the Prandtl value 2k(1+π /2). The punch pressure for an anisotropic solid is less than this when c is positive and greater than this when c is negative. It may be noted that the punch pressure is the same for orientations ψ and π /2 – ψ of the axes of anisotropy, which is a consequence of the symmetry of the anisotropy about the directions equally inclined to the anisotropic axes. The incipient deformation mode consists of plastic flow parallel to the sliplines CDB and CD B , the material below these boundaries being held rigid. The velocity is of magnitude U sin λ parallel to CDB, and of magnitude U cos λ parallel to CD’B’, where U is the downward speed of the punch. These are also the magnitudes of the velocity discontinuities across A C and AC, respectively. In an alternative solution, valid only for a smooth punch (Hill, 1950a), the bounding sliplines meet at a point S on the punch face. The magnitudes of the velocity discontinuity initiated at this point are then modified to U cosec λ and U sec λ, while the punch pressure is still given by (6.54).

6.4.4 Indentation of a Finite Medium Consider the indentation of the plane surface of a medium of finite depth h resting on a rigid smooth foundation. When the ratio h/a is sufficiently small, where a is the semiwidth of the punch, the plastic zone spreads downward from the punch face to reach the foundation at the yield point state. The slipline field shown in Fig. 6.11 (a) is developed from the triangle ABC, which is not an isosceles triangle unless AB coincides with an anisotropic axis. For an arbitrary angle ψ made by the x-axis of anisotropy with the free surface, the centered fans ACE and BCD are of unequal radii, their angular spans being θ and δ, respectively. The angle λ which the β-line makes with the punch face is given by (6.53). Since the foundation is smooth, the bounding β-line BDF meets the foundation at the same angle λ. The existence of an incipient velocity field at the yield point can be easily established following the method used for isotropic materials. By the analogue of Hencky’s first theorem, which is obtainable from (6.49), we have ωD = ωF + ωC − ωE , where the subscripts refer to the points in the slipline field. The right-hand side of the above equation can be evaluated for any assumed angle θ, using (6.50) and the relations

6.4

Plane Strain in Anisotropic Metals

439

Fig. 6.11 Indentation of a block of anisotropic material of finite depth. (a) Slipline field, (b) variation of punch pressure with block thickness

φ F = φC = −

π 2

−λ+ψ ,

φC = φE = θ .

The other fan angle δ is then computed from (6.50) by a trial-and-error procedure, using the fact that φ D –φ D –φ C = δ. The slipline field is generated from the given intersecting sliplines CD and CE by the usual small arc process of approximation. The construction is similar to that for isotropic solids (Chakrabarty, 1987), except that the value of φ at each unknown nodal point must be determined from the computed slipline variable ω. The mean compressive stress pF at F must be determined from the condition of zero horizontal force transmitted across the bounding sliplines AF and BF, Considering the α-line AEF, and a pair of rectangular axes (ξ , η) through the midpoint of AB, the resultant horizontal thrust exerted on AEF by the rigid material on its left may be written as F=−

τ dξ +

σ dη = 2 k

sdξ +

pdη − k

ds dφ

dη,

where τ = τ αβ and σ = –σβ , denoting the tangential stress and normal pressure on a slipline element, and the integrals are taken along the entire slipline AEF. Integrating the second term on the right-hand side by parts, and noting that dp = –2 k dω along

440

6 Plastic Anisotropy

an α-line, the boundary condition F = 0, which holds when point F lies on the foundation, is reduced to h

p F

2k

=

F A

1 sdξ + 2

F A

ds dφ

dη −

F

ηdω

(6.55)

E

where s(φ) is given by (6.48). The integrals can be evaluated numerically for a selected value of θ since the values of φ, ω, ξ , and η are known along the slipline, to obtain a value of pF using (6.55). As h increases to a critical value h∗ , the pressure pF decreases to the value – p0 , where p0 is given by (6.52). The normal pressure on the foundation then vanishes at F, an element at F being under a uniaxial tension 2p0 parallel to the foundation. For h > h∗ , it is necessary to introduce a triangle FGH of height t with the inclined sides tangential to the sliplines at F. The triangle contributes a horizontal tensile force 2p0 t, and the condition of zero resultant force across the boundary AEFH again leads to (6.55), which furnishes h (and hence t) for a given θ . The indentation pressure q, uniformly distributed over the punch face, is given by q pC + p0 pF + p0 = = + (ωF + ωC − 2ωE ) . 2k 2k 2k The first term on the right-hand side is positive for h 0

(7.53)

to a close approximation, the integral being extended over the entire volume of the material of the beam. The uniqueness is guaranteed when (7.53) is satisfied for all possible velocity fields consistent with (7.52). A sufficiently wide class of admissible velocity fields, representing simultaneous bending in two orthogonal planes together with nonuniform twisting and warping of the cross sections, would be too complicated for practical purposes. On the other hand, the velocity gradients appearing in (7.53) can be determined with sufficient accuracy from the relatively simple field

502

7

˙ υy = υ − φx ˙ , υx = u − φy,

Plastic Buckling

du dυ υz = x + y , dz dz

(7.54)

where (u, υ) denotes the velocity of the axis of the beam, and φ˙ denotes the rate of twist about this axis, all these quantities being functions of z only. It follows from (7.54) that 2 ⎫ d2 υ ∂υz d u ⎪ ⎪ =− x 2 +y 2 , ⎬ ∂z dz dz ∂υx du dφ ∂υy dυ dφ˙ ⎪ ⎪ = −y , = + x .⎭ ∂z dz dz ∂z dz dx

(7.55)

The assumed velocity field (7.54) is consistent with the fact that ε˙ xy = 0 throughout the beam, but it does not give a nonzero value of ε˙ xx = ε˙ yy. The field is also unsuitable for calculating the shear rates ε˙ xz and ε˙ yz , since the rate of warping of the cross section has been disregarded. It is reasonable, however, to assume the relations ε˙ xz = 0,

ε˙ yz = x

dφ˙ , dz

(7.56)

which apply to the elastic torsion of bars of narrow rectangular cross section. The usefulness of such simplified velocity fields with appropriate adjustments has been noted by Pearson (1956) in the context of elastic buckling. The substitution from (7.55) and (7.56) into (7.53) gives

2 2 d2 υ dφ˙ d2 u + 4G x (T − σ ) x 2 + y dz dz dz $ 2 2 % ˙ dφ dφ˙ du dυ −y +x dx dy dz > 0, + σ + dz dz dz dz

where x varies between –b and b, y varies between –h and h, and z varies between 0 and l. The evaluation of the above integral is greatly simplified by the fact that σ (–y) = –σ (y) and T(–y) = T(y), giving

h −h

σ dy =

h

−h

σ y dy = 2

h −h

Tydy = 0.

Since σ and T are independent of x, and the stress distribution across each vertical section is statically equivalent to a bending moment equal to M, the condition for uniqueness becomes

7.3

Lateral Buckling of Beams

l 0

αEIy

du dz

503

2 + βEIx

dυ dz

2 + GJ

dφ˙ dz

2 + 2M

du dφ˙ dz dz

dz > 0, (7.57)

where Ix and Iy are the principal moments of inertia of the cross section about the x- and y-axes, respectively, and GJ is the torsional rigidity with J = 4Iy , the constants α and β being defined as 1 αEh = 2

h

3 Tdy = Rσ0 , βEh = 2 −h 3

h −h

Ty2 dy,

where R is the radius of curvature of the bent axis at the incipient buckling, and σ 0 the magnitude of the numerically largest stress that occurs at y = ±h. The deformation prior to buckling is assumed to be small, so that the longitudinal strain is equal to –y/R throughout the elastic/plastic bending. The minimum value of the functional in (7.57) with respect to the variations of u, υ, and φ˙ must vanish at the point of bifurcation. The Euler–Lagrange differential equations characterizing this variational problem are easily established as ⎫ d2 φ˙ d4 υ d4 u ⎪ ⎪ − M = 0, = 0, ⎬ dz4 dz2 dz4 ⎪ d2 φ˙ d2 u ⎪ ⎭ GJ 2 + M 2 = 0. dz dz

αEIy

(7.58)

The ends of the beam are assumed to be supported in such a way that they cannot rotate about the z-axis. Then the curvature rate in the xz-plane also vanishes at these sections, and the boundary conditions become φ˙ =

d2 u = 0 at z = 0 and z = l. dz2

Due to the symmetry of the loading, the curvature rate in the yz-plane must have identical values at the sends of the beam. The first two equations of (7.58) may therefore be integrated twice to give αEIy =

d2 u d2 υ − M φ˙ = 0, = constant. 2 dz dz2

(7.59)

The second equation of (7.59) merely states that the axis of the beam continues to bend into a circular arc at the point of bifurcation. The elimination of d2 u/dz2 between the first equation of (7.59) and the last equation of (7.58) results in d2 φ˙ MI + k2 φ˙ = 0, k = EIy dξ 2

1+ν , 2α

(7.60)

504

7

Plastic Buckling

where ξ = z/l and ly = 4bh3 /3. The solution to this differential equation, subject to the boundary conditions φ = 0 at ξ = 0 and ξ = 1, is readily obtained as φ˙ = A sin π ξ , k = π , where A is a constant. Over the elastic range, we √ have σ 0 = Eh/R giving α = 1, and the critical moment is given by Ml/EIy = π 2/ (1 + υ), which is in agreement with the well-known result for elastic buckling of narrow rectangular beams. When the buckling occurs in the plastic range, α depends on the magnitude of the critical moment through the strain-hardening characteristic of the material. Assuming the power law σ /Y = (Eε/Y)n for the uniaxial stress–strain curve in the plastic range, the stress distribution for y ≥ 0 may be written as y σ = − (0 ≤ y ≤ c) , Y c

y n σ =− (c ≤ y ≤ h) . Y c

The radius of curvature of the bent axis, when the elastic/plastic boundary is at y = c, is R = Ec/Y. The bending moment across any section is

h

M = −4b 0

n h 3 4 2 1 − n c 2 . σ ydy = bh Y − 3 2+n c 2+n h

Introducing the initial yield moment Me = 4bh2 Y/3, and using the fact that h/c is equal to the curvature ratio k/ke , the moment–curvature relationship may be expressed in the dimensionless form M 3 = Me 2+n

κ κe

n −

1 − n κ e 2 . 2+n κ

(7.61)

Since the greatest bending stress at any stage is σ o = Y(h/c)n during the elastic/plastic bending, we have α=

Rσ0 = (c/h)1−n = (κe /κ)1−n . Eh

Substituting for α and M in (7.60) and setting k = π , the equation for the critical curvature ratio for lateral buckling is obtained as 3 2+n

κ κe

(1+n)/2

1 − n κe (3+n)/2 π Eh − = 2+n κ Yl

2 = λ (say) . 1+ν

(7.62)

When κ/κ e is computed from (7.62) for any given values of n and Eh/Yl, the critical bending moment follows from (7.61). This solution is valid for κ/κ e ≥ 1, which is equivalent to the condition λ ≥ 1. In the case of λ ≤ 1, the buckling will occur in the elastic range when M/Me = λ, obtained by setting n = 1 in (7.62). Figure 7.8 shows the variation of the dimensionless critical bending moment with the parameter μ for several values of n.

7.3

Lateral Buckling of Beams

505

Fig. 7.8 Dimensionless critical bending couple for the lateral buckling in the plastic range as a √ function of a parameter λ (Eh/Yl) 2/(1 + ν)

Equations (7.59) are equally applicable to thelateral buckling of beams of arbitrary doubly symmetric cross sections J = 4Iy subjected to terminal couples in the yz-plane, which is considered as the plane of maximum flexural rigidity. The last equation of (7.58) must be modified, however, by the inclusion of the term ˙ 4 on the left-hand side to take account of the warping rigidity, the −ECw d4 φ/dz warping constant Cw being identical to that for elastic buckling (Timoshenko and Gere, 1961). The governing differential equation (7.60) is then replaced by a fourthorder equation for φ˙ , which can be solved in terms of trigonometric and hyperbolic functions.

7.3.2 Buckling of Transversely Loaded Beams We begin with a cantilever beam of narrow rectangular cross section carrying a terminal load P applied in the yz-plane, as shown in Fig. 7.9. As the load is increased to its critical value, the beam can buckle laterally in the xz-plane, and this is accompanied by the rotation of cross sections by varying degrees along the beam. Following the customary treatment for the elastic buckling, the basic equations for the plastic analysis under transverse loads are assumed to be the same as those in pure bending, provided M is regarded as the local bending moment equal to –P(l – z). The basic differential equations involving u and φ˙ therefore become αEIy

d2 u + P (l − z) φ˙ = 0, dz2

GJ

d2 φ˙ d2u − P − z) = 0. (l dz2 dz2

506

7

Plastic Buckling

Fig. 7.9 Lateral plastic bucking of an end-loaded cantilever of narrow rectangular cross section

Eliminating d2 u/dz2 between these two relations and setting z = l(1–ξ ), we obtain the differential equation for φ˙ as d2 φ˙ Pl2 α 2 + β 2 ξ 2 φ˙ = 0, β = EIy dξ

1+ν , 2

(7.63)

where α is a function of ξ in the elastic/plastic portion of the beam. Evidently, α = 1 over the part of the beam that is entirely elastic. If the length of the elastic portion of the beam is denoted by a, then α=1

(0 ≤ ξ ≤ a/l) , α = ρ −(1−n)

(a/l ≤ ξ ≤ 1) ,

where ρ denotes the absolute value of the curvature ratio κ/κ e , the variation of ρ with ξ in the elastic/plastic portion of the beam being given by (7.61) with the substitution Plξ for M. Since Pa = Me , the result is l s= ξ = a

1−n 3 n ρ − ρ −2 . 2+n 2+n

(7.64)

It is convenient at this stage to change the independent variable from ξ to s. Since Me = 4bh2 Y/3 and Iy = 4hb3 /3, the differential equation (7.63) is transformed into d2 φ˙ ˙ + k2 λ2 φ, ds2

Yah k= Eb2

1+v , 2

(7.65)

7.3

Lateral Buckling of Beams

λ=

507

3 1−n ρ (1+n)/2 − ρ −(3+n)/2 . 2+n 2+n

(7.66)

Since elastic bending corresponds to n = 1, it is only necessary to set λ = s in the above differential equation over the elastic portion (0 ≤ s ≤ 1). At the built-in end of the beam, the angle of twist vanishes, requiring φ˙ = 0, while ˙ at the free end of the beam, the torque is zero requiring dφ/ds = 0. The boundary conditions may therefore be written as dφ˙ =0 ds

at s = 0, φ˙ = 0 at s =

l . a

Over the elastic portion of the beam, where λ = s, the solution to the differential equation (7.65) can be expressed in terms of Bessel functions. In view of the first ˙ boundary condition dφ/ds = 0at s = 0, the solution is easily shown to be φ˙ =

√ sJ−1/4

1 2 ks , 2

0 ≤ s ≤ 1.

(7.67a)

The constant of integration has been arbitrarily set to unity for convenience. By a well-known formula for the first derivative of the Bessel functions, we have dφ˙ = −ks3/ 2 J3/4 ds

1 2 ks , 2

0 ≤ s ≤ 1.

(7.67b)

Over the elastic/plastic portion of the beam (1 ≤ s ≤ l/a) the relationship between λ and s is given by (7.64) and (7.66) parametrically through ρ. For a given b/h and a selected value of a/h, the differential equation (7.65) can be integrated numerically ˙ starting from s = 1, the values of φ˙ and d φ/ds at this section being those given by (7.67). The remaining boundary condition φ˙ = 0 at s = l/a finally gives the value l/a, furnishing the solution for a definite value of l/h. The corresponding value of the critical load is obtained from the relation hIy h 4 Me . (7.68) = bhY =Y P= a 3 a ab2 The greatest value of hl/b2 for which the elastic/plastic analysis is applicable corresponds to a = l, for which φ˙ = 0 at s = 1 giving J–1/4 (k/2) = 0 or k = 4.013. Buckling will therefore occur in the plastic range if hl E ≤ 4.013 2 b Y

2 . 1+ν

For higher values of hl/b2 , buckling will occur in the elastic range, and the critical load is then obtained by setting a = l in the last expression of (7.68) and using the greatest value of hl/b2 defined by the above inequality.

508

7

Plastic Buckling

In the case of a simply supported beam of length 2 l bent in the yz-plane by a concentrated load 2P acting at the midspan, the governing equations for lateral buckling are still (7.64), (7.65), and (7.66), provided z is measured from the center of the beam and a denotes the length of the elastic part on each side of the central section. Assuming that the ends of the beam are prevented from rotation about the z-axis by appropriate constraints, the boundary conditions may be written as φ˙ = 0 at s = 0,

l dφ˙ = 0 at s = . ds a

The solution for the elastic portion on each side is obtained by setting λ = s in the differential equation (7.65) and using the boundary condition φ˙ = 0 at s = 0. The rate of twist and its derivative over the elastic region (0 ≤ s≤ 1) therefore become φ˙ =

√

sJ1/4

1 2 ks , 2

dφ˙ = ks3/2 J−3/4 ds

1 2 ks . 2

(7.69)

The integration of the differential equation (7.65) over the elastic/plastic lengths (1 ≤ s ≤ l/a) can be carried out in the same way as that indicated in the case of the cantilever, using the continuity conditions k dφ˙ = kJ−3/ 4 at s = 1, ds 2 ˙ in view of (7.69). The symmetry condition dφ/ds = 0 at s = –l/a finally gives the ratio l/h for any assumed value of a/h, while the critical value of P follows from (7.68). When a = l, the plastic zones disappear, and (7.69) holds for the entire beam. ˙ Then dφ/ds must vanish at s = 1, requiring J–3/4 (k/2) = 0, which gives k = 2.117. It follows that the elastic/plastic analysis applies only over the range φ˙ = J1/4

k , 2

E hl ≤ 2.117 2 b Y

2 . 1+v

Outside this range, buckling would occur while the beam is still elastic, and the corresponding value of P is obtained by inserting in the last expression of (7.68) the value of h/ab2 that corresponds to a = l. The solution to the plastic buckling problem for other kinds of loading may be obtained in a similar manner. The analysis can be extended to the buckling of beams of arbitrary doubly symmetric cross sections in the manner indicated for pure bending.

7.4 Buckling of Plates Under Edge Thrust 7.4.1 Basic Equations for Thin Plates Consider a thin plate of arbitrary shape in which the material is bounded between the planes z = ±h/2, and the middle surface z = 0 is bounded by a closed curve

7.4

Buckling of Plates Under Edge Thrust

509

that defines the edge of the plate. The bounding planes are unstressed, while uniform compressive stresses of magnitudes σ 1 and σ 2 act in the x-and y-directions, respectively, to represent the plastic state. If the transverse shear rates on the incipient deformation mode at bifurcation are disregarded, the admissible velocity field may be written as υx = u − z

∂w , ∂x

υy = υ − z

∂w , ∂y

υz = w,

(7.70)

where u, ν, and w are functions of x and y, representing the components of the velocity of the middle surface. The velocity field (7.70) is adequate for expressing all the strain rate components except the through-thickness one, which follows from the relation ε˙ zz = −η ε˙ xx + ε˙ yy , where η is the contraction ratio in the current state. If the yield surface is assumed to be that of von Mises, √ the relevant components of√the associated unit normal are nxx = − (2σ1 − σ2 )/ 6σ¯ and nyy = − (2σ1 − σ2 )/ 6σ¯ , where σ¯ is the equivalent stress defined as σ¯ 2 = σ12 − σ1 σ2 + σ22 . Using the rate form of the constitutive law defined by (1.37), and considering the linearized solid for which the Jaumann stress rate is denoted by τ˙ij we obtain the relations 3σ22 T τ ˙ = 1− − η− xx 4σ¯ 2 H 3σ12 T = 1− τ˙yy − η − 4σ¯ 2 H

T ε˙ xx T ε˙ yy

3σ1 σ2 T 4σ¯ 2 H 3σ1 σ2 T 4σ¯ 2 H

τ˙yy , (7.71)

τ˙xx ,

where T is the tangent modulus related to the plastic modulus H by T = EH/(E+ H), while the contraction ratio η satisfies the relation 1 – 2η = (1 – 2v)T/E. The constitutive relations (7.71) are readily inverted to express the stress rates in terms of the strain rates, the result being τ˙xx = E α ε˙ xx + β ε˙ yy ,

τ˙yy = E β ε˙ xx + γ ε˙ yy ,

τ˙xy = 2G˙εxy ,

where the last relation follows from the fact that the shear strain rate is purely elastic, the parameters α, β, and γ being given by

510

7

Plastic Buckling

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

2 α = ρ −1 4 − 3 (1 − T/E) σ12 / σ¯ , β = ρ −1 2 − 2 (1 − 2v) T / E − 3 (1 − T / E) σ1 σ2 /σ¯ 2 , γ = ρ −1 4 − 3 (T / E) σ22 /σ¯ 2 ,

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2⎭ ρ = (5 − 4v) − (1 − 2v) T / E − 3 (1 − 2v) (1 − T / E) σ1 σ2 /σ¯ .

(7.72)

These quantities evidently depend on the current state of stress. It follows from the preceding expressions for the stress rates that

2 2 2 + 4G˙εxy + 2β ε˙ xx ε˙ yy + γ ε˙ yy . τ˙ij ε˙ ij = E α ε˙ xx

(7.73)

To obtain the condition for bifurcation of the plate in the elastic/plastic range, consider the uniqueness criterion in the form (7.5). Since the only nonzero components of the stress tensor are σ xx = –σ 1 and σ yy = –σ 2 , which are small compared to the modulus of elasticity E, the condition for uniqueness becomes $

τ˙ij ε˙ ij − σ1

∂υy ∂x

2 +

∂υz ∂x

2 %

$ − σ2

∂υx ∂y

2 +

∂υz ∂y

2 % dV > 0.

Inserting (7.73) into the above inequality and substituting for the strain rates and the velocity gradients which are given by ∂ 2w ∂ 2w ∂ 2w ∂u ∂υ ∂u ∂υ − z 2 , ε˙ yy = −z 2, + − 2z , 2˙εxy = ∂x ∂y ∂y ∂x ∂x∂y ∂x ∂y ∂υy ∂υ ∂ 2w ∂υx ∂u ∂ 2w = −z , = −z , ∂x ∂x ∂x∂y ∂y ∂y ∂x∂y

ε˙ xx =

in view of (7.70), and integrating through the thickness of the plate, the condition for uniqueness is reduced to 2 2 ∂υ 2 ∂u ∂υ ∂ 2u ∂ 2υ G ∂υ + α dx dy + 2β 2 2 + γ + ∂x ∂y E ∂y ∂x ∂x ∂y 2 2 2 2 2 ∂ w ∂ 2w ∂ 2w 4G ∂ 2 w ∂ w h2 + 2β 2 2 + y + α + dx dy 12 E ∂x∂y ∂x2 ∂x ∂y ∂y2 $ 2 % $ 2 2 % ∂w ∂w ∂υ ∂u 2 σ1 σ2 + + − dx dy > 0, E ∂x ∂x E ∂y ∂y with a minor approximation that is perfectly justified for thin plates. All the integrals appearing in the above expression extend over the middle surface of the plate. The left-hand side of the above inequality is seen to have a minimum value when

7.4

Buckling of Plates Under Edge Thrust

511

u = v = 0, and the Euler–Lagrange differential equation, associated with the minimization with respect to arbitrary variations of w, is easily shown to be α

∂ 4w ∂ 4w 12 ∂ 4w + 2 (β + μ) 2 2 + γ 4 = − 2 4 ∂x ∂x ∂y ∂y h

σ1 ∂ 2 w σ2 ∂ 2 w , + E ∂x2 E ∂y2

(7.74)

where μ = 1/(1 + v). The solution to the bifurcation problem is therefore reduced to the solution of the differential equation (7.74) under appropriate boundary conditions. When the bifurcation occurs in the elastic range (T = E), we have α = β = μ = γ = 1/(1–v2 ) and (7.74) reduces to the well-known governing equation for elastic buckling. In order to establish the static boundary conditions in terms of w, it is convenient to take the components of the nominal stress rate s˙ij as approximately equal to those of τ˙ij . This is justified by the fact that the stresses at bifurcation will be small compared to the elastic and plastic moduli. Then the rates of change of the resultant bending and twisting moments per unit length are given by 2 ⎫ ∂ w Eh2 ∂ 2w ⎪ ˙x = ⎪ α 2 + β 2 ,⎪ M τ˙xx zdz = − ⎪ ⎪ 12 ∂x ∂y ⎪ −h/2 ⎪ ⎪ h/2 ⎬ 3 2 2 ∂ w Eh ∂ w ˙ β 2 +γ 2 , τ˙yy zdz = − My = ⎪ 12 ∂x ∂y −h/2 ⎪ ⎪ ⎪ h/2 ⎪ 2 2 ⎪ Eh ∂ w ⎪ ⎪ ˙ xy = τ˙xy zdz = − M , ⎭ 12 (1 + v) ∂x∂y −h/2

h/2

(7.75)

in view of the relations (7.72) and (7.70) with u = v = 0. These expressions furnish ˙ nt across a typical ˙ n and the twisting moment rate M the bending moment rate M ˙ n must vanish along boundary element by the rule of tensor transformation. While M ˙ n + ∂M ˙ nt /∂S is required to vanish along a a simply supported edge, the quantity Q ˙ n is the transverse shear force rate per unit length and ds is an arc free edge, where Q element of the boundary.

7.4.2 Buckling of Rectangular Plates To illustrate the preceding theory, consider a rectangular plate whose sides are of lengths a and b, the origin of coordinates being taken at one of the corners of the rectangle as shown in Fig. 7.10(a). The plate is assumed to be simply supported along two opposite sides x = 0 and x = a, while different edge conditions may apply to the remaining two sides y = 0 and y = b. The plate is compressed by equal and opposite forces in the direction perpendicular to the simply supported sides. ˙ x must vanish along the Since the deflection rate w and the bending moment rate M simply supported edges, the boundary conditions are

512

7

Plastic Buckling

Fig. 7.10 Buckling of uniformly compressed rectangular flat plates. (a) Uniaxial edge thrust, (b) biaxial edge thrust

w = 0,

∂ 2w =0 ∂x2

on x = 0 and x = a.

It may be noted that w = 0 implies ∂ 2 w/∂y2 = 0 along these sides. Assuming that the plate buckles in m sinusoidal half-waves, the deflection rate is taken in the form

mπ x w = f (y) sin , a where f(y) is an unknown function of y. The boundary conditions along x = 0 and x = a are identically satisfied, and the substitution for w in the differential equation (7.74) with σ 1 = σ and σ 2 = 0 results in the ordinary differential equation d2 f 2φ d2 f ψf − 2 2 − 4 = 0, 4 dy b dy b

(7.76)

where φ and ψ are dimensionless parameters expressed by the relations ⎫ ⎪ 7 − 2v mπ b 2 1 − 2v T ⎪ ⎪ −3 , ⎪ ⎬ a 1+v 1+v E $ % 2 2 2 ⎪ mπ b 12ρσ b 3T 1 mπ b ⎪ ,⎪ − 1+ ψ= ⎪ ⎭ 4 a E h E a

1 φ= 4

(7.77)

in view of (7.72). The physical constraints along the edges y = 0 and y = b are usually such that ψ > 0 at bifurcation. The solution to (7.76) may then be expressed in terms of two parameters k1 and k2 which are defined as k1 =

−φ +

φ2

+ ψ,

k2 =

φ+

φ 2 + ψ.

(7.78)

7.4

Buckling of Plates Under Edge Thrust

513

The general solution to the above fourth-order differential equation can be written in the form k1 y k2 y k2 y k1 y + B sin + C cosh + D sinh , (7.79) f (y) = A cos b b b b where A, B, C, and D are constants of integration, the ratios of which can be determined from the boundary conditions on y = 0 and y = b, which also furnish the critical stress σ for bifurcation. An incremental theory for the plastic buckling of plates has been discussed earlier by Pearson (1950). The analysis given here is essentially due to Sewell (1963, 1964). As a first example, consider the situation where the sides y = 0 and y = b are also simply supported, so that all four sides of the plate have identical supports. Then the additional boundary conditions are w = 0,

∂ 2 w/ ∂y2 = 0 on y = 0 and y = b.

In view of the assumed expression for w, these conditions are equivalent to f = d2 f/dy2 = 0 along y = 0 and y = b. They are satisfied by taking A = C = D = 0 in the general solution (7.79) and by setting k1 = nπ where n is an integer. The first relation of (7.78) therefore gives ψ − 2π 2 n2 φ − π 4 n4 = 0, and the substitution from (7.77) then furnishes the critical compressive stress for bifurcation as 2 σ 3T 3 1 − 2ν mb π 2 h2 a 2 T 2+ 1+ , + = 1− + 2 E 3ρb E 2a 2 1+ν E mb (7.80) ρ = 3 + (1 − 2ν) 2 − (1 − 2ν) T/E where we have set n = 1 to minimize σ . It remains to choose the value of m for given a/b, h/b, and T/E ratios, so that the right-hand side of (7.80) is a minimum. When the ratio a/b is not too large, it is natural to expect the bifurcation mode to involve a√single half-wave in the direction of compression (m = 1), which requires 2a2 /b2 ≤ 1 + 3T/E. For sufficiently large values of a/b, the critical stress is closely approximated by π 2 h2 σ = E 3ρb2

2+

3T 3 1+ + E 2

1 − 2ν 1+ν

T 1− E

,

(7.81)

√ which is obtained by setting 2a2 /m2 b2 = 1 + 3T/E in (7.80), although the corresponding value of m is not generally an integer. The graphical plot in Fig. 7.11

514

7

Plastic Buckling

Fig. 7.11 Dimensionless critical stress for buckling of rectangular plates under unidirectional compression. The parameter c denotes the exponent of the Ramberg–Osgood stress–strain law

shows how the critical stress varies with the ratio a/b, when the stress-strain curve is represented by (7.9) with m being replaced by c. The critical stress is considerably lowered by using the rate form of the Hencky stress-strain relation, as has been shown by Shrivastava (1979), Durban and Zuckerman (1999), and Wang et al. (2001). As a second example, suppose that the sides y = 0 and y = b are rigidly clamped so that these edges are prevented from rotation. The boundary conditions along these edges therefore take the form w = 0,

∂w/ ∂y = 0

on y = 0 and y = b.

These conditions are evidently equivalent to f = df/dy = 0 along y = 0 and y = b where f(y) is given by (7.79). The consideration of the side y = 0 indicates C = –A and D = –(k1 − k2 ) B, and the function f then becomes

7.4

Buckling of Plates Under Edge Thrust

k1 y f (y) = A cos b

515

k2 y − cosh b

k1 y + B sin b

k2 y k1 − sinh . k2 b

The application of the boundary conditions to the remaining side y = b leads to the pair of equations k1 A (cos k1 − cosh k2 ) + B sin k1 − sinh k2 = 0, k2 − A (k1 sin k1 + k2 sinh k2 ) + Bk1 (cos k1 − cosh k2 ) = 0,

which can be satisfied by nonzero values of A and V if the determinant of their coefficients is zero. The relationship between k1 and k2 at the point of bifurcation therefore becomes k2 k1 sin k1 sinh k2 . 2 (1 − cos k1 cosh k2 ) = − (7.82) k2 k1 For a given stress–strain curve and the ratio a/b, we may assume a value of σ /E, guided by the corresponding elastic solution (Timoshenko and Gere, 1961). The parameters k1 and k2 are then computed from (7.77) and (7.78) with an appropriate value of m. If the computed values do not satisfy (7.82), the initial assumption must be altered and the procedure repeated until a consistent value is obtained for the critical stress. As a final example, we consider the buckling of a rectangular plate in which the sides x = 0, x = a, and y = 0 are simply supported, while the remaining side y = b is free. The boundary conditions along the edges y = 0 and y = b may be written as w = 0, ∂ 2 w/∂y2 = 0 along y = 0 ˙ y + ∂M ˙ y = 0, Q ˙ xy /∂x = 0 along y = b M The condition of moment equilibrium of a typical element of the plate, the shear ˙ y , is given by the differential equation force rate Q ˙y ˙ xv ∂M ∂M ∂ 2w ∂ 2w Eh3 ∂ ˙ Qy = + =− (β + μ) 2 + γ 2 , ∂x ∂y 12 ∂y ∂x ∂y in view of (7.75). Using (7.72), the free-edge boundary conditions may be expressed in terms of the deflection rate w in the form ∂ 2w ∂ ∂ 2w + η = 0, ∂y ∂y2 ∂x2

∂ 2w ∂ 2w + ξ ∂y2 ∂x2

= 0 on y = b,

(7.83)

where η is the contraction ratio, and ξ = (1 + η) (2 − ν)/(1 + ν). Due to the simply supported edge conditions along y = 0, it is necessary to set A = C = 0 in (7.79), and the function f(y) then becomes

516

7

k1 y f (y) = B sin b

Plastic Buckling

k2 y + D sinh . b

The remaining boundary conditions (7.83), corresponding to the free edge y = b, furnish the relations m 2 π 2 b2 m 2 π 2 b2 2 2 − B k1 + η sin k1 + D k2 − η sin k2 = 0, a2 a2 m 2 π 2 b2 m2 π 2 b2 2 + Dk − ξ − Bk1 k12 + ξ cos k k cosh k2 = 0. 1 2 2 a2 a2 Setting the determinant of the coefficients of A and B in the above equations to zero, which is required by nonzero values of these constants, the relationship between k1 and k2 is obtained as m 2 π 2 b2 m2 π 2 b2 2 k tan k1 − ξ k2 k12 + η 2 a2 a2 m 2 π 2 b2 m2 π 2 b2 2 2 k1 + ξ tanh k2 . = k2 k2 − η a2 a2

(7.84)

For specified values of the ratios a/b and h/b, and a given stress–strain curve, the least value of the critical compressive stress that satisfies (7.84) can be determined by a trial-and-error procedure, following the same method as explained before. The direction of the yield surface normal has a marked influence on the critical stress, as has been shown by Sewell (1973).

7.4.3 Rectangular Plates Under Biaxial Thrust A rectangular plate, which is simply supported along all its edges, is subjected to compressive stresses σ 1 and σ 2 uniformly distributed along the sides perpendicular to the x- and y-axes, respectively, Fig. 7.10(b). All the boundary conditions are identically satisfied by taking the deflection rate in the form w = w0 sin

mπ x a

sin

nπ y b

,

where w0 is a constant. The substitution for w into the differential equation (7.74) furnishes 2

2 σ

2 π 2 h2 2 2 σ1 2 a 4 b 2 2 4 a + 2 (β + μ) m n + γ n m +n = αm . E b E a b 12b2 (7.85) This is the required relationship between the applied stresses σ 1 and σ 2 at the point of bifurcation. The integers m and n should be such that for a given value of one of these stresses, the other one is a minimum. When the tangent modulus is

7.4

Buckling of Plates Under Edge Thrust

517

independent of the stress, as in the case of an elastic material, the critical combination of stresses defined by (7.85) lie on a concave polygon in the (σ 1 , σ 2 )-plane, as has been shown by Timoshenko and Gere (1961). Consider any sequence of states satisfying (7.85) and lying in the neighborhood of the state σ 2 = 0, so that the mode of bifurcation corresponds to n = 1. Then for a given value of σ 2 , the critical value of σ 1 is established by the equation ⎧ ⎫ ⎛ ⎞ ⎪ ⎪ ⎪ ⎪ 2 ⎨

2σ ⎟ 2⎬ ⎜ σ1 mb 12b π 2 h2 a 2 ⎜γ − ⎟ α + 2 + μ) + = (β ⎝ E 12b2 ⎪ a π 2 h2 E ⎠ mb ⎪ ⎪ ⎪ ⎩ ⎭

(7.86)

for an appropriate value of m that minimizes the right-hand side of this equation. When σ 2 /E ≤ γ (π 2 h2 /12b2 ), the range of values of a/b, for which m = 1 is applicable, is given by a2 ≤ b2

α , λ

λ=γ −

12b2 σ2 ≥ 0. π 2 h2 E

For higher values of the ratio a/b, the critical state corresponds to m = 2. When a/b is sufficiently large, the minimization is very closely achieved by set√ ting (a/mb)2 = α/λ. The critical state is then independent of a/b and is given by ⎧' ⎫ 2 2 πh ⎨ πh⎬ σ1 γπ h 3σ2 = + (β + μ) . α − E 3b ⎩ E 2b ⎭ 4b2

(7.87)

This expression is an immediate generalization of (7.81) for bifurcation under biaxial compression satisfying the condition λ ≥ 0. For λ < 0, the bifurcation state will correspond to n = 1 only for a certain range of values of the aspect ratio a/b. For arbitrary combinations of σ 1 and σ 2 , the conditions under which the critical state corresponds to m = n = 1, so that the bifurcation mode involves only one halfwave in both directions of compression, can be established by using (7.85). The bifurcation state in this case is given by σ1 a 2 σ2 π 2 h2 + = E b E 12b2

a 2 b 2 α . + 2 (β + μ) + γ a b

(7.88)

The validity of (7.88) requires that for m = 2, n = 1 and for m = 1, n = 2 the equality sign in (7.85) must be replaced by an inequality in which the left-hand side is less than the right-hand side. Thus,

518

7

Plastic Buckling

2

a 2 b + 8 (β + μ) + γ 16α , a b

a 2

a 2 σ b 2 π 2 h2 σ1 2 α . + 8 (β + μ) + 16γ +4 ≤ E b E 12b2 a b

σ1 a 2 σ2 π 2 h2 4 + ≤ E b E 12b2

These inequalities together with (7.88) lead to the necessary restrictions on σ 1 and σ 2 for which the bifurcation state is defined by (7.88), the continued inequalities to be satisfied by the stress σ 1 being

α − 4y

a 4 b

≤

a 2 12a2 σ1 ≤ 5α + 2 + μ) . (β b π 2 h2 E

The value of σ 1 defined by the lower limit and the value of σ√2 furnished by the 2 2 upper √ limit in the above inequalities are negative when 2(a/b) > α/γ and (a/b) < 2 α/γ , respectively. When the state of stress is an equibiaxial compression defined by σ 1 = σ 2 = σ , the parameters ρα, ρ(β +μ), and ργ are each equal to 1+3T/E in view of (7.72) with σ¯ = σ , and the expression for the critical stress then becomes σ π 2 h2 = E 12a2

(1 + 3T / E) 1 + a2 / b2 . 2 (1 + v) [1 + (1 − 2ν) T / E]

(7.89)

For a given aspect ratio a/b, the bifurcation stress in equibiaxial compression is found to be considerably lower than that in unidirectional compression. Over the elastic range of buckling (T = E), the equibiaxial critical stress for a square plate (a = b) is exactly one-half of the unidirectional critical stress. The bifurcation stress predicted by (7.89) is plotted in Fig. 7.12 as a function of the aspect ratio, assuming the stress–strain curve to be given by (7.9) with different values of the exponent m, which is here replaced by c. The influence of edge restraints produced by friction on the critical stress has been investigated by Gjelsvik and Lin (1985) and Tugcu (1991). Solutions for the critical stress based on the Hencky stress–strain relations have been presented by Illyushin (1947), Bijlaard (1949, 1956), Gerard (1957), and El-Ghazaly and Sherbourne (1986). The buckling loads predicted by the total strain theory, despite its physical shortcomings, are found to be in better agreement with experiments, whereas those based on the incremental theory are found to be significantly higher than the experimental ones. This apparent paradox is partly due to the presence of geometrical and other imperfections which are not considered in the theory. It is also possible for some kind of non-associated flow rule, resembling the Hencky relations, to apply at the point of bifurcation. The plastic buckling of plates based on the slip theory of plasticity has been discussed by Batdorf (1949) and by Inoue and Kato (1993). A detailed investigation of the plastic buckling of relatively thick plates has been carried out by Wang et al. (2001).

7.4

Buckling of Plates Under Edge Thrust

519

Fig. 7.12 Variation of critical stress with aspect ratio for the buckling of rectangular plates under equibiaxial compression

7.4.4 Buckling of Circular Plates A circular plate of thickness h and radius a is submitted to a radial compressive stress σ uniformly distributed around the periphery. The incipient deformation mode at the point of bifurcation is assumed to be such that the deformed middle surface is a surface of revolution in which the deflection rate w is a function of the radius r only. Since the radial velocity of particles on the middle surface may be set to zero in the investigation for bifurcation, the velocity field may be written as υr = −z

d2 w , dr2

υθ = 0,

υz = w,

(7.90a)

in cylindrical coordinates (r, θ , z). The shear strain rates are identically zero, and the relevant components of the velocity gradient are d2 w ∂υr = −x 2 , ∂r dr

dυz dw ∂υr =− =− . ∂z dr dr

(7.90b)

520

7

Plastic Buckling

The radial and circumferential components of the linearized constitutive relation, which are similar to (7.71), can be written as τ˙rr = E (α ε˙ rr + β ε˙ θθ ) ,

τ˙θθ = E (β ε˙ rr + γ ε˙ θθ ) ,

where α, β, and γ are given by (7.72) with σ1 = σ2 = σ , the result being ⎫ 1 ⎪ , γ = α,⎪ β =α− ⎬ 1+ν ⎪ T ⎪ ⎭ ρ = 4 (1 + ν) (1 − η) , 2η = 1 + (1 − 2ν) . E

1 α= ρ

3T 1+ , E

(7.91)

Since the radial and circumferential strain rates are ε˙ rr = ∂νr /∂r and ε˙ θθ = νr /r, the criterion for uniqueness in this case takes the form % $

υ 2 ∂υz 2 υr ∂υr ∂υr 2 r −σ dV > 0, + 2β +γ E α ∂r r ∂r r ∂r where the integral extends over the entire volume of the plate. Substituting from (7.90) and integrating through the thickness, we get

a 0

2 2 d2 w 1 dw 2 12σ dw 2 1 dw d w α rdr > 0. +γ + 2β − 2 dr2 r dr dr2 r dr h E dr

The deflection rate w should be such that the left-hand side of the above inequality is a minimum. The associated variational problem is characterized by the Euler– Lagrange differential equation α

d2 dr2

2 1 dw 2 12σ d d w dw r 2 −γ + 2 r = 0. r dr dr dr h E dr

Since the coefficients of this equation have constant values throughout the plate at the point of bifurcation, the first integral of the above equation can be immediately written down, and the constant of integration can be set to zero. Introducing the notation 2r 3σ dw , ξ= , φ= dr h αE the governing differential equation for the occurrence of bifurcation is easily shown to be ξ2

d2 φ dφ 2 + ξ − 1 φ = 0, + ξ dξ dξ 2

(7.92)

which is recognized as Bessel’s differential equation having the general solution

7.4

Buckling of Plates Under Edge Thrust

521

φ = AJ1 (ξ ) + BY1 (ξ ) ,

(7.93)

where J1 (ξ ) and Y1 (ξ ) are Bessel functions of the first order, and of the first and second kinds, respectively. Since φ must be zero at the center of the plate ξ = 0, we must set B = 0 in (7.93). The critical stress for bifurcation evidently depends on the condition of support of the circular edge r = a. As a first application of the preceding theory, consider the plastic buckling of a circular plate which is fully clamped around its edge. Then the boundary condition is dw/dr = 0 along r = a, which is equivalent to φ = 0 along ξ = k, and the critical stress is then given by k2 h2 σ αk2 h2 = = E 12a2 12a2

1 + 3T/E , 2 (1 + ν) 1 + (1 − 2ν) T/E

(7.94)

in view of (7.91). It follows from the boundary condition applied to (7.93), where B is identically zero, that k is the smallest root of the equation J1 (k) = 0. Since T/E is a function of σ /E for any given stress–strain curve of the material, the solution must be found by a trail-and-error procedure. In the case of elastic buckling, we have T = E and α = 1/(1 – v2 ), giving k ≈ 3.832 and (1 – v2 )σ /E ≈1.224h2 /a2 at the point of bifurcation. As a second example, let the circular plate be simply supported along its edge, so that the rate of change of the radial bending moment vanishes along r = a. Since the bending moment rate is very closely given by the expression ˙r = M

h/2 −h/2

τ˙rr zdz = −

Eh3 12

2 d w β dw α 2 + , r dr dr

˙ r = 0 can be written down as the simply supported edge condition M α

φ dφ + β = 0 or dr r

dφ βφ + =0 dξ αξ

at r = a or ξ = k, where φ is given by (7.93) with B = 0. Using the well-known derivative formula ξ J0 (ξ ) − J1 (ξ ) = J1 (ξ ) and applying the boundary condition to the above expression for φ, the bifurcation state is given by β 4 (1 − η) 2a kJ0 (k) =1− = , k= J1 (k) α 1 + 3T/E h

3σ , αE

(7.95)

in view of (7.91). Since the right-hand sides of these equations are functions of σ /E, the critical stress must be computed by trial and error, using a table of Bessel functions. When the bifurcation occurs in the elastic range, T = E, and

522

7

Plastic Buckling

k2 = 12(1 – v2 )a2 σ /h2 E, and the critical stress is then given by (1 – v2 ) σ /E ≈ 0.35h2 /a2 when v = 0.3. The bifurcation stress is therefore strongly dependent on the edge condition. A detailed investigation of the plastic buckling of circular plates has been carried out by Hamada (1985). The influence of the transverse shear on the critical stress has been examined by Wang et al. (2001). If a radially compressed circular plate has a concentric circular hole, which is assumed as stress free, a uniform radial compressive stress applied around the outer edge produces a nonuniform distribution of stress within the plate. The differential equation (7.92) is therefore modified, and the critical stress for bifurcation then depends on the ratio b/a, where b denotes the radius of the hole. The plastic buckling of annular plates under pure shear has been discussed by Ore and Durban (1989). An analysis for the plastic buckling of relatively thick annular plates under uniform compression has been presented by Aung et al. (2005).

7.5 Buckling of Cylindrical Shells 7.5.1 Formulation of the Rate Problem Consider a circular cylindrical shell of uniform thickness h and mean radius a, subjected to the combined action of an axial compressive stress σ and a uniform lateral pressure p. The cylinder is supported in such a way that it is free to expand or contract radially during the loading. For certain critical values of the applied loads, a point of bifurcation is reached, and the shell no longer retains its cylindrical form. Let (x, θ , r) be a right-handed system of cylindrical coordinates, in which the x-axis is taken along the axis of the shell, the origin of coordinates being taken at one end of the shell. At a generic point in the material of the shell, situated at a radially outward distance z from the middle surface, the components of the velocity vector may be written as υx = u + zωθ ,

υθ = υ − zωx ,

υx = w,

(7.96)

where (u, ν, w) are the velocities at the middle surface, and ωx , ωθ are the rates of rotation of the normal to the middle surface about the positive x- and θ -axes, respectively. Within the framework of thin-shell theory, the latter quantities are directly obtained from the fact that the through-thickness shear rates ε˙ rx and ε˙ rθ are identically zero. Denoting the remaining component of the spin vector by ωr , we have

ωx =

1 a

∂w −υ , ∂θ

ωθ = −

∂w , ∂x

ωr =

1 2

∂υ 1 ∂u − . ∂x a ∂θ

(7.97)

The nonzero components of the anti-symmetric spin tensor ω ij are related to the components of the spin vector as

7.5

Buckling of Cylindrical Shells

− ωxθ = ωθx = ωr ,

523

− ωrx = ωxr = ωθ ,

− ωθr = ωrθ = ωx .

The nonzero components of the strain rate, except the through-thickness one, that are associated with the velocity field (7.96), may be written as ε˙ xx = λ˙ x − zκ˙ x , ε˙ θθ = λ˙ θ − zκ˙ θ , ε˙ xθ = λ˙ xθ − zκ˙ xθ ,

(7.98)

where λ˙ x , λ˙ θ , and λ˙ xθ are the rates of extension and shear of the middle surface, while κ˙ x , κ˙ θ, and κ˙ xθ are the rates of change of curvature and twist of the middle surface. It is easily shown that λ˙ x =

∂u , ∂x

κ˙ x =

∂ 2w , ∂x2

⎫ 1 ∂υ ∂υ 1 ∂u ⎪ ⎪ ˙ +w , λxθ = + , ⎪ ⎬ ∂θ 2 ∂x a ∂θ (7.99) ⎪ 1 ∂ ∂w 1 ∂ ∂w ⎪ ⎪ κ˙ θ = 2 −υ , κ˙ xθ = − υ .⎭ a ∂θ ∂θ a ∂x ∂θ

λ˙ θ =

1 a

The strain rates given by (7.98) and (7.99) are consistent with the customary thin-shell approximation and are adequate for the investigation of bifurcation. The material is assumed to obey the von Mises yield criterion and the associated Prandtl–Reuss flow rule. Since the nonzero components of the current stress tensor σ ij are σ xx = –σ and σ θθ = –pa/h, the outward drawn unit normal nij to the yield surface has the nonzero components nxx = −

2σ − pa/ h 2pa/ h − σ , nθθ = − √ , √ 6σ¯ 6σ¯

nrr =

σ + pa/ h , √ 6σ¯

where σ¯ is the equivalent stress, which is equal to the current yield stress in simple compression, and is given by σ¯ 2 = σ 2 − σ (pa/h) + (pa/h)2 . Introducing the linear comparison solid, for which the Jaumann stress rate is denoted by τ˙ij , the constitutive equation may be written as ε˙ ij =

3 1 (1 + v) τ˙ij − vτ˙kk δij + E 2

1 1 − τ˙kl nkl nij , T E

(7.100)

where T is the tangent modulus in the current state of hardening, E is Young’s modulus, and v is Poisson’s ratio for the material. Since τ˙rr is identically zero, the axial and circumferential components of the rate of extension are given by vT 3 1 T pa 2 T 3σ pa τ˙xx − T ε˙ xx = 1 − + 1− 1− 1− τ˙θθ , 4 E σ¯ h E 2 E 2σ¯ 2 h vT 3 1 T 3σ pa T σ 2 T ε˙ θθ = + 1 − + 1− 1− 1 − τ ˙ τ˙θθ . xx E 2 E 4 E σ¯ 2σ¯ 2 h

524

7

Plastic Buckling

The nonzero shear strain rate ε˙ xθ is purely elastic and is equal to (1 + v) τ˙xθ /E by Hooke’s law. The preceding pair of equations can be solved for τ˙xx and τ˙θθ to give τ˙xx =

E E E˙εxθ , (α ε˙ xx + β ε˙ θθ ) , τ˙θθ = (β ε˙ xx + γ ε˙ θθ ) , τ˙xθ = 1+v 1+v 1+v (7.101)

where ⎫

⎪ α = ρ −1 (1 + v) 4 − 3 (1 − T / E) σ / σ¯ 2 , ⎪ ⎪ ⎪

⎪ ⎪ ⎪ −1 2 ⎪ β = ρ (1 + v) 2 − 2 (1 − 2ν) T / E − 3 (1 − T / E) σ pa/ σ¯ h , ⎪ ⎬

2 (7.102) ⎪ ⎪ , γ = ρ −1 (1 + v) 4 − 3 (1 − T / E) pa/ σ¯ 2 h ⎪ ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ρ = (5 − 4v) − (1 − 2v)2 T / E − 3 (1 − 2v) (1 − T / E) σ pa/ σ¯ 2 h ,⎭ The parameters α, β, γ , and ρ are easily calculated for any given state of stress and rate of hardening. Using (7.101) for the nonzero stress rates, we get τ˙ij ε˙ ij =

E 2 2 2 . + 2˙εxθ α ε˙ ij + 2β ε˙ xx ε˙ θθ + γ ε˙ θθ 1+v

(7.103)

The complete rate problem also involves the rate equations of equilibrium in which geometry changes are duly allowed for. A set of equilibrium equations in terms of the rate of change of the stress resultants have been developed by Batterman (1964). Such equation are, however, not required in the present analysis, which involves a variational principle based on the appropriate criterion for uniqueness.

7.5.2 Bifurcation Under Combined Loading A sufficient condition for uniqueness of the deformation in an elastic/plastic body under the combined action of incremental dead loading and uniform fluid pressure is given by inequality (1.82). Since the plastic modulus H would be large compared to the applied stresses at the onset of buckling, the quantity σij ε˙ jk ε˙ ik may be omitted in the uniqueness criterion, which therefore becomes

τ˙ij ε˙ ij + σij ωik ωjk − 2˙εik ωjk dV − p

lk ε˙ kj + ωkj υj dSf > 0,

(7.104)

for all continuous velocity fields vanishing at the constraint. The volume integral extends throughout the material of the body, while the surface integral extends over the boundary that is submitted to fluid pressure. Using (7.103), and remembering that the only nonzero stress components are σ xx = – σ and σ θθ = – pa/h, the condition for uniqueness is reduced to

7.5

Buckling of Cylindrical Shells

525

2 2 2 α ε˙ xx dx dθ dz + 2β ε˙ xx ε˙ θθ + γ ε˙ θθ + 2˙εxθ σh ωr2 + ωθ2 + 2ωr λ˙ xθ dx dθ − (1 + v) E pa ωx2 + ωθ2 − 2ωθ λ˙ xθ dx dθ − (1 + v) E p λ˙ x + λ˙ θ w + uωθ − νωx dx dθ > 0, + (1 + v) E with sufficient accuracy, since σ /E and p/E are small compared to unity. Substituting from (7.97), (7.98), and (7.99) and introducing the dimensionless parameters s = (1 + v)

σ , E

q = (1 + v)

pa , Eh

k=

h2 , 12a2

ξ=

x , a

and integrating through the thickness of the shell, we obtain 2 ∂u 1 ∂υ ∂υ ∂u 2 ∂u ∂υ α dξ dθ +w +γ +w + + + 2β ∂ξ ∂ξ ∂θ ∂θ 2 ∂ξ ∂θ 2 2 2 2 ∂ w ∂ 2 w ∂ 2 w ∂υ ∂ w ∂υ +k α + 2β 2 − − +γ ∂ξ 2 ∂ξ ∂θ 2 ∂θ ∂θ 2 ∂θ 2 2 2 2 ∂υ ∂υ ∂ w ∂w − dξ dθ − s dξ dθ +2 + ∂ξ ∂θ ∂ξ ∂ξ ∂ξ 2 2 ∂w ∂u ∂w ∂u + −w +w +u dξ dθ > 0, −q ∂θ ∂θ ∂ξ ∂ξ where use has been made of the fact that the velocities are single-valued functions of θ . The terms containing the quantities sλ˙ 2x˙ θ and qλ˙ 2x˙ θ have been neglected in the last two integrals to be consistent with the basic approximation. The occurrence of bifurcation is marked by the vanishing of the above functional, which must be minimized with respect to the admissible velocities u, υ, and w. The Euler–Lagrange differential equations associated with this variational problem are easily found to be α

∂ 2u 1 ∂ 2u 1 ∂ 2υ ∂w ∂ 2 u ∂w = 0, + + β + + β + q − ∂ξ 2 2 ∂θ 2 2 ∂ξ ∂θ ∂ξ ∂ξ ∂θ 2

2 2 ∂ υ ∂w ∂ u ∂ 2υ 1 1 ∂ 2υ + γ + − s β+ + 2 ∂ξ 2 ∂θ 2 ∂ξ 2 ∂θ ∂θ 2 ∂ξ 2 2 2 ∂ υ ∂ 3w ∂ υ ∂ 3w = 0, − + 2) +k 2 2 +γ − (β ∂ξ ∂θ 2 ∂θ 3 ∂ξ 2 ∂θ

526

β

7

∂u +γ ∂ξ

Plastic Buckling

∂u ∂υ ∂ 2υ ∂ 2w +w +s 2 +q +w+ 2 ∂θ ∂ξ ∂ξ ∂θ 3 4 ∂ υ ∂ υ ∂ 4w ∂ 4w + k − (β + 2) 2 − γ 4 + 2 (β + 1) 2 2 + γ 4 = 0. ∂ξ ∂θ ∂θ ∂ξ ∂θ ∂θ (7.105)

In the case of elastic buckling, we have α = γ = 1/(1 – ν) and β = v/(l – v), and (7.105) reduce to those given by Timoshenko and Gere (1961), except for the coefficients of certain small-order terms, the effects of which are insignificant in the final result. The above equations provide a systematic generalization of the eigenvalue problem when buckling occurs in the plastic range (Chakrabarty, 1973). The class of admissible velocity fields for the investigation of bifurcation, which is characterized by a nonuniform mode of deformation, may be considered as that in which the radial velocity vanishes at the ends of the shell. Denoting the length of the shell by l, the solution of (7.105) may therefore be sought in the form u = U cos λξ cos mθ , υ = V sin λξ sin mθ , w = W sin λξ cos mθ ,

(7.106)

where λ = rπ a/1, and m and r are integers, while U, V, and W are arbitrary constant velocities. The virtual velocity field (7.106) evidently satisfies the boundary conditions w = ∂ 2 w/ ∂ξ 2 = 0

at

ξ = 0 and ξ = 1,

which correspond to a shell with simply supported edges. For sufficiently long shells, the results based on (7.106) can be applied to other types of edge condition without appreciable error. The velocity field (7.106) implies that the generator of the shell is subdivided into r half-waves and the circumference into 2m half-waves at the onset of bifurcation. Substituting (7.106) into (7.105), these equations are found to be satisfied everywhere in the shell if

⎫ 1 1 2 ⎪ ⎪ −q m U− + β λmV − (β + q) λW = 0, αλ + ⎪ ⎪ 2 2 ⎪ ⎪ ⎪

⎪ ⎪ 1 1 2 ⎪ 2 2 2 2 ⎪ V − + β λmU + λ + γ m − λ s + k 2λ + γ m ⎪ ⎪ 2 2 ⎬ + γ m + km (2 + β) λ2 + γ m2 W = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ − (β + q) λU + γ m + km (2 + β) λ + γ m V ⎪ ⎪ ⎪

⎪ ⎪ 2 2 4 2 2 4 W = 0.⎭ + γ − λ s − m − 1 q + k αλ + 2 (1 + β) λ m + γ m (7.107)

2

This is a system of three linear homogeneous equations for the unknown velocities U, V, and W. For nontrivial solutions to these quantities, the determinant of

7.5

Buckling of Cylindrical Shells

527

their coefficients must vanish. It is interesting to note that the matrix of this determinant is symmetric. Expanding the determinant, and neglecting the small-order terms involving the squares and products of s, q, and k, the result may be expressed in the form A + Bk = Cs + Dq,

(7.108)

where δ = αγ − β 4 = (4/ ρ) (1 + v)2 (T / E) , A = δλ4 , B = αλ4 + 2 (δ − β) λ2 m2 + γ m4 αλ4 + 2 (1 + β) λ2 m2 + γ m4 − 2m2 (2 + β) λ2 + γ m2 (2δ − β) λ2 + γ m2 ,

C = λ2 αλ4 + 2 (δ − β) λ2 m2 + γ m2 + λ2 2δλ2 + γ m2 ,

D = m2 − 1 αλ4 + 2 (δ − β) λ2 m2 + γ m4 + λ2 2βλ2 − γ m2 .

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (7.109)

For a given material and shell geometry, and with selected values of r and m, (7.108) defines the relationship between the critical values of s and q for bifurcation. Using a constant value of r and successive values of m, a series of curved segments may be obtained in a graphical plot of s against q. The value of m appropriating over a certain range is that which provides the smallest ordinate for a given abscissa. A set of curves may be constructed in this way for various values of λ = rπ a/l, similar to those presented by Flügge (1932) for the buckling of an elastic shell.

7.5.3 Buckling Under Axial Compression Consider the special case where the lateral pressure is absent and the axial compressive stress σ is increased to a critical value to cause buckling in the plastic range. The critical stress is then obtained by setting q = 0 in (7.108) and using the values of λ and m which correspond to a minimum value of s. When the cylindrical shell is relatively short, we may expect it to buckle into short longitudinal waves, so that λ2 is sufficiently large. Retaining only the first terms in the expressions for B and C in (7.109), the critical stress may be written in the simplified form k αλ4 + 2 (1 + β) λ2 m2 + γ m4 δλ2 + , (7.110) S= αλ4 + 2 (δ − β) λ2 m2 + γ m4 λ2 where δ = 4(1 + v)2 (T/ρE), the coefficients α, β, γ , and ρ being given by (7.102) with σ¯ = σ and ρ = 0. The value of λ2 which makes the right-hand side of (7.110) a minimum is found to be given by

528

7

αλ4 −

√

Plastic Buckling

δ / k − 2 (δ − β) m2 λ2 + γ m4 = 0,

and the corresponding expression for the critical stress for buckling then becomes s=2

kg + (1 − δ + 2β) m2 .

Since δ – 2β ≤ 1 for 0 < T/E < 1, the smallest value of s for plastic buckling corresponds to m = 0, which represents a symmetrical mode of buckling. The preceding relations therefore reduce to αλ2 =

√

√ s = 2 δk.

δ / k,

Substituting for k, α, and δ, the critical values of σ /E and λ2 for plastic buckling are finally obtained as 2h σ = E a

' T , 3ρE

4a 3T −1 3ρT λ = 1+ h E E 2

(7.111)

This is the true tangent modulus formula for the plastic buckling of relatively short cylindrical shells under axial compression. The same formula has been obtained by Batterman (1965) using a different method. Setting T/E = 1 and ρ = 4(1 – v2 ) reduces (7.111) to the well-known formulas for elastic buckling in which case m need not be zero. The influence of a singular yield criterion in lowering the critical stress has been examined by Ariaratnam and Dubey (1969). Plastic buckling formulas based on the total strain theory together with some experimental results have been given by Bijlaard (1949) and also by Jones (2009). The effect of an initial imperfection on the critical stress based on the total strain theory has been investigated by Lee (1962) and by Bardi et al. (2006). Since the eigenmode is symmetrical, the second equation of (7.107) is identically satisfied, and either of the two remaining equations furnishes W/U = αλ/β. The eigenfield may therefore be written in terms of an arbitrary constant velocity u0 as u = u0 β cos λξ ,

υ = 0,

w = u0 αλ sin λξ .

Since a uniform axial compression is always a possible mode, the nonzero components of the actual velocity at bifurcation may be expressed as u = −u0

λx x + cβ 1 − cos , l l

w = u0

λx ηa + cαλ sin , l l

(7.112)

where c is a constant and η is the contraction ratio, the axial velocity being assumed to vanish at the end x = 0. The condition of no instantaneous unloading of the plastically compressed cylinder may be written as ε˙ xx < 0, and it follows from (7.98) and (7.112) that

7.5

Buckling of Cylindrical Shells

529

cλl z δ 1+ β− sin λξ > 0 a a k √ in view of the result αλ2 = δ/k. This inequality will be satisfied throughout the shell if ' a 3T a − < β + 2 (1 + v) λc < , l ρE l where β and ρ are given by (7.102). The buckling may therefore occur in a range of possible modes, when the axial stress σ attains the value given by (7.111), with the load increasing as the shell continues to deform in the postbuckling range. In the case of long cylindrical shells, the generators are expected to buckle into relatively long waves in the longitudinal direction, so that λ is sufficiently small. We may then neglect in the expressions for B and C all powers of λ higher than the second to obtained the result

C = γ λ2 m2 m2 + 1 , A = δλ4 ,

2 γ m2 + 2 (1 + δ) λ2 . B = γ m2 m2 − 1 Substituting in (7.108), where q is set to zero, the critical stress for buckling may be expressed as 2 k m2 − 1 δλ2 γ m2 + s= + 2 (1 + δ) . m2 + 1 λ2 γ m2 m2 + 1

(7.113)

Minimizing this expression with respect to λ, the associated values of λ2 and s are found to be √

k/ δ, λ2 = γ m2 m2 − 1

m2 − 1 √ 2 s=2 kδ + k + δ) − 1 . m (1 m2 + 1 Evidently, the smallest value of the critical stress corresponds to m = 2, giving the final results ' ' 4 h 3E 6h h 1+δ T σ 2 = + , (7.114) , λ = E 5a 3ρE 4a 1 + ν a ρT where δ is given by the first equation of (7.109), and ρ by the last equation of (7.102) with the last term set to zero. For an elastic material, T = E,δ = (1 + v)/(1–ν), and

530

7

Plastic Buckling

ρ = 4(1 – v2 ), reducing the above expressions to those given by Timoshenko and Gere (1961). A limit of applicability of (7.114) is marked by the buckling of the shell as a column, the critical stress then being given by the formula σ = π 2 a2 T/2l The plastic buckling of axially compressed cylindrical shells has also been considered by Gellin (1979), Shrivastava (1979), and Reddy (1980). The preceding discussion is based on the assumption that the shell is thick enough for the critical stress σ to exceed the yield stress of the material. The results for the whole range of values of h/a may, however, be presented on the basis of the plastic buckling formula if we adopt the relations σ ε= E

3 1+ 7

σ σ0

n−1 ,

3n E =1+ T 7

σ σ0

n−1 ,

(7.115)

for the uniaxial stress–strain curve, where σ 0 and n are empirical constants. The assumed curve has an initial slope E, but the tangent modulus T steadily decreases as the stress increases from zero. Inserting (7.115) into the critical stress formula (7.111) or (7.114), a relationship between σ /E and h/a is obtained for any given values of σ 0 /E and n. The relationship between λ and h/a then follows from the corresponding formula for λ2 . The results are displayed in Fig. 7.13 for several values of n, assuming v = 0.3 and σ 0 /E = 0.002. Due to geometrical imperfections and other uncertainties, the experimental buckling loads are found to be appreciably lower than the theoretical ones. A useful review of the subject has been reported by Babcock (1983). The buckling problem for a square-section tube, based on the total strain theory, has been investigated by Li and Reid (1992).

7.5.4 Influence of Frictional Restraints When a cylindrical shell is axially compressed between a pair of rigid platens, lateral displacement of the elements in contact with the platens is prevented by friction, and the shell is therefore subjected to simultaneous bending and compression right from the beginning of the loading. The bending moment at the crests of the longitudinal waves nearest to the platens rapidly increases with increasing load until the yield point state is reached. The first convolution that appears at one end of the cylinder, where the frictional restraint is more predominant, then collapses under decreasing load. When the first convolution nearly flattens out, a second convolution begins to form on top of the other one, and the load starts to increase again with further compression. If the shell is relatively thick, the result is a concertina-type of deformed shape, for which an average buckling load has been given by Alexander (1960b). For thinner shells, there is generally a polygonal-type of final configuration treated by Pugsley and Macualay (1960), and Pugsley (1979). An experimental investigation on the energy absorption of tubular structures during buckling has been made by Balen and Abdul-Latif (2007).

7.5

Buckling of Cylindrical Shells

531

Fig. 7.13 Critical stress for plastic bucking of an axially compressed cylindrical shell as a function of the wall thickness

Consider the concertina-type of collapse in which each convolution is approximated by two identical conical surfaces as shown in Fig. 7.14. In this simplified model, the convolutions are taken to be purely external to the cylinder, though in actual practice they are formed partly outward and partly inward. The formation of each convolution is associated with three circular hinges which allow rotation of the conical bands, each having a constant slant height b. For simplicity, the small change in thickness that occurs in the formation of convolution will be disregarded. Neglecting elastic strains and work-hardening, and denoting the current semivertical angle of the conical bands by ψ the increment of plastic energy dissipation in the hinge circles during a small change in angle dψ is found as b sin ψ dψ, dW1 = 4π bM0 (2a + b sin ψ) dψ = 2π ah2 Y 1 + 2a the influence of the meridional force on the fully plastic moment being disregarded. Since the mean circumferential strain in the material between the hinge circles during the incremental change in angle dψ is dεθ = b cosψdψ(2a + bsinψ), the increment of plastic work done during the circumferential stretching of the material is

532

7

Plastic Buckling

Fig. 7.14 Concertina-type of plastic collapse of a circular cylindrical shell under axial compression with frictional restraint

dW2 = 2π bN0 (2a + b sin ψ) dεθ = 2π hb2 Y cos ψdψ. This expression involves the approximate yield condition Nθ ≈ N0 and the fact that the meridional strain increment is assumed to be zero. The total plastic work done in collapsing a typical convolution, as ψ increases from 0 to π/2, is W=

(dW1 + dW2 ) = π h2 (π a + b) Y + 2π b2 hY.

(7.116)

The applied compressive stress decreases from an initial value σ 0 during the collapse of a convolution, the stress at a generic stage being denoted by σ . Available theoretical and experimental evidences (Chakrabarty and Wasti, 1971) tend to suggest that the variation of the stress may be written as

σ = σ0 cos ψ + 2

h 2 sin ψ . a

The total work done by the external load must be equal to W. Since the amount of axial compression corresponding to a change dψ in the semi-vertical angle is equal to 2b sinψ, dψ, we have W = 4π ahbσ0 0

π/2

cos ψ + 2

h 2 sin ψ sin2 ψdψ. a

7.5

Buckling of Cylindrical Shells

533

Integrating, and inserting the expression (7.116) for W, the critical stress is found to be given by σ0 Y

h 2b 3 h πa +1+ 1+2 = . a 4a b h

The quantity b√is still an unknown but can be determined by minimizing σ 0 the result being b = π ah/2 . The critical stress formula therefore becomes 3h σ0 =3 Y 4a

πa 2b +1+ b h

5

h . 1+2 a

(7.117)

The critical stress for the initiation of the first convolution would be somewhat smaller than (7.117) since the hinge circle at the base of the deformation zone is absent during its formation. The analysis has practical importance in the design of buffers bringing moving bodies to a stop without appreciable damage. Further results on the axial crushing of cylindrical shells have been given by Mamalis and Johnson (1983). The influence of the meridional stress on the magnitude of the buckling load has been examined by Wierzbicki and Abramowicz (1983).

7.5.5 Buckling Under External Fluid Pressure We begin with the situation where a circular cylindrical shell is submitted to a uniform external pressure acting on the lateral surface only. The critical hoop stress at the incipient buckling is then directly given by (7.108), where s must be set to zero. The parameters α, β, γ , and δ appearing in (7.109) are obtained from (7.102) on setting σ = 0 and pa/h = σ¯ . If the length of the cylinder is greater than about twice its diameter, the ratio λ2 /m2 will be a small fraction, and we may omit all terms containing λ2 and λ4 in the expressions for B and D in (7.109). Then the result for the critical pressure is easily shown to be q≈

δλ4 + kγ m2 − 1 . γ m4 m2 − 1

(7.118)

Since q increases with λ, the least value of the critical hoop stress corresponds to r = 1, giving λ = π a/l. Substituting for γ , δ, λ, and k into the above expression, we finally obtain the buckling formula 2 h / 12a2 (1 + 3T / E) m2 − 1 (4T / E) (π a/ l)4 pa + = . Eh (1 + 3T / E) m4 m2 − 1 (5 − 4ν) − (1 − 2ν)2 T / E

(7.119)

In the case of elastic buckling (T = E), this formula reduces to that originally given by Southwell (1913). Equation (7.119), which is due to Chakrabarty (1973), represents a rigorous extension of the result when buckling occurs in the plastic

534

7

Plastic Buckling

range. For a very long tube, the first term on the right-hand side of (7.119) may be disregarded, and the least value of the critical stress then corresponds to m = 2. In a wide range of values of l/a and h/a would generally require m = 3 for a lower value of the critical stress, and the condition for this to happen is 27 h2 π 4 a4 (E/ T) (1 + 3T / E)2 . > l4 5a2 (5 − 4ν) − (1 − 2ν)2 T / E For shorter tubes, the critical stress corresponding to m = 4 would be lower than that given by m = 3. For exceptionally short tubes, however, the approximation leading to (7.118) is no longer valid. For the numerical evaluation of the critical stress, T/E may be eliminated from (7.119) by using (7.115), to obtain the relationship between the critical hoop stress pa/h and the shell parameters. The resulting equation can be most conveniently solved by selecting a value of pa/Eh for a given value of l/a, and calculating the corresponding value of h2 /a2 required for bifurcation. The computation may be carried out with a suitable value of m, changing it if necessary to obtain the least value of the critical stress. Figure 7.15 shows the results of the computation based on v = 0.3, n = 3.0, and σ 0 /E = 0.001. The curves for plastic buckling are represented by the solid lines, while those for elastic buckling are indicated by the broken lines. Only two of the equations in (7.107) are independent when U, V, and W correspond to the eigenfield. Setting s = 0 in the last two equations of (7.107), and using (7.118), it is easily shown that U = λw0 ,

V = mw0 ,

W = −m2 w0 ,

to a close approximation, where w0 is an arbitrary constant velocity. Introducing a constant parameter c, the components ν and w of the actual velocity at bifurcation may be written as ⎫

πx ⎪ sin mθ , υ = mw0 c sin ⎬ l

πx ⎭ w = −m2 w0 1 + c sin cos mθ ,⎪ l

(7.120)

the radial velocity at each end of the shell being taken as equal to –m2 wo. The substitution from (7.120) into the second equation of (7.98) furnishes the hoop strain rate ε˙ θθ = −

πx m2 w0 cz 2 1+ m − 1 sin cos mθ , a a l

which must be negative for no incipient unloading at the point of bifurcation. Since the value of the second term in the curly brackets of the above expression varies between –(m2 – 1)(ch/2a), and (m2 – 1)(ch/2a), the loading condition ε˙ θθ < 0 gives

7.5

Buckling of Cylindrical Shells

535

Fig. 7.15 Critical stress for plastic buckling of a circular cylindrical shell under uniform external pressure on the lateral surface

−

2a 2 2a < m −1 c< . h h

The nonlinear elastic/plastic solid may therefore buckle in a range of possible ways, and the external pressure must continue to increase from its critical value (7.118) as the deformation continues in the post-buckling range. Consider now the situation where a circular cylindrical shell is closed at both ends by rigid plates, and is submitted to an all-round external pressure of intensity √ ρ. The uniform deformation that precedes buckling involves σ = pa/2 h = σ¯ / 3, and (7.102) become α = (1 + ν) (3 + T / E) ρ −1 , γ = (1 + ν) (4T / ρE) = δ / (1 + ν),

⎫ β = ν (1 + ν) (4T / ρE) ,⎬

ρ = 3 + 1 − 4ν 2 T / E.⎭

(7.121)

536

7

Plastic Buckling

The critical pressure for buckling is obtained by setting s = q/2 in (7.108), the coefficient of q in the resulting expression being D + C/2. Neglecting small-order terms, and using the fact that δ = (1 + v)γ and δ – β = γ in view of (7.121), we obtain the simplified formula k m2 − 1 γ m2 + 2 (1 + δ) λ2 λ4 pa + = 2 2 Eh m m − 1 m2 + 2.5λ2 (1 + ν) m2 + 2.5λ2

(7.122)

to a close approximation, where λ = π a/l. For given l/a and h/a ratios, the value m that makes the right-hand side of (7.122) a minimum should be used for calculating the critical pressure. In Fig. 7.16, the results for a closed-ended cylinder under an all-round hydrostatic pressure are compared with those for an open-ended cylinder under a radial pressure alone, using (7.115) with n = 3 and σ 0 /E = 0.002. For relatively thick tubes, the critical values of the equivalent stress in the two cases are found to be approximately the same. The plastic buckling of an initially imperfect cylindrical shell under internal pressure and axial compression, based on the total strain theory, has been considered by Paquette and Kyriakides (2006).

Fig. 7.16 Comparison of critical external pressure for the plastic buckling of cylindrical shell with open and closed end conditions

7.6

Torsional and Flexural Buckling of Tubes

537

7.6 Torsional and Flexural Buckling of Tubes 7.6.1 Bifurcation Under Pure Torsion In the case of a thin cylindrical shell subjected to equal and opposite twisting moments at its ends, the deformation mode ceases to be uniform when the applied shear stress τ attains a certain critical value. The thickness h and length l in relation to the mean radius a are assumed to be such that the bifurcation occurs beyond the elastic limit. Since the only nonzero stress at the onset of buckling is σ xθ = τ , the rates of extension in the longitudinal and circumferential directions are purely elastic, and the corresponding normal components of the stress rate are τ˙xx =

E (˙εxx + ν ε˙ θθ ) , 1 − ν2

τ˙θθ =

E (˙εθθ + ν ε˙ θθ ) . 1 − ν2

The only nonzero rate of shear is ε˙ xθ whose plastic component is (3/2H) τ˙xθ and consequently, ε˙ xθ =

1 3 + 2G 2H

τ˙xθ =

3 1 − 2ν − 2T 2E

τ˙xθ ,

where T is the tangent modulus at the current state of hardening of the material. The shear stress rate may therefore be written as τ˙xθ

2Eα = ε˙ xθ , 1 − ν2

1 − ν2 T/ E α= . 3 − (1 − 2ν) T / E

(7.123)

During the uniform twisting, α progressively decreases from the elastic value (1 – v) / 2 with increasing plastic strain. It follows from the above expressions that τ˙ij ε˙ ij =

E 2 2 2 ε˙ xx + 2ν ε˙ xx ε˙ θθ + ε˙ θθ . + 4α ε˙ xθ 2 1−ν

The condition for uniqueness of the deformation mode is given by inequality (7.104) with ρ = 0. Since σ xθ = τ , while all other components are identically zero, the inequality becomes 2 2 2 ε˙ xx + 2ν ε˙ xx ε˙ θθ + ε˙ θθ + 4α ε˙ xθ τ

+2 1 − ν 2 (ωxr ωθr + ε˙ θθ ωθx ) dx dθ dz > 0 E

(7.124)

to a close approximation, the possibility of sideways buckling of the tube being excluded. In terms of the velocities of the middle surface, the components of the strain rate are given by (7.98) and (7.99), while those of the rate of spin are given by (7.97), in view of the relations ωxr = ωθ , ωθr = −ωx , and ωθx = ωr . Substituting into (7.124), and integrating through the shell thickness, we obtain

538

7

∂u ∂ξ

2

Plastic Buckling

2 ∂υ ∂u ∂υ ∂υ ∂u 2 +w + +w +α + ∂ξ ∂θ ∂θ ∂ξ ∂θ $ 2 2 2 ∂ 2w ∂ 2 w ∂ 2 w ∂υ ∂ w ∂υ +k + 2ν − − + ∂ξ 2 ∂ξ 2 ∂θ 2 ∂θ ∂θ 2 ∂θ % 2 2 ∂w ∂w ∂ w ∂υ + 2φ + 4α − −υ ∂ξ ∂θ ∂ξ ∂ξ ∂θ ∂υ ∂υ + +w dξ dθ > 0, ∂ξ ∂θ

+ 2ν

on neglecting certain small-order terms and on introducing the dimensionless quantities τ

, φ = 1 − ν2 E

k=

h2 , 12a2

ξ=

x . a

The bifurcation would occur when the functional on the left-hand side of the above inequality vanishes for some distribution of the velocities u, υ, and w that makes the functional a minimum. The Euler–Lagrange differential equations associated with this variational problem are easily shown to be ∂ 2u ∂ 2u ∂ 2υ ∂w + α + + ν) +υ = 0, (α ∂ξ 2 ∂θ 2 ∂ξ ∂θ ∂ξ 2 ∂ 2υ ∂ 2υ ∂ υ ∂w ∂w ∂ 2υ + α + + ν) + + 2φ + (α ∂θ 2 ∂ξ 2 ∂ξ ∂θ ∂θ ∂ξ ∂θ ∂ξ (7.125) 2 2 3 3 ∂ w ∂ υ ∂ w ∂ υ − = 0, + 4α − + ν) +k (4α ∂θ 2 ∂ξ 2 ∂ξ 2 ∂θ ∂θ 3 ∂υ ∂υ ∂u ∂ 2υ +υ + w + 2φ − ∂θ ∂ξ ∂ξ ∂ξ ∂θ 4 4 ∂ 3w ∂ w ∂ 4w ∂ 3w ∂ w + 2υ 2 2 + 2 − (4α + ν) 2 − 3 = 0. +k ∂ξ 2 ∂ξ ∂θ ∂θ ∂ξ ∂θ ∂θ In the case of an elastic material, (7.125) reduce to those given by Timoshenko and Gere (1961), except for certain small-order terms which do not affect the final results significantly. The problem of the buckling of a cylindrical shell under pure torsion is reduced to the integration of (7.125), using the boundary conditions. In each of the three equations of (7.125), there are both odd- and even-order derivatives of a given velocity component with respect to the same independent variable. These equations cannot therefore be satisfied by assuming solutions in the form of products of sines and cosines of angles involving ξ and θ . When the cylinder is sufficiently large, so that the critical stress is practically independent of the edge constraints, we may assume the simple velocity field

7.6

Torsional and Flexural Buckling of Tubes

u = A cos (λξ − mθ) , υ = B cos (λξ − mθ ) , W = C sin (λξ − mθ ) ,

539

(7.126)

where λ = rπ a/l, with r denoting the number of longitudinal waves. The mode of buckling involves m circumferential waves which run helically along the cylinder. The substitution of (7.126) into (7.125) results in

− λ2 + αm2 A + (α + ν) λmB + νλC = 0, (α + ν) λmA − (1 + k) m2 + (1 + 4 k) αλ2 − 2λmφ B − m + km3 + k (4α + ν) λ2 m − 2λφ C = 0, νλA − m + km3 + k (4α + ν) λ2 m − 2λφ B

− 1 + k λ4 + 2νλ2 m2 + m4 − 2λmφ C = 0. This system of linear homogeneous equations can have nontrivial solutions for A, B, and C only if the determinant of their coefficients vanishes. Equating the determinant to zero, and neglecting terms containing k2 , kφ, φ 2 , etc., the result may be expressed as Tφ = R + kQ,

(7.127)

where R, Q, and T are dimensionless parameters given by the expressions

R = 1 − ν 2 αλ4 ,

2

Q = m2 m2 − 1 αm2 + 1 + 4α 2 − ν 2 λ2 + 2λ2 m4 1 − ν 2 (α + ν) λ2 − 2α 2 m2 ,

T = 2λm m2 − 1 αm2 + 1 − 2αν − ν 2 λ2 + αλ4 . The terms involving higher powers of λ have been omitted in the above expression for Q. Substituting for R, Q, and T into (7.127), the expression for the critical shear stress may be written with sufficient accuracy as km m2 − 1 λ3 τ + , = E 2m m2 − 1 m2 + βλ2 2λ 1 − ν 2

(7.128)

where β = (1 – v2 )/α – 2v. The value of λ that minimizes the critical stress may be approximately taken as that for elastic buckling, in which case βλ2 is negligible in comparison with m2 . The optimum value of λ is then given by

540

7

m2 m2 − 1 h λ = √ . 6 1 − ν2 a 2

Plastic Buckling

(7.129)

The critical stress formula for plastic buckling, obtained by omitting the quantity βλ2 in the first term of (7.128) and substituting from (7.129), may be written as τ = E

−3/4 h 3/2 3 2 1−ν 2 3a

where we have set m = 2 to minimize τ . The critical stress for plastic buckling can be computed from (7.128) and (7.129) by setting m = 2 and using the relation ⎧ √ n−1 ⎫ ⎬ ⎨ 9n 3τ , α = 1 − ν 2 / 2 (1 + ν) + ⎩ ⎭ 7 σ0

(7.130)

√ which is obtained from (7.123) and (7.115) with the substitution σ = 3τ . The results for plastic buckling are displayed as solid curves in Fig. 7.17, using v = 0.3,

Fig. 7.17 Critical shear stress for plastic buckling of a long cylindrical tube under pure torsion

7.6

Torsional and Flexural Buckling of Tubes

541

σ 0 /E = 0.002, and two different values of n. The broken curve represents the elastic buckling formula which is valid only for very small thicknesses. In the case of relatively short cylindrical shells, the conditions of support at the ends of the cylinder must be taken into account for the estimation of the critical stress. For the elastic range of buckling, an approximate analysis of the problem has been discussed by Donnell (1933), who introduced several simplifications into the basic differential equations (7.125). By applying appropriate boundary conditions on v and w, while permitting some axial motion of the ends of the cylinder, Donnell obtained buckling formulas that were found to be in good agreement with experiment. An empirical extension of the elastic buckling formula to cover the plastic range of loading has been suggested by Gerard (1962). Assuming the cylindrical shell to be clamped at both ends, the critical stress given by Gerard may be written as

−5/8 τ = 0.82 1 − η2 S

5/4 h a 1/2 , a l

(7.131)

where S is the secant modulus at the onset of buckling and η is the contraction ratio given by the relation 1 – 2η = (1 – 2v)S/E. For a given h/a ratio, (7.131) should be used for the range of values of l/a, which corresponds to a lower value of the critical stress than that given by (7.128). Adopting the Ramberg–Osgood equation for the stress–strain curve, the values S and η are easily computed by using (7.115). A modified buckling formula that includes the minor effect of preventing the axial strain in an approximate manner has been proposed by Rees (1982), who also produced some experimental evidence in support of the theory.

7.6.2 Buckling Under Pure Bending When an initially straight cylindrical tube is subjected to a gradually increasing bending moment, the circular cross section becomes increasingly oval until the applied moment starts to decrease after attaining a maximum, which constitutes a state of collapse due to buckling of the tube. This phenomenon is particularly significant for sufficiently long tubes, and the maximum compressive stress at the point of buckling is found to be considerably lower than the critical stress in an axially compressed cylindrical shell. The problem has been investigated by Brazier (1926) for the elastic range of buckling, and by Ades (1957), Gellin (1980), and Zhang and Yu (1987) for the plastic range of buckling. The effect of an internal pressure on the plastic collapse of a bent tube has been examined by Corona and Kyriakides (1988). Figure 7.18 shows the deformed configuration of the cross section of the middle surface, in which the position of a typical particle is specified by the angular coordinate θ measured from the crown of the circle. The circumferential and radial components of the displacement of any particle will be denoted by v and w, respectively, and the curvature of the neutral surface of the bent tube will be denoted by

542

7

Plastic Buckling

Fig. 7.18 Buckling of a cylindrical tube under pure bending. (a) Overall shape of the bent tube, (b) cross section before and after buckling

v, which is measured positive as indicated in the figure. By simple geometry, the distance of any point on the deformed middle surface from the neutral plane is y = (a + w) cos θ − υ sin θ ,

(7.132)

where α is the mean radius of the cylindrical tube. If the deformation of the middle surface is assumed to be inextensional in the circumferential direction, and the deformation and rotation are everywhere small, the longitudinal and circumferential components of the strain are ⎫ εx = −yκ, εθ = −zκθ , ⎪ ⎬ dw 1 d υ− ,⎪ κθ = − 2 ⎭ dθ a dθ

(7.133)

where κ θ is the circumferential curvature of the middle surface, and z is the radially outward distance from the middle surface. The velocity components υ and w are related to one another through the equation dυ/dθ + w = 0, which is the condition of zero circumferential strain at z = 0. Since the variation of the stress ratio during the prebuckling deformation is expected to be sufficiently small, the total strain theory of Hencky may be used without introducing significant error. Denoting the axial and circumferential stresses by σ x and σ θ , respectively, the stress–strain relations may be written as Sεx = σx − ησθ ,

Sεθ = σθ − ησx ,

where S is the secant modulus of the effective stress–strain curve corresponding to the current effective stress σ¯ , and

7.6

Torsional and Flexural Buckling of Tubes

η=

S 1 1 − (1 − 2ν) , 2 E

543

E =1+ S

m−1 3 σ¯ , 7 σ0

(7.134)

when the stress–strain curve is represented by the Ramberg–Osgood equation (7.115). The stress–strain relations are readily inverted to express the stresses as

σx =

S (εx + ηεθ ) , 1 − η2

σθ =

S (εθ + ηεx ) , 1 − η2

(7.135)

The significant components of the resultant force and moment are the axial force Nx and the circumferential bending moment Mθ acting per unit perimeter. They are defined as Nx =

h/2 −h/2

σx dz, Mθ = −

h/2 −h/2

σθ z dz,

(7.136)

where h is the uniform wall thickness of the bent tube. The variation of the internal virtual work per unit length of the shell may be written to a close approximation in the form δU ≈ a

2π

0

(Nx δεx + Mθ δκθ ) dθ ,

(7.137)

where δε x and δκ θ are the variations of εx and κ θ , respectively. The contributions to the work from the remaining stress resultants Nθ and Mx are neglected. For a prescribed curvature κ of the neutral surface, the displacements v and w should be such that δU = 0, which is a variational statement of equilibrium of the bent tube. While evaluating Nx , and Mθ , it would be a good approximation to assume a constant value of S equal to that on the middle surface. Then, in view of (7.135) and (7.136), and the fact that εθ = –zkθ, we have Nx =

Sh 1 − η2

εx ,

Mθ =

Sh 1 − η2

h2 κθ . 12

Inserting in (7.137), and setting δU = 0, the variational relation is reduced to the compact form

2π 0

S h2 2 κ dθ = 0. y δy + δκ κ θ θ 1 − η2 12

(7.138)

To evaluate the integral, the secant modulus S must be determined as a function of θ . Since σ x = Sεx / (1–η2 ) and σ θ = ησ x at z = 0, we have S 1 − η + η2 2 2 |εx | σ¯ = σx − σx σθ + σθ = 1 − η2

544

7

Plastic Buckling

at z = 0. The substitution for σ¯ and η in terms of S/E into the above relation, using (7.134), leads to the equation

μ2 S 3E +μ− 4S E

3 μ2 S2 + 2 4 E

1/(m−1) −1/2 Eaκ 7 E = −1 3 S σ0

y , (7.139) a

where μ = 12 − v, and y/a is a known function for any given displacement field. The applied bending moment M increases with the curvature κ in the prebuckling stages, the magnitude of the bending couple being given by

2π

M = −a

2π

Nx y dθ = a3 hk

0

0

S y dθ , 1 − η2 a

(7.140)

where S/(1—η2 ) corresponds to z = 0 as before. The critical moment Mc is the maximum value of M for varying κ during the bending. The components of the displacement of the middle surface, satisfying the inextensibility condition dυ/dθ = –w and the fourfold symmetry condition, may be expressed in the general form

w=−

N 1

wm cos 2mθ ,

m=1

υ=

N 1

(wm /2m) sin 2mθ .

(7.141)

m=1

The analysis for the buckling problem will be given here for the special case of N = 2, which is a good approximation for practical purposes. We therefore write w = − (w1 cos 2θ + w2 cos 4θ ) ,

υ=

1 1 w1 sin 2θ + w2 sin 4θ , 2 2

where w1 and w2 are constants. Considering only the first quadrant, υ is seen to vanish at θ = 0 and θ = π /2, the value of w at these two points being –(w1 + w2 ) and (w1 —w2 ), respectively. The substitution for υ and w into (7.132) and (7.133) gives ⎫ y = cos θ a − [w1 − (2 − 3 cos 2θ ) w2 ] cos2 θ ,⎪ ⎬ κθ =

3 (w1 cos 2θ + 5w2 cos 4θ ) . a2

⎪ ⎭

(7.142)

The constant displacements w1 and w2 can be determined by inserting (7.142) into the variational equation (7.138). It is convenient at this stage to set λ = (S/E) / (1—η2 ) and denote the various integrals as

7.6

Torsional and Flexural Buckling of Tubes

π/2

A1 =

⎫ ⎪ ⎪ λ (2 − 3 cos 2θ ) cos6 θ dθ , ⎪ ⎪ ⎪ 0 ⎪ ⎪ π/2 ⎪ ⎪ ⎪ 6 ⎪ B2 = λ (2 − 3 cos 2θ ) cos θ dθ , ⎪ ⎬

λ cos4 θ dθ ,

π/2

B1 = 0

λ cos6 θ dθ ,

0

(7.143) ⎪ 3 15 π/2 ⎪ ⎪ λ cos2 2θ dθ , C2 = λ cos 2θ cos 4θ dθ , ⎪ ⎪ ⎪ 4 0 4 0 ⎪ ⎪ ⎪ π/2 π/2 ⎪ ⎪ 75 2 2 6 ⎭ C3 = λ cos 4θ dθ , B3 = λ (2 − 3 cos 2θ ) cos θ dθ ,⎪ 4 0 0

C1 =

π/2

A2 =

0

545

π/2

where λ is a known function of θ in view of (7.139) and (7.142). Using (7.138) and (7.142), and the fact that δw1 and δw2 are arbitrary variations of w1 and w2 , respectively, we thus obtain the pair of equations

κ 2 a4 w 2 w1 + C2 − B2 2 = A1 , a h a κ 2 a4 w1 κ 2 a4 w 2 C 2 − B2 2 + C3 + B3 2 = −A2 , h a h a C 1 + B1

κ 2 a4 h2

for the two dimensionless constants w1 /a and w2 /a. Setting ρ = κa2 /h, the solution can be written in the form ⎫ A1 C3 + B3 ρ 2 + A2 C2 − B2 ρ 2 w1 ⎪ = ⎪ 2 , ⎪ ⎪ ⎬ 2 2 2 a C1 + B1 ρ C3 + B3 ρ − C2 − B2 ρ ⎪ A1 C2 − B2 ρ 2 + A2 C1 + B1 ρ 2 w2 ⎪ ⎪ = − 2 .⎪ ⎭ 2 2 2 a C1 + B1 ρ C3 + B3 ρ − C2 − B2 ρ

(7.144)

When the ratios w1 /a and w2 /a have been computed from (7.143) and (7.144) for a selected value of ρ, the corresponding bending moment can be determined from (7.140). In view of the relations (7.142) and (7.143), the bending moment can be expressed in the dimensionless form 4hEρ M = M∗ π aσ ∗

0

π/2

λ cos2 θ dθ − 2A1

w 1

a

+ 2A2

w

w w 1 2 + B3 −2B2 a2

2

+ B1

a

w 2 2

a

w 2 1

a

(7.145)

,

where M∗ = π a2 hσ ∗ , with σ ∗ denoting the critical stress E (h/a) / 3 1 − v2 for the elastic buckling of the shell under uniform axial compression (Section 7.5). The critical bending moment Mc under pure bending is the maximum value of M predicted by (7.145) for varying values of ρ. Over the elastic range, S = E and

546

7

Plastic Buckling

η = v, giving the critical moment Mc ≈ 9.545M∗ as the maximum bending moment that corresponds to p ≈ 0.495 when v = 0.3. The solution is marginally improved if the nonlinearity of the strain– displacement relations is taken into consideration. This has been demonstrated by Gellin (1980), who computed the bending moment and curvature of the tube at the onset of buckling for different values of n, using a series of values of the parameter σ0 = σ∗

aσ0 3 1 − ν2 hE

and assuming ν = 0.3. His results are displayed in Fig. 7.19, which indicates that the critical curvature is relatively insensitive to the variation of σ 0 /σ ∗ for usual values of n. The theory has been found to be in reasonable agreement with available experimental data.

Fig. 7.19 Dimensionless bending moment and curvature at the onset of plastic buckling as functions of the ratio σ 0 /σ ∗ (after S. Gellin, 1980)

When the tube is not too long, the prebuckling ovalization of the cross section may be neglected, as has been shown by Seide and Weingarten (1961), Akserland (1965), and Reddy (1979) in their bifurcation analyses for buckling of the shell. The critical value of the greatest compressive stress in this case is found to be only slightly higher than the critical stress σ ∗ in pure axial compression. The same conclusion may be assumed to hold for the flexural buckling in the plastic range when the tube is relatively short.

7.7 Buckling of Spherical Shells 7.7.1 Analysis for a Complete Spherical Shell A complete spherical shell of uniform small thickness h is subjected to a uniform external pressure of intensity p per unit area of the middle surface. When the

7.7

Buckling of Spherical Shells

547

pressure is increased to a certain critical value, the spherical form of equilibrium is no longer guaranteed, and a nonuniform mode of deformation is possible as a result of buckling. It is assumed at the outset that the buckled shape of the middle surface is symmetrical with respect to a diameter of the sphere. The velocity field at the incipient buckling, referred to the spherical coordinates (r, φ, θ ), where φ is measured from one of the poles, may be written as υr = w,

υφ = u + zω,

υθ = 0,

where z is the radially outward distance from the middle surface, and ω is the rate of spin about the θ -axis. The thin shell approximation ε˙ rφ = 0 gives ω=

dw 1 u− , a dφ

where a denotes the radius of curvature of the middle surface. The nonzero components of the strain rate at a generic point of the shell material are ε˙ θθ = λ˙ θ − zκ˙ θ ,

ε˙ φφ = λ˙ φ − zκ˙ φ .

(7.146)

The quantities λ˙ θ ,λ˙ φ are the rates of extension of the middle surface, and κ˙ θ ,κ˙ φ are the rates of curvature of the middle surface. By (5.67), with the necessary change in sign for w, we have ⎫ 1 1 du ˙ ⎪ + w ,⎪ λφ = (u cot φ + w) , ⎬ a α dφ ⎪ 1 dw 1 d dw ⎭ κ˙ θ = 2 − u cot φ, κ˙ φ = 2 − u .⎪ a dφ a dφ dφ λ˙ θ =

(7.147)

The current state of stress prior to buckling is a balanced biaxial compression represented by σ θθ = σ φφ = –pa / 2h the remaining stress components being identically zero. The rate of change of the stress at the incipient buckling varies, however, along the meridian due to the nonuniformity of the mode of deformation. We consider a regular isotropic yield surface for the material which is stressed in the plastic range with a current uniaxial yield stress σ and tangent modulus T. The unit normal to the yield surface, considered in a nine-dimensional space, has the nonzero components nθθ = nφφ

1 = −√ , 6

nrr =

2 . 3

Considering a linearized elastic/plastic solid having the constitutive equation (7.100), the expressions for the circumferential and meridional rates of extension can be written as

548

7

Plastic Buckling

3T T 4T ε˙ θθ = 1 + τ˙θθ + 1 − (1 + 4ν) τ˙φφ , E E T 3T 4T ε˙ φφ = 1 + (1 + 4ν) τ˙θθ + 1 + τ˙φφ . E E Setting T = E in these equations, we recover the rate form of stress–strain relations for an isotropic elastic material. The above equations are easily solved for the stress rates to give τ˙θθ =

E α ε˙ θθ + β ε˙ φφ , 1+ν

τ˙φθ =

1 + 3T/E , 2 1 + (1 − 2ν) T/E

β=

E β ε˙ θθ + α ε˙ φφ , 1+ν

(7.148)

where α=

−1 + (1 + 4ν) T/E . 2 1 + (1 − 2ν) T/E

(7.149)

The remaining components of the stress rate are identically zero. It follows from (7.148) that τ˙ij ε˙ ij =

E 2 2 , α ε˙ θθ + 2β ε˙ θθ ε˙ φφ + α ε˙ φφ 1+ν

which gives the leading term in the uniqueness functional (7.104), the strain rates appearing in this expression being given by (7.146) and (7.147). The condition for uniqueness of the deformation mode is easily established by using the facts that the only nonzero components of the stress tensor are σ θθ and σ φφ , each being equal to —pa / 2h, and that the only nonzero components of the spin tensor are ωrφ = –ω and ωφr = ω. Since the shear components of the strain rate are identically zero, the uniqueness criterion (7.104) furnishes

h/2

−h/2 0

π

2z 2 2 sin φdφdz α ε˙ θθ + 2β ε˙ θθ ε˙ φφ + α ε˙ φφ 1+ a pa π 2 − (1 + ν) ω sin φdφ E 0 π p + (1 + ν) λ˙ θ + λ˙ φ w + uw sin φdφ > 0 E 0

to a close approximation. Substituting for the strain rates and the rate of spin, integrating through the thickness of the shell, and omitting the quantity d(uw sin φ) which does not contribute to the integral, the inequality may be expressed as

7.7

Buckling of Spherical Shells

549

du +w α (u cot φ + w) + 2β (u cot φ + w) dφ 0 % 2 du +α +w sin φdφ dφ 2 2 π$ dw dw du d w h2 2 α − − u cot φ + 2β −u cot φ + 12a2 0 dφ dφ dφ 2 dφ 2 2 % d w du sin φdφ +α − dφ dφ % π $ 2 pa dw 2 − (1 + ν) − 2w sin φdφ > 0 2Eh 0 dφ π

2

on neglecting terms of order (h/a)2 times the square of the rate of extension of the middle surface. The bifurcation would occur when the left-hand side of the above inequality is zero, the corresponding velocity field being that which minimizes the functional. Introducing the dimensionless quantities q = (1 + ν)

pa , 2Eh

k=

h2 , 12a2

the Euler–Lagrange differential equations satisfied by the two velocity components u and w can be easily written down following the standard technique of the calculus of variations. Taking due account of the factor sinφ that appears outside the square brackets in each integral, we obtain 2 du dw d u 2 + cot φ − β + α cot φ u + (α + β) (1 + k) α 2 dφ dφ dφ (7.150) 3 d2 w dw d w 2 + cot φ 2 − β + α cot φ −k α = 0, dφ 3 dφ dφ 3 d2 u d u du + 2 cot φ 2 + u cot φ + 2w + k −α (α + β) dφ dφ 3 dφ du

− cot φ α − β + αcosec2 φ u + β + αcosec2 φ dφ 4 (7.151) d2 w d3 w d w 2 + 2 cot φ 3 − β + αcosec φ +α dφ 4 dφ dφ 2 2

dw dw d w + cot φ α − β + αcosec2 φ + cot φ + 2w = 0. +q dφ dφ dφ 2 An immediate simplification of (7.150) is achieved by omitting k in the first factor, since it is small compared to unity. The above equations can be expressed in

550

7

Plastic Buckling

a more convenient form by setting u = dυ/dφ where υ is a new variable, and by introducing an operator H defined as H=

d d2 + cot φ + 2. 2 dφ dφ

In terms of the dependent variables υ and w, the differential equation (7.150) then becomes d [αH (υ) + (α + β) (w − υ) − kαH (w) + k (α + β) w] = 0 dφ The last term in the square brackets of this equation is seen to be negligible compared to the second. Integrating this equation, and setting the constant of integration to zero, we have αH (υ) + (α + β) (w − υ) − kαH (w) = 0.

(7.152)

Equation (7.151) can be similarly expressed in terms of the operator H and the new variable υ. The analysis is considerably simplified by noting the fact that the expressions in υ and w appearing in the square brackets of (7.151) are identical to one another except for the sign. The final result is easily shown to be (α + β) [H (υ) + 2 (w − υ)]+qH (w)+kαHH (w − υ)−(α + β) H (w − υ)] = 0. (7.153) on neglecting a term of order k compared to unity. The analysis for the bifurcation problem is therefore reduced to the solution of the pair of differential equations (7.152) and (7.153), the only restriction on the admissible velocity field being dυ/dφ = dw/dφ = 0 at φ = π and 0 = π in view of the symmetry of the field about the diameter passing through these points.

7.7.2 Solution for the Critical Pressure The nature of the differential equations (7.152) and (7.153) indicates that the solution may be expressed in terms of Legendre functions Pm of integer orders m. These functions are defined as m (m − 1) (2m)! Pm (cos φ) = (cos φ)m−2 (cos φ)m − 2 m 2. (2m − 1) 2 (m!) (7.154) m (m − 1) (m − 2) (m − 3) m−4 + − ··· . (cos φ) 2.4. (2m − 1) (2m − 3) The expression on the right-hand side of (7.154) is a polynomial of degree m in the variable cos φ. The Legendre functions of the first few orders are

7.7

Buckling of Spherical Shells P0 (cos φ) = 1, P3 (cos φ) =

551 1 P2 (cos φ) = 3 cos2 φ − 1 , 2 5 3 4 7 cos φ − 6 cos2 φ + . P4 (cos φ) = 8 5

P1 (cos φ) = cos φ,

1 cos φ 5 cos2 φ − 3 , 2

All these functions are found to satisfy a second-order linear differential equation, known as Legendre’s equation, which is d 2 Pm dPm + m (1 + m) Pm = 0. + cot φ 2 dφ dφ

(7.155)

Applying the operator H on the Legendre function Pm , and using (7.155), it is readily shown that H (Pm ) = −λm Pm ,

HH (Pm ) = λ2m Pm ,

λm = m (1 + m) − 2.

(7.156)

The usefulness of the operator H when dealing with Legendre functions now becomes obvious. To obtain a general solution to the eigenvalue problem, it is convenient to express the quantities v and w in the form of the infinite series υ=

∞ 1 m=0

Am Pm (cos φ) ,

w=

∞ 1

Bm Pm (cos φ) ,

m=0

where Am and Bm are arbitrary constants. Substitute these expressions into the differential equations (7.152) and (7.153), and using (7.156), we obtain the relations ∞ 1 m=0 ∞ 1 m=0

{[α + β + αλm ] Am − [α + β + kαλm ] Bm } Pm = 0, {[(α + β) (2 + λm ) + kλm (α + β + αλm )] Am 6 − 2 (α + β) − qλm + kλm (α + β + αλm ) Bm Pm = 0.

These equations will be satisfied only if the expressions appearing on the lefthand side for each value of m individually vanish. Hence, omitting the summation sign, we obtain the pair of equations ⎫ (α + β + αλm ) Am − (α + β + kαλm ) Bm = 0, ⎪ ⎬ [(α + β) (2 + λm ) + kλm (α + β + αλm )] Am ⎪ ⎭ − 2 (α + β) − qλm + kλm (α + β + αλm ) Bm = 0,

(7.157)

for the two typical constants Am and Bm . For the bifurcation to occur, (7.157) must admit nonzero values of these constants for some value of m, in which case the

552

7

Plastic Buckling

determinant of the coefficients of Am and Bm must vanish. Neglecting the small terms involving qk and k2 , the result may be expressed as

qλm (α + β + αλm ) − λm α 2 − β 2 − kαλm αλ2m − 2 (α + β) = 0 Since λm = 0 implies m = 1, representing a rigid-body motion, only λm = 0 will be relevant for bifurcation. Using the fact that α—β = 1, the dimensionless critical pressure is obtained as (α + β) + kα αλ2m − 2 (α + β) . q= (α + β) + αλm

(7.158)

To obtain the smallest value of q for which bifurcation may occur, it is convenient to regard the right-hand side of (7.158) as a continuous function of λm . Then setting dq/dλm = 0 to minimize q, the result is easily shown to be ⎫ ⎪ α+β ⎬ , αλm = − (α + β) + (7.159) k ⎪ ⎭ q = 2 k (α + β) − 2 k (α + β) , to a close approximation. Since λm is a large number, the critical pressure given by (7.159) cannot differ significantly from the smallest pressure based on successive integer values of m. For practical purposes, the second term on the right-hand side for q in (7.159) may be disregarded without any significant error. Using (7.149) for α and β we obtain the critical compressive stress for bifurcation in the form √ pa σ 2 (h/ a) , = =√ E 2Eh 3 (1 + ν) (1 − 2ν + E/ T)

(7.160)

which is the true tangent modulus formula for the plastic buckling of a spherical shell under external pressure. The same formula has been obtained earlier by Batterman (1969) using the rate equations of equilibrium instead of a variational principle. For any given material, the critical stress can be easily computed from (7.160) and (7.115) as a function of the ratio h/a, the results of the computation being presented graphically in Fig. 7.20. The deformation mode at the incipient buckling consists of a uniform radial contraction superposed on the eigenmode with m representing the nearest integer corresponding to the value of λm predicted by (7.159). Since Am / Bm ≈ k (α + β) in view of (7.157) and (7.159), the actual velocity field at the point of bifurcation may be written as w = −w0 (1 + cPm ) , (7.161) u = −cw0 k (α + β)Pm ,

7.7

Buckling of Spherical Shells

553

Fig. 7.20 Variation of critical stress with wall thickness for the plastic buckling of a complete spherical shell under uniform external pressure

where c is a constant and the prime denotes differentiation with respect to φ the quantity (1 + c)w0 being the radially inward velocity at φ = 0. The condition of continued loading of the radially compressed shell may be expressed as ε˙ θθ + ε˙ θθ < 0, which is equivalent to the restriction λ˙ θ + λ˙ φ − z κ˙ θ + κ˙ φ < 0 for all values of φ and z. Substituting from (7.147) and (7.161), and using the Legendre equation (7.155), this inequality is reduced to h cm (1 + m) μ + (1 − μ) Pm < 2 (1 + cPm ) , a √ where μ = k (α + β). This inequality can be satisfied for a range of values of c, giving the possible modes of deformation at bifurcation which occurs under increasing external pressure. It is interesting to note that the critical stress for buckling of an externally pressurized spherical shell is identical to that of an axially compressed short cylindrical

554

7

Plastic Buckling

shell, not only for an elastic material but also for an incompressible elastic/plastic material (v = 0.5). As in the case of cylindrical shells, slight geometrical imperfections in a spherical shell have a significant effect of lowering the critical pressure from the theoretical value (Hutchinson, 1972). An analysis for the plastic buckling of spherical shells based on the total strain theory has been discussed by Gerard (1962). The plastic buckling of shells of revolution has been investigated by Bushnell and Galletly (1974) and Bushnell (1982).

7.7.3 Snap-Through Buckling of Spherical Caps Consider a deep spherical cap of thickness h and radius a, which is subjected to an inward load P at the apex applied through a rigid circular boss of radius b, as shown in Fig. 7.21(a). As the load is gradually increased from zero, the material response is initially elastic, and the deflection at the apex increases monotonically with the intensity of loading. Over the practical range of values of h/a, plastic deformation inevitably begins before the failure occurs due to buckling. As the loading is continued in the plastic range, the load–deflection curve continues to rise with decreasing slope until the load attains a maximum, the corresponding deflection being about half the shell thickness. Subsequently, the load falls rapidly with increasing deflection and reaches a minimum when the deflection is about twice the shell thickness. Thereafter, the load increases again with deflection due to the strengthening effect of the membrane forces. If the shell is subjected to an incremental dead loading, a snap-through type of buckling would occur when the maximum value of the load is reached.

Fig. 7.21 Plastic collapse of a spherical cap centrally loaded through a rigid boss. (a) Geometry and loading, (b) conditions at incipient collapse

A complete elastic/plastic analysis of the problem, taking due account of the geometry changes, is actually required to obtain the load–deflection behavior of the shell, in order to predict the initiation of the snap-through action. For practical purposes, however, the maximum load can be approximately estimated by finding

7.7

Buckling of Spherical Shells

555

the point of intersection of the load–deflection curves obtained on the basis of purely elastic and rigid/plastic behaviors of the shell. The rigid/plastic curve begins with the theoretical collapse load, which is most conveniently determined by using the limited interaction yield condition. The analysis is similar to that used for a spherical cap under external pressure (Section 5.4) and involves the yield condition in the dimensionless form nφ = −1, −1 < nθ < 0 (φ0 ≤ φ ≤ α) , mθ = 1 (φ0 ≤ φ ≤ β) , mθ − mφ = 1 (β ≤ φ ≤ α) , where φ 0 = sin-1 (b/a). The angles α and β represent the extent of the deformation zone and the interface between the two plastic regimes, respectively. The shearing force s and the circumferential force nθ in the dimensionless form are easily shown to be ⎫

s = 1 − k¯qcosec2 φ tan φ,⎬ φ0 ≤ φ ≤ α, ⎭ n = − (1 − k¯q) sec2 φ, θ

where q¯ = P/2π M0 and k = h/4a. These relations follow from the equations of force equilibrium under zero surface loading. In view of the preceding relations, the equation of moment equilibrium becomes 1 d 1 − q¯ sec φ − − 1 cos φ, mφ sin φ = dφ k k dmφ 1 = − q¯ tan φ − (¯q − 1) cot φ, dφ k

⎫ ⎪ φ0 ≤ φ ≤ β,⎪ ⎬ ⎪ ⎭ β ≤ φ ≤ α. ⎪

(7.162)

The deformation mode at the incipient collapse involves hinge circles at φ = φ 0 and φ = α, requiring mφ = 1 and mφ, = –1 at these two sections, respectively. Using these as boundary conditions, (7.162) can be readily integrated to given the distribution of mφ. Since mφ must vanish at φ = β, we obtain

sec β + tan β (1 − k) sin β − sin φ0 = (1 − k¯q) ln sec φ0 + tan φ0 sin α cos β k (¯q − 1) ln − 1 = (1 − k¯q) ln . sin β cos α

⎫ ⎪ ,⎪ ⎬ ⎪ ⎪ ⎭

(7.163)

The angle α, defining the position of the plastic boundary, must be such that the bending moment mφ is a minimum at φ = α. The second equation of (7.162) therefore gives the relation q¯ = cos2 α +

1 2 sin α, k

(7.164)

from which the collapse load parameter q¯ can be determined for any assumed α. The corresponding values of φ 0 and β then follow from (7.163). For a given value

556

7

Plastic Buckling

of k, the collapse load increases as the boss size is increased, approaching a limiting value equal to 1/k as α tends to π/2. Since nθ < 1 for α < π /2 the preceding solution is statically admissible. The results of the calculation are shown in Fig. 7.21(b), which indicates that the region of plastic deformation is increasingly confined in the neighborhood of the loading boss as the radius of the boss is increased. The solution is also kinematically admissible, the associated velocity field being similar to that for a clamped spherical cap under uniform external pressure. Denoting the axial speed of the boss by δ˙ , the normal component of velocity of the material around it can be written as cos φ sec φ + tan φ , φ0 ≤ φ ≤ β, (7.165) w = δ˙ 1 − η ln sec φ0 + tan φ0 cos φ0 where sec β + tan β tan α −1 η = ln + sin β ln . sec φ0 + tan φ0 tan β In the outer plastic region (β ≤ φ ≤ α), the normal component of velocity is obtained by replacing the expression in the curly brackets of (7.165) with η sin β ln(tan α/tan φ). The velocity field implies the formation of hinge circles at φ = φ 0 and φ = α with appropriate discontinuities in the velocity gradient. The slope of the load–deflection curve at the incipient collapse has been determined by Leckie on the basis of a method due to Batterman (1964). A graphical plot of the large initial slope, included in Fig. 7.21(b), indicates that the carrying capacity would be considerably lowered due to the effect of the elastic deformation of the shell. The load–deflection curves shown in Fig. 7.22 have been obtained by Leckie and Penny (1968) using a series of carefully controlled tests. While these curves have the same general trend, the variation of load with√deflection is seen to be more pronounced for higher values of the parameter ρ = b/ ah. The theoretical collapse loads and the associated initial slopes of the load–deflection curves, predicted by the rigid/plastic analysis, are indicated by broken lines in the figure. We may consider a typical elastic load–deflection curve in which the plastic yielding is disregarded, the derivation of such a curve being that discussed by Ashwell (1959). The point of intersection of this curve with the corresponding rigid/plastic line is seen to provide a reasonable approximation for the maximum load at which the snap-through action is initiated. The snap-through buckling of a spherical √ cap results in an inversion of a central portion of the cap, having a radius of order hw0 , where w0 is the central deflection of the cap, and h the shell thickness The inverted cap has the same geometrical configuration as that obtained by removing a central portion of the cap, turning it over, and reuniting it with the parent cap around its edge. In actual practice, the transition between the inverted cap and the parent cap occurs through a narrow region √ in the form of a toroidal knuckle the width of which is of the order ah, where a is the mean radius of the shell. Over the initial stages of the post-buckling behavior, the variation of the load P with the central deflection w0 may be expressed by

Problems

557

Fig. 7.22 Experimental load–deflection curves for a spherical cap loaded inwardly through a rigid boss (after Leckie and Penny, 1968)

2.5 the formula Pa = 1.7λEw0.5 0 h , where E denotes Young’s modulus, while λ is a dimensionless parameter whose value is unity for purely elastic buckling (Calladine, 2001) Over the plastic range of buckling„ λ should depend on the ratio T/ E, where T is the tangent modulus at the incipient buckling of the shell.

Problems 7.1 A uniform straight column of slenderness ratio k is pin-supported at both ends, and is subjected to axial compression by equal and opposite forces P applied along the centroidal axis, The stress–strain curve of the material in the plastic range may be expressed by the equation

558

7

ε=

m αY σ σ + −1 , E E Y

Plastic Buckling

σ ≥ Y,

where α and m are dimensionless constants, the slope of the stress–strain curve being equal to E at the yield point. If the cross section of the column is rectangular, and the critical stress is denoted by σ according the tangent modulus theory, and by σ ∗ according to the reduced modulus theory, show that σ Y

1 + mα

σ Y

−1

m−1

π2 = 2 k

E , Y

σ∗ Y

1+

1 + mα

σ∗ Y

−1

m−1

=

2π k

E Y

7.2 A work-hardening elastic/plastic material yields according to the von Mises yield criterion, but obeys the non-associated flow rule furnished by the rate form of the Hencky stress–strain relation. Show that the constitutive equations for a balanced biaxial state of stress may be written as σ˙ xx = S α ε˙ xx + β ε˙ yy ,

σ˙ yy = S β ε˙ xx + α ε˙ yy ,

τ˙xy = S˙εxy / 1 + η ,

where S is the current secant modulus of the uniaxial stress–strain curve of the material, η is the modified contraction ratio obtained by replacing T with S in the expression for η, the parameters α and β being given by the expressions α=

1 ρ

3T 1+ , S

β =α−

1 , 1 + η

ρ = 4 (1 − η) 1 + η

7.3 Considering a state of plane stress, in which the current state is defined by σ x = –σ 1 , σ y = –σ 2 , and τ xy = 0, and assuming the rate form the Hencky stress–strain relation, derive the constitutive equations for an isotropic elastic/plastic material in the form σ˙ xx = S α ε˙ xx + β ε˙ yy ,

σ˙ yy = S β ε˙ xx + γ ε˙ yy ,

τ˙xy = S˙εxy / 1 + η

where S is the secant modulus of the uniaxial stress–strain curve, while α, β, and γ are dimensionless parameters expressed in terms of the ratios of the applied stresses to the effective stress σ¯ as T σ12 T σ1 σ 2 T ρα = 4 − 3 1 − , ρβ = 4η − 3 1 − , ργ = 4 − 3 1 − S σ¯ 2 S S σ¯ 2 S S T σ1 σ 2 , 2η = 1 − (1 − 2ν) ρ = 3 + (1 − 2ν) (1 + 2η) − 3 1 − E S E σ¯ 2

σ22 σ¯ 2

7.4 A rectangular plate, whose sides are of lengths a and b, respectively, where b ≤ a, is subjected to normal compressive stresses of magnitude σ along the shorter sides of the rectangle. Using the non-associated constitutive equations given in the preceding problem, show that the critical compressive stress for buckling is given by σ π 2 h2 = S 3ρb2

3T 2+ 1+ S

S a 2 mb 2 3 1 − 2ν T , + + − 2a 2 1+η E E mb

where m is an integer that minimizes the right-hand side of the above equation. Obtain the corresponding that holds for sufficiently large values of the aspect ratio a/b.

Problems

559

7.5 A rectangular plate of sides a and b is subjected to equibiaxial compressive stresses of magnitude σ normal to the edges of the plate. Using the non-associated constitutive equations of Prob.7.2, show that the critical stress for buckling is given by the expression

⎫ ⎧ ⎬ (1 + 3T/S) 1 + a2 /b2 π 2 h2 ⎨ σ = S 3a2 ⎩ 3 − (1 − 2ν) S/E 1 + (1 − 2ν) T/E ⎭ Using the Ramberg–Osgood equation with α = 3/7, E/σ 0 = 750 and c = 3, draws a graph of σ b2 /Eh2 against a/b when ν = 0.3, and compare it with that given by the Prandtl–Reuss theory. 7.6 A solid circular plate of radius a is subjected to a uniform radial compressive stress σ along its boundary. Using the constitutive equations given in Prob.7.2, obtain he critical stress for buckling of a clamped plate in the form σ k 2 h2 = S 12a2

1 + 3T/S , 4 (1 − η) 1 + η

2η = 1 − (1 − 2ν)

S E

where k is the smallest root of the equation J1 (k) = 0. Adopting the Ramberg–Osgood stress–strain curve of the preceding problem, and using ν = 0.3, obtain the variation of σ /E with h/a, and compare it with that given by the Prandtl–Reuss theory. 7.7 If the circular plate of the preceding problem is simply supported along its boundary .r = a, show that the critical stress for buckling under a uniform radial compression σ is given by 4 (1 − η) kJ1 (k) , = J0 (k) 1 + 3T/S

4a k= h

3σ S

1/2 (1 − η) 1 + η 1 + 3T/S

in view of the results given in Prob. 7.1. Assuming the Ramberg–Osgood equation of the preceding problem, compute the critical value of σ /E when ν = 0.3 and h/a = 0.05, and compare the result with that given by the Prandtl–Reuss theory.

Chapter 8

Dynamic Plasticity

In this chapter, we shall be concerned with the class of problems in which the plastic deformation is so rapid that the inertia effects cannot be disregarded. Problems of dynamic plasticity arise in the high-velocity forming of metals, penetration of highspeed projectiles into fixed targets, enlargement of cavities by underground explosion, and design of crash barriers related to collisions, to name only a few. The rate of loading and the size of the components are usually such that the deformation process can be described in terms of the propagation of elastic/plastic waves. However, simplified theories which disregard the wave propagation phenomenon are generally capable of providing useful information for practical purposes. In the case of structural members subjected to impact loading, the mode of plastic deformation can be most conveniently represented by the existence of discrete yield hinges that rapidly move away from the point of loading. The concept of moving yield hinges is a useful device for the dynamic analysis of structures.

8.1 Longitudinal Stress Waves in Bars 8.1.1 Wave Propagation Without Rate Effects The propagation of longitudinal elastic/plastic waves in thin rods or wires was first discussed independently by von Karman (1942), Taylor (1942), and Rakhmatulin (1945), although a theoretical treatment in a restricted sense was given earlier by Donnel (1930). Following von Karman and Duwez (1950), the theory will be developed here in terms of the nominal stress and strain, denoted here by σ and ε, respectively, and the initial coordinate x measured along the axis of the bar, which is assumed to have a uniform cross section. If the longitudinal velocity of the particle at any instant is denoted by v, and transverse plane sections are assumed to remain plane, the equation of motion is ∂υ ∂σ =ρ , ∂x ∂t where ρ is the initial density of the material, and σ is a known function of ε, tensile stress being taken as positive. Since υ = ∂u/∂t and ε = ∂u/∂x, where u is the J. Chakrabarty, Applied Plasticity, Second Edition, Mechanical Engineering Series, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77674-3_8,

561

562

8 Dynamic Plasticity

longitudinal displacement, the differential equation for a rate insensitive material becomes 2 ∂ 2u 2∂ u = c , ∂t2 ∂x2

c2 =

1 dσ , ρ dε

(8.1)

which is a quasi-linear wave equation governing the motion of the particle, with c = c(ε) denoting the speed of propagation of the wave. Equation (8.1) applies only during the loading process in which the stress continuously increases with time, and for common engineering materials, the wave speed decreases with increasing strain. If the loading is tensile, a critical stage would be reached when the total strain is equal to that at the onset of necking, and the plastic wave speed is then reduced to zero. The solution of the differential equation (8.1) is simplified by the fact that this equation is actually hyperbolic. Indeed, along any curve considered in the (x, t)plane, the variation of the velocity v is given by dυ =

∂ε ∂υ ∂υ ∂ε dx + dt = dx + c2 dt, ∂x ∂t ∂t ∂x

(8.2)

in view of the identity ∂υ/∂x = ∂ε/∂t, and the relation ∂υ/∂t = c2 (∂ε/∂x) which follows from (8.1). The variation of the strain ε along this curve is dε =

∂ε ∂ε dt + dx. ∂t ∂x

For given values of dυ and dε along the curve, the derivatives ∂ε/∂t and ∂ε/∂x can be uniquely determined from the last two equations unless the determinant of their coefficients vanishes. The considered curve will therefore be a characteristic if (dx)2 = c2 (dt)2 . Equation (8.1) is therefore hyperbolic, the characteristic directions and the differential relations holding along them being dx = ±c, dt

dυ = ±cdε.

(8.3)

where the second result follows on substitution for dx and dt from the first into the second expression of (8.2). Since c is a function of ε, the characteristic lines generally consist of two families of curves corresponding to the upper and lower signs in (8.3). When the characteristic is a straight line, c is a constant, which means that the stress, strain, and velocity remain constant along its length. Unloading begins in a given cross section as soon as the stress begins to decrease after reaching a certain maximum value σ ∗ , corresponding to a strain ε ∗ . The stress– strain relation for the unloading process may be written as σ − σ ∗ = E ε − ε∗ ,

8.1

Longitudinal Stress Waves in Bars

563

where E is Young’s modulus for the material. Substituting in the equation of motion, and using the fact that υ = ∂u/∂t, we obtain the differential equation 2 ∂ 2u d 2∂ u = c + 0 2 2 dx ∂t ∂x

σ∗ − c20 ε∗ , ρ

(8.4)

√ where c0 = E/ρ representing the speed of propagation of the elastic wave. Equation (8.4) is also hyperbolic with two families of straight characteristics in the (x, t)-plane, the characteristic relations being obtained in the same way as before with the result dx = ±c0, dt

ρc0 dυ = ±dσ .

(8.5)

Since σ ∗ and ε ∗ are functions of x, and are not known in advance, the shape of the loading/unloading boundary must be determined as a part of the solution. For a bar of finite length, waves are reflected from the ends of the bar, and the solution is strongly dependent on the nature of the boundary conditions. If a bar is subjected to impact loading at one end x = –l, the first wave to reach the other end x = 0 is always an elastic wave. The displacement of the particles during the propagation of the direct elastic wave is governed by (8.1) with c replaced by the elastic wave speed c0 . A general solution of this wave equation is u1 = f1 (c0 t − x) , where f1 is an arbitrary function that can be determined from the prescribed initial conditions. For the reflected wave, which propagates in the opposite direction, the displacement is u2 = f2 (c0 t + x) . The total displacement of a particle traversed by the incident and reflected waves is u = u1 +u2 , and the corresponding stress is σ =E

6 7 ∂ (u1 + u2 ) = E f2 (c0 t + x) − f1 (c0 t − x) , ∂x

where the prime denotes differentiation with respect to the argument c0 t– x. When the end x = 0 is free, the stress vanishes there for all t, and consequently f1 (c0 t) = f2 (c0 t), which gives f1 = f2 = f . The displacement and velocity at the free end therefore become u = 2f (c0 t) ,

υ = 2c0 f (c0 t) .

Thus, the displacement and velocity at the free end are doubled due to the reflection, a compression wave being reflected as a tension wave and vice versa. If the

564

8 Dynamic Plasticity

end x = 0 is fixed, the total displacement u1 +u2 vanishes there for all t, and we have f2 = −f1 = f and the stress at the fixed end becomes σ = 2Ef (c0 t). The stress is therefore doubled in magnitude due to the reflection, and an elastic wave may therefore be reflected as a plastic wave. Across a wave front, the first derivatives of v and ε are necessarily discontinuous. In the case of weak waves, υ and ε are themselves continuous, and the wave front then coincides with a characteristic. In problems of plastic wave propagation, one frequently encounters shock waves in which the wave fronts are surfaces of discontinuity even for υ and ε. Shock waves are usually generated by a sudden change in velocity imposed at one end of the bar. If a shock wave front moves through a distance dx during a time interval dt, the displacement on either side of the wave front changes by the amount du = υ dt+ε dx. The condition of continuity of the displacement across the wave front therefore gives [υ] + cs [ε] = 0,

(8.6a)

where cs is the speed of propagation of the shock wave, and the square brackets represent the jump in the enclosed quantity when the front has passed a given cross section. The momentum equation for the element of length dx traversed by the wave front during the time dt yields ρcs [υ] + [σ ] = 0.

(8.6b)

The two jump conditions established in (8.6), together with the stress–strain equation, are sufficient to study the propagation of shock waves, provided the change in temperature due to impact is negligible. The elimination of [υ] between the two relations of (8.6) furnishes ρc2s = [σ ]/[ε]. The speed of propagation of the shock wave coincides with that of the weak wave only when the stress–strain curve is linear.

8.1.2 Simple Wave Solution with Application Consider the situation where ε and v are functions of the ratio x/t, but not of x and t individually. Then all the characteristics of positive slope are straight lines passing through the origin, the result being a centered fan of linear characteristics, having the equation x/t = c, known as simple waves. Since v and ε are functions of c only, we have ∂υ ∂ε dυ =t =t , dc ∂x ∂t the right-hand side of this expression being –c times the derivative of ε with respect to c. Thus

8.1

Longitudinal Stress Waves in Bars

565

dε 1 dσ dυ = −c =− . dc dc ρc dc Using the boundary condition υ = υ 0 along the characteristic that corresponds to σ = Y, the above equation is immediately integrated to give σ ρc0 (υ − υ0 ) = −

(c0 /c) dσ .

(8.7)

γ

Simple waves generally appear under instantaneous loading by a uniform stress or velocity at one end of the bar and exist in a region adjacent to one of constant stress or strain, which evidently satisfies (8.1) and the boundary condition. The initial loading of the bar is generally followed by an unloading process in which the stress progressively decreases in magnitude. To illustrate the basic principles, consider the normal impact of a cylindrical bar of length l, which is moving parallel to its longitudinal axis with a velocity –U against a stationary rigid target x = 0 (Lee, 1953). For mathematical convenience, we superimpose a constant velocity U to the whole system in the opposite direction and write the initial and boundary conditions as υ = 0, σ = 0 at t = 0, 0 < x ≤ t, υ = U, at x = 0, σ = 0 at x = l, t > 0. The stress and strain will be considered positive in compression. The characteristic field is shown in Fig. 8.1(a), where OA is the elastic wave front across which the stress rises instantaneously from zero to the yield stress Y. The fan OAD consists of plastic waves with OD representing the characteristic for υ = U. Within the triangle ODE, the material has a constant velocity U and is subjected to a constant stress that corresponds to the characteristic OD. At the free end x = l, the elastic loading wave is reflected as an unloading shock wave which propagates with a speed c0 and has the equation x = 2 l − c0 t. At A, the value of ρc0 υ is equal to Y before the reflection, in view of (8.6b) with cs = co and a change in sign for σ . The physical quantities along ABC after the reflection will be denoted by a prime. Since the velocity at the free end is doubled by the reflected wave, ρc0 υ is equal to 2Y at A, and the second relation of (8.5) applied to the boundary wave AC gives ρc0 υ − 2Y = σ as the stress vanishes at the free end after the reflection. At a generic point of AC, we may apply the jump condition (8.6b) with cs = –c0 and –σ written for σ , the result being − ρc0 υ − υ = σ − σ

566

8 Dynamic Plasticity

Fig. 8.1 Longitudinal impact on a bar of finite length. (a) Characteristics in the plastic region and (b) distribution of plastic strain along its length

Eliminating υ between the above relations, the amount of stress discontinuity across the unloading wave may be expressed in terms of the stress and velocity before the reflection as σ − σ = Y −

1 (ρc0 υ − σ ) . 2

(8.8a)

The relationship between υ and σ is obtained from (8.7) by setting ρc0 υ 0 = Y and changing the sign of σ . Thus σ ρc0 υ = Y +

(c0 − c) dσ ,

(8.8b)

Y

where ρc2 = dσ/dε, given by the stress–strain curve. Since c ≤ co , the integral in (8.8b) is increasingly greater than σ – Y, and the expression in the parenthesis of (8.8a) steadily increases along AC from its minimum value zero at A. Thus the unloading shock wave is progressively absorbed during its propagation. If the impact velocity U is sufficiently high, the unloading wave will be completely

8.1

Longitudinal Stress Waves in Bars

567

absorbed at a point C, where the stress discontinuity is reduced to zero. When the impact velocity is below a certain critical value, the unloading wave can continue through to the impact face at E, and the material is unloaded throughout the length of the bar. In the case of a supercritical impact velocity that terminates the unloading wave at C, the plastic region spreads from this point into an area above AE. The solution to this part of the impact problem involves discontinuities in stress and velocity derivatives, which must be admitted for the continuation of the loading–unloading boundary. For a subcritical impact velocity, which ensures that the unloading wave traverses the entire length of the bar, a new plastic region is initiated at a point

Fig. 8.2 Characteristic field in the propagation of longitudinal plastic waves involving regions of loading and unloading (after E. H. Lee, 1953)

568

8 Dynamic Plasticity

where the unloaded bar has a stress equal to the maximum stress previously attained in the same section during loading. The complete characteristic field for this particular case obtained by Lee is shown in Fig. 8.2, where D is the point of initiation of the new plastic zone DEFGK. At point K, the loading–unloading boundary is intersected by the unloading wave reflected from the impact face. The permanent strain produced by the secondary plastic region is found to be small compared to that due to the primary plastic loading wave, as may be seen from Fig. 8.1(b), which displays the permanent strain distribution in the bar.

8.1.3 Solution for Linear Strain Hardening The problem of plastic wave propagation is greatly simplified by the assumption of a linear strain-hardening law, which makes the stress–strain curve consist of a pair of straight lines of slopes E and T in the elastic and plastic ranges. The elastic and plastic with constant speeds c0 and cp , respectively, √ waves then propagate √ where c0 = E/ρ and cρ = T/ρ. To illustrate the simplicity of this approach, we consider the same problem as that discussed before, a uniform bar of length l being assumed to strike a rigid wall x = 0 with a velocity U from right to left at time t = 0. For sufficiently large values of U, the elastic and plastic waves are simultaneously generated at x = 0 and are propagated along the length of the bar, the positions of the wave fronts at subsequent times being shown in Fig. 8.3. The solution, which is due to Lensky (1949), has been discussed by Cristescu (1967). During the time interval 0 ≤ t l/c0 , there are three distinct regions in the bar as indicated in Fig. 8.3(a). In the outer region, the material is undisturbed with υ1 = −U and σ1 = ε1 = 0, while in the central region that is traversed by the elastic waves only, we have σ2 = Y, ε2 = Y/E = ε0 (say), and υ2 = −U + c0 ε0 , in view of the jump conditions (8.6). In the region adjacent to the wall, where the material is brought to rest (υ 3 = 0) by the plastic shock wave, the compressive stress and strain are similarly obtained as σ3 = Y + ρcp (U − c0 ε0 ) ,

ε3 = ε0 + (U − c0 ε0 )/cρ .

It follows that the bar will be rendered plastic only if the velocity of impact satisfies the condition U > c0 ε0 . For t > 1/c0 , the elastic wave front moves backward after reflection from the free end of the bar, while the plastic wave front advances further to the right as indicated in Fig. 8.3(b). The reflected wave front completely unloads the region traversed by it, giving σ 4 = ε 4 = 0, and its velocity is reduced in magnitude to υ4 = 2c0 ε0 − U. At time t = ts the reflected elastic wave front meets the advancing plastic wave front at some section S, which is at a distance xs from the wall. Then ts = xs/cρ = 2(2 l − xs )/c0 , giving the relations xs = 2lcp / c0 + cp ,

ts = 2 l/ c0 + cp .

8.1

Longitudinal Stress Waves in Bars

569

Fig. 8.3 Propagation of shock waves in an elastic/plastic bar of finite length striking a rigid target and the associated reflection and interaction of elastic and plastic waves

If the impact velocity U is not too large, the plastic wave will not propagate any further for t > ts , and reflected elastic waves will spread in both directions from S, as shown in Fig. 8.3(c). In the region between the two reflected waves, the particle velocities on both sides of section S must be the same, but the strains in the two portions will be different. By (6), the associated stresses and strains are given by σ5 = ρc0 (υ5 − υ4 ) , σ5 = σ3 − ρc0 υs , ε = (υ5 − υ4 )/c0 , ε5 = ε3 − ((υ5 − υ3 )/c0 ) . Substituting for σ3 , ε 3 , υ 3 , and υ 4 and noting the fact that σ5 = σ’5 to a close approximation for longitudinal equilibrium, we obtain cp 1 , υ5 = U − (U − c0 ε0 ) 3 − 2 c0

570

8 Dynamic Plasticity

ε5 = (U − c0 ε0 )

c0 + cp 2c20

,

ε5

= (U − c0 ε0 )

c0 + cp 2c0 − cp 2c20 cp

.

(8.9) If the plastic strain stops at x = xs , the region to the right of section S must remain elastic, which requires ε5 < ε0 . The preceding solution is therefore valid for 1

(3c0 + cp )(c0 + cp ), the advancing plastic wave will be intercepted either by the elastic wave which propagates to the left and is subsequently reflected at x = 0 or by the elastic wave which propagates to the right and is subsequently reflected at x = l. The sequence of events leading to the first of these two possibilities is represented in the characteristic plane shown in Fig. 8.3(e). The reflected wave front propagating from x = 0 meets the plastic wave front at a section R when t = tr , the distance of the section from the rigid wall being x = xr . It is easily shown that xr = 2lcp / c0 − cp ,

te = 2 l/ c0 − cp .

In the case of the second possibility, the total distance traveled by the elastic wave front before it meets the plastic wave front is 2(2l – xs ) – xr , which leads to the relations 2 xr = 4lc0 cp / c0 + cp ,

2 tr = 4lc0 / c0 + cp .

8.1

Longitudinal Stress Waves in Bars

571

In the critical situation, when both the elastic waves meet the plastic waves simultaneously, we have c20 − 4c0 cp − c2p = 0

or

c0 /cp = 2 +

√ 5 ≈ 4.24.

Thus, the leftward moving elastic wave from S meets the plastic wave on reflection when c0 /cp > 4.24, and the rightward moving elastic wave from S meets the plastic wave on reflection when c0 /cp > 4.24. During the time interval ts < t < 2l/c0 , the stress, strain, and velocity in regions 1–4 are identical to those previously given, while in region 5, these quantities are σ5 = Y, ε5 = ε0 , and υ5 = 3c0 ε0 − U. In the remaining regions 6 and 7, the stresses and strains are given in terms of the velocities as σ6 − Y = ρcp (υ6 − υ5 ) , σ7 − σ3 = −ρc0 υ7 , ε6 − ε0 = (υ6 − υ5 )/cp , ε7 − ε3 = −υ7 /c0 . Substituting for υ 5 , σ 3 , and ε3 and using the equilibrium and continuity conditions σ6 ≈ σ7 and υ6 = υ7 , respectively, we obtain the relations υ6 = υ7 =

2c0 cp ε0 , c0 + cp

ε7 =

ε0 c2 + c2p c0 U , ε6 = ε7 −2 − 0 − 1 ε0 . (8.11) cp cp c0 + cp cp

The solution can be continued in a similar manner for any given value of c0 /cp . The final configuration of the bar will contain two stationary discontinuities occurring at x = xs and x = xr , where there are abrupt changes in the cross section. Other examples of longitudinal wave propagation in bars have been discussed by White and Griffis (1947), Rakhmatulin and Shapiro (1948), De Juhasz (1949), Lebedev (1954), Ripperger (1960), Clifton and Bodner (1966), and Cristescu (1970), among others. The problem of combined longitudinal and torsional plastic waves in thin-walled tubes has been discussed by Clifton (1966), Goel and Malvern (1970), Ting (1972), and Wu and Lin (1974). The dynamic plastic behavior of extensible strings has been examined by Craggs (1954) and Cristescu (1964), and discussed at great length by Cristescu (1967).

8.1.4 Influence of Strain-Rate Sensitivity It is well known that the yield stress of engineering materials can be considerably higher under dynamic loading than under quasi-static loading. For example, the dynamic yield stress of annealed mild steel has been found to be more than double the quasi-static yield stress. The relevant experimental evidence has been provided by Duwez and Clark (1947), Campbell (1954), Goldsmith (1960), and Davies and Hunter (1963), along with a number of other investigators. The dynamic elastic modulus is essentially the same as the quasi-static elastic modulus, but the overall

572

8 Dynamic Plasticity

dynamic stress–strain curve is generally much higher than the quasi-static curve as shown in Fig. 8.4, which is based on the experimental results of Kolsky and Douch (1962). The magnitude of the strain rate at which a given material begins to be rate sensitive varies from material to material, but rate dependence of the stress–strain curve is an important factor which must be included in the theoretical framework for a realistic prediction of the dynamic behavior (Campbell, 1972).

Fig. 8.4 Comparison of static and dynamic stress–strain curves for (a) copper and (b) aluminum (due to Kolsky and Douch, 1962)

For a work-hardening material having its uniaxial stress–strain curve given by σ = f(ε) under quasi-static conditions, the simplest constitutive equation relating the stress and strain to the rate of straining may be obtained on the assumption that the plastic part of the strain rate is a function of the overstress σ – f(ε), which is the difference between the dynamic and quasi-static yield stresses corresponding to a given strain. Since the elastic part of the strain rate is related to the stress rate by Hooke’s law, the constitutive equation becomes E˙ε = σ˙ + F σ − f (ε) .

(8.12)

The function F(z) must be positive for z > 0 and zero for z ≤ 0. Equation (8.12), which has been proposed by Malvern (1951), is a generalization of one due to Sokolovsky (1948b), who assumed the material to be ideally plastic in the quasi-static state. The expression F(z) = kzn , where k and n are positive constants,

8.1

Longitudinal Stress Waves in Bars

573

has been found to fit experimental data reasonably well by Kukudjanov (1967). Lindholm (1964) has experimentally verified the constitutive relation σ = σ0 (ε) + σ1 (ε) 1n˙ε , where σ 0 (ε) defines the stress–strain curve under a unit strain rate. A different type of constitutive equation based on microstructural considerations has been examined by Steinberg and Lund (1989). In the high-velocity impact or explosive loading, of components, most of the heat generated by the plastic deformation remains in the specimen, causing a thermal softening of the material. Using a Hopkinson pressure bar recovery technique, the isothermal stress–strain curves for materials at high strain rates have been obtained by Nemat-Nasser et al. (1991), who also estimated the effect of the adiabatic rise in temperature on the dynamic response. Their experimental data over the plastic range of strains can be fitted by the constitutive equation m ε˙ T exp −λ −1 σ = σ0 ε 1 + ε˙ 0 T0 n

(8.13)

where σ 0 , ε0 , and T0 are reference values of the stress, strain rate, and temperature, respectively, while m, n, and λ are material parameters. The above equation is very similar to that suggested by Nemat-Nasser et al. (1994) and is found to be sufficiently accurate over the practical range. In general, both the elastic and plastic strain increments would be involved in a prescribed stress increment, whether or not the material is rate sensitive. Considering this fact, a constitutive equation has been proposed by Cristescu (1963) and Lubiner (1965), in which the instantaneous strain rate is expressed in terms of the stress rate in the generalized form E

∂σ ∂ε = φ (σ ,ε) + ψ (σ ,ε) . ∂t ∂t

(8.14)

When the material is not rate sensitive, ψ = 0 and φ is independent of t, while the derivatives in (8.14) may be considered with respect to any monotonically increasing parameter. For a highly rate-sensitive material, on the other hand, it is a good approximation to set φ = 1, and (8.14) then reduces to one of type (8.12). The problem of longitudinal wave propagation in an elastic/plastic bar with an arbitrary material response is therefore governed by (8.14), as well as the relations ∂υ ∂σ =ρ , ∂x ∂t

∂υ ∂ε = , ∂x ∂t

for the three unknowns σ , ε, and υ. The first equation above is the equation of motion, while the second is the equation of compatibility. This system of equations is hyperbolic, as may be shown by considering the variation of σ , ε, and v along any curve in the (x, t)-plane. The conditions under which the first derivatives of these

574

8 Dynamic Plasticity

quantities may be discontinuous across the curve furnishes three families of real characteristics (Hopkins, 1968), which are given by dx = 0,

dx/dt = ±c,

c=

E/ρφ,

(8.15)

where c denotes the wave speed which is generally much higher than that given by (8.1). The differential relations holding along the characteristic curves dx/dt = ±c are easily shown to be dσ = ±ρcdυ − (ψ/φ) dt,

(8.16a)

where the upper and lower signs correspond with those in (8.15). The relationship that holds along the remaining family of characteristics dx = 0 is found to be Edε = φdσ + ψdt.

(8.16b)

Across a stationary wave front dx = 0, only ∂ε/∂x may be discontinuous, while all other first-order derivatives of σ , ε, and υ must be continuous. This is easily established from the condition of continuity of these quantities across the wave front, and the fact that dε = (∂ε/∂t) dt and so on along this line. The continuity of the time derivatives ∂ε/∂t and ∂υ/∂t implies the continuity of ∂υ/∂x and ∂σ/∂x, but no information is available for ∂ε/∂x which may therefore be discontinuous. Across a moving wave front defined by the second relation of (8.15), all the first-order derivatives of σ , ε, and υ may become discontinuous. The method of numerical integration of (8.16a) and (8.16b) along the characteristics has been expounded by Cristescu (1967). In the special case when ψ = 0 and φ is independent of t, the results are similar to those for the rate-independent material, the only difference being the existence of a third family of characteristics arising from the differential form of the constitutive relation. In the other extreme case√of φ = 1, the plastic wave speed becomes identical to the elastic wave speed c0 = E/ρ, while the characteristic lines (which are straight) and the differential relations along them are still given by (8.15) and (8.16) with c = c0 . It may be noted that the first term on the right-hand side of (8.13), when φ = 1, represents the elastic strain increment, and the second term represents the plastic strain increment. The stress and elastic strain increase as a result of the wave propagation, causing the plastic strain to increase along the stationary characteristics. More refined constitutive equations for the propagation of one-dimensional plastic waves with strain rate effects have been examined by Cristescu (1974). Generalized constitutive equations for viscoplastic solids, from the standpoint of continuum mechanics, have been presented by Perzyna (1963) and Mandel (1972). Other types of constitutive equations for the dynamic behavior of materials, based on the dislocation motion, have been examined by Nemat-Nasser and Guo (2003).

8.1

Longitudinal Stress Waves in Bars

575

8.1.5 Illustrative Examples and Experimental Evidence Some quantitative results for the plastic wave propagation in a rate-sensitive material have been given by Malvern (1951), who discussed an example in which the end x = 0 of a semi-infinite bar lying along the positive x-axis is instantaneously pulled with a longitudinal velocityv = −U at t = 0. The constitutive law adopted by Malvern corresponds to (8.14) with ψ = k σ − f (ε) ,

φ = 1,

f (ε) = Y (2 − Y/Eε)

(8.17)

where k is a material constant. The numerical computation is based on the values k = 106 /s, Y/E = 10–3 , E = 70 GPa, c = c0 = 5 km/s, and U = 15 m/s. The predicted dynamic yield stress for this material is about 10% higher than the quasi-static value for a constant strain rate of 200/s. Along the leading wave front x = c0 t, the stress and strain are not constant in the solution with strain rate effect, which allows the stress to rise above Y without exceeding the elastic limit. The stress attained in an element just after the passage of the leading wave front is given by dσ = −

k (σ − Y)2 dt 2σ

in view of (8.16a) and (8.17), together with the relations Eε = σ and ρc0 υ = −σ which hold along this shock wave. Integrating, and using the initial condition σ = ρc0 U at t = 0, which follows from (8.6b), we obtain the solution t=

2 k

(ρc0 U − Y) (ρc0 U − σ ) Y + 1n . (σ − Y) (ρc0 U − Y) (σ − Y)

(8.18)

It may be noted that ρc0 U = 3Y according to the assumed numerical data. Following the impulsive loading, the stress jumps instantaneously from 0 to 3Y, but rapidly falls to a value very nearly equal to Y as the wave propagates along the bar. If, on the other hand, the strain rate effect is disregarded, the stress would have a constant value Y all along the leading wave front. The solid curves in Fig. 8.5(a) represent the lines of constant strain in the (x, t)plane according to rate-dependent solution, while the broken straight lines passing through the origin correspond to the solution that neglects strain rate effects. The region of constant strain, that exists in the rate-independent solution above the wave front of the maximum strain ε = 0.0074, disappears in the rate-dependent solution, which predicts a gradual transition from plastic to elastic response across the chaindotted line. Along this unloading boundary, the quasi-static stress–strain relation σ = f(ε) holds, where σ and ε slowly decrease with increasing x and t. The stress–strain relations at various sections of the bar are displayed in Fig. 8.5(b), where the broken curve represents the quasi-static stress–strain relation. It is seen that the plastic deformation occurs with decreasing stress in sections closer to the impact end x = 0, and with increasing stress in sections farther from x = 0. This is a

576

8 Dynamic Plasticity

Fig. 8.5 Calculated results for the plastic wave propagation along a long bar made of rate-sensitive material. (a) Lines of constant strain and (b) stress–strain relations at different sections of the bar (after L.E. Malvern, 1951)

consequence of the high initial stress instantaneously attained at x = 0, and the fact that the strain rate at any section rapidly decreases with time. At sections sufficiently far away from the end x = 0, the stress–strain curves resemble the quasi-static curve. The strain distribution shown in Fig. 8.6 (a) indicates that the plateau of constant strain near the impact end disappears, which is contrary to that experimentally observed (Bell, 1968). The existence of the strain plateau can be theoretically predicted, however, by using a suitable modification of the constitutive equation. The plastic wave propagation in a bar of finite length, initially prestressed to the yield point with a longitudinal stress σ = Y, has been treated by Cristescu (1965), using the relations (8.17) with the same material constants as those employed in the preceding example. The bar is assumed to have an initial length l = 7.5 cm with one end fixed and the other end prescribed to move with a velocity υ = −Ut/t0 (0 < t ≤ t0 ) , υ = −U (t ≥ t0 ) , where U = 30 m/s and t0 = l2 μs. The solution has been obtained numerically by using the finite difference form of the differential relations (8.16) along the characteristics. Some of the computed results are displayed in Fig. 8.7(a) and (b), which exhibit the same general trends as those in the previous example. There is, of course, no strain plateau near the impact end of the bar.

8.1

Longitudinal Stress Waves in Bars

577

Fig. 8.6 Strain rate effects on the longitudinal plastic wave propagation. (a) Strain distribution at t = 102.4 μs and (b) stress–time relation at a distance x = 16.2 cm from impact end

Fig. 8.7 Results for a bar of finite length subjected to a prescribed velocity at x = 0. (a) Time dependence of physical quantities at x = 3 cm and (b) dynamic stress–strain relations (after N. Cristescu, 1967)

578

8 Dynamic Plasticity

Several other problems involving the strain rate effect have been discussed by Cristescu (1972), Banerjee and Malvern (1975), and Cristescu and Suliciu (1982). The problem of plastic wave propagation due to the longitudinal impact of two identical bars, one of which is at rest and the other one fired from an air gun, has been considered by Cristescu and Bell (1970). The effects of strain rate history on the dynamic response have been examined by Klepaczko (1968) and Nicholas (1971). The application of the endochronic theory to the propagation of one-dimensional plastic waves has been discussed by Lin and Wu (1983). One of the earliest experimental investigations on the propagation of plastic waves, reported by von Karman and Duwez (1950), consisted in measuring the longitudinal strain in annealed copper wires subjected to impact loading by means of a drop hammer. Although strain rate effects were disregarded in their theoretical calculations, the overall agreement between theory and experiment was found to be reasonably good. Sternglass and Stuart (1953) applied small amplitude strain pulses to prestressed copper wires to study the propagation of plastic waves. The speed of propagation of the wave front was found to be that of elastic waves, and the wave speed of any part of the pulse was found to be much higher than the value given by the rate-independent theory. Further experimental supports for the rate-dependent theory have been provided by Alter and Curtis (1956), Malyshev (1961), Bianchi (1964), Dillons (1968), and others, and the results obtained by them were consistent with the differential constitutive law of type (8.12). Critical reviews of the published experimental work on the plastic wave propagation have been reported by Bell (1973), Clifton (1973), and Nicholas (1982). The development of plastic strains with time at various distances from the impact end of bars, made of mild steel and aluminum, has been experimentally determined by Bell (1960) and Bell and Stein (1962), using a diffraction grating method. The split Hopkinson pressure bar has been used by Lindholm (1964), following a method proposed by Kolsky (1949), to establish the dynamic properties and stress–strain curves for lead, aluminum, and copper under different values of the rate of strain. An electromagnetic method of measurement of the particle velocity at any section of the bar during the propagation of plastic waves along its length has been discussed by Malvern (1965) and Efron and Malvern (1969). Both these methods are quite accurate and can be used for obtaining the overall dynamic response from the moment of impact to the moment when the unloading begins. Further references to the theoretical and experimental investigations on the plastic wave propagation have been given by Nicholas (1982).

8.2 Plastic Waves in Continuous Media 8.2.1 Plastic Wave Speeds and Their Properties Consider an isolated wave front, not necessarily plane, which is advancing through an elastic/plastic material that work-hardens isotropically without exhibiting strain rate effects. We shall be concerned here with the propagation of weak waves across

8.2

Plastic Waves in Continuous Media

579

which discontinuities in the derivatives of stress, strain, and particle velocity may exist. Since the governing equations are quasi-linear, the weak waves are necessarily characteristics of the hyperbolic system. Any physical quantity in the immediate neighborhood of the wave front may be regarded as a function of the surface coordinates, the surface normal, and time t. Since the surface derivatives of a typical variable f must be continuous across the wave front, which has a velocity c in the direction of the normal n, the jump in the space and material derivatives of f across the wave front may be written as ∂f ∂f ∂f ni c, (8.19) = f˙ = − ∂xi ∂n ∂n where ni denotes the unit vector in the direction of n. The second expression follows from the fact that the rate of change of f relative to the motion of the wave front in the xi space is continuous, since f itself must be continuous. Each pair of square brackets in (8.19) will be taken to represent the value of the enclosed quantity behind the wave front in excess of that ahead of the front. In the analysis of the problem of plastic wave propagation, all rotational effects will be disregarded. Consequently, the stress rate entering into the constitutive equation will be taken as the material derivative of the true stress σ ij . If the material is prestressed in the plastic range, and obeys the von Mises yield criterion and the associated Prandtl–Reuss flow rule, the constitutive equation may be written in the form of equation (1.38), where nij = σ˙ ij = 2G ε˙ ij +

3 σ 2 sij/¯

The stress rate is therefore given by

ν 3α , ε˙ kl skl ≥ 0, ε ˙ s s ε˙ kk δij − kl kl ij 1 − 2ν 2σ¯ 2

(8.20)

where G is the shear modulus, ν is Poisson’s ratio, σ¯ is the equivalent stress, sij is the deviatoric stress tensor, and α = 3G/(H + 3G) with H denoting the current plastic modulus. In the case of unloading, indicated by ε˙ kl skl < 0, it is only necessary to set α = 0 in (8.20). Introducing the expression ∂υj 1 ∂υi ε˙ ij = + 2 ∂xj ∂xi for the true strain rate, where υ i denotes the particle velocity, (8.20) may be expressed as 2ν ∂υk ∂υi ∂υi 3α ∂υk σ˙ ij = G + + ∂ij − 2 skl sij . ∂xj ∂xi 1 − 2ν ∂xk σ¯ ∂xl Admitting possible discontinuities in the stress rate and the velocity gradient across the wave front, and using (8.19), we obtain the discontinuity relation ∂σij 2ν 3α λk nk δij − 2 λk nl skl sij , = G λi nj + λj ni + −c ∂n 1 − 2ν σ¯

580

8 Dynamic Plasticity

where λi , denotes the quantity ∂υi /∂n . Multiplying the preceding equation by nj following the summation convention, and using the relations σij nj = σi ,

sij nj = si ,

nj nj = 1,

where si and σ i are the deviatone and actual stress vectors acting across the wave front, we get −c

3α ∂σi = G λi + (1 − 2ν)−1 λj nj ni − 2 λk sk si . ∂n σ¯

(8.21)

The scalar parameter λk sk , which represents the discontinuity in the deviatoric work rate, must be positive for continued loading, in view of the sign convention for λi . When λk sk is negative, the work rate in the element is decreased by the passage of the wave front, and unloading would occur as a result. In order to complete the analysis, we must consider the equation of motion of the material element. Denoting the density of the material by ρ, which may be assumed constant since only elastic changes in volume are involved, the equation of motion may be written as ∂σij − ρ υ˙ i = 0. ∂xj

(8.22)

Using (8.19), the discontinuity relation corresponding to (8.22) is readily obtained, and the substitution σij nj = σi then results in

∂σi + ρcλi = 0. ∂n

The elimination of the stress gradient discontinuity ∂σi /∂n between this equation and (8.21) furnishes the connection equation (Jansen et al., 1972) as

c2 − c22 λi − c21 − c22 λj nj ni + c22 α/k2 λk sk si = 0,

(8.23)

√ where k = σ/ ¯ 3 is the current yield stress in shear, and c1 , c2 are the speeds of propagation of elastic dilatational and shear waves, given by c21

=

1−ν 1 − 2ν

2G , ρ

c22 =

G ρ

Equation (8.23) constitutes a set of three linear homogeneous equations in the unknown component of the discontinuity vector λi . The existence of nontrivial solutions for these unknowns requires the determinant of their coefficients in these equations to vanish. The result is a cubic equation for c2 having three distinct roots (Craggs, 1961; Mandel, 1962). One of these roots is obtained by a mere inspection of (8.23), the corresponding solution being

8.2

Plastic Waves in Continuous Media

c2 = c22 ,

581

λj nj = 0,

λk sk = 0.

The wave front therefore advances at the speed of the elastic shear wave. The associated discontinuity in the velocity gradient vector is tangential to the wave front and is perpendicular to the plane containing the deviatoric stress vector and the normal to the wave front. The other two roots of c2 are most conveniently obtained by forming the scalar products of (8.23) with the vectors ni and si , in turn. Setting λ j n j = λn ,

si ni = sn ,

λk sk = ω,

and denoting the magnitude of the deviatoric stress vector by s, the two scalar equations involving the unknown quantities λn and ω are found as ⎫

⎪ ⎬ c2 − c21 λn + c22 α/k2 sn ω = 0,

⎭ − c21 − c22 sn λn + c2 − c22 1 − αs2 /k2 ω = 0.⎪

(8.24)

If τ denotes the magnitude of the shear stress transmitted across the wave front, then s2 = s2n + τ 2 , and the two equations in (8.24) will be simultaneously satisfied for nonzero values of λn and ω if

c2 − c22

c2 − c21 + c22 c2 − c22 αs2n /k2 + c2 − c21 ατ 2 /k2 = 0 (8.25)

This equation indicates that there are two real roots for c2 , one of which lies between 0 and c22 , and the other between c22 and c21 . The former corresponds to slow waves and the latter to fast waves, their speeds of propagation being denoted by cs and c f, respectively. The roots of the above quadratic may be expressed as ⎫

√ 1 2 ⎬ c2f ,c2s = c1 + c22 − c22 αs2 /k2 ± n , ⎪ 2

⎪ η = c21 − c22 − c22 αs2 /k2 + 4c22 c21 − c22 ατ 2 /k2 .⎭

(8.25a)

In the special case of an elastic material (α = 0), the wave speeds cf and cs reduce to c1 and c2 , respectively, the discontinuities in the velocity gradient in the two cases being normal and tangential, respectively, to the wave front. The related problem of acceleration waves in solids has been studied by Thomas (1961) and Hill (1962).

8.2.2 A Geometrical Representation For an elastic/plastic material, a useful geometrical interpretation of the wave speed equation (8.25) is obtained by writing it in the alternative form

582

8 Dynamic Plasticity

c21 c21 − c2

αs2n + k2

c22 c22 − c2

ατ 2 =1 k2

(8.26)

√ √ This relation indicates that if αsn and ατ are plotted as rectangular coordinates, the locus of constant wave speed is an ellipse when c = cs and is a hyperbola

when c = cf . The semifocal distance in each case is equal to k c21/c22 − 1, and consequently, the family of ellipses is orthogonal to the family of hyperboles, as shown in Fig. 8.8 Moreover, it followsfrom (8.25a) that the sum of the squares of cs and cf is constant along any circle α s2n + τ 2 = constant, having its center at the origin of the stress plane (Ting, 1977).

Fig. 8.8 Elastic/plastic wave speeds in a continuous medium and their dependence on the deviatoric normal and shear stresses acting across the wave front

For a given deviatoric stress tensor sjj and the direction of propagation ni , the normal and tangential components of the deviatoric stress vector acting across the wave front can be found from the relations sn = sij ni nj ,

s2 = sij sjk ni nk , τ 2 = s2 − s2n

The plastic wave speeds then follow from (8.25a), and the associated directions of the velocity gradient discontinuity are given by (8.23). When the deviatoric stress vector is tangential to the wave front (sn = 0, s = τ ), (8.25a) furnishes

8.2

Plastic Waves in Continuous Media

c2f = c21 ,

583

c2s = c22 1 − ατ 2 /k2 .

Since ω vanishes for the fast wave and λn vanishes for the flow wave in view of (8.24), the discontinuities in the velocity gradient in the two cases are directed along the normal and tangent to the wave front, respectively. When the deviatoric stress vector is normal to the wave front (τ = 0), the wave propagates along a principal axis of the stress, and we have c2s = c22 ,

c2f = c21 − c22 αs2n /k2

αs2n /k2 ≤ c21 /c22 − 1.

It follows from (8.23) and the first equation of (8.24) that the directions of the velocity gradient discontinuity again coincide with the normal and tangent to the wave front for the fast and slow waves, respectively. When the above inequality is reversed, the values of cf and cs and the associated discontinuity directions are simply interchanged. When the rectangular axes of reference are taken along the principal axes of the stress, the deviatoric state of stress at any point can be represented by the appropriate Mohr circle. If the deviatoric principal stresses are denoted by s1 , s2 , and s3 , with s2 assumed to lie between s1 and s3 , we have the identity s1 + s2 + s3 = 0,

s1 > s2 > s3 .

Evidently, s1 > 0 and s3 < 0, while s2 can be either positive or negative. For given values of s1 , s2 , and s3 , the stress point (sn , τ ) lies within a region bounded by three circles, the largest of which has the equation sn −

s1 + s2 2

2 + τ2 =

s1 − s2 2

2 .

(8.27)

The two principal components s1 and s3 appearing in this equation cannot be chosen arbitrarily, since they are required to satisfy the yield criterion which may be expressed as 3 (s1 + s3 )2 + (s1 − s3 )2 = 4 k2 This restriction makes the family of Mohr’s circles (8.27) bounded by an envelope, which is easily shown to be an ellipse (Ting, 1977) having the equation 3 s n 2 τ 2 + = 1. 4 k k

(8.28)

The broken curve in Fig. 8.8 represents this ellipse for a typical value of α. The focal points A and B of the ellipse defined by (8.26) with c = cs lie inside or outside the defined by (8.28) depending on whether 4α/3 is greater or smaller than 2 ellipse c1/c22 − 1 . The situation cf = cs = c2 can arise only when the former condition is satisfied.

584

8 Dynamic Plasticity

The smallest value of c2f , which corresponds to τ = 0, is equal to c22 when the focal points are inside the envelope, and to c21 − 43 α22 when they are outside the envelope. The smallest value of c2s , on the other hand, corresponds to the point of tangency of the largest possible Mohr circle with the appropriate ellipse of constant cs . It can be obtained as a function of s2 by solving (8.26) and (8.27) simultaneously for sn and τ , after setting c = cs ,s1 + s3 = −s2 , (s1 − s3 )2 = 4 k2 − 3s22 , and establishing the condition for the quadratics to have equal roots. When s2 = 0, the smallest value of c2s is equal to (1 − α) c22 , corresponding to sn = 0.

8.2.3 Plane Waves in Elastic/Plastic Solids The preceding results are directly applicable to the propagation of plane waves in which the wave front is a plane surface advancing in a uniformly prestressed elastic/plastic medium. When the state of stress is a pure shear in which the normal stress vanishes across the plane (sn = 0,τ = k), the normal and tangential√components of the velocity gradient discontinuity propagate with speeds c1 and c2 1 − α respectively, in the direction of the normal to the plane. The strength of the former remains unchanged, while that of the latter steadily decreases as the plastic √ strain increases. In the case of a uniaxial prestress normal to the plane (sn = 2 k/ 3,τ = 0), the discontinuities in the tangential and normal velocity gradients propagate with speeds

c2 and c21 − 4αc22/3, respectively. The latter wave speed depends on the value of α, which increases with increasing plastic strain. Consider now the rectilinear propagation of a plane wave in an isotropic elastic/plastic body, which is in the form of a thick plate whose lateral dimensions are infinitely large. The wave travels through the thickness of the plate along the x-axis, the kinematical restrictions being ε˙ y = ε˙ z = 0,

ε˙ y e = −˙εy p =

1 p ε˙ x 2

which follow from the relation σ˙ y = σ˙ z holding throughout the body. The elastic stress–strain relations therefore give the elastic and plastic parts of ε˙ x E˙εx e = σ˙ x − 2ν σ˙ y ,

E˙εx p = 2 (1 − ν) σ˙ y − ν σ˙ x

Combining these two relations, the total rate of extension in the x-direction is expressed as E˙εx = (1 − 2ν) σ˙ x + 2σ˙ y

(8.29)

The material is assumed to be rate sensitive, with the plastic part of the strain rate satisfying the quasi-linear constitutive equation

8.2

Plastic Waves in Continuous Media

585

E˙εxp = φ1 σ˙ x + φ2 σ˙ y + ψ, where φ 1 and φ 2 are functions of stress and strain, and ψ is a function of the ρ dynamic overstress. In view of the expression for ε˙ x given above, the preceding relation becomes (2ν + φ1 ) σ˙ x − 2 (1 − ν) − φ2x σ˙ y + ψ = 0 The elimination of σ˙ y between (8.29) and the preceding equation leads to the relationship between σ˙ x and ε˙ x in the form 2 (1 − ν) − φ2x E˙εy = (1 − 2ν) {[2 (1 + ν) + 2φ1 − φ2 ] σ˙ x + 2ψ}

(8.30)

The characteristics of this hyperbolic system are given by dx/dt = ±c, where c is the speed of propagation of the wave. Along the characteristics, we have σ˙ x =

∂σx ∂σx ∂υx ∂υx = ±c = ±ρc = ±ρc2 = ρc2 ε˙ x . ∂t ∂x ∂t ∂x

It should be noted that ∂σx/∂t = σ˙ x when geometry changes are disregarded. Since ε˙ x and σ˙ x are not uniquely determined along the characteristics, the preceding two relations involving ε˙ x and σ˙ x furnish the result c2 =

E , ρ (1 − 2ν) (1 + 2λ)

λ=

2ν + φ1 2 (1 − ν) − φ2

(8.31)

The range of values of φ 1 and φ 2 for which the wave speed is real must satisfy the conditions φ2 = 2 (1 − ν) , φ2 − 2φ1 < 2 (1 + ν) as well as those obtained by reversing these inequalities. It is easy to see that ν/(1 − ν) < λ < 1,

K/ρ < c < c1

where K is the bulk modulus for the material. When φ 1 = φ 2 = 0, there is no instantaneous plastic strain in the material response, and (8.31) gives c = c1– . When ψ = 0, implying the absence of strain rate effects, the constitutive law gives φ1 = −φ2 = E/H, and the plastic wave speed given √ by (8.31) coincides with the value of cf given by (8.25a) with τ = 0 and s = 2k/ 3, where k is the yield stress in shear. In the (φ 1 , φ 2 )-plane, λ (and hence c) remains constant along straight lines passing through the point [−2v,2 (1 − v)], where the wave speed is indeterminate. It may be noted that the relation c = c1 holdsall along the straight line vφ2 +(1 − v) φ1 = 0, not merely at φ1 = φ2 = 0. In the σ˙ x ,σ˙ y -plane, (8.30) represents a straight line for given values of φ1 ,φ2 , and ψ. The slope of this straight line is equal to λ, and it intersects the axis σ˙ y = 0 at a point that depends on φ 1 . No plastic flow is possible

586

8 Dynamic Plasticity

along the straight line (1 − ν) σ˙ y − ν σ˙ x = 0, which therefore represents elastic unloading. The propagation of plane waves excluding the strain rate effects, but including the variation of the bulk modulus with the hydrostatic pressure, has been considered by Morland (1959), who also studied the interaction of loading and unloading waves when a pressure pulse is applied on the free surface. The propagation of cylindrical waves in an infinite medium has been discussed by Cristescu (1967) under torsional loading and by Jansen et al. (1972) under radially symmetric loading. The propagation of spherically symmetric waves in an unbounded medium has been investigated by Hunter (1957) and Hopkins (1960).

8.3 Crumpling of Flat-Ended Projectiles One of the simplest methods of studying the effect of high strain rates on the dynamic yield strength of metals consists in firing flat-ended cylindrical projectiles against rigid targets. The axial stress at the impact end immediately attains the yield limit, and a plastic wave moves away from the target plate rendering the projectile partially plastic. An elastic wave front initiated at the same time moves ahead of the plastic wave front, the region between the two wave fronts being stressed to the yield point. Due to the reflections of the elastic wave front from the free end of the bar and the advancing plastic wave front, the rear part of the projectile rapidly decelerates and comes to rest within a distance equal to the difference between the initial and final lengths of the cylinder. The extent of the deformed and undeformed portions of the projectile after impact depends on its kinetic energy before impact, as well as on the dynamic yield strength of the material.

8.3.1 Taylor’s Theoretical Model Let U denote the velocity of normal impact of a cylindrical projectile having an initial length L and an initial cross-sectional area A0 . At a generic stage of the dynamic process, the overall length of the projectile is reduced to l due to the piling up of material over a length h, leaving a nonplastic rear part of length x as shown in Fig. 8.9(b). For simplicity, the material is assumed to have a constant dynamic yield stress Y, and the radial inertia is disregarded so that the stress distribution may be considered as uniform over any given cross section (Taylor, 1948b). Within a small time interval dt, a nonplastic element of length –dx and area A0 passes through the advancing plastic boundary to come to rest as a plastic element of length dh and area A. Neglecting elastic strains, the continuity equation may be written as A dh =–A0 dx, which is equivalent to Aυ = A0 (u + υ) ,

(8.32)

8.3

Crumpling of Flat-Ended Projectiles

587

Fig. 8.9 Deformation of a flat-ended projectile fired at a speed U against a flat rigid target

where u is the current velocity of the rear part of the projectile, and υ is the velocity of the plastic boundary, the velocity of the nonplastic part relative to the plastic boundary being u + υ, Thus υ=

dh , dt

u+υ =−

dx . dt

(8.33)

The momentum of the elemental volume –A0 dx changes from −ρA0 udx to zero during the time dt, where ρ is the density of the material. Since the net force acting on the element is of magnitude (A–A0 )Y, the equation of motion becomes ρA0 (u + υ)u = (A − A0 ) Y.

(8.34)

Equations (8.32) and (8.34) are sufficient to express ρu2 /Y and υ/u in terms of a variable e = 1 − (A0/A), the result being e2 ρu2 = , Y 1−e

υ 1−e = . u e

(8.35)

To obtain the expressions for x and h at any instant, we consider the equation of motion of the rear portion of the projectile, which moves as a rigid body with a velocity u under an opposing force equal to A0 Y. Since the change in momentum of this portion is ρA0 xdu during the time interval dt, the equation of motion is ρx

du = −Y. dt

(8.36)

Combining this equation with the second equation of (8.33), and using (8.35), we obtain the differential equation ρux dx = du Ye

or

dx x (2 − e) . = de 2 (1 − e)2

Let e0 be the value of e at the moment of impact when x = L. Using this initial condition, the last equation is readily integrated to give

588

8 Dynamic Plasticity

x 2 L

=

1 − e0 1−e

e − e0 exp . (1 − e0 ) (1 − e)

(8.37)

The quantity e0 depends on the impact velocity U according to the first equation of (8.35) with e = e0 and u = U. When the projectile is brought to rest, e = 0 and x = x∗ (say), the relationship between x∗ /L and ρU2 /Y being obtained as

x∗ L

eo , = (1 − e0 ) exp − 1 − e0

e20 ρU 2 , = Y 1 − eo

(8.38)

in view of (8.37) and (8.35). It may be noted that x∗ /L decreases and e0 increases as the parameter ρU2 /Y is increased. The shape of the deformed part of the projectile at any instant can be determined by the integration of the equation dh υ =− = − (1 − e) , dx u+υ which is obtained from (8.33) and (8.35). In view of the initial conditions h = 0 and e = e0 at x = L, the solution may be written as h x = (1 − e0 ) − (1 − e) + L L

eo

x

e

L

de.

(8.39)

The integral is evaluated numerically for any given value of e0 and selected val√ ues of e, using (8.37) for x/L. Each value of e furnishes a radius a = a0/ 1 − e of the deformed part corresponding to a distance h from the target plate, where a0 is the initial radius of the cylinder. The shapes of the projectile after impact, predicted by the present theory for e0 = 0.5, 0.7 and e = 0.8, are shown in Fig. 8.9(c)–(e) for the case where the diameter was initially 0.3 of the height. The calculated value of h∗ /L and l∗ /L, where h∗ is the final length of the deformed part and l∗ is the final overall length, is plotted against ρU 2/Y in Fig. 8.10, which includes some experimental points obtained by Whiffen (1948). To determine how the various physical quantities vary with time t, measured from the beginning of the impact, it is necessary to integrate the differential equation for de/dt, which is most conveniently obtained from (8.36) and the first equation of (8.35). Indeed, the time derivative of the latter equation gives

2−e ρ du de = . 3/2 Y dt 2 (1 − e) dt

√ Eliminating du/dt by means of (8.36), and substituting for Y/ρ obtained by setting u = U and e = e0 in the first equation of (8.35), we obtain the differential equation de 2U =− dt x

√ 1 − e0 (1 − e)3/2 . eo 2−e

8.3

Crumpling of Flat-Ended Projectiles

589

Fig. 8.10 Results of calculation for the longitudinal 8impact of a flat-ended projectile. The measures values of l∗ /L and h∗ /L are indicated by • and . respectively

Since e = e0 at the moment of impact t = 0, the time interval t can be found numerically from the relation

Ut eo =√ L (1 − eo )

eo e

x (2 − e) de 2L (1 − e)3/2

,

(8.40)

where x/L is given by (8.37) as a function of e and e0. The parameter Ut/L, computed by numerical integration for the cases e0 = 0.5 and 0.7, is plotted against h/L in Fig. 8.11, which indicates how the plastic boundary moves away from the target plate with the time interval. The same problem has been analyzed by Lee and Tupper (1954) on the basis of the elastic and plastic waves in the projectile, taking into account the strain hardening of the material. The Taylor anvil test has been used to establish the constitutive modeling of materials by Johnson and Holmquist (1988) and Nemat-Nasser et al. (1994).

590

8 Dynamic Plasticity

Fig. 8.11 Advancement of the plastic boundary during the longitudinal impact of a flat-ended projectile

8.3.2 An Alternative Analysis The preceding theory, developed by Taylor (1948b), has been found to be in reason able agreement with experiment for relatively low-impact velocities ρU 2/Y ≤ 0.5 . The shapes of the slugs fired at greater speeds generally have a concave profile over the plastically strained part, instead of a convex profile predicted by Taylor’s theory. To explain this mushrooming effect associated with the high-speed impact of projectiles, Hawkyard (1969) proposed an alternative method in which the rate of plastic work done on the projectile is equated to the total external energy supplied to it. The resultant equations defining the final geometry of the projectile are fairly simple, and the predicted profiles are found to be in closer agreement with experiment. The amount of plastic work dW, which is done on an element of length –dx in changing its cross-sectional area from A0 to A before it comes to rest, is equal to −A0 Y ln (A/A0 ) dx. Since the time taken for this change is dt, the rate of plastic work done is ˙ = A0 Y (u + υ) 1n (A/A0 ) W in view of (8.33). During this time interval, the kinetic energy of the element is reduced from 12 ρu2 (−A0 dx) to zero, while the work done by the external force becomes equal to −A0 Y (dx + dh). Hence, the rate at which the total energy is

8.3

Crumpling of Flat-Ended Projectiles

591

supplied to the system is ρu2 . E˙ = A0 Y u + (u + υ) 2Y Neglecting the loss of kinetic energy due to impact, we may equate the rate of ˙ to the rate of supply of energy E˙ to give plastic work W

ρu2 A (u + υ) 1n − A0 2Y

= u.

(8.41)

This equation replaces (8.34) obtained on the basis momentum equilibrium. ( of Solving (8.32) and (8.41), and setting e = 1 − A0 A as before, we obtain the relations ρu2 1 = 1n − e, 2Y 1−e

1−e υ = . u e

(8.42)

The equation of motion (8.36) for the undeformed part of the projectile is now combined with the time derivative of the first equation of (8.42) to give ρu de = dt Y

1−e e

1−e u du =− . dt e x

(8.43)

In view of the second equation of (8.42), (8.43) can be combined with (8.33) to give dx x = , de 1−e

dh = −x. de

Integrating these two equations in succession, and using the initial conditions x = L and h = 0 when e = e0 , we obtain the solution x 1 − e0 = , L 1−e

1−e h . = (1 − e0 ) 1n L 1 − e0

(8.44)

The values of x and h at the end of the impact, denoted by x∗ and h∗ , respectively, are given by (8.44) on setting e = 0. The impact velocity U is related to e0 by the equation 1 ρU 2 − eo, = 1n 2Y 1 − e0 which follows from (8.42). The calculated shape of the projectile after impact, corresponding to e0 = 0.8 ρU 2 /Y = 1.62 according to (8.44), is shown in Fig. 8.9(f). The profile is seen to be concave in form, resembling what is experimentally observed. The calculated values of h∗ /L and l∗ /L predicted by the above analysis

592

8 Dynamic Plasticity

are compared with those given by Taylor’s theory in Fig. 8.10. The influence of strain hardening of the material has also been examined by Hawkyard (1969). The variation of the strain parameter e with time can be determined by the integration of (8.43) after substitution for u/U given by (8.42), the final differential equation being U (1 − e)2 de =− dt L e (1 − e0 )

'

e + 1n (1 − e) . eo + 1n (1 − e0 )

In view of the initial condition e = e0 at t = 0, the solution may be written as Ut = (1 − e) L

eo

'

e

eo + 1n (1 − eo ) ede . e + 1n (1 − e) (1 − e)2

(8.45)

The integral can be evaluated numerically for any given value of e0 . The results corresponding to e0 = 0.5 and e0 = 0.7 are compared in Fig. 8.11 with those predicted by Taylor’s theory. The two solutions do not seem to differ substantially from one another, though the shape of the projectile is predicted more ( realistically by the analysis based on the energy principle, particularly for ρU 2 Y ≥ 0.5.

8.3.3 Estimation of the Dynamic Yield Stress The measurement of the final overall length and the position of the plastic boundary, after the impact, provides a convenient means of estimating the dynamic yield stress of the material. For practical purposes, a simple formula for the yield point Y can be developed on the assumption that the velocity of the undeformed part of the projectile ( relative to the advancing plastic boundary has a constant magnitude c. Then dx dt = −c in view of (8.33), and the equation of motion (8.36) gives Y du = dx ρcx

or

u=U−

L Y ln , cρ x

(8.46)

in view of the condition u = U when x = L. If the distance traveled by the rear of the projectile at any instant is denoted by du/dt = u (du/ds), then (8.36) gives u

cρ du Y Y =− =− exp − (u − U) ds ρx ρL Y

in view of (8.46). Integrating, and using the conditions u = U when s = 0, and u = 0 when s = L − l∗ , we obtain the solution ρc2 Y

1−

l∗ L

=

cρU cρU − 1 + exp − . Y Y

8.4

Dynamic Expansion of Spherical Cavities

593

The parameter cρU/Y is equal to ln (L/x∗ ) in view of (8.46) with u = 0 and x = x∗ . The elimination of c from the above relation therefore gives Y ln (L/x∗ ) − (1 − x∗ /L) = 2 . ρU 2 (1 − l∗ /L) ln (L/x∗ )

(8.47)

If the deceleration of the rear of the projectile is assumed uniform, the formula for Y/ρU 2 would become that given by Taylor. Equation (8.47) should be sufficiently accurate for the estimation of the dynamic yield stress Y for a given impact velocity U, using the measured values of x∗ /L and l∗ /L. The experimental results of Whiffen (1948) and Hawkyard (1969) indicate that the values of Y computed from (8.47) for different sets of values of U, x∗ /L and l∗ /L are approximately the same for a given material. It is possible to define a mean strain rate as the ratio of the overall longitudinal plastic strain to the duration of the impact. If the rear portion of the projectile is assumed to move with a constant deceleration, the duration of impact is equal to 2(L — l∗ )/U by the simple kinematics of the rigid-body motion. Since the initial and final lengths of the plastically deformed part of the projectile are L – x∗ and l∗ – x∗ , respectively, the mean strain rate λ˙ may be written as λ˙ =

U L − x∗ ln . 2 (L − l∗ ) l∗ − x ∗

(8.48)

The value of λ˙ in each particular case may therefore be obtained from the direct measurement of the final undeformed length and the final overall length of the projectile. There seems to be very little variation of the duration of impact with the impact velocity, although there are fairly large variations in the other physical quantities. The buckling that would occur in a sufficiently long projectile due to the impact has been examined by Abrahamson and Goodier (1966) and Jones (1989). The related problem of dynamic plastic buckling of columns has been treated by Lee (1981).

8.4 Dynamic Expansion of Spherical Cavities The formation of spherically symmetric cavities in an infinitely extended medium under quasi-static conditions, originally discussed by Bishop et al. (1945), has been presented elsewhere (Chakrabarty, 2006). In this section, the problem of spherical cavity formation under dynamic conditions will be discussed for a material which obeys an arbitrary regular yield condition. The pressure applied at the cavity surface is supposed to be a given function of the current cavity radius. In the practical situation of cavity formation caused by high explosives, some kind of return motion following the expansion phase would be expected. Since the expansion process involves large plastic strains, it would be reasonable to begin by neglecting the elastic compressibility not only in the plastic region but also in the elastic region

594

8 Dynamic Plasticity

(Hopkins, 1960). The compressibility of the material will be allowed for in a subsequent treatment of the cavity expansion process.

8.4.1 Purely Elastic Deformation A spherical cavity of initial radius a0 is expanded into an infinitely extended medium which is assumed to be completely incompressible. The internal pressure p = p(t) is supposed prescribed at each instant of the expansion. Since the density ρ of the material remains constant by hypothesis, the equation for the conservation of mass may be written as ∂ 2 r υ = 0, ∂r where ν is the radial velocity of a typical particle currently situated at a radius r. This equation is immediately integrated to give

υ = a2 /r2 a˙ , a˙ = da/dt.

(8.49)

The particle velocity is therefore everywhere determined in terms of the velocity of the cavity surface. The associated components of the strain rate are 2 2a ∂υ ε˙ r = =− a˙ , ∂r r3

υ ε˙ θ = ε˙ φ = = r

a2 r2

a˙ .

(8.50)

For an isotropic material, the result ε˙ θ = ε˙ φ implies σθ = σφ throughout the medium, and the equation of motion in terms of the stresses and velocity becomes ∂σr 2 + (σr − σθ ) = ρ ∂r r

∂υ ∂υ +υ ∂t ∂r

.

(8.51)

The convective term represented by the second term in parenthesis will be retained even when the deformation is small, as it is not necessarily negligible under conditions of high-speed cavity formation. In the case of purely elastic deformation of the entire medium, the stress difference σθ − σr in (8.51) must be expressed as a function of r using the stress–strain relations before the integration can be carried out. Since the displacement is small in the elastic range, we may write υ ≈ ∂u/∂t, where u is the radial displacement, and (8.49) then integrates to

u = a3 − a30 /3r2 ≈ a2 /r2 ua

(8.52)

in view of the initial condition u = 0 when a = a0 , the quantity ua being the displacement a–a0 of the cavity surface. The strain components corresponding to (8.52) are

8.4

Dynamic Expansion of Spherical Cavities

εr = −

2a2 r3

595

ua ,

εθ = ε φ =

a2 r3

ua .

These results may be obtained by the direct integration of (8.50) using the same order of approximation. By the elastic stress–strain relations for an incompressible material (ν = 0.5), we have 2 2 a σθ − σr = E (εθ − εr ) = 2E 3 ua , 3 r

(8.53)

where E is Young’s modulus of the material. Substituting this into (8.51), and using (8.49), the equation of motion is reduced to 2 2 a ∂σr 2a a3 a = 4E 4 ua + ρ a ¨ + 1 − a2 . ∂r r r2 r2 r3 Integrating, and using the condition that the stress vanishes at infinity, the solution for the radial stress is obtained in the form 2 a a3 4 ρa a¨a + 2 − 3 a˙ 2 , σr = − E 3 u a − 3 r r 2r

(8.54)

which involves the velocity a˙ and acceleration ä of the cavity surface. The applied pressure p at the cavity surface is given by the boundary condition σr = −p at r = a, the result being p=

a0 3 4 E 1− + ρ a¨a + a˙ 2 . 3 a 2

(8.55)

If the initial pressure is denoted by p0 , the initial acceleration is equal to p0 /ρa0 , the initial velocity of the cavity being assumed to vanish. Multiplying both sides of (8.55) by a2 da, and using the identity 2a¨a + 3˙a2 a2 da = d a3 a2 , the resulting expressions can be integrated between the limits a0 and a to obtain

a

ao

pa2 da =

2 a0 2 1 2 3 + ρ a˙ a . E 1− 3 a 2

(8.56)

with a minor approximation. The evaluation of the integral is straightforward when p(a) is prescribed. A further quadrature of (8.56) after multiplying it by da furnishes the relationship between a˙ and a, permitting the cavity radius a(t) to be determined by integration. Plastic yielding begins when the yield condition σθ − σr = Y is first satisfied during the elastic expansion. Since σθ − σr has its greatest value at r = a in view of (8.53), yielding first occurs at the cavity surface when a = a1 and a˙ = a˙ 1 , such that

596

8 Dynamic Plasticity

a1 ≈ ao

Y , 1+ 2E

ρ a˙ 21

Y Y ≈ po − . E 3

The last result follows from (8.56) with the approximation p ≈ p0 during the elastic loading. It may be noted that the radial displacement at the cavity surface at the onset of yielding is quite independent of inertial effects. Although the displacement at this stage is quite small, the corresponding expansion velocity may be high.

8.4.2 Large Elastic/Plastic Expansion Subsequent to the commencement of yielding, an elastic/plastic boundary defined by r = c spreads outward from the cavity surface. The material outside this radius is elastic, while that within this radius is rendered plastic with the stresses satisfying the yield criterion σθ − σr = Y,

a ≤ r ≤ c.

The material is assumed to be nonhardening, so that the dynamic yield stress Y has a constant value throughout the plastic region. The velocity is still given by (8.49) everywhere in the medium, but the displacement in the elastic region is now given by ∂u = ∂t

c2 r2

υc

or

u=

c2 r2

r ≥ c,

uc ,

where uc is the displacement of the particles that are currently at the elastic/plastic boundary. The stress difference in the elastic region is σθ − σr = 2E

c2 r3

uc = Y

c3 r3

r ≥ c,

,

(8.57)

where the last expression follows from the fact that material at the elastic/plastic boundary is just at the point of yielding. Denoting by c0 the initial radius to the particle which is currently at radius c, the condition of incompressibility may be written as a3 − a30 = c3 − c30 ≈ 3c2 uc = (3Y/2E) c3 .

(8.58)

This equation relates the current cavity radius to the radius of the elastic/plastic boundary. The strains in the plastic region are given by

∂r0 εr = − ln ∂r

,

r εθ = εφ = ln r0

,

8.4

Dynamic Expansion of Spherical Cavities

597

where r0 is the initial radius to a typical particle, the relationship between r and r0 being obtained by integrating the equation dr/dt = ν along the path of the particle. The stress distribution in the elastic and plastic regions must be determined by integrating the equation of motion (8.51). Using (8.49) to eliminate ν, this equation is reduced to ∂σr 2 + (σr − σθ ) = ρ ∂r r

a2 r2

a¨ +

2a a3 1 − 3 a˙ 2 , r r

(8.59)

where σθ − σr is given by (8.53) in the elastic region (r ≥ c), and by (8.57) in the plastic region (r ≤ c). The integration in the elastic region is based on the condition that σ r is finite at r = ∞, while that in the plastic region involves the boundary condition σ r = –p at r = a. The result is easily shown to be a a3 −ρ r ≥ c, a¨a + 2 − 3 a˙ 2 , r 2r

r a a a3 3 2 2 +ρ 1− a¨a − 2 − 3 a˙ + a˙ , σr = −p + 2Y ln a r r 2 2r

2 σr = − Y 3

c3 r3

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ r ≤ c.⎪ ⎭ (8.60)

The fact that the stress in the plastic region contains a logarithmically divergent term indicates that the plastic region is always finite in extent. The condition of continuity of σ r across r = c furnishes

c 2 3 2 p = Y + 2Y ln + ρ a¨a + a˙ 3 a 2 in view of (8.60). Eliminating c/a from the above equation by means of (8.58), the applied pressure can be finally expressed as a30 2E 2 2 3 + Y ln 1 − 3 + ρ a¨a + a˙ 2 . p = Y 1 + ln 3 3Y 3 2 a

(8.61)

The first two terms on the right-hand side of (8.61) correspond to the quasi-static result for an ideally plastic incompressible material. Multiplying (8.61) by a2 da, integrating between the limits a1 and a, and using (8.56) corresponding to a = a1 , we get

a a0

2 2E 3Y 3 3 pa2 da = Y a0 + a − a30 1 + ln 9 4E 3Y a30 a3 1 3 3 + a ln 1 − 3 − a0 ln 3 − 1 + ρa3 a˙ 2 . 2 a a0

(8.62)

598

8 Dynamic Plasticity

This equation expresses the energy balance from the commencement of cavity expansion. The process comes to a stop when a = a2 and a = 0, the maximum cavity radius a2 being given by a a0

2 pa2 da = Y 9

2E 3Y 3 3 3 3 1 + ln + a2 ln 1 − a + a − a0 4E 0 3Y

a30

a3

− a30 ln

a32 a30

−1

(8.63) p

p

Since ε˙ r = −2˙εθ , and ε˙ θe − ε˙ re = 0 in the plastic region, the rate of plastic work done per unit volume is equal to (2Y/3) (˙εθ − ε˙ r ),which is nonnegative for a˙ ≥ 0 in view of (8.50). The preceding analysis is therefore valid at all stages of the expansion. When the expansion is so large that a0/a is small compared to unity, 3 the ratio c3/a is approximately equal to 2E/3Y, which is independent of the cavity radius. In this case, (8.63) reduces approximately to U=

8 2E π Ya32 1 + ln , 9 3Y

(8.64)

where U denotes the total internal energy of the explosion products utilized in the cavity formation. Equation (8.64) is a particularly simple result that furnishes the maximum cavity radius a2 from known values of U, Y, and E, when the cavity expansion is sufficiently large. The problem of fragmentation of a spherical shell due to an internal explosion has been discussed by Al-Hassani and Johnson (1969). It may be noted that (8.63) involves the total work done by the applied pressure and is quite independent of the rate of doing this work. Let the law of expansion of the volume of the explosion products be assumed isentropic, according to the relation p = p0 (a0 /a)3γ , where p0 is the initial pressure and γ > 1 is the index whose value is assumed constant over the considered range of expansion. The substitution of the above expression into the left-hand side of (8.63) results in

a2

a0

3(γ −1) p0 a30 a0 pa da = 1− . 3 (γ − 1) a2 2

(8.65)

Using (8.63) and (8.65), the ratio a2/a0 can be calculated for any given values of p0/Y, E/Y and γ . In actual practice, γ generally decreases during the expansion, and consequently a suitable mean value of γ should be used in (8.65). In the case of TNT, it is a good approximation to assume γ ≈ 3 for moderate expansions (Johnson, 1972). The variation of a2/a0 with p0/Y is shown graphically in Fig. 8.12 for E/Y = 450 and different values of γ . It is important to note that the final radius of the cavity achieved under dynamic conditions is the same as that under quasi-static conditions

8.4

Dynamic Expansion of Spherical Cavities

599

Fig. 8.12 Dependence of the maximum cavity radius on the intensity of initial pressure generated by an explosion

for the same expenditure of energy. An analysis for the cavity formation by deep underground explosion has been presented by Chadwick et al. (1964). Consider now the nature of the return motion following the end of the first expansion phase. Due to the isochronous nature of the process, the entire plastic region instantaneously unloads, and consequently, the initial part of the first contraction phase is purely elastic. This part of the return motion is terminated when plastic yielding again occurs at the cavity surface r = a, the yield condition then being σr − σθ = Y in the absence of the Bauschinger effect. Since σθ − σr = Y in the plastic region at the instant of unloading, and σθ − σr = 2Eu0 , at r = a due to the superposed elastic stresses during the unloading, plastic flow at the cavity surface requires ua = −Y/E, giving a cavity radius a3 = (1 − Y/E) a2 at the end of the elastic unloading. In the subsequent part of the first contraction phase, a new elastic/plastic boundary spreads outward creating a plastic zone where the yield condition is σr − σθ = Y. Since the return motion is very limited in extent, a detailed analysis of the contraction phase is of minor practical interest.

600

8 Dynamic Plasticity

The internal energy of the explosion products is mostly dissipated into plastic work, and the available energy at the end of the first expansion phase is relatively small. The details of the analysis for the subsequent motion, consisting of alternate contraction and expansion phases, becomes progressively more complicated. Eventually, a final shakedown state of purely elastic oscillating motion results. The initial motion is so highly damped by the plastic deformation produced in the first expansion phase that the subsequent motion is reduced to one of small oscillations with minor significance. It should be noted that in the hypothetical case of cavity formation from zero radius, only the first expansion phase is geometrically similar, while the subsequent phases are not.

8.4.3 Influence of Elastic Compressibility To simplify the analysis for the dynamic cavity expansion, when the compressibility of the elastic/plastic material is duly taken into account, we assume a constant speed U of the cavity surface which is expanded from zero radius. The plastic region at any instant is defined by a ≤ r ≤ c, where b = Ut, and the elastic region is defined by c ≤ r ≤ b, where b = Vt, with V denoting the speed of the elastic dilatational wave in the material. In terms of the radial displacement u, the stress–strain relations in the elastic region may be written as E

∂u = σr − 2νσθ, ∂r

E

u = (1 − ν) σθ − νσr, r

in view of the symmetry condition σθ = σφ , holding everywhere in the material. From the above relations, the elastic stress components are expressed as 3K ∂u u + 2ν , σr = (1 − ν) 1+ν ∂r r

3K σθ = 1+ν

∂u u + ν , ∂r r

(8.66)

where K denotes the elastic bulk modulus, equal to E/[3 (1 − 2ν)]. Substituting from (8.66) into the equation of motion (8.51), where υ is replaced by ∂u/∂t, and neglecting the convective term υ (∂u/∂r), we obtain the wave equation ∂ 2 u 2 ∂u 2u 1 ∂ 2u − + = , r ∂r ∂r2 r2 V 2 ∂t2 due to Forrestal and Luk (1988). The quantity V denotes the speed of propagation of the elastic wave front of radius b and is given by V = 2

1−ν 1 − 2ν

2G = ρ

1−ν 1+ν

3K ρ

8.4

Dynamic Expansion of Spherical Cavities

601

where ρ denotes the initial density of the material. Since the configuration maintains geometrical similarity during the expansion process, the dimensionless stresses σr/Y and σθ/Y, and the dimensionless displacement, u/c must be functions of the single variable r/c. Setting r ξ= , c

' 1+ν ρ c˙ u¯ = = c , V 1 − ν 3K

and using the fact that the first derivatives of the displacement are du¯ ∂u = , ∂r dξ

∂u du¯ = c˙ u¯ − ξ , ∂t dξ

(8.67)

where c˙ is a constant, the wave equation is transformed into the ordinary differential equation

d2 u¯ 2 du¯ 2¯u 1 − α2ξ 2 + − 2 = 0. dξ 2 ξ dξ ξ

(8.68)

The integration of (8.68) is facilitated by the substitution u¯ = ξ ψ, where ψ is a function of ξ , resulting in

d2 ψ 2 2 dψ + 2 2 − α ξ = 0. ξ 1 − α2ξ 2 dξ dξ 2 This is a first-order differential equation in dψ/dξ , which is easily integrated under the boundary condition dψ/dξ = 0 at the wave front r = b or ξ = 1/α, sin ce u¯ and du¯ /dξ both vanish at this boundary. A second integration then furnishes the quantity ψ in view of the boundary condition ψ = 0 at ξ = 1/α. Consequently, D 1 − α2ξ 2 dψ , =− dξ ξ4

ψ=

D (1 − αξ )2 (1 + 2αξ ) , 3ξ 3

where D is a constant of integration. These expressions yield the function u¯ and its derivative as 2D 1 − α 3 ξ 3 du¯ D (1 − αξ )2 (1 + 2αξ ) , . (8.69) =− u¯ = 3ξ 2 dξ 3ξ 3 The constant D is now obtained from the condition that the material at the elastic/plastic boundary r = c is at the point of yielding. Substituting from (8.69) into (8.66) with u = c¯u, and setting σθ − σr = Y at ξ = 1, we get

D = (Y/E) (1 + ν) / 1 − α 2

602

8 Dynamic Plasticity

The stress distribution in the elastic region (1 ≤ ξ ≤ 1/α) therefore becomes ⎫ 2 (1 − αξ ) (1 − 2ν) (1 + αξ ) + (1 + ν) α 2 ξ 2 ⎪ σr ⎪ ,⎪ =− ⎪ ⎬ 2 3 Y 3 (1 − 2ν) 1 + α ξ (1 − αξ ) (1 − 2ν) (1 + αξ ) − 2 (1 + ν) α 2 ξ 2 ⎪ σθ ⎪ ⎪ , ⎪ = ⎭ Y 3 (1 − 2ν) 1 + α 2 ξ 3

(8.70)

It may be noted that both the stress components vanish at ξ = 1/α as expected. The velocity distribution in the elastic region (1 ≤ ξ ≤ 1/α), by the second equation of (8.67), is given Y (1 + ν) 1 − α 2 ξ 2 dψ υ = −ξ 2 = c dξ E 1 − α2 ξ 2

(8.71)

The elastic/plastic interface velocity c˙ , which is c/a times the cavity velocity U, must be determined by considering the plastic region. In an incompressible material, the elastic wave front does not exist (α = 0), and the elastic region is then defined by ξ ≥ 1. The effect of compressibility on the dynamic expansion was first considered by Hunter and Crozier (1968). The dynamic expansion of an elastic/plastic spherical shell under blast loading has been examined by Baker (1960). Since the change in density is only of the elastic order, a constant value of ρ will be assumed in the equation of motion (8.51). The other equation necessary for the analysis in the plastic region is the compressibility equation 2υ ∂υ + = ∂c r

∂ ∂ +υ ∂t ∂r

σr + 2σθ 3K

.

(8.72)

The left-hand side of this equation is equal to −ρ/ρ, ˙ where the dot denotes the rate of change following the particle. In view of the yield criterion σθ − σr = Y, the two governing equations in the plastic region become ⎫ ∂υ ∂σr 2Y ∂υ ⎪ − =ρ +υ ,⎪ ∂r r ∂t ∂r ⎬ a≤r≤c ⎪ ∂υ 2υ 1 ∂σr ∂σr ⎪ ⎭ + = +υ . ∂R r K ∂t ∂r Using the similarity transformation as before with the help of the dimensionless quantities r ξ= , c

σr s= , Y

υ w= , c

ρ β=c , K

and noting the fact that c (∂υ/∂t) = −c2 ξ (dw/dξ ), the preceding equations can be expressed as

8.4

Dynamic Expansion of Spherical Cavities

dw ⎫ 2 K 2 2 ds ⎪ − = β β −ξ ,⎪ ⎬ dξ ξ Y dξ dw 2w Y ds ⎪ ⎭ + = (w − ξ ) , ⎪ dξ ξ K dξ

603

(8.73)

It has been shown by Forrestal and Luk (1988) that the velocity distribution can be estimated with insignificant error by omitting the convective terms on the righthand side of (8.73). Introducing this approximation, and eliminating s between the two equations of (8.73), we obtain the differential equation

dw 2w 2Y + =− . 1 − β2ξ 2 dξ ξ K

(8.74)

Integrating, and using the boundary condition w = (1 + ν) Y/E at ξ = 1, in view of (8.71), the solution for the dimensionless velocity is obtained as w=

Y 1 1 − β 2 ξ 2 1 + μβ 2 1 (1 + βξ ) (1 − β) , (8.75) − + + ln Kβ 2 ξ ξ2 1 + β2 2β (1 − βξ ) (1 + β)

where μ = (1 + ν)/[3 (1 − 2ν)]. The elimination of ξ (dw/dξ ) from the first equation of (8.73) with the help of (8.74) leads to the differential equation Kβ 2 ds = dξ Y

2w dw 2 . + w + 2 2 dξ 1−β ξ ξ 1 − β 2ξ 2

On substituting for w, in the second term of the parenthesis, and integrating under the boundary condition s = s0 and w = w0 at ξ = 1, the solution is found as Kβ 2 2 s = s0 + w − w20 , 2Y 1 − β2ξ 2 2 1 + μβ 2 (1 − ξ ) 1 (1 + βξ ) (1 − β) + ln ln . + − βξ (1 − βξ ) (1 + β) 1 − β2 ξ 1 − β2 ξ 2 (8.76) The quantities s0 and w0 , obtained by setting ξ = 1 in (8.70) and (8.76), respectively, may be written as 2 Y (1 + ν) μY (1 + ν) α 2 , w0 = s0 = − = . 1+ 3 (1 − 2ν) (1 + α) 3 K (1 − 2ν) K Since the boundary condition ν = U at the cavity surface r = a is equivalent to w = a/c at ξ = a/c in view of the relation c˙ = (c/a) U, the ratio c/a is obtained as the solution of the transcendental equation c a

a2 + μκ 2 c2 a2 − λ2 c2 /a2

+

1 + ρU 2 /Y 1 (a − λc) (1 + λ) . ln = 2λ 1 − λ2 (a + λc) (1 − λ)

(8.77)

604

8 Dynamic Plasticity

√ where λ = U ρ/K. This equation can be solved for λc/a with any given value of λ, ν and Y/K, and the ratio c/a then follows as the end result. The cavity pressure p is finally obtained by setting s = −p/Y and ξ = a/c in (8.76), the result being 1 ρU 2 2 λ2 c2 p (1 + ν) μλ2 c2 − = − + ν) 1+ (1 Y 3 2 Y (1 − ν) (1 + α) α 2 3a2

c a2 + μλ2 c2 1 − λ2 c2 (a − λc) (1 + λ) . +2 + ln + ln −1 a a2 − λ2 c2 a2 − λ2 c2 (a + λc) (1 − λ) (8.78) Although this expression is more complicated than the corresponding formula for an incompressible material, the evaluation of the cavity pressure from (8.78) is straightforward, once c/a has been computed from (8.77) by a trial-and-error procedure. The dynamic expansion of a cylindrical cavity in a compressible elastic/plastic medium has been treated by Luk and Amos (1991). The results for the cavity expansion from zero radius with a constant velocity U in an incompressible elastic/plastic material may be directly obtained by setting a0 = 0, a˙ = U, and a¨ = 0 in the more general solution given earlier, the cavity pressure in this case being expressible in the form 3 c 2 9ρU 2 c3 p = 1 + ln 3 + , Y 3 4E a a3

c3 2E . = 3Y a3

(8.79)

In view of (8.61) and (8.58), the relationship between the interface velocity c˙ = (c/a) U and the cavity velocity U for a compressible elastic/plastic material is compared with that for the incompressible material in Fig. 8.13(a), assuming ν = 13 and Y/K = 0.00435. The cavity pressures in the two cases are compared with one another in Fig. 8.13(b) for identical material properties. The cavity pressure obtained by replacing in (8.79) the ratio c3 /a3 by its quasi-static value E/[3 (1 − ν) Y] for a compressible material, almost coincides with the exact numerical solution represented by the broken curve. A numerical solution for a compressible work-hardening material has been given by Luk et al. (1991) assuming a power law of hardening.

8.5 Mechanics of Projectile Penetration The problem considered here is that of penetration into sufficiently thick targets caused by the motion of rigid projectiles striking at normal incidence with a velocity v0 . The depth of penetration that occurs before the projectile comes to rest depends on the mechanical properties of the target material as well as on the geometry of the projectile. The projectile produces a cylindrical tunnel in the target, about the size of the shank diameter, and a thin layer of material between the target and the nose of the projectile is melted during the penetration. Consequently, the resistance to penetration due to friction is usually small compared to that due the normal pressure

8.5

Mechanics of Projectile Penetration

605

Fig. 8.13 Expansion of a spherical cavity in an elastic/plastic medium. (a) Interface velocity against cavity velocity and (b) cavity pressure against cavity velocity

exerted on the surface of the projectile. The problem has been treated by Backman and Goldsmith (1978) for an incompressible target material using the results for the expansion of spherical cavities in an elastic/plastic medium. The elastic compressibility of the material has been taken into account by Forrestal et al. (1988), who developed empirical formulas for the penetration mechanics based on spherical and cylindrical cavity expansion processes, and performed terminal-ballistic experiments to support their theory.

8.5.1 A Simple Theoretical Model We begin by considering an ideally plastic target material, and an ogival shape of the projectile nose defined by a pair of circular arcs meeting the shank tangentially as shown in Fig. 8.14(b). The radius of each circular arc is denoted by na, where a is the radius of the shank and n is a constant factor. A generic point on the nose surface is subjected to a normal pressure p and a tangential traction μp, where μ is the coefficient of friction between the nose and the surrounding material. The resultant axial thrust acting on the ogival nose in the positive z-direction is given by the expression P = 2π an

π/2 θ0

(cos θ + μ sin θ ) prdθ ,

606

8 Dynamic Plasticity

where θ denotes the angle of inclination of the normal to the surface with the axis of symmetry, and θ = θ 0 corresponds to the apex O. From simple geometry, the coordinates of a generic point on the nose are r = a [n sin θ − (n − 1)] ,

z = 1 − na cos θ .

Fig. 8.14 Various shapes of projectile. (a) Spherical nose projectile, (b) ogival nose projectile, and (c) conical nose projectile

The first of these relations immediately gives sinθ0 = (n − 1)/n, and the preceding expression for the axial force becomes P = 2π a n

2 2

π/2

θ0

(cos θ + sin θ ) (sin θ − sin θ0 ) pdθ ,

(8.80)

where p is an unknown function of θ having its greatest value at the apex θ = θ 0 . The height of the ogival nose is √ l = na cos θ0 = a 2n − 1. The hemispherical nose shown in Fig. 8.14(a) is the limiting case of an ogival nose, and corresponds to n = 1 giving θ 0 = 0. The conical nose shown in Fig. 8.14(c) cannot be obtained by a limiting process, since the nose in this case meets the shank with a discontinuous slope. To obtain an approximate solution to the penetration problem, it is assumed that the normal pressure p acting on the nose is equal to the internal pressure necessary to expand a spherical cavity from zero radius in the same medium with a radial

8.5

Mechanics of Projectile Penetration

607

velocity equal to the local particle velocity in the direction normal to the nose surface. The normal pressure distribution on the nose penetrating into a compressible elastic/plastic target is therefore obtained from (8.79) on setting U = υ cos θ , where ν is the current projectile velocity, and replacing the quantity c3 /a3 by its quasi-static value, the result being expressed in the dimensionless form p E 2 3ρυ 2 cos2 θ = , 1 + ln + Y 3 3 (1 − υ) Y 4 (1 − ν) Y

(8.81)

where ρ is the density of the target material. Equation (8.81) implies that the pressure acting at the base θ = π/2 is equal to the quasi-static expansion pressure. Substituting (8.81) into (8.80) and integrating, the penetration force is obtained as 2 ρυ P = π a2 Y A + B , Y

(8.82)

where A and B are dimensionless constants, given in terms of the mechanical properties of the target material, the geometry of the projectile nose and the friction coefficient as ⎫ l E 2 ⎪ 2 π − θ0 − (n − 1) 1 + ln 1+μ n ,⎪ A= ⎪ ⎬ 3 3 (1 − ν) Y 2 a .

⎪ l 3 q 4n − 1 μ 3 2 π l2 ⎪ ⎪ − − l) + B= n − θ + (n ⎭ 0 1−ν 8n2 8 2 2 a 2 n2 a2 (8.82a) The parameters θ0 and l/a in (8.82a) are defined by the value of n. When the frictional effect is disregarded (μ = 0), the constant A reduces to the quasi-static value of p/Y for the spherical cavity expansion process. The variation of the projectile velocity ν with time t and the depth of penetration h can be determined by considering the equation of rigid-body motion of the projectile. Since the rate of change of momentum is m(dυ/dt), where m is the mass of the projectile, the equation of motion is P = −m

dυ dυ = −mυ . dt dh

Substituting P from (8.82), and integrating under the initial condition υ = υ 0 when t = 0 and h = 0, we obtain % $ % ⎫ ⎪ ⎪ Bρ Bρ ⎪ t= tan−1 υ0 − tan−1 υ ,⎪ ⎪ ⎬ AY aY ⎪ ⎪ A + B ρυ02 /Y πρa2 P0 1 1 ⎪ ⎪ ln ln h= , = ⎪ ⎭ m 2B 2B P A + B ρυ 2 /Y

π a2 Y m

'

Y Bρ

$

(8.83)

608

8 Dynamic Plasticity

where P0 denotes the initial value of the force. It follows that the force decreases exponentially with the increase in depth of penetration. The projectile comes to rest when υ = 0, the corresponding depth of penetration h∗ being given by

πρa2 m

2 Bρυ 1 0 ln 1 + h∗ = , 2B AY

(8.84)

where A and B are obtained from (8.82a) for any given n. When the nose is hemispherical (n = 1), these expressions are simplified to A=

2 E μπ 1+ , 1 + ln 3 2 3 (1 − ν) Y

B=

3 (1 + μπ/4) . 8 (1 − ν)

The total time of penetration t∗ is obtained from the first relation of (8.83) by setting υ = 0, and using the appropriate value of n to compute the constants A and B. When the nose of the projectile is a circular cone of vertex angle 2α, the normal component of velocity at any instant has a constant value equal to νsin α. The normal pressure p, which is also a constant, is given by (8.81) with θ = π/2 − α, and the axial force acting on the conical surface is P = π a2 p (1 + μ cot α) for a coefficient of friction μ. The substitution of (8.81) with θ = π/2 − α into the above relation results in (8.82), where the constants are now given by ⎫ 2 E ⎪ A = (1 + μ cot α) 1 + ln ,⎪ ⎬ 3 3 (1 − ν) Y . ⎪ 3 E ⎭ B = (1 + μ cot α) ln tan2 α. ⎪ 4 3 (1 − ν) Y

(8.85)

The variations of the projectile velocity and depth of penetration with time are defined by (8.83) and (8.85). The final penetration is still given by (8.84) as a function of the striking velocity v0 , provided A and B are determined from (8.85). In Fig. 8.15, the theoretical predictions for the ogival nose (n = 6) and conical nose (α = 18.4◦ ) projectiles are compared with some available experimental results (Forrestal et al., 1988). The mechanical properties of the target material, which is prestrained aluminum, consist of v = 13 , Y = 400 Mpa, E = 68.9 Gpa, and ρ = 2.71 103 kg/m3 , the projectiles being made of managing steel with a = 15.2 mm, l = 3a, and m = 0.024 kg. The theoretical calculations are based on the assumption that the coefficient of friction μ would lie between 0.02 and 0.1 in the dynamic penetration. The agreement between theory and experiment is seen to be reasonably good, considering the uncertainty that exists in selecting the appropriate value of μ.

8.5

Mechanics of Projectile Penetration

609

Fig. 8.15 Depth of penetration of rigid projectiles into thick metallic targets as a function of the striking velocity. (a) Ogival nose projectile (n = 6) and (b) conical nose projectile (α = 18.4◦ )

The simplest way of taking into account the work-hardening property of the target material is to assume that the internal pressure in the dynamic cavity expansion is increased by the same amount as that for the quasi-static expansion in an incompressible elastic/plastic medium. If the stress–strain curve in the plastic range is approximated by a straight line of slope T, then (8.81), for the normal pressure on the projectile nose, is modified to E 2 π 2T 3ρυ 2 cos2 θ p = 1 + ln + + Y 3 3 (1 − ν) Y 9Y 4 (1 − ν) Y

(8.86)

in view of a known result for the quasi-static expansion of spherical cavities (Chakrabarty, 2006). A similar expression can be written for a nonlinear hardening. Empirical formulas based on a numerical solution to the spherical cavity expansion process, using a power law of hardening, have been given by Forrestal et al. (1991), who obtained good agreement of their theoretical predictions with experimental results. The high-speed impact and penetration of long rods has been investigated by Tate (1969) and Rosenberg and Dekel (1994), while the associated ricochet problem has been examined by Johnson et al. (1982). Explicit formulas for the penetration dynamics of rigid projectiles into thick plates have been presented by Chen and Li (2003).

610

8 Dynamic Plasticity

8.5.2 The Influence of Cavitation At sufficiently high velocities of penetration, the target material sometimes flows away due to its own inertia to produce a hole which is larger in diameter than that of the projectile, Fig. 8.16. This phenomenon, known as cavitation, has the effect of enhancing the resistance to penetration, since a part of the kinetic energy is absorbed by the plastic deformation involved in the enlargement of the hole. The problem of cavitation in the penetration process has been discussed by Hill (1980), who assumed the normal pressure distribution over the projectile nose, defined by a convex function r = r(z), in the form p = q + kρυ 2

d dr r , dz dz

(8.87)

where q is the pressure corresponding to the quasi-static process, and k is a dimensionless constant that depends on the shape of the nose. Cavitation would occur over the region where the value of p given by (8.87) is found to be negative. The resistive force acting on the projectile in the absence of cavitation is P = 2π

a

prdr = 2π

0

0

l

dr dz = π qa2 . p r dz

provided r dr/dz vanishes at both z = 0 and z = l. The resistance to penetration in this case is therefore independent of the dynamic factor. Since for a conventional nose, p decreases monotonically from z = 0 to z = l, cavitation would begin at the base of the nose when the velocity attains a critical value υ c . If the radius of curvature of the profile at this point is denoted by na, then d 2 r/dz2 = −1/na at z = 1, where p = 0 at the incipient cavitation. When d 2 r/dz2 = 0 at z = 1, the critical velocity is υc =

nq/kp.

Fig. 8.16 Projectile penetration with cavitation. (a) Steady-state cavity formation and (b) geometry of conventional projectiles

8.5

Mechanics of Projectile Penetration

611

For υ > υ c the target loses contact with the nose at a point which progressively moves toward the tip, and the resistive force becomes P=

π r02

q + kρυ

2

dr dz

2 ,

υ ≥ υc ,

(8.88a)

r=r0

where r = r0 corresponds to the point on the nose at which p = 0 according to (8.87). Consequently, d dr = 0, r q + kρυ 2 dz dz r=r0

υ ≥ υc ,

(8.88b)

Let the local radius of the hole due to cavitation be denoted by λa. Since the enlarged cavity may be imagined to have been produced by a projectile of radius λa without cavitation, we may write p = π λ2 a2 q. Inserting this expression in (8.88a), and using the relation nq = kpυ c , we get

dr n dz

2 =

λ 2 a2 υc 2 −1 , 2 ro υ

n

d2 r λ2 a 2 υ c 2 = , dz2 ro3 υ

(8.89)

at the point where the material breaks contact with the nose. The second equation of (8.89) follows from the first on combining it with (8.88b). Equations (8.89) are sufficient to determine r0 /a and λ in terms of the velocity ratio v/vc for a given shape of the nose. In the case of an ogival nose, the profile of the nose has a constant radius of curvature na, the equation of the circular profile being [r + (n − 1) a]2 + (z − l)2 = n2 a2 , √ where l is the height of the nose, equal to a 2n − 1. In view of the above relation, equation (8.89) yields ⎫ % 1 λυc 2/3 ⎪ ⎪ ⎪ = n− ⎪ ⎬ υ n−1 4/3 2 $ 2 % # 2/3 ⎪ ⎪ λυc a λυc λυc ⎪ ⎪ n− = n− ⎭ υ r0 υ υ

λυc υ

2/3

a r0

$

(8.90)

The elimination of a/r0 between these two equations furnishes the relationship between λ and υ/υ c as

612

8 Dynamic Plasticity

λυc υ

2/3

= n − (n − 1)2/3 n −

υc2 υ2

1/3 , υ ≥ υc

(8.91)

As υ tends to infinity, the parameter λυ c /υ tends to a limiting value that depends on n, the limiting position r = r0∗ of the point where cavitation is initiated being given by

r0∗ = na

n−1 n

1/3 n − 1 2/3 1− n

in view of (8.90). Consequently, the contact is always maintained over a finite part of the nose. The pressure distribution on the nose over the region of contact according to (8.87) is given by (n − 1) n2 a3 p−q = − 1, kρυ 2 [r + (n − 1) a]3

0 ≤ r ≤ r0 .

(8.92)

The pressure decreases monotonically from its greatest value at the tip, vanishing at r = r0 ≤ a, when υ ≥ υc . The cavitation velocity υ c can be determined by an optimal fit of (8.92) with the experimentally measured value λ for a given material and any particular value of n defining the ogival nose. When n = l, the nose is hemispherical, and the projectile could be a spherical ball. Considering this as a limiting case of an ogival nose, the pressure distribution is uniform and can be written as p = q − kρυ 2 , 0 ≤ r ≤ a, √ so long as there is no cavitation (υ < υc ). When v exceeds vc = q/kρ the pressure is zero everywhere except at the pole, where there is a concentrated force P = π λ2 a2 q with λ = υ/υc . The nose therefore makes contact with the target only at the pole over the cavitation range (υ > υ c ). The results for a conical nose cannot be obtained from those for the ogival nose by a limiting process, because of the sharp corner existing at the base r = a. The pressure distribution is, however, uniform according to (8.87) and is given by p = q + kρυ 2 tan α, where α is the semiangle at the vertex. The cavitation in this case is only possible at the base, where there is a singularity in pressure and may occur at any velocity of the projectile. Since the resistive force is pA = qλ2 A where A = π a2 is the area of the base, we have p λ = =1+ q 2

kp υ 2 tan2 α. q

8.5

Mechanics of Projectile Penetration

613

Assuming q/Y to be given by the first term on the right-hand side of (8.81), the constant k can be determined from the experimentally measured values of the cavity radius λa. It remains to establish how the velocity υ of the projectile varies with the depth of penetration h, the striking velocity υ 0 being given. For a mass m of the projectile, the equation of motion is mυ

dυ = −λ2 qA. dh

For a conventional nose, λ = 1 when υ ≤ υ c but is a function of υ when υ > υ c . A direct integration of the above equation therefore results in m h −h= qA ∗

υ 0

ξ dξ mυc2 1+ , υdυ = 2 2qA 1 λ

υ ≥ υc,

(8.93)

where ξ = (υ0 /υc )2 and h∗ are the final penetration when the projectile comes to rest. Since υ = υ 0 when h = 0, we have h∗ =

β mn dξ , 1+ 2 2 kρA 1 λ

υ0 ≥ υc ,

where β = (υ0 /υc )2 . When the incidence velocity υ 0 is less than υ c , the quantity in the curly brackets must be replaced by β. Indeed, in the absence of friction and cavitation, qah∗ = mυ02 /2 for a conventional nose. In the particular case of an ogival nose, the integral in (8.93) is evaluated by using (8.91) for λ.(ξ ), and the results are displayed graphically in Fig. 8.17. For a hemispherical nose, on the other hand, we have λ2 = ξ , and the integral is then equal to In β. For a conical nose, an analysis similar to above gives h∗ =

mn cρ 2 ln 1 + υ0 , 2 kρA q

where c = k tan2 α. The results for a composite nose, in which an ogival frustum is surmounted by a circular cone, can be similarly obtained. Some experimental evidence in support of the theoretical prediction involving cavitation has been reported by Hill (1980), who used copper targets and steel bullets with various head shapes. In the case of hypervelocity impact, for which the speed of the projectile exceeds the elastic wave speed in the target material, the pressure generated on the cavity surface is large compared to the yield stress of the material. Since the rise in temperature caused by the impact is extremely high, the material can be treated as a fluid for the analysis of the penetration problem. When a meteor traveling in a highly eccentric orbit strikes the surface of a planet, there is complete volatilization of some material during the formation of the crater, and the situation is effectively similar to that encountered in a subsurface explosion (Johnson, 1972).

614

8 Dynamic Plasticity

Fig. 8.17 Relationship between depth of penetration and squared striking velocity for projectiles with ogival heads

8.5.3 Perforation of a Thin Plate A projectile having a conical nose or a sharp ogival nose strikes a target in the form of a thin plate of uniform thickness h0 with a sufficiently high velocity v0 , the axis of the projectile being normal to the plane of the plate, Fig. 8.18(a). It is assumed that the projectile penetrates the plate without shattering it and produces a lip of height b as it leaves behind a circular hole of radius equal to the shank radius a. In the simplified model, each element of the lip is assumed to reach its final position through rotation under a uniaxial tension in the circumferential direction. If the initial radius to a typical element is denoted by s, and the final distance of the element from the outer edge of the lip is denoted by x, then the condition of plastic incompressibility requires h0 s ds = ha dx

or

dx/ds = h0 s/ha,

where h is the local thickness of the lip. In view of the assumed uniaxial state of stress, we have the additional relation

h ln h0

1 a = − ln 2 s

or

h = h0

s . a

8.5

Mechanics of Projectile Penetration

615

Fig. 8.18 Perforation of a flat plate by a projectile with pointed nose. (a) Geometry of lip formation and (b) fractional change in velocity against square of the striking velocity

Eliminating h/h0 between the preceding relations, the resulting differential equation for x can be integrated to give x 2 s 3/2 , = a 3 a

h = h0

3x 2a

1/3 ,

(8.94)

in view of the boundary condition x = 0 when s = 0. The remaining boundary condition x = b at s = a indicates that the total height of the lip is b = 2a/3. The radial pressure exerted by the projectile when the lip is fully formed is given by p=

h0 x 1/3 h , Y= Y a a b

when the material is nonhardening with a uniaxial yield stress equal to Y. The pressure varies, therefore, from zero at x = 0 to h0 Y/a at x = b. It may be noted that the strain at x = 0 is infinitely large. Suppose that the material work-hardens according to the power law σ = σ 0 εk , where σ 0 and k are constants. Then the plastic work done per unit volume in a typical element is σ ε/(1 + k), where σ is the hoop stress corresponding to the final hoop strain ε = ln(a/s). Hence, the total plastic work expended during the formation of the lip is

616

8 Dynamic Plasticity

W1 =

2π h0 σ0 1+k

1

ε1+k sds =

0

2π a2 h0 σ0 1+k

∞

e−2e ε1+k dε.

0

The integral on the right-hand side is equal to (2 + k)/22+k , where (ξ) is the well-known gamma function of a positive variable |. The expression for the plastic work therefore becomes Γ (1 + k) . (8.95a) W1 = π a2 h0 σ0 21+k To obtain the work done by the inertia forces, let r denote the radius of the hole at any instant of time t. Since the mass of material displaced at time t is equal to π ρh0 r2 , where ρ is the density of the material, the work done by the distribution of the radial accelerating force F is W2 =

a

Fdr = πρh0

z

0

a

d 2 dr r dr. dt dt

The work expended in overcoming the frictional resistance will be disregarded, and W2 will be evaluated on the basis of a constant projectile velocity equal to υ 0 (Thomson, 1955). Considering an ogival nose defined by r = na (sin θ − sin θ0 ) ,

sin θ0 = (n − 1) /n,

n > 1,

and setting dr/dt = υ 0 cot θ and dθ /dt = (υ 0 /na) cosec θ , we obtain the expression W2 = πρho υ02

π/2

θ

2nar cot θ cos θ − r2 cos ec2 θ cot θ dθ .

Inserting the expression for r(θ ), and carrying out the integration, the result is found to be 1 n − (1 + 2n) . (8.95b) W2 = πρh0 υ02 n2 ln n−1 2 Let the speed of the projectile decrease from υ 0 to υ f during the perforation. Equating the loss of kinetic energy of the projectile to the total work done on the material, we have

m υ02 − υf2 = 2 (W1 + W2 ) , where m0 is the mass of the projectile. The substitution of W1 and W2 from (8.95) into the preceding relation furnishes the square of the velocity ratio as

υf υ0

2 =1−

m0 m

n (1 + k) 2 ln + 2n − + 2n) , (1 n−1 2k η

(8.96)

8.6

Impact Loading of Prismatic Beams

617

where η = ρυ02/σ0 and m0 = π ρh0 a2 is the mass of the displaced material. When the projectile has a conical nose, dr/dt = υ 0 a/l, and an independent analysis leads to the formula

υf υ0

2

m0 =1− m

(1 + k) 2a2 + . 2k η l2

The residual velocity υ f for a standard ogival nose is somewhat higher than that for a conical nose having the same a/1 ratio. Figure 8.18(b) shows how the ratio (υ 0 – υ f )/υ 0 varies with the parameter η in the case of an ogival nose projectile for different values of m0 /m. Due mainly to the neglect of plastic bending of the plate beyond the radius r = a, the theoretical prediction is found to underestimate the velocity change (Goldsmith and Finnegan, 1971) An experimental investigation on the perforation of target plates by the normal and oblique impact of projectiles has been reported by Piekutowski et al (1996). In the case of perforation of a thin plate by the normal impact of a flat-ended cylinder, a plate plug is generally formed by shearing and is ejected from the target as the projectile passes through the plate. The diameter of the plug is approximately equal to that of the projectile, and the velocity of its ejection differs only marginally from the residual velocity of the projectile (Recht and Ipson, 1963). The analysis of the perforation problem involving truncated projectiles has been discussed by Zaid and Paul (1958). Useful experimental results for impact on finite plates have been reported by Calder and Goldsmith (1971). For very high velocities of impact, the effect of the rate of straining on the resistance to shear becomes significant, as has been shown by Chou (1961). The problem of ricochet of the deforming projectile after impact with plates has been investigated by Zukas and Gaskill (1996).

8.6 Impact Loading of Prismatic Beams 8.6.1 Cantilever Beam Struck at Its Tip Consider a uniform cantilever of length l which is struck transversely at the end by an object of mass m0 moving with velocity U, the mass being assumed to be attached to the beam during the plastic deformation that follows. The material is assumed to be rigid/plastic with constant uniaxial yield stress Y. The kinetic energy of the moving object is absorbed by a plastic hinge which is initially formed at the tip of the cantilever. As the inertia effect progressively decreases, the plastic hinge moves along the beam toward the built-in end, causing a permanent change in curvature over the distance traversed by the hinge. This problem has been investigated theoretically and experimentally by Parkes (1955) based on the rigid/plastic model and by Symonds and Fleming (1984) on an elastic/plastic model. At any instant of time t measured from the moment of impact, let the plastic hinge H be situated at a distance ξ l from the loaded end of the cantilever, Fig. 8.19(a). The bending moment at the hinge is equal to –M0 , where M0 is the yield moment that

618

8 Dynamic Plasticity

Fig. 8.19 Uniform cantilever struck at its tip. (a) Deformed configuration at any instant and (b) position of plastic hinge as a function of time

depends on the yield stress, the area of cross section, and the shape factor. Since the shearing force is zero at the hinge, where the bending moment is a relative maximum, we may analyze the portion of the beam between the hinge and the tip as a rigid body. Since the inertia force per unit length acting at a generic point of the beam is (m/l)(∂ 2 w/∂t2 ) acting in the upward sense, where m is the total mass of the beam and w the downward deflection, the application of D’Alembert’s principle for the dynamic equilibrium of forces and moments gives ⎫ m ξ t d2 w d 2 w0 ⎪ ⎪ dx = 0,⎪ m0 2 + ⎬ 2 l 0 dt dt ξt ⎪ x d2 w d 2 w0 ⎪ M 0 + m0 ξ l 2 + m ξ− dx = 0, ⎪ ⎭ 2 dt l dt 0

(8.97)

where w0(t) denotes the deflection at x = 0 and is assumed to be sufficiently small. Since the deformed part of the beam between the tip and the hinge at any instant rotates about x = ξ l as a rigid body, the velocity of a generic point is given by x dw0 ∂w = 1− . ∂t ξ l dt

(8.98)

Substituting in (8.97), and introducing a constant deflection δ together with a set of dimensionless quantities ρ, z, and τ , which are defined as δ=

m0 lU 2 , 2M0

ρ=

m x , z= , 2m0 l

τ=

tU 2δ

we obtain a pair of ordinary differential equations for w0 in a mathematically convenient form. The result may be expressed as

8.6

Impact Loading of Prismatic Beams

619

⎫ w ¨ 0 + ρ ξw ¨ 0 + ξ˙ w ˙ 0 = 0, ⎬ ρ 2 2δ + ξ w ¨0 + ¨ 0 + ξ ξ˙ w˙ 0 = 0,⎭ 2ξ w 3

(8.99)

where the dot denotes differentiation with respect to the dimensionless time τ . Since the expression in parenthesis of the first equation of (8.99) forms an exact differential, this equation is immediately integrated once to give (1 + ρξ ) w˙ 0 = 2δ in view of the initial conditions ξ = 0 and dw0 /dt = U (or w˙ 0 = 2δ ) when τ = 0. A suitable combination of the two equations in (8.99) gives

¨ 0 + 2ξ ξ˙ w˙ 0 = 6δ ρ ξ 2w The expression in parenthesis of the above equation is an exact differential, and integration gives ρξ 2 w˙ 0 = 6δτ , on using the initial condition ξ = 0 when τ = 0. The relationship between the velocity of the tip of the cantilever and the time interval over the range 0 ≤ ξ ≤ 1 is therefore given parametrically as w˙ 0 = 2δ/ (1 + ρξ ) , 3τ = ρξ 2 / (1 + ρξ ) ,

0 ≤ τ ≤ τ0 = ρ/ (3 + 3ρ) ,

(8.100)

where τ 0 is the dimensionless time when the plastic hinge reaches the built-in end of the cantilever. Figure 8.19(b) indicates how the hinge position varies with time during the initial phase of bending. For τ > τ 0 , the plastic hinge remains fixed at the built-in end, and the hinge moment is no longer a relative maximum. The nonzero shear force that exists at the hinge can be determined from the equation of vertical motion. Considering the angular motion, we set ξ = 1 and ξ˙ = 0 in the second equation of (8.99) to have 2 ¨ 0 + 2δ = 0. 1+ ρ w 3 Integrating, and using the condition that w0 is continuous at τ = τ 0 , we obtain the solution w˙ 0 =

6δ (1 − τ ) , τ0 ≤ τ ≤ 1. 3 + 2ρ

(8.101)

Since w ˙ 0 = 0 when τ = 1, the motion stops at this value of τ . Hence the duration of the motion is t∗ = 2δ/U = m0 lU/M0 , which is independent of the mass of the beam.

620

8 Dynamic Plasticity

The shape of the deformed beam at any instant can be determined by solving the appropriate differential equation for w. It follows from (8.98), (8.100), and (8.101) that ⎫ 2δ (ξ − z) ∂w ⎪ ⎪ = , 0 ≤ τ ≤ τ0 , ⎬ ∂τ ξ (1 + ρξ ) ∂w 6δ (1 − τ ) (1 − z) ⎪ ⎭ = , τ0 ≤ τ ≤ 1,⎪ ∂τ 3 + 2ρξ

(8.102)

the relationship between τ and ξ in the interval 0 ≤ τ ≤ τ 0 being given by the second equation of (8.100). The differentiation of this equation with respect to ξ gives ρξ (2 + ρξ ) dτ , = dξ 3 (1 + ρξ )2

0 ≤ τ ≤ τ0 .

The above relation can be used to change the independent variable from τ to ξ in the first equation of (8.102), the result being ∂w 2δρ (ξ − z) (2 + ρξ ) , = ∂ξ 3 (1 + ρξ )3

0 ≤ z ≤ ξ.

Integrating, and using the boundary condition w = 0 at the plastic hinge z = ξ , we obtain the solution w 1 + ρξ 2 1 ξ −z ρ (ξ − z) , 0 ≤ τ ≤ τ0 , = ln − 2− δ 3ρ 1 + ρz 3 1 + ρξ (1 + ρξ ) (1 + ρz) (8.103) which holds over the length 0 ≤ z ≤ ξ. The remainder of the beam is undeformed, giving w = 0 for ξ ≤ z ≤ 1. When τ > τ0 , the expression for w is readily obtained by integrating the second equation of (8.102), and using the condition of continuity of w at τ = τ 0 (ξ = 1). It is easily shown that w 1+ρ 1−z 2 = 3τ (2 − τ ) ln + δ 3 + 2ρ 3ρ 1 + ρz 1 1−z ρz 3ρ − 2+ + , 3 1+ρ (1 + ρz) (3 + 2ρ)

(8.104) τ0 ≤ τ ≤ 1,

Equations (8.103) and (8.104) furnish the vertical displacement of the bent beam throughout the motion following the impact. The slope of the deflection curve at z = 1 is nonzero for τ 0 ≤ τ ≤ 1, implying a discontinuity which occurs in the same sense as that permitted by the plastic hinge. The final shape of the deformed cantilever and the deflection of the tip of the beam as a function of time are of special practical interest. Setting τ = 1 in (8.104), and denoting the limiting deflection by w∗ , we obtain the relation

8.6

Impact Loading of Prismatic Beams

621

1+ρ 2 1−z w∗ = ln + δ 3ρ 1 + ρz 3 (1 + ρ) (1 + ρz)

(8.105)

giving the shape of the cantilever when it has come to rest. The deflection at the tip of the cantilever during its motion is obtained by setting z = 0 in (8.103) and (8.104), the result being ⎫ 3 ξ (2 + ρξ ) w0 ⎪ ⎪ = ln (1 + ρξ ) − , 0 ≤ τ ≤ τ , 0 ⎬ δ 2ρ 3 (1 + ρξ )2 ⎪ 3τ (2 − τ ) 2 2 + 7ρ/3 w0 ⎭ , τ0 ≤ τ ≤ 1⎪ = + ln (1 + ρ) − δ 3 + 2ρ 3ρ (1 + ρ) (3 + 2ρ) (8.106) where ξ is given by (8.100) as a function of τ < τ0 . If the mass of the beam is small compared to that of the striking object, so that ρ tends to zero, the plastic hinge moves almost instantaneously to the built-in end, and (8.104) reduces to w/δ = τ (2 − τ ) (1 − z) ,

0 ≤ τ ≤ 1.

It follows that δ is identical to the final deflection of the tip of the cantilever when the mass of the beam is negligible in comparison with the striking mass. The beam rotates in this case as a rigid body with the plastic hinge at the built-in end, the time taken by the striking object to come to rest being twice the time required by it to travel the same distance with a constant velocity U. The influence of the value of ρ on the final shape of the beam and the time dependence of the tip deflection are shown in Figs. 8.20 and 8.21, respectively. The problem of central impact of a simply supported beam can be similarly treated (Ezra, 1958). The transverse impact of a beam built-in at both ends has been considered by Parkes (1958) and Jones (1989). The influence of elastic deformation of the beam has been investigated by Symonds and Fleming (1984).

8.6.2 Rate Sensitivity and Simplified Model When the ratio of the kinetic energy input to the greatest possible elastic energy in the beam is sufficiently large, the strain-rate dependence of the yield stress must be included in the analysis for a realistic prediction of the dynamic behavior. Since the strain-rate influence generally changes the mode of deformation, a simple correction factor for the yield stress could not be applied to the rate-independent theory without discrimination. To illustrate the procedure, we consider the cantilever beam of Fig. 8.19(a) and assume the mass m0 to be attached to the tip instead of being dropped on the beam. Following Bodner and Symonds (1962), the relationship between the uniaxial stress σ and the plastic strain rate ε˙ will be taken in the form of (8.13), specialized by setting n = λ = 0, and by replacing m with 1/n. The power law,

622

8 Dynamic Plasticity

Fig. 8.20 Final shape of the cantilever beam due to impact loading with different ratios of the attached mass to the mass of the beam

Fig. 8.21 Deflection of the cantilever tip as a function of time for various mass ratios

8.6

Impact Loading of Prismatic Beams

623

σ =1+ Y

1/n ε˙ , α

(8.107)

where α and n are empirical constants, has been found to fit with Manjoine’s experimental data (1944) reasonably well by taking a ≈ 400/s, n ≈ 5 for mild steel, and a ≈ 6500/s, n ≈ 4 for aluminum. In the case of high strain rates with mild steel specimens, different values of these constants have been suggested by Hashmi (1980). In view of (8.107), and the fact that ε˙ varies linearly with the vertical distance y measured from the axis of the beam, the bending moment for a beam of rectangular section of width b and depth h is given by

h/2

M = 2b

σ dy = M0

0

2 1+ 2 ε˙ 0

ε0 0

1/n ε˙ ε˙ dε˙ , α

where M0 is the quasi-static value of the fully plastic moment, equal to bh2 Y/4, and ˙ with κ˙ denoting ε˙ 0 is the maximum strain rate in the cross section, equal to κh/2, the curvature rate of the bent axis. The relationship between the bending moment M and the curvature rate κ˙ may therefore be written as M =1+ M0

1/n κ˙ , β

β=

2α 1 n . 1+ h 2n

(8.108)

The nature of the (M, κ) ˙ curve is therefore identical to that of the (σ, ε˙ ) curve and is completely defined by the empirical constants α and n for a given depth of the beam. Due to the effect of strain hardening of the material, the actual bending moment will depend not only on the curvature rate but also on the curvature of the beam, as has been shown experimentally by Apsden and Campbell (1966). For the cantilever beam subjected to a tip impulse, as we have seen, a plastic hinge starts at the tip and moves toward the built-in end, when the material is ideally plastic. A rate-sensitive material, on the other hand, requires the plastic region to initially extend over the whole length of the beam without the formation of a localized plastic hinge. This is a consequence of the fact that the bending moment is a continuous function of the curvature rate, which vanishes at the tip of the cantilever and has its greatest value at the built-in end. The plastic zone continually shrinks during the motion and becomes zero when the beam comes to rest. The shape of the ratesensitive cantilever at an intermediate stage of motion is compared with that of the ideally plastic cantilever during its first phase of motion in Fig. 8.22. For simplicity, the analysis will be carried out on the assumption that the outer portion of the cantilever is one of constant slope ψ that varies with time t as the motion continues. It would be instructive at first to derive the rate-independent solution corresponding to the above simplified model (Mentel, 1958). Since the total momentum of the beam together with the attached mass has a constant value equal to the applied tip impulse I, while the resultant angular momentum about the built-in end is equal to the angular impulse Il – M0 t, we have

624

8 Dynamic Plasticity

Fig. 8.22 Simplified deformation mode for a cantilever subjected to tip impulse. (a) Perfectly plastic material and (b) rate-sensitive material

⎫ 1 ˙ 2 ⎪ ˙ l = I, ⎪ mψξ l + m0 ψξ ⎬ 2 1 ⎪ ˙ 2 l − ξ l2 + m0 ψξ ˙ l2 = Il − M0 t,⎪ mψξ ⎭ 2 3

(8.109)

where m is the mass of the beam, m0 is the attached mass, and the dot denotes the time derivative. These equations are immediately solved for ψ˙ and t as functions of ξ, the result being ψ˙ =

1 , m0 ξ (1 + ρξ ) l

Il t= 3M0

ρξ 2 1 + ρξ

,

where ρ denotes the ratio m/2m0. The time taken by the hinge to reach the built-in end (ξ = 1) and the corresponding angle of rotation are found to be t0 =

ρ 1+ρ

Il ρ (4 + 3ρ) I 2 , ψ0 = . 3M0 6 (1 + ρ)2 M0 m0

(8.110)

It is interesting to note that the duration of the first phase is identical to that given by (8.100), if we set I = m0 U. Since the second equation of (8.109) continues to hold (with ξ = 1) in the second phase, during which the whole beam rotates about the fixed end, the angular velocity is ψ˙ =

3 (Il − M0t ) , (3 + 2ρ) m0 l2

t ≥ t0 .

In view of the initial condition ψ = ψ 0 when t = t0 , this equation is readily integrated to give

8.6

Impact Loading of Prismatic Beams

625

3 (t − t0 ) M0 (t + t0 ) ψ − ψ0 = 1− , 2l (3 + 2ρ) m0 l

t ≥ t0

(8.111)

The beam comes to rest when ψ˙ = 0, the total time of impact t∗ and the final angle of rotation ψ∗ being found as t∗ =

Il , M0

ψ∗ =

I2 . 2M0 m0

Surprisingly, both t∗ and ψ∗ are independent of the mass ratio ρ according to this simplified analysis, the result for t∗ being in agreement with that obtained in the previous solution. The plastic work done during the second phase is M0 (ψ ∗ — ψ 0 )> which is used up in absorbing a part of the kinetic energy input equal to I2 /2m0 . For sufficiently small values of ρ, most of the kinetic energy is absorbed in the second phase.

8.6.3 Solution for a Rate-Sensitive Cantilever The inclusion of rate dependence of the yield moment completely changes the kinematics of the dynamic response of the beam, Fig. 8.22(b). The outer portion CA of constant slope ψ, at any instant, has unloaded from the plastic state, while the inner segment OC has its bending moment increasing from M0 at C to a magnitude M0 at 0. Let M denote the magnitude of the bending moment at a typical section in the plastic region at a distance sl from the free end. If the inertia effects in the plastic region are disregarded, the shearing force has a constant value R, and the bending moment distribution may be written as M = M0 + R (s − ξ ) l = M0 − R (1 − s) l. Eliminating R and M between these relations in turn, and using (8.108) to express the ratios M/M0 and M0 /M0 in terms of the corresponding curvature rates, we have κ˙ = κ˙ 0

s−ξ 1−ξ

n ,

Rl (κ˙ 0 /β)1/n = . M0 1−ξ

(8.112)

The curvature rate increases from zero at s = ξ to attain its greatest value κ 0 at the fixed end s = 1. Since ∂θ/∂s = −lκ, where θ is the local slope of the plastic segment, the local angular velocity is θ˙ = l

1 s

κ˙ ds = lκ˙ 0

1−ξ 1+n

1−

s−ξ 1−ξ

1+n (8.113)

in view of the boundary condition θ˙ = 0 at s = 1. Setting s = ξ in (8.113) furnishes the angular velocity ψ˙ of the rigid outer segment as

626

8 Dynamic Plasticity

1−ξ . 1+n Similarly, integrating the equation ∂ w/∂s ˙ = −lθ˙ , where w ˙ is the particle velocity, and using the boundary condition w˙ = 0 at s = 1, we obtain the velocity distribution in the plastic region as ψ˙ = lκ˙ 0

w˙ = l κ˙ 0 2

1−ξ 1+n

$ % 1−ξ s − ξ 2+n 1− . (1 − s) − 2+n 1−ξ

(8.114)

In particular, the velocity w ˙ c at the rigid/plastic interface s = ξ and the free-end velocity w˙ a which exceeds w˙ c by the amount ξ ψ˙ are given by w ˙ c = l2 κ˙ 0

(1 − ξ )2 , 2+n

w˙ a = l2 κ˙ 0

1−ξ 2+n

1+

ξ . 1+n

(8.115)

In all the preceding relations, ξ is a function of t to be determined. Once this is known, along with κ˙ 0 , the physical quantities κ, θ , and w can be found by time integration, the initial conditions being κ = θ = w = 0 at t = 0. The two other equations necessary for the mathematical formulation of the problem are furnished by the principle of impulse and momentum. The equation of linear momentum in the vertical direction and the equation of angular momentum about the built-in end are easily shown to be ⎫ 1 ⎪ ⎪ Rdt = m0 w˙ a + mξ (w˙ c + w˙ a ) cos ψ + G, ⎬ 2 0 , t ⎪ ξ 1 ⎭ ˙ a + 2w˙ c ) + m0 w˙ a l + H ⎪ Il − Rdt = mlξ (w˙ c + w ˙ a ) − (w 2 3 0

I−

t

(8.116)

where G and H represent the linear and angular momentum, respectively, of the plastic region OC and are given by G≈m

1 ξ

w ˙ ds,

H ≈ ml

ξ

1

w˙ (1 − s) ds.

It turns out that G and H make only minor contributions to the total momentum during the motion. For practical purposes, it is therefore a good approximation to take G≈

ml2 κ˙ 0 (1 − ξ )3 , 3 (2 + n)

H≈

ml3 κ˙ 0 (1 − ξ )4 . 3 (2 + n)

(8.117)

The first expression of (8.117) is obtained by assuming w˙ to be given by the righthand side of the first equation of (8.115) with j written for ξ, so that the fixed-end conditions w˙ = ∂ w/∂s ˙ = 0 at s = 1 are identically satisfied. The expression for H is also obtained in the same way, but the numerical factor in the denominator is

8.6

Impact Loading of Prismatic Beams

627

adjusted in such a way that both equations in (8.116) furnish identical initial values of κ˙ 0 . Indeed, setting t = ξ = 0 in (8.116) results in (κ˙ 0 )t=0 =

3 (2 + n) l = cnβ (3 + 2) m0 l2

(say) ,

(8.118)

in view of (8.115) and (8.117). The integrals appearing on the left-hand side of (8.116) may be evaluated in an approximate manner, without introducing significant errors, by using the initial value of κ˙ 0 for expressing the integrands, which are then obtained from (8.112) as c Rl ≈ , M0 1−ξ

M ≈ 1 + c. M0

Introducing this approximation, and using (8.115) for the velocities, the momentum equations (8.116) are easily reduced to λ−c

τ

dτ =φ 1 −ξ 0

1−ξ 2+n

λ − (1 + c) τ = φ

2ρ ρ n−1 1 2ρ 1+ ξ − ξ 2 cos ψ + + 3 3 1+n 3 n+1

1−ξ 2+n

1+

2ρ ξ ρ + + 3 1+n 3

(8.119)

3−ξ ξ2 , 1+n

the nondimensional quantities λ, τ , and φ introduced here being defined as I t λ= √ , τ= l M0 m

M0 m , φ = l2 κ˙ 0 . m M0

By eliminating the common factor φ(1 — ξ)/(2 + n) between the two equations (8.119), and solving the resulting equation numerically under the initial condition ξ = 0 when τ = 0, we obtain τ as a function of ξ. The second equation of (8.119) then gives φ as a function of ξ, and the shape of the deformed cantilever is finally obtained from the distribution of θ determined by the integration of (8.113). The free-end deflection of the cantilever is given by wa = l

t 0

w˙ a l

t

dt = 0

φ

1−ξ 2+n

ξ 1+ dτ , 1+n

(8.120)

˙ of the rigid portion is similarly computed which follows from (8.115). The slope ψ as a function of time. The total time t∗ required by the beam to come to rest corresponds to φ = 0, or τ = λ/(l + c) in view of the second equation of (8.119). Hence

Il t∗ = M0

1/n −1 3 (2 + n) lh 2n . 1+ 1 + 2n 2α (3 + 2ρ) m0 l2

(8.121)

628

8 Dynamic Plasticity

Some of the computed results, obtained by Ting (1964) using λ = 293, ρ = √ 0.305, n = 5.0, α = 1036/s, and η = l m/M0 = 0.039s, are displayed graphically in Fig. 8.23, the duration of impact in this case being t∗ = 0.064 s (ψ∗ = 1.033 radians). The broken curves are based on the approximation cos ψ ≈ 1, which neglects the change in geometry. The theory has been found to be in good overall agreement with experiment by Bodner and Symonds (1962). The agreement is not so good, however, when the rate-independent theory is used. ˙ of the rigid segment The preceding theory indicates that the angular velocity ψ decreases almost linearly with time. Using the linear relationship between ψ and t as an approximation (Ting, 1964), and assuming an initial value of ψ equal to 3l/(3 + 2ρ)m0 l, it is easily shown that

Fig. 8.23 Variation of curvature rate and slope at the built-in end of an impulsively loaded ratesensitive cantilever which is progressively rendered plastic (after Bodner and Symonds, 1962)

8.6

Impact Loading of Prismatic Beams

ψ≈

3It t 2− ∗ , 2 (3 + 2ρ) m0 l t

629

ψ∗ =

3I 2 , 3 (3 + 2ρ) (1 + c) M0 m0

(8.122)

in view of the terminal condition ψ˙ = 0 at t = t∗ and the initial condition ψ = 0 at t = 0. The terminal slope ψ∗ predicted by (8.122) is somewhat smaller than that given by (8.111). Further solutions on the dynamics of beams including strain-rate effects, using simplified models, have been discussed by Perrone (1965) and Lee and Martin (1970), among others. The influence of strain hardening of the material has been considered by Jones (1967) and Perrone (1970). As a consequence of the elastic response of the material, an elastic flexural wave develops at the point of impact and propagates along the length of the cantilever before it is reflected from the fixed end. The reflected wave moves back toward the tip of the cantilever and meets the traveling plastic hinge midway along the beam. The interaction between the primary bending wave and the reflected wave significantly modifies the deformation mode from that predicted by the rigid/plastic theory. The subsequent deformation of the cantilever depends to a large extent on the ratio of the tip mass to the mass of the beam material. At the base of the cantilever, the bending moment oscillates in magnitude and sense, producing some reversed bending in this region. The final deflection of the tip of the cantilever is found to be about the same as that predicted by the rigid/plastic theory (Stronge and Yu, 1993).

8.6.4 Transverse Impact of a Free-Ended Beam A uniform beam of mass 2m and length 2l, initially at rest, is subjected to a concentrated impact load at its midpoint, such that the central section instantaneously attains a velocity U which is subsequently maintained constant. For simplicity, we propose to analyze the equivalent problem of a beam moving with a uniform normal velocity U, the central section of the beam being suddenly brought to rest by mean of a rigid stop, Fig. 8.24(a). Over a sufficiently small time interval after the beam strikes the stop, plastic hinges occur not only at the central section but also at two other sections each at a distance ξ l from the center. The problem has been treated by Symonds and Leth (1954) for a beam of finite length, the corresponding problem for infinitely extended beams having been discussed by Lee and Symonds (1952), Conroy (1952, 1956), and Shapiro (1959). Because of symmetry, it is only necessary to consider one-half of the beam with segments OH and HA subjected to end moments of magnitude M0 , which is assumed to have a constant value. The shear force vanishes at the moving hinge H, where the bending moment has a relative maximum, but a shear force of magnitude R/2 exists at O, where R denotes the reaction exerted by the stop at any instant t. The segment OH rotates about O with an angular velocity dφ/dt while HA rotates with an angular velocity dψ/dt and translates with zero acceleration of its mass center. OH acquires a permanent deformation from the traveling plastic hinge H, but the material to the right of this hinge at any instant is undeformed. Since the moments of inertia of

630

8 Dynamic Plasticity

Fig. 8.24 Beam of infinite length subjected to transverse impact. (a) Deformation mode following impact and (b) graphical presentation of results

OH and HA about the respective mass centers are mξ 3 l2 /12 and m(1 – ξ )3 l2 /12, respectively, the equations of angular rigid-body motion of these two segments may be written as 1 2 2 d2 φ mξ l = −2M0 , 3 dt2

d2 ψ 1 m (1 − ξ )3 l2 2 = M0 . 12 dt

(8.123)

Since the deformed segment OH is actually curved, φ must be interpreted as the angle made by the chord joining O and H. The velocity of the mid-section of the outer segment is always a constant, and its value must be equal to the impact velocity U. It follows from the kinematics of the motion that ξ

1 U dψ dφ + (1 − ξ ) = . dt 2 dt l

(8.124)

It may be noted that the linear acceleration is discontinuous across the moving hinge H. Equations (8.123) and (8.124) form the basis for finding the three unknown quantities φ, ψ, and ξ . Introducing the dimensionless quantities ω=

l dψ M0 t l dφ , λ= , τ= , U dt U dt mlU

and using a dot to denote differentiation with respect to τ , the preceding equations (8.123) and (8.124) can be expressed in the more convenient form ⎫ λ˙ = 12/ (1 − ξ )3 ,⎬ ω˙ = −6/ξ 3 , 1 ⎭ ξ ω + (1 − ξ ) λ = 1 2

(8.125)

8.6

Impact Loading of Prismatic Beams

631

At the moment of impact, H coincides with O, and the angular velocity of HA vanishes. Hence the initial conditions are ξ = 0,

λ = 0,

φ=ψ =0

when τ = 0.

The last equation of (8.125) therefore indicates that ω tends to infinity as ξ tends to zero, such that ξω = 1 in the initial state, implying a singularity in the angular velocity at the point of impact. To obtain the solution of (8.125), we begin by differentiating the last of these ˙ equations with respect to τ , and using the other two equations to eliminate ω ˙ and λ, the results being easily shown to be 6 (1 − 2ξ ) 1 . ω − λ ξ˙ = 2 ξ 2 (1 − ξ )2

(8.126)

Differentiating this equation with respect to τ , and substituting again for ω ˙ and λ˙ using (8.125), we obtain the differential equation for ξ as

ξ (1 − ξ ) (1 − 2ξ ) ξ¨ + 1 − 3ξ + 3ξ 2 ξ 2 = 0. Since the independent variable τ does not appear explicitly, a first integration can be easily carried out by setting ξ˙ = η, so that ξ¨ = η(dη/dξ). The preceding equation then becomes 1 − 3ξ + 3ξ 2 η dn + = 0. dξ ξ (1 − ξ ) (1 − 2ξ ) Since we are concerned only with the situation ξ < above equation results in η=

1 2,

the integration of the

√ dξ 1 − 2ξ = , dτ Cξ (1 − ξ )

(8.127)

where C is a constant to be determined later. Integrating again, and using the initial condition ξ = 0 when τ = 0, we obtain the solution τ=

C 1 − 2ξ . 1 − 1 + ξ − ξ2 5

(8.128)

Substituting (8.127) for ξ˙ into (8.126), and combining the resulting expression with the last equation of (8.125), the dimensionless angular velocities are found as √ ω =1+

1 − 2ξ , ξ

√ 1 − 2ξ λ=2 1− , 1−ξ

(8.129)

632

8 Dynamic Plasticity

on setting C = 16 to satisfy the condition λ = 0 when τ = 0. The angles φ and ψ at any instant during the motion of the outer hinge are ⎫ ξ (2 − ξ ) ⎪ mU 2 ⎪ τ+ ωdτ = ,⎪ ⎬ M 12 0 0 0 t ⎪ ξ2 mU 2 τ mU 2 U ⎪ ⎪ 2τ − λdt = λdτ = ψ= , ⎭ l 0 M0 0 M0 16

U φ= l

t

mU 2 ωdt = M0

τ

(8.130)

in view of (8.127) and (8.129), the dimensionless timeτ being given by (8.128) with C = 16 . The reaction R at the stop is most conveniently obtained from the condition of moment equilibrium of the segment OH about its mass center. Thus 1 3 2 d2 φ = Rξ l − 8M0 . mξ l 3 dt2 The elimination of d2 φ/dt2 between this equation and the first equation of (8.123) immediately furnishes R = 6M0 /ξ l. For sufficiently small values of ξ , the beam behaves as being infinitely long. In 2 this case, (8.127) and (8.128) √ give ξ ≈ 6/ξ and τ ≈ ξ /12, so that the parameter Rl/M0 becomes equal to 3/τ approximately. An elastic/plastic analysis for an infinitely long beam with an arbitrary moment–curvature relation has been given by Duwez et al. (1950). The preceding analysis remains valid until the angular velocities of the inner and outer segments of the beam become equal to one another. Setting ω = λ, and using (8.129) and (8.128), the corresponding values of ξ and τ are found to be ξ0 =

√ 1/2 5−2 ≈ 0.486,

τ0 ≈ 0.0265.

During the subsequent motion (τ > τ 0 ), there is a single plastic hinge occurring at the central section of the beam, and the two halves rotate as rigid bodies with an angular velocity dφ/dt, the relevant equations in dimensionless form being ω˙ = −3,

φ˙ = mU 2 /M0 ω,

τ ≥ τ0 .

The first equation defines the angular motion, while the second equation follows from the definition of ω. The integration of the above equations gives ω = A − 3τ ,

φ=

mU 2 M0

3 B + Aτ − τ 2 , 2

τ ≥ τ0 ,

(8.131)

where A and B are constants. Since ω = ω0 ≈ 1.3456 and φ = φ 0 ≈ 0.0877mU2 /Mo when τ = τ 0 , in view of (8.129) and (8.130), we have A ≈ 1.425 and B ≈ 0.051.

8.7

Dynamic Loading of Circular Plates

633

The motion stops when ω = 0, and this corresponds to τ ≈ s 0.495, giving φ ≈ 0.389mU2 /M0 as the limiting angle of rotation. The reaction R at the support for τ > τ 0 discontinuously changes to a constant value equal to 3M0 /l. Figure 8.24(b) shows the variation of φ, ψ, and R with time over the range 0 ≤ τ ≤ 0.05. The neglect of geometry changes, which is implicit in the analysis, would be justified if the value of φ 0 is sufficiently small, preferably less than about 0.15 radians (say). The assumption of a rigid/plastic material, on the other hand, requires that the kinetic energy absorbed in the plastic deformation greatly exceeds the elastic energy stored in the beam. The range of validity of the solution may therefore be approximately defined as kM0 l 5 mU 2 < , < El M0 3 where EI is the flexural rigidity of the beam, and k is a numerical factor (presumably of the order of 10) that may be found from experiments. Outside this range, the elastic deformation of the beam must be considered for a realistic prediction of the dynamic behavior. A variety of related problems on the dynamic plastic behavior of beams have been considered by Lee and Symonds (1952), Symonds (1953), and Seiler and Symonds (1954) using force pulses; by Symonds (1954), Johnson (1972), and Nonaka (1977) for blast loading; by Cotter and Symonds (1955), Martin and Symonds (1966), and Yu and Jones (1989) for impulsive loading. The dynamics of elastic/plastic beams has been treated by Bleich and Salvadori (1953), Seiler et al. (1956), Martin and Lee (1968), Symonds and Fleming (1984), and Reid and Gui (1987). The influence of axial restraints has been examined by Symonds and Mentel (1958), and that of shear has been considered by Karunes and Onat (1960). The dynamic load characteristics of curved bars have been discussed by Owens and Symonds (1955), Perrone (1970), and Stronge et al. (1990). The dynamic plastic response of frames has been considered by Rawlings (1964), Symonds (1980), and Raphanel and Symonds (1984). The dynamic plastic behavior of a circular beam subjected to impact loading has been discussed by Yu et al. (1985), and that of a right-angled bent cantilever loaded at its tip has been examined by Reid et al. (1995). Lower and upper bound principles in the dynamic loading of structures have been discussed by several authors including Kalisky (1970) and Stronge and Yu (1993). A great deal of published work on the dynamic failure of structural members caused by severe plastic deformations has been reported by Wierzbicki and Jones (1989).

8.7 Dynamic Loading of Circular Plates 8.7.1 Formulation of the Problem The theory of plastic bending of circular plates, developed in Chapter 4, can be extended to include inertia effects which arise under dynamic loading conditions. In accordance with the usual assumption for thin plates, shearing stresses normal to the

634

8 Dynamic Plasticity

plate surface are neglected in comparison with the bending stresses parallel to the surface. The resultant shearing force must be included, however, in the equation of dynamic equilibrium. For a rotationally symmetric state of stress, the nonzero stress resultants Mr , Mθ , and Q, denoting the radial and circumferential bending moments and the transverse shearing force, respectively, are functions of the radial coordinate r and time t. If the downward deflection of the plate is denoted by w, which is also a function of r and t, the inertia force per unit area acting in the upward sense is −μ(∂ 2 w/∂t2 ), where μ is the surface density of the plate. Then the equations of dynamic equilibrium may be written as ∂ (rMr ) − Mθ − rQ = 0, ∂r

∂ ∂ 2w (rQ) + rp = μr 2 , ∂r ∂t

(8.132)

where ρ denotes the local intensity of the normal pressure acting on the plate. Eliminating Q from the first equation of (8.132) by means of the integrated form of the second equation, we obtain the governing differential equation ∂ (rMr ) − Mθ = − ∂r

r 0

∂ 2w p − μ 2 r dr, ∂t

(8.133)

which is independent of the mechanical properties of the plate material. For a plastically deforming plate made of an ideally plastic isotropic material, these properties are specified by the yield condition involving Mr , Mθ , and a fully plastic moment M0 , and the associated flow rule defining the ratio of the rates of change of the principal curvatures κ r and κ θ , which are given by κr = −

∂ 2w , ∂r2

κθ = −

1 ∂w . r ∂r

(8.134)

The rate of change of all physical quantities is specified by the partial derivative with respect to t, the effect of geometry changes being disregarded. If the material yields according to the maximum shear stress criterion of Tresca, the yield locus in the moment plane is the hexagon ABCDEF shown in Fig. 8.25(a). The generalized strain rate having components κ˙ r = ∂ κ˙ r /∂t and κ˙ θ = ∂ κ˙ θ /∂t, corresponding to each side of the hexagon, is represented by a vector directed along the exterior normal to the side. If the stress state is represented by a vertex of the hexagon, the flow vector may have any direction between those defined by the two limiting normals. In most physical problems, the plate may be divided into a central circular region together with surrounding annular regions, each of which corresponds to a different plastic regime. Across a circle separating two plastic regimes, the radial moment Mr and the shear force Q must be continuous but the circumferential moment Mg may be discontinuous. The deflection w and the velocity ∂w/∂t must also be continuous across , but all the second derivatives of w may become discontinuous as will be shown later. When the velocity slope ∂ 2 w/∂r ∂t and hence the circumferential curvature rate κ˙ θ are discontinuous across , it is called a hinge circle, which may be either stationary or moving with respect to the plate. Since the discontinuity

8.7

Dynamic Loading of Circular Plates

635

Fig. 8.25 Dynamic of pulse-loaded circular plates. (a) Yield condition and (b) growth of central deflection with time (p ≤ 2 p0 )

must be considered as the limit of a narrow annulus of rapid change in slope, the ratio κ˙ r /˙κθ becomes infinite at the hinge circle. It follows that a hinge circle can be associated only with the side CD and FA, as well as the corners A, C, D, and F of the yield hexagon (Hopkins and Prager, 1954). To establish the relations between the possible discontinuities across a hinge circle , we represent this circle by the equation r = ρ(t). Since w and ∂w/∂t are continuous across , the derivative of these quantities along this curve is also continuous. Consequently,

∂w dp ∂w + = 0, ∂t dt ∂t

dp ∂ 2 w ∂ 2w + = 0, ∂t2 dt ∂r∂t

(8.135)

where the square brackets denote the jump in the enclosed quantities. Since ∂w/∂t is continuous, the first equation of (8.135) indicates that the slope ∂w/∂r can be discontinuous only across a stationary hinge circle (dρ/dt = 0). The second equation of (8.135), on the other hand, shows that the acceleration ∂ 2 w/∂t2 must be continuous across a stationary hinge circle, but discontinuous across a moving hinge circle. Since ∂w/∂t is continuous across a moving hinge circle, its tangential derivative gives

dp ∂ 2 w ∂ 2w = 0, + ∂r∂t dt ∂ 2 r

dp

= 0. dt

The radial curvature ∂ 2 w/∂t2 is therefore discontinuous across a moving hinge circle. Due to the continuity of the radial bending moment, the space and time derivatives of Mr are both discontinuous across a moving hinge circle.

636

8 Dynamic Plasticity

8.7.2 Simply Supported Plate Under Pressure Pulse A circular plate which is simply supported round its edge r = a is subjected to a uniformly distributed normal pressure of intensity ρ that is brought on suddenly at t = 0. The pressure is maintained constant during a small time interval 0 < t < t0 , and then suddenly removed at t = t0 . The dynamic plastic action will occur only if ρ exceeds the value ρ 0 necessary for quasi-static plastic collapse (Section 4.1). We are therefore concerned here with p > p0 = 6M0 /a2 . For ρ = ρ 0 , the inertia forces do not arise because the deformation occurs indefinitely slowly. The dynamic behavior of the plate for ρ > ρ 0 is found to depend on whether or not ρ is greater or less than 2ρ 0 . Following Hopkins and Prager (1954), the former will be referred to as high load and the latter as medium load. We begin by considering the medium load, characterized by ρ 0 ≤ ρ ≤ 2ρ 0 The first phase of the dynamic behavior of the plate corresponds to the time interval 0 ≤ t ≤ t0 , during which the applied pressure is held constant throughout the plate. The equilibrium equation (8.133) therefore simplifies to ∂ (rMr ) = Mθ − M0 ∂r

3pr2 − λ , p0 a2

(8.136)

where λ (r,t) =

μ M0

r 0

∂ 2w rdr ∂t2

(8.137)

Here, use has been made of the relation p0 a2 = 6M0 . Equations (8.136) and (8.137) hold for all values of the ratio p/p0 . Under medium loads, the middle surface of the entire plate may be assumed to deform into a conical surface, as in the case of plastic collapse under the pressure p = p0 . We therefore take w (r,t) =

r p0 1− f (t) , μ a

(8.138)

where f(t) is a function of t to be determined. This expression for w satisfies the boundary condition w = 0 at r = a for all t ≥ 0. The rates of curvature associated with (8.138) are κ˙ r = −

∂ 3w = 0, ∂r2 ∂t

κ˙ θ = −

1 ∂ 2w p0 = f (t) . r ∂r∂t μar

For f (t) > 0, these relations correspond to the flow rule associated with side AB of the yield hexagon, requiring Mθ = M0 and 0 ≤ Mr ≤ M0 . It follows from (8.137) and (8.138) that

8.7

Dynamic Loading of Circular Plates

λ (r,t) =

637

r2 a2

2r 3− f (t) . a

Substituting this into (8.133), and using the yield condition Mθ = M0 , the resulting equation can be readily integrated. Since Mr = M0 at r = 0, we get r pr2 r2 Mr 1 − =1− + f (t) . M0 p0 a2 a2 2a The boundary condition Mr = 0 at r = a will be satisfied for all t if f”(t) is a constant equal to 2(p/p0 – 1). Since w = ∂w/∂t = 0 at t = 0, this gives f (t) =

p − 1 t2 . p0

(8.139)

The deflection of the plate as a function of r and t is completely defined by (8.138) and (8.139). The bending moment distribution is, however, independent of t, the radial moment being finally given by 2 Mr r r r p 1− =1− 2 2− − . M0 a a p0 a

(8.140)

The bending moment Mr according to (8.140) monotonically decreases from M0 at r = 0 when p0 < p < 2p0 , and the yield condition is therefore nowhere violated. This completes the solution for the first plastic phase of the motion. At t = t0 , the load is suddenly removed, but the motion continues until the kinetic energy acquired during the application of the load is dissipated by plastic work. The second plastic phase therefore corresponds to p = 0 and t > t0 . All conditions of the problem in the second phase can be satisfied by assuming w to be given by (8.138) together with f”(t) = –2. In view of the yield condition Mθ = M0 , and the conditions of continuity of w and ∂w/∂t at t = t0 , we have ⎫ pt0 (2t − t0 ) − t2 ,⎪ ⎪ ⎬ p0 t ≥ t0 r ⎪ r2 Mr ⎪ ⎭ =1− 2 2− . M0 a a

f (t) =

(8.141)

Since Mr lies between 0 and M0 , the yield condition is nowhere violated, the only restriction on the solution being pt0 f (t) = 2 − t ≥ 0, p0

or

t ≤ t∗ =

pt0 . p0

At the instant t = t∗ , the entire plate comes to rest, since f’(t) = 0 implies that the velocity w is identically zero. The final deflection of the plate is found from (8.138) and (8.141) as

638

8 Dynamic Plasticity

p w = μ ∗

p r − 1 t02 1 − , p0 a

(8.142)

which indicates that the deflection of the plate in the final stage at any radius r is p/p0 times that at t = t0. Figure 8.25(b) shows the manner in which the ratio w/w∗ varies with t/t∗ for given values of p/p0 in the medium load range. The dynamic bending problem for circular and annular plates under a linearly distributed pressure pulse has been considered by Jones (1968a,b).

8.7.3 Dynamic Behavior Under High Loads The bending moment distribution (8.140) is not admissible for p > 2p0 , since the predicted value of Mr then exceeds M0 in the neighborhood of the plate center. Indeed, ∂ 2 Mr/ ∂r2 is positiveat r = 0 according to (8.140) when p > 2p0 , indicating that Mr is a relative minimum at r = 0, when Mr = M0 . We may therefore expect Mr = Mθ = M0 in a central region of the plate within some radius p0 , which corresponds to the corner A of the yield locus. It follows from (8.136) and (8.137) that this region moves down with a constant acceleration equal to p/μ. Thus ∂ 2 w/∂t2 = p/μ,

0 ≤ r ≤ ρ0 .

In the remainder of the plate, the plastic regime AB should apply, and a conical mode of deflection may be assumed as before. In view of the initial conditions w = ∂w/∂t = 0 at t = 0, and the condition of continuity of w across r = ρ 0 , the expression for the deflection during the initial phase of the motion may be written as ⎧ 2 pt ⎪ ⎪ 0 ≤ r ≤ ρ0 , ⎨ , w (r,t) = 2μ2 pt ⎪ a−r ⎪ ⎩ , ρ0 ≤ r ≤ a, 2μ a−ρ0

(8.143)

where ρ 0 is independent of t but depends on the applied pressure p. The boundary condition w = 0 at r = a is identically satisfied, and the curvature rates associated with the velocity field (8.143) are κ˙ r = κ˙ θ = 0, κ˙ r = 0,

0 ≤ r ≤ ρ0 , pt κ˙ θ = , μr (a − ρ0 )

ρ0 ≤ r ≤ a.

Since κ θ is discontinuous across r = ρ 0 , this radius coincides with a stationary hinge circle. The above relations are seen to be compatible with the assumed plastic regimes inside and outside the circle of radius ρ 0 . It may be noted that the slope ∂w/∂r is discontinuous across the hinge circle. To obtain the bending moment distribution in the region ρ 0 ≤ r ≤ a, we observe that the acceleration ∂ 2 w/∂t2 is equal to p/μ inside the circle r = ρ 0 , and equal to

8.7

Dynamic Loading of Circular Plates

639

(p/μ)(a – r)/(a – p0 ) outside the circle r = ρ 0 . The substitution in (8.137) therefore results in p 3r2 − (r − ρ0 )2 (2r + ρ0 ) , ρ0 ≤ r ≤ a. λ (r,t) = p0 a2 (r − ρ0 ) Equation (8.136) is now readily integrated to obtain an expression for Mr outside the circle r = ρ 0 . Since Mr = M0 at r = ρ 0 in view of the continuity of the bending moment, and Mr = 0 at r = a in view of the boundary condition, we get 2a3 p = , 0 ≤ t ≤ t0 , p0 (a − ρ0 )2 (a + ρ0 ) Mr a (r − ρ0 )3 (r + ρ0 ) , ρ0 ≤ r ≤ a. =1− M0 r (a − ρ0 )3 (a + ρ0 )

(8.144) (8.145)

The last equation indicates that Mr decreases monotonically from M0 at r = p0 to 0 at r = a, and the yield condition is nowhere violated. The central deflection of the plate at t = t0 is δ0 = pt02 /2μ. It may be noted that ρ 0 /a tends to unity as p/p0 tends to infinity. A second plastic phase begins with the sudden removal of the load at t = t0 . During this phase, the radius of the hinge circle decreases from p0 to some radius p = p(t) at any instant, so that the kinetic energy of the plate decreases due to dissipation by plastic work. The velocity field in the second phase may therefore be taken in the form ⎧ pt ⎪ 0 , 0 ≤ r ≤ ρ, ∂w ⎨ μ = pt a−r ⎪ ∂t ⎩ 0 , ρ ≤ r ≤ a. μ a−ρ

(8.146)

Since ρ(t0 ) = ρ 0 , the velocity is automatically continuous at t = t0 . The acceleration ∂ 2 w/∂t2 vanishes inside the circle r =ρ and assumes the value (pt0 /μ)ρ (a – r)/(a – p)2 outside this circle. It follows from (8.137) that 3a r2 − ρ 2 − 2 r3 − ρ 3 p , ρ ≤ r ≤ a. λ = t0 ρ p0 a2 (a − ρ)2 Setting Mθ = M0 and p = 0 in (8.136), substituting the above expression for λ, and using the boundary conditions Mr = M0 and ∂Mr /∂r = 0 at r = ρ and Mr = 0 at r = a, the integration of the resulting equation furnishes (a − ρ) (a + 3ρ)

dρ 2p0 a3 , =− dt pt0

t0 ≤ t ≤ t1 ,

(8.147)

where t = t1 represents the instant when the hinge circle reaches the center of the plate. The function ρ(t) is obtained by integrating (8.147) under the initial condition ρ = ρ 0 at t = t0 , the result being

640

8 Dynamic Plasticity

M a =1− M0 r

r−ρ a−ρ

2

ρ (4a − 3ρ) + 2 (a − ρ) r − r2 , ρ ≤ r ≤ a, (8.148) (a − ρ) (a + 3ρ)

in view of (8.144). Evidently, t1 = pt0 /2p0 , corresponding to ρ = 0. It may be noted that (8.149) establishes the same relationship between ρ/a and t/t1 as (8.144) does between ρ 0 /a and p/2p0. Since 0 < Mr < M0 according to (8.148), the yield condition is nowhere violated. Consider now the deflection of the plate during the time interval t0 < t < t1 . Since the region inside the circle r = ρ has been associated with the single plastic regime A, the deflection in this region is found by a straightforward integration of the first equation of (8.146), using the initial condition w = pt02 /2μ at t = t0 . Thus

2p0 t t ρ ρ2 = , 1− 1− 2 = a a pt0 t1

t0 ≤ t ≤ t1,

(8.149)

in view of (8.149). For the region outside the circle of radius ρ, it is convenient to change the independent variable from t to ρ in the second equation of (8.146) using

w=

pt2 pt0 (2t − t0 ) = 0 2μ 2μ

ρ p ρ2 1− 1− 2 −1 , p0 a a

0 ≤ r ≤ ρ, (8.150)

(8.147), the result being p2 t02 3ρ ∂w r =− , 1+ a 1− ∂ρ 2μp0 a a

ρ ≤ r ≤ a.

For the annular region ρ 0 ≤ r ≤ a, which has been associated only with the plastic regime AB, the preceding equation is readily integrated under the initial condition w = (pt02 /2μ)(a − r)/(a − ρ0 ) at ρ = ρ 0 to obtain the solution pt2 a r 3 ρ0 + ρ p ρ0 − ρ 1+ , ρ0 ≤ r ≤ a. w= 0 1− + 2μ a a − ρ0 p0 a 2 a (8.151) Let τ denote the time when the contracting hinge circle coincides with a given radius r. Setting t = τ and ρ = r in (8.149), we get r pt0 r2 1− τ= 1− 2 . 2p0 a a The material at radius r belongs to plastic regime A for t ≤ τ and to plastic regime AB for t > τ , the deflection at t = τ being pt0 w (r,τ ) = 2μ

r p r2 1− 1− 2 −1 , p0 a a

8.7

Dynamic Loading of Circular Plates

641

which is obtained by setting ρ = r in (8.150). The integration of the deflection equation under the condition w = w(r, τ ) when ρ = r then furnishes pt2 w= 0 2μ

p r+ρ p2 r−ρ r 1+ −1 , 1− 2 + 1− p0 a a 2a a

ρ ≤ r ≤ ρ0 . (8.152)

Equations (8.150), (8.151), and (8.152) provide the complete solution for the deflection of the plate during the second plastic phase. The deflection at the end of this phase within the circle of radius ρ 0 is pt0 w (r,t1 ) = 2μ

r p r2 r 1− 1+ + 2 −1 , p0 a a 2a

0 ≤ r ≤ ρ0 .

(8.153)

The slope ∂w/∂r is continuous across r = ρ but discontinuous across r = ρ 0 , the discontinuity being of amount pt02 /2μ(a − ρ0 ). The jump condition (8.135) is found to be satisfied across r = ρ at each stage of the interval t0 ≤ t ≤ t1 . The central deflection of the plate is δ1 = (pt02 /2μ)(p/p0 − 1) when t = t1 , obtained by setting r = 0 in the preceding equation. The shape of the deflected circular plate at the instant t = t1 for different values of p/p0 is displayed in Fig. 8.26.

Fig. 8.26 Deformed shape of a pulse-loaded simply supported circular plate at the end of the second dynamic phase for different values of p/p0 ≥2

For t > t1 , there is a third plastic phase during which the entire plate is in the regime AB, and the velocity field may then be written as ∂w p0 r = 1− f (t) , ∂t μ a

t1 ≤ t ≤ t ∗ ,

(8.154)

which is identical in form to that given by (8.146), the instant when the plate comes to rest being denoted by t∗ . The integration of the above equation results in w (r,t) = w (r,t1 ) +

p0 r 1− f (t) , μ a

642

8 Dynamic Plasticity

where f (t) is defined in such a way that f ( t1 ) = 0. The equation of dynamic equilibrium (8.136), where p = 0 and Mθ =M0 , can be satisfied along with the boundary conditions by taking f”(t) = –2. Integrating, and using the initial conditions f(t1 ) = pt0 /p0 = 2ti and f(t1 )=0, which ensure the continuity of the velocity and deflection at t = t1 , we get f (t) = (t − t1 ) (3t1 − t) ,

f (t) = 2 (2t1 − t) ,

t∗ = 2t1

The bending moment distribution during this phase is the same as that given by (8.148) with ρ = 0, while the deflection of the plate is expressed as w (r,t) = w (r,t1 ) +

r p0 1− (t − t1 ) (3t1 − t) , μ a

t1 ≤ t ≤ t∗ .

(8.155)

The final deflection w∗ (which corresponds to t = t∗ ), considered over the region inside the circle r = ρ 0 , is obtained from (8.153) and (8.155) as w∗ =

pt02 2μ

r p r2 2r 1− + 2 −1 3+ 2p0 a a a

0 ≤ r ≤ ρ0 .

(8.156)

Outside this circle, the shape of the deformed plate is conical, and the deflection at any radius r is (a – r)/(a — p0 ) times that at r = ρ 0 , which is directly obtained from (8.156). The central deflection of the plate in the final stage is δ∗ = (pt02 /2μ)(3p/2p0 − 1). Figure 8.27 shows the variation of w∗ /δ∗ with r/a for several values of the ratio 2p0 /p and indicates how the central curved part of the plate increases with increasing pressure. The corresponding problem for a built-in circular plate has been analyzed by Perzyna (1958) and Florence (1966). The dynamic behavior of circular plates under a central circular loading has been investigated by Conroy (1969) and Liu and Strange (1996).

Fig. 8.27 Final shape of the deformed plate in relation to the central deflection for p/p0 ≥ 2. The solid circles indicate the positions of hinge circles

8.7

Dynamic Loading of Circular Plates

643

8.7.4 Solution for Impulsive Loading A simply supported circular plate is subjected to a blast-type loading which instantaneously imparts a uniform transverse velocity U to the entire plate except at r = a. The impulsive action is immediately withdrawn so that the plate is free from transverse loads thereafter. During the first phase of the dynamic plastic deformation, a central part of the plate of steadily decreasing radius ρ continues to move with velocity U, while the surrounding annulus involves a conical flow field with w = U at r = ρ. The velocity field may therefore be written as ∂w U, 0 ≤ r ≤ ρ = ∂t U (a − r) / (a − ρ) , ρ ≤ r ≤ a.

(8.157)

There is a plastic moving hinge at r = ρ across which the velocity slope is discontinuous. Both the principal curvature rates vanish in the circular region 0 ≤ r ≤ ρ, which corresponds to the plastic regime A, giving Mr = Mθ = M0. Since kr = 0 and κ˙ θ > 0 in the annular region ρ ≤ r ≤ a, it corresponds to the plastic regime AB for which Mθ = M0 , and the differential equation (8.133) for the bending moment becomes μρU ∂ (rMr ) = M0 + ∂r (a − ρ)2

r

ρ

(a − r) rdr,

ρ ≤ r ≤ a.

The bending moment must satisfy the boundary conditions Mr = 0 at r = a and Mr = M0 at r = p, while ∂Mr /∂r must vanish at r = ρ for the shearing force to be continuous. The integration of the above equation under these conditions furnishes Mr , which is the same as (8.148), while the differential equation for ρ is found to be

(a − ρ) (a + 3ρ)

2M0 a dp =− . dt μU

(8.158)

Integrating, and using the initial condition ρ = a when t = 0, the solution is obtained as

12M0 t ρ ρ2 t 1− 1− 2 = = , a t1 a μa2 U

(8.159)

where t1 denotes the time corresponding to ρ = 0. The defection of the plate during the time interval 0 ≤ t ≤ t1 is obtained by the integration of (8.157). Using (8.158), to change the independent variable from t to ρ, the second equation of (8.157) may be rewritten as ∂w Ut1 r 3ρ =− 1− , 1+ ∂ρ a a a

ρ ≤ r ≤ a.

644

8 Dynamic Plasticity

If t = τ denotes the time at which the moving plastic hinge coincides with a given circle of radius r, then

r 3ρ 1+ τ = t1 1 − a a in view of (8.159). Since w = Uτ when t = τ or ρ = r, the integration of the above differential equation for w furnishes

ρ2 r−ρ r+ρ r , ρ ≤ r ≤ a. 1− 2 + w = Ut1 1 − a a 2a a

(8.160)

Within the circle 0 ≤ r ≤ ρ, the deflection at any instant has a constant value equal to Ut. The slope ∂w/∂r is continuous across r = p, although the velocity gradient is not. The deflection at the end of this phase is obtained by setting p = 0 in (8.160). The rest of the analysis for impulsive loading is essentially the same as that for the high-pressure pulse considered before. The deflection during the second phase may therefore be written as

1 t r t r2 r , w = Ut1 1 − −1 3− 1+ + 2 + a a 2a 2 t1 t1

t1 ≤ t ≤ t∗ (8.161)

where t1 is given by (8.159). The acceleration at r = 0 has a constant value equal to –U/t1 , and the bending moment distribution satisfying the differential equation and the boundary conditions is given by (8.148) with ρ = 0. The motion stops when t = t∗ , where t∗ = 2t1 =

μa2 U . 6M0

Setting t = 2t1 in (8.161), the shape of the deformed plate after it has finally come to rest is obtained as (Wang, 1955) w∗ =

r 1 r2 2r Ut1 1 − + 2 .β 3+ 2 a a a

(8.162)

It may be noted that the slope of the deflected plate is discontinuous at r = 0 over the range t1 ≤ t ≤ t∗ . The central deflection of the plate finally attains the value δ∗ = 3Ut1 /2. The ratio w/δ∗ is plotted as a function of r/a in Fig. 8.28 for t/t1 = 0.5, 1.0, and 2.0, the last two values defining the ends of the two plastic phases. The corresponding solution for a clamped circular plate has been given by Wang and Hopkins (1954). The solution for impact loading of an annular plate clamped at the inner radius has been discussed by Shapiro (1959), Florence (1965), and Johnson (1972). An analysis for a clamped circular plate impulsively loaded over a central circular area has been presented by Weirzbicki and Nurick (1996).

8.8

Dynamic Loading of Cylindrical Shells

645

Fig. 8.28 Deformed shape of a simply supported circular plate under blast loading at different instants of time (t1 = μa2 U/12M0 )

The influence of the shape of the pressure pulse on the dynamic behavior has been examined by Youngdahl (1971) and Krajcinovic (1972). The dynamic behavior of circular plates under a central pulse loading has been investigated by Florence (1977). The effect of transverse shear on the dynamic plastic response has been studied by Kumar and Reddy (1986). An analysis for the dynamic bending problem of square plates has been presented by Cox and Morland (1959), and that of rectangular plates by Jones (1970). The use of mode approximation in predicting the dynamic plastic response of plates has been discussed by Chon and Symonds (1977). The influence of rate sensitivity on the dynamic behavior has been considered by Perrone (1967) and Perrone and Bhadra (1984). The effect of membrane forces on shape changes in dynamically loaded plates has been investigated by Jones (1971) and Symonds and Wierzbicki (1979). The dynamic buckling of rectangular plates in the plastic range has been treated by Goodier (1968). A variety of other problems on the dynamic plastic behavior of plates have been considered by Nurick et al. (1987), Jones (1989), Yu and Chen (1992), and Zhu (1996).

8.8 Dynamic Loading of Cylindrical Shells 8.8.1 Defining Equations and Yield Condition Consider a circular cylindrical shell of radius a and thickness h, subjected to a uniform radial pressure p, whose initial value is greater than the quasi-static collapse pressure under identical boundary conditions. Such a loading will produce accelerated plastic flow that requires the inclusion of inertia effects in the theoretical framework. The state of stress in the shell is characterized by the axial bending moment Mx and the circumferential force Nθ acting per unit length of the circumference. The state of strain rate, on the other hand, is defined by the radially

646

8 Dynamic Plasticity

inward velocity w. As in the case of static analysis (Section 5.1), Mx will be taken as positive if it corresponds to tensile stresses on the inner surface, while Nθ will be reckoned positive when it is tensile in nature. The duration of the applied pressure is assumed small enough to justify the neglect of geometry changes in the analysis. If the surface density of the material of the shell is denoted by μ, the inertia force per unit area of the middle surface is μ(∂ 2 w/∂t2 ) acting in the radially outward sense, and the equations of dynamic equilibrium are ∂Mx − Q = 0, ∂x

∂Q Nθ ∂ 2w + +p=μ 2 , ∂x a ∂t

where Q is the shearing force per unit circumference and x is the distance measured along the length of the shell. The elimination of Q between the above equations gives the differential equation Nθ ∂ 2w ∂Mx + +p=μ 2 ∂x a ∂t

(8.163)

for simplicity, the material is assumed to have a constant uniaxial yield stress Y, the fully plastic values of the resultant force and moment being N0 = Yh and M0 = Yh2 /4, respectively. It is convenient at this stage to introduce the dimensionless qualities Mx mx = , M0

Nθ nθ = , N0

pa q= , Yh

ξ =x

2 , ah

τ=

t . t0

Denoting the differentiation with respect to τ by a superimposed dot, (8.163) can be expressed in the dimensionless form w ¨ ∂ 2 mx + 2 (nθ + q) = , 2 δ ∂ξ

(8.164)

where δ = Yht02 /2μa denotes a representative constant deflection of the middle surface. This equation can be integrated with the help of an appropriate yield condition and a suitable choice of the deflection function w(ξ , τ ). The analysis of the dynamic problem is greatly simplified by the use of the square yield condition shown in Fig. 8.29(a). It is an approximation not only to the Tresca yield condition (shown broken) but also to the von Mises yield condition (not shown). Referred to the dimensionless variables ξ and τ , the generalized strain rates may be defined as w˙ λ˙ θ = − , a

κ˙ x = −

2 ∂ 2w ˙ . ah ∂ξ 2

(8.165)

8.8

Dynamic Loading of Cylindrical Shells

647

Fig. 8.29 Dynamic loading of a cylindrical shell. (a) Yield condition and (b) clamped shell with a pressure pulse

When the stress point lies on one of the sides of the yield locus, the vector (N0 λ˙ θ ,M0 κ˙ x ) is directed along the outward normal to this side. At a corner of the yield locus, the vector must lie between the two extreme normals defined there. In any particular problem, the stress profile generally includes two or more plastic regimes, and the flow rule in each case can be easily established (Hodge, 1955). Considerations of equilibrium require the bending moment and the shearing force to be continuous, while cohesion of the material demands that the deflection and ˙ and w are all continuous across the velocity are continuous. Thus, mx ,∂mx /∂ξ,w, any boundary, moving or stationary. The velocity slope ∂ w/∂ξ ˙ may, however, be discontinuous across a hinge circle, which corresponds to a finite value of w˙ and an 2 . Such conditions can only hold at the corners A infinitely large value of ∂ 2 w/∂ξ ˙ and B of the field locus when nθ is compressive.

8.8.2 Clamped Shell Loaded by a Pressure Pulse A cylindrical shell of length 2 l is rigidly clamped at both ends, Fig. 8.29(b), and is instantaneously loaded at t = 0 by a uniform radial pressure p which is held constant for a sufficiently small time interval t0 . The static collapse pressure p0 , which must be exceeded for dynamic actions, can be determined in the same way as that using the hexagonal yield condition (Section 5.1). Considering one-half of the shell defined by 0 ≤ x = l, the incipient velocity at collapse is taken as w˙ = ξ, which gives κ˙ x = 0 and λ˙ < 0. The stress profile is therefore entirely on side AB, with x = 0 corresponding to point A and x = l to point B. Setting nθ = – 1, q = q0 , and w = 0 in (8.164), and integrating it under the boundary conditions mx = 1 at ξ = 0, and mx = 1 at ξ = ω, we obtain the solution

648

8 Dynamic Plasticity

mx = (q0 − 1) ξ (ω − ξ ) + 2 (ξ/ω) − 1. Since the bending moment is a relative maximum at x = l, the derivative ∂mx /∂ξ must vanish at ξ = ω, giving the dimensionless collapse pressure 2 q0 = 1 + 2 , ω

ω=l

2 . ah

For a range of values of q > q0 , it is reasonable to suppose that the same plastic regime AB is applicable for dynamic loading. Since the velocity at each instant is then proportional to ξ , the expressions for w˙ and w may be written as w˙ = w˙ 0 (ξ/ω) ,

w = w0 (ξ/ω) ,

(8.166)

where w0 (t) is the deflection at the central section ξ = ω. Substituting into the equilibrium equation (8.164), and setting nθ = – 1, we obtain the differential equation for mx as d 2 mx wξ ¨ . = −2 (q − 1) + 2 δω dξ

(8.167)

The assumed mode of deformation implies the formation of hinge circles at ξ = 0 and ξ = ω. The preceding equation may therefore be integrated under the boundary conditions ∂mx /∂ξ = 0 at ξ = ω and mx = –1 at ξ = 0, resulting in w ¨0 ξ2 ξ ω− . mx = −1 + (q − 1) ξ (2ω − ξ ) − 2δ 3ω The remaining boundary condition mx = 1 at ξ = ω furnishes the central acceleration w ¨ 0 = 3 (q − q0 ) δ,

0 ≤ τ ≤ 1,

(8.168)

in view of the expression for q0 , and the bending moment distribution in the first plastic phase becomes 1 ξ2 mx = −1 + (q − 1) ξ (2ω − ξ ) − (q − q0 ) ξ 3ω − , 2 ω

0 ≤ τ ≤ 1 (8.169)

Evidently, mx = mx (ξ ) in this phase, being independent of time. Integrating (8.168), and using the initial conditions w˙ 0 = w0 = 0 at τ = 0, we get w˙ = 3 (q − q0 ) τ , δ

3 w0 = (q − q0 ) τ 2 , δ 2

0 ≤ τ ≤ 1.

(8.170)

The preceding solution will be acceptable if the maximum bending moment does occur at ξ = ω. Since mx (ω)has the value q – (3q0 – 2), which is negative for q < 3q0 – 2, the applied pressure must satisfy the inequalities

8.8

Dynamic Loading of Cylindrical Shells

q0 = 1 +

649

2 6 < q < 1 + 2 = qc ω2 ω

(say).

(8.171)

When q exceeds the upper limit qc over this range, considered as the medium load range, mx is a relative minimum at ξ = ω, and its value exceeds unity in the neighborhood of this section, thereby violating the yield condition. Suppose that the applied pressure is instantaneously removed at t = t0 , which marks the beginning of a second plastic phase. For a sufficiently short shell subjected to medium load, the stress profile should continue to be on side AB. The central acceleration and the bending moment distribution are therefore given by (8.168) and (8.169), respectively, with q = 0, the latter quantity being

mx = − 1 + 2ωξ − ξ

2

1 ξ2 . + q0 ξ 3ω − 2 ω

(8.172)

The yield condition will not be violated in the neighborhood of ξ = 0 so long as m x ≤ 0. By (8.172), this condition is equivalent to q0 ≥

4 3

or

ω≤

√ 6.

Such shells will be regarded as short shells, as opposed to long shells for which the above inequalities are reversed. Since m x (ω) < 0 during the second plastic phase, the bending moment distribution (8.172) is acceptable for short shells. Integrating (8.168) after setting q = 0, and using the conditions of continuity of w˙ 0 and w0 at τ = 1, we get w˙ 0 = 3 (q − q0 τ ) δ, 3 w˙ 0 = q (2τ − 1) − q0 τ 2 δ, 2

⎫ ⎬ 1 ≤ τ ≤ q/q0 ,⎭

(8.173)

in view of (8.170). The shell comes to rest (w˙ = 0) when τ = τ ∗ = q/q0. The final deflection of the shell is w∗ = (ξ/ω)w∗0 , and it follows from (8.173) that w∗ 3q = δ 2

ξ q −1 , q0 < q < qc , q0 ω

ω2 ≤ 6.

(8.174)

The final value of the central deflection is therefore equal to 1.5δ when q and ω √ have their limiting values of 2 and 6, respectively. For long shells (ω2 > 6) under medium loads (q < qc ), the preceding solution still holds for τ < 1, but that for τ > 1 the solution is modified since the plastic regime does not apply throughout the shells. It is natural to expect that a region 0 < ξ < ρ near the built-in end would correspond to the plastic regime AD, for which mx = –1 and w˙ = 0. This portion of the shell therefore becomes rigid after being previously deformed. The section ξ = ρ defines the instantaneous position of the hinge circle which requires mx = –1 there. The remaining portion ρ ≤ ξ ≤ ω of the half-shell corresponds to the plastic regime AB and

650

8 Dynamic Plasticity

involves continuation of the plastic deformation. The velocity in this region may be written as ξ −ρ , ρ ≤ ξ ≤ ω, τ ≥ 1, (8.175) w˙ = w˙ 0 ω−ρ so that κ˙ x = 0, and w˙ is automatically made continuous across ξ = ρ. The distribution of mx in the deforming region can be determined by integrating (8.167) with q = 0, nθ = –1, and the expression w ¨ =w ¨0

ξ −ρ ω−ρ

,−

ρ˙ w ¨0 ω−ρ

ω−ξ ω−ρ

,

ρ ≤ ξ ≤ ω.

Using the boundary conditions ∂mx /∂ξ = 0 and ξ = ρ at the central section £ = co, the solution to the differential equation (8.167) in the region ρ < ξ < ω is obtained in the form 1 w ¨0 ρ˙0 w˙ 0 ω − ξ − w ¨0 + . (8.176) mx = 1 + (ω − ξ )2 1+ 2δ 6δ ω−ρ ω−ρ Since the bending moment and its derivative must be continuous across the hinge circle, the conditions mx = –1 and ∂mx /∂ξ = 0 at ξ = ρ must also be satisfied. Hence ρ˙ w˙ 0 2 ρ˙ w˙ 0 . −w ˙ 0 = 4δ, − 2w˙ 0 = 6δ 1 + ω−ρ ω−ρ (ω − ρ)2 These two relations may be combined together to express w˙ 0 and ρ˙ w ˙ 0 in terms of ρ, the result being w ¨ 0 = −2δ 1 +

6 (ω − ρ)2

,

6 ρ˙ w˙ 0 = 2δ 1 − . ω−ρ (ω − ρ)2

(8.177)

The substitution from (8.177) into (8.176) finally gives the bending moment distribution in the form

ω−ξ mx = 1 − 2 3 − 2 ω−ρ

ω−ξ ω−ρ

2 ,

ρ ≤ ξ ≤ ω.

(8.178)

It follows from the boundary conditions on mx that –1 ≤ mx ≤ 1 for ρ < ξ < ω. Since w ¨ 0 = ρ˙ (dw/dρ), ˙ the elimination of ρ˙ between the two relations of (8.177) leads to the differential equation d w˙ 0 + dρ

(ω − ρ)2 + 6 (ω − ρ)2 − 6

w˙ 0 = 0. ω−ρ

8.8

Dynamic Loading of Cylindrical Shells

651

Since ρ = 0 at τ = 1, when w˙ 0 = 3(q−q0 )δ, the integration of the above equation results in 3ω (q − q0 ) (ω − ρ)2 − 6 w˙ 0 , = δ ω2 − 6 (ω − ρ)

0≤ρ ≤ω−

√ 6.

(8.179)

To determine the variation of ρ with time, we substitute (8.179) into the second equation of (8.177) and obtain the differential equation 2 dρ = dτ 3ω

ω2 − 6 q − q0

2ω = 3

4 − 3q0 q − q0

.

Thus ρ˙ is a constant, which means that ρ varies linearly with the time, the result of integration of the above equation being ρ 2 = ω 3

4 − 3q0 q − q0

(τ − 1) ,

1 ≤ τ ≤ τ ∗.

(8.180)

√ The velocity everywhere vanishes when ρ = ω − 6, and the motion is terminated, the duration of the motion τ ∗ being obtainable from (8.180). The central deflection of the shell for τ ≥ 1 can be found by the integration of (8.179), using the fact that w˙ = ρ˙ (dw0 /dρ), and substituting for ρ. The result is easily shown to be $ %

6 ρ w0 3 3 (q − q0 ) ρ (2ω − ρ) + ln 1 − , = (q − q0 ) 1 + δ 2 4 − 3q0 ω 2 ω2 − 6 ω2 − 6

ω2 > 6

(8.181) The deflection at a generic section can be determined by integrating a similar equation obtained from (8.175) and (8.179), together with the change √ of variable to ρ. The final shape of the shell obviously corresponds to ρ = ω − 6. The ratio w0 /δ is plotted against t/t∗ in Fig. 8.30 for ω2 = 3 and 6, and for three different values of P/P0 The derivation of the uppermost broken curve is based on the analysis for high loads which is given below.

8.8.3 Dynamic Analysis for High Loads Consider the range of loads for which q > qc , applied to sufficiently short shells characterized by ω2 ≤ 6. In this case, a central portion of the shell is in regime B, while the remainder of the shell is in regime AB, the two portions being separated by a hinge circle. During the first plastic phase (0 ≤ τ ≤ 1), the hinge circle is fixed at a section ξ = α 0 , the velocity and deflection of the outer portion of the shell being given by w˙ = w˙ 0 ξ/α0 ,

w = w0 ξ/α0,

0 ≤ ξ ≤ α0 ,

(8.182)

652

8 Dynamic Plasticity

Fig. 8.30 Central deflection of a clamped cylindrical shell as a function of time under a uniform pressure pulse

where w0 represents a uniform deflection of the central portion α 0 ≤ ξ ≤ ω. Since mx = 1 and nθ = –1 over the length α 0 ≤ ξ ≤ ω, it follows from (8.164) that w ¨ 0 = 2 (q − 1) δ, which gives on integration w˙ 0 = 2 (q − 1) τ , w

w0 = (q − 1) τ 2 , δ

0 ≤ τ ≤ 1.

(8.183)

The bending moment distribution in the region 0 ≤ ξ ≤ α 0 and the quantity α 0 are determined by the integration of (8.167), where α 0 is written for ω, using the boundary conditions mx = –1 at ξ = 0 and mx = 1, ∂mx /∂ξ = 0 at ξ =α 0. Thus mx = 1 − 2 (1 − ξ/α0 )3 , mx = 1,

nθ = −1,

nθ = −1,

(q − 1) α02 = 6,

or

0 ≤ ξ ≤ α0 ,

α0 ≤ ξ ≤ ω,

α0 =

(8.184)

6/ (q − 1).

For τ > 1, the load is absent, and the hinge circle separating the two regions progressively moves toward the central section, its position at any instant being denoted by ξ = α. The velocity (but not the deflection) is still given by (8.182) with a written for α 0 , the central acceleration being given by w0 = −2δ. Integrating, and using the initial conditions w0 = 2(q – 1)δ at τ = 1, we get

8.8

Dynamic Loading of Cylindrical Shells

653

w˙ 0 = 2 (q − τ ) δ,

1 ≤ τ ≤ τ1 ,

where τ 1 is the value of r when the hinge circle reaches ξ = ω. The velocity and displacement in the central region during the second plastic phase are given by w˙ = 2 (q − τ ) , δ

w = (2τ − 1) q − τ 2 , α ≤ ξ ≤ w, δ

(8.185)

The generalized stresses in this region are mx , = 1 and nθ = – 1. In the region 0 ≤ ξ ≤ α, the bending moment distribution can be directly written from the conditions mx (0) = −1,mx (α) = 1,mx (α) = 0, and mx (0) = 2, the result being α2 ξ α2 ξ 3 2 , +ξ − 1+ mx = −1 + 3 − 2 α 2 α3

0 ≤ ξ ≤ α.

(8.186)

Inserting this expression into the differential equation (8.164), setting nθ = – 1 and q = 0, and using the fact that w˙ ξ w˙ α˙ ξ = 2 (q − τ ) , = −2 1 + (q − τ ) , δ α δ α α

0 ≤ ξ ≤ α,

(8.164) is found to be satisfied if α is given by the differential equation (q − τ )

6 + α2 dα = , dτ 2α

which is readily integrated under the initial condition α = α 0 when τ = 1 to give q−1 6 + α2 = 2 q −τ 6 + α0

' or

α=

6τ , q−τ

(8.187)

in view of the last equation of (8.184). Since α = ω when τ = τ 1 , (8.187) furnishes

τ1 = qω2 / 6 + ω2 = q/qc in view of (8.171). The instant τ = τ 1 marks the end of the second plastic phase, since the hinge circle reaches the central section and can go no further. In order to complete the solution for the second plastic phase (1 ≤ τ < τ 1 ), it is necessary to find the deflection in the region 0 ≤ ξ ≤ α by the integration of the differential equation ∂w 2 q ξ = 2δ (q − τ ) = δ ξ (q − τ ) − 1, ∂τ α 3 τ

0 ≤ ξ ≤ α.

654

8 Dynamic Plasticity

The solution is straightforward for the region 0 ≤ ξ ≤ α 0 , which has always been in the plastic regime AB, the initial condition for this region being w0 /δ = (q – 1) ξ /α at t = 1, in view of (8.182) and (8.183). The integration of the above equation therefore gives τ 3 2 w ξ −1 −1 1 =√ q sin − sin √ δ q q 6 2 3 5 τ (q − τ ) − q q − 1 , 0 ≤ ξ ≤ α0 . q−τ + 2 2

(8.188)

The elements in the region α 0 ≤ ξ ≤ α have passed from regime B to regime AB at different instants as they have been traversed by the moving hinge circle. Let τ be the value of τ for which a typical section of the shell coincides with the hinge circle. Setting α = ξ and τ = τ in (8.187), we have

τ = qξ 2 / 6 + ξ 2 . The deflection of the element at this instant is given by (8.185) with τ = τ . Using this as the initial condition, the solution for the deflection in the region α 0 ≤ ξ ≤ α is easily shown to be τ 3 2 5 −1 −1 ξ τ (q − τ ) q sin − sin √ q−τ + 2 q 2 6 0.5ξ 2 −q 1+ , α0 ≤ ξ ≤ α. 6 + ξ2

ξ w =√ δ 6

(8.189)

It is readily verified that the deflection is continuous at ξ = α 0 . The continuity of the deflection at ξ = α is also ensured by the fact that the right-hand side of (8.189) at ξ = a coincides with that given by (8.185). For τ > τ 1 , the entire stress profile is in plastic regime AB, and the velocity distribution throughout the shell is given by the first equation of (8.182) with w to written for α 0 . The bending moment distribution over the entire shell becomes ω2 ξ ω2 ξ 2 , 0 ≤ ξ ≤ ω, + ξ2 − 1 + mx = −1 + 3 3 − 2 ω 2 ω3

(8.190)

obtained by simply replacing α by w in (8.186). The substitution in the differential equation (8.167) with q = 0 then gives the central acceleration as w ¨ 2 = − 1 + 2 = −3q0 . δ ω In view of the continuity of the velocity w ˙ at τ = τ 1 , the integration of the above equation results in the velocity field

8.8

Dynamic Loading of Cylindrical Shells

w˙ ξ = 3 (q − q0 τ ) , δ ω

655

τ1 ≤ τ ≤ τ ∗ , 0 ≤ ξ ≤ ω,

where τ ∗ = q/q0 , representing the instant when the motion is terminated. A straightforward integration furnishes the deflection at any point as 3 w ξ = τ (2q − q0 τ ) + λ (ξ ) , δ 2 ω

q q ≤τ ≤ , qc q0

(8.191)

where λ(ξ ) must be determined from the condition of continuity of w at τ = τ = q/qc . Using (8.188) and (8.189), it is easily shown that ⎫ ⎪ 3 ⎪ −1 1 −1 1 ⎪ − sin √ q sin − q − 1 , 0 ≤ ξ ≤ α0 , ⎪ ⎬ 2 qc q ⎪ 3 ω ξ qξ qξ ⎪ ⎪ ⎪ − q, α − ≤ ξ ≤ ω. λ (ξ ) = q tan−1 √ − tan−1 √ 0 ⎭ 2 2 2 6 + ξ 6 6 (8.192) qξ λ (ξ ) = 2

Setting τ = q/q0 in (8.191), the final shape of the deformed middle surface of the shell is obtained as w∗ 3q2 ξ = + λ (ξ ) , δ 2q0 ω

q ≥ qc ,

ω2 ≤ 6.

(8.193)

When q=qc , we have α 0 –w and λ(ξ ) = –3qξ/2ω, which reduces (8.193) to (8.174) as expected. Setting ξ = ω in (8.193) and using (8.192), the central deflection in the final phase is found to be given by q 3 w0 = q 3τ − − 1 − q0 τ 2 , δ 2qc 2

q q ≤τ ≤ , qc q0

(8.194)

where q0 and qc depend only on ω and are given by (8.171). Figure 8.31 displays the final deformation pattern of the clamped shell when ω2 = 3, each curve being based on a definite value q. For longer shells (ω2 > 6), the stress and velocity distributions are identical to those for short shells during the period of application of the load, but the subsequent part of the solution, following the load removal, is modified, due to the presence of a second hinge circle which begins at ξ – 0 and moves along the length of the shell. The effects of blast loading under different end conditions on the dynamic plastic behavior have been discussed by Hodge (1956b, 1959). The dynamic plastic response of cylindrical shells under a band of pressure has been discussed by Eason and Shield (1956), Kuzin and Shapiro (1966), Youngdahl (1972), and Li and Jones (2005). The influence of membrane forces has been examined by Jones (1970) and Galiev and Nechitailo (1985). The dynamic plastic response of spherical caps under pulse and impact, loadings has been investigated by Sankaranarayanan (1963, 1966). The dynamics of an impulsively loaded

656

8 Dynamic Plasticity

Fig. 8.31 Final shape of the deformed meridian of a clamped cylindrical shell subjected to a highpressure pulse

cylindrical shell based on a generalized yield condition has been investigated by Lellep and Torn (2004). The progressive crumpling of cylindrical tubes under axial compression has been discussed by Abramowicz and Jones (1984) and Jones (1989). The dynamic plastic buckling of cylindrical shells has been considered by Vaughan and Florence (1970) and Jones and Okawa (1976) and that of a complete spherical shell by Jones and Ahn (1974). A simplified method of analysis for the plastic buckling, based on energy considerations, has been discussed by Gu et al. (1996).

8.9 Dynamic Forming of Metals 8.9.1 High-Speed Compression of a Disc Consider the rapid compression of a short circular cylinder between a pair of parallel platens, or dies, the speed of compression being such that the inertia effects are significant. The lower die z = 0 is stationary, while the upper die z = h is assumed to move down with a constant speed U, the coefficient of friction μ between the dies and the cylindrical block being assumed to be constant (Haddow, 1965). The elastic and plastic stress waves initiated at the upper die travel up and down the block several times during the compression process. The load acting on the upper platen, which is significantly in excess of the quasi-static value in the early part of the process, appreciably decreases during the final stages. Only the incipient com-

8.9

Dynamic Forming of Metals

657

pression of a thin block of uniform diameter will be considered in what follows, ignoring the effect of wave propagation on the dynamic process. Since the distribution of stresses and strains is symmetrical about the axis of the block, which is assumed to coincide with the z-axis as shown in Fig. 8.32(a), the equation of radial motion of a typical element in cylindrical coordinates (r, θ , z) may be written as σr − σθ ∂τrz ∂σr + + =ρ ∂r r ∂z

∂u ∂u ∂u +u +ω , ∂t ∂r ∂z

where u and w are the radial and axial components of the velocity, and ρ is the density of the material. If the influence of barreling is disregarded, σ r , σ θ , σ z , and u are independent of z, but the presence of die friction requires τ rz to vary with z so that τ rz = μp at z = 0 and τ rz = –μp at z = h, where ρ is the die pressure. The multiplication of the preceding equation by dz and integration between the limits 0 and h therefore result in ∂u σr − σθ 2μp ∂u ∂σr + − =ρ +u . (8.195) ∂r r h ∂t ∂r

Fig. 8.32 High-speed compression of short cylinders. (a) Condition of loading and (b) mean die pressure against kinetic energy of impact

In the absence of barreling, the velocity field corresponds to a uniform compression of the block and is consequently given by

658

8 Dynamic Plasticity

u=

Ur Uz , w=− , 2h h

(8.196)

satisfying the condition of plastic incompressibility. Since ε˙ r = ε˙ θ according to (8.196), we have σ r = σ θ for an isotropic material, and the yield criteria of both Tresca and von Mises reduce to σr − σz = σθ − σz = Y, where σ z = –p, and the material is considered as ideally plastic with a uniaxial yield stress Y. Substituting from (8.196) into (8.195), setting σ e = σ r , and using the yield criterion, we obtain the differential equation dp 2μp 3ρU 2 r . + =− dr h 4 h2 Since σ r must vanish along the cylindrical surface, the boundary condition is p = Y at r = a. The integration of the above equation therefore furnishes p 2μ (a − r) 3ρU 2 1 a r 3ρU 2 1 = 1− − − exp + . (8.197) Y 8Yμ 2μ h h 8Yμ 2μ h This equation predicts the die pressure distribution as a function of r. The inertia effect represented by the parameter ρU2 /Y is therefore to increase the die pressure at any given radius. The mean die pressure p¯ corresponding to (8.197) is easily shown to be given by p¯ 2μa h h a 3ρU 2 1 h = − exp 1− − 1+ Y qμ 8Yμ 2μ h 2μa h 2μa (8.198) 2 3 a ρU − . + 4Yμ 4μ h When the ratio ρU2 /Y is vanishingly small, this formula reduces to (3.57), obtained for the quasi-static compression. Expanding exp(2μa/h) in ascending powers of μa/h, and neglecting terms containing powers of μ higher than the fourth, (8.198) can be reduced to 2μa 3ρU 2 a 2 p¯ , =1+ + Y 3h 16Y h which is sufficiently accurate for small values of μ. Figure 8.32(b) shows the variation of the mean die pressure with ρU2 /Y for different values of a/h in the special case of μ = 0.1. In the case of steel, for instance, the inertia effect is significant when the speed of compression exceeds about 50 m s–1 . The situation where the speed of compression varies with time has been considered by Lippmann (1966) and Dean (1970).

8.9

Dynamic Forming of Metals

659

The inclusion of the strain rate sensitivity of the material, based on the homogeneous deformation mode (8.196), is quite straightforward. Since the effective strain rate ε˙ has a constant value equal to U/h, it is only necessary to replace Y in the preceding analysis by the modified yield stress n n U ε˙ =Y 1+ , Y 1+ α αh where α and η are material constants to be determined by experiment. In particular, the simplified formula for small μ is easily shown to be modified to n U 2μa p¯ 3ρU 2 a 2 . = 1+ 1+ + Y 3h αh 16Y h

(8.199)

This is a complete generalization of the well-known Siebel formula for the plastic compression of short cylinders. It involves the estimation of the coefficient of die friction, the speed of compression, and the empirical constants characterizing the dynamic response of the material. A upper bound analysis for the dynamics of a closed die forging process has been discussed by Scrutton and Marasco (1995).

8.9.2 Dynamic Response of a Thin Diaphragm A thin circular diaphragm of initial thickness h0 is rigidly held along its periphery r = a and is subjected to a uniform velocity U normal to its plane (Hudson, 1951). An elastic wave front immediately sweeps inward from the edge, producing a radially outward motion of the material particles. At any later instant, a plastic bending wave generated at the edge travels some distance toward the center, producing a bulged shape of the diaphragm as shown in Fig. 8.33(a). The annular region swept over by the bending wave forms a surface of revolution, which is assumed to have come to rest, while the flat central region yet unaffected by the wave retains its normal velocity U. Since we are dealing with large plastic strains, all elastic effects other than those of the initial stress wave will be neglected, the material being effectively considered as rigid/plastic in the dynamic analysis of the process. At any time t after the beginning of the process, let b denote the radius of the central flat portion of the diaphragm that has been uniformly deformed to a thickness h under the action of a constant normal velocity U and a variable radial velocity υ induced at r = b. During a time internal dt, an elemental ring of width ds just ahead of the bending wave impulsively rotates to form a part of the bulge after being swept over by the wave. Since the radial velocity of the ring relative to that of the wave is equal to υ–b, we have ds = (υ –b) dt. The radial and transverse components of the displacement of the inner edge of the ring are –b dt and U dt, respectively, giving √ ds = U 2 + b2 dt. Consequently,

660

8 Dynamic Plasticity

Fig. 8.33 Impact loading of a circular diaphragm. (a) Geometry of deformation and (b) deformed shape for different initial velocities

ds = υ − b˙ = U 2 + b˙ 2 dt

or

υ 2 − U2 b˙ = . 2υ

(8.200)

Since b˙ is negative, it follows from (8.200) that v < U throughout the deformation. The assumed uniformity of thickness in the flat central region requires a state of balanced biaxial tension σ r = σ θ to exist in this region. The radial and circumferential strain rates are therefore equal to one another, and the rate equation of incompressibility is h˙ + 2˙εθ = 0 h

or

h˙ 2υ + = 0. h b

(8.201)

when considered at the interface r = b. If r0 denotes the initial radius to a particle that is currently at a radius r in the interior of the central region, then the integrated form of the flow rule gives εθ = ε r = 12 ln (h0 /h). Hence ∂r r = = r0 ∂r0

h0 . h

The assumed uniformity of the stress and strain in the central region is incompatible with the existence of inertia forces, the effect of which is disregarded in the yielding and flow of the material. Consequently, σ r is equal to the current yield stress σ , which is a function of the total compressive thickness strain equal to ln(h0 /h). Let q denote the resultant radial stress, including the inertia effect, in the material just ahead of the bending wave at r = b. As the wave sweeps inward over an annular element of width ds, the work done by the stress during the time interval dt is equal to –qhbdθ (υ dt), where dθ is the angle subtended by the element at the center of the disc. Since the corresponding change in kinetic energy of the element is –(ρ/2)h ds dθ (υ 2 + U2 ), we have

8.9

Dynamic Forming of Metals

q=p

661

υ 2 + U2 2υ

ds = ρ U 2 + b˙ 2 dt

(8.202)

in view of (8.200). If the radially outward accelerating force acting on a typical element of mass phr dr dθ in the central region at any instant is denoted by dF dθ = (∂F/∂r), dr dθ , then the equation of motion for this element may be written as ' ∂F h d2 h0 ∂ 2r 2 = ρhr 2 = ρhr . 2 ∂r ∂t h0 dt h It is reasonable to suppose that the rate of work done by the distribution of this force over the central region is equal to that produced by the stress difference q – σ r occurring at r = b. Then

υ ∂r dF = ∂t b

(q − σr ) hbυ =

b

r 0

∂F dr. ∂r

Substituting for ∂F/∂r and integrating, we obtain the result ' 1 q = σ + ρb2 4

h d2 h0 dt2

h0 h

.

(8.203)

The right-hand side of this equation depends only on the thickness ratio h/h0 . The solution to the system of equations (8.201) to (8.203) requires the specification of an initial value of v at the bending wave r = b. Assuming a linear distribution of velocity in the central region, Hudson (1951) obtained the initial condition ' υ = 2Y

1−ν ρE

at t = 0,

where E is Young’s modulus, and ν is Poisson’s ratio for the material of the diaphragm. The motion begins at t = 0 when the stress at the clamped edge rises suddenly from zero to the initial yield stress Y. We begin with the situation where the material is nonhardening, so that σ – Y throughout the motion. It is convenient to introduce the dimensionless parameters Ut z b ξ= = , α= , a a a

υ β= , η= U

h0 , h

λ=

4Y , ρU 2

where z is the current height of the central flat part above the initial plane of the diaphragm. Equations (8.200) and (8.201) immediately become 2β

dα = − 1 − β2 , dξ

α

dn = ξ η, dξ

(8.204)

662

8

Dynamic Plasticity

while the elimination of q between (8.202) and (8.203) leads to the differential equation α2 d2 η dα 2 = (λ − 4) + , 4 dξ η dξ which can be combined with the second equation of (8.204) to eliminate η. After some algebraic manipulation using the first equation of (8.204), the resulting equation is reduced to 1 + β2 2 + β2 dβ . (8.205) = −λ + α dξ 2β 2 Equations (8.204) and (8.205) form a set of three basic differential equations for the three unknowns α, η, and β as functions of ξ , the initial conditions being α = η = 1,

β = β0 =

λ (1 − ν) Y/E

when ξ = 0.

To carry out the integration, we combine (8.205) with the two equations of (8.204) in turn to obtain the results 1 − β 2d β dα , =− α 2 − (2λ − 3) β 2 + β 4

dη 2β 3 d β . = η 2 − (2λ − 3) β 2 + β 4

Although these equations can be integrated exactly, it is more convenient for practical purposes to introduce a minor approximation (since β 4 is usually a small fraction) to express the solution in the form (m−1)/4m2 ⎫ ⎪ β02 − β 2 ⎪ mβ 2 − 1 ⎪ exp α= ,⎪ ⎪ ⎪ 2 ⎬ 4m mβ0 − 1 ⎫ 2 1/2m2 ⎪ mβ0 − 1 β 2 − β02 ⎬ ⎪ ⎪ ⎪ ⎪ , η= exp ⎭ ⎭ ⎪ 2m mβ 2 − 1

(8.206)

where m = λ − 32 . These relations furnish η, as a function of α parametrically through β. Finally, ξ can be found as a function of β by the numerical integration of (8.205). When λ is sufficiently large, so that m ≈ λ, the thickness h and the height z of the central flat part are closely approximated by the relations h ≈ h0

4/λ b , a

ρ z b ≈U 1− . a Y a

(8.207)

The diaphragm is therefore deformed into a conical shape, the central deflection being proportional to the initial velocity √ U. The radial velocity v of the flat part rapidly increases to its terminal value U/ λ. When the diaphragm is completely formed

8.9

Dynamic Forming of Metals

663

(α = 0), thickness vanishes at the center, no doubt as a result of the mathematical idealization. The total time √ for the deflection, which is known as the swing time, is shows the shape of the profiles of the approximately equal to a ρ/Y. Figure 8.33(b)√ completely deformed diaphragm for β 0 = 0.04 λ with different values of λ. The simplest way of taking into account the effect of work-hardening of the material is to consider a mean yield stress based on the given stress–strain curve. It is therefore only necessary to replace Y in the expression for λ by a quantity that depends on ε = ln(h0 /h). The general effect of work-hardening is to decrease the central deflection for a given initial velocity U. The predicted shape of the profile is found to be in complete qualitative agreement with observation. Figure 8.34, which has been obtained experimentally by Keil (1960), indicates the difference between the static and dynamic behaviors of a clamped circular diaphragm. An energy method of analysis for a circular membrane subjected to impact loading has been discussed by Boyd (1966). The propagation of plastic waves in an impulsively loaded circular membrane has been discussed by Munday and Newitt (1963) and Cristescu (1967).

Fig. 8.34 Comparison of static and dynamic responses of clamped circular diaphragms (after A.H. Keil, 1960)

664

8 Dynamic Plasticity

8.9.3 High-Speed Forming of Sheet Metal A variety of techniques have been developed in recent years for the forming of sheet metal using loading rates that are large enough to have a dominating effect on the deformation process. Chemical explosives are commonly used to generate the energy which is transmitted to the workpiece through an intervening medium such as water. The forming die, together with the workpiece, is immersed in water contained in a forming tank, and the air in the die cavity is often evacuated with the help of a vacuum pump. The explosive charge is located at a suitable stand-off distance from the workpiece, and the kinetic energy released by the detonation of the charge is utilized in forming the component. Detailed accounts of the high-rate forming of metals have been presented by Rinehart and Pearson (1965). One of the common methods of explosive forming of sheet metals is indicated in Fig. 8.35(a). An air-backed circular blank is clamped around the periphery and is subjected to a shock wave generated by the detonation of an explosive at a stand-off distance approximately equal to the blank radius. The shock wave provides the necessary kinetic energy which is dissipated in doing the plastic work. The mean initial velocity necessary for the material to attain a given polar strain in the deformed state can be approximately estimated from an assumed distribution of the thickness strain in the bulge (Johnson, 1972). Experiments tend to indicate that the final hoop strain is approximately equal to the radial strain, which means that the equivalent strain is approximately equal to the compressive thickness strain whose greatest value occurs at the pole. In the explosive-forming process, the final shape of the blank depends on such factors as the hydrostatic head, the stand-off distance, the type of explosive, and the

Fig. 8.35 Explosive forming of circular blanks. (a) Experimental setup and (b) experimental results for stand-off distances of 3, 6, and 9 inches (after Travis and Johnson, 1962)

Problems

665

weight of the charge. Figure 8.35(b), which is due to Travis and Johnson (1962), indicates how the hoop strain and thickness strain distributions in the deformed blank are affected by the ratio of the stand-off distance to the blank radius. When the size of the blank is relatively small, a closed system in which the water is contained in the die that is closed by a plate is sometimes used for maintaining the pressure for a longer interval of time. The related problem of deep drawing of cylindrical cups using an explosive device has been investigated by Johnson et al. (1965). An interesting situation arises in the high-speed blanking of sheet metal, using a typical setup of die and punch. The width of the zone of shearing, which is confined near the area of clearance between the die and punch, decreases as the speed of the operation increases. The energy required for the separation of the blank from the stock generally increases with an increase in the speed of blanking, and this phenomenon may be attributed to the strain rate sensitivity of the material. The nonuniformity of the sheared edge, which is an essential feature of the process, constitutes a major challenge for its improvement. Useful experimental results on high-speed blanking have been reported by Johnson and Travis (1966), Balendra and Travis (1970), and Dowling et al. (1970). An electromagnetic method of highspeed bulging of tubes has been investigated by Aizawa et al. (1990).

Problems 8.1 A vertical wire of length l and cross-sectional area A is made of a rigid-perfectly plastic material with a rate-sensitive yield stress. σ . The wire is suspended from a ceiling and has an attached mass m at the lower end which is given an initial axial velocity u0 . The axial displacement and velocity of the mass at any instant t are denoted by x and u, respectively. Neglecting the mass of the wire, form the equation of motion, and using (8.107) show that the final displacement δ when the motion comes to a stop is given by

δ =μ l

ξ0 0

ξ dξ , 1 + ξ 1/n

ξ=

u , αl

μ=

mlα 2 YA

where ξ 0 denotes the initial value of ξ . Assuming n = 5 and ξ 0 = 32, determine the numerical value of δ/l in terms of μ Obtain also the result based on the approximation ξ ≈ ξ0 in the denominator of the integrand and verify that it differs only marginally from the exact value. 8.2 A thick-walled spherical shell made of a nonhardening rigid/plastic material with a uniaxial yield stress Y, and having an internal radius a0 and external radius b0 , is subjected to an explosive internal pressure that produces large plastic expansion of the shell. Let p denote the magnitude of the pressure at any instant when the radii of the shell have increased to a and b respectively. Using the equation of motion and the yield criterion, together with the incompressibility condition, show that

dλ a2 a (p/Y) − 2 ln (b/a) λ= 1+ 2 + 4− 1+ , dξ b 1 − a/b b

λ=

ρυa 2 , 2Y

ξ = ln

a a0

666

8 Dynamic Plasticity

where ρ is the density of the material and υ a = da/dt is the radial velocity at r = a. Using the relation p = pe (a0 /a)3γ , this equation can be integrated numerically with the initial condition λ = 0 when ξ = 0 for any given values of b0 /a0 , p0 /Y, and γ . Verify that the hoop stress is tensile throughout the shell, vanishing at r = a when p = Y, which may be regarded as the criterion for the dynamic rupture. 8.3 A thin spherical shell having an initial thickness h0 , mean radius a0 , and density ρ is subjected to an internal explosion by detonating a concentric spherical charge. The explosion pressure is large enough to be approximated by the relation p = ρhυ, ˙ where υ is the radial velocity and h the instantaneous thickness when the shell radius is a. Using the relation p = p0 (a0 /a)3γ , where p0 is the effective detonation pressure and γ is a constant isentropic expansion index, show that

ν 2 U

'

a 3(γ −1) 2p0 , = 1− 0 3 (γ − 1) Y a

U=

Ya0 ρh0

Note that the velocity υ rapidly approaches a constant value obtained by omitting the second term in the curly brackets. Based on this constant velocity, show the time taken by the shell to rupture (defined by p = Y) and the corresponding shell radius are given by ' U tf = a0

(γ − 1)

3Y 2p0

af a0

−1 ,

p0 = Y

af 3γ a0

8.4 A lateral load P is suddenly applied at the tip of a cantilever of length l and mass m at t = 0, and held constant for a short period. For moderate values of P exceeding the static collapse load P0 = M0 /l, the beam rotates as a rigid body about a plastic hinge formed at the built-in end. Considering the equations of motion of a segment of length x measured from the tip, and using th boundary conditions, obtain the distribution of shearing force Q and bending moment M in terms of p = P/P0 as x 3x Q 2− , = −p + (p − 1) P0 2l l

M x x x 3− , 1≤p≤3 = − p − (p − 1) M0 l 2l l

Verify that the reaction at the built-in end vanishes when p = 3, and must continue to do so when p > 3, for which the plastic hinge is located at a distance b0 < l from the tip of the cantilever. Considering the equations of motion of the segment of length b0 x, and using the boundary conditions, show that Q x 2 = −p 1 − , P0 b0

M x =− M0 l

x2 3x + 2 3− b0 b0

,

b0 3 = , l p

p≥3

over the length 0 ≤ x ≤ b0 , the remainder of the cantilever being at rest. Verify that the tip acceleration is constant during this phase, giving a tip deflection of w0 = p2 t2 (P0 /3m) when p ≥ 3. 8.5 Considering the range of loading p ≥ 3, applied over a short period 0 ≤ t ≤ t0 , suppose that the load is suddenly removed. The plastic hinge then moves away from x = b0 toward the built-in end to assume a position x = b at any instant t, the part of the cantilever traversed by the plastic hinge being curved. Form the equations of linear and angular momentum to show that b = 3t/pt0 during this phase. Setting δ = P0 t0 2 /m, prove that the tip deflection and the shape of the curved portion are given by

Problems

667

p2 w0 = δ 3

b 1 + 2 ln , b0

2p2 w = δ 3

ln

b x + −1 , x b

(b0 ≤ x ≤ b)

On reaching the built-in end x = l, the plastic hinge becomes anchored at the root, while the cantilever rotates as a rigid whole with a constant angular acceleration until the motion ceases. Show that the tip velocity during this phase and the associated tip deflection are given by w ˙0 2 = δ t0

3t 2p − , t0

w0 = δ

p2 t 3t p + −p p− 1 + 2 ln , t0 t0 3 3

2p t p ≤ ≤ 3 t0 3

8.6 A free-ended beam of length 2l, made of a rigid/plastic material with a fully plastic moment M0 , is subjected to a central impact load P which increases with time. Show that a plastic hinge begins to form at the middle of the beam when q = Pl/4 M0 is equal to unity. For q > 1, each half of the beam rotates as a rigid body with an angular acceleration α, while the central hinge moves on with a linear acceleration a. Considering the equations of motion of one-half of the beam, show that μl2 a = 2 (4q − 3) , M0

μl3 α = 12 (q − 1) M0

where μ is the mass per unit length of the beam. Derive an expression of the bending moment at a distance x from the middle of the beam, and show that the maximum value of the moment has the magnitude M0 when 3x/l = q/(q – 1), the value of q at this stage being 5.725 given by the cubic 4

q 3 3

− 10.5

q 2 3

+6

q 3

−1=0

8.7 An annular plate, made of a rigid/plastic Tresca material, is clamped along its inner edge r = a and is given a constant normal velocity U along its free outer edge r = b. A plastic hinge initiated along the outer edge at t = 0 travels inward to some radius ρ = λb at a generic instant. Assuming a velocity field w ˙ = U (r − ρ)/(b − ρ) for ρ ≤ r ≤ b, using the equations of dynamic equilibrium with Mθ = - M0 , and setting t0 = μ b2 U/12 M0 , where μ denotes the surface density of the material, show that t0 λ˙ = −

2 (1 + 2λ) 1 , q=− (1 − λ) (1 + 3λ) (1 − λ) (1 + 3λ)

where q is the value of Q/bM0 at r = b. Show also that the time interval t and the deflection w at any radius r can be expressed in terms of the parameter λ as

1 t = t0 (1 − λ) 1 − λ2 , w = Ut0 (ξ − λ)2 (1 + ξ + 2λ) , 2

λ≤ξ ≤1

where ξ = r/b. Derive also the radial bending moment distribution in this part of the plate in the form Mr = −1 + M0

⎫ ⎧ 2 2 2 ξ − λ ⎨ (2 − λ − ξ ) λ + ξ + 2λξ − 2 (3 − 2λ) λ ⎬ , ⎭ 1−λ ⎩ (1 − λ)3 (1 + 3λ)

λ≤ξ ≤1

668

8 Dynamic Plasticity

8.8 In the preceding problem, let the imposed velocity U be suddenly removed at the instant t = t1 , when the plastic hinge reaches the clamped edge r = a. The plate then rotates about this edge with a deceleration, the deflection of the plate at any subsequent instant being expressed in the form w (ξ ,t) = w1 + U

ξ −α 1−α

t ≥ t1 = t0 (1 − α)2 (1 + α) ,

f (t) ,

where α = a/b, and w1 is given by the preceding expression of w considered at λ = α. The function f (t) must satisfy the initial conditions f (t1 ) = 1 and f (t1 ) = 0. Show that the equation of dynamic equilibrium for the bending moment, along with the boundary conditions, can be satisfied if

1 f (t) = (t − t1 ) 1 − 2

t − t1 t22 − t1

,

t2 − t1 = t0 (1 − α)2 (3 + α)

where t2 denotes the instant when the motion stops. Verify that the final deflection of the plate exceeds w1 by the amount (Ut0 /2) (ξ – α) (1 – α) (3 + α), and hence obtain the final shape of the plate. 8.9 A rigid/plastic circular cylindrical shell of length 2l, mean radius a, and thickness h, is clamped at both ends and subjected to an explosive blast loading, in which the external pressure p rises instantaneously to a peak value and then decays exponentially with time according to the law q=

pa = λq0 e−τ , Yh

q0 = 1 +

2 , ω2

ω=l

2 , ah

τ=

t t0

where λ > 1 is a constant, and t0 is a constant time. Let x = ξ l denote the axial distance measured from the left-hand edge of the shell. Using the square yield condition, show that the axial bending moment, radial velocity, and deflection of the shell for moderate values of λ are given by λq e−τ − 1

2 0 mx = −1 + ξ 3 − ξ 2 + ξ (1 − ξ )2 , 1 ≤ λ ≤ 3 − q0 − 1 q0 3 3 w˙ τ w − λ 1 − e−τ ξ = q0 λ 1 − e−τ − τ ξ , = q0 τ λ − δ 2 δ 2 2 where δ = Yht0 2 /2aμ with μ denoting the surface density, while the dot denotes the derivative with respect to τ . Determine the central deflection of he shell when the motion comes to a stop. 8.10 Consider the range λ ≥ 3 − 2/q0 , for which there is a moving hinge circle located at ξ = ρ. The stress point remains on the same side of the locus over the region 0 ≤ ξ ≤ ρ, while the remainder of the shell corresponds to the corner of the yield locus. Integrating the equation of motion, obtain the radial acceleration for the two portions of the shell, and hence show that 3ξ w ˙ = δ 2

τ

0

q0 − 1 dτ λq0 e−τ − 1 − ρ ρ2

w ˙ = λq0 1 − e−τ − τ δ

(ρ ≤ ξ ≤ 1)

(0 ≤ ξ ≤ ρ) ,

Problems

669

Invoking the continuity of the radial velocity at ξ = ρ, dividing the resulting equation by ρ, and then differentiating with respect to τ , show that the parameter ρ, the time τ 1 for which ρ = 1, and the final deflection of the central section are given by 2 3 (q0 − 1) τ 1 − e−τ1 (0 ≤ τ ≤ τ =3− ), λ 1 τ1 q0 λq0 1 − e−τ − τ

3 3 w0 τ τ τ + = q0 λ τ2 − 1 − τ2 1 + 2 q0 − 1 τ1 1 + 1 δ 2 3 2 2 2

ρ2 =

where τ2 > τ1 denotes the instant when the motion stops. Assuming q0 = 1.4, obtain a graphical plot for the variation of w0 /δ with λ, covering the range 1 ≤ λ ≤ 10.

Chapter 9

The Finite Element Method

In the numerical solution of engineering problems, it is often convenient to assume the physical domain to consist of an assemblage of a finite number of subdomains, called finite elements, which are connected with one another along their interfaces. The distribution of a governing physical parameter within each element is approximated by a suitable continuous function, which is uniquely defined in terms of its values at a specified number of nodal points that are usually located along the boundary of the element. The solution to the original boundary value problem is often reduced to that of a variational problem involving the nodal point values of the unknown parameter. In this chapter, we shall be concerned with a rigid/plastic formulation of the finite element method, a complete elastic/plastic formulation of the problem being available elsewhere (Chakrabarty, 2006).

9.1 Fundamental Principles The technological forming of metals generally involves plastic strains which dominate over the elastic strains, and the rigid/plastic approximation of material response is therefore appropriate in most cases. At any stage of the deformation process, the current shape of the workpiece and the associated strain distribution are supposed to be known from the previous computation. It is further assumed that the influence of geometry changes during an incremental loading of the workpiece may be disregarded in a variational formulation for the estimation of the strain increment suffered by each material element. The geometry of the workpiece can be subsequently updated on the basis of the computed strain increment, and the solution can be continued in a stepwise manner. The simplest rigid/plastic formulation of the finite element method, widely used for the analysis of metal-forming processes (Kobayashi et al., 989), will be described in what follows.

9.1.1 The Variational Formulation In a typical boundary value problem, the surface traction Fj is prescribed over a part of the boundary, and the velocity vj is prescribed over the remainder. Let ε˙ ij denote J. Chakrabarty, Applied Plasticity, Second Edition, Mechanical Engineering Series, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77674-3_9,

671

672

9 The Finite Element Method

the true strain rate corresponding to any kinematically admissible velocity field, and let σij denote the associated true stress that is not necessarily in equilibrium. Among a sufficiently wide class of admissible velocity fields, the actual field corresponds to a stationary value of the functional U=

σij ε˙ ij dV −

Fj vj dS.

The variational principle is analogous to the conventional upper bound technique for the estimation of the yield point load. The finite element approach, because of its discretization, allows the consideration of a much wider class of velocity fields than that possible in the usual upper bound analysis. The restriction imposed on the admissible velocity field by the incompressibility of the material can be removed by introducing a large positive constant , known as the penalty constant, which allows the variational functional to be written in the modified form 1 2 dV − Fj vj dS. (9.1) U = σij ε˙ ij dV + ε˙ kk 2 ˙ and setting the first variation δU to Denoting the volumetric strain rate ε˙ kk by λ, zero, we get

σ¯ δ ε˙¯ dV +

λ˙ δ λ˙ dV −

Fj δvj dS = 0,

(9.2)

where σ¯ and ε¯˙ are the effective stress and strain rate, respectively. The parameter , which is similar to the elastic bulk modulus, must be carefully chosen, since too large a value of would cause difficulties in the convergence, while too small a value of would result in unusually large changes in volume. An appropriate choice of seems to be that for which λ˙ is restricted to the order of 10–3 times the mean effective strain rate in the material (Kobayashi et al., 1989). Rigid zones, which generally coexist with plastically deforming zones in most metal-forming processes, may be identified by the occurrence of effective strain rates that are smaller than a certain limiting value ε˙ 0 . These regions can be approximately included in the analysis by setting σ¯ = hε˙¯ in the first integral of (9.2), where h is a constant. Such a modification of the variational equation is necessary only over those regions which are considered as nearly rigid. Realistic estimates of the extent of the deforming zone can be achieved, without adversely affecting the convergence of the numerical analysis, by taking ε¯ to have an assigned limiting value of 10–2 approximately.

9.1.2 Velocity and Strain Rate Vectors The finite element analysis of the boundary value problem begins with the specification of a velocity distribution within each element. The velocity must have continuous first derivatives within the element and must satisfy the condition of

9.1

Fundamental Principles

673

continuity across its interfaces with the adjacent elements. The components of the velocity at any point within an element are completely defined by those at a sufficient number of nodal points, which are generally located along the boundary of the element. Considering a three-dimensional velocity field with rectangular components u, v, and w, it is possible to express them in the form u=

1

N α uα ,

v=

1

Nα vα ,

w=

1

Nα wα

(9.3)

where uα, vα , and wα are the velocity components at the αth node, Nα is the associated shape function, and the summation extends over all the nodal points of the element. The explicit forms of the shape function in specific cases will be discussed later. Equation (9.3) can be conveniently written in the matrix form v = Nq

(9.4)

where N is the shape function matrix, v is the velocity vector for a generic particle, and q is the nodal velocity vector. These vectors are defined as vT = {u, v, w} ,

qT = {u1 , v1 , w1 , u2 , v2 , w2 , . . .} ,

where the superscript T denotes the transpose. The shape function matrix assumes the form ⎤ N1 0 0 N2 0 0 N3 0 0 . . . N = ⎣ 0 N1 0 0 N2 0 0 N3 0 . . . ⎦ 0 0 N1 0 0 N3 0 0 N3 . . . ⎡

(9.5)

The total number of columns in the N-matrix is equal to the nodal degrees of freedom, defined by the number of nodal velocity components. The components of the true strain rate within the element can be expressed in terms of the nodal velocities and the derivatives of the shape functions. If a typical component of the velocity at the αth node is denoted by qi (α) then vi = Nα qi (α) in view of (9.4), and the expression for the true strain rate tensor becomes ε˙ ij =

∂vj 1 ∂vi 1 ∂Nα ∂Nα + = qi (α) + qj (α), 2 ∂xj ∂xi 2 ∂xi ∂xi

Denoting the rates of extension in the coordinate directions by ε˙ x , ε˙ y , and ε˙ z , and the associated rates of engineering shear by γ˙xy , γ˙yz , and γ˙yz , the strain rate vector ε˙ may be defined by its transpose 7 6 ε˙ T = ε˙ x , ε˙ y , ε˙ z , ε˙ xy , ε˙ yz , ε˙ zx .

674

9 The Finite Element Method

In view of the preceding expression for the strain rate, the components of the vector are given by 1 1 Pα uα , ε˙ y = Qα uα, ε˙ z = Rα w α , 1 1 γ˙xy = (Qα uα + Pα vα ) , γ˙yz = (Rα vα + Qα wα ) ,

ε˙ x =

1

(9.6)

together with a similar expression for γ˙zx , where we have introduced the notation Pα =

∂Nα , ∂x

Qα =

∂Nα , ∂y

Rα =

∂Nα . ∂z

(9.7)

It may be noted that the parameters Pα , Qα , and Rα are generally functions of the space variables. Equation (9.6) may be written in the matrix form ε˙ = Bq

(9.8)

where B denotes the strain rate matrix, which can be written explicitly as ⎡

p1 ⎢ 0 ⎢ ⎢ 0 B=⎢ ⎢ Q1 ⎢ ⎣ 0 R1

0 Q1 0 P1 R1 0

0 0 R1 0 Q1 P1

P2 0 0 Q2 0 R2

0 Q2 0 P2 R2 0

0 0 R2 0 Q2 P2

P3 0 0 Q3 0 R3

0 Q3 0 P3 R3 0

0 0 R3 0 Q3 P3

⎤ ... ...⎥ ⎥ ...⎥ ⎥. ...⎥ ⎥ ...⎦ ...

The number of columns in the B-matrix is evidently identical to the nodal degrees of freedom of the considered element. The equivalent or effective strain rate ε˙¯ and the volumetric strain rate λ, which occur in the finite element analysis, need to be expressed in matrix forms involving the nodal velocity vector q. Since ε˙¯ =

1/2 2 ε˙ ij ε˙ ij 3

(9.9)

when the material is isotropic, it follows from (9.8) that ε¯˙ 2 = ε˙ T D˙ε = qT Sq., S = BT DB

(9.10)

For a three-dimensional deformation mode, D is a diagonal matrix whose first three diagonal elements are equal to 23 and the last three diagonal elements are equal to 13 . In the special case of plane stress, as we shall see later, the matrix D is not diagonal. In view of the first three relations of (9.6), the volumetric strain rate is λ˙ = CT q

9.1

Fundamental Principles

675

where CT is a row vector obtained by adding the vectors represented by the first three rows of B. Thus CT = {P1 , Q1 , R1 , P2 , Q2 , R2 , . . .} .

(9.11)

It is interesting to note that this vector is also obtained by premultiplying the matrix B by a row vector whose first three elements are unity and the remaining three elements are zero.

9.1.3 Elemental Stiffness Equations The global integral appearing in the variational equation (9.2) is actually an assembly of integrals taken over the individual elements in the deforming body. The derivation of the stiffness equation in the matrix form at the elemental level is therefore essential for the establishment of the global stiffness equation. Denoting typical elements of the matrices N, S, and C by the quantities Nij , Sij , and Cj , respectively, and using (9.4), (9.10), and (9.11), we have σ¯ σ¯ 1 σ¯ 1 ˙2 1 ˙ σ¯ δ ε¯ = δ ε¯ = δ Sij qi qj = Sij qi δqj , 2 2 ε˙¯ ε˙¯ ε˙¯ 2 λ˙ δ λ˙ = Cj qj δ (Ci qi ) = Ci Cj qj δqi , Fj δvj = Fj δ Nji qi = Nji Fj δqi . The expressions on the left-hand side of the above relations are actually the successive integrands in the variational equation (9.2) in relation to a typical element. Since δqj is arbitrary, the result becomes ∂U/∂qi =

( σ¯ ε˙¯ Sij qj dV +

Ci Cj qj dV −

Nji Fj dS = 0,

In matrix notation, the equation for the minimization of the functional U therefore takes the form (9.12) (σ¯ /ε˙¯ )Sq dV + CCT q dV − NT F dS = 0 where F is a column vector representing the applied force (Fx , Fy , Fz ) on the considered element, the last integral being the equivalent nodal point force. The preceding result represents a set of nonlinear simultaneous equations for the nodal point velocities. By assembling equations of type (9.12) over all the elements in the deforming body, we obtain the global equation for the boundary value problem. The solution to the stiffness equation (9.12) is generally obtained by an iterative procedure based on a linearization with the help of Taylor’s expansion of the functional U in the neighborhood of an assumed solution point q = q0 . The condition δU = 0 may therefore be written as

676

9 The Finite Element Method

∂ 2U ∂U + qj = −fi + kij qj = 0 ∂qi ∂qi ∂qj where the derivatives are considered at q = q0 , while q denotes the first-order correction to the nodal velocity q0 . The matrix form of the preceding equation is kq = f

(9.13)

where k denotes the elemental stiffness matrix, and f is the residual of the nodal point force vector obtained by setting q = q0 on the left-hand side of (9.12). Since the first derivative ∂U/∂qi is represented by (9.12), it is easily shown that σ¯ 1 ∂ σ¯ ∂ 2U Sij dV + Sik qk Sjm qm dV + ci cj dV = ∂qi ∂qj ε˙ ∂ ε˙¯ ε˙¯ ε˙¯ ( in view of (9.10). Setting ∂ σ¯ /ε˙¯ ε˙¯ = η, the elemental stiffness matrix can be expressed as k=

(σ¯ /ε˙¯ )S dV +

(η/ε˙¯ )SqqT ST dV +

CCT dV

(9.14)

The stiffness equations are most conveniently solved by an iterative method in which σ¯ /ε˙¯ is assumed constant (η = 0 ) during each iteration (Oh, 1982). The computation may begin by assuming that that the effective strain rate ε˙¯ in each element is the same as that in the previous step. Since σ¯ then follows from the computed value of ε¯ , the ratio σ¯ /ε˙¯ is easily evaluated in each element, leading to the stiffness matrix k. The elemental stiffness equations are then assembled to form the global stiffness equations (Section 9.4), which are solved under the prescribed boundary conditions. The solution for the velocity correction furnishes an updated nodal velocity field, and a modified strain rate in each element, which can be used to test the convergence of the solution. When the effective strain rate ε˙¯ in a given element is found to be less than a preassigned value, it may be considered as nonplastic with an effective stress proportional to ε˙¯ . The stiffness equation for such an element should be modified by replacing σ¯ /ε˙¯ in the leading integral of (9.12) with a constant h. The stiffness matrix k is similarly obtained by setting σ¯ /ε˙¯ = h and η= 0 in (9.14). The penalty function therefore enables us to separate the deforming region from the undeforming one.

9.2 Element Geometry and Shape Function 9.2.1 Triangular Element It is evident from the preceding discussion that the shape function Nα for any given geometry of the element is a fundamental quantity in the finite element analysis. Equation (9.3) indicates that if xβ ,yβ denote the rectangular coordinates of the βth node, then

9.2

Element Geometry and Shape Function

677

Nα xβ ,yβ = δαβ , where δαβ is the familiar Kronecker delta. It should be noted that any scalar function f(x, y) can be expressed in the same way as the velocity components are, using the same shape functions. In the case of two-dimensional problems, the simplest finite element is a triangle whose vertices are defined by the coordinates (x1 , y1 ), (x2 , y2 ), and (x3 , y3 ). The coordinates of any point P within the triangle can be expressed in terms of those of its vertices using the transformation x = L1 x1 + L2 x2 + L3 x3 , y = L1 y1 + L2 y2 + L3 y3 ,

(9.15)

where L1 , L2 , and L3 are the ratios of the areas of the three triangles, formed by joining the generic point P to the vertices 1, 2, and 3, to the total area A of the triangular element shown in Fig. 9.1. It follows from this definition that L1 = 0 along the side 2–3, L2 = 0 along the side 3–1, and L3 = 0 along the side 1–2. In view of the identity L1 +L2 + L3 = 1, (9.15) can be solved for the area coordinates L1 , L2 and L3 to give Lα = (aα + bα x + cα y) /2A, 2A = (x1 − x2 ) (y2 − y3 ) − (x2 − x3 ) (y1 − y2 ) ,

(9.16)

where aα , bα , and cα depend on the coordinates of the vertices of the triangle 1–2–3 and are given by ⎫ a1 = x2 y3 − x3 y2 , b1 = y2 − y3 , c1 = x3 − x2 , ⎬ a2 = x3 y1 − x1 y3 , b2 = y3 − y1 , c2 = x1 − x3 , ⎭ a3 = x1 y2 − x2 y1 , b3 = y1 − y2 , c3 = x2 − x1 .

Fig. 9.1 Triangular elements. (a) Linear element and (b) quadratic element

(9.17)

678

9 The Finite Element Method

It is important to note that Lα xβ ,yβ = δαβ , which represents a fundamental property of the area coordinates, similar to that for the shape functions. A linear triangular element consists of three nodes located at its vertices, as shown in Fig. 9.1 (a), the velocity components at any point within the triangle being assumed to vary linearly with x and y. The continuity of the velocity at the nodal points therefore ensures its continuity along the sides of the triangle. The velocity distribution within the triangle may be written as u = N1 u1 + N2 u2 + N3 u3 , v = N1 v1 + N2 v2 + N3 v3 , where the shape functions N1 , N2 , and N3 are linear functions of x and y. It follows from (9.15) and the linearity of the area coordinates that these functions are identical to L1 , L2 , and L3 respectively. Thus N1 = (a1 + b1 x + c1 y) / 2A,

N2 = (a2 + b2 x + c2 y) / 2A,

N3 = (a3 + b3 x + c3 y) / 2A. (9.18)

Elements which involve shape functions that are identical to the functions defining the coordinate transformation of type (9.15) are known as isoparametric elements. The linear triangular element is therefore isoparametric. The relevant components of the strain rate matrix B for the linear triangular element are y2 − y3 y3 − y1 , P2 = , 2A 2A x3 − x2 x1 − x3 Q1 = , Q2 = , 2A 2A

P1 =

P3 = − (P1 + P2 ) , (9.19) Q3 = − (Q1 + Q2 ) ,

in view of (9.7) and (9.18). It may be noted that the components of the strain rate corresponding to the linear triangular element are constant over each element. A useful integral involving exponents of the area coordinates taken over the area of the triangle is

p

L1m L2n L3 dxdy =

m! n! p!(2A) (m + n + p + 2)!

(9.20)

where m, n, and p are integers. The result follows from the fact that the Jacobian of the transformation of coordinates is J=

∂x ∂y ∂x ∂y − = 2A ∂L1 ∂L2 ∂L2 ∂L1

The linear triangular element can be used for treating finite deformation problems by using a linear interpolation of the Lagrangian strains from the midpoints of the sides of the triangle, as has been shown by Flores (2006). In a quadratic triangular element, there are three primary nodes located at the vertices of the triangle, and three secondary nodes at the midpoints of the sides of the triangle, as shown in Fig. 9.1(b). The velocity components u and v, which are

9.2

Element Geometry and Shape Function

679

assumed to be quadratic functions of x and y, are again continuous across the sides of the triangle and are expressed in terms of the nodal values as u = N1 u1 + N2 u2 + N3 u3 + N4 u4 + N5 u5 + N6 u6 v = N1 v1 + N2 v2 + N3 v3 + N4 v4 + N5 v5 + N6 v6 where (uα ,vα ) denote the velocity vector at a typical nodal point α. The shape functions associated with the six nodal velocities can be expressed in terms of the area coordinates L1 , L2 , and L3 , the result being easily shown to be N1 = L1 (2L1 − 1) , N4 = 4L1 L2 ,

N2 = L2 (2L2 − 1) , N5 = 4L2 L3 ,

N3 = L3 (2L3 − 1) , N6 = 4L3 L1 .

(9.20)

The elements of the strain rate matrix can be determined in the same way as that for the linear triangle. It can be seen that the quadratic element with straight sides is not isoparametric. It is possible, however, to construct curvilinear triangles in this case to form isoparametric elements (Zienkiewicz, 1977).

9.2.2 Quadrilateral Element In the solution of special problems, it is often convenient to use quadrilateral elements with nodal points located at the corners. The element is generally defined parametrically in terms of auxiliary coordinates (ξ , η), known as natural coordinates, so that the quadrilateral is transformed into a square defined by ξ ± 1 and η = ± 1, as shown in Fig. 9.2. The shape functions are bilinear in ξ and η according to the relations

Fig. 9.2 Linear quadratic element. (a) Natural coordinates and (b) physical coordinates

680

9 The Finite Element Method

N1 = N3 =

1 4 1 4

(1 − ξ ) (1 − η) ,

N2 =

(1 + ξ ) (1 + η) ,

N4 =

1 4 1 4

(1 + ξ ) (1 − η) , (1 − ξ ) (1 + η) .

(9.21)

It is easy to see that Nα (ξα ,ηα ) = δαβ , where (ξα ,ηα ) are the natural coordinates of a typical node. The transformation of the natural coordinates (ξ ,η) into the physical coordinates (x, y) is given by x = N1 x1 + N2 x2 + N3 x3 + N4 x4 , y = N1 y1 + N2 y2 + N3 y3 + N4 y4 .

(9.22)

The condition of constancy of the slope dy/dx of each side of the quadrilateral is therefore identically satisfied. Since the coordinate transformation involves the same shape functions as those in the velocity relations u=

1

Nα uα ,

v=

1

Nα vα ,

where α varies from 1 to 4, the linear quadrilateral element is isoparametric. If we consider an 8-node isoparametric element in which the secondary nodes are located at the midpoints of the sides of the square in the (ξ , η)-plane, the shape of the element in the physical plane is a curvilinear quadrilateral defined by shape functions that are obtained on multiplying the right-hand side of (9.23) by suitable linear functions of ξ and η. The elements of the strain rate matrix B involve the derivatives of the shape functions with respect to x and y. Since the corresponding derivatives with respect to ξ and η are given by

∂Nα /∂ξ ∂Nα /∂η

∂x/∂ξ ∂y/∂ξ = ∂x/∂η ∂y/∂η

∂Nα /∂ξ ∂Nα /∂y

,

where the square matrix is the well-known Jacobian matrix having a determinant J, which is the Jacobian of the transformation, and is given by J=

∂x ∂y ∂x ∂y − ∂ξ ∂η ∂η ∂ξ

(9.23)

A straightforward process of inversion of the preceding matrix equation furnishes 1 ∂y/∂η −∂y/∂ξ ∂Nα /∂x ∂Nα /∂ξ = . ∂Nα /∂y ∂Nα /∂η J −∂x/∂η ∂x/∂ξ In the case of a liner quadrilateral element, the partial derivatives appearing in the Jacobian matrix and its inverse are readily obtained from (9.22) and (9.23), the result being 8 J = (x13 y24 − x24 y13 ) + (x34 y12 − x12 y34 ) ξ + (x23 y14 − x14 y23 ) η, (9.24)

9.2

Element Geometry and Shape Function

681

where xij = xi − xj and yij = yi − yj . Since ∂Nα /∂x and ∂Nα /∂y are equal to Pα and Qα , respectively, according to (9.7), we have ⎧ ⎫ ⎪ ⎪ P1 ⎪ ⎪ ⎨ ⎬ P2 = ⎪ ⎪ P3 ⎪ ⎪ ⎩ ⎭ P4 ⎧ ⎫ Q1 ⎪ ⎪ ⎪ ⎨ ⎪ ⎬ Q2 = Q3 ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ Q4

1 8J

1 8J

⎧ y24 ⎪ ⎪ ⎨ −y13 −y24 ⎪ ⎪ ⎩ y13 ⎧ −x24 ⎪ ⎪ ⎨ x13 x24 ⎪ ⎪ ⎩ −x13

−y34 ξ +y34 ξ +y12 ξ −y12 ξ +x34 ξ −x34 ξ −x12 ξ +x12 ξ

⎫ −y23 η ⎪ ⎪ ⎬ +y14 η , −y14 η ⎪ ⎪ ⎭ +y23 η ⎫ +x23 η ⎪ ⎪ ⎬ −x14 η . +x14 η ⎪ ⎪ ⎭ −x23 η

(9.25)

The results for the 8-node isoparametric quadrilateral element are evidently more complex. It is more convenient in this case to evaluate Pα and Qα numerically for selected values of ξ and η, using the method employed for the linear element.

9.2.3 Hexahedral Brick Element For treating three-dimensional problems, the quadrilateral element must be replaced by a brick element with eight corners. The simplest isoparametric element involves a node at each of the eight corners of the element, which assumes the form of a cube defined by ξ = ± 1, η = ± 1, and ζ = ± 1, in the associated natural coordinate system (ξ , η, ζ). The shape functions are defined as Nα =

1 (1 + ξα ξ ) (1 + ηα η) (1 + ζα ζ ) , 8

(9.26)

where (ξ α , ηα , ζα ) are the natural coordinates of a typical node α. Since ξα2 = ηα2 = ζα2 − 1, the above expression indicates that Nα = 1 at the αth node, while Nα = 0 at all other nodes due to the vanishing of at least one of its factors. The velocity distribution is given by (9.3), while the coordinate transformation is x=

1

Nα xα ,

y=

1

Nα yα ,

z=

1

N α zα ,

(9.27)

where (xα , yα , zα ) are the rectangular coordinates of the αth node. Figure 9.3 shows the brick element defined in both the natural coordinates and the physical coordinates. The velocity field for the brick element can be expressed as u=

1

Nα uα ,

v=

1

Nα vα ,

w=

1

Nα wα

where (uα , vα , wα ) are the rectangular components of the velocity vector at the αth node. The 8-node brick element is therefore isoparametric. The Jacobian matrix

682

9 The Finite Element Method 8

8

7

7

5 4

ζ

4 3

z

3 y

η ξ

6

5

6

x

1

1

2

2

(a)

(b)

Fig. 9.3 Three–dimensional brick element depicted in (a) natural coordinate system, and (b) rectangular coordinate system

⎡

∂x/∂ξ J = ⎣ ∂x/∂η ∂x/∂ζ

∂y/∂ξ ∂y/∂η ∂y/∂ζ

⎤ ∂z/∂ξ ∂z/∂η ⎦ ∂z/∂ζ

(9.28)

for the transformation of coordinates is easily formed for any selected values of (ξ , η,ζ), using (9.26) and (9.27). This matrix and can be inverted numerically to determine the quantities Pα , Qα , and Rα , using the expression ⎫ ⎫ ⎧ ⎫ ⎧ ⎧ ⎨ Pα ⎬ ⎨ ∂Nα /∂x ⎬ ⎨ ∂Nα /∂ξ ⎬ Qα = ∂Nα /∂y = j−1 ∂Nα /∂η ⎭ ⎭ ⎩ ⎭ ⎩ ⎩ Rα ∂Nα /∂z ∂Nα /∂ζ

(9.29)

Since the column vector on the right-hand side of (9.29) is easily obtained using (9.26), the column vector on the left-hand side can be evaluated for any selected node α. The determinant of the Jacobian matrix (9.28) furnishes the value of J in each particular case.

9.3 Matrix Forms in Special Cases 9.3.1 Plane Strain Problems We begin with the situation where the resultant velocity of each particle is parallel to a given plane, which is considered as the (x, y)-plane, the rectangular components of the velocity of a typical particle being denoted by (u, v). Adopting the 4-node isoparametric quadrilateral element shown in Fig. 9.2, the nodal velocity vector q and the shape function matrix N may be written as

9.3

Matrix Forms in Special Cases

683

qT = {u1 v1 u2 v2 u3 v3 u4 v4 } , N1 0 N2 0 N3 0 N4 0 , N= 0 N 1 0 N2 0 N3 0 N4

(9.30)

where N1, N2 , N3 , and N4 are given by (9.21) in terms of the natural coordinates (ξ , η). The vectors representing the particle velocity and the strain rate are u v= = Nq, v

⎫ ⎧ ⎨ ε˙ x ⎬ = Bq, ε˙ = ε˙ y ⎭ ⎩ γ˙xy

where B is the strain rate matrix having the form ⎡

⎤ P1 0 P2 0 P3 0 P4 0 B = ⎣ 0 Q1 0 Q2 0 Q3 0 Q4 ⎦ , Q1 P 1 Q2 P 2 Q3 P 3 Q4 P 4

(9.31)

where Pε and Q4 are given by (9.25) as functions of (ξ , η) in terms of the global coordinates of the nodal points. The equivalent strain rate ε˙¯ and the volumetric strain rate λ˙ can be evaluated from (9.10) and (9.11), respectively, the relevant matrix and vector being of the form ⎡ ⎤ 100 2⎣ 0 1 0 ⎦, D= 3 0 0 12

⎧ ⎫ ⎨1⎬ C = BT 1 . ⎩ ⎭ 0

(9.32)

The set of nonlinear equations for the unknown nodal velocities are finally obtained by considering (9.12) for each individual element. The solution is most conveniently obtained by using linearized stiffness equations of type (9.13) as explained before in general terms.

9.3.2 Axially Symmetrical Problems In problems of axial symmetry, the finite element is taken in the form of a ring whose cross section is identical to the two-dimensional element. For an isoparametric ring element, the global coordinates (r, z) of a generic point inside the element are related to the natural coordinates (ξ , n) by the transformation r=

1

Nα r α ,

z=

1

N α zα ,

(9.33)

where (rα , zα ) are the coordinates of a typical node, and Nα is the corresponding shape function that depends on (ξ , η). For the 4-node quadrilateral element shown in Fig. 9.4, the associated shape functions are given by (9.30). The radial and circumferential components of the velocity of a typical particle are given by

684

9 The Finite Element Method

Fig. 9.4 Axisymmetric ring element having the cross section of a 4-node quadrilateral in the meridian plane

u=

1

w=

Nα u α ,

1

Nα wα ,

where (uα wα ) are the nodal velocity components, the shape function matrix N being identical to that in (9.27). The vector representing the strain rate then becomes ⎧ ⎫ ⎡ ∂/∂r ε˙ r ⎪ ⎪ ⎪ ⎬ ⎢ ⎨ ⎪ ε˙ z 0 =⎢ ε˙ = ⎣ 1/r ε ˙ ⎪ ⎪ θ ⎪ ⎭ ⎩ ⎪ γ˙rz ∂/∂z

⎤ 0 ∂/∂z ⎥ ⎥ u = Bq. 0 ⎦ w ∂/∂r

(9.34)

The nodal velocity vector q is given by (9.30) with vα replaced by wα , and the strain rate matrix B has the modified form ⎡

P1 ⎢ 0 B=⎢ ⎣ B1 Q1

0 Q1 0 P1

P2 0 B2 Q2

0 Q2 0 P2

P3 0 B3 Q3

0 Q3 0 P3

P4 0 B4 Q4

⎤ 0 Q4 ⎥ ⎥, 0 ⎦ P4

(9.35)

where Pα and Qα are given by (9.26) with xij and yij replaced rij by and zij , respectively, the expression for J in (9.24) being similarly modified, while the third row of (9.35) is given by Bα = Nα /r = Nα /(N1 r1 + N2 r2 + N3 r3 + N4 r4 ) . The evaluation of ε˙¯ and λ˙ is identical to that for the plane strain case, the matrix D and the vector C being similar to those given by (9.32). The final stiffness equations can be handled in the same way as those for plane strain.

9.4

Sheet Metal Forming

685

9.3.3 Three-Dimensional Problems In the three-dimensional finite element analysis, it is generally convenient to use the 8-node isoparametric brick element shown in Fig. 9.3. In the natural coordinate system (ξ , η, ζ ), this element is a cube, and the deformation mode is specified by the nodal velocity vector q, where qT = [u1

v1

w1

u2

...

v2

u8

v8

w8 ]

The transformation between the natural and global coordinates defined by (9.27) in terms of the shape functions are given by (9.26), the shape function matrix being ⎡

N1 N = ⎣0 0

0 N1 0

0 0 N1

N2 0 0

0 N2 0

0 0 N2

. . . . N8 .... 0 .... 0

0 N8 0

⎤ 0 0 ⎦ N8 ..

(9.36)

The velocity field within the element and the associated strain rate vector, having the six rectangular components given by (9.6), may be written as ⎧ ⎫ ⎨u ⎬ u = v = Nq, ε˙ = Bq ⎩ ⎭ w where B is the strain rate matrix for the 8-node brick element, in which the components of train rate are given by (9.26). It is easily shown that ⎡

P1 ⎢ 0 ⎢ ⎢ 0 B=⎢ ⎢ Q1 ⎢ ⎣ 0 R1

0 Q1 0 P1 R1 0

0 0 R1 0 Q1 P1

P2 0 0 Q2 0 R2

0 Q2 0 P2 R2 0

0 0 R2 0 Q2 P2

. .. .. .. .. ..

P8 0 0 Q8 0 R8

0 Q8 0 P8 R8 0

⎤ 0 0 ⎥ ⎥ R8 ⎥ ⎥ 0 ⎥ ⎥ Q8 ⎦ P8

(9.37)

The Jacobian of the transformation of coordinates, given by the determinant of the matrix (9.28), is easily evaluated numerically in each particular case. The square matrix S, which defines the stiffness matrix, is given by (9.10), where D is a diagonal matrix in which each of the six diagonal element is 2/3.

9.4 Sheet Metal Forming 9.4.1 Basic Equations for Sheet Metals In the plastic forming of sheet metal, the stress component in the thickness direction is generally disregarded. A state of plane stress therefore exists in each element of the sheet, which is assumed to be orthotropic with the anisotropic axes coinciding

686

9 The Finite Element Method

with the rolling, transverse, and thickness directions. Considering a biaxial loading of the sheet, we choose a set of rectangular axes in which the x- and y-axes are directed along the rolling and transverse directions, respectively, the z-axis being taken along the normal to the sheet. The effective stress in a material element is given by the yield criterion, and may be defined in such a way that it reduces to the current uniaxial yield stress in the rolling direction. Using the quadratic yield criterion for simplicity (Section 6.2), we write 1/2 2H F+H 2N σx σy + σy2 + τxy 2 σ¯ = σx 2 − G+H G+H G+H

(9.38)

where F, G, H, and N are parameters defining the state of planar anisotropy of the sheet metal. When the hypothesis of strain equivalence is adopted for the hardening of the material, the effective strain rate in a deforming element may be written as ε˙¯ = (G + H)

2 /4 ε˙ x2 + ε˙ x ε˙ y + ε˙ y2 + γ˙xy

1/ 2 ,

G2 + GH + H 2

so that ε˙¯ becomes identical to the longitudinal strain rate ε˙ x in the case of a simple tension applied in the rolling direction. This is easily verified by setting σ = σ as the only nonzero stress in the associated flow rule, which may be written as ε˙ y γ˙xy ε˙ x = = = λ˙ + H) σ + H) σ (G (F N x − Hσy y − Hσx

(9.39)

where λ˙ is a positive scalar. The thickness strain rate follows from the incompressibility condition ε˙ z = −(˙εx + ε˙ y ) Since the rate of plastic work per unit volume is equal to σ¯ λ˙ in view of (9.39) and (9.38), the effective strain rate according to the hypothesis of work equivalence is equal to λ˙ , giving ε˙¯ =

√

G+H

(F + H) ε˙ x2 + 2H ε˙ x ε˙ y + (G + H) ε˙ y2 FG + GH + HF

+

2 γ˙xy

2N

1/2 ,

(9.40)

which is obtained by expressing the stresses in terms of the strain rates using the flow rule (9.39), and substituting them into the field criterion (9.38). The ratios of the anisotropic parameters F, G, H, and N can be determined from the measured R-values of the sheet in the rolling, transverse, and at 45◦ to the rolling direction. The strain rate vector in the plane stress formulation is identical to those for plane strain. It is convenient to express the effective strain rate in the matrix form ε˙ =

ε˙ T D˙ε

as before, but the forms of the square matrix D for the anisotropic material depends on the hardening hypothesis and is expressed by

9.4

Sheet Metal Forming

⎡

2 1 (G + H) ⎣ 1 2 D= 2S 0 0 2

687

⎤ 0 0 ⎦, 1 2

⎡

⎤ F+H H 0 G+H ⎣ H G + H 0 ⎦ , (9.41) D= T 0 0 T/2 N

where S = G2 +GH+H2 and T = FG+GH+HF. The first expression in (9.41) corresponds to the hypothesis of strain equivalence and the second expression to that of work equivalence. When the material exhibits normal anisotropy with a uniform R-value, it is only necessary to set F = G, H = RG, N = (1 +2R)G, and T = NG in the preceding relations. The shape function matrix and the strain rate matrix for a given shape of the element are identical to those for plane strain. The application of the preceding theory to the flange drawing and bore-expanding processes has been reported by Lee and Kobayashi (1975). A finite element formulation of the problem based on the biquadratic yield criterion (Section 6.2) has been discussed by Gotoh (1978, 1980).

9.4.2 Axisymmetric Sheet Forming In the case of out-of-plane deformations of the sheet metal, such as in the hydraulic bulging and punch stretching, the deformed sheet at each stage may be regarded as a membrane with a state of plane stress existing in each element. Additional equations are obviously necessary to determine the deformed shape of the sheet metal, and the distribution of stress and strain in the workpiece. When the deformed sheet forms a surface of revolution, it may be approximated by a succession of conical frustums, each frustum being treated as a finite element, Fig. 9.5. Consider a line element along the meridian, extending between the nodal points 1 and 2 with coordinates (r1 , z1 ) and (r2 , z2 ) respectively. If the radial and axial coordinates of a generic particle are denoted by r and z, and the corresponding components of the velocity are denoted by u and w, respectively, then

z

2

Fig. 9.5 Finite element approximation of a surface of revolution developed in the axisymmetric forming of a sheet metal

2 (ξ = 1) 1 (ξ = – 1)

1

r

688

u=

9 The Finite Element Method

1 1 1 1 (u1 + u2 ) + (u2 − u1 ) ξ , r = (r1 + r2 ) + (r2 − r1 ) ξ , 2 2 2 2

−1 ≤ ξ ≤ 1

where u1 and u2 are the radial velocities at the nodal points 1 and 2, respectively. Similar relations may be written down for the components w and z. If φ is the angle made by the surface normal with the axis of symmetry, which coincides with the z-axis, then the circumferential and meridional components of the strain rate are given by (u2 + u1 ) + (u2 − u1 ) ξ u = r (r2 + r1 ) + (r2 − r1 ) ξ 1/2 z2 − z1 2 ∂u u2 − u1 + α, ˙ α = 1+ + φ˙ tan φ = ε˙ φ = ∂r r2 − r1 r2 − r1 ε˙ =

(9.42)

The effective stress and strain rate for a uniform R-value material with normal anisotropy, according to the hypothesis of work equivalence, may be written as 1/2 2R 2 2 σ¯ = σθ − σθ σφ + σφ 1+R 1/2 1+R 2R ε˙ θ2 + ε˙¯ = √ ε˙ θ ε˙ φ + ε˙ φ2 1+R 1 + 2R in view of (9.38) and (9.40). The effective strain can be expressed in a matrix form as before in terms of a square D, the result being ˙ ε¯ = ε˙ T Dε,

ε˙ =

ε˙ θ ε˙ φ

,

l+R l+R D= l + 2R R

R l+R

(9.43)

The finite element procedure is based on a variational principle similar to (9.1). Since the condition of incompressibility need not be dealt with, the penalty function may be omitted for the variational formulation. It is convenient in this case to multiply the integrands in ( ) by a small time increment Δt and write the functional in the modified form U = σ (ε) dV − Fj (uj ) dS, ˙ and uj = vj t, not to be confused with the radial velocity. where ¯ε = εt, Considering the variation of U, and using the fact that δ σ¯ = Hδ (¯ε ,), where H is the plastic modulus, we have

δU =

h

σ + H (¯ε) δ (¯ε) dA − Fj Nij δ(qj ) dS ε

(9.44)

where h is the local sheet thickness, A is the surface area of the element, Nji is a typical element of the shape function transpose matrix of which the nonzero components

9.4

Sheet Metal Forming

689

are N11 = N22 = (1 – ξ )/2 and N31 = N42 = (1 + ξ )/2, and Δqj is a typical component of the nodal displacement increment vector, similar to the nodal velocity vector. The effective strain increment ¯ε can be expressed in terms of the components of the strain increment by replacing the strain rates appearing on the right-hand side of (9.39) by the corresponding strain increments. In matrix notation, the effective strain increment becomes 1/2 , ¯ε = (εT )D(ε)

εT = εθ εφ ,

in view of (9.43). The strain increments are evidently given by the strain rates in (9.43) multiplied by Δt. It follows from (9.43) that (Toh and Kobayashi, 1985) ∂U = ∂qi,

h

σ¯ + H ai dA − Fj Nji dS, ¯ε

ai =

∂ T ε D (ε) (9.45) ∂qi

The second derivative of the functional U with respect to a nodal velocity component follows from (9.45) and is easily shown to be ∂ 2U = ∂qi, ∂qj bij =

h

σ¯ hσ¯ ai aj dA, +H bij + cij dA − 2 ¯ε (¯ε)

∂ 2 T ∂ T ∂ (ε) ε ε D(ε), cij = D ∂qi ∂qj ∂qi ∂qi

(9.46)

The element stiffness equations for the sheet metal-forming process now given by kij qj = fi where kij and –fi are given by the first equations of (9.46) and (9.45), respectively, while Δqj represents the displacement correction vector. The formation of the global stiffness equations and their solution can be carried out in the same way as described before. A finite element formulation of the problem based on a nonlinear membrane shell theory has been considered by Wang and Budiansky (1978). An implementation of the nonquadratic yield criterion (Section 6.2) into the finite element formulation has been presented by Wang (1984).

9.4.3 Sheet Forming of Arbitrary Shapes In general, the out-of-plane deformation of a sheet metal is complicated by the fact that the principal axes of the stress and strain increments are not known in advance. It is therefore necessary to extend the analysis given above to deal with the general sheet forming process. Assuming a state of normal anisotropy of the sheet metal as

690

9 The Finite Element Method

before, the effective stress and strain increments are obtained from (9.38) and (9.40) in the form 1/2 2R 1 + 2R 2 σx σy + σy2 + 2 τxy σ¯ = σx2 − 1+R 1+R ¯ε =

2 2 1 2 1/2 1+R γxy (1 + R) (εx )2 + εy + 2R (εx ) εy + εy + 1 + 2R 2 (9.47)

according to the hypothesis of work equivalence. Introducing a square matrix D, the effective strain increment can be expressed in the matrix form 61/2 7 ¯ε = εT D(ε) ,

⎡ l+R R l+R ⎣ R l+R D= l + 2R 0 0

⎤ 0 0 ⎦ 1/2

(9.48)

It is convenient to discretize the sheet metal into an assemblage of linear triangular elements and consider a set of rectangular axes in which the x- and y-axes are taken along the plane of the sheet, and the z-axis along the normal. The components of the increment of displacement at any point of the element during a time increment Δt may be expressed in terms of the nodal values in the matrix form ⎧ ⎫ ⎤ ⎡ N1 0 0 N2 0 0 N3 0 0 ⎨ u ⎬ (9.49) u = v = Nq, N = ⎣ 0 N1 0 0 N2 0 0 N3 0 ⎦ ⎩ ⎭ 0 0 N1 0 0 N2 0 0 N3 w where N1 , N2 , and N3 are the shape functions, which are identical to the area coordinates and are given by (9.19), while q is the nodal displacement increment vector given by qT = [u1

v1

w1

u2

v2

w2

u3

v3

w3 ]

Each normal component of the true strain in the coordinate directions is the logarithm of the ratio of the final and initial material line elements originally coinciding with each coordinate axis. These are the logarithmic normal components of the Lagrangian strain tensor, and their sufficiently small increments in the surface, if the deforming sheet may be written with sufficient accuracy as εx =

2 ∂ 1 ∂ (u) + (w) , ∂x 2 ∂x

εy =

2 ∂ 1 ∂ (v) + (w) ∂y 2 ∂y

(9.50a)

where {u, v, w} are the components of the displacement of a generic particle. The increment of the surface shear strain, to the same order of approximation, may be written as γxy =

∂ ∂ ∂ ∂ (v) + (u) + (w) (w) ∂x ∂y ∂x ∂y

(9.50b)

9.5

Numerical Implementation

691

In view of (9.49), the various derivatives appearing in the above equations may be expressed as 1 ∂ (u) = Pα uα , ∂x

1 ∂ (v) = Pα vα , ∂x

1 ∂ (w) = Pα wα ∂x

1 ∂ (u) = Qα uα , ∂y

1 ∂ (v) = Qα vα , ∂y

1 ∂ (w) = Qα wα ∂y

where Pα and Qα are given by (9.19), where α varies from 1 to 3. Substituting from the above into equation (9.49), the vector representing the strain increment may be written in the matrix form ⎧ ⎫ ⎫ ⎧ ⎨ εx ⎬ ⎨ εx ⎬ , ε = εy = Bq + εy ⎩ ⎭ ⎩ ⎭ γxy γxy

⎡

⎤ P1 0 0 P2 0 0 P3 0 0 B = ⎣ 0 Q1 0 0 Q2 0 0 Q3 0 ⎦ Q 1 P1 0 Q 2 P2 0 Q 3 P3 0 (9.51) where B is the strain increment matrix. The second column vector in (9.50) arises from the change of the deforming sheet metal and is given by εx =

2 1 1 Pα w α , 2

εy =

2 1 1 Qα wα , 2

= γxy

1

P α wα

1

Qα wα

Using the same variational principle as that for the axisymmetric forming process, we arrive at the element stiffness equation, which is still governed by (9.45) and (9.46), but the strain increment vector ε and the associated matrix appearing in these equations now correspond to (9.51) and (9.48), respectively. It may be noted that the first derivatives appearing in (9.44 ) can be expressed in terms of Pα , Qα , and the associated components of the nodal displacement increment. A finite element formulation for sheet metal forming, including planar anisotropy of the sheet, has been presented by Yang and Kim (1987). A simplified method of finite element analysis based on the total strain theory of plasticity has been discussed by Majlessi and Lee (1988). The influence of bending of the sheet, which is locally important in a variety of sheet-forming processes, has been incorporated in the finite element formulation by Huh et al. (1994).

9.5 Numerical Implementation 9.5.1 Numerical Integration The elemental stiffness equation involves volume and surface integrals which generally require some kind of numerical integration in which the integrand is evaluated at a finite number of points, called integration points, within the limits of integration. We begin with the one-dimensional situation in which a scalar function f (x) is to be integrated over the range a ≤ x ≤ b. Introducing the natural coordinate ξ , such

692

9 The Finite Element Method

that 2x = (b + a) + (b – a)ξ , the formula for the numerical integration can generally be expressed as

b a

1 f (x)dx = (b − a) 2

1

−1

1 1 (b − a) wi F(ξi ) 2 n

F(ξ )dξ =

(9.52)

i=1

where wi is a weight factor associated with the integration point ξ = ξ i and n is the number of integration points. Simpson’s one-third rule of integration is a special case of (9.52), where n = 3, and w1 = w3 = 1/3, w2 = 4/3, the integration points being ξ 1 = –1, ξ 2 = 0, ξ 3 = 1. In the finite element analysis, it is customary to employ the Gaussian quadrature, as it requires the minimum number of integration points for the same degree of accuracy, The Gaussian quadrature formula for n integration points gives the exact result when F (ξ ) is a polynomial of degree less than or equal to 2n – 1. Setting F (ξ ) = ξ s in (9.52), and integrating, we have n 1

wi ξis = 0 (s = 1, 3, 5,..., 2n−1),

i=1

n 1 i=1

wi ξis =

2 s+1

(s = 0, 2, 4,...,2n−2)

(9.53) These relations enable us to determine the integration points and weight factors for any selected value of n. Considering n = 2, and setting s = 0,. . .,3, we have w1 + w2 = 2, w1 ξ 1 + w2 ξ 2 = w1 ξ 1 3 + w2 ξ 2 3 = 0, w1 ξ 1 2 + w2 ξ 2 2 = 2/3, giving the solution w1 = w2 = 1,

1 − ξ1 = ξ2 = √ (n = 2) 3

Similarly, considering n = 3, and setting s = 0,. . .,5 in (9.53), we obtain a set of six equations which are easily solved to give 5 w1 = w3 = , 9

8 w 2 = , − ξ1 = ξ 3 = 9

3 , 5

ξ2 = 0

From the geometrical point of view, the Gaussian integration formulas corresponding to n = 2 and n = 3 are equivalent to linear and quadratic approximations, respectively, of the given function F (ξ ). The integration points in the Gaussian quadrature for any given value of n are in fact the roots of the equation Pn ( ξ ) = 0, where Pn ( ξ ) denotes the Legendre polynomial of degree n. Consider now a scalar function f (x, y), which is defined over a two-dimensional isoparametric element with natural coordinates (ξ , η). If the number of integration points in the ζ and η directions be taken as m and n respectively, the integral of f (x, y) over the area of the element may be written as

9.5

Numerical Implementation

I=

f (x, y)dx dy =

1

693 1

−1 −1

F(ξ , η) J (ξ , η) dξ dη =

m 1 n 1

wi wj F(ξi , ηj ) J(ξi , ηj )

i=1 j=1

(9.54) where J (ξ , η) is the Jacobian of the transformation given by (9.23), while wi and wj are the weight factors corresponding to the integration points ξ i and ηj , respectively. In the case of axial symmetry, involving the volume integration of a function f (r, z) defined over a ring element shown n Fig. 9.4, the integration formula becomes I = 2π

f (r, z) r dr dz = 2π

m 1 n 1

wi wj F(ξi , ηj ) r(ξ , η) J(ξi , η)

(9.55)

i=l j=l

where J (ξ , η) is given by (9.23) with x and y replaced by r and z, respectively, while r = N1 r1 + N2 r2 + N3 r3 + N4 r4 . The numerical integration formula for the general three-dimensional situation can be written down as a straightforward extension of (9.55). In the Gaussian quadrature for two- or three-dimensional cases, the integration points and the weight factors in each coordinate direction for a given number of integration points are the same as those in the one-dimensional case. Setting φ(ξ , η) = F(ξ , η)J(ξ , η), in the double integral (9.55), and assuming m = n =2, we have four Gaussian integration points, each having a weight factor of unity, and the integration formula then becomes I = φ(− α, − α) + φ(− α, α) + φ(α, − α) + φ(α, α),

√ α = 1/ 3

(9.56a)

The assumption m = n = 3, on the other hand, gives us nine Gauss points, one of which is located at the center of the element with a weight factor of 64/81, four of which are located along the diagonals η = ±ξ with a weight factor of 25/81 for each one, the remaining four being along the axes ξ = 0 and η = 0 each with a weight factor of 40/81. The Gaussian integration formula then becomes I=

64 40 [φ(− α, 0) + φ(0, − α) + φ(α, 0) + φ(0. α)] φ(0, 0) + 81 81 25 [φ(− α, − α) + φ(− α, α) + φ(α, − α) + φ(α, α)] , + 81

α=

√ 0.6

(9.56b)

In problems of axial symmetry using quadrilateral ring elements, the Gaussian integration formula for the volume integral can be expressed exactly in the same forms as (9.56), provided we set φ(ξ ,η) = 2π F(ξ ,η) r(ξ ,η) J(ξ ,η),

694

9 The Finite Element Method

In the case of a linear triangular element, for which J = 2A, it is customary in the finite element analysis to consider a single integration point located at the centroid of the triangle, and the integral becomes I = 2A F0 , where F0 is the value of F at the centroid, which corresponds to L1 = L2 = L3 = 1/3. When the same element is used as an axisymmetric ring element, a similar approximation to the volume integral gives I = 4π Ar0 F0 , where r0 is the radius to the centroid of the triangle In a linear quadrilateral element, the condition of constancy of volume cannot be satisfied at all points except for a uniform mode of deformation. In the finite element formulation, this difficulty is usually overcome by using a single-point integration scheme for dealing with the volumetric strain rate term. If, on the other hand, four linear triangular elements are arranged to form a quadrilateral, then the plastic incompressibility condition can be satisfied over the entire quadrilateral (Nagtegaal et al., 1974). In the solution of metal-forming problems, the reduced integration scheme is frequently used, particularly for the evaluation of the stresses

9.5.2 Global Stiffness Equations The finite element analysis of a physical problem is based on dividing the body into a large number of finite elements which are joined together at their nodal points. It is customary to assign the global node numbers and the element numbers sequentially as shown in Fig. 9.6(a). The physical constraints require the velocity vector at any nodal point to be identical to that of the individual elements sharing the same nodal point. The force vector at a given nodal point, on the other hand, is the sum of the forces associated with the elements having this nodal point in common (Desai and Abel, 1972). 1

5

9

13

17 o

1 2 2 3

7

4 6 5 7

12 18

14

10 8 11

13

K= 19

15

o 3 4

6 8

14

9 12

(a)

16

20

(b)

Fig. 9.6 Finite element mesh and associated global stiffness matrix. (a) Element node numbering, (b) typical banded matrix

It is customary to have the elemental nodes numbered in the same sequence for each element in the assemblage. Adopting the elemental node numbering to be

9.5

Numerical Implementation

695

indicated in Fig. 9.2, the elemental stiffness equation (9.13) may be expressed in terms of 2 × 2 submatrices in the form ⎫ ⎧ e ⎫ ⎡ e e e e ⎤ ⎧ e ⎪ {q1 e } ⎪ ⎪ ⎪ ⎪ ⎪ {f1 } ⎪ k11 e k12 e k13 e k14 e ⎪ ⎨ ⎥ {q2 } ⎬ ⎨ {f2 e } ⎬ ⎢ k21 e e T k k k 22 23 24 ⎢ e e e e ⎥ = , kji = kij e e ⎦ ⎣ k31 {q3 } ⎪ {f3 } ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ e k32 e k33 e k34 e ⎪ ⎩ ⎭ ⎭ ⎩ {q4 e } {f4 e } k41 k42 k43 k44 (9.57) where the last relation follows from the symmetry of the stiffness matrix. The submatrices introduced in (9.57) are defined as e e k 2i−1, 2j−1 kij = ke 2i, 2j−1

ke 2i−1, 2j , ke 2i, 2j

7 e6 qj =

uj e , vj e

7 e6 fj =

fj e gj e

(9.58)

The global stiffness equation is formed by a suitable combination of the elemental stiffness equations, taking into account the connectivity of the elements, and may be written as KV = T

(9.59)

where K is the global stiffness matrix, T is the global load vector, and V the global velocity change vector, which differs from the elemental vector q only in the node numbering. Referring to Fig. 9.6(a), and considering, for example, element 2, whose connectivity with the neighboring elements is defined by the nodal points (2, 3, 7, 6), the submatrices of K associated with these nodes are found as [K22 ] = [k11 1 ] + [k44 2 ],

[K23 ] = [k23 2 ],

[K26 ] = [k12 1 ] + [k43 2 ]

[K 33 ] = [k11 2 ] + [k44 3 ],

[K36 ] = [k36 2 ],

[K37 ] = [k12 2 ] + [k43 3 ]

[K66 ] = [k22 1 ] + [k33 2 ] + [k11 4 ] + [k44 5 ],

[K67 ] = [k32 2 ] + [k41 5 ]

[K77 ] = [k22 2 ] + [k33 3 ] + [k11 5 ] + [k44 6 ],

[K27 ] = [k27 2 ]

The remaining submatrices of the global stiffness matrix can be similarly established by considering the connectivity of the other elements. When any two nodal points do not belong to the same element, the corresponding submatrix becomes a null matrix. Due to the limited influence of the element connectivity, the global stiffness matrix is a sparse matrix, which can be arranged in a banded form, as indicated in Fig. 9.6(b). With the help of an appropriate node numbering, the band width can be kept down to a minimum. The global stiffness equations are most conveniently solved by the Gaussian elimination technique using a linear equation solver. In a skyline solver, the matrix coefficients are stored column-wise, starting from the first diagonal element and ending with the last nonzero element. The computational time required to solve the matrix equation is found to be proportional to the square of the semi-bandwidth of the matrix. It is therefore necessary to number the nodes in such a way that the band width is a minimum.

696

9 The Finite Element Method

9.5.3 Boundary Conditions The solution of the global stiffness equations requires due consideration of the boundary conditions. In general, the boundary surface of the workpiece consists of a part on which the traction is prescribed, a part Sv on which the velocity is prescribed, and a part ST which is the tool–workpiece interface. The imposition of the traction boundary condition on SF in the form of nodal point forces is straightforward. Consider, for example, the three-dimensional brick element shown in Fig. 9.3, and suppose that the lower surface 1–2–3–4 is subjected to normal and tangential tractions specified by F. Since ζ = 1 over this surface, the nonzero shape functions are given by (9.21), and the shape function matrix reduces to

N1 N= 0

0 N1

N2 0

0 N2

N3 0

0 N3

N4 0

0 N4

The associated nodal point force vector is easily determined from the expression (NT F) dx dy = (NT F) J dξ dη (9.60) f0 = where J is the Jacobian of the transformation, given by (9.28), and the integral extends over the area of the entire surface SF For a nodal point on Sv over which the velocity is prescribed, the velocity correction is zero, and the corresponding stiffness equation needs to be omitted. In the finite element solution, the simplest way to impose the velocity boundary condition ΔVm = 0 at a nodal point m is to set the diagonal element of the mth row of the stiffness matrix to unity and replace the remaining elements in the corresponding row and column by zeros, as indicated below. ⎫ ⎧ ⎫ ⎤⎧ ⎡ T1 ⎪ K11 K12 . . 0 . . K1n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ V1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ K21 K22 . . 0 . . K2n ⎥ ⎪ V T2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎥⎨ ⎪ ⎢ ⎨ ⎬ ⎬ ⎥ ⎢ . . . . . . . . . . ⎥ ⎢ = ⎢ 0 0 ; . 1 . . 0 ⎥ Vm ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ . . . . . . . . . ⎦⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ⎪ ⎭ ⎩ Vn Kn1 Kn2 . . 0 , . Knn Tn On the remainder of the surface, ST , representing the interface between the die and the workpiece, the boundary condition is of the mixed type, as the velocity is prescribed along the normal to the interface of contact, while the tangential traction is prescribed in the direction of relative sliding between the die and the workpiece. The tangential stress is usually specified in terms of a constant coefficient of friction, or as a constant frictional stress mk, where 0 ≤ m ≤ 1. When the element surface does not conform with the die surface, an additional approximation is necessary to obtain the associated distribution of nodal forces. Once the velocity solution is obtained for the entire workpiece, the geometry of the workpiece must be updated by changing the coordinates of the various nodes.

9.6

Illustrative Examples

697

For a two-dimensional problem, the rectangular coordinates of a typical nodal point (xj ,yj ) are changed by the amounts xj = uj t, yj = vj t where t denotes the increment of time scale. The nodal point strains are similarly updated from the available values of the strain rate. In metal-forming analysis, the time increment may be taken as that for which the next free node of the workpiece comes in contact with the die surface. Since the deformation that occurs in metal-forming processes is generally large, the size and shape of the element soon become unacceptable as the deformation continues, and the imposition of the boundary conditions also becomes increasingly difficult. In order to overcome theses difficulties, it is necessary to modify the mesh system periodically, so that the mesh size remains sufficiently small, and also to transfer the information from the old mesh system to the new one through interpolation. The difficulty can be largely overcome by using a spatially fixed meshing scheme, as has been discussed by Derbalian et al. 1978) and by Mori et al. (1983). An area-weighted averaging method of evaluating a parameter at a node internal to an original linear quadrilateral element, which is found to be sufficiently accurate in metal-forming analysis, has been discussed by Kobayashi et al. (1989). The complete elastic/plastic formulation for large strain finite element analysis has been discussed by McMeeking and Rice (1975) and Nagtegaal and DeJong (1981). A useful discussion of the elastic/plastic formulations in relation to metal-forming problems has been made by Rebelo and Wertheimer (1986).

9.6 Illustrative Examples Numerous solutions to the metal-forming problems, based on the various types of finite element formulation presented in Section 9.1, have been given in detail by Kobayashi et al. (1989). A number of these solutions are based on the variational method, and a few of these will be briefly discussed in what follows in order to illustrate the application of the preceding theory.

9.6.1 Compression of a Cylindrical Block In the axial compression of a cylindrical block between a pair of flat dies, the plastic deformation is inhomogeneous due to the presence of friction at the interfaces, and the mean compressive stress exceeds the uniaxial yield stress of the material (Section 3.4). The deformation of the block is characterized by a barreling of the free surface, a part of which comes in contact with the die during the compression. When the ratio of the height of the block to its diameter is sufficiently small, the barreling consists of a single bulge in which the maximum diameter occurs at the central cross section of the

698

9 The Finite Element Method

block. For fairly large values of the height/diameter ratio, a double bulge is sometimes observed. The analysis may be based on a constant frictional stress equal to mk along the interface between the die and the workpiece. The finite element analysis carried out by Lee and Kobayashi (1971) for a cylindrical block with an initial height/diameter ratio of 2.5 reveals the formation of a double bulge, as depicted in Fig. 9.7. The double bulge gives way to a single bulge as the height/diameter ratio progressively decreases to sufficiently small values during the continued compression.

Fig. 9.7 Grid distortion patterns in the axial compression of a cylinder with an initial hight/diameter ratio of 2.5 (after Lee and Kobayashi, 1971)

The analysis may also be used to predict the limit of workability of ductile materials due to the formation of surface cracks during the compression. This has been investigated experimentally by Kudo and Aoi (1967), who measured the equatorial surface stains in upsetting solid cylindrical specimens under various frictional conditions until the surface cracks were observed. The computed strain paths of the critical element, obtained from the finite element analysis for a block of a unit initial height/diameter ratio under various frictional conditions, are plotted in Fig. 9.8, which also includes the experimental results referred to above. The limit set by the occurrence of surface cracks, based on the experimental data, may be approximated by the criterion 2ε θ + εz = 0.8 to a close approximation. Finite element solutions to the axial compression of hollow cylinders have been discussed by Chen and Kobayashi (1978) and also by Hartley et al. (1979). A finite. element analysis of the upsetting process based on the total strain theory of plasticity has been reported by Vertin and Majlessi (1993).

9.6.2 Bar Extrusion Through a Conical Die Consider the axisymmetric extrusion of a cylindrical billet through a conical die, along which the frictional stress has a constant value equal to mk, the container wall being assumed to be perfectly smooth. The material in the container approaches the die with a uniform unit speed, and it leaves the die with a uniform speed equal to

9.6

Illustrative Examples

699

1.0 Theory Experiment

lubricated

Fracture

=0 .2

.1

m

m=

Hoop Strain

grooved dies

0.6

0.4 5 m= 0.3

unlubricated

0.8

m

=0

m=

0

0.4

0.2

0

–0.4

–0.2

–0.6 Axial Strain

–0.8

–1.0

Fig. 9.8 Strain paths of an equatorial element during axial compression of a cylinder with an initial height/diameter ration of unity

b2 /a2 , where a and b denote the final and initial radii of he billet. The assumed finite element mesh for the extrusion problem is shown in Fig. 9.9, where the corners of the die have been slightly modified by straight lines joining he nodal points closest to the corners, in order to avoid singularities of velocity components near the edge of the die. The origin of the coordinate (r, z) is taken on the axis of symmetry at O with the r-axis coinciding with the exit plane of the die, as shown in the figure. Along the die face AB, the boundary conditions are τ = mk,

w = u cot α,

along

z = (r − a) cot α,

a≤r≤b

where α denotes the semiangle of the die. The extruded part of the billet, which moves as a rigid body, is entirely free of surface tractions. The remaining boundary conditions may be written as u = 0, w = −1,

Fz = 0,

along

Fr = 0 on

r=0 z = c;

and

r=b

w = −(b2 /a2 )

on

z=0

700

9 The Finite Element Method

Fig. 9.9 Geometry and finite element grid pattern for the axisymmetric extrusion of a cylindrical billet through a conical die

r

B α τ

C

A

1 b

b 2/a 2 a

z O

The problem can be treated as one of steady state in which the geometrical configuration does not change with time. A complete elastic/plastic analysis for extrusion through a sigmoidal die until the attainment of the steady state has been discussed by Lee et al. (977). The results presented here have been obtained by the rigid/plastic method by Chen and Kobayashi (1978) and Chen et al. (1979). In the finite element analysis for the extrusion of a work-hardening material, the components of the strain rate at the center of each element are initially assumed to be same as those in a nonhardening material. Starting from a selected point on the plane of entry, where the effective strain is zero, the rate of change of the effective strain is determined from the known values at the surrounding element centers. Since the velocity of the selected point is found from the element interpolation formula, the effective strain and the new position of the particle are then easily obtained from the increment of time. This procedure is sequentially repeated, following the path of the particle, until the exit plane is reached. The flow lines emanating from different points on the entry plane determined this way furnish the shape of the distorted grid, and also the distribution of the effective strain throughout the deforming region. Since the distribution of the effective stress follows from the given stress– strain curve of the material, the new distribution of nodal point velocities can be computed in order to carry out the next iteration. When the velocity solution converges after a few iterations, the mean extrusion pressure and the distribution of the die pressure can be determined for the given frictional condition, die angle, and fractional reduction in area. ◦ The steady-state grid distortion pattern for frictionless extrusion through a 90 conical die, obtained by the finite element solution, is displayed in Fig. 9.10, where the upper half holds for a nonhardening material and the lower half for a workhardening material (SAE 1112 steel). The difference between the two patterns is due to the restriction of metal flow that occurs in a work-hardening material. The ◦ distribution of radial and axial velocities within the die for α = 45 and b/a = 2,

9.6

Illustrative Examples

701

Fig. 9.10 The distortion of an initial square grid in a cylindrical billet extruded through a 90◦ conical die (after Chen et al., 1978)

furnished by the finite element solution, is shown in Fig. 9.11. The material adopted in this solution is the same as that used in an experimental investigation by Shabaik and Thomsen (1968), who obtained remarkably similar results for the velocity distribution. The computation also reveals that the hydrostatic part of the stress in a region near the center of the deforming region becomes tensile at sufficiently large reductions, leading to the possibility of a central crack which is frequently observed.

9.6.3 Analysis of Spread in Sheet Rolling In the rolling of sheets and slabs, in which the width of the workpiece is less than about five times the length of the arc of contact, the usual assumption of plane strain is not justifiable. The amount of lateral spread that occurs in such cases is quite appreciable, and must be taken into consideration in the analysis of the rolling process. A finite element analysis of the problem using the three-dimensional brick element has been carried out by Li and Kobayashi (1982), who adopted the nonsteadystate approach for the solution. Figure 9.12 shows a narrow strip of the workpiece with the arrangement of an element in the upper half of the material within the arc of contact. The deformation of the material entering the roll gap with a bite is considered in a step-by-step manner based on a constant frictional stress, while updating the material properties and the coordinates of the nodal points at the end of each step. A steady state is assumed to be reached when the associated roll torque has attained a steady value, and the spread contour has become stationary. The computation has been carried out with a friction stress equal to 0.5 k, and using R/h0 = 160, two different values of w0 /h0 , and several values of the reduction in thickness, where R denotes the roll radius, 2 h0 is the initial slab thickness, and 2w0 is the initial slab width, the material used in the analysis being annealed

702

9 The Finite Element Method 0

0.5

z / b = 1.0

0

z / b = 1.0 –1.0

0.8 0.8 0.6

0.6

–0.5

0.4

w

u

0 –2.0

–1.0

0.3

0.4

0.2

–3.0

–1.5

0.1

0.3 0.2

0.1

0

–2.0

–4.0 –4.5 0

0.5

1.0 r/ b

–2.5

0

1.0

0.5 r/ b

Fig. 9.11 Distribution of radial and axial velocities of particles moving through a 90◦ conical die with b / a = 2 (after Chen and Kobayashi, 1982)

AISI 1018 steel. The final values of the thickness and width of the rolled stock are denote by 2hf and 2wf , respectively. The computed value of the mean lateral spread is plotted against the final reduction in height in Fig. 9.13a, which shows excellent agreement with some experimental results reported by Kobayashi et al. (1989). During the rolling process, not only the thickness but also the cross-sectional area of the rolled stock progressively decreases due to the effect of the lateral spread. The solid curves in Fig. 9.13(b) show the variation of the reduction in cross section of the rolled stock with the reduction in height for w0 /h0 =1 and 3, while the broken straight line indicates the plane strain situation in which the reduction in cross-sectional area is equal to the reduction in height. The spread in rolling has also been investigated approximately by Lahoti and Kobayashi (1974), and, by a finite element analysis, by Kanazawa and Marcal (1982). The finite element solution for the compression of a rectangular block has been discussed by Park and Kobayashi (1984), and that of a ring of square cross section has been considered by Park and Oh (1987). The shape rolling of bars of various cross sections, using the finite element method, has been investigated by Park and Oh (1990).

9.6

Illustrative Examples

703 y

x

2w f

ne it pla

ex ne

y pla

entr

O 2hf

2w o

z 2ho

Fig. 9.12 A schematic view of sheet rolling with lateral spread indicating the location of a typical finite brick element

25

10

[(w f –w 0) / w 0] × 102

8

w 0 / h 0 =1.0

6 4 3.0 2 0

0

20 10 [(h 0 –h f) / h 0] × 102

(a)

Reduction in area, percent

Theory Experiment

3.0

20

w0 / h0 15 1.0 10 5 0

0

10 20 Reduction in height, percent

(b)

Fig. 9.13 Results for sheet rolling with lateral spread. (a) Variation of overall spread with reduction in height, (b) variation of change in cross section with change in height

704

9 The Finite Element Method

9.6.4 Deep Drawing of Square Cups As a final example, consider the deep drawing of a square cup using a flat punch, the base of the punch being a square of sides 2a The schematic view of the process is similar to that shown in Fig. 2.28(a). The cup is drawn from a square blank whose sides have an initial length equal to 2b0 , the initial blank thickness being denoted by h0 . In the finite element formulation, the continuous blank holding force is replaced by a set of concentrated forces acting at the nodal points along the periphery of the blank. The frictional condition at the interfaces between the tools and the sheet metal is assumed to be governed by Coulomb’s law with a constant coefficient of friction. Denoting the die and punch profile radii by rd and rp respectively, and the punch corner radius by rc , the geometry of the process is defined as b0 = 2.75, a

h0 = 0.043, a

rp rd = = 0.25, a a

rc = 0.16. a

The material is aluminum killed steel having a uniform R-value equal to 1.6, the planar stress–strain curve being given by the power law σ = Cε n , where n = 0.228 and C = 739 MPa. The blank-holding force is taken as 4.9 kN, the friction coefficient being 0.2 over the punch and 0.04 over the die.

y

x

0 Under Punch

Under Blankholder

Fig. 9.14 Finite element mesh in a square blank to be drawn into a cup with square base

9.6

Illustrative Examples

705

The finite element mesh used in the analysis of the square cup drawing is shown in Fig. 9.14, which indicates a choice of finer mesh over the region where the thickness is expected to vary rapidly. The thinning of the sheet is found to have maximum values over the punch and die profile radii, particularly along the diagonal of the square. In Fig. 9.15, the computed distribution of the thickness strain across the diagonal of the formed cup, based on a = 20 mm, is compared with that obtained experimentally by Thomson (1975) for a given punch load. The predicted strain distribution has the same trend as that of the experimental one, though there is an appreciable difference in the magnitude of the strain. The discrepancy is due to the fact that the observed punch penetration is 1.50 a, which is significantly higher than the value 1.01 a predicted by the finite element solution and may be attributed to the difference in the frictional conditions existing in the experiment. Similar results based on a simplified finite element analysis have been reported by Majlessi and Lee (1993). Rigid/plastic finite element solutions for the hydrostatic bulging of circular diaphragms have been given by Lee and Kobayashi 1975) and Kim and Yang (1985a), and of elliptical diaphragms by Chung et al. (1988). The bulging of rectangular diaphragms has been investigated experimentally by Duncan and Johnson (1968) and numerically by Yang and Kim (1987). The large strain elastic/plastic

–80

y

0

x

Theory (d = a) Experiment (d = 1.5 a)

Thickness strain, percent

–60

–40

–20

Under punch

Under blankholder

0

Fig. 9.15 Comparison of theoretical and experimental distributions of thickness strain along the diagonal of a square blank (after Toh and Kobayashi, 1985)

+20

0

40 60 20 Initial distance from blank center, mm

706

9 The Finite Element Method

finite element formulation has been applied to the axisymmetric punch stretching problem by Kim et al. (1978) and to the plane strain bending of sheets by Oh and Kobayashi (1980). A rigid/plastic finite element solution to the punch stretching problem, based on the nonquadratic yield criterion, has been presented by Wang (1984). An elastic/plastic finite element analysis for the deep drawing of anisotropic cups has bee reported by Saran et al. (1990), and the associated problem of flange wrinkling has been investigated by Kim et al. (2000) and Correia et al. (2003).

Appendix: Orthogonal Curvilinear Coordinates

Cylindrical Coordinates The position of a typical particle is defined by the coordinates (r, θ , z) taken in the radial, circumferential, and axial directions, respectively. If the associated components of the velocity are denoted by (u, v, w), respectively, then the components of the true strain rate are ε˙ r = ε˙ θ =

∂u ∂r

1 ∂v u+ , r ∂θ ε˙ z =

∂w , ∂z

∂v v 1 ∂u − + , ∂r r r ∂θ 1 ∂v 1 ∂w + γ˙ θz = , 2 ∂z r ∂θ 1 ∂u ∂w γ˙ rz = + . 2 ∂z ∂r

γ˙ rθ =

1 2

If σr , σθ , and σz denote the normal stresses and τrθ , τθz , and τrz the shear stresses, then the equations of equilibrium in the absence of body forces are 1 ∂τrθ ∂τrz σr − σθ ∂σr + + + = 0, ∂r r ∂θ ∂z r 1 ∂σθ ∂τθz 2τrθ ∂τrθ + + + = 0, ∂r r ∂θ ∂z r ∂τrz 1 ∂τθz ∂σz τrz + + + = 0. ∂z r ∂θ ∂z r

Spherical Coordinates The coordinate system is defined by (r, φ, θ ), where r is the length of the radius vector, φ is the angle made by the radius vector with a fixed axis, and θ is the angle measured round this axis. If the velocity components in the coordinate directions are denoted by (u, v, w), then the components of the true strain rate are J. Chakrabarty, Applied Plasticity, Second Edition, Mechanical Engineering Series, C Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77674-3,

707

708

Appendix: Orthogonal Curvilinear Coordinates

∂u 1 ∂v v 1 ∂u ε˙ r = , γ˙rφ = − + , ∂r 2 ∂r r r ∂φ 1 1 1 ∂w w 1 ∂v ∂v ε˙ φ = − cot φ + u+ , γ˙rφ = , r ∂φ 2 r ∂φ r r sin φ ∂θ 1 1 ∂w w 1 ∂u ∂w ε˙ θ = − + u + v cot φ+cosec φ , γ˙rθ = . r ∂θ 2 ∂r r r sin φ ∂θ Denoting the normal stresses by σr , σφ , and σθ and the shear stresses by τrφ , τφθ , and τrθ , the equations of equilibrium in the absence of body forces can be written as ∂σr 1 ∂τrφ 1 ∂τrθ 1 + + 2σr − σφ − σθ + τrφ cot φ = 0, ∂r r ∂φ r sin φ ∂θ r 6 ∂τrφ 1 ∂σφ 1 ∂τφθ 1 7 + + σφ − σθ cot φ + 3τrφ = 0, ∂r r ∂φ r sin φ ∂θ r 1 ∂τφθ 1 ∂σθ 1 ∂τrθ + + + 3τrθ + 2τφθ cot φ = 0. ∂r r ∂φ r sin φ ∂θ r When the deformation is infinitesimal, the preceding expressions for the components of the strain rate may be regarded as those for the strain itself, provided the components of the velocity are interpreted as those of the displacement.

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Name Index

A Abrahamson, G.R., 593 Abramowicz, W., 533, 656 Ades, C.S., 541 Adie, J.F., 198 Ahn, C.S., 656 Aizawa, T., 665 Akserland, E.L., 546 Alexander, J.M., 94, 97, 103, 112, 184, 186, 198, 445, 530 Alter, B.E.K., 578 Amos, D.E., 604 Aoki, I., 447 Apsden, R.J., 623 Ariaratnam, S.T., 528 Asaro, R.J., 1 Ashwell, D.G., 556 Astuta, T., 487, 499 Atkins, A.G., 203, 205 Avitzur, B., 169, 177, 198 Azrin, M., 448 B Backman, M.E., 605 Backofen, W.A., 10, 103, 448 Bailey, J.A., 430 Baker, J.F., 487 Baker, W.E., 602 Balendra, R., 665 Baltov, A., 28 Banerjee, A.K., 578 Barlat, F., 421 Batdorf, S.R., 518 Batterman, S., 524, 528, 552, 556 Bell, J.F., 576, 578 Belytschko, T., 268, 270 Bhadra, P., 645 Bianchi, G., 578 Bijlaard, P.P., 518, 528

Biron, A., 323, 379 Bishop, J.F.W., 2, 15, 69, 167, 593 Bland, D.R., 24 Blazynski, T.Z.V., 181 Bleich, H.H., 633 Bodner, S.R., 571, 621, 628 Bourne, L., 465 Boyce, W.E., 233 Boyd, D.E., 663 Bramley, A.N., 412, 416 Brazier, L.G., 541 Bridgman, P.W., 11, 162, 163 Brooks, G.M., 323, 327, 358 Budiansky, B., 2, 26, 103, 455, 689 Bushnell, D., 554 C Caddell, R.M., 203, 205, 419 Calder, C.A., 617 Calladine, C.R., 198, 255 Campbell, J.D., 571, 572, 623 Casey, J., 26 Chaboche, J.L., 38 Chadwick, P., 599 Chakrabarty, J., 6, 12, 39, 43, 44, 45, 72, 76, 95, 97, 103, 109, 123, 126, 166, 185, 189, 192, 202, 206, 257, 275, 276, 278, 418, 420, 428, 439, 444, 446, 461, 466, 489, 526, 532, 533, 593, 609, 671 Chan, K.C., 61, 416, 451 Chan, K.S., 451 Chater, E., 451 Chawalla, E., 487 Chen, C.C., 198, 205, 698, 700, 701 Chen, F.K., 461 Chen, F.L., 645 Chen, W.F., 18, 487, 496, 499 Cheng, S.Y., 166 Chenot, J.L., 188, 200

743

744

Name Index

Chern, J., 94 Chiang, D.C., 122, 463 Chitkara, N.R., 443 Chon, C.T., 645 Chou, P.C., 617 Chu, C.C., 131, 451 Chung, S.Y., 112, 184 Chung, W.J., 705 Cinquini, C., 237, 302 Clark, D.S., 571 Clifton, R.J., 571 Cockroft, M.G., 177 Cole, I.M., 181 Collins, I.F., 169, 181, 203, 270, 275, 443, 444 Conroy, M.F., 629, 642 Coon, M.D., 323 Corona, A., 541 Cotter, B.A., 633 Cox, A.D., 645 Craggs, J.W., 571, 580 Cristescu, N., 568, 571, 573, 574, 576, 577, 578, 586, 663 Crossland, B., 11 Crozier, R.J.M., 602 Curtis, C.W., 578

Dugdale, D.S., 157 Duncan, J.L., 451, 705 Durban, D., 148, 198, 445, 514, 522 Duszek, M., 237, 243 Duwez, P.E., 561, 571, 578, 632

D Dafalias, Y.F., 29, 33 Danyluk, H.T., 147, 157 Davidenkov, N.N., 162 Davies, C.J., 11 Davies, R.M., 571 De Juhasz, K.J., 571 Dean, T.H., 658 Dekel, E., 609 Demir, H.H., 327 Denton, A.A., 184 DePierre, V., 177 DeRuntz, J.A., 379 Desdo, D., 151 Dillamore, I.L., 414 Dillons, O.W., 578 Ding, J.L., 422 Dinno, K.S., 385 Dodd, B., 419 Donnell, L.H., 541 Dorn, J.F., 407 Douch, L.S., 572 Dowling, A.R., 665 Drucker, D.C., 16, 19, 23, 26, 38, 209, 211, 294, 315, 381 Dubey, R.N., 528 Duffill, A.W., 185

G Galiev, S.U., 655 Galletly, G.D., 554 Gaskill, B., 617 Gaydon, P.A., 85 Gellin, S., 530, 546 Gerard, G., 518, 541, 554 Gere, J.M., 496, 505, 515, 517, 526, 530, 538 Ghosh, A.K., 451 Gill, S.S., 323, 385 Gjelsvik, A., 518 Goel, R.P., 571 Goldsmith, W., 571, 605, 617 Goodier, J.N., 593, 645 Gotoh, M., 424, 465, 687 Graf, A., 451 Green, A.P., 17, 62, 73, 77, 80 Griffis, Le Van, 571 Gu, W., 656 Gui, X.G., 633 Gunasekera, J.S., 220 Gurson, A.L., 2

E Eason, G., 155, 248, 297, 324, 326, 655 Efron, L., 578 Eisenberg, M.A., 38 El-Ghazaly, H.A., 518 El-Sebaie, M.S., 122, 463 Engesser, F., 486 Ewing, D.J.F., 72, 73 Ezra, A.A., 621 F Finnegan, S.A., 617 Fleming, W.T., 617, 621, 633 Fliigge, W., 343, 348, 370 Florence, A.L., 642, 644, 645, 656 Fogg, B., 128 Ford, H., 15, 18, 62, 94, 95 Forrestal, M.J., 600, 603, 605, 608, 609 Freiberger, W., 297, 393 Fukui, S., 128

H Haar, A., 148 Haddow, J.B., 144, 147, 157, 169, 211, 243, 656

Name Index Hailing, J., 198 Hamada, H., 522 Han, C.H., 198 Han, D.J., 18 Hart, E.W., 10 Hashmi, M.S.J., 623 Hassani, H.A., 131, 445 Hawkyard, J.B., 172, 176, 220, 590, 592, 593 Haydi, H.A., 367 Haythornthwaite, R.M., 233, 244, 255 Hecker, S.S., 447, 448 Hector, L.G., 111 Hencky, H., 26 Hetnarski, R.B., 29, 265 Hill, R., 1, 2, 15, 17, 19, 20, 21, 25, 42, 43, 45, 52, 59, 62, 69, 91, 97, 102, 108, 126, 140, 144, 151, 160, 161, 169, 170, 177, 205, 212, 214, 406, 407, 409, 418, 421, 425, 435, 438, 465, 480, 484, 581, 610, 613 Hillier, M.J., 65 Hodge, P.G., 21, 28, 85, 89, 97, 237, 243, 260, 264, 265, 268, 269, 270, 284, 320, 321, 323, 326, 328, 334, 339, 340, 341, 350, 351, 355, 356, 358, 361, 372, 376, 379, 647, 655 Hoffman, G.A., 402 Hong, H.K., 33 Hopkins, H.G., 229, 232, 240, 260, 297, 574, 586, 594, 635, 636, 644 Horne, M.R., 494 Horner, M.R., 323 Hosford, W.F., 419, 451 Hu, L.W., 263, 472 Hu, T.C., 391 Huang, S., 2 Hudson, G.E., 659, 661 Huh, H., 691 Hundy, B.B., 17, 62 Hunter, S.C., 571, 586, 602 Hutchinson, J.W., 2, 166, 412, 451, 484, 487, 554 I Ilahi, M.F., 97, 445 Ilyushin, A.A., 26 Inoue, T., 518 Ipson, T.W., 617 Ishlinsky, A., 26, 157 Issler, W., 402 Ivanov, G.V., 349 J Jackson, L.R., 407, 409 Jaeger, T., 238, 265, 302

745 Jansen, D.M., 580, 586 Jaumann, J.J., 25 Jiang, W., 28, 34 Johansen, K.W., 265, 281, 293 Johnson, R.W., 203, 220, 705 Johnson, W., 76, 112, 131, 169, 172, 176, 184, 193, 203, 211, 220, 265, 270, 279, 288, 440, 533, 589, 598, 609, 613, 633, 644, 664, 665, 705 Jones, L.L., 281, 629 Jones, N., 593, 621, 629, 633, 645, 655, 656 Juneja, B.L., 220 K Kachanov, A., 26 Kachi, Y., 33 Kaftanoglu, B., 103, 445 Kalisky, S., 633 Kamalvand, H., 499 Kamiya, N., 258 Karunes, B., 633 Kasuga, Y., 122 Kato, B., 518 Kawashima, I., 258 Keck, P., 456 Keeler, S.P., 103, 447 Keil, A.H., 663 Ketter, R.L., 496 Khan, A.S., 2 Kim, J.H., 103, 220, 445, 705, 706 Kim, M.U., 220 Klepaczko, J., 578 Klinger, L.G., 412 Kliushnikov, V.D., 26 Kobayashi, S., 103, 112, 122, 172, 177, 181, 188, 196, 208, 214, 217, 220, 419, 463, 671, 672, 687, 689, 705, 706 Koide, M., 445 Koiter, W.T., 24 Kojic, H., 410 Kolsky, H., 572, 578 Kondo, K., 255 K¨onig, J.A., 302 Koopman, D.C.A., 270 Kozlowski, W., 294 Krajcinovic, D., 237, 645 Kudo, H., 169, 177, 196, 698 Kuech, R.W., 379 Kukudjanov, V.N., 573 Kumar, A., 645 Kummerling, R., 214 Kuzin, P.A., 655

746 Kwasczynska, K., 169 Kyriakides, A., 536, 541 L Lahoti, G.D., 177, 181, 217, 702 Lakshmikantham, C., 361, 372 Lamba, H.S., 33 Lambert, E.R., 188 Lance, R.H., 233, 270, 347, 375 Landgraf, R.W., 30 Lange, K., 168 Lebedev, N.E., 571 Leckie, F.A., 385, 556, 557 Lee, C.C., 33 Lee, C.H., 26, 172, 220, 698, 706 Lee, C.W., 37 Lee, D., 451, 691, 705 Lee, E.H., 26, 28, 220, 362, 366, 367, 375, 379, 565, 568, 589, 629 Lee, L.C., 366 Lee, L.H.N., 131, 528, 593, 700 Lee, L.S.S., 629 Lee, S.H., 687, 705 Lee, S.L., 379 Lee, W.B., 2, 61, 416, 451 Lengyel, B., 186 Lensky, V.S., 568 Leth, C.F., 629 Leung, C.P., 358 L´evy, M., 16 Li, S., 530, 609, 655, 701 Lian, J., 421, 451 Lianis, G., 15, 18, 62 Lin, G.S., 518 Lin, H.C., 571, 578 Lin, S.B., 422 Lin, T.H., 2 Lindholm, U.S., 573, 578 Ling, F.F., 198 Liou, J.H., 33 Lippmann, H., 151, 214, 658 Liu, D., 642 Liu, J.H., 388 Liu, T., 642 Liu, Y.H., 384 Lockett, F.J., 156, 157 Lode, W., 13 Logan, R.W., 419 Lu, L.W., 499 Lubiner, J., 26, 573 Ludwik, P., 7 Luk, V.K., 600, 603, 604 Lund, O., 573

Name Index M Macdonald, A., 168 Maclellan, G.D.S., 203 Male, A.T., 177 Malvern, L.E., 571, 572, 575, 576, 578 Malyshev, V.M., 578 Mamalis, A.G., 220, 533 Mandel, J., 26, 574, 580 Manjoine, H.J., 623 Manolakos, D.E., 265 Mansfield, E.H., 270 Marcal, A.V., 297, 702 Marciniak, Z., 448, 451 Markin, A.A., 444 Markowitz, J., 472 Marshall, E.R., 163 Martin, J.B., 629, 633 Massonnet, C.E., 270, 281, 286, 297, 402, 475 Mazumdar, J., 248 McCrum, A.W., 85 McDowell, D.L., 33 Megarefs, C.J., 393 Meguid, S.A., 444 Mellor, P.B., 97, 112, 122, 123, 184, 185, 265, 412, 416, 419, 445, 448, 450, 451, 463 Mentel, T.J., 623, 633 Meyer, C.E., 158 Miles, J.P., 43, 166 Mitchell, L.A., 198 Miyauchi, K., 453 Montague, P., 323 Moore, G.G., 181, 184, 414 Mori, K., 697 Morland, L.W., 586, 645 Morrison, A.L., 205 Mr¨oz, Z., 28, 29, 33, 169 Munday, G., 663 Murakami, S., 253, 258 Myszkowsky, S., 258 N Nadai, A., 31 Naghdi, P.M., 24, 26, 28, 97, 258 Nagpal, V., 220 Nakamura, T., 348, 370 Nardo, S.V.N., 323 Naruse, K., 422 Naziri, H., 460 Neal, B.G., 80 Neale, K.W., 131, 445, 451 Nechitailo, N.V., 655 Needleman, A., 166, 414 Nemat-Nasser, S., 94, 97, 573, 574, 589

Name Index Nemirovsky, U.V., 237 Newitt, D.M., 663 Newmark, N.M., 97 Nicholas, T., 578 Nieh, T.G., 11 Nimi, Y., 198 Nine, H.D., 126 Nonaka, T., 633 Norbury, A.L., 159 Nordgren, R., 97 Novozhilov, V.V., 28 Nurick, G.N., 645 O Oblak, M., 258 Oh, S.I., 214, 676, 702, 706 Ohashi, Y., 253, 258, 323 Ohno, N., 33 Okawa, D.M., 656 Okouchi, T., 323 Onat, E.T., 233, 255, 297, 299, 331, 344, 347, 366, 367, 374, 393, 633 Osakada, K., 198 Osgood, W., 9 Owens, R.H., 633 P Padmanabhan, K.A., 11 Palgen, L., 38 Palusamy, S., 358 Park, J.J., 220, 702 Parkes, E.W., 617, 621 Parmar, A., 419, 445, 450 Paul, B., 340, 341, 617 Payne, D.J., 385 Pearce, R., 418, 460 Pearson, C.E., 502, 513 Pell, W.H., 238 Penny, R.K., 556, 557 Perrone, N., 89, 629, 633, 645 Perzyna, P., 574, 642 Phillips, A., 28, 37 Pian, T.H.H., 255 Popov, E.P., 33, 248 Prager, W., 25, 26, 28, 35, 97, 228, 229, 233, 238, 297, 302, 303, 344, 391, 393, 635, 636 Prandtl, L., 4, 16 Presnyakov, A.A., 11 Pugh, H., 186 Pugsley, A., 530 R Raghavan, K.S., 448 Rakhmatulin, H.A., 561, 571

747 Ramberg, W., 9 Ranshi, A.S., 76, 81 Raphanel, J.L., 633 Rawlings, B., 633 Recht, R.F., 617 Reddy, B.D., 198, 530 Reddy, V.V.K., 645 Rees, D.W.A., 29, 541 Reid, S.R., 530, 633 Reiss, R., 393, 402 Reuss, E., 16 Rice, J.R., 1, 66, 413, 434, 697 Richards, C.E., 73 Richmond, O., 205, 402 Rinehart, J.S., 664 Ripperger, E.A., 571 Robinson, M., 349, 358 Rogers, T.G., 97 Rosenberg, Z., 609 Ross, E.W., 97 Rowe, G.W., 203 Rychlewsky, R., 302 S Sachs, G., 412 Sagar, R., 220 Salvadori, N.G., 633 Samanta, S.K., 169, 208 Samuel, T., 159 Sanchez, L.R., 126 Sankaranarayanan, S., 97, 243, 339 Save, M.A., 281, 297, 402 Sawczuk, A., 28, 237, 238, 243, 265, 270, 284, 286, 289, 292, 302, 323, 328, 466 Sayir, M., 339 Schumann, W., 260 Seide, P., 546 Seiler, J.A., 633 Senior, B.W., 129 Sewell, M.J., 480, 484, 486, 513, 516 Shammamy, M.R., 97, 102, 445 Shanley, F.S., 482 Shapiro, G.S., 349, 571, 629, 644, 655 Shaw, M.C., 163 Sherbourne, A.N., 258, 367, 518 Sheu, C.Y., 302 Shield, R.T., 26, 145, 148, 152, 155, 209, 211, 233, 294, 297, 305, 324, 326, 381, 390, 391, 393, 655 Shrivastava, H.P., 514, 530 Shull, H.F., 263 Sidebottom, O.M., 33 Siebel, E., 167, 199, 659

748 Sinclair, G.B., 161 Skrzypek, J.J., 29, 265 Sobotka, Z., 238, 281 Sokolovsky, V.V., 59, 572 Sortais, H.C., 208 Southwell, R.V., 533 Sowerby, R., 451 Spencer, A.J.C., 193 Spurr, C.E., 72 Srivastava, A., 258 Stein, A., 578 Steinberg, D., 573 Sternglass, E.J., 578 Storakers, B., 102 Storen, S., 66 Stout, M.G., 415, 447 Stronge, W.J., 258, 629, 633 Stuart, D.A., 578 Suliciu, I., 578 Swift, H., 8, 65, 112, 177, 184, 430 Symonds, P.S., 617, 621, 629, 633, 645

T Tabor, D., 158 Tadros, A.K., 448 Tan, Z., 444 Tanaka, M., 258 Tate, A., 609 Taylor, G.I., 1, 92, 561, 586, 589, 590, 593 Tekinalp, B., 297 Thomas, H.K., 581 Thomsen, E.G., 196, 701 Thomson, T.R., 705 Thomson, W.T., 616 Timoshenko, S., 496, 505, 515, 517, 526, 530, 538 Ting, T.C.T., 571, 582, 583, 628 Tirosh, J., 122, 161, 198 Toh, C.H., 689, 705 Toth, L.C., 430 Travis, F.W., 664, 665 Tresca, H., 14 Triantafyllidis, N., 126, 414 Tseng, N.T., 33 Tsuta, T., 2 Tsutsumi, S., 112, 122 Tugcu, P., 518 Tupper, S.J., 589 Turvey, G.J., 258

Name Index U Unksov, L.P., 168 V Valanis, K.C., 38 Van Rooyen, G.T., 172 Vaughan, H., 656 Venter, R., 440 Vial, C., 419 Voce, E.B., 8 von Karman, Th., 148, 486, 487, 561, 578 von Mises, R., 1, 13, 16 W Wagoner, R.H., 415 Wallace, J.F., 122, 184, 414 Wang, A.J., 240, 644 Wang, N.M., 97, 445, 689, 706 Wang, X., 131 Wasti, S.T., 532 Weil, N.A., 97 Weingarten, V.l., 546 Weiss, H.J., 89 Weng, G.J., 28 Wertheimer, T.B., 697 Whiffen, A.C., 588, 593 White, M.P., 571 Whiteley, R.L., 460 Wierzbicki, T., 533, 633, 645 Wifi, S.A., 103, 112 Williams, B.K., 181, 203 Wilson, D.V., 451, 460 Wilson, W.R.D., 111 Wistreich, J.G., 203 Woo, D.M., 97, 103, 112 Wood, R.H., 281 Woodthorpe, J., 418 Wu, H.C., 2, 32, 39, 412, 422, 431, 578 X Xu, B.Y., 379 Y Yakovlev, S., 444 Yamada, Y., 445, 447 Yamaguchi, K., 451 Yang, D.Y., 97, 198, 220, 691, 705 Yeom, D.J., 358 Yin, Y., 2 Yoshida, K., 38 Yossifon, S., 122 Youngdahl, C.K., 645, 655 Yu, T.X., 131, 258, 541, 629, 633, 645

Name Index Z Zaid, M., 617 Zanon, P., 237 Zaoui, A., 2 Zaverl, R., 451

749 Zhang, L.C., 258, 541 Zhao, I., 451 Zhu, L., 645 Ziegler, H., 26, 28, 402 Zukas, J.A., 617

Subject Index

A Admissible fields, 42, 213, 481 Angular velocity hodograph, 288, 310 Anisotropic hardening, 33 Anisotropic material flow rule, 407, 413 work-hardening, 409, 413 yield criterion, 405–407 Anisotropic parameters, 408–410 Anisotropy effects on plane plastic flow, 405–407 on plastic torsion, 424 on plates and shells, 466 on sheet metal forming, 444, 445 on slipline fields, 438–440 on stress-strain curves, 414–416 Annular plates, 372 Associated flow rule, 15 Axisymmetric problems compression of blocks, 169, 175 conical flow field, 147 extrusion of billets, 184 indentation, 155 necking in tension, 161–163 tube sinking, 177 wire drawing, 198–199 yield point in tubes, 144 B Back pull factor, 204 Bar drawing, 214–218 Bauschinger effect, 6, 26 Beam columns, 500 Bending of plates and shells, 227, 394 of prismatic beams, 73–76 Bending moments, 76, 228, 303 Bifurcation, 43, 480–484 Blanking, 664

Blast loading, 633 Bounding surface, 33–36 Brinell hardness, 158 Buckling of circular plates, 519–522 of cylindrical shells, 522–524, 656 of eccentric columns, 489 of narrow beams, 500 of rectangular plates, 511–516 of spherical shells, 546–550 of straight columns, 482 C Cantilevers, 73–76, 506, 625–629 Cavitation, 610–614 Cavity expansion compressible material, 604 incompressible material, 602 Characteristics axial symmetry, 164 plane plastic strain, 413 plane stress, 49–52 propagation of waves, 564, 569 torsion of bars, 502 Circular plates limit analysis, 240, 466 optimum design, 294 Collapse load, 19 circular plates, 227–229, 472 conical shells, 379 cylindrical shells, 321 hollow square plates, 87 noncircular plates, 258–261 spherical shells, 355–358, 471 Column buckling, 479–480, 487 Combined hardening, 28–30 Compression of hollow cylinders, 175, 698 of rectangular blocks, 212

751

752 Compression (cont.) of solid cylinders, 166–170 Compression test, 6 Conical shells, 367–369 Constitutive equations, 39 Constraint factor, 68–71 Contraction ratio, 4–5 Converging flow, 145 Crumpling of projectiles, 586–593 Curvature rates, 228, 260, 295 Curvilinear coordinates, 707 Cyclic plasticity, 35 Cylindrical coordinates, 707 Cylindrical shells limit analysis, 323 plastic buckling, 522–524 Cylindrical tubes combined loading, 167 flexural buckling, 537–541 torsional buckling, 537–541 D Deep drawing of anisotropic blanks, 460 of isotropic blanks, 452 Deflection of beams, 618, 621 of circular plates, 243–253, 634 of cylindrical shells, 379, 647 Deformation theory, 253, 476 Deviatoric plane, 12 Deviatoric stress, 11 Diffuse necking, 65 Direct forces, 332 Discontinuity in stress, 63–64 in velocity, 59–61 Dissipation function, 260, 261, 267 Double modulus, 571 Drawbeads, 125 Drawing processes bar drawing, 214 cup drawing, 704 tube drawing, 179 wire drawing, 198–208 Drawing stress, 179 Drucker’s postulates, 26 Dynamic analysis of cavity formation, 593, 600 of circular diaphragms, 659 of projectile penetration, 604 of structural members, 487, 633 Dynamic expansion of cavities, 593–604

Subject Index Dynamic forming of metals, 656–665 Dynamic loading of cantilevers, 617, 622 of circular plates, 633 of cylindrical shells, 645 of free-ended beams, 629 E Earing of deep-drawn cups, 464 Effective strain, 129 Effective stress, 21 Elastic/plastic analysis of circular plates, 243 of cylindrical shells, 554 Elastic/plastic material, 9 Elemental stiffness equation, 675 Elliptic plates, 308 Empirical stress-strain equations, 7 Endochronic theory, 38 Engineering strain, 3 Equation of motion, 563, 602, 613 Equilibrium equations axial symmetry, 139 circular plates, 229 cylindrical shells, 315 noncircular plates, 260 plane stress, 51 shells of revolution, 342 Equivalent strain, 60, 408, 422 Equivalent stress, 21, 408, 422 Explosive forming, 664 Extrusion of metals through conical dies, 186, 196 through square dies, 189, 193 Extrusion pressure, 187 Extrusion ratio, 188 F Fatigue failure, 30 Finite element method, 265, 671 element shape functions, 673 elemental stiffness equations, 675 global stiffness equations, 695 Finite elements isoparametric elements, 678 quadrilateral elements, 679 three-dimensional brick element, 679 triangular elements, 676 Finite element solutions axisymmetric extrusion, 698 compression of a cylinder, 697 lateral spread in rolling, 701 plastic collapse of plates, 265 square cup drawing, 704

Subject Index Finite expansion of a hole in infinite plate, 91 of a spherical cavity, 593 Flange drawing, 130, 687 Flange wrinkling, 126 Flexural buckling, 537 Flow rules L´evy-Mises, 55 Prandtl-Reuss, 17 Tresca’s associated, 17 Flush nozzles, 385 Forging of metals high speed forming, 664 quasi-static forgoing, 479, 571 Forming limit diagram, 447 Fractional reduction, 188 Friction coefficient, 188 Friction factor, 200 G Generalized strain rates, 228, 259, 349 Generalized stresses, 228, 259, 349 Geometry changes, 20 Global stiffness equation, 675 Grooved sheet in tension, 61 H Hardening rules combined or mixed, 28 isotropic, 21 kinematic, 26 Hardness of metals, 157 Head shape, 613 Hencky theory, 26, 66, 258 Hinge circle, 229, 321, 338 Hinge rotation, 277 Hodograph, 272 Hole expansion, 89, 97 Homogeneous work, 188 Hooke’s law, 16 Hydrostatic stress, 11 I Ideal die profile, 205 Ideally plastic material, 20 Impulsive loading, 633, 643, 644 Incompressibility, 15, 20 Indentation of anisotropic medium, 436 by circular flat punch, 152 by conical indenter, 155 by rectangular flat punch, 208 by spherical indenter, 157 Instability in tension

753 of circular diaphragms, 134 of cylindrical bars, 31, 161 of plane sheets, 97, 413, 445 Interaction curves, 143, 241, 320 Invariants, 11 Ironing of cups, 126 I-section beams, 81 Isotropic hardening, 21 Isotropic material, 17 J Jacobian matrix, 682 Jacobian of transformation, 678 Jaumann stress rate, 24 Jump conditions, 564, 641 K Kinematically admissible, 20 Kinematic hardening, 26 Kinetic energy, 586 Kronecker delta, 169, 677 L Lateral buckling, 499 Lateral spread, 212 Length changes in torsion, 430 L´evy-Mises equations, 55 Limit analysis circular plates, 227 conical shells, 367 cylindrical shells, 313 extrusion of billets, 184 hollow square plates, 81 rectangular plates, 261, 270 spherical shells, 353 triangular plates, 279 Limited interaction, 350 Limiting drawing ratio, 460 Limit strains, 451 Limit theorems, 18 Linearized yield condition, 319, 456 Linear programming, 270 Load–deflection relations, 248 Loading surface, 33 Localized necking, 60, 64, 412 Lode variables, 17, 62 Logarithmic strain, 690 Longitudinal stress waves, 561 Lower bound loads, 82, 317, 351 Lower bound theorem, 20 L¨uders bands, 7 Ludwik equation, 9 M Material derivative, 93

754 Maximum work principle, 69 Membrane forces, 253, 346 Metal forming problems bar drawing, 214 deep drawing, 111, 452 extrusion, 184 forging, 212, 214, 659 steckel rolling, 217 tube sinking, 178 wire drawing, 198 Micromechanical model, 29 Minimum weight design of circular plates, 284 of cylindrical shells, 382 of elliptic plates, 308 Mixed hardening, 28 Moment-curvature relation, 489 N Necking in tension, 163 Neutral loading, 23 Nodal forces, 696 Nodal velocities, 679 Nominal stress, 3 Nominal stress rate, 39 Nonquadratic yield function, 421, 445 Normality rule, 16 Normal stress, 138 Notched strip in tension, 67 Numerical integration, 691 O Objective stress rate, 25 Ogival nose, 611 Optimum design, 296, 393 Optimum die angle, 128 Orthotropic material, 410 P Penalty constant, 672 Perfectly plastic material, 624 Perforation of a plate, 617 Plane plastic strain, 272 Plane strain analogy, 270 Plane strain compression, 434 Plane stress anisotropic material, 392, 420 isotropic material, 47 Plastic anisotropy, 405 Plastic buckling, 484 Plastic collapse, 73 Plastic instability, 65, 132, 445, 450 Plastic modulus, 4, 29 Plastic wave propagation, 568, 579

Subject Index Plate bending, 265 Poisson’s ratio, 4 Polycrystalline aggregate, 2 Prandtl–Reuss relations, 17 Preferred orientation, 405 Pressure vessels, 382 Principal curvatures, 272 Principal line theory, 151 Principal stresses, 11 Projectile crumpling, 586 Projectile penetration, 610 Proportional limit, 3 Pulse loading of circular plates, 635 of cylindrical shells, 645 Punch load, 106, 122, 462 Punch stretching, 107 Punch travel, 122, 125, 459 Pure bending, 487, 545 Pure shear, 13 R Ramberg-Osgood equation, 9 Rate of extension, 46, 413 Rate of rotation, 25 Rate sensitive material, 576, 624 Recrystallization, 10 Rectangular plates, 261, 274 Reduced modulus, 486 Redundant work, 202 Residual stress, 94, 186 Reversed loading, 29 Rigid body rotation, 25 Rigid/plastic material, 16 Rotational symmetry, 227, 230 R-value of sheet metal, 416 S Sandwich approximation, 319, 320 Shape functions, 681 Shear force, 306 Shear modulus, 16 Sheet metal forming deep drawing, 111, 452 explosive forming, 664 stretch forming, 97 Shell buckling, 522 Shells of revolution, 342 Shock waves, 568 Simple shear, 73, 425 Simple waves, 564 Slipline fields, 150, 440 Slip planes, 2 Snap through buckling, 554

Subject Index Spherical cavity, 594 Spherical coordinates, 707 Spherical shell, 353, 552 Spin tensor, 45 Stability criterion, 43 Statically admissible, 19 Stiffness matrix, 685 Strain equivalence, 414 Strain hardening isotropic hardening, 21 kinematic hardening, 26 mixed hardening, 28 Strain hardening exponent, 454 Strain rate, 10 Strain rate effect, 577, 578 Strain rate sensitivity, 10 Stress discontinuity, 64, 185 Stress profile, 249, 357 Stress rate, 24 Stress resultant, 272, 315, 393 Stress space, 12, 334 Stress-strain curves, 3, 30, 416, 559 Stress–strain relations, 26 Stretch forming by hydrostatic pressure, 144, 272 by rigid punch head, 447 Strong discontinuity, 579 Strong support, 78 Superplasticity, 10 Swaging process, 140 Swift equation, 8 T Tangent modulus, 4 Tensile test, 3 Three-dimensional problems, 208 Torque–twist relation, 31 Torsional buckling, 586 Torsion test, 431 Tresca criterion, 14 Tresca’s associated flow rule, 17 True strain, 3 True strain rate, 24 True stress, 3 True stress rate, 39 Tube buckling, 542 Tube extrusion, 193 Tube sinking, 178 Two-surface theory, 33 U Uniqueness criterion, 41 Uniqueness of stress, 39

755 Unloading process, 32, 95, 565 Unloading waves, 568 Upper bound loads, 219, 292 Upper bound theorem, 20 V Variational principle, 215, 524, 552, 672, 688 Velocity discontinuity, 64 Velocity field, 172, 196, 213, 267 Virtual velocity, 20, 170 Virtual work principle, 19 Voce equation, 8 Volume constancy, 6, 118 Von Mises criterion, 14 W Warping function, 426 Wave front, 574 Wave propagation longitudinal waves, 571 planes waves, 563 three-dimensional waves, 681 Weak support, 77 Weak waves, 564 Wire drawing conical die profile, 199 ideal die profile, 205 Work equivalence, 415 Work-hardening, 21 Y Yield condition bending of plates, 229, 390 cylindrical shells, 316, 319 shells of revolution, 342 Yield criterion anisotropic, 407 regular, 13 singular, 13 Tresca, 14 von Mises, 13 Yield function, 11, 407 Yield hinge, 261, 324 Yield line solutions distributed loading, 286 elliptical plates, 289 rectangular plates, 264 semi-circular plates, 293 Yield locus, 13, 58, 228, 320 Yield point, 3, 19 Yield surface, 12, 33, 333, 336 Young’s modulus, 4

Mechanical Engineering Series (continued from page ii) D. Gross and T. Seelig, Fracture Mechanics with Introduction to Micro-mechanics K.C. Gupta, Mechanics and Control of Robots R. A. Howland, Intermediate Dynamics: A Linear Algebraic Approach D. G. Hull, Optimal Control Theory for Applications J. Ida and J.P.A. Bastos, Electromagnetics and Calculations of Fields M. Kaviany, Principles of Convective Heat Transfer, 2nd ed. M. Kaviany, Principles of Heat Transfer in Porous Media, 2nd ed. E.N. Kuznetsov, Underconstrained Structural Systems P. Ladevèze, Nonlinear Computational Structural Mechanics: New Approaches and Non-Incremental Methods of Calculation P. Ladevèze and J.-P. Pelle, Mastering Calculations in Linear and Nonlinear Mechanics A. Lawrence, Modern Inertial Technology: Navigation, Guidance, and Control, 2nd ed. R.A. Layton, Principles of Analytical System Dynamics F.F. Ling, W.M. Lai, D.A. Lucca, Fundamentals of Surface Mechanics: With Applications, 2nd ed. C.V. Madhusudana, Thermal Contact Conductance D.P. Miannay, Fracture Mechanics D.P. Miannay, Time-Dependent Fracture Mechanics D.K. Miu, Mechatronics: Electromechanics and Contromechanics D. Post, B. Han, and P. Ifju, High Sensitivity and Moiré: Experimental Analysis for Mechanics and Materials R. Rajamani, Vehicle Dynamics and Control F.P. Rimrott, Introductory Attitude Dynamics S.S. Sadhal, P.S. Ayyaswamy, and J.N. Chung, Transport Phenomena with Drops and Bubbles A.A. Shabana, Theory of Vibration: An Introduction, 2nd ed. A.A. Shabana, Theory of Vibration: Discrete and Continuous Systems, 2nd ed. Y. Tseytlin, Structural Synthesis in Precision Elasticity