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Continuum Mechanics and Plasticity © 2005 by Chapman & Hall/CRC Press © 2005 by Chapman & Hall/CRC Press CRC SERIES: MODERN MECHANICS AND MATHEMATICS PUBLISHED TITLES BEYOND PERTURBATION: INTRODUCTION TO THE HOMOTOPY ANALYSIS METHOD by Shijun Liao MECHANICS OF ELASTIC COMPOSITES by Nicolaie Dan Cristescu, Eduard-Marius Craciun, and Eugen Soós CONTINUUM MECHANICS AND PLASTICITY by Han-Chin Wu FORTHCOMING TITLES HYBRID INCOMPATIBLE FINITE ELEMENT METHODS by Theodore H.H. Pian, Chang-Chun Wu MICROSTRUCTURAL RANDOMNESS IN MECHANICS OF MATERIALS by Martin Ostroja Starzewski Series Editors: David Gao and Ray W. Ogden CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. Han-Chin Wu Continuum Mechanics and Plasticity © 2005 by Chapman & Hall/CRC Press This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. © 2005 by Chapman & Hall/CRC Press No claim to original U.S. Government works International Standard Book Number 1-58488-363-4 Library of Congress Card Number 2004055118 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper Library of Congress Cataloging-in-Publication Data Wu, Han-Chin. Continuum mechanics and plasticity / Han-Chin Wu. p. cm. — (Modern mechanics and mathematics series ; no. 3) Includes index. ISBN 1-58488-363-4 (alk. paper) 1. Continuum mechanics. 2. Plasticity. I. Title. II. CRC series—modern mechanics and mathematics ; 3. QA808.2.W8 2004 531—dc22 2004055118 © 2005 by Chapman & Hall/CRC Press Visit the CRC Press Web site at www.crcpress.com Contents Preface xiii Author xvii Part I Fundamentals of Continuum Mechanics 1 Cartesian Tensors 3 1.1 Introduction 3 1.1.1 Notations 3 1.1.2 Cartesian Coordinate System 4 1.1.3 Special Tensors 4 1.2 Vectors 5 1.2.1 Base Vectors and Components 5 1.2.2 Vector Addition and Multiplication 5 1.2.3 The e–δ Identity 6 1.3 The Transformation of Axes 8 1.4 The Dyadic Product (The Tensor Product) 12 1.5 Cartesian Tensors 13 1.5.1 General Properties 13 1.5.2 Multiplication of Tensors 16 1.5.3 The Component Form and Matrices 18 1.5.4 Quotient Law 19 1.6 Rotation of a Tensor 20 1.6.1 Orthogonal Tensor 20 1.6.2 Component Form of Rotation of a Tensor 22 1.6.3 Some Remarks 23 1.7 The Isotropic Tensors 28 1.8 Vector and Tensor Calculus 34 1.8.1 Tensor Field 34 1.8.2 Gradient, Divergence, Curl 34 1.8.3 The Theorem of Gauss 37 References 40 Problems 41 2 Stress 45 2.1 Introduction 45 2.2 Forces 45 v © 2005 by Chapman & Hall/CRC Press vi Contents 2.3 Stress Vector 46 2.4 The Stress Tensor 47 2.5 Equations of Equilibrium 50 2.6 Symmetry of the Stress Tensor 52 2.7 Principal Stresses 53 2.8 Properties of Eigenvalues and Eigenvectors 55 2.9 Normal and Shear Components 59 2.9.1 Directions Along which Normal Components of σ ij Are Maximized or Minimized 60 2.9.2 The Maximum Shear Stress 60 2.10 Mean and Deviatoric Stresses 64 2.11 Octahedral Shearing Stress 65 2.12 The Stress Invariants 66 2.13 Spectral Decomposition of a Symmetric Tensor of Rank Two 69 2.14 Powers of a Tensor 71 2.15 Cayley–Hamilton Theorem 72 References 73 Problems 74 3 Motion and Deformation 79 3.1 Introduction 79 3.2 Material and Spatial Descriptions 80 3.2.1 Material Description 80 3.2.2 Spatial Description 81 3.3 Description of Deformation 83 3.4 Deformation of a Neighborhood 83 3.4.1 Homogeneous Deformations 85 3.4.2 Nonhomogeneous Deformations 86 3.5 The Deformation Gradient 88 3.5.1 The Polar Decomposition Theorem 88 3.5.2 Polar Decompositions of the Deformation Gradient 90 3.6 The Right Cauchy–Green Deformation Tensor 98 3.6.1 The Physical Meaning 98 3.6.2 Transformation Properties of C RS 101 3.6.3 Eigenvalues and Eigenvectors of C RS 103 3.6.4 Principal Invariants of C RS 104 3.7 Deformation of Volume and Area of a Material Element 105 3.8 The Left Cauchy–Green Deformation Tensor 108 3.9 The Lagrangian and Eulerian Strain Tensors 108 3.9.1 Definitions 108 3.9.2 Geometric Interpretation of the Strain Components 112 3.9.3 The Volumetric Strain 115 3.10 Other Strain Measures 118 3.11 Material Rate of Change 119 3.11.1 Material Description of the Material Derivative 119 3.11.2 Spatial Description of the Material Derivative 120 © 2005 by Chapman & Hall/CRC Press Contents vii 3.12 Dual Vectors and Dual Tensors 122 3.13 Velocity of a Particle Relative to a Neighboring Particle 124 3.14 Physical Significance of the Rate of Deformation Tensor 125 3.15 Physical Significance of the Spin Tensor 128 3.16 Expressions for D and W in Terms of F 129 3.17 Material Derivative of Strain Measures 131 3.18 Material Derivative of Area and Volume Elements 132 References 133 Problems 134 4 Conservation Laws and Constitutive Equation 141 4.1 Introduction 141 4.2 Bulk Material Rate of Change 142 4.3 Conservation Laws 145 4.3.1 The Conservation of Mass 145 4.3.2 The Conservation of Momentum 146 4.3.3 The Conservation of Energy 148 4.4 The Constitutive Laws in the Material Description 150 4.4.1 The Conservation of Mass 150 4.4.2 The Conservation of Momentum 151 4.4.3 The Conservation of Energy 163 4.5 Objective Tensors 164 4.6 Property of Deformation and Motion Tensors Under Reference Frame Transformation 166 4.7 Objective Rates 169 4.7.1 Some Objective Rates 169 4.7.2 Physical Meaning of the Jaumann Stress Rate 172 4.8 Finite Elasticity 174 4.8.1 The Cauchy Elasticity 175 4.8.2 Hyperelasticity 177 4.8.3 Isotropic Hyperelastic Materials 181 4.8.4 Applications of Isotropic Hyperelasticity 185 4.9 Infinitesimal Theory of Elasticity 193 4.9.1 Constitutive Equation 193 4.9.2 Homogeneous Deformations 195 4.9.3 Boundary-Value Problems 197 4.10 Hypoelasticity 197 References 200 Problems 200 Part II Continuum Theory of Plasticity 5 Fundamentals of Continuum Plasticity 205 5.1 Introduction 205 © 2005 by Chapman & Hall/CRC Press viii Contents 5.2 Some Basic Mechanical Tests 209 5.2.1 The Uniaxial Tension Test 209 5.2.2 The Uniaxial Compression Test 216 5.2.3 The Torsion Test 219 5.2.4 Strain Rate, Temperature, and Creep 225 5.3 Modeling the Stress–Strain Curve 231 5.4 The Effects of Hydrostatic Pressure 234 5.5 Torsion Test in the Large Strain Range 237 5.5.1 Introduction 237 5.5.2 Experimental Program and Procedures 241 5.5.3 Experimental Results and Discussions 246 5.5.4 Determination of Shear Stress–Strain Curve 256 References 260 Problems 263 6 The Flow Theory of Plasticity 265 6.1 Introduction 265 6.2 The Concept of Yield Criterion 265 6.2.1 Mathematical Expressions of Yield Surface 269 6.2.2 Geometrical Representation of Yield Surface in the Principal Stress Space 271 6.3 The Flow Rule 274 6.4 The Elastic-Perfectly Plastic Material 276 6.5 Strain-Hardening 286 6.5.1 Drucker’s Postulate 287 6.5.2 The Isotropic-Hardening Rule 290 6.5.3 The Kinematic-Hardening Rule 296 6.5.4 General Form of Subsequent Yield Function and Its Flow Rule 301 6.6 The Return-Mapping Algorithm 306 6.7 Combined Axial–Torsion of Strain-Hardening Materials 308 6.8 Flow Theory in the Strain Space 314 6.9 Remarks 316 References 317 Problems 318 7 Advances in Plasticity 323 7.1 Introduction 323 7.2 Experimenal Determination of Yield Surfaces 324 7.2.1 Factors Affecting the Determination of Yield Surface 325 7.2.2 A Summary of Experiments Related to the Determination of Yield Surfaces 328 7.2.3 Yield Surface Versus Loading Surface 333 7.2.4 Yield Surface at Elevated Temperature 335 7.3 The Direction of the Plastic Strain Increment 336 7.4 Multisurface Models of Flow Plasticity 340 © 2005 by Chapman & Hall/CRC Press Contents ix 7.4.1 The Mroz Kinematic-Hardening Model 340 7.4.2 The Two-Surface Model of Dafalias and Popov 344 7.5 The Plastic Strain Trajectory Approach 351 7.5.1 The Theory of Ilyushin 351 7.5.2 The Endochronic Theory of Plasticity 356 7.6 Finite Plastic Deformation 357 7.6.1 The Stress and Strain Measures 358 7.6.2 The Decomposition of Strain and Strain Rate 358 7.6.3 The Objective Rates 364 7.6.4 A Theory of Finite Elastic–Plastic Deformation 368 7.6.5 A Study of Simple Shear Using Rigid-Plastic Equations with Linear Kinematic Hardening 374 7.6.6 The Yield Criterion for Finite Plasticity 384 References 393 Problems 397 8 Internal Variable Theory of Thermo-Mechanical Behaviors and Endochronic Theory of Plasticity 399 8.1 Introduction 399 8.2 Concepts and Terminologies of Thermodynamics 399 8.2.1 The First Law of Thermodynamics 399 8.2.2 State Variables, State Functions, and the Second Law of Thermodynamics 400 8.3 Thermodynamics of Internal State Variables 403 8.3.1 Irreversible Systems 403 8.3.2 The Clausius–Duhem Inequality 405 8.3.3 The Helmholtz Formulation of Thermo-Mechanical Behavior 406 8.3.4 The Gibbs Formulation of Thermo-Mechanical Behavior 408 8.4 The Endochronic Theory of Plasticity 410 8.4.1 The Concepts of the Endochronic Theory 410 8.4.2 The Simple Endochronic Theory of Plasticity 412 8.4.3 The Improved Endochronic Theory of Plasticity 421 8.4.4 Derivation of the Flow Theory of Plasticity from Endochronic Theory 425 8.4.5 Applications of the Endochronic Theory to Metals 427 8.4.6 The Endochronic Theory of Viscoplasticity 441 References 450 Problems 452 9 Topics in Endochronic Plasticity 455 9.1 Introduction 455 9.2 An Endochronic Theory of Anisotropic Plasticity 455 9.2.1 An Endochronic Theory Accounting for Deformation Induced Anisotropy 455 9.2.2 An Endochronic Theory for Anisotropic Sheet Metals 461 © 2005 by Chapman & Hall/CRC Press x Contents 9.3 Endochronic Plasticity in the Finite Strain Range 468 9.3.1 Corotational Integrals 469 9.3.2 Endochronic Equations for Finite Plastic Deformation 474 9.3.3 Application to a Rigid-Plastic Thin-Walled Tube Under Torsion 476 9.4 An Endochronic Theory for Porous and Granular Materials 487 9.4.1 The Endochronic Equations 490 9.4.2 Application to Concrete 500 9.4.3 Application to Sand 502 9.4.4 Application to Porous Aluminum 503 9.5 An Endochronic Formulation of a Plastically Deformed Damaged Continuum 506 9.5.1 Introduction 506 9.5.2 The Anisotropic Damage Tensor 507 9.5.3 Gross Stress, Net Stress, and Effective Stress 512 9.5.4 An Internal State Variables Theory 516 9.5.5 Plasticity and Damage 521 9.5.6 The Constitutive Equations and Constraints 523 9.5.7 A Brief Summary of Wu and Nanakorn’s Endochronic CDM 526 9.5.8 Application 530 9.5.9 Concluding Remarks 535 References 537 Problems 541 10 Anisotropic Plasticity for Sheet Metals 543 10.1 Introduction 543 10.2 Standard Tests for Sheet Metal 545 10.2.1 The Uniaxial Tension Test 545 10.2.2 Equibiaxial Tension Test 545 10.2.3 Hydraulic Bulge Test 545 10.2.4 Through-Thickness Compression Test 545 10.2.5 Plane-Strain Compression Test 546 10.2.6 Simple Shear Test 546 10.3 Experimental Yield Surface for Sheet Metal 546 10.4 Hill’s Anisotropic Theory of Plasticity 548 10.4.1 The Quadratic Yield Criterion 548 10.4.2 The Flow Rule and the R-Ratio 550 10.4.3 The Equivalent Stress and Equivalent Strain 552 10.4.4 The Anomalous Behavior 553 10.5 Nonquadratic Yield Functions 555 10.6 Anisotropic Plasticity Using Combined Isotropic–Kinematic Hardening 558 10.6.1 Introduction 558 © 2005 by Chapman & Hall/CRC Press