Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity: Multiplicative Decomposition with Subloading Surface Model [1 ed.] 0128194286, 9780128194287

Table of contents :
Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity
Copyright
Contents
Preface
1 Mathematical fundamentals
1.1 Matrix algebra
1.1.1 Summation convention
1.1.2 Kronecker’s delta and alternating symbol
1.1.3 Matrix notation and determinant
1.2 Vector
1.2.1 Definition of vector
1.2.2 Operations of vector
1.2.2.1 Scalar product
1.2.2.2 Vector product
1.2.2.3 Scalar and vector triple products
1.2.2.4 Primary and reciprocal vectors
1.2.2.5 Tensor product
1.3 Definition of tensor
1.4 Tensor operations
1.4.1 Properties of second-order tensor
1.4.2 Tensor components
1.4.3 Transposed tensor
1.4.4 Inverse tensor
1.4.5 Orthogonal tensor
1.4.6 Tensor decompositions
1.4.6.1 Symmetric and skew-symmetric tensors
1.4.6.2 Spherical and deviatoric tensors
1.4.7 Axial vector
1.4.8 Determinant
1.4.9 Simultaneous equation for vector components
1.5 Representations of tensors
1.5.1 Notations in tensor operations
1.5.2 Operational tensors
1.5.3 Isotropic tensors
1.6 Eigenvalues and eigenvectors
1.6.1 Eigenvalues and eigenvectors of second-order tensors
1.6.2 Spectral representation and elementary tensor functions
1.6.3 Cayley–Hamilton theorem
1.6.4 Scalar triple products with invariants
1.6.5 Second-order tensor functions
1.6.6 Positive-definite tensor and polar decomposition
1.6.7 Representation theorem of isotropic tensor-valued tensor function
1.7 Differential formulae
1.7.1 Partial derivatives of tensor functions
1.7.2 Time derivatives in Lagrangian and Eulerian descriptions
1.7.3 Derivatives of tensor field
1.7.3.1 Gradient
1.7.3.2 Divergence
1.7.3.3 Rotation (or curl)
1.7.4 Gauss’ divergence theorem
1.7.5 Material-time derivative of volume integration
1.8 Variations of geometrical elements
1.8.1 Deformation gradient and variations of line, surface and volume elements
1.8.2 Velocity gradient and rates of line, surface and volume elements
2 Curvilinear coordinate system
2.1 Primary and reciprocal base vectors
2.2 Metric tensor and base vector algebra
2.3 Tensor representations
3 Tensor operations in convected coordinate system
3.1 Advantages of description in embedded coordinate system
3.2 Convected base vectors
3.3 Deformation gradient tensor
3.4 Pull-back and push-forward operations
3.5 Convected time-derivative
3.5.1 General convected derivative
3.5.2 Corotational rate
3.5.3 Objectivity of convected rate
4 Deformation/rotation (rate) tensors
4.1 Deformation tensors
4.2 Strain tensors
4.2.1 Green and Almansi strain tensors
4.2.2 General strain tensors
4.2.3 Logarithmic strain tensor
4.3 Volumetric and isochoric parts of deformation gradient tensor
4.4 Strain rate and spin tensors
4.4.1 Strain rate and spin tensors based on velocity gradient tensor
4.4.2 Strain rate tensor based on general strain tensor
5 Conservation laws and stress tensors
5.1 Conservation laws
5.1.1 Conservation law of physical quantity
5.1.2 Conservation law of mass
5.1.3 Conservation law of linear momentum
5.1.4 Conservation law of angular momentum
5.2 Cauchy stress tensor
5.2.1 Definition of Cauchy stress tensor
5.2.2 Symmetry of Cauchy stress tensor
5.3 Balance laws in current configuration
5.3.1 Translational equilibrium
5.3.2 Rotational equilibrium: symmetry of Cauchy stress tensor
5.3.3 Virtual work principle
5.3.4 Conservation law of energy
5.4 Work-conjugacy
5.4.1 Kirchhoff stress tensor and work-conjugacy
5.4.2 Work-conjugate pairs
5.4.3 Physical meanings of stress tensors
5.4.3.1 Two-point contravariant pull-back: first Piola–Kirchhoff and Nominal stress tensors
5.4.3.2 Contravariant pull-back: second Piola–Kirchhoff stress tensor
5.4.3.3 Covariant–contravariant pull-back: Mandel stress tensor
5.4.4 Relations of stress tensors
5.4.5 Relations of stress tensors to traction vectors
5.5 Balance laws in reference configuration
5.5.1 Translational equilibrium
5.5.2 Virtual work principle
5.5.3 Conservation law of energy
5.6 Simple shear
6 Hyperelastic equations
6.1 Basic hyperelastic equations
6.2 Hyperelastic constitutive equations of metals
6.2.1 St. Venant–Kirchhoff elasticity
6.2.2 Modified St. Venant–Kirchhoff elasticity
6.2.3 Neo-Hookean elasticity
6.2.4 Modified neo-Hookean elasticity (1)
6.2.5 Modified neo-Hookean elasticity (2)
6.2.6 Modified neo-Hookean elasticity (3)
6.2.7 Modified neo-Hookean elasticity (4)
6.3 Hyperelastic equations of rubbers
6.4 Hyperelastic equations of soils
6.5 Hyperelasticity in infinitesimal strain
7 Development of elastoplastic and viscoplastic constitutive equations
7.1 Basis of elastoplastic constitutive equations
7.1.1 Fundamental requirements for elastoplasticity
7.1.1.1 Decomposition of deformation/rotation (rate) into elastic and plastic parts
7.1.1.2 Incorporation of yield surface
7.1.1.3 Stress rate versus strain rate relation
7.1.2 Requirements for elastoplastic constitutive equation
7.1.2.1 Continuity condition
7.1.2.2 Smoothness condition
7.2 Historical development of elastoplastic constitutive equations
7.2.1 Infinitesimal hyperelastic-based plasticity
7.2.2 Hypoelastic-based plasticity
7.2.3 Multiplicative hyperelastic-based plasticity
7.3 Subloading surface model
7.4 Cyclic plasticity models
7.4.1 Cyclic kinematic hardening models with yield surface
7.4.2 Ad hoc Chaboche model and Ohno-Wang model excluding yield surface
7.4.3 Extended subloading surface model
7.5 Formulation of (extended) subloading surface model
7.5.1 Normal-yield and subloading surfaces
7.5.2 Evolution rule of elastic-core
7.5.3 Plastic strain rate
7.5.4 Strain rate versus stress rate relations
7.5.5 Calculation of normal-yield ratio
7.5.6 Improvement of inverse and reloading responses
7.5.7 Cyclic stagnation of isotropic hardening
7.6 Implicit time-integration: return-mapping
7.6.1 Return-mapping formulation
7.6.2 Loading criterion
7.6.3 Initial value of normal-yield ratio in plastic corrector step
7.6.4 Consistent tangent modulus tensor
7.7 Subloading-overstress model
7.7.1 Constitutive equation
7.7.2 Defects of past overstress model
7.7.3 On irrationality of creep model
7.7.4 Implicit stress integration
7.7.5 Temperature dependence of isotropic hardening function
7.8 Fundamental characteristics of subloading surface model
7.8.1 Distinguished abilities of subloading surface model
7.8.2 Bounding surface model with radial-mapping: Misuse of subloading surface model
8 Multiplicative decomposition of deformation gradient tensor
8.1 Elastic-plastic decomposition of deformation measure
8.1.1 Necessity of multiplicative decomposition of deformation gradient tensor
8.1.2 Isoclinic concept
8.1.3 Uniqueness of multiplicative decomposition
8.1.4 Embedded base vectors in intermediate configuration
8.2 Deformation tensors
8.2.1 Elastic and plastic right Cauchy−Green deformation tensor
8.2.2 Strain rate and spin tensors
8.2.2.1 Strain rate and spin tensors in current configuration
8.2.2.2 Strain rate and spin tensors in intermediate configuration
8.2.2.3 Substructure spin
8.3 On limitation of hypoelastic-based plasticity
8.4 Multiplicative decomposition for kinematic hardening
9 Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations
9.1 Stress measures
9.2 Hyperelastic constitutive equations
9.3 Conventional elastoplastic model
9.3.1 Flow rules for plastic strain rate and plastic spin
9.3.2 Confirmation for uniqueness of multiplicative decomposition
9.3.3 Plastic strain rate
9.4 Continuity and smoothness conditions
9.5 Initial subloading surface model
9.6 Multiplicative extended subloading surface model
9.6.1 Multiplicative decomposition of plastic deformation gradient for elastic-core
9.6.2 Normal-yield, subloading, and elastic-core surfaces
9.6.3 Plastic flow rules
9.6.4 Plastic strain rate
9.7 Material functions of metals and soils
9.7.1 Metals
9.7.1.1 Hyperelastic equation
9.7.1.2 Hyperelastic equation for kinematic hardening variable
9.7.1.3 Hyperelastic equation for elastic-core
9.7.1.4 Yield function
9.7.2 Soils
9.7.2.1 Hyperelastic equation
9.7.2.2 Yield function
9.8 Calculation procedure
9.9 Implicit calculation by return-mapping
9.9.1 Return-mapping
9.9.2 Loading criterion
9.9.3 Initial value of normal-yield ratio in plastic corrector step
9.10 Cyclic stagnation of isotropic hardening
9.11 Multiplicative subloading-overstress model
9.11.1 Constitutive equation
9.11.2 Calculation procedure
9.11.3 Implicit calculation by return-mapping
9.12 On multiplicative hyperelastic-based plastic equation in current configuration
10 Subloading-friction model: finite sliding theory
10.1 History of friction models
10.2 Sliding displacement and contact traction vectors
10.3 Hyperelastic sliding displacement
10.4 Normal-sliding and subloading-sliding surfaces
10.5 Evolution rule of friction coefficient
10.6 Evolution rule of sliding normal-yield ratio
10.7 Plastic sliding velocity
10.8 Calculation procedure
10.9 Return-mapping
10.9.1 Return-mapping formulation
10.9.2 Loading criterion
10.10 Subloading-overstress friction model
10.11 Implicit stress integration for subloading-overstress friction model
10.12 On crucially important applications of subloading-friction model
10.12.1 Loosening of screw
10.12.2 Deterministic prediction of earthquake occurrence
11 Comments on formulations for irreversible mechanical phenomena
11.1 Utilization of subloading surface model
11.1.1 Mechanical phenomena described by subloading surface model
11.1.2 Standard installation to commercial software
11.2 Disuses of rate-independent elastoplastic constitutive equations
11.3 Impertinence of formulation of plastic flow rule based on second law of thermodynamics
Appendix 1 Proofs for formula of scalar triple products with invariants
Appendix 2 Convective stress rate tensors
Appendix 3 Cauchy elastic and hypoelastic equations
A3.1 Cauchy elastic equation
A3.2 Hypoelastic equation
Bibliography
Index

Citation preview

NONLINEAR CONTINUUM MECHANICS FOR FINITE ELASTICITY-PLASTICITY

NONLINEAR CONTINUUM MECHANICS FOR FINITE ELASTICITYPLASTICITY Multiplicative Decomposition With Subloading Surface Model

KOICHI HASHIGUCHI Technical Adviser, MSC Software Ltd. (Emeritus Professor of Kyushu University), Tokyo, Japan

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, urther information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-819428-7 For Information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Dennis McGonagle Editorial Project Manager: Joshua Mearns Production Project Manager: Sojan P. Pazhayattil Cover Designer: Greg Harris Typeset by MPS Limited, Chennai, India

Contents Preface

xi

1. Mathematical fundamentals

1

1.1 Matrix algebra 1.1.1 Summation convention 1.1.2 Kronecker’s delta and alternating symbol 1.1.3 Matrix notation and determinant 1.2 Vector 1.2.1 Definition of vector 1.2.2 Operations of vector 1.3 Definition of tensor 1.4 Tensor operations 1.4.1 Properties of second-order tensor 1.4.2 Tensor components 1.4.3 Transposed tensor 1.4.4 Inverse tensor 1.4.5 Orthogonal tensor 1.4.6 Tensor decompositions 1.4.7 Axial vector 1.4.8 Determinant 1.4.9 Simultaneous equation for vector components 1.5 Representations of tensors 1.5.1 Notations in tensor operations 1.5.2 Operational tensors 1.5.3 Isotropic tensors 1.6 Eigenvalues and eigenvectors 1.6.1 Eigenvalues and eigenvectors of second-order tensor 1.6.2 Spectral representation and elementary tensor functions 1.6.3 Cayley Hamilton theorem 1.6.4 Scalar triple products with invariants 1.6.5 Second-order tensor functions 1.6.6 Positive-definite tensor and polar decomposition 1.6.7 Representation theorem of isotropic tensor-valued tensor function 1.7 Differential formulae 1.7.1 Partial derivatives of tensor functions 1.7.2 Time-derivatives in Lagrangian and Eulerian descriptions 1.7.3 Derivatives of tensor field 1.7.4 Gauss’ divergence theorem 1.7.5 Material-time derivative of volume integration

v

1 1 2 2 6 7 7 15 18 18 19 20 21 22 24 25 27 30 31 31 32 34 35 35 37 38 39 39 40 42 43 43 48 49 51 52

vi

CONTENTS

1.8 Variations of geometrical elements 1.8.1 Deformation gradient and variations of line, surface and volume elements 1.8.2 Velocity gradient and rates of line, surface and volume elements

53 56

2. Curvilinear coordinate system

61

2.1 Primary and reciprocal base vectors 2.2 Metric tensor and base vector algebra 2.3 Tensor representations

61 65 68

3. Tensor operations in convected coordinate system

77

3.1 3.2 3.3 3.4 3.5

77 79 80 83 88 89 92 95

Advantages of description in embedded coordinate system Convected base vectors Deformation gradient tensor Pull-back and push-forward operations Convected time-derivative 3.5.1 General convected derivative 3.5.2 Corotational rate 3.5.3 Objectivity of convected rate

4. Deformation/rotation (rate) tensors 4.1 Deformation tensors 4.2 Strain tensors 4.2.1 Green and Almansi strain tensors 4.2.2 General strain tensors 4.2.3 Logarithmic strain tensor 4.3 Volumetric and isochoric parts of deformation gradient tensor 4.4 Strain rate and spin tensors 4.4.1 Strain rate and spin tensors based on velocity gradient tensor 4.4.2 Strain rate tensor based on general strain tensor

5. Conservation laws and stress tensors 5.1 Conservation laws 5.1.1 Conservation law of physical quantity 5.1.2 Conservation law of mass 5.1.3 Conservation law of linear momentum 5.1.4 Conservation law of angular momentum 5.2 Cauchy stress tensor 5.2.1 Definition of Cauchy stress tensor 5.2.2 Symmetry of Cauchy stress tensor 5.3 Balance laws in current configuration 5.3.1 Translational equilibrium 5.3.2 Rotational equilibrium: symmetry of Cauchy stress tensor 5.3.3 Virtual work principle

53

101 101 106 106 109 113 114 117 117 120

123 123 123 124 125 126 127 127 130 132 133 133 134

CONTENTS

5.3.4 Conservation law of energy 5.4 Work-conjugacy 5.4.1 Kirchhoff stress tensor and work-conjugacy 5.4.2 Work-conjugate pairs 5.4.3 Physical meanings of stress tensors 5.4.4 Relations of stress tensors 5.4.5 Relations of stress tensors to traction vectors 5.5 Balance laws in reference configuration 5.5.1 Translational equilibrium 5.5.2 Virtual work principle 5.5.3 Conservation law of energy 5.6 Simple shear

6. Hyperelastic equations 6.1 Basic hyperelastic equations 6.2 Hyperelastic constitutive equations of metals 6.2.1 St. Venant Kirchhoff elasticity 6.2.2 Modified St. Venant Kirchhoff elasticity 6.2.3 Neo-Hookean elasticity 6.2.4 Modified neo-Hookean elasticity (1) 6.2.5 Modified neo-Hookean elasticity (2) 6.2.6 Modified neo-Hookean elasticity (3) 6.2.7 Modified neo-Hookean elasticity (4) 6.3 Hyperelastic equations of rubbers 6.4 Hyperelastic equations of soils 6.5 Hyperelasticity in infinitesimal strain

7. Development of elastoplastic and viscoplastic constitutive equations 7.1 Basis of elastoplastic constitutive equations 7.1.1 Fundamental requirements for elastoplasticity 7.1.2 Requirements for elastoplastic constitutive equation 7.2 Historical development of elastoplastic constitutive equations 7.2.1 Infinitesimal hyperelastic-based plasticity 7.2.2 Hypoelastic-based plasticity 7.2.3 Multiplicative hyperelastic-based plasticity 7.3 Subloading surface model 7.4 Cyclic plasticity models 7.4.1 Cyclic kinematic hardening models with yield surface 7.4.2 Ad hoc Chaboche model and Ohno-Wang model excluding yield surface 7.4.3 Extended subloading surface model 7.5 Formulation of (extended) subloading surface model 7.5.1 Normal-yield and subloading surfaces 7.5.2 Evolution rule of elastic-core

vii 135 135 136 137 138 141 142 145 145 146 146 147

151 151 155 155 156 157 157 158 158 159 159 160 161

163 163 164 166 168 168 178 181 182 188 188 191 192 195 195 198

viii

CONTENTS

7.5.3 Plastic strain rate 7.5.4 Strain rate versus stress rate relations 7.5.5 Calculation of normal-yield ratio 7.5.6 Improvement of inverse and reloading responses 7.5.7 Cyclic stagnation of isotropic hardening 7.6 Implicit time-integration: return-mapping 7.6.1 Return-mapping formulation 7.6.2 Loading criterion 7.6.3 Initial value of normal-yield ratio in plastic corrector step 7.6.4 Consistent tangent modulus tensor 7.7 Subloading-overstress model 7.7.1 Constitutive equation 7.7.2 Defects of past overstress model 7.7.3 On irrationality of creep model 7.7.4 Implicit stress integration 7.7.5 Temperature dependence of isotropic hardening function 7.8 Fundamental characteristics of subloading surface model 7.8.1 Distinguished abilities of subloading surface model 7.8.2 Bounding surface model with radial-mapping: Misuse of subloading surface model

205 206 207 208 209 213 213 221 224 227 229 230 238 240 243 249 249 250

8. Multiplicative decomposition of deformation gradient tensor

255

8.1 Elastic-plastic decomposition of deformation measure 8.1.1 Necessity of multiplicative decomposition of deformation gradient tensor 8.1.2 Isoclinic concept 8.1.3 Uniqueness of multiplicative decomposition 8.1.4 Embedded base vectors in intermediate configuration 8.2 Deformation tensors 8.2.1 Elastic and plastic right Cauchy-Green deformation tensor 8.2.2 Strain rate and spin tensors 8.3 On limitation of hypoelastic-based plasticity 8.4 Multiplicative decomposition for kinematic hardening

256

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations 9.1 Stress measures 9.2 Hyperelastic constitutive equations 9.3 Conventional elastoplastic model 9.3.1 Flow rules for plastic strain rate and plastic spin 9.3.2 Confirmation for uniqueness of multiplicative decomposition 9.3.3 Plastic strain rate 9.4 Continuity and smoothness conditions 9.5 Initial subloading surface model 9.6 Multiplicative extended subloading surface model 9.6.1 Multiplicative decomposition of plastic deformation gradient for elastic-core

252

256 259 262 263 264 264 265 269 271

273 273 275 277 277 281 281 283 284 286 286

CONTENTS

9.7

9.8 9.9

9.10 9.11

9.12

9.6.2 Normal-yield, subloading and elastic-core surfaces 9.6.3 Plastic flow rules 9.6.4 Plastic strain rate Material functions of metals and soils 9.7.1 Metals 9.7.2 Soils Calculation procedure Implicit calculation by return-mapping 9.9.1 Return-mapping 9.9.2 Loading criterion 9.9.3 Initial value of normal-yield ratio in plastic corrector step Cyclic stagnation of isotropic hardening Multiplicative subloading-overstress model 9.11.1 Constitutive equation 9.11.2 Calculation procedure 9.11.3 Implicit calculation by return-mapping On multiplicative hyperelastic-based plastic equation in current configuration

10. Subloading-friction model: finite sliding theory 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

ix 289 291 294 296 296 300 306 309 309 312 314 317 320 320 326 328 330

335

History of friction models Sliding displacement and contact traction vectors Hyperelastic sliding displacement Normal-sliding and subloading-sliding surfaces Evolution rule of friction coefficient Evolution rule of sliding normal-yield ratio Plastic sliding velocity Calculation procedure Return-mapping 10.9.1 Return-mapping formulation 10.9.2 Loading criterion 10.10 Subloading-overstress friction model 10.11 Implicit stress integration 10.12 On crucially important applications of subloading-friction model 10.12.1 Loosening of screw 10.12.2 Deterministic prediction of earthquake occurrence

335 336 339 340 341 342 344 348 349 349 353 356 362 363 363 364

11. Comments on formulations for irreversible mechanical phenomena

365

11.1 Utilization of subloading surface model 11.1.1 Mechanical phenomena described by subloading surface model 11.1.2 Standard installation to commercial software 11.2 Disuses of rate-independent elastoplastic constitutive equations 11.3 Impertinence of formulation of plastic flow rule based on second law of thermodynamics

365 365 367 368 369

x

CONTENTS

Appendix 1: Proofs for formula of scalar triple products with invariants Appendix 2: Convective stress rate tensors Appendix 3: Cauchy elastic and hypoelastic equations Bibliography Index

371 373 377 379 393

Preface The elastoplasticity theory is now faced to the epoch-making development that the exact description of the finite irreversible (plastic or viscoplastic) deformation/sliding behavior under the monotonic/cyclic loading in the general rate of deformation/sliding from the static to the impact loading is attained as the subloading multiplicative hyperelastic based plasticity and viscoplasticity. This is the first book on this theory, comprehensively describing the underlying concepts and the formulations for the subloading surface model and for the multiplicative decomposition of deformation gradient tensor into the elastic and the plastic (or viscoplastic) parts and their combination. The precise description of the plastic strain rate induced by the rate of stress inside the yield surface is inevitable for the prediction of cyclic loading behavior, which is crucial for the accurate mechanical design of solids and structures in engineering. A lot of works have been executed and various unconventional plastic constitutive (cyclic plasticity) models, named by Drucker (1998), have been proposed aiming at describing the plastic strain rate caused by the rate of stress inside the yield surface after 1960s when the demands of mechanical designs of solids and structures for the mechanical vibration and the seismic vibrations have been highly raised responding to the high development of machine industries and the frequent occurrences of earthquakes, e.g. Chile (1960) and Niigata (Japan) (1964). Among various unconventional models the multi surface model (Mroz, 1967; Iwan, 1967), the two surface model (Dafalias and Popov, 1975; Krieg, 1975; Yoshida and Uemori, 2002), and the superposed-kinematic hardening model (Chaboche et al., 1979; Ohno and Wang, 1993) are well known. However, they assume a surface enclosing a purely elastic domain and are based on the premise that the plastic strain rate develops with the translation of the small yield surface so that they are called the cyclic kinematic hardening model. Therefore they possess various defects, for example, (1) the abrupt transition from the elastic to the plastic state violating the continuity and the smoothness conditions (Hashiguchi, 1993a,b, 1997, 2000), (2) the incorporation of the offset value of the plastic strain at yield, which is accompanied with the unreality and the arbitrariness, (3) the incapability of cyclic loading behavior for the stress amplitude less than the small

xi

xii

Preface

surface enclosing an elastic domain, (4) the incapability of the nonproportional loading behavior, (5) the incapability of extension to the ratedependency at high rate of deformation up to the impact loading behavior, (6) the limitation to the description of deformation behavior in metals, and (7) the necessity of the additional cumbersome operation to pull back the stress to the yield surface or the small surface enclosing an elastic domain. In particular, it is quite pitiful from the scientific point of view that the superposed cyclic plasticity model, i.e. the Chaboche model and the Ohno-Wang model are diffused widely, which are the most primitive ad hoc cyclic plasticity models ignoring the historical development of the plasticity but regressing to the easy going way by the empirical method as will be explained in Section 7.4. Now, it should be noted that the plastic strain rate is not induced abruptly but develops gradually as the stress approaches the yield surface. In fact, the mutual slips of material particles, for example, crystal particles in metals and soil particles in sands and clays leading to the plastic deformation is not induced simultaneously but induced gradually from parts in which mutual slips can be induced easily, exhibiting the smooth transition from the elastic to the plastic transition. The subloading surface model (Hashiguchi, 1978, 1980, 1989, 2017a; Hashiguchi and Ueno, 1997) is free from the existence of the stress region enclosing the purely elastic domain, while the existence has been postulated in the other elastoplasticity models. The subloading surface, which passes through the current stress and is similar to the yield surface, is assumed inside the yield surface, and then it is postulated that the plastic strain rate is not induced suddenly at the moment when the stress reaches the yield surface but it develops as the stress approaches the yield surface, that is, as the subloading surface expands. Therefore the smooth transition from the elastic to the plastic state, that is, the smooth elastic-plastic transition leading to the continuous variation of the tangent stiffness modulus tensor is described in this model. The subloading surface model has been applied to the descriptions of the elastoplastic deformation behaviors of various solids, for example, metals, soils, concrete, etc. Further, it has been extended to describe the viscoplastic deformation by incorporating the concept of the overstress. The subloading surface model would be regarded to be the governing law of the irreversible mechanical phenomena of solids. The subloading surface model has been incorporated into the commercial software “Marc” in MSC Software Corporation as the standard installation by the name “Hashiguchi model,” which can be used by all Marc users (contractors) since October, 2017. Therefore it is explained in the Marc user manual (MSC Software Corporation, 2017) in brief. Further, the function for the automatic determination of material parameters was installed into the Marc as the standard function in June

Preface

xiii

2019. Furthermore, the subloading-friction model will also be incorporated into the Marc as the standard installation until the end of 2020. The mechanisms of the elastic deformation and the plastic deformation in the solids consisting of material particles are physically different from each other such that the former is induced by the deformation of material particles themselves (e.g., crystal particles in metals and soil particles in sands and clays) but the latter is induced by the mutual slips between the material particles. Further, note that all the deformation measures, for example, the infinitesimal and the finite-strain tensors and the strain rate tensor (skew-symmetric part of velocity gradient tensor) are defined by the deformation gradient tensor. Therefore the exact description of finite elastoplastic deformation requires the exact decomposition of the deformation gradient tensor into the elastic and the plastic parts. Furthermore, note that the deformation gradient tensor is defined by the ratio (note: not difference) of the current infinitesimal line element vector to the initial one. Then, the multiplicative decomposition of the deformation gradient tensor has been introduced for the exact description of finite elastoplastic deformation by the leading scholars (Kroner, 1960; Lee and Liu, 1967; Lee, 1969; Mandel, 1971, 1972, 1973a; Kratochvil, 1973). However, it now passed already more than a half century after the proposition of the multiplicative decomposition of deformation gradient tensor. In the meantime, unfortunately the hypoelastic-based plasticity has been studied enthusiastically by numerous workers represented by Rodney Hill and James R. Rice after the proposition of the hypoelasticity by Truesdell (1955), which is not based on the multiplicative decomposition so that it is limited to the infinitesimal elastic deformation and accompanied with the cumbersome timeintegration procedure of the corotational rates of the stress and tensorvalued internal variables. In addition, the concept of the multiplicative decomposition has not been delineated properly even in the notable books referring to this concept (cf. Lubliner, 1990; Simo, 1998; Simo and Hughes, 1998; Lubarda, 2002; Haupt, 2002; Nemat-Nasser, 2004; Asaro and Lubarda, 2006; Bonet and Wood, 2008; de Sauza Neto et al., 2008; Gurtin et al., 2010; Hashiguchi and Yamakawa, 2012; Belytshko et al., 2014, etc.). The multiplicative hyperelastic based plasticity has been studied centrally by Simo and his colleagues (e.g., Simo, 1985, 1988a,b, 1992; Simo and Ortiz, 1985) in the last century, in which the logarithmic strain has been used mainly leading to the coaxiality of stress and strain rate so that it has been limited to the isotropy. It has been developed actively from the beginning of this century by Lion (2000), Menzel and Steinmann (2003a,b), Wallin et al. (2003), Dettmer and Reese (2004), Menzel et al. (2005), Wallin and Ristinmaa (2005), Gurtin and Anand (2005), Sansour et al. (2006, 2007), Vladimirov et al. (2008, 2010),

xiv

Preface

Henann and Anand (2009), Brepols et al. (2014), etc., in which constitutive relations are formulated in the intermediate configuration imagined fictitiously by the unloading to the stress-free state along the hyperelastic relation, based on the isoclinic concept (Mandel, 1971). However, the plastic flow rule with the generality unlimited to the elastic isotropy remains unsolved and only the conventional plasticity model, named by Drucker (1998), with the yield surface enclosing the elastic domain have been incorporated so that only the monotonic loading behavior of elastically isotropic materials is concerned in them. The subloading multiplicative hyperelastic based plastic model has been formulated by the author recently (Hashiguchi, 2018c), which is capable of describing the finite elastoplastic deformation/rotation rigorously under the monotonic/cyclic loading process. Further, it has been extended to the subloading-multiplicative hyperelastic-based viscoplasticity recently, which is capable of describing the rate-dependent elastoplastic deformation behavior at the general rate from the static to the impact loading. It is to be the best opportunity to review the multiplicative hyperelastic based plasticity comprehensively and explain the detailed formulation of the subloading multiplicative hyperelastic based plastic model systematically. This is the first book on the subloading multiplicative hyperelastic based plasticity and viscoplasticity for the description of the general irreversible deformation/sliding behavior. The subloading surface model and the multiplicative hyperelastic based plasticity are explained comprehensively providing the detailed physical interpretations for all relevant concepts and the deriving processes of all equations. Further, the incorporation of the subloading surface model to the multiplicative hyperelastic plastic relation is described in detail. Further, it is extended to the description of the viscoplastic deformation by incorporating the concept of overstress, which is capable of describing the general rate of deformation ranging from the quasistatic to the impact loading behaviors (Hashiguchi, 2016a, 2017a). In addition, the exact hyperelastic based plastic and viscoplastic constitutive equation of friction (Hashiguchi, 2018c) is formulated rigorously, while the hypoelastic-based plastic constitutive equation of friction has been formulated formerly (Hashiguchi et al., 2005; Hashiguchi and Ozaki, 2008; Hashiguchi, 2013a). The aim of this book is to give a comprehensive explanation of the finite elastoplasticity theory and viscoplasticity under the monotonic and the cyclic loading processes. The incorporation of the Lagrangian tensors is required originally in the formulation of finite elastoplasticity and viscoplasticity, since the deformation of the material involved in the reference configuration, which is invariant through the deformation, is physically relevant. Therefore the necessity and the meanings of the

Preface

xv

Lagrangian tensors and the transformations rules between the Eulerian and the Lagrangian tensors, that is, the pull-back and push-forward operations are explained concisely. Various Lagrangian stress tensors are derived based on the requirement of the work-conjugacy from the Cauchy stress tensor in the current configuration. To this end, the descriptions of physical quantities and relations in the embedded (convected) coordinate system, which turns into the curvilinear coordinate system under the deformation of material, are required, since their physical meanings can be captured clearly by observing them in the coordinate system that not only moves but also deforms and rotates with material itself. In other words, the essentials of continuum mechanics cannot be captured without the incorporation of the general curvilinear coordinate system, to which the embedded coordinate system changes, although the explanation only in the rectangular coordinate system is given in a lot of books entitled “continuum mechanics.” The author expects that the readers of this book will capture the fundamentals in the finite-strain elastoplasticity theory and they will contribute to the development of mechanical designs of machinery and structures in the field of engineering practice by applying the theories addressed in this book. A reader is apt to give up reading through a book if one encounters a matter that is uneasy to understand by insufficient explanation. For this reason, the detailed explanations of physical concepts in elastoplasticity are delineated, and the derivations/transformation processes of all equations are given with detailed proofs but without abbreviation. The author wishes to express cordial thanks to his colleagues at Kyushu University, who have discussed and collaborated over several decades: Prof. M. Ueno (currently Emeritus Professor at University of the Ryukyus) in particular, and Dr. T. Okayasu (currently Associate Professor at Kyushu University), Dr. S. Tsutsumi (currently Associate Professor at Osaka University), Dr. T. Ozaki of Kyushu Electric Eng. Consult. Inc., Dr. S. Ozaki (currently Associate Professor at Yokohama National University), and Dr. T. Mase of Tokyo Electric Power Services Co., Ltd. (currently Professor of Tezukayama Gakuin Univ.) Furthermore, the author is thankful to Dr. K. Okamura, Dr. N. Suzuki, and Dr. R. Higuchi, Nippon Steel & Sumitomo Metal Corporation, Dr. M. Oka and Mr. T. Anjiki, Yanmar Co. Ltd., for the collaborations on constitutive relations of metals and the numerical calculations. In particular, the numerical calculations performed by Mr. T. Anjiki was quite effective for the improvement of the subloadingoverstress model. The author is also grateful to Mr. T. Kato (President) and Dr. M. Tateishi (Fellow), MSC Software, Ltd., Japan for the standard implementation of the Hashiguchi (subloading surface) model to the commercial FEM (Finite Element Method) software Marc.

xvi

Preface

The heartfelt gratitude of the author is dedicated to Prof. Yuki Yamakawa of Tohoku University, for various advices and close collaborations with detailed discussions on elastoplasticity theory, particularly on the finite-strain theory and the numerical method. In addition, the author acknowledges the great gratitude to Prof. Yamakawa for critical reading of the original manuscript and then suggesting various precious elaborations. The author expresses his sincere gratitude to Prof. Genki Yagawa, Emeritus Professor, The University of Tokyo, for encouraging always the author with undeserved high appreciation of research contributions, and thus the author was stimulated to the publication of this book. The author is convinced that this book will contribute substantially to the steady developments of solid mechanics and the manufacturing and constructing industries through the readers. Finally, the author would like to acknowledge the enthusiastic supports by the editor Mr. Dennis Mcgonagle, the editorial project manager Mr. Joshua Mearns, and the project manager: Mr. Sojan P. Pazhayattil, Elsevier, for the generous corporations on the publication of this book. Koichi Hashiguchi June 2020

C H A P T E R

1 Mathematical fundamentals The mathematical fundamentals are addressed in this chapter, which are required to understand sufficiently the elastoplasticity theory described in the subsequent chapters. First, the basics of vector and tensor algebra are explained and then the differential formula and the variations of the geometrical elements are described comprehensively. Readers are tempted to skip to study these mathematical fundamentals but they are explained concisely by showing the derivation processes for almost all equations. Component descriptions of vectors and tensors in this chapter are limited in the normalized rectangular coordinate system, that is, rectangular coordinate system with unit base vectors, while the terms orthogonal, orthonormal, and Cartesian are often used instead of rectangular. However, these tensor relations hold even in the general curvilinear coordinate system of the Euclidian space described in the subsequent chapters.

1.1 Matrix algebra The basic matrix algebra with some conventions and symbols appearing in the continuum mechanics are described in this section.

1.1.1 Summation convention The Cartesian summation convention is first introduced in which repeated suffix in a term is summed over numbers that the suffix can take, for example,

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity DOI: https://doi.org/10.1016/B978-0-12-819428-7.00001-8

1

© 2020 Elsevier Inc. All rights reserved.

2

1. Mathematical fundamentals

8 3 X > > > u v 5 ur vr 5 u1 v1 1 u2 v2 1 u3 v3 ; > r r > > > r51 > > < 3 X Trr 5 T11 1 T22 1 T33 Trr 5 > > r51 > > > 3 X > > > > Tir vr 5 Tir vr 5 Ti1 v1 1 Ti2 v2 1 Ti3 v3 ; :

(1.1)

r51

A letter of the repeated suffix is arbitrary and thus it is called the dummy index as known from ur vr 5 us vs ;

T rr 5 Tss ; Tir vr 5 Tis vs :

(1.2)

This rule is also called Einstein’s summation convention. A repeated index obeys this convention unless otherwise specified by the additional remark “(no sum)” after an equation.

1.1.2 Kronecker’s delta and alternating symbol The Kronecker’s delta δij ði; j 5 1; 2; 3Þ is defined as follows:  1: i 5 j δij 5 0 : i 6¼ j

(1.3)

fulfilling δir δrj 5 δij 5 δji ;

δii 5 3

(1.4)

Further, the alternating (or permutation) symbol or Eddington’s epsilon εijk is defined as follows: 8 > < 1: even permutation of ijk from 123 ð123; 231; 312Þ εijk 5 2 1: odd permutation of ijk from 123 ð213; 321; 132Þ (1.5) > : 0: others fulfilling the following relation for the product. εijk εijk 5 3!

(1.6)

1.1.3 Matrix notation and determinant Let the quantity T possessing nine (3 3 3) components Tij be expressed in the arrangement

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

3

1.1 Matrix algebra

2

T11 T 5 ½Tij  5 4 T21 T31

T12 T22 T32

3 T13 T23 5 T33

(1.7)

which is called a matrix notation. The matrix I possessing the components of the Kronecker’s delta is given by 2 3 1 0 0 I 5 ½δij  5 4 0 1 0 5 (1.8) 0 0 1 The quantity v possessing three (3 3 1) components is expressed as 2 3 v1   v 5 ½vi  5 4 v2 5 5 v1 v2 v2 (1.9) v3 The multiplications of the quantity v and the matrix B by the matrix A are denoted as Av and AB and defined as ðAvÞi 5 Air vr 5 vr Air 5 ðvAT Þi

(1.10)

ðABÞij 5 Air Brj 6¼ Bir Arj 5 ðBAÞij

(1.11)

where ð ÞT stands for the transpose of the row and the column in the matrix. The quantity defined by the following equation is called the determinant of T and is shown by the symbol det T, that is,    T11 T12 T13    (1.12) detT 5 εijk T1i T2j T3k 5 εijk Ti1 Tj2 Tk3 5  T21 T22 T23   T31 T32 T33  with detTT 5 detT;

detðsTÞ 5 s3 detðTÞ

(1.13)

Here, the number of permutations that the suffixes i, j, and k in εijk can take is 3!. Therefore Eq. (1.12) can be written as detT 5

1 εijk εpqr Tip Tjq Tkr 3!

(1.14)

Eq. (1.14) is rewritten as detT 5

1 Trs ðcof TÞrs ; 3

detT 5

1 1 T: ðcof TÞ 5 trðTðcofTÞT Þ 3 3

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.15)

4

1. Mathematical fundamentals

or detT 5 T1s ðcofTÞ1s 5 T2s ðcofTÞ2s 5 T3s ðcofTÞ3s 5 Tr1 ðcofTÞr1 5 Tr1 ðcofTÞr1 5 Tr2 ðcofTÞr2 5 Tr3 ðcofTÞr3

(1.16)

where ðcofTÞip  noting

1 εijk εpqr Tjq Tkr 2!

(1.17)

  1 1 1 1 εijk εpqr Tip Tjq Tkr 5 Tip εijk εpqr Tjq Tkr 5 Tip ðcofTÞip 3! 3 2! 3

ðcofTÞij is called the cofactor for the i-column and the j-row. The cofactor is obtained through multiplying the minor determinant lacking the ith row and jth column components by the sign ð21Þi1j . The following lemmas for the properties of the determinant hold. Lemma 1.1: If the first and the second rows are same, that is, T2j 5 T1j for instance, we have εijk T1i T1j T3k 5 εjik T1j T1i T3k 5 2 εijk T1i T1j T3k . Therefore we have the lemma “the determinant having same lines or rows is zero.” Therefore the following relation is obtained from Eq. (1.16) that Tis Δjs 5 Tri Δrj 5 δij detT

(1.18)

Lemma 1.2: If the first and the second lines are exchanged, that is, 122 for instance, we have εijk T2i T1j T3k 5 εjik T1i T2j T3k 5 2 εijk T1i T2j T3k . Therefore we have the lemma “the determinant changes only its sign by exchanging lines (or rows).” By multiplying εijk to both sides in Eq. (1.12), we have εijk detT 5 εijk εpqr T1p T2q T3r 5 εpqr Tip Tjq Tkr

(1.19)

The transformation from the second side to the third side in Eq. (1.19) is resulted from the abovementioned Lemmas 1.1 and 1.2. Here, note that the expression of the determinant in Eq. (1.14) is derived also by multiplying εijk to both sides in Eq. (1.19) and noting Eq. (1.6). The additive decomposition of the components T2j into T2j 5 A2j 1 B2j leads to εijk T1i ðA2j 1 B2j ÞT2k 5 εijk T1i A2j T2k 1 εijk T1i B2j T2k

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.20)

1.1 Matrix algebra

5

Therefore the value of determinant in which components in a line (or row) are decomposed additively is the sum of the two determinants made by exchanging the line (or row) of the original determinants into the decomposed components. The determinant of the multiplication of tensors is given from Eqs. (1.12) and (1.19) as follows: detðABÞ 5 εijk ðA1p Bpi ÞðA2q Bqj ÞðA3r Brk Þ 5 A1p A2q A3r εijk Bpi Bqj Brk 5 A1p A2q A3r εpqr detB noting εijk Bpi Bqj Brk 5 εpqr detB due to Eq. (1.19), and thus one has the following product law of determinant. detðABÞ 5 detAdetB

(1.21)

The partial derivative of determinant is given from Eq. (1.14) as @detT 5 @Tij

@

1 εabc εpqr Tap Tbq Tcr 3! @Tij

5

1 εabc εpqr ðδia δjp Tbq Tcr 1 Tap δib δjq Tcr 1 Tap Tbq δic δjr Þ 3!

5

1 εibc εjqr Tbq Tcr 2!

which leads to @detT 5 cofT ; @T

@detT 5 ðcofTÞij @Tij

(1.22)

The permutation symbol in the third order, that is, εijk appears often hereinafter. It is related to Kronecker’s delta by the determinants as      δ1i δ1j δ1k   δ1i δ2i δ3i      (1.23) εijk 5  δ2i δ2j δ2k  5  δ1j δ2j δ3j   δ3i δ3j δ3k   δ1k δ2k δ3k  which can be proved is expanded as   δ1i   εijk 5  δ2i   δ3i

as follows: Note that the second side in Eq. (1.23) δ1j δ2j δ3j

 δ1k   δ2k  5 δ1i δ2j δ3k 1 δ1k δ2i δ3j 1 δ1j δ2k δ3i  δ3k  2 δ1k δ2j δ3i 2 δ1i δ2k δ3j 2 δ1j δ2i δ3k

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

6

1. Mathematical fundamentals

Then, one has ε123 5 δ11 δ22 δ33 1δ13 δ21 δ32 1δ12 δ23 δ31 2δ13 δ22 δ31 2δ11 δ23 δ32 2 δ12 δ21 δ33 5 1 ε213 5 δ12 δ21 δ33 1δ13 δ22 δ31 1δ11 δ23 δ32 2δ13 δ21 δ32 2δ12 δ23 δ31 2 δ11 δ22 δ33 52 1 for instance. The third side in Eq. (1.23) can be confirmed as well. The following relations are obtained from Eqs. (1.23) and (1.21).   2  3 2 3  δ1i δ2i δ3i  δ1p δ1q δ1r   δ1i δ2i δ3i δ1p δ1q δ1r        6  7 6 7 εijk εpqr 5  δ1j δ2j δ3j  δ2p δ2q δ2r  5  4 δ1j δ2j δ3j 5 4 δ2p δ2q δ2r 5        δ1k δ2k δ3k  δ3p δ3q δ3r   δ1k δ2k δ3k δ3p δ3q δ3r    (1.24)  δip δiq δir      5  δjp δjq δjr     δkp δkq δkr  from which we have   δip δiq   εijk εpqk 5  δjp δjq   δkp δkq

      δkk  δik δjk

5 δip δjq δkk 1 δiq δjk δkp 1 δik δjp δkq 2 δik δjq δkp 2 δip δjk δkq 2 δiq δjp δkk 5 δip δjq 2 δiq δjp    δii δij δiq      εijp εijq 5  δji δjj δjq     δpi δpj δpq  5 δii δjj δpq 1 δij δjq δpi 1 δiq δji δpj 2 δii δjq δpj 2 δij δji δpq 2 δiq δjj δpi 5 9δpq 1 δiq δpi 1 δiq δip 2 3δpq 2 3δpq 2 3δpq 5 2δpq εijk εijk 5 2δkk 5 6 Eventually, one obtains the following relations. εijk εpqk 5 εkij εkpq 5 δip δjq 2 δiq δjp εijp εijq 5 2δpq ; εijk εijk 5 6

(1.25)

The last equation is no more than Eq. (1.6).

1.2 Vector The definitions and the operations of the vector will be delineated in this section, which are required for the study of the continuum mechanics. Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7

1.2 Vector

1.2.1 Definition of vector The quantity having only magnitude is defined as a scalar. On the other hand, a quantity having direction and sense in addition to magnitude is defined as a vector. The following commutative, distributive, and the associative laws hold for a vector.  a 1 b 5 b 1 a; ða 1 bÞ 1 c 5 a 1 ðb 1 cÞ (1.26) aðbvÞ 5 ðabÞv 5 bðavÞ; ða 1 bÞv 5 ðb 1 aÞv; aða 1 bÞ 5 aa 1 ab where v designates a vector and a and b are arbitrary scalars. The magnitude of vector is denoted by jjvjj. In particular, the vector whose magnitude is zero is called the zero vector and is shown as 0. The vector whose magnitude is unity, that is, jjvjj 5 1, is called the unit vector.

1.2.2 Operations of vector Basic operations of vectors are delineated in this section, which are required for the formulations of ingredients in the continuum mechanics. 1.2.2.1 Scalar product Denoting the angle between the two vectors a; b by θ, the scalar or inner product is defined as :a::b:cosθ where : : designates the magnitude (length or norm) and it is denoted by the symbol a b, that is,





a b  :a::b:cosθ

(1.27)

Consider the normalized orthogonal coordinate system, where the word “normalized” means to adjust the magnitude to unity and the word “orthogonal” means to adopt the mutually orthogonal base vectors. Let it be denoted as fO 2 xi g, while the unit vectors are denoted by the triad fei g. The scalar product between the base vectors is given from Eqs. (1.3) and (1.27) as follows:



ei ej 5 δij

(1.28)

Vector v is described in the linear associative form as follows: v 5 vr er ð 5 v1 e1 1 v2 e2 1 v 3 e3 Þ

(1.29)

where v1, v2, and v3 are the components of v. Denoting the angle of the direction of vector v from the direction of the base vector ei by θi , cosθi 5 n ei is called the direction cosine by which the component of v is given as



Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

8

1. Mathematical fundamentals





vi 5 v ei 5 :v:n ei 5 :v:cosθi

(1.30)

The magnitude of vector v and its unit direction vector n are given from Eq. (1.30), noting cos2 θ1 1 cos2 θ2 1 cos2 θ3 5 1 as follows: :v: 



pffiffiffiffiffiffiffiffi vr vr ;

n

v vr 5 er :v: :v:

(1.31)



Because of a b 5 ar er bs es 5 ar bs δrs , the scalar product is expressed by using the components as



a b 5 ar br

(1.32)

The magnitude of vector is also expressed by setting θ 5 0 in Eq. (1.27) as follows: pffiffiffiffiffiffiffiffiffiffi :v: 5 v v (1.33)



The quantity obtained by the scalar product is a scalar and the following commutative, distributive, and associative laws hold.







a b5b a





(1.34)



a ðb 1 cÞ 5 a b 1 a c

   ðaa 1 bbÞ  c 5 aa  c 1 bb  c

sða bÞ 5 ðsa bÞ 5 a ðsbÞ 5 ða bÞs

(1.35) (1.36) (1.37)

for arbitrary scalars s, a, and b. 1.2.2.2 Vector product The operation obtaining a vector having (1) magnitude identical to the area of the parallelogram formed by the two vectors a and b, provided that they are translated to the common initial point, and (2) direction of the unit vector n which forms the right-hand bases a; b; n in this order is defined as the vector (or cross) product and is denoted by the symbol a 3 b as shown in Fig. 1.1. Therefore denoting the angle between the two vectors a and b by θ when they are translated to the common initial point, it holds that a 3 b  :a::b:sinθn

ð:n: 5 1Þ

(1.38)

Incidentally, the surface vector is defined by the vector product. Then, it follows for the normalized orthonormal base vectors Figure 1.1 that ei 3 ej 5 εijk ek

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.39)

9

1.2 Vector

a × b = an

v = (a × b) · c =[abc]

c n (|| n|| = 1)

b θ

a = || a × b|| =|| a || ||b|| sin θ

a

FIGURE 1.1 Vector product (surface vector) and scalar triple product (volume) with area.

and thus one has a 3 b 5 ai ei 3 bj ej 5 εijk ai bj ek 5 ða2 b3 2 a3 b2 Þe1 1 ða3 b1 2 a1 b3 Þe2 1 ða1 b2 2 a2 b1 Þe3 which is expressed in the matrix form   e1  a 3 b 5  a1  b1

as follows:  e2 e3  a2 a3  b2 b3 

(1.40)

(1.41)

The following equations hold for the vector product. a3a50

(1.42)

a3b52b3a

(1.43)

a 3 ðb 1 cÞ 5 a 3 b 1 a 3 c

(1.44)

sða 3 bÞ 5 ðsa 3 bÞ 5 a 3 ðsbÞ 5 ða 3 bÞs

(1.45)

ðaa 1 bbÞ 3 c 5 aa 3 c 1 bb 3 c

(1.46)

1.2.2.3 Scalar and vector triple products The operation defined by the following equation for the three vectors is called scalar triple product.



½abc  ða 3 bÞ c

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.47)

10

1. Mathematical fundamentals

which is rewritten as follows:





½abc 5 ða 3 bÞ c 5 εijr ai bj er ck ek 5 εijr ai bj ck δrk 5 εijk ai bj ck      a1 b1 c1   a1 a2 a3          5  a2 b2 c2  5  b1 b2 b3       a3 b3 c3   c1 c2 c3 

(1.48)

fulfilling ½abc 5 ½bca 5 ½cab 5 2 ½bac 5 2 ½cba 5 2 ½acb

(1.49)

Denoting the vectors a; b; c as v1 ; v2 ; v3 , it follows from Eq. (1.47) that     vi vj vk 5 εijk v1 v2 v3 (1.50) noting the fact that the term in the right-hand side of this equation is 1 ½v1 ; v2 ; v3 , 2½v1 ; v2 ; v3 , and 0 when indices i; j; k are even and odd permutations and two of indices coincide with each other, respectively. Here, note that the scalar triple product [v1 v2 v3] designates the volume of the parallelopiped formed by the vectors v1, v2, v3 in this order. Here, the following equations hold for the scalar triple product. ½ei ej ek  5 εijk

(1.51)

½sa; b; c 5 ½a; sb; c 5 ½a; b; sc 5 s½abc

(1.52)

½aa 1 bb; c; x 5 a½ucx 1 b½bcx

(1.53)

The vector triple product is defined as follows: a 3 ðb 3 cÞ 5 εijk εkpq aj bp cq ei

(1.54)

noting a 3 ðb 3 cÞ 5 aj ej 3 ðεkpq bp cq ek Þ 5 aj εkpq bp cq ej 3 ek 5 εkpq aj bp cq εkij ei It follows from Eqs. (1.25) and (1.54) that ½a 3 ðb 3 cÞi 5 εkij εkpq aj bp cq 5 ðδip δjq 2 δiq δjp Þaj bp cq 5 bi ðaj cj Þ 2 ci ðaj bj Þ Then, the following relations hold for the vector triple product. a 3 ðb 3 cÞ ða 3 bÞ 3 c

 

 

5 ða cÞb 2 ða bÞc 5 ða cÞb 2 ðb cÞa

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.55)

11

1.2 Vector

Permuting the letters and taking the summation in Eq. (1.55), it follows that a 3 ðb 3 cÞ 1 b 3 ðc 3 aÞ 1 c 3 ða 3 bÞ 5 0

(1.56)

noting a 3 ðb 3 cÞ 1 b 3 ðc 3 aÞ 1 c 3 ða 3 bÞ

















5 ½ða cÞb 2 ða bÞc 1 ½ða bÞc 2 ða cÞb 1 ½ðc bÞa 2 ðc aÞb 5 0 One has







ða 3 bÞ ðc 3 dÞ 5 ða cÞðb dÞ 2 ðb cÞða dÞ

(1.57)

noting











ða 3 bÞ ðc 3 dÞ 5 ½ða 3 bÞ c d 5 fða 3 bÞ 3 cg d 5 fða cÞb 2 ðb cÞag d by virtue of Eq. (1.55),





Setting t 5 a 3 b in t 3 ðc 3 xÞ 5 ðt dÞc 2 ðt cÞd due to Eq. (1.55)1, one has





ða 3 bÞ 3 ðc 3 dÞ 5 ½ða 3 bÞ dc 2 ½ða 3 bÞ cd 5 ½abdc 2 ½abcd

(1.58)

It follows from Eq. (1.58) setting c-b and d-c for the particular case that ða 3 bÞ 3 ðb 3 cÞ 5 ½abcb 2 ½abbc 5 ½abcb

(1.59)

which leads to





ða 3 bÞ 3 ðb 3 cÞ ðc 3 aÞ 5 ½abcb ðc 3 aÞ so that ½a 3 b

b 3 c c 3 a 5 ½abc2

(1.60)

Because of





ða 3 bÞ ½ðc 3 xÞ 3 ðy 3 zÞ 5 ða 3 bÞ ð½cxzy 2 ½cxyzÞ 5 ½aby½cxz 2 ½abz½cxy due to Eq. (1.58), it follows that ½a 3 b; c 3 x; y 3 z 5 ½aby½zcx 2 ½abz½ycx

(1.61)

1.2.2.4 Primary and reciprocal vectors Arbitrary vector v is expressed by the linear combination of the independent vectors a; b; c as follows: Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

12

1. Mathematical fundamentals

v 5 va a 1 vb b 1 vc c

(1.62)

where the coefficients va ; vb ; vc are given by operating the scalar products of a 3 b, b 3 c, and c 3 a to Eq. (1.62) as follows: ½abv 5 vc ½abc;

½bcv 5 va ½abc;

½cav 5 vb ½abc

that is, va 5

½bcv ; ½abc

vb 5

½cav ; ½abc

vc 5

½abv ½abc

(1.63)

The vector v is rewritten by substituting Eq. (1.63) into Eq. (1.62) as follows: v5

½bcv ½cav ½abv a1 b1 c ½abc ½abc ½abc

(1.64)

Then, the components va ; vb ; vc in Eq. (1.63) are rewritten by va 5

b3c v 5 a v; ½abc





vb 5

c3a v 5 b v; ½abc





c3a ; ½abc

c 

vc 5

a3b v 5 c v ½abc (1.65)





where a  a

b3c ; ½abc

b 

b 3 c  ; ½a b c 

b

c  3 a ; ½a b c 

a3b ½abc

c

a  3 b ½a b c 

(1.66)

ða ; b ; c Þ are called the reciprocal vectors of the primary vectors ða; b; cÞ. The lower part of Eq. (1.66) is verified as b 3 c  5 ½a b c  5

ðc 3 aÞ 3 ða 3 bÞ ½abca a 5 5 2 2 ½abc ½abc ½abc



ðb 3cÞ 3ðc3 aÞ ða 3bÞ ½bca½bca2½bcb½aca ½abc2 1 5 5 5 3 3 3 ½abc ½abc ½abc ½abc

by using Eqs. (1.59) and (1.60), leading to the first equation for a, for example. Here, the following relations hold.

 



 

a a 5 1; b b 5 1; c c 5 1 a b  5 a c  5 b c  5 b a 5 c a 5 c b  5 0



  







½abc½a b c  5 1

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.67)

13

1.2 Vector

Further, it follows from Eqs. (1.56) and (1.66) that a 3 a 1 b 3 b  1 c 3 c  5 0

(1.68)

Next, decompose the vector v into the surface vectors b 3 c, c 3 a, and a 3 b, that is, v 5 v a ðb 3 cÞ 1 v b ðc 3 aÞ 1 v c ða 3 bÞ

(1.69)

where the coefficients v a ; v b ; v c are given by operating the scalar products of a; b; c as follows:



a v 5 v c ½abc; leading to v a 5





b v 5 v a ½abc;



a v ; ½abc

v b 5



b v ; ½abc

c v 5 v c ½abc

v c 5



c v ½abc

(1.70)

The substitution of Eq. (1.70) into Eq. (1.69) reads: v5







a v b v c v ðb 3 cÞ 1 ðc 3 aÞ 1 ða 3 bÞ ½abc ½abc ½abc

(1.71)

An arbitrary vector v is expressed from Eqs. (1.64) and (1.71) with Eq. (1.67) as follows:













v 5 ða vÞa 1 ðb vÞb 1 ðc vÞc 5 ða vÞa 1 ðb vÞb 1 ðc vÞc (1.72) where the third side is obtained from the second side by exchanging a; b; c and ða ; b ; c Þ. Eq. (1.72) is rewritten as v 5 gv

(1.73)

where g 5 aa 1 bb 1 cc 5 a a 1 b b 1 c c ð5 gT Þ

(1.74)

g is regarded as the generalized identity tensor and will be called the metric tensor in the general coordinate system in Section 2.2. In particular, g is reduced in the normalized rectangular coordinate system as follows: I 5 δij ei ej 5 ei ei ;



δij 5 ei ej

(1.75)

which is called the identity tensor transforming the vector to itself. Besides, the permutation tensor is defined by ε 5 εijk ei ej ek ;

εijk 5 ½ei ej ek 

noting Eq. (1.51). The following relation holds.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.76)

14

1. Mathematical fundamentals

ε:ðuvÞ 5 u 3 v

(1.77)

noting ðεijk ei ej ek Þ:ður er vs es Þ 5 εijk ei ur vs δjr δks 5 εijk ei uj vk 5 uj ej 3 vk ek

1.2.2.5 Tensor product Based on the vectors vð1Þ ; vð2Þ ; ?; vðmÞ , one can make the mth order tensor as follows: ð2Þ ðmÞ vð1Þ vð2Þ . . .vðmÞ 5 vð1Þ p1 vp2 . . .vpm ep1 ep2 . . .epm

(1.78)

For two vectors, one has the second-order tensor ab 5 ai ei bj ej 5 ai bj ei ej which is expressed in the matrix form 2 a1 b1 a1 b2 ½ab 5 4 a2 b1 a2 b2 a3 b1 a3 b2

(1.79)

3 a1 b3 a2 b3 5 a3 b3

(1.80)

As described above, one can make a tensor from vectors as shown for the second-order tensor by ab, which is called the tensor (cross) product or dyad, meaning “one set by two.” Particularly, it holds for three arbitrary vectors



abc 5 aðb cÞ

(1.81)

because of ðai ei bj ej Þck ek 5 ai ei ðbj ck δjk Þ 5 ai ei ðbj cj Þ Therefore abc is the vector possessing the direction of a, while b is projected to the direction of c resulting in a scalar. Here, we find



trðabÞ 5 b a

(1.82)

ðabÞT 5 ba

(1.83)

aðb 1 cÞ 5 ab 1 ac

(1.84)





ðbc 2 cbÞa 5 a 3 ðb 3 cÞ 5 ða cÞb 2 ða bÞc

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.85)

15

1.3 Definition of tensor

8 xðabÞy 5 ða xÞðb yÞ > > > > > ðabÞ:ðcdÞ 5 ðacÞðbdÞ > > > < ðxyÞ:ðabcdÞ 5 ða xÞðb yÞcd > ðxyÞ:ðabcÞz 5 ða xÞðb yÞðc zÞ > > > > > ðabcdÞ:ðxyÞ 5 ðc xÞðd yÞab > > : ðwxÞ:ðabcdÞ:ðyzÞ 5 ða wÞðb xÞðc yÞðd zÞ

























(1.86)



The vector product is represented in the direct notation of the alternating tensor ε as follows: a 3 b 5 ε:ðabÞ;

ei 3 ej 5 ε:ðei ej Þ

(1.87)

because of





a 3 b 5 εijk aj bk ei 5 εijk ei ar ðej er Þbs ðek es Þ 5 εijk ei ej ek :ðar er bs es Þ (1.88) noting Eq. (1.40).

1.3 Definition of tensor Let the set of nm functions in the coordinate system fO-xi g with the origin O and the axes xi ði 5 1; 2; . . .; nÞ in the n-dimensional space be designated by Tðp1 ; p2 ; . . .; pm Þ, where each of the indices p1 ; p2 ; . . .; pm takes the number 1; 2; . . .; n. This set of functions is defined as the mth-order tensor in the n-dimension, if the set of functions is observed in the other coordinate system fO-xi g with the origin O and the axes xi as follows: T ðp1 ; p2 ; . . .pm Þ 5

@xp1 @xp2 @xq1 @xq2

?

@xpm @xqm

Tðq1 ; q2 ; . . .qm Þ

(1.89)

provided that only the directions of axes are different and the relative motion between the axes does not exist. Here, we introduce the notation Qij 5

@xi @xj

(1.90)

which fulfills Qir Qjr 5 δij

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.91)

16

1. Mathematical fundamentals

because of Qir Qjr 5

@xi @xr @xr @xj

Eq. (1.89) is written using Qij as follows: T  ðp1 ; p2 ; . . .pm Þ 5 Qp1 q1 Qp2 q2 . . .Qpm qm Tðq1 ; q2 ; . . .qm Þ

(1.92)

Designating Tðp1 ; p2 ; . . .; pm Þ by the symbol Tp1 p2 ?pm , Eq. (1.89) is expressed as Tp1 p2 ...pm 5 Qp1 q1 Qp2 q2 . . .Qpm qm T q1 q2 ... qm

(1.93)

Because of Qp1 r1 Qp2 r2 ?Qpm rm T p1 p2 ?pm 5 Qp1 r1 Qp2 r2 ?Qpm rm Qp1 q1 Qp2 q2 ?Qpm qm T q1 q2 ?qm 5 ðQp1 r1 Qp1 q1 ÞðQp2 r2 Qp2 q2 Þ?ðQpm rm Qpm qm ÞTq1 q2 ?qm 5 δr1 q1 δr2 q2 ?δrm qm Tq1 q2 ?qm with Eq. (1.91), the inverse relation of Eq. (1.93) is given by Tr1 r2 ?rm 5 Qp1 r1 Qp2 r2 ?Qpm rm Tp1 p2 ?pm

(1.94)

Indices put in a tensor take the dimension of the space in which the tensor is based. The number of indices which is equal to the number of operators Qij is called the order of tensor. For instance, the transformation rules of the first-order tensor, that is, vector vi and the second-order tensor Tij are given by   vi 5 Qir vr ; vi 5 Qri vr (1.95) Tij 5 Qir Qjs Trs ð 5 Qir Trs Qjs Þ; T ij 5 Qri Qsj T rs ð 5 Qri Trs Qsj Þ Consequently, in order to prove that a certain quantity is a tensor, one needs only to show that it obeys the tensor transformation rule (1.93). The coordinate transformation rule in the form of Eq. (1.93) or (1.94) is called the objective transformation. Tensor obeying the objective transformation rule even between the coordinate systems with the relative rate of motion, that is, the relative parallel and rotational velocities, is called an objective tensor. Vectors and tensors without the time dimension, for example, force, displacement, rotational angle, stress, and strain are objective vector and tensors. On the other hand, time-rate quantities, for example, rate of force, velocity, spin, and the materialtime derivatives of physical quantities, for example, stress are not

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

1.3 Definition of tensor

17

objective vectors and tensors usually; they are influenced by the relative rate of motion between the coordinate systems, that is, the rigid-body rotational rate of material. Constitutive equations of materials must be formulated in terms of objective tensors, since material properties are not influenced by the rigid-body rotation of material and therefore must be described in the identical form independent of the coordinate systems. The following linearity is satisfied in tensors.

8 > < Additivity: Tp1 p2 ?pm Ap1 p2 ?pl 1 Bp1 p2 ?pl 5 Tp1 p2 ?pm Ap1 p2 ?pl 1 T p1 p2 ?pm Bp1 p2 ?pl (1.96) >

: Homogeneity: Tp1 p2 ?pm sAp1 p2 ?pl 5 sT p1 p2 ?pm Ap1 p2 ?pl where s is an arbitrary scalar variable. Therefore the tensor transforms linearly a tensor to the other tensor and thus it is called also the linear transformation. The operation to lower the order of tensor by multiplication of tensors is called the contraction. Introducing the notation

( Q1TU p p ...p  Qp1 q1 Qp2 q2 . . .Qpm qm T q1 q2 ...qm T

1 2 m (1.97) Q 1TU p p ...p  Qq1 p1 Qq2 p2 . . .Qqm pm Tq1 q2 ...qm 1 2

m

for the component transformation of the general tensor, let Eqs. (1.93) and (1.94) be represented formally by the symbolic (direct) notation for convenience as follows: T 5 Q1TU;

T 5 QT 1T U

(1.98)

In particular, Eq. (1.95) as the simple cases of Eq. (1.98) is denoted in the direct notation as follows: v 5 Qv; T 5 QTQT ;

v 5 QT v T 5 QT T  Q

(1.99)

Here, it should be emphasized that the equations described in this subsection do not express relations between different vectors or tensors but they express the relations between components of a certain vector or tensor described by the two different coordinate systems with the bases, say fei g and fei g. Then, note that Eqs. (1.98) and (1.99) are the specious direct notations that imitate tensor relations between two different vectors or tensors. The components observed by the base fei g and fei ðtÞg are denoted by ð Þ and ð Þ , respectively, for the sake of brevity (it should be emphasized that the equation, which involves the quantity denoted by ð Þ , represents the relation between the components).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

18

1. Mathematical fundamentals

Denoting the unit base vectors e1 ; e2 ; . . .; em of the coordinate axes the quantity Qij in Eq. (1.90) is represented in terms of the base vectors as follows:

x1 ; x2 ; . . .; xm ,



Qij 5 ei ej

(1.100)

noting Qij 5

@xi 5 ei @xj



@xs s  es 5 ei  es 5 ei  δjs es  @x @xj @xj

where the coordinate transformation operator Qij is interpreted as Qij  cosðangle between ei and ej Þ

(1.101)

which fulfills Eq. (1.91), that is, Qir Qjr 5 Qri Qrj 5 δij

(1.102)

which can be also verified by





 

Qir Qjr 5 ðei er Þðej er Þ 5 ei ðej er Þer 5 δij The transformation rule of the base vectors is given by ei 5 Qri er ;

ei 5 Qir er

(1.103)

noting



ei 5 ðei er Þer ;



ei 5 ðei er Þer

The nth order tensor is described in terms of the components with the base vectors as follows: T 5 T 12...n e1 e2 . . .en 5 T12...n e1 e2 . . .en

(1.104)

1.4 Tensor operations Various tensors and their algebra are addressed in this section, which are used often throughout this book.

1.4.1 Properties of second-order tensor The general definition of tensor was given in Section 1.3. Here, based on the definition, basic properties of second-order tensor are described below.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

19

1.4 Tensor operations

Two tensors A and B are same when they yield same transformation of an arbitrary vector v, that is, A 5 B when Av 5 Bv

(1.105)

The following equations hold by virtue of the linear transformation in Eq. (1.96). Tðaa 1 bbÞ 5 aðTaÞ 1 bðTbÞ

(1.106)

A1B5B1A

(1.107)

sðABÞ 5 ðsAÞB 5 AðsBÞ

(1.108)

ðA 1 BÞv 5 Av 1 Bv

(1.109)

ðABÞv 5 AðBvÞ

(1.110)

AðB 1 CÞ 5 AB 1 AC;

ðA 1 BÞC 5 AC 1 BC

AðBCÞ 5 ðABÞC

(1.111) (1.112)

where a; b, and s are arbitrary scalar variables. The following relations hold for the trace. trðABÞ 5 trðBAÞ

(1.113)

trðabÞ 5 a b

(1.114)

trT  T:I 5 Tij δij 5 Tii

(1.115)



The magnitude of tensor is given by pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi :T: 5 Trs T rs 5 T:T

(1.116)

The power of tensors is defined as follows: T0 5 I;

zfflffl}|fflffl{ n Tn 5 T. . .T

(1.117)

which obey the rules of exponentiation: Tm Tn 5 Tm1n 5 Tn Tm ;

ðaTÞn 5 an Tn ;

ðTm Þn 5 Tmn

(1.118)

for arbitrary integers m and n. The zero tensor O possessing the zero component Oij 5 0 and thus it transforms an arbitrary vector v to the zero vector 0, that is, Ov 5 0

(1.119)

1.4.2 Tensor components The component of vector is given in Eq. (1.30). The second-order tensor in terms of the components and the base vectors is described from Eq. (1.104) as follows: Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

20

1. Mathematical fundamentals

T 5 Tij ei ej

(1.120)

the components of which are given by



T ij 5 ei Tej

(1.121)

by virtue of Eq. (1.86)1. The component Ξijk of the third-order tensor Ξ 5 Ξ ijk ei ej ek

(1.122)

Ξ ijk 5 ðei ej Þ:Ξek

(1.123)

is given by

by virtue of Eq. (1.86)3. The component Tijkl of the fourth-order tensor T 5 Tijkl ei ej ek el

(1.124)

Tijkl 5 ðei ej Þ:T:ðek el Þ

(1.125)

is given by

by virtue of Eq. (1.86)5.

1.4.3 Transposed tensor The tensor TT satisfying the following equation for bay arbitrary vectors a and b is defined as the transposed tensor of a tensor T.





a Tb 5 b TT a

(1.126)

Noting





a ðuvÞb 5 b ðvuÞa it follows from Eq. (1.126) that ðuvÞT 5 vu

(1.127)

Further, comparing the equation





ei ðTrs er es Þej 5 Tij 5 ej ðT rs es er Þei with Eq. (1.126), one has ðT rs er es ÞT 5 Trs es er 5 T sr er es

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.128)

1.4 Tensor operations

21

that is, ðTT Þij 5 ðTÞji

(1.129)

If T satisfies the following equation, that is, if TT 5 T holds, T is called the symmetric tensor.





a Tb 5 b Ta

(1.130)

Therefore the components satisfy the following relation.





Tij 5 ei Tej 5 ej Tei 5 Tji

(1.131)



T TT 5 T ij ei ej 5 T ij ej ei 5 Tji ei ej

(1.132)

Then, it follows that

The transposed tensor is given by simple exchange of suffices in components in the Cartesian coordinate system. However, the transposed tensor in the mix-variant expressions in the general coordinate system is not given by the exchange of suffices in components but given by the exchange of the base vectors as will be shown in Eq. (2.64). The following equations hold for the transposed tensor.



ðTT ÞT 5 T

(1.133)

jjTT jj 5 jjTjj

(1.134)

ðA1BÞT 5 AT 1 BT

(1.135)

ðABÞT 5 BT AT ðAjr Bri 5 Bri Ajr Þ

(1.136)

Tv 5 vTT ðT ir vr 5 vr Tir Þ

(1.137)

ðabÞT 5 ba

(1.138)

trðABÞ 5 A:BT 5 AT :B

(1.139)

trðABÞT 5 trðBT AT Þ

(1.140)





Ta b 5 a TT b 5 a bT ððTri ai Þbr 5 ai ðTri br Þ 5 ai ðbr T ri ÞÞ ( Tab 5 aTT b ððT ir ar Þbi 5 ðar T ir Þbi Þ aTb 5 abTT ðai ðTjr br Þ 5 ai ðbr T jr ÞÞ

(1.141) (1.142)

1.4.4 Inverse tensor The tensor T fulfilling detT 6¼ 0 is called the nonsingular tensor, for which there exists the tensor, called the inverse tensor and designated by T21 , satisfying the relation

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

22

1. Mathematical fundamentals

TT21 5 T21 T 5 I;

Tir Trj21 5 Tir21 Trj 5 δij

(1.143)

where the components are denoted as Tij21  ðT21 Þij for brevity. Then, there follows: ðABÞ21 5 B21 A21

(1.144)

because of ðABÞðB21 A21 Þ ð 5 ABB21 A21 5 AIA21 5 AA21 Þ 5 I It follows also that T2n  ðT21 Þn 5 ðTn Þ21

(1.145)

and thus the rule of exponentiation extends to the negative power.

1.4.5 Orthogonal tensor The orthogonal tensor, that is, coordinate transformation tensor is defined as the tensor which keeps a scalar product of vectors to be constant and thus it fulfills





ðQaÞ ðQbÞ 5 a b

(1.146)

designating the orthogonal tensor by Q. By virtue of Eq. (1.141), the lefthand side in Eq. (1.146) becomes





ðQaÞ ðQbÞ 5 a ðQT QbÞ

(1.147)

Comparing Eq. (1.147) with Eq. (1.146), the orthogonal tensor must fulfill QQT 5 QT Q 5 I

(1.148)

QT 5 Q21

(1.149)

leading to

The component description of Eq. (1.148) is given in Eq. (1.102). The particular selection b 5 a 5 v in Eq. (1.146) gives :Qv: 5 :v:

(1.150)

Therefore the magnitude of vector does not change by the orthogonal transformation. This fact along with Eq. (1.146) means that the angle formed by vectors also does not change by the orthogonal transformation.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

23

1.4 Tensor operations

The magnitude of the orthogonal tensor is given from Eq. (1.148) as pffiffiffi :Q: 5 3 (1.151) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi noting :Q: 5 trðQQT Þ 5 trI. While the scalar product of two vectors does not change by the orthogonal transformation as described in Eq. (1.146), the trace of two tensors also does not change by it, that is, tr½ðQAQT ÞðQBQT Þ 5 trðABÞ

(1.152)

While the magnitude of vector does not change by the orthogonal transformation as described in Eq. (1.150), the magnitude of tensor also does not change by the transformation, that is, :jQTQT : 5 :T:

(1.153)

noting Eqs. (1.113), (1.148), and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tr ½QTQT ðQTQT ÞT  5 tr ðQTQT QTT QT Þ 5 tr ðTTT Þ Then, the following tensor possessing the components Qij in Eq. (1.100) satisfies Eq. (1.102) the direct notation of which is identical to Eq. (1.148) and thus the orthogonal tensor is expressed with the components and the base vectors as follows: Q 5 Qij ei ej 5 Qij ei ej 5 ei ei

(1.154)

noting





Qij ei ej 5 ei ðei ej Þej 5 ei ei 5 ðei er Þer ei 5 Qri er ei QQT 5 ðei ei Þðej ej ÞT 5 ei ei ej ej 5 δij ei ej by use of Eq. (1.103). Furthermore, because of ei 5 er δir 5 er er ei ;

ei 5 er δir 5 er er ei

one has the expressions: ei 5 Qei ;

ei 5 QT ei

(1.155)

The equations on the coordinate transformation tensor Q are collectively shown from Eqs. (1.100), (1.103), (1.154), and (1.155).



Qij 5 ei ej 5 cosðangle between ei and ej Þ ei 5 QT ei 5 Qir er ;

ei 5 Qei 5 Qri er ;  Q 5 Qij ei ej 5 Qij ei ej 5 ei ei

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.156)

24

1. Mathematical fundamentals

The orthogonal tensor for the rotation by θ in the anticlockwise direction around the base vector e3 is represented as follows: 2 3 2 3 Q11 Q12 Q13 cosθ sinθ 0 Q 5 4 Q21 Q22 Q23 5 5 4 2 sinθ cosθ 0 5 (1.157) 0 0 1 Q31 Q32 Q33 that is, Q 5 Q11 e1 e1 1 Q12 e1 e2 1 Q21 e2 e1 1 Q22 e2 e2 1 e3 e3 5 cosθe1 e1 1 sinθe1 e2 2 sinθe2 e1 1 cosθe2 e2 1 e3 e3 5 ðe1 e1 1 e2 e2 Þcosθ 1 ðe1 e2 2 e2 e1 Þsinθ 1 e3 e3

(1.158)

for which we can write 9 2 8 9 8 38 9 38 9 2 Q11 Q12 Q13 > cosθ sinθ 0 > > = > = = = < e1 > < Q1r er > < e1 > < e1 > 6 7 6 7  e2 5 Q2r er 5 4 Q21 Q22 Q23 5 e2 5 4 2sinθ cosθ 0 5 e2 > > > > > > ; > ; ; : > : : ; : e3 0 0 1 Q3r er Q31 Q32 Q33 e3 e3 (1.159) Eq. (1.157) is referred to as the canonical expression for orthogonal tensor.

1.4.6 Tensor decompositions Several types of decompositions of tensor are used often for convenience as will be described below. 1.4.6.1 Symmetric and skew-symmetric tensors The tensor T is additively decomposed into the symmetric tensor S and the skew-(or anti-) symmetric tensor Ω as follows: T5S1Ω

(1.160)

where 1 S  sym½T  ðT 1 TT Þ; 2

1 Ω  ant½T  ðT 2 TT Þ 2

(1.161)

which satisfy 

ST 5 S;

ΩT 5 2 Ω

  a  ðΩbÞ 5 2 b  ðΩaÞ

a Sb 5 b Sa

(1.162) (1.163)

noting Eq. (1.126). Eq. (1.160) is called the Cartesian decomposition. Here, it follows that

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

25

1.4 Tensor operations

SST 5 S2

(1.164)

trðSΩÞ 5 trðSΩ Þ 5 0 T

(1.165)

because of trðSΩÞ 5 trððSΩÞT Þ 5 trðΩT SÞ 5 2 trðΩSÞ 5 2 trðSΩÞ, noting Eqs. (1.133) and (1.162). The skew-symmetric tensor fulfills trΩ 5 0;



v ðΩvÞ 5 0

(1.166)

exploiting Eqs. (1.162)2 and (1.163). By virtue of Eq. (1.165), it follows that trðABÞ 5 trðsym½Asym½BÞ 1 trðant½Aant½BÞ

(1.167)

1.4.6.2 Spherical and deviatoric tensors Tensor T can be decomposed as follows: T 5 Tm 1 T0 1 Tm  T m I; Tm  trT; T0  T 2 T m I 3

(1.168) 0

0

0

ðT n 6¼ Tn : trT0  0:trT n 6¼ 0Þ (1.169)

where Tm and T0 are called the spherical (or mean) part and the deviatoric part, respectively, of the tensor T. The prime ð Þ0 is used for the deviatoric part throughout this book.

1.4.7 Axial vector The antisymmetric tensor Ω is represented in the matrix form from Eq. (1.162)2 as follows: 2 3 0 Ω12 Ω13   Ωij 5 4 0 Ω23 5 (1.170) ant: 0 Thus the antisymmetric tensor possesses only three components, and thus we can infer that it can be related to a vector uniquely. Then, we examine this fact below. The axial vector ω is defined as the vector that fulfills the following equation for the skew-symmetric tensor Ω and an arbitrary vector a. Ωa 5 ω 3 a

(1.171)

Choosing a as the base vector ej and making the scalar product with ei , we find

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

26

1. Mathematical fundamentals









Ω ij 5 ei Ωej 5 ei ω 3 ej 5 ei ωr er 3 ej 5 ei ωr εrjk ek 5 ωr εrik δik 5 2 εijr ωr (1.172)

that is,

Ω 12 5 2 ω3 ;

Ω 23 5 2 ω1 ;

Ω 13 5 ω2

(1.173)

Then, Ω is expressed as follows: Ω 5 2 εω 5 2 εijk ωk ei ej that is, 



2

Ω ij 5 4

3 ω2 2 ω1 5 0

2 ω3 0

0 ant:

(1.174)

(1.175)

The inverse relation of Eq. (1.174) is given by 1 1 1 ω 5 2 εΩ 5 2 Ω ij ei 3 ej 5 2 εirs Ω rs ei 2 2 2

(1.176)

because of ωp ep 5 δpq ωq ep 5 εijp εijq ωq ep =2 5 2 εijp Ω ij ep =2 5 2 Ω ij ei 3 ej =2 by virtue of Eqs. (1.25), (1.39), and (1.172). It follows from Eq. (1.174) that Ω5I3ω5ω3I because of



2εijk ωk ei ej 5

(1.177)

ei ωk εikj ej 5 ei ei 3 ωk ek 5 ei ei 3 ω 2 ej 3 ek ωk ej 5 ek ωk 3 ej ej 5 ω 3 ej ej

noting Eq. (1.39). It follows in view of (1.55)2 that





ðuv 2 vuÞa 5 ðv aÞu 2 ðu aÞv 5 ðv 3 uÞ 3 a

(1.178)

Then, it is known by virtue of Eq. (1.171) that v 3 u is the axial vector of the antisymmetric second-order tensor uv 2 vu. Now consider the rotation of the Cartesian coordinate system. The time differentiation of the base vector ei in Eq. (1.155)2 leads to

_

_ei 5 Q_ T ei 1 QT_ei 5 Q_ T Qei 1 QT_ei

(1.179)

If we put ei 5 0 in Eq. (1.179), we have the variation of ei observed from the coordinate system with the base fei g leads to

_ei 5 Ωei Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.180)

27

1.4 Tensor operations

where

_T

_

Ω  Q Q 5 er er ;

_  _ 

Ω ij 5 ðer ei Þðer ej Þ

(1.181)

Ω designates the spin of the base vector ei observed from the coordinate system with the base fei g, bearing in mind that only the direction is  changeable in the base vector because of the unit vector. ei can be rewritten as

_

_er 5 ω 3 er

(1.182)

by virtue of the property in Eq. (1.171), where ω is the angular velocity of ei , which is related to Ω in Eq. (1.176), where Ω rs is specified in Eq. (1.181)2. It follows from Eqs. (1.176) and (1.181) with Eq. (1.157) for the rotation around the axis e3 that 8 9 2 3  > > 3 > > < εijk sT1i sT2j sT3k 5s εijk T1i T2j T3k 1 1 1 det T5 εabc εpqr Tap Tbq Tcr 5 εpqr εabc Tpa Tqb Trc 5 εabc εpqr Tpa Tqb Trc > > 3! 3! 3! > > > : detðABÞ5ε ðA B ÞðA B ÞðA B Þ5ε A A A B B B 5A A A ε detB pqr 1a ap 3c cr pqr 1a 2b 3c ap bq cr 1a 2b 3c abc 2b bq

    expðT 11 Þ 0 0    5expðT11 1T 22 1T 33 Þ  expðT22 Þ 0 detðexpTÞ5  0  0 0 expðT 33 Þ 

(1.192)

(1.193)



detðabÞ5εijk ða1 bi Þða2 bj Þða3 bk Þ5a1 a2 a3 εijk bi bj bk 5ða1 a2 a3 Þðb3bÞ b (1.194) detTdet½T21 5det½TT21 51

(1.195)

noting Eqs. (1.12), (1.14) and (1.21). The following equations hold for the cofactor, noting Eq. (1.185) together with Eq. (1.191).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

29

1.4 Tensor operations

8 cofðsTÞ 5 s2 cofT > > > > > ðcofTÞT 5 cofðTT Þ; ðcofTÞ21 5 cofðT21 Þ; ðcofTÞ2T 5 cofðT2T Þ > > > < cofðABÞ 5 cofðAÞcofðBÞ 1 > > > trðcofTÞ 5 ðtr2 T 2 trT2 Þ 5 II > > 2 > > > : detðcofTÞ 5 1

(1.196)

noting 1 1 1 trðcofTÞ 5 εabc εaqr Tbq Tcr 5 ðδbq δcr 2 δbr δcq ÞTbq T cr 5 ðTqq Trr 2 Trq T qp Þ 2 2 2 II is the second principal invariant as will be defined in Section 1.6.1. The vector product in Eq. (1.38) and the scalar triple product in Eq. (1.48) are described by the determinant as follows:    e1 e2 e3           a2 a3   a3 a1   a1 a2  e 1  e 1  e 5  a a2 a3  (1.197) a 3 b 5   b2 b3  1  b3 b1  2  b1 b2  3  1 b1 b2 b3  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     ffi  a2 a3 2  a3 a1 2  a1 a2 2  1    jja 3 bjj 5  (1.198)  b3 b1  1  b1 b2  b2 b3 

   a3   a e1 a e2 a e3  (1.199) b3  5  b e1 b e2 b e3  c 3   c e1 c e2 c e3    a c b c    ða 3 bÞ ðc 3 dÞ 5  (1.200) a d b d       a2 a3  p1 q2 r3   ai pi ai qi ai ri   a p a q a r  b2 b3  p1 q2 r3  5  bi pi bi qi bi ri  5  b p b q b r  c2 c3  p1 q2 r3   ci pi ci qi ci ri   c p c q c r 

  a1  ½abc 5  b1  c1

  

a2 b2 c2

 



  a1  ½abc½pqr5  b1  c1

  

  

 

  

  

  

(1.201) noting Eq. (1.189)4. The following equation is derived as the special case of Eq. (1.201) for the three vectors v1 ; v2 ; v3 .    v1 v1 v1 v2 v1 v3      2 v2 v2 v2 v3  (1.202) v2 5 v1 v2 v3 5   sym: v3 v3 



 

  

where

  v 5 v1 v2 v3 5 εijk ðv1 Þi ðv2 Þj ðv2 Þk ð 5 εijk ðv1 ei Þðv2 ej Þðv3 ek ÞÞ







Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.203)

30

1. Mathematical fundamentals

is the volume of the parallelopiped formed by the line elements v1 ; v2 ; v3 . By introducing the symbol



vij  vi vj Eq. (1.202) can be written as   v11  2 v 5   sym:

(1.204)

 v13  v23  5 detðvij Þ v33 

v12 v22

(1.205)

It follows from Eq. (1.201) that detT 5 ½Te1 noting

  T11   detT 5  T21   T31 5 ½ e1

T12 T22 T32 e2



Te3  5 Te1 ðTe2 3 Te2 Þ

Te2

(1.206)

   T13   ðe1 Te1 Þ ðe1 Te2 Þ ðe1 Te3 Þ     T23  5  ðe2 Te1 Þ ðe2 Te2 Þ ðe2 Te3 Þ     T33   ðe3 Te1 Þ ðe3 Te2 Þ ðe3 Te3 Þ 

  

e3 ½Te1

Te2

  

Te3  5 ½Te1

  

Te2

Te3 

Applying Eq. (1.189) to Eqs. (1.148) and (1.162)2, the determinants of the orthogonal tensor and the skew-symmetric tensor are given as detQ 5 detQT 5 6 1

(1.207)

detΩ 5 0

(1.208)

1.4.9 Simultaneous equation for vector components Now, when we regard the transformation of the vector v to the vector u by the tensor T, that is, Tv 5 u;

T ij vj 5 ui

(1.209)

as the inhomogeneous simultaneous equation in which the components of v are the unknown numbers, solution exists for u 6¼ 0 if detT 6¼ 0 and is given by v 5 T21 u, noting Eq. (1.186), as follows: v5

ðcofTÞT u; detT

vi 5

ðcofTÞji detT

uj

(1.210)

Here, T must be the nonsingular tensor fulfilling detT 6¼ 0 in order that the nontrivial solution v 6¼ 0 exists for u 6¼ 0. On the other hand, in order that there exists the nonzero solution of v in the homogeneous simultaneous equation

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

31

1.5 Representations of tensors

Tv 5 0;

Tij vj 5 0

(1.211)

the following equation must hold. detT 5 0

(1.212)

1.5 Representations of tensors Various notations are used to represent tensors in continuum mechanics. They are collectively shown in this section.

1.5.1 Notations in tensor operations The following notations are used throughout this book for the vectors a; b; c; d, and v, the second-order tensors A; B, and T, the third-order tensor Ξ, and fourth-order tensor T. 8 > < a b 5 trðabÞ for ai bi a 3 b 5 ε: ðabÞ for εijk aj bk (1.213) > : ab for ai bj with ab c for ai bj cj  Tv for Tij vj ; vT for vi T ij (1.214) v 3 T for ðv 3 TÞij 5 εikl vk Tlj ; T 3 v for ðT 3 vÞij 5 εjkl Tik vl ! v 3 T 5 vk ek 3 Tlj el ej 5 vk Tlj εkli ei ej 5 εkli vk Tlj ei ej 5 εikl vk Tlj ei ej T 3 v 5 T ik ei ek 3 vl el 5 Tik vl ei εklj ej 5 εklj Tik vl ei ej 5 εjkl Tik vl ei ej





8 > > > > > > > > > > > > > >
T:ðabÞ 5 a Tb for Tij ai bj > > > > > ðTaÞb 5 ðabÞT for T ij ar br ; ðaTÞb 5 aðTbÞ > > > > > ðabÞ:ðcdÞ 5 ða cÞðb dÞ for ai ci bj dj > > > : ðabcÞv 5 ðc vÞab for ai bj cr vr



(





for ai Tjr br

(1.215) ΞT

for Ξ ijk Tkl ;

TΞ

Ξ:T

for Ξ ijk Tjk ;

T:Ξ for Tij Ξ ijk

for Tij Ξ jkl

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.216)

32

1. Mathematical fundamentals

 (

TT for Tijkl T ln ; TT for Tij Tjkln T:T for Tijkl T kl ; T:T for T ij Tijkl for Tijkl Akl Bij 5 Bij Tijkl Akl

ðT:AÞ:B 5 B: T:A

ðT :AÞ:B 5 A: T:B for Tklij Akl Bij 5 Akl Tklij Bij ( pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi :v: 5 v v for vi vi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi :T: 5 T:T 5 trðTTT Þ for Tij T ij T



@f 2 ðTÞ @T@T

for

@f 2 ðTÞ @Tij @T kl

(1.217) (1.218)

(1.219)

(1.220)

The following symbols for the tensor products of second-order tensors A, B, and C are defined (del Piero, 1979; Steinmann et al., 1997; Kintzel and Bazar, 2006; Wang and Dui, 2008). 8 ðABÞijkl 5Aij Bkl with AB:C5AðB:CÞððAB:CÞij 5Aij ðBkl Ckl ÞÞ and > > > > > C:AB5ðC:AÞBððC:ABÞkl 5Cij Aij Bkl Þ > > > > > > ðABÞijkl 5Aik Bjl with AB:C5ACBT ððAB:CÞij 5Aik Bjl Ckl 5Aik Ckl Bjl Þ and > > > > > > C:AB5AT CBððC:ABÞkl 5Cij Aik Bjl 5Aik Cij Bjl Þ > > > > > > ðABÞijkl 5Ail Bjk with AB:C5ACT BT ððAB:CÞij 5Ail Bjk Ckl 5Ail Ckl Bjk Þ and > < C:AB5BT CT AððC:ABÞkl 5Cij Ail Bjk 5Bjk Cij Ail Þ > > > e e e ijkl 5Aik Blj with AB:C5ACBððA > B:CÞ > ðABÞ ij 5Aik Blj Ckl 5Aik Ckl Blj Þ and > > > T T > e e kl 5Cij Aik Blj 5Aik Cij Blj Þ > C:AB5A CB ððC:ABÞ > > > > T > > ðABÞijkl 5Ail Bkj with AB:C5AC BððAB:CÞij 5Ail Bkj Ckl 5Ail Ckl Bkj Þ and > > > e e e > > > T > > C:AB5BC AððC:ABÞkl 5Cij Ail Bkj 5Bkj Cij Ail Þ : e e (1.221) from which the following equations are derived. 8 AB 5 ABI 5 A I:B > > < BAT 5 IBAT 5 IA:B ABT 5 ABT I 5 AI:B > > : T T A B 5 IAT BT 5 IB:A

(1.222)

1.5.2 Operational tensors The fourth-order identity tensor I and the transposing tensor T are defined by

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

1.5 Representations of tensors

33

8 eI < I 5 δik δjl ei ej ek el 5 ei ej ei ej 5 II 5 I : T 5 δil δjk ei ej ek el 5 ei ej ej ei 5 II 5 II e

(1.223)

It holds that I:T 5 T:I 5 T; T:T 5 T:T 5 TT ; I:T 5 T:I 5 T and @T=@T 5 I; @TT =@T 5 T. Moreover, the fourth-order tracing tensor is defined by I  δij δkl ei ej ek el 5 ei ei ej ej 5 II

(1.224)

The spherical part of the tensor defined in Eq. (1.169) is described as follows: Tm  I:T

(1.225)

The components given in Eqs. (1.223) and (1.224) are confirmed with the aid of Eq. (1.124) as follows: 8 > < Iijkl 5 ðei ej Þ: ðea eb ea eb Þ: ðek el Þ 5 δia δjb δka δlb 5 δik δjl Tijkl 5 ðei ej Þ: ðea eb eb ea Þ: ðek el Þ 5 δia δjb δkb δla 5 δil δjk > : Iijkl 5 ðei ej Þ: ðea ea eb eb Þ: ðek el Þ 5 δia δja δkb δlb 5 δij δkl The symmetrizing tensor S and the skew-(or anti-)symmetrizing tensor A are defined by 1 1 S  ðδik δjl 1 δil δjk Þei ej ek el 5 ðI 1 TÞ 2 2 1 1 A  ðδik δjl 2 δil δjk Þei ej ek el 5 ðI 2 TÞ 2 2 leading to S:T 5 ðT 1 TT Þ=2; expressions hold.

(1.226)

A:T 5 ðT 2 TT Þ=2. Then, the following

sym½T 5 S:T;

ant½T 5 A:T

The deviatoric projection tensor is defined by   1 1 I0  δik δjl 2 δij δkl ei ej ek el 5 I 2 I 3 3

(1.227)

(1.228)

The deviatoric part of the tensor defined in Eq. (1.169) is described as follows: T0  I0 :T The following relations hold in these tensors.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.229)

34

1. Mathematical fundamentals

8 I:I 5 I; T:T 5 I; I:T 5 T:I 5 T > > > < I:I 5 3I > I:I 5 I:I 5 T:I 5 I:T 5 I > > : 0 I:I 5 I0 ; T:I0 5 T 2 I=3; I:I0 5 O

(1.230)

S:S 5 S; I0 :I0 5 I0 ; I0 :S 5 S:I0 5 O

(1.231)

The detailed proofs are given in the literature (Hashiguchi, 2017a). The symmetrizing tensor S can be used instead of the identity tensor I for symmetric tensors.

1.5.3 Isotropic tensors Isotropic tensor is defined as a tensor possessing components that are unchanged by arbitrary rotation of coordinate system and thus it must satisfy T 5 Q1TU

(1.232)

where use is made of the notation for objective transformation in Eq. (1.98) for general tensor. As a trivial case, all tensors possessing zero component are isotropic tensors. We consider nontrivial isotropic tensors for the first to fourth-order tensors for which Eq. (1.232) is described as 8 S5S > > > > < vi 5 Qir vr T ij 5 Qir Qjs T rs (1.233) > > T 5 Q Q Q T > ir js kt rst ijk > : T ijkl 5 Qir Qjs Qkt Qlu Trstu Here, note that the permutation of indices does not have influence on the values of components by virtue of the isotropy. The zero to fourth-order isotropic tensors are given as follows (cf. Hashiguchi, 2017a): 1. All scalars (zero-order tensors) are isotropic tensor. 2. There is no first-order isotropic tensor. 3. The second-order isotropic tensor is given by sδij ei ej 5 sI

(1.234)

4. The third-order isotropic tensor is given by sεijk ei ej ek 5 sε

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.235)

35

1.6 Eigenvalues and eigenvectors

5. The fourth-order isotropic tensor is given by J  ðλIijkl 1 μSijkl 1 νAijkl Þei ej ek el 5 λI 1 μS 1 νA

(1.236)

1.6 Eigenvalues and eigenvectors The tensor is expressed in the component notation having only normal components by choosing the coordinate base into special directions. In what follows, consider these special directions for the second-order tensor.

1.6.1 Eigenvalues and eigenvectors of second-order tensors The unit vector e fulfilling Te 5 Te ;

Tij ej 5 Tei

(1.237)

that is, ðT 2 ΤIÞe 5 0;

ðTij 2 Τδij Þej 5 0

(1.238)

for the second-order tensor is called the eigenvector (or principal or characteristic or proper vector) and the scalar T is called the eigenvalue (or principal or characteristic or proper value). In order that the linear simultaneous Eq. (1.238) possesses a nonzero solution e 6¼ 0, the determinant of the coefficient must be zero as described in Eq. (1.212) for Eq. (1.211), that is,     T13     T11 2 T T12  5 0 (1.239)    det½T 2 TI 5 0 ; Tij 2 Tδij 5  T21 T22 2 T T23   T31 T32 T33 2 T  Eq. (1.239) is called the characteristic equation of the tensor, which is regarded as the cubic equation of T. Three eigenvectors e1 ; e2 ; e3 are derived by solving Eq. (1.238) for each of three solutions T1 ; T2 ; T3 of T obtained from Eq. (1.239). It can be verified that the eigenvectors are mutually orthogonal for eigenvalues that differ from each other in the second-order real symmetric tensor fulfilling T 5 TT (cf. Hashiguchi, 2017a). Eq. (1.239) leads to the characteristic equation ð2 det½T 2 ΤI 5 Þ T 3 2 IT 2 1 IIΤ 2 III 5 0

(1.240)

I  T11 1 T22 1 T33 5 Tii 5 trT

(1.241)

where

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

36

1. Mathematical fundamentals

      T T  T T  T T  1 1 II   22 23  1  11 13  1  11 12  5trðcofTÞ5 ðTrr Tss 2Trs Tsr Þ5 ðtr2 T2trT2 Þ T32 T33 T31 T33 T21 T22 2 2 (1.242)

   T11 T12 T13    1 1 1 III   T21 T22 T23  5detT5εrst Tr1 Ts2 Tt3 5 tr3 T2 trTtrT2 1 trT3 6 2 3  T31 T32 T33 

(1.243) The direct notation in the last side of Eq. (1.243) will be shown later in Eq. (1.261) based on the Cayley 2 Hamilton theorem in Section 1.6.3. On the other hand, the characteristic Eq. (1.240) is expressed by use of the principal values as follows: ðT 2 T1 ÞðT 2 T2 ÞðT 2 T3 Þ 5 0

(1.244)

Comparing Eqs. (1.240) and (1.244), coefficients I, II, and III are described as I 5 T1 1 T2 1 T3 II 5 T1 T2 1 T2 T3 1 T3 T1 III 5 T1 T2 T3

(1.245)

Eq. (1.245) can also be derived by inserting T11 5 T1 ; T 22 5 T2 ; T33 5 T3 ; T 12 5 T23 5 T31 5 0 into Eqs. (1.241)(1.243). Since I, II, and III are the symmetric functions of principal values, they are the invariants under the rotation of coordinate system and are called the principal invariants. The following invariants are called the moments. I  trT;

II  trT2 ;

III  trT3

(1.246)

The principal invariants are described in terms of these moments from Eqs. (1.241)(1.243) as follows: 8 > I5I > > > > > 1 > < II 5 ðI 2 2 IIÞ 2 (1.247) > > > 1 3 1 1 > > > III 5 I 2 I II 1 III > : 6 2 3 Next, consider the deviatoric tensor T0 . The characteristic equation of T is given by replacing T with T0 in Eq. (1.240) as follows: 0

T03 1 II0 T 0 2 III0 5 0

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.248)

37

1.6 Eigenvalues and eigenvectors

noting I0  trT0 5 0

(1.249)

where 1 1 2 II0  trT0 5 T0rs T0sr 2 2 5

1 02 2 2 2 2 2 ðT 1 T 0 22 1 T 0 33 Þ 1 T 0 12 1 T 0 23 1 T 0 31 2 11

5

 1 2 2 2 ðT11 2T22 Þ2 1 ðT22 2T33 Þ2 1 ðT33 2T11 Þ2 1 T 0 12 1 T 0 23 1 T0 31 6

5

 1 02 1 2 2 ðT 1 1 T0 2 1 T 0 3 Þ 5 ðT1 2T2 Þ2 1 ðT2 2T3 Þ2 1 ðT3 2T1 Þ2 2 6 (1.250)

III0  detT0 5

1 1 3 trT0 5 T 0rs T 0sr T0rt 5 T011 T 022 T033 3 3 2 T 011 T0 23 2 T022 T 0 31 2 T 033 T0 12 1 2T 012 T023 T031  1  03 3 3 T 1 1 T0 2 1 T0 3 5 T01 T 02 T 03 5 3 2

2

2

(1.251)

1.6.2 Spectral representation and elementary tensor functions By choosing the base vectors to the principal directions eP of the symmetric tensor, the following expression holds: T5

3 X

TP eP eP

(1.252)

P51

where TP are the principal values, fulfilling TeP 5 TP eP ðP 5 1; 2; 3; no sumÞ

(1.253)

Eq. (1.252) is called the spectral decomposition (or representation). In general, the second-order tensor is represented as follows: T5

3 X

TP uP vP

(1.254)

P51

where uP and vP are the left and the right eigenvectors, respectively.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

38

1. Mathematical fundamentals

Further, it follows for the power of symmetric tensor T that Tn eP 5 T nP eP Tn 5 Tn

3 X

eP eP 5

P51

3 X

(1.255) T nP eP eP

(1.256)

P51

by the repeated applications of T to Eq. (1.237), that is, n21 n22 n22 2 n23 n TnP e 5 Tn21 P T P e 5 T P Te 5 TT P T P e 5 TT P Te 5 T T P T P e 5 ? 5 T e

Eq. (1.255) means that the principal values of Tn are TPn where TP are principal values of T and the principal directions of the tensor Tn are identical to those of T, while the tensors having an identical set of principal directions are called to be coaxial or said to fulfill the coaxiality. Then, the second-order tensor power function fðTÞ of only T is coaxial with T and the principal values are given by fðTP Þ, and thus it follows that fðTÞ 5

3 X

fðTP ÞeP eP

(1.257)

P51

1.6.3 CayleyHamilton theorem Multiplication to the characteristic Eq. (1.240) by the principal vector e with the aid of Eq. (1.255) yields T3 2 IT2 1 IIT 2 IIII 5 O

(1.258)

Eq. (1.258) is referred to as the CayleyHamilton theorem. By virtue of Eq. (1.258), any power function of T can be expressed in polynomial equation of T2 ; T; I with coefficients consisting of the invariants. For instance, the fourth power of T reduces to



T4 5 IT2 2IIT1IIII T5IT3 2IIT2 1IIIT5I IT2 2IIT1IIII 2IIT2 1IIIT

5 I2 2II T2 2 ðI II2IIIÞT1I III I (1.259) and the inverse of T is expressed from Eq. (1.258) by T21 5

ðT2 2 IT 1 IIIÞ III

(1.260)

It is concluded that the power of the tensor T is expressed by the linear combination of T2 ; T; I with coefficients consisting of the principal invariants. Furthermore, any function other than power function can be

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

39

1.6 Eigenvalues and eigenvectors

described by them, exploiting the Maclaurin expansion, which will be described in Section 1.6.5. Besides, third principal invariant III is described from Eq. (1.258) with Eqs. (1.241) and (1.242) as follows: III 5

1 3 1 1 tr T 2 trTtrT2 1 trT3 6 2 3

(1.261)

1.6.4 Scalar triple products with invariants The following formulae of the scalar triple products related to the principal invariants hold. ½T a b c 1 ½a T b c 1 ½a b Tc 5 trT½abc 5 I½abc ½a Tb Tc 1 ½Ta b Tc 1 ½Ta Tb c 5 trðcofTÞ½abc 5 II½abc ½Ta Tb Tc 5 detT½abc 5 III½abc

(1.262)

which will be used often in the subsequent chapters in the formulation of constitutive relations. Eq. (1.206) is the special case of Eq. (1.262)3. The detailed proof of Eq. (1.262) is given in Appendix 1.

1.6.5 Second-order tensor functions Applying the Maclaurin expansion fðTP Þ 5

N ðnÞ X f ð0Þ n50

n!

TPn 5 fð0Þ 1 f 0 ð0ÞTP 1

fvð0Þ 2 fwð0Þ 3 T 1 T 1? 2! P 3! P

(1.263)

to the scalar function fðTP Þ in Eq. (1.257), various second-order tensor functions are defined for general (not limited to symmetric tensor) second-order tensor as follows: expT 5

N X 1 n 1 1 T 5 I 1 T 1 T2 1 T3 1 ? n! 2! 3! n50

lnðI 1 TÞ 5

N X ð21Þn11 n51

ðI1TÞm 5

n

Tn 5 T 2

T2 T3 1 2? 2 3

(1.264)

(1.265)

N X

m! mðm21Þ 2 mðm21Þðm22Þ 3 Tn 5I1mT1 T 1 T 1? ðm2nÞ!n! 2! 3! n50 (1.266) 21

21

T 5½I1ðT2IÞ 5I2ðT2IÞ1ðT2IÞ 2? 2

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.267)

40

1. Mathematical fundamentals

aT 5

N X ðlnaÞn n50

n!

Tn 5I1ðlnaÞT1

sinT5

ðlnaÞ2 2 ðlnaÞ3 3 T 1 T 1? 2! 3!

(1.268)

N X ð21Þn 2n11 1 1 T 5T2 T3 1 T5 2? 3! 5! ð2n11Þ! n50

cosT5

N X ð21Þn

ð2nÞ! n50

(1.269)

1 1 T2n 5I2 T2 1 T4 2? 2! 4!

(1.270)

These functions satisfy basic properties of each relevant scaler functions as follows: ðexpTÞT 5 exp ðTT Þ ðexpΩÞT 5 ðexpΩT Þ 5 exp ð2 ΩÞ

(1.271) for ΩT 5 2 Ω

(1.272)

n

expðnTÞ 5 ðexpTÞ

(1.273)

expðA 1 BÞ 5 expA expB

(1.274)

det½expðA 1 BÞ 5 exp ðtrAÞ exp ðtrBÞ

(1.275)

T

ðexpΩÞðexpΩÞ 5 ðexpΩÞðexpΩT Þ 5 expðΩ 2 ΩÞ 5 I

(1.276)

noting N X 1 n50

n!

n

ðA1BÞ 5

N X 1 n50

n!

ðAn 1nAn21 B1?1nABn21 1Bn Þ5

N X 1 n50

n!

! An

N X 1 n50

n!

! Bn

det½expT5ðexpTÞ1 ðexpTÞ2 ðexpTÞ3 5expT 1 expT 2 expT3 5expðT1 1T2 1T3 Þ for Eqs. (1.274) and (1.275).

1.6.6 Positive-definite tensor and polar decomposition The second-order symmetric tensor P fulfilling



Pv v . 0

(1.277)

for an arbitrary vector vð6¼ 0Þ is called the positive-definite tensor. Here, denoting the principal value and vector of P as PJ and eJ , respectively, it follows that





2

PeJ eJ 5 PJ eJ eJ 5 PJ :eJ : . 0 ðno sumÞ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.278)

1.6 Eigenvalues and eigenvectors

41

Therefore the principal values of the positive-definite tensor are positive, that is,PJ . 0 and further it follows noting Eq. (1.245)3 that detP 5 III . 0

(1.279)

The following positive-definiteness holds for the symmetric tensors TT T and TTT . ( 2 TT Tv v 5 Tv Tv 5 :Tv: . 0 (1.280) 2 TTT v v 5 TT v TT v 5 :TT v: . 0

 





noting Eq. (1.141). Therefore TT T and TTT are the positive-definite ten2 sors. Then, designating their principal values by λ2α and λα and their T T eigenvectors N α and nα ðα 5 1; 2; 3Þ for T T and TT , respectively, the following representations in the spectral decomposition hold. 8 3 X > > > TT T 5 λ2α N α N α > < α51 (1.281) 3 X > 2 > T > > TT 5 λα nα nα : α51

Further, introduce the following positive-definite tensors. 8 3 X > > 1=2 T > λα N α N α ð 5 U T Þ ðU 2 5 TT TÞ > < U 5 ðT TÞ 5 α51

3 > X > > T 1=2 > Þ 5 λα nα nα ð 5 V T Þ ðV 2 5 TTT Þ V 5 ðTT :

(1.282)

α51

which gives UN ðαÞ 5 λα N α ;

Vnα 5 λα nα ðno sumÞ

(1.283)

Further, let the following tensor be introduced. R 5 TU 21 ; R 5 V 21 T for which one has  RRT 5 ðTU 21 ÞðTU 21 ÞT 5 TðU 2 Þ21 TT 5 TT21 T2T TT 5 I RT R 5 ðV 21 TÞT ðV 21 TÞ 5 TðV 2 Þ21 TT 5 TT T2T T21 T 5 I

(1.284)

(1.285)

Therefore R is the orthogonal tensor. It follows from Eq. (1.284) that T 5 RU 5 VR

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.286)

42

1. Mathematical fundamentals

which is called the polar (spectral) decomposition, and RU and VR are called the right and the left polar decompositions, respectively. One has the following expression from Eq. (1.286). U 5 RT VR;

V 5 RURT

(1.287)

The substitution of Eq. (1.282)1 into Eq. (1.287)2 gives V 5 RURT 5 R

3 X

λα N α N α RT 5

α51

3 X

λα RN α RN α

α51

The coincidence of this equation with Eq. (1.282)2 requires the following relations. ( λα 5 λα ; (1.288) nα 5 RN α ; N α 5 RT nα resulting in 1

V 5 ðTTT Þ /2 5

3 X

λα nα nα

(1.289)

α51

R is expressed from Eq. (1.288)1 that R5

3 X

nα N α

(1.290)

α51

The substitution of Eqs. (1.282)1 and (1.290) into Eq. (1.286) gives T5

3 X

λα nα N α ;

T21 5

α51

3 X 1 N α nα λ α51 α

(1.291)

1.6.7 Representation theorem of isotropic tensor-valued tensor function The general representation of isotropic tensor-valued tensor function of a single second-order tensor is given below, to which isotropic elastic constitutive equations belong. Now, letting A and T be symmetric second-order tensors, consider the following isotropic tensor function A of T. A 5 fðTÞ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.292)

43

1.7 Differential formulae

where f fulfills fðQTQT Þ 5 QfðTÞQT

(1.293)

It is verified that the function fðTÞ fulfilling Eq. (1.293) is given by fðTÞ 5 φ0 I 1 φ1 T 1 φ2 T2

(1.294)

where φ0 ; φ1 ; φ2 can be expressed by the invariants I, II, and III of T in Eq. (1.245), because they are scalar functions (cf. Hashiguchi, 2017a). In the particular case in which f is the linear function of the tensor T, Eq. (1.294) reduces to fðTÞ 5 λðtrTÞI 1 2μT

(1.295)

where λ and μ are the material constants, called the Lame´ constants, regarding f and T as the stress and the strain, respectively, in the linear elastic equation. Eq. (1.295) is rewritten as fðTÞ 5 C : T

(1.296)

where C  λI 1 μS

ðCijkl  λδij δkl 1 μðδik δjl 1 δil δjk Þ=2Þ

(1.297)

Eq. (1.296) is obtained also by multiplying the fourth-order isotropic tensor in Eq. (1.236) by T, noting that the term with the antisymmetrizing tensor A vanishes because of the symmetry of the tensor T.

1.7 Differential formulae Partial differential formulae are shown in this section, which will be used often in the solid mechanics.

1.7.1 Partial derivatives of tensor functions Partial derivatives which will appear in the subsequent chapters are listed below. 1. n @Tn X e n2k e 1 Tn21 I e e n21 1 TT e n22 1 ? 1 Tn22 T 5 Tk21  0 T1 5 IT @T k51 n    n X @@T A 5 Tk21 Tn2k ik lj @T k51 ijkl

(1.298)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

44

1. Mathematical fundamentals

Examples for low-order tensors are shown below. 8 @Tij @Tji @T @TT > > e 5 T 5II 5I I; 5δik δjl ; 5δil δjk > 5 I5 II5II; > > @T @T @T @T kl kl > e > > > < @T2 @Tir Trj e 1 TI; e 5IT 5δik Tlj 1 Tik δlj @T @Tkl > > >

> > 3 > @ Tir Trs Tsj > @T 2 2 > e e e 5 δik Tls Tsj 1Tik Tlj 1Tir Trk δlj > : @T 5IT 1 TT 1 T I; @Tkl (1.299) For symmetric tensor T 5 TT , one can write @T 5 S; @T

@Tij 1 5 ðδik δjl 1 δil δjk Þ 2 @Tkl

(1.299) ’

2. @trTn 5 nðTn21 ÞT @T @ðTr0 r1 Tr1 r2 Tr2 r3 . . .Trn22 rn21 Trn21 r0 Þ @Tij 5 δir0 δjr1 T r1 r2 T r2 r3 . . .Trn22 rn21 T rn21 r0 1 T r0 r1 δir1 δjr2 T r2 r3 . . .T rn22 rn21 T rn21 r0 1 . . . 1 T r0 r1 T r1 r2 T r2 r3 . . .T rn22 rn21 δirn21 δjr0 5 T jr1 T r1 r2 T r2 r3 . . .T rn23 rn22 T rn22 i 1 T r0 i T jr3 T r3 r4 . . .T rn22 rn21 T rn21 r0 1 . . . 1 T jr1 T r1 r2 T r2 r3 . . .T rn22 i 5 nT jr1 T r1 r2 T r2 r3 . . .T rn23 rn22 T rn22 i (1.300)

Examples: @trT @I @I 5 5 5 I; @T @T @T

@trT2 @II 5 2TT ; 5 @T @T

@trT3 5 3ðT2 ÞT 5 3ðTT Þ2 @T (1.301)

3. @T 0ij @T0 5 I0 ; 5 I0 ijkl @T @Tkl 0 1 @T0ij @ðT 2 T δ 1 ij m ij @ 5 5 δik δjl 2 δij δkl A 3 @Tkl @Tkl

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.302)

1.7 Differential formulae

4.

pffiffiffiffiffiffiffiffiffiffiffiffi Tij @:T: T @ Trs Trs  t ðt: t 5 1Þ; 5 5 pffiffiffiffiffiffiffiffiffiffiffiffi 5 tij @T @T Trs Trs :T: ij pffiffiffiffiffiffiffiffiffi 2 2 @:T: @ trT 1 1 @trT 1 2T 5 pffiffiffiffiffiffiffiffiffi 5 pffiffiffiffiffiffiffiffiffi 5 2 trT2 @T 2 trT2 @T @T

45

(1.303)

5. @tij @t 1 1 5 ðI 2ttÞ; 5 pffiffiffiffiffiffiffiffiffiffiffiffi ðδik δjl 2 tij tkl Þ @T :T: @Tkl Trs Trs 0 1 1 0 pffiffiffiffiffiffiffiffiffiffiffiffi @tij Tij @ 1 @ @Tij pffiffiffiffiffiffiffiffiffiffiffiffi @ Trs Trs A C B pffiffiffiffiffiffiffiffiffiffiffiffi 5 5 Trs Trs 2 Tij C B @Tkl @Tkl Trs Trs Trs Trs @Tkl @Tkl C B C B     B p ffiffiffiffiffiffiffiffiffiffiffiffi Tij Tkl C 1 A @5 1 21=2 δik δjl Trs Trs 2Tij ðTrs Trs Þ Tkl 5 pffiffiffiffiffiffiffiffiffiffiffiffi δik δjl 2 Trs Trs Trs Trs Trs Trs  @t  : t50 @T (1.304) 6.  @t0ij @t0ij @t0 @t0 1 1  5 0 5 0 ðI0 2t0 t0 Þ; 5 0 5 0 I0 ijkl 2 t0ij t0kl @T @T :T : @Tkl @T kl :T : 0 1 0 0 0 @ @t 5 @t : @T 5 1 ðI0 2 t0 t0 Þ:I0 5 1 ðI0 2 t0 t0 ÞA @T @T0 @T :T0 : :T0 :

(1.305)

7. pffiffiffi 2 @cos 3θ pffiffiffi @cos 3θ 5 3 6t 0 ; 5 3 6t0ir t0rj 0 0 @t @t ij  @cos 3θ pffiffiffi @trt0 3  5 6 @t0 @t0 where θ is called the Lode’s angle defined as pffiffiffi cos 3θ  6trt03

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.306)

(1.307)

46

1. Mathematical fundamentals

8.

1 pffiffiffi 0 @cos 3θ 3 6 @1 0 2 1 2 52 0 :t : I1 pffiffiffi cos 3θt0 2 t0 A @T :T : 3 6 1 pffiffiffi 0 @cos 3θ 3 6 @1 0 0 1 trt ttr δij 1 pffiffiffi cos3θt0ij 2 t0tr t0rj A 52 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi @Tij 6 T 0pq T 0pq 3 0 @cos3θ B @T B B B @

1 pffiffiffi 0 2 1  1 @cos 3θ @t 0 0 5 II 2t 53 6 t : : t I2 C @t0 3 @T :T0 : C C pffiffiffi h i C 3 6 02 1 02 A 03 0 5 0 t 2 ðtrt ÞI2 ðtrt Þt 3 :T :

(1.308)

0

9. @T21 e 21 5 2 T21 IT21 ; 5 2 T21 T @T

@T21 ij @Tkl

21 21 21 5 2 T21 ik T lj 5 2 T ir I rskl T sj

(1.309) 10. 21 21 Trs_ Tsj _T021 5 2 T21_TT21 ; _T21 ij 5 2 Tir _

_

_

_

21

@I 5 ðT21 TÞ 5 T T 1 T21 T 5 O;

@T21 ij

1

A Tkl T21 _Tkl 5 2 T21 ik _ lj @Tkl

(1.310)

11. @II @trðcofTÞ 5 5 ðtrTÞI 2 TT ; @T @T

@II @trðcofTÞ 5 5 Trr δij 2 Tji @T ij @T ij

(1.311)

noting Eqs. (1.17), (1.25), and (1.242). 12. @ðcofTÞij @T kl

5 εikq εjls Tqs

0 B@ðcofTÞ ij B 5 B @ @T kl

1 @ εipq εjrs Tpr Tqs 2 @Tkl

1 5

C 1 C ðεipq εjrs δkp δlr Tqs 1 εipq εjrs Tpr δkq δls ÞC A 2 (1.312)

13. @III @detT 5 5 cofT 5 ðdetTÞT2T ; @T @T

@III @detT 5 5 ðcofTÞij 5 ðdetTÞT21 ji @T ij @T ij (1.313)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

47

1.7 Differential formulae

which is derived already in Eq. (1.22) but can be derived in the alternative way as follows: 2 0 13 @III @ 1 1 1 : T 5 4 @ tr3 T 2 trTtrT2 1 trT3 A5 : T @T @T 6 2 3 2 3 3 1 T 5 4 ðtr2 TÞI 2 ItrT2 2 ðtrTÞTT 1 ðT2 Þ 5 : T 6 2   3 1 2 2 3 5 tr2 TtrTT 2 ðtrTÞtrTT 2 ðtrTÞtrTT 1 trTT 6 2 5

3 3 3 tr T 2 ðtrTÞtrT2 1 trT3 5 3III 5 IIIT2T : T 6 2

noting Eq. (1.243). It follows from Eq. (1.313) that pffiffiffiffiffiffiffiffiffiffiffi @ detT 1 pffiffiffiffiffiffiffiffiffiffiffi 2T 5 detT T @T 2 14.

(1.314)

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi @ln detT 1 21 @ln detT 1 21 5 T ; 5 Tij @T 2 @Tij 2 0 pffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffi @ln detT 1 @ detT 1 1 @ 5 pffiffiffiffiffiffiffiffiffiffiffi 5 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ðdetTÞT21 A @T detT @T detT 2 detT

(1.315)

15. N n N n X @expT X 1X @expT 1X e n2k ; ð 5 Þijkl 5 Tk21 T ðTk21 Þik ðTn2k Þlj @T n! @T n! n51 n51 k51 k51

(1.316) noting

@expT @ I 1 T 1 5 @T

1 @T

1 2 2! T

1 3 3! T

1?



due to Eq. (1.264) with Eq. (1.298). 16. @½ fðAÞB @T

0 1 @fðAÞ @B @ @fðAÞ @BA 5 B 1 fðAÞ 6¼ B 1 fðAÞ @T @T @T @T 0 1 @@½ fðAÞBij  5 Bij @fðAÞ 1 fðAÞ @Bij A @Tkl @Tkl @Tkl

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.317)

48

1. Mathematical fundamentals

1.7.2 Time derivatives in Lagrangian and Eulerian descriptions The physical quantity ψ in materials can be described by the position vector X of material particle in the reference (initial) state (configuration) and the current time t as ψðX; tÞ or by the position vector x of material particle in the current state and the current time t as ψðx; tÞ. The former and the latter are called the Lagrangian (material) description (representation) and the Eulerian (spatial) description, respectively. Now, the rate of the physical quantity ψ in a particular material particle can be observed, moving with material particle and it is referred to as the material-time derivative, which is given by the total differentiation with respect to time. The material-time derivative in the Lagrangian description is defined by @ψðX; tÞ  _ψ  Dψ Dt @t

(1.318)

On the other hand, the material-time derivative in the Eulerian description is given by dψðx; tÞ @ψðx; tÞ @ψðx; tÞ @x @ψðx; tÞ  5 1  @t 5 @t 1 vðx; tÞ  gradψðx; tÞ _ψ  Dψ Dt dt @t @x (1.319) The first term in Eq. (1.319) signifies the nonsteady (or local time derivative) term describing the variation of the quantity in a spatially fixed point x with time, so that it is called the spatial-time derivative. The second term signifies the convective term describing the variation attributable to the movement of material particle under the existence of the gradient @ψðx; tÞ=@x due to the heterogeneity in distribution of ψ in the space. The symbol ð Þ or Dð Þ=Dt is used to specify the materialtime derivative, that is, the total time derivative. The material-time derivatives of vector a and second-order tensor T in the Eulerian and the Lagrangian descriptions are given from Eq. (1.319) as



@aðX; tÞ daðx; tÞ @aðx; tÞ @aðx; tÞ @aðx; tÞ 5 5 5 1v 5 1 v  grada _a 5 Da Dt @t dt @t @x @t (1.320) @TðX;tÞ dTðx; tÞ @Tðx;tÞ @Tðx;tÞ @Tðx; tÞ 5 5 5 1v  5 1 v  gradT _T 5 DT Dt @t dt @t @x @t (1.321) Motion of perfect fluid is independent of a deformation history from the initial configuration and thus only a current spatial flow (motion) of physical quantity is meaningful. Then, the Eulerian description based

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

1.7 Differential formulae

49

on the spatial-time derivative @ψðx; tÞ=@t may be employed in fluid mechanics. On the other hand, the physical properly in solids depend on the deformation history except for the isotropic rigid-perfectly plastic material and thus they must be described moving with a material particle in a space. Then, in solid mechanics the material-time derivative ψ must be used. Further, however, the convected derivative based on the rate of physical quantity observed moving and deforming with a material, which will be described in Section 3.5 must be used for the tensorvalued variables: the stress and internal variables in constitutive relations of solids. On the other hand, the material-time derivative can be used for constitutive relations formulated in the configuration which is independent of the rigid-body rotation of material as will be described in the multiplicative hyperelastic-based plasticity formulated in the intermediate configuration in Chapters 8 and 9.

_

1.7.3 Derivatives of tensor field Various derivatives in the tensor field are shown collectively below, where use is made of the operator r5

@ @ 5 er @x @xr

(1.322)

which is called the nabla or del or Hamilton operator. 1.7.3.1 Gradient Scalar field: grads 5 rs 5 Vector field: 8 > > vr > < gradv 5 > > > rv :

@s ei @xi

@ @vi ej 5 ei ej : right form @xj @xj @vj @ ei vj ej 5 ei ej : left form 5 @xi @xi

(1.323)

5 vi e i 

Second-order tensor field: 8 @ > > Tr 5 Tij ei ej  ek 5 > < @xk gradT 5 @ > > > : rT 5 @x ei Tjk ej ek 5 i

(1.324)

@Tij ei ej ek : right form @xk (1.325) @Tjk ei ej ek : left form @xi

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

50

1. Mathematical fundamentals

1.7.3.2 Divergence Vector field:





divv 5 r v ð 5 v rÞ 5 vi ei Second-order tensor field: 8 @ > > Tr 5 Tij ei ej ek 5 > > < @xk divT 5 > @ > > rT 5 ei Tjk ej ek 5 > : @xi

i  @x@ j ej 5 @v @xi

(1.326)

@Tij ei : right form @xj @Tji ei : left form @xj

1.7.3.3 Rotation (or curl) Vector field: 8 @ @vi > > v 3 r 5 vi ei 3 ej 5 εijk ek : right form > > @xj @xj < rotv 5 @ @vi > > > r3v5 ei 3 vj ej 5 εjik ek 5 2 v 3 r : left form > : @xj @xj Second-order tensor field: 8 @ > > T 3 r 5 Tij ei ej 3 ek > > @xk > > > > > > > @Tij @Tij > > 5 ei ðej 3 ek Þ 5 εjkr ei er : right form > > @xk @xk < rotT 5 @ > > > ei 3 Tjk ej ek r3Τ5 > > @x > i > > > > > @Tjk @Tjk > > > 5 ðei 3 ej Þek 5 εijr er ek : left form > : @xi @xi

(1.327)

(1.328)

(1.329)

The symbol r is regarded as a vector, and the scalar product of itself, that is,



Δ  r2  r r 5

@ er @xr

 @x@ s es 5 @x@r @xr 2

(1.330)

has the meaning of r2( )div(grad( )). The symbol Δ is called the Laplacian or Laplace operator, which is often used for scalar or vector fields as

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

1.7 Differential formulae

Δs 5

@2 s ; @xr @xr

Δv 5

@2 v s es @xr @xr

51 (1.331)

Various formulae are derived from Eqs. (1.323)(1.329). The ones appearing often in continuum mechanics are shown below, while Eqs. (1.25), (1.39), and (1.40) are used for the derivations.  8 vgrads 1 svr : right form > > gradðsvÞ 5 > > > gradsv 1 srv : left form > > > > gradðu vÞ 5 ðgraduÞ v 1 ðgradvÞ u > > > > > > divðsvÞ 5 sdivv 1 v grads > > > > > divðu 3 vÞ 5 v ðr 3 uÞ 1 u ðr 3 vÞ > >  > < ðsTÞr 5 Τgrads 1 sðTrÞ : right form divðsTÞ 5 rðsTÞ 5 ΤT grads 1 sðrΤÞ : left form > > > > > > divðTvÞ 5 ΤT : gradv 1 ðrΤÞ v > > 8 > > > ðu 3 vÞ 3 r > > > > > > > < > 5 ðdivuÞv 2 divvÞu 2 ðgraduÞv 1 ðgradvÞu: right form > > rotðu 3 vÞ 5 > > > > r 3 ðu 3 vÞ 5 2 ðu 3 vÞ 3 r > > > > : : 5 2 ðdivvÞu 1 divvÞu 1 ðgraduÞv 2 ðgradvÞu: left form















(1.332) Hereinafter, only the symbols grad( ) and div( ) are used for the ones in which there does not exist the difference between the right and left forms.

1.7.4 Gauss’ divergence theorem The transformation rule of the volume integration of physical quantity to the surface integration is shown below. Suppose the infinitesimally slender prism cut by the four planes perpendicular to the x2 -axis and x3 -axis from a zone inside the material. The following equation holds for the scalar- or the vector- or the tensorvalued quantity ψ(x) in the prism. ð ð @ψ @ψ x1 dv 5 dx1 dx2 dx3 5 ½ψx211 dx2 dx3 5 ðψ1 2 ψ2 Þ dx2 dx3 (1.333) @x @x 1 1 v v where ( )1 and ( )2 designate the values of physical quantity at the maximum and the minimum x1 -coordinates, respectively. The following relations hold. 1 2 2 n1 1 da 5 dx2 dx3 ; n1 da 5 2 dx2 dx3

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.334)

52

1. Mathematical fundamentals

ð

@ψ 2 1 1 1 2 2 2 1 2 2 dv 5 ψ1 n1 1 da 2 ψ ð2 n1 da Þ 5 ψ n1 da 1 ψ n1 da v @x1

Then, the following relation holds for the whole zone. ð ð @ψ dv 5 ψn1 da v @x1 a

(1.335)

(1.336)

Similar equations are obtained also for the x2 - and x3 -directions, and one has ð ð @ψ dv 5 ψni da (1.337) v @xi a which is referred to as the Gauss’ theorem or simply divergence theorem. The following equations for the scalar s, the vector v and the second tensor T are derived from Eq. (1.337). ð ð ð ð @s dv 5 sni da; divsdv 5 snda (1.338) v @xi a v a ð ð ð ð @vi dv 5 vi ni da; divvdv 5 v nda (1.339) v @xi a v a ð ð ð ð @Tij dv 5 Tij ni da; divTdv 5 Tnda (1.340) v @xi a v a



1.7.5 Material-time derivative of volume integration The zone of material occupying the volume v at the current moment ðt 5 tÞ changes to occupy the volume v 1 δv after the infinitesimal time ðt Ð 5 t 1 δtÞ. Then, the material-time derivative of the volume integration v ψðx; tÞdv of the scalar- or tensor-valued physical quantity ψðx; tÞ involved in the volume is represented by the following equation. ð 

ð  ð 1 ψðx; tÞdv 5 lim ψðx; t 1 δtÞdv 2 ψðx; tÞdv δt-0 δt v v1δv v

ð  ð 1 5 lim fψðx; t 1 δtÞ 2 ψðx; tÞgdv 1 ψðx; t 1 δtÞdv δt-0 δt v δv (1.341)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

1.8 Variations of geometrical elements

which leads to

ð

 ψðx; tÞdv

v

ð 5

_

½ψðx; tÞ 1 ψðx; tÞdivvdv v

53

(1.342)

which is referred to as the Reynolds’ transportation theorem. Eq. (1.342) can also be obtained by the following manner. ð  ð  ð _ ψðx; tÞdv 5 ψðX; tÞJdV 5 ðψðX; tÞJ 1 ψðX; tÞJÞdV v V v 2 3 ð @v _ tÞ 1 ψðx; tÞ r 5dv 5 4ψðx; @xr v

_

where V is the initial volume of zone and it is set that J  dv=dV, exploiting Eq. (1.367) which will be formulated in the next section.

1.8 Variations of geometrical elements Variations of line, surface, and volume elements and their rates under the deformation are described in this section.

1.8.1 Deformation gradient and variations of line, surface and volume elements Variations of line, surface, and volume elements are described below. Relation of the current infinitesimal line element dx to the initial infinitesimal line element dX is represented as dx 5 FdX; dX 5 F21 dx

(1.343)

where F

@x @X

(1.344)

which is referred to as the deformation gradient tensor. The following expression holds by introducing the polar decomposition for the deformation gradient tensor in Eq. (1.286) into Eq. (1.343). dx 5 RUdX 5 VRdX;

dX 5 U21 RT dx 5 RT V21 dx

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.345)

54

1. Mathematical fundamentals

The indices in the transposed and the inversed tensors of the deformation gradient tensor are expressed as follows: 8 @x @xi > 5 F5 ei ej > > > @X @Xj > > > 0 1T 0 1T > > > > > @xj @x @xi @xi > > FT 5 @ A 5 @ ei ej A 5 ej ei 5 ei ej > > < @X @Xj @Xj @Xi (1.346) @X @Xi > 21 > 5 F 5 e e > i j > > @x @xj > > > 0 1T > > > > > @Xj @X > 2T 21 T @ i A 5 > ei ej > : F 5 ðF Þ 5 @xj ei ej @xi noting FF21 5

@xi @Xr @xi @Xj ei ej er es 5 ei es 5 δis ei es 5 I @Xj @xs @Xj @xs

Incidentally, it follows for the time-derivative of the deformation gradient from Eq. (1.346) that 0 1 8 > @x @v @vi > > > 5 F5@ A 5 ei ej > > @X @X @X > j > > > 0 1T 0 1T > > > > @vj > @v @v T i > F 5 ðFÞT 5 @ A 5 @ > ei ej A 5 ei ej > > @X @X @X < j i 0 121 0 121 (1.347) > > > 21 @v @v @X i i 21 > > F 5 ðFÞ 5 @ A 5 @ ei ej A 5 ei ej > > @X @Xj @vj > > > > 0 1T > > > > > 2T @X > F 5 ðF21 ÞT 5 @ i e e A 5 @Xj e e > > i j i j : @vj @vi

_ _

_

_

_

_

noting @X

@vi @Xr @vi j ei ej er es 5 ei es 5 δis ei es 5 I _F_F21 5 @X @vs @Xj @vs j

(1.348)

The current position vector is related to the reference position vector by use of the displacement vector u as x 5 X 1 u 5 ðXi 1 ui Þei

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.349)

55

1.8 Variations of geometrical elements

Then, the deformation gradient is represented as follows:   @ðX 1 uÞ @u @ui 5I1 5 δij 1 F5 ei ej @X @X @Xj

(1.350)

from which it follows that dF 5

@du @dui 5 ei ej @X @Xj

(1.351)

The current and the reference infinitesimal volume elements dv and dV formed by the infinitesimal line elements dxa ; dxb ; dxc and dXa ; dXb ; dXc are related by dv5½dxa dxb dxc 5½FdXa FdXb FdXc 5detF½dXa dXb dXc 5ðdetFÞdV (1.352) by virtue of Eq. (1.262)3. Then, the ratio of the current to the reference infinitesimal volume elements, that is, Jacobian is given by J 5 detF 5 detU 5 detV 5

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi dv ρ 5 0 detC 5 detb 5 dV ρ

(1.353)

pffiffiffiffi pffiffiffiffiffiffiffiffi noting Eq. (1.189), while the physical meanings of U 5 C 5 FT F and pffiffiffi pffiffiffiffiffiffiffiffi V 5 b 5 FFT which are defined by substitution of F to T in Eq. (1.282) will be delineated in Section 4.1. ρ0 and ρ are the mass densities in the reference and the current configurations, respectively. The following equalities hold for the infinitesimal volume elements, designating the infinitesimal reference and current surface element vectors as dA 5 dXa 3 dXb and da 5 dxa 3 dxb , respectively, noting Eq. (1.141).  da dxc dv 5 (1.354) JdV 5 JdA dXc 5 JdA F21 dxc 5 JF2T dA dxc









where the area vectors da and dA in the current and the reference configurations are given by da 5 dan;

dA 5 dAN

(1.355)

designating the infinitesimal area and the unit outward normal as ðda; nÞ and ðdA; NÞ in the current and the reference configurations, respectively. The following relations are obtained from Eq. (1.354).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

56

1. Mathematical fundamentals

n = F –T N J dA da F

N

da = F –T N · n JdA

dA

dx = Fd X dv= JdV

dX

dV

Current configuration Reference configuration

FIGURE 1.2 Variation of line, area, volume, and unit normal vector.

da 5 JF2T dA 5 ðcof FÞdA;

dA 5 FT da=J 5 ðcof FÞ21 da

(1.356)

noting cof F 5 JF2T

(1.357)

by virtue of Eq. (1.185), leading to

n 5 F2T NJdA=da;



FT n Nda J

(1.358)

N 5 FT nda=ð JdAÞ

(1.359)



da 5 F2T N nJdA;

dA 5

Eq. (1.358) is referred to as the Nanson’s formula. The variations of the infinitesimal line element vector, the unit normal vector, the infinitesimal surface area element, and the infinitesimal volume element are illustrated in Fig. 1.2. Here, the following Euler’s formula holds for the cofactor. rX ðcofFÞ 5

@cofF 5 0; @X

@ðcofFÞap @Xp

5

@ 12 εabc εpqr Fbq Fcr 50 @Xp

(1.360)

noting Eq. (1.17) and that Xp is not contained in the component ðcofFÞap .

1.8.2 Velocity gradient and rates of line, surface and volume elements Rates of line, surface, and volume elements are described below.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

57

1.8 Variations of geometrical elements

Differentiating Eq. (1.343), we have

_





ðdxÞ 5 FdX; ðdxÞ 5 ldx

(1.361)

where l  gradv 5 vrx 5

@v 5 FF21 @x

_

(1.362)

noting

_

_

@v @x @x @X 5 5 @x @x @X @x

(1.363)

The tensor l is called the velocity gradient tensor. The following relation

_

_21 5 l 1 F_F21 5 O that

is derived from ðFF21 Þ 5 FF21 1 FF 

_21 5 2 l 5 l21

FF

(1.364)

It follows from Eqs. (1.326) and (1.353) that

_



ðdvÞ 5 ðtrlÞdv 5 J dV

(1.365)

trl 5 divv

(1.366)

_J 5 Jtrl

(1.367)

noting :_ F 5 ðdetFÞF2T :_ F 5 ðdetFÞtrðF21_ FÞ 5 ðdetFÞtrð_ FF21 Þ _J 5 ðdetFÞ 5 @detF @F 

with Eq. (1.313). Further, it follows from Eq. (1.357) that ðcof FÞ 5 elcof F

(1.368)

el  ðtrlÞI 2 lT

(1.369)



where

noting

_

_2T J 5 _JJ cof F 1 _F2T FT cof F 5 _JJ cof F 2 F2T_FT cof F

ðcof FÞ 5 J F2T 1 F 

el is referred to as the surface strain rate tensor. Further, differentiating Eq. (1.356) and noting Eq. (1.368), we have

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

58

1. Mathematical fundamentals   ðdaÞ 5 ðcof FÞ dA 5 elðcof FÞdA

which is rewritten as ðdaÞ 5 elda 

(1.370)

in the spatial description. Now, we express the surface vectors as follows: da  nda;

dA  NdA

(1.371)

where da and dA are the infinitesimal areas and n and N are the unit normal vectors of the current and the reference infinitesimal surface vectors, respectively. Here, noting n n 5 0 from n n 5 1 for the unit vector n, it follows that

_ 











_



ðdaÞ 5 n nðdaÞ 5 n ½ðndaÞ 2 nda 5 n ðdaÞ



(1.372)

Substituting Eq. (1.370) into Eq. (1.372), one obtains the rate of the current infinitesimal area as follows:



 ðdaÞ 5 n elda

(1.373)

or ðdaÞ 5 ðtrl 2 n dnÞda



(1.374)

d  sym½l

(1.375)



where



noting n wn 5 0 because of Eq. (1.166)2, where w  ant½l

(1.376)

Here, d and w are called the strain rate tensor and the continuum spin tensor, respectively. Further, the rate of the unit normal of the current surface element is given from Eqs. (1.369), (1.373), and (1.374) as follows:

_n 5 ½ðn  dnÞI 2 lT n 5 ½ðn  lnÞI 2 lT n noting

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(1.377)

59

1.8 Variations of geometrical elements

_nda 5 ðndaÞ 2 nðdaÞ 





5 ½ðtrlÞI 2 lT nda 2 n½ðtrlÞ 2 n dnda T



5 2 l nda 1 n dnda The formula for the physical quantities on the geometrical variation of material formulated in this section will be often used in the subsequent chapters.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

C H A P T E R

2 Curvilinear coordinate system The physical meanings of finite strain (rate) tensors are captured clearly by their representations in the embedded (convected) coordinate system as will be described in the subsequent chapters, since the components in the primary bases are kept constant. The embedded coordinate system changes to a curvilinear coordinate system even if it is the rectangular coordinate system at the initial state of deformation process. Then, mathematics in the general curvilinear coordinate system, that is, the primary and the reciprocal base vectors, the representations and the coordinate transformations of vector and tensor, the metric tensor, the scalar, the vector, and the tensor products, etc. are addressed in this chapter prior to the explanation for the description of deformation in the convected coordinate system in the subsequent chapters.

2.1 Primary and reciprocal base vectors The following reciprocal base vectors g1 ; g2 ; g3 are defined by the primary base vectors g1 ; g2 ; g3 as follows: gi 5

g j 3 gk 1 εijk ; that is; 2 vg

g1 5

g2 3 g3 g 3 g1 g 3 g2 ; g2 5 3 ; g3 5 1 ½g1 g2 g3  ½g1 g2 g3  ½g1 g2 g3  (2.1)

noting Eq. (1.66) and ε1 jk gj 3 gk 5 2g2 3 g3 for example, where vg is the volume of parallelopiped made up of the primary vectors g1 ; g2 ; g3 in the right-hand coordinate system, that is, vg  ½g1 g2 g3 

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity DOI: https://doi.org/10.1016/B978-0-12-819428-7.00002-X

61

(2.2)

© 2020 Elsevier Inc. All rights reserved.

62

2. Curvilinear coordinate system

noting Eq. (1.50). The inverse expression of Eq. (2.1) is given by gi 5

1 εijk vg gj 3 gk ; 2

g1 5

i:e:;

g2 3 g3 ; ½g1 g2 g3 

g2 5

g3 3 g1 ; ½g1 g2 g3 

g3 5

g1 3 g2 ½g1 g2 g3  (2.3)

noting Eq. (1.66), where it is satisfied noting Eq. (1.67)3 that ½g1 g2 g3  ½g1 g2 g3  5 1

(2.4)


 1 5 g1 g2 g3 vg

(2.5)

leading to

Note Consider the simplest case that the primary vectors g1 ; g2 ; and g3 are orthogonal and have the length 2 m each. Then, the volume of the rectangular parallelopiped formed by them is V 5 8 m3 and the reciprocal vectors g1 ; g2 ; and g3 are also orthogonal and have the length 1=2 m each, which form the rectangular parallelopiped possessing the volume ½g1 g2 g3  5 1=8 m3 , so that Eq. (2.5) is satisfied. Here, note that the primary and the reciprocal basis in Eqs. (2.1) and (2.3) satisfy the following relations.





(2.6)

gi 3 gi 5 0

(2.7)

j

gi gj 5 δi ;

gi gi 5 3

which are equivalent to Eqs. (1.67)1,2 and (1.68). Now, we consider the general curvilinear coordinate system fθi g with the primary base fgi g. The infinitesimal line element dx at the material point vector x is described as follows: dx 5 dxi ei 5 dθi gi

(2.8)

where





dxi 5 dx ei ; dθi 5 dx gi

(2.9)

by virtue of Eq. (1.72). It follows from Eq. (2.8) with Eq. (2.6) that





dx dx 5 dxi dxi 5 dθi dθj gi gj

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(2.10)

2.1 Primary and reciprocal base vectors

63

FIGURE 2.1 Curvilinear coordinate system with primary base fgi g.

Further, one has the expression dx 5

@x @x dxi 5 i dθi @xi @θ

(2.11)

Then, the following expressions are derived from Eqs. (2.8) and (2.11). ei 5

@x ; @xi

gi 5

@x @θi

(2.12)

The reciprocal base vector gk is always perpendicular to the coordinate plane formed by the primary base vectors ðgi ; gj Þ ðk ¼ 6 i; k 6¼ jÞ. The reciprocal base vectors g1 and g2 are illustrated in Fig. 2.2A as the twodimensional observation in the state that the base vectors g3 and g3 are taken to be perpendicular to the common coordinate plane ðg1 ; g2 Þ or ðg1 ; g2 Þ. The relations of the base vectors are illustrated in Fig, 2.2B. Note, however, that fgi g and fθi g are merely defined momentarily from the primary base vector fgi g. Therefore a continuous coordinate system fθi g with the reciprocal vector base fgi g does not exist when the coordinate system fθi g with the primary base fgi g is a curvilinear coordinate system. The transformation rules of the primary and the reciprocal base vectors between the two different primary coordinate systems ðgi ; θi Þ and i ðgi ; θ Þ are given by 8 j > @θj @θ > > > gi 5 i gj ; gi 5 i gj > > @θ < @θ (2.13) i > @θ j @θi j > i > i >g 5 j g ; g 5 jg > > : @θ @θ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

64

2. Curvilinear coordinate system

FIGURE 2.2 Reciprocal base vectors. (A) Reciprocal base vectors g1 and g2 in two dimensional representation g3 =:g3 : 5 g3 =:g3 : . (B) Reciprocal base vector g3.

noting @x

@x @θj @θj gi 5 i 5 j i 5 gj i ; @θ @θ @θ @θ

j

j

@x @x @θ @θ gi 5 i 5 j i 5 gj i @θ @θ @θ @θ

for the primary base by virtue of Eq. (2.12) and 8 i i > @θ @θ j > j j > g g 5 g g 5 g g  gi gi > i i > < j @θj @θj







> @θi > j > g g 5 g gj 5 gi > j i > : j @θ







(2.14)



@θi @θ

j



(2.15)

g  gi g j

i

for the reciprocal base which is required to satisfy this reciprocal relation to the primary base with





gj gj 5 gi gi 5 3

(2.16)

The following expressions hold from Eq. (2.13). 8 ir i > < gi 5 gir er ; g 5 g~ er ~ ri r > : ei 5 g~ gr 5 gri g ~ where

(2.17)

8 ir i > < gir  gi er ; g~  g er ~ j jr j j j > : gir g~ 5 ðgi er Þðg er Þ 5 gi ðg er Þer 5 gi g 5 δi ~









 



Further, introducing the second-order tensor

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(2.18)

65

2.2 Metric tensor and base vector algebra

it can be written that

8 > g  gi  ei 5 gir er  ei g21 5 ei  gi 5 gir ei  er : ~ ~

(2.19)

8 < gi 5 g ei ; ei 5 g21 g ~ ~ i : gi 5 ei g21 ; ei 5 gi g ~ ~

(2.20)

The tensors in Eq. (2.19) satisfy ( gi  ei ej  gj 5 gi  gi 5 g 21 gg 5 g e  e g~ js e  e 5 δs e  e 5 e  e 5 I s s r r ir r i j r r ~~ ~ where g is the metric tensor, which will be defined in Eq. (2.28).

2.2 Metric tensor and base vector algebra Now, we introduce the scalar quantities



gij  gi gj ;

j



j

gi  gi gj 5 δi ;





gij  gi gj 5 δij ;

gij  gi gj

(2.21)

noting Eqs. (2.6) and (2.13), where the symmetries gij 5 gji ; gij 5 gji ; j gij 5 gi hold. Here, note that gij and gij are necessary to express the length of the line element and the angle between line elements, which are the most basic geometrical ingredients as will be shown in Eqs. (2.53) and (2.54). Then, they are called the metric tensors. It follows from Eq. (2.21) that j

j

gir grj 5 δi ; ½gir ½grj  5 ½gir grj  5 ½δi 

(2.22)

noting







j

gir grj 5 ðgi gr Þ ðgr gj Þ 5 gi gj 5 δi or gir grj 5

@θr @θj @θj j 5 5 δi @θi @θr @θi

Hence, we have ij 21 gij 5 g21 ij ; ½g  5 ½gij 

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(2.23)

66

2. Curvilinear coordinate system

from which it follows that       det gij det gij 5 1 ðno sumÞ; det gij 5

1 detðgij Þ

(2.24)

and   1 1 g  det ð gij Þ 5 ½g1 g2 g3 2 5 v2g ; det gij 5 5 2 g vg

(2.25)

noting Eqs. (1.21) or (1.189)4, (1.202), and (1.205). Eq. (2.23) means that the matrices with the components gij and gij are inverse tensors to each other, while this property is based on their non-singularities, that is, ½gij  6¼ 0 and ½gij  6¼ 0. Further, the following relation holds. @g 5 ggji @gij

(2.26)

noting @g 5 gg21 ji @gij by virtue of Eqs. (1.313) and (2.23).By use of the notation in Eq. (2.21), the following transformation rules between the primary and the reciprocal base vectors hold. gi 5 gir gr ;

gi 5 gir gr

(2.27)

It is known from Eq. (2.27) that gij and gij play the role of the shifter that makes the index go up and down, respectively. The generalized identity tensor in the general coordinate system is given from Eq. (1.74) as follows: gð5 ei  ei Þ 5 gi  gi 5 gi  gi 5 gij gi  gj 5 gij gi  gj ð5 gT Þ fulfilling jjgjj 5

pffiffiffi 3; detg 5 1;

g 5 gT 5 g21

(2.28)

(2.29)

The fact that the tensor g is the generalized identity tensor is confirmed noting Eq. (1.72) as follows:   gi  gi v 5 ðv gi Þgi gv 5 5v (2.30) gi  gi v 5 ðv gi Þgi

 

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

67

2.2 Metric tensor and base vector algebra

The components of g shown in Eq. (2.28) are necessary for the description of the geometrical elements, for example, the length of line element and the angle between line elements in the general space as will be shown in Section 2.3. Then, it is called the metric tensor. It holds that 8 

i 1 ij i i > > > sym g  gi 5 sym gi  g 5 2 g g  gj < (2.31)

i  1 > i j i > sym g  g  g g g  g 5 sym g 5 > : i i 2 ij

_

_

_

_

_

_

8
i 
1 i j i i j i > > > ant g  gi 5 2 ant gi  g 5 2 ðg g 2 g g Þg  gj < 
i  1
> i j i > > : ant gi  g 5 2 ant g  gi 5 2 ðgi gj 2 gi gj Þg  g

_ _

_

_ _ 

_

_ _ 

(2.32)

noting

 _ _ _ _ _ 2sym½_ gi  gi 5 _ gj  gj 1gi  _ gi 5ð_ gj  gi Þgi  gj 1gi  ð_ gi  gj Þgj 5ðgi  gj Þ gi  gj 2sym½gi  gi 5 gj  gj 1gi  gi 5ðgj gi Þgi  gj 1gi  ðgi gj Þgj 5ðgi gj Þ gi  gj 



_ _ _ _ _ i j i i j i 2ant½_ gi  g 5 _ gj  g 2g  _ gi 5ð_ gj  gi Þg  g 2g  ð_ gi  gj Þgj 2ant½gi  gi 5 gj  gj 2gi  gi 5ðgj gi Þgi  gj 2gi  ðgi gj Þgj

Further, Eqs. (2.1) and (2.3) are written as gi 5

1 ijk e gj 3 gk ; 2

gi 5

1 eijk gj 3 gk 2

(2.33)

and one has the equations from Eqs. (2.2) and (2.5) ½gi gj gk  5 eijk ;

½gi gj gk  5 eijk

(2.34)

introducing the symbols eijk  εijk vg ;

eijk 

εijk vg

(2.35)

where the following transformation rules exist. eijk 5 gir gjs gkt erst ;

eijk 5 gir gjs gkt erst

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(2.36)

68

2. Curvilinear coordinate system

noting εijk 5 vg detðgpq Þεijk 5 vg gir gjs gkt εrst 5 gir gjs gkt vg εrst vg vg εijk 5

1 1 1 detðgpq Þεijk 5 gir gjs gkt εrst 5 gir gjs gkt εrst vg vg vg

with the aid of Eqs. (1.19) and (2.25). The following relations hold by virtue of Eq. (1.25). 8 ijk > < e eirs 5 ejki ersi 5 δjr δks 2 δjs δkr eijk eijs 5 2δks > : ijk e eijk 5 3! 5 6

(2.37)

Further, the vector products of base vectors which are the generalization of Eq. (2.7) are given by gi 3 gj 5 eijk gk ;

gi 3 gj 5 eijk gk

(2.38)

noting 8 1 1 1 > > g 3gj 5 ðgi 3gj 2gj 3gi Þ5 ðδir δjs 2δis δjr Þgr 3gs 5 ekij ekrs gr 3gs 5ekij gk > < i 2 2 2 1 i 1 1 kij > j j i r s r s kij i j > > : g 3g 5 2 ðg 3g 2g 3g Þ5 2 ðδir δjs 2δis δjr Þg 3g 5 2 e ekrs g 3g 5e gk by virtue of Eqs. (1.43), (2.33), and (2.37).

2.3 Tensor representations The vector is described in Eq. (1.72) as





v 5 ðv gi Þgi 5 ðv gi Þgi ð 5 gi  gi v 5 gi  gi vÞ

(2.39)

v 5 vi gi 5 vi gi

(2.40)

leading to

with





vi 5 v gi ; vi 5 v gi

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(2.41)

69

2.3 Tensor representations

v2 g2 θ2 g 2 ( ⊥ g1 ) v 2 g2

v

v 2 a2 g2 a 2 11g a a1 v 1 g1

g 3 (// g 3)

θ1

g1 ( ⊥ g 2 )

v1 g1

FIGURE 2.3 Vector representation in general coordinate system (two-dimensional observation in the state g3 =:g3 : 5 g3 =:g3 :).

which is shown in Fig. 2.3 in the two-dimensional state. The relation of vi and vi are given as vi 5 gir vr ;

vi 5 gir vr

(2.42)

noting



vi 5 vr gr gi ;



vi 5 vr gr gi

The metric tensor component plays the role of shifter as known from Eq. (2.42). The vector is described by the other bases as follows: v 5 vi gi 5 vi gi which is rewritten by Eq. (2.15) as 8 j > > > vi g 5 vi @θ g 5 > j > < i @θi v5 > @θi j > > vi gi 5 vi g 5 > j > : @θ and

v5

(2.43)

i

@θ j v gi ð 5 vi gi Þ @θj @θj

(2.44)

v gi ð 5 vi gi Þ i j @θ

8 @θj @θi > > > vi gi 5 vi i gj 5 j vj gi ð 5 vi gi Þ > > < @θ @θ i j > @θ j @θ > i > > v g 5 vi j g 5 i vj gi ð 5 vi gi Þ > : i @θ @θ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(2.45)

70

2. Curvilinear coordinate system

The comparisons of these equations with Eq. (2.40) lead to the following transformation rules of components. 8 i i > > > vi 5 @θ vj ; vi 5 @θ vj > > j j > @θ < @θ (2.46) j > @θj @θ > > > vi 5 i vj ; vi 5 i vj > > : @θ @θ where the right equations are derived analogously to the left equations but they can be derived also from the left ones as follows: 8 j

> @θi r @θi @θ r > ir r i > > v 5 δ v 5 v 5 rv > j > @θr < @θ @θ (2.47)

j j > @θr @θr @θ @θ @θr > ir > > vr vi 5 δ vr 5 i vr 5 j i vr 5 i > > : @θ @θ @θj @θ @θ Here, note that the primary base vector gi and the component vi are transformed by the same rule and the reciprocal base vector gi and the component vi are transformed by the same rule as seen in Eqs. (2.13) and (2.46). The former gi and vi obey the transformation rule of the variation of the function f, i.e. @f=@θi along the coordinate system with the primary base vectors, which is called the covariant derivative conventionally as known j

j

j

j

from @f=@θi 5 ð@f=@θ Þð@θ =@θi Þ 5 ð@θ =@θi Þð@f=@θ Þ by the chain rule. Then, gi and vi are called the covariant base vector and the covariant component, respectively. The general covariant derivative of tensor is referred to Hashiguchi and Yamakawa (2012). On the other hand, gi and vi are called the contravariant base vector and the contravariant component, respectively. Incidentally, note that a component and a base vector with opposite variants to each other are coupled as a pair as shown in Eqs. (2.40) and (2.43), where vi gi is called the contravariant description and vi gi is called the covariant description, respecting a variant of component than a variant of base vector. The subscript and the superscript are added to the covariant and the contravariant component and base vector, respectively. The scalar, the vector, and the tensor products are given by Eqs. (2.38) and (2.40) as



u v 5 ui vi 5 ui vi 5 gij ui vj 5 gij ui vj pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi jjvjj 5 v v 5 vi vi 5 gij vi vj 5 gij vi vj

(2.49)

u 3 v 5 eijk ui vj gk 5 eijk ui vj gk

(2.50)





ðu 3 vÞ w 5 eijk ui vj wk 5 eijk ui vj wk Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(2.48)

(2.51)

71

2.3 Tensor representations

u  v 5 ui vj gi  gj 5 ui vj gi  gj 5 ui vj gi  gj 5 ui vj gi  gj

(2.52)

The square of the length of an infinitesimal line element vector dx is expressed using the metric tensor in Eq. (2.49) as follows: 2



:dx: 5 dx dx 5 gij dθi dθj 5 gij dθi dθj 5 dθi dθi

(2.53)

noting









dx dx 5 dθi gi dθj gj 5 dθi gi dθj gj 5 dθi gi dθj gj Further, the cosine of angle between two infinitesimal line element vectors dx and dy is expressed using Eqs. (2.48) and (2.53) as follows: 8 gij dxi dyj > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > grs dxr dxs gpq dyp dyq > > > > < gij dxi dyj dx dy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2.54) 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðdx dyÞ 5 grs dxr dxs gpq dyp dyq :dx::dy: > > > > > > dxi dyi > > pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi > > r : dx dxr dys dys





gij is called the covariant metric tensor and gij is the contravariant metric tensor, which are the most basic geometrical ingredients for the description of the length of line element and the angle between line elements. The infinitesimal surface vector dak of the surface formed by the line elements dθi gi ðno sumÞ and dθj gj ðno sumÞ along the primary base fgi g is given from Eq. (2.38) by dak 5 dθi gi 3 dθj gj 5 eijk dθi dθj gk

ðno sumÞ

(2.55)

where i; j; k are different from each other and are ordered in an even permutation, which is called the surface element vector (Fig. 2.4). The volume dV of the infinitesimal parallelopiped (Fig. 2.4) formed by the sides along the primary base fgi g is given from Eq. (1.352) with Eq. (1.52) by dv 5 ½dθ1 g1 dθ2 g2 dθ3 g3  leading to dv 5 ½g1 g2 g3 dθ1 dθ2 dθ3 5 vg dθ1 dθ2 dθ3

(2.56)

The second-order tensor T is expressed in four types as j

T 5 T ij gi  gj 5 Ti j gi  gj 5 Ti gi  gj 5 Tij gi  gj

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(2.57)

72

2. Curvilinear coordinate system

FIGURE 2.4 Surface element vector and volume element of parallelopiped formed by the side elements in the direction of primary base vectors.

exploiting the tensor product in (2.52). We cannot write Tij because of j Ti j 6¼ Ti . The components of T are given as follows:



Tij 5 gi Tgj ;



T i j 5 gi Tgj ;

j





T i 5 gi Tgj ;

Tij 5 gi Tgj

(2.58)

by virtue of Eqs. (1.121) and (2.6). Then, the relations between the components in the different variance are given by substituting Eq. (2.57) into Eq. (2.58) as follows: ( j  Tij 5 Ti r grj 5 gir Tr 5 gir T rs gsj ; Ti j 5 T ir grj 5 gir T rs gsj 5 gir T rj (2.59) j  Ti 5 gir Trj 5 gir Tr s gsj 5 T ir grj ; Tij 5 gir Trs gsj 5 gir T r j 5 Ti r grj The scalar product of second-order tensors A and B is expressed in components as follows: j

j

A:B 5 Aij Bij 5 Ai j Bi 5 Ai Bi j 5 Aij Bij 5 gir gjs Aij Brs 5 girgjs Aij Brs noting

8 ij ðA gi  gj Þ:ðBrs gr  gs Þ 5 ðgi gr Þðgj > > > > >  > ðAi j gi  gj Þ:ðBrs gr  gs Þ 5 ðgi gr Þðgj > > > > > < ðA j gi  g Þ:ðBr g  gs Þ 5 ðgi g Þðg s r j r j i rs i j i > > ðAij g  g Þ:ðB gr  gs Þ 5 ðg gr Þðgj > > > > ij > > ðA gi  gj Þ:ðBrs gr  gs Þ 5 gir gjs Aij Brs > > > : ðAij gi  gj Þ:ðBrs gr  gs Þ 5 girgjs Aij Brs

   

 gsÞAijBrs  gsÞAi jBrs  gsÞAi jBrs  gsÞAijBrs 

 



Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(2.60)

73

2.3 Tensor representations

The magnitude of tensor T is given by setting A 5 B 5 T in Eq. (2.60) as pffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi jffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :T: 5 T:T 5 T ij Tij 5 T i j T i 5 gir gjs Tij T rs (2.61) The definition of the trace trT  T:I in the Cartesian space is extended as the scalar product with the metric tensor in the Riemann space, leading to the sum of the diagonal components of the mixed variant form, that is, 

trT  T:g 5 g:T 5 Tii 5 Ti i 5 T i i 5 T ii noting

(2.62)

8 > ðgi  gi Þ:ðT rs gr  gs Þ 5 T ii > > > < ðgi  g Þ:ðT r g  gs Þ 5 Ti s r i i s r  i > ðgi  g Þ:ðT r g  gs Þ 5 Ti i > > > : ðgi  gi Þ:ðT rs gr  gs Þ 5 T ii

where use is made of Eq. (1.215)7 The following relations hold. 8 i g ðTrs gr  gs Þgj 5 Tij 5 gj > > > > < gi ðTr g  gs Þg 5 Ti 5 g s r j j j j s r > > g ðTr g  gs Þgj 5 Ti 5 gj > > : ii g ðTrs gr  gs Þgj 5 Trs 5 gj

   

 ðTrsgs  gr Þgi  ðTrsgs  grÞgi  ðTrsgs  grÞgi  ðTrs gs  grÞgi  

Comparing these relations with Eq. (1.126), the transposed tensor TT is given by j



TT 5 T ji gi  gj 5 Tj i gi  gj 5 T i gi  gj 5 Tji gi  gj

(2.63)

Therefore the transposed tensor is given by exchange of the front and the rear base vectors. However, it is not obtained by the exchange of front and rear suffices in components, differing from the tensor expressions in the Cartesian coordinate system, as known from ( i ) ( j T j gi  gj T i gi  gj T T 5 ¼ 6 (2.64)  j Tj i gi  gj T  i gi  gj If T is the symmetric tensor, it follows from Eqs. (2.57) and (2.63) that 

j

j

Tij 5 Tji ; Ti j 5 Tj i ðTi 5 T  i Þ; Tij 5 Tji

(2.65)

However, the indices cannot be exchanged between the contravariant and the covariant ones, that is,

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

74

2. Curvilinear coordinate system j

j



Ti j 6¼ T i ; Ti 6¼ Tj i

(2.66)

Therefore the mixed components Ti j and Tj i possess the nine independent values even in the symmetric tensor, leading to the inconvenience of analysis. The representations of the tensor T on the combinations of the covariant base fgi g and the contravariant base fgi g are given as follows: ij

j

i

T 5 T gi  gj 5 T  j gi  gj 5 T i gi  gj 5 Tij gi  gj

(2.67)

On the other hand, the substitution of Eq. (2.15) into Eq. (2.57) reads: r

T 5 Tij j

5 Ti

s

r

@θ @θ @θ @θj g  j gs 5 Ti j i gr  s gs i r @θ @θ @θ @θ s

@θi @θ

r rg 

@θ @θi @θj g 5 T ij r gr  s gs j s @θ @θ @θ

Comparing this equation with Eq. (2.67), the transformation rules of components are given by 8 i j > @θ @θ rs @θi @θj rs > ij > > T 5 r s T ; T ij 5 r s T > > @θ @θ > @θ @θ > > > > i s > > @θ @θs r @θi @θ r i > i > > T 5 T ; T 5 T s j  j r > > @θr @θj s < @θ @θj (2.68) j r > j @θr @θ  s @θ @θj  s > j > > Ti 5 i s Tr ; Ti 5 i s Tr > > > @θ @θ @θ @θ > > > > r s > > @θr @θs @θ @θ > > > T 5 T ; T 5 T rs rs ij ij > i j > @θi @θj : @θ @θ n The component T qp11?q ?pm in the general tensor fulfilling the transformation rule

i ...im

T jl1 ...jn 5

i1

im

@θ @θ @θs1 @θsn r1 ...rm ? ? T jn s1 ...sn @θr1 @θrm @θj1 @θ

(2.69)

is called the ðm 1 nÞth order mixed variant components, consisting of mth order contravariant and nth order covariant. The base vectors gi and gi are not the unit vectors, possessing the magnitudes

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

2.3 Tensor representations

:gi : 5

pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi gi gi 5 gii ;



:gi : 5

qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffi gi gi 5 gii ðno sumÞ



75 (2.70)

by virtue of Eqs. (2.21) and (2.49). gi and gi possess 2 m and 1=2 m length, respectively, so that the unit components vi 5 1 and vi 5 1 lead to the actual length 2 m and 1/2 m, respectively, for the example described in (Note) in Section 2.1. The components for the normalized base vectors gi =:gi : and gi =:gi : are called the physical components, which do not obey the orthogonal transformation, are explained in the literature (Hashiguchi and Yamakawa, 2012). The covariant derivatives of the vector and the tensor with the Christoffel symbol are also delineated in that literature.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

C H A P T E R

3 Tensor operations in convected coordinate system Tensors describing physical quantities [e.g., the Cauchy stress tensor and the strain rate tensor (the symmetric part of velocity gradient tensor)] are first defined in the current (actual) configuration which one can recognize (observe) actually. They are referred to as the spatial or Eulerian tensors. On the other hand, constitutive equations describing mechanical responses in each material point must be described by tensors defined in the reference configuration, which are referred to as the material or Lagrangian tensors. Here, the inverse transformation of the Lagrangian tensors to the Eulerian tensors is required, since one can actually recognize only the Eulerian tensors in the current configuration. The transformation from the Eulerian tensor to the Lagrangian tensor is called the pull-back operation and the inverse transformation from the Lagrangian tensor to the Eulerian tensor is called the push-forward operation. These operations can be clearly interpreted by the description of tensors in the embedded coordinate system. The description of tensors in the embedded coordinate system and the pull-back, push-forward operations are described comprehensively in this chapter. The embedded derivatives, the corotational rates, and the objectivity of rate tensors are also described in detail.

3.1 Advantages of description in embedded coordinate system The representation of tensors in the convected (convective or embedded or intrinsic) coordinate system falling within the framework of the general curvilinear coordinate system is indispensable in order to understand and describe exactly the finite deformation by the following reasons.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity DOI: https://doi.org/10.1016/B978-0-12-819428-7.00003-1

77

© 2020 Elsevier Inc. All rights reserved.

78

3. Tensor operations in convected coordinate system

1. All tensors describing deformation (rate) can be expressed exactly by the variations of the base vectors embedded in a material itself. Then, it is fundamentally noticeable that the deformation gradient tensor is represented as the tensor possessing the components of the Kronecker’s delta in the convected coordinate system. Then, physical meaning of strain (rate) tensors can be captured clearly as a variation of the metric tensor in the embedded coordinate system. By virtue of this benefit, all the strain and strain rate tensors will be shown in terms of the metric tensor in addition to the description in terms of the deformation gradient tensor. 2. Lagrangian tensor and Eulerian tensor can be defined independently and their difference or mutual relation can be captured systematically and explicitly by representing them referring to the reference base vectors and the current base vectors, respectively, of the embedded coordinate system. In addition, the feature of the twopoint tensor, which extends over the reference and the current base vectors, appears explicitly in this representation. 3. Transformation between the Lagrangian tensor and the Eulerian tensor, that is, the pull-back and push-forward operations of tensors can be performed through multiplying the tensor by the deformation gradient tensor and its inverse and transpose. On the other hand, their mathematical and physical meanings can be captured clearly as the exchange of the reference and the current embedded base vectors. There exist two and four types of transformation rules from the Eulerian vector and tensor to the Lagrangian vector and tensor (or their inverse transformation rules) for the vector and secondorder tensor, respectively. 4. The variation of physical quantities can be described exactly in the convected coordinate system so that the rates of physical quantities have to be given by the convected time-derivatives in general. 5. The general meaning of the Lie derivative of tensor can be captured comprehensively as the convected derivative, that is, the rate of variation observed by the material element itself moving, deforming and rotating with material. The convected derivative can be attained by the push-forward operation of the material time differentiation of the quantity pulled-backed from the current configuration. 6. Physical meaning of corotational derivatives can be captured clearly as the convected derivative taken account of only rotation of material element. It is noteworthy that the base vectors vary with a deformation but instead the components of material line-element vector are kept constant in the embedded coordinate system, although the components change in the Cartesian coordinate system with the fixed base vectors.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

3.2 Convected base vectors

79

3.2 Convected base vectors Consider the coordinate system fΘi gði 5 1; 2; 3Þ, which is printed (etched) inside the material itself at the reference time t0 , possessing the origin at the material point, changes to the coordinate system fθi ðtÞg under the deformation of material as shown in Fig. 3.1. Therein, the embedded primary base fGi g are taken to be tangent to the embedded coordinates fΘi g at the reference time t0 , which changes to fgi ðtÞg at the current time t. Here, it should be noted that θi 5 Θi holds for the material point under the deformation because they are the coordinates in the embedded coordinate system, although the primary base fgi ðtÞg changes. The coordinate system fΘi g (and fθi ðtÞg) is called the convected (or embedded) coordinate system and the primary base fGi g (and fgi ðtÞg) is called the convected (or embedded) base. On the other hand, the reciprocal bases in the reference configuration and in the current configuration are denoted by fGi g and fgi ðtÞg, respectively, and their components are designated as fΘi g and fθi ðtÞg, respectively, as shown by the dashed arrows in Fig. 3.2. Here, it should be noted that the reciprocal bases must be reformed at every moment with a deformation in order to keep the mathematical relation in Eq. (2.1) to the primary bases so that they cannot be embedded. Here, the covariant components change in general as known from dθi ðtÞ 5 gij ðtÞdθj 5 gij ðtÞdΘj ¼ 6 Gij dΘj 5 dΘi in Eq. (2.42) because of gij ðtÞ ¼ 6 Gij . Consequently, it follows that θi 5 dΘi ;

dθi 5 dΘi

ðdθi ðtÞ 6¼ dΘi Þ

(3.1)

The following relations hold for the primary and the reciprocal base vectors by virtue of Eq. (2.21)2,3.

FIGURE 3.1 Deformation of parallelopiped element with sides parallel to the embedded coordinate system.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

80

3. Tensor operations in convected coordinate system



j

Gi Gj 5 δ i ;



j

gi ðtÞ gj ðtÞ 5 δi

Gi 5 Gir Gr ; Gi 5 Gir Gr gi ðtÞ 5 gir ðtÞgr ðtÞ; gi ðtÞ 5 gir ðtÞgr ðtÞ

(3.2) (3.3)

where



Gij 5 Gi Gj ;





Gij 5 Gi Gj

gij ðtÞ 5 gi ðtÞ gj ðtÞ;



gij ðtÞ 5 gi ðtÞ gj ðtÞ

(3.4)

The ratio of the volume in the current configuration to that in the reference configuration is given by ½g1 g2 g3 dθ1 dθ2 dθ3 ½g1 g2 g3 dΘ1 dΘ2 dΘ3 dv 5 5 dV ½G1 G2 G3 dΘ1 dΘ2 dΘ3 ½G1 G2 G3 dΘ1 dΘ2 dΘ3 leading to dv dV


5 J 5 detF 5

 vg ½g1 g2 g3  5 ½G1 G2 G3  VG

(3.5)

Therefore the ratio of the current volume to the reference volume is identical to the ratio of the volume of the parallelopiped formed by the current primary base vectors to that of the ratio of the parallelopiped formed by the reference primary base vectors. The following tensors are nothing but the metric tensors defined by Eq. (2.28). ( G  Gi  Gi 5 Gi  Gj 5 Gij Gi  Gj 5 Gij Gi  Gj ð 5 GT Þ gðtÞ  gi ðtÞ  gi ðtÞ 5 gi ðtÞ  gi ðtÞ 5 gij ðtÞgi ðtÞ  gj ðtÞ 5 gij ðtÞgi ðtÞ  gj ðtÞð 5 gT ðtÞÞ (3.6)

The vector and the tensor based in the reference and the current configuration are called the Lagrangian and the Eulerian vector and tensor, respectively. In principle, the Lagrangian and the Eulerian vector and tensor are denoted by the uppercase and the lowercase letters, respectively, throughout this book. Further, the tensor based in both of the reference and the current configurations is called the Lagrangian 2 Eulerian or Eulerian 2 Lagrangian two-point tensor and denoted by the uppercase letter.

3.3 Deformation gradient tensor The infinitesimal line-element dX in the initial configuration and dxðtÞ in the current configuration are described as follows:

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

81

3.3 Deformation gradient tensor

(

dX 5 dΘi Gi 5 dθi Gi ð 5 dΘi Gi 6¼ dθi Gi Þ dxðtÞ 5 dθi gi ðtÞ 5 dΘi gi ðtÞ ð 5 dθi gi ðtÞ 6¼ dΘi gi ðtÞÞ

(3.7)

The relation of the primary base vectors gi ðtÞ and Gi can be described by replacing dxðtÞ and dX, respectively, in Eq. (1.343) as follows: gi ðtÞ 5 FðtÞGi Gi 5 F21 ðtÞgi ðtÞ or noting

(3.8)

8 @xðtÞ @xðtÞ @xðtÞ @X > > gi ðtÞ 5 5 5 > > @X @Θi < @θi @Θi > @X @X @X @xðtÞ > > > : Gi 5 @Θi 5 @θi 5 @xðtÞ @θi

with the aid of Eq. (2.12). It follows from Eq. (3.8) that FðtÞ 5 gi ðtÞ  G 5 F j gi ðtÞ  G ; i

i

F j 5 δij

j

i

F21 ðtÞ 5 Gi  gi ðtÞ 5 ðF21 Þi j Gi  gj ðtÞ;

@θi @θi 5 j 5 j @Θ @θ

ðF21 Þi j 5 δij

!

@Θi @θi 5 j 5 j @θ @θ

!

(3.9) because of

(

(

FðtÞ 5 FðtÞG 5 FðtÞGi  Gi F21 ðtÞ 5 F21 ðtÞg 5 F21 ðtÞgi ðtÞ  gi ðtÞ





Fi j 5 gi ðtÞ FðtÞGj 5 gi ðtÞ ðgr ðtÞ  Gr ÞGj 5 δir δrj 21 i

ðF Þ j ðtÞ 5 G

i

F

21



ðtÞgj ðtÞ 5 Gi ðGr  gr ðtÞÞgj ðtÞ 5 δir δrj

noting Eqs. (3.1) and (3.8). The fact that the deformation gradient is the EulerianLagrangian two-point tensor based in the initial and the current base vectors can be understood by the representation in the embedded coordinate system [Eq. (3.9)], although this fact is hidden in the rectangular coordinate system. Obviously, the deformation gradient is the EulerianLagrangian two-point tensor as shown in Eq. (3.9)1. Further, it is noteworthy that the deformation gradient tensor is regarded as the identity tensor possessing the Kronecker’s delta in the convected coordinate system as known from Eq. (3.9). Here, it follows from Eq. (3.9) that

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

82

3. Tensor operations in convected coordinate system 

FT ðtÞ 5 Gi  gi ðtÞ 5 ðFT Þj i Gj  gi ðtÞ;



ðFT Þj i 5 δij

j

F2T ðtÞ 5 gi ðtÞ  Gi 5 ðF2T Þi gi ðtÞ  Gj ;

j

ðF2T Þi 5 δi j

(3.10)

from which we have gi ðtÞ 5 F2T ðtÞGi Gi 5 FT ðtÞgi ðtÞ

(3.11)

for the reciprocal base vectors, noting ( i g ðtÞ 5 gj ðtÞδij 5 ðgj ðtÞ  Gj ÞGi 5 F2T ðtÞGi Gi 5 Gj δij 5 ðGj  gj ðtÞÞgi ðtÞ 5 FT ðtÞgi ðtÞ In what follows, we eliminate the symbol (t) for the time dependence. The rates of the current base vectors are given in terms of the velocity gradient tensor l in Eq. (1.362) from Eqs. (3.8) and (3.11) as follows:

noting

_gi 5 _FGi 5 lgi _gi 5 _F2T Gi 5 2 lT gi

(3.12)

_F 5 _gi  Gi 5 lF; _FT 5 Gi  _gi 5 FT lT _F21 5 Gi  _gi 5 2 F21 l; _F2T 5 _gi  Gi 5 2 lT F2T

(3.13)

(

_FF21 FGi 5 lgi _F2T Gi 5 _F2T FT F2T Gi 5 ðF_F21 ÞT gi 5 ð2_FF21ÞT gi 5 2 lT gi 21 G 5 ðFF21 Þ 5 F_ F 1_ FF21 5 O. with _ 

Further, it follows from Eqs. (3.9) and (3.12) that

_

_

l 5 gi  gi 5 FF21

_

_T

l 5 gi  gi 5 F2T F T

(3.14)

noting

_

_

lgi  gi 5 gi  gi 5 FGi  gi The convected coordinate systems choosing the rectangular-normal coordinate system for the reference or the current states, respectively, are called the material and current coordinate systems, respectively, in Bazar and Weichert (2000). Incidentally, needless to say, the relative

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

3.4 Pull-back and push-forward operations

83

position of other material points from the certain material point within a material is described by the convected coordinate system. On the other hand, the absolute position of material point cannot be described by the convected coordinate system but has to be described by the coordinate system fixed in the space.

3.4 Pull-back and push-forward operations The Eulerian tensor defined by the current base fgi g or fgi g is transformed to the Lagrangian tensor by replacing these bases to the reference base fGi g or fGi g. This operation is called the pull-back. Its inverse operation is called the push-forward. These operations can also be performed by the multiplication of the deformation gradient as will be explained below. The Eulerian vector v is expressed by the following contravariant and the covariant forms in the current configuration by Eq. (2.40). v 5 vi gi 5 vi gi

(3.15)

The pull-back transformations from the Eulerian vector to the Lagrangian vectors are performed by replacing the current base fgi g or fgi g to the reference base fGi g or fGi g and designated as follows: ~

~

v G  vi G i ;

v G  vi G i

(3.16)

FIGURE 3.2 Pull-back and push-forward operations illustrated in the two-dimensional space.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

84

3. Tensor operations in convected coordinate system

where the over arrow directed to the left ð’Þ is added (the right arrow ð-Þ for the push-forward operation described later). Further, the uppercase index G is added in order to stipulate the replacement of the current base to the reference base (the lowercase index g for the pushforward operation in order to stipulate the replacement of the reference base to the current base described later) and it is put in a lower or upper position of suffix of the transposed base vector. These symbols for the pull-back and the push-forward operations have been introduced by Hashiguchi (2011). Eq. (3.16) is rewritten as ~

~

v G 5 vi F21 gi 5 F21 vi gi ;

v G 5 vi FT gi 5 FT vi gi

using Eqs. (3.8) and (3.11), and thus the following direct notations are obtained. ~

~

v G 5 vi Gi 5 F21 v;

v G 5 vi Gi 5 FT v

(3.17)

which are called the contravariant, and the covariant pull-back operations, respectively. In particular, for the pure rotation with F 5 R leading to Gi 5 RT gi ; Gi 5 RT gi , the two relations in Eq. (3.17) unified to the single relation. ~

~

v R ð 5 vi RT gi Þ 5 v R ð 5 vi RT gi Þ 5 RT v

(3.18)

in which the distinction between the covariant and the contravariant transformation diminishes. In the inverse to the above, the Lagrangian vector defined on the reference base fGi g or fGi g, that is, V 5 V i Gi ;

V 5 V i Gi

(3.19)

~ g 5 Vi gi V

(3.20)

is transformed to the Eulerian vectors ~ g 5 Vi g ; V i

by the push-forward operations replacing the reference base fGi g or ~ g and V ~ g are called the confGi g to the current base fgi g or fgi g, where V travariant and the covariant push-forward operation, respectively. Here, because of ~ g 5 Vi FGi ; V

~ g 5 Vi F2T Gi V

noting Eq. (3.11), the following direct notations hold. ~ g 5 V i g 5 FV; V i

~ g 5 Vi gi 5 F2T V V

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(3.21)

85

3.4 Pull-back and push-forward operations

Further, for the pure rotation with gi 5 RGi ; gi 5 RGi , it follows from Eq. (3.21) that

F5R

leading

~ R ð 5 Vi RGi Þ 5 RV ~ R ð 5 V i RGi Þ 5 V V

to

(3.22)

Next, the second-order tensor t in the current configuration is represented by the following four types from Eq. (2.57). j

t 5 tij gi  gj 5 ti j gi  gj 5 ti gi  gj 5 tij gi  gj j

(3.23)

j

Here, when tij 5 tji ; ti j 5 t i ; ti 5 tj i ; tij 5 tji , t is the symmetric tensor, and thus t 5 tT holds by virtue of Eq. (2.65). The Eulerian tensor in Eq. (3.23) is replaced to the Lagrangian tensors by the pull-back operations replacing both of the current base vectors to the reference base vectors as follows: 8 > > t GG  tij Gi  Gj 5 tij F21 gi  F21 gj 5 F21 tij gi  gj F2T > > > > > > > G < t G  ti j Gi  Gj 5 ti j F21 gi  FT gj 5 F21 ti j gi  gj F ~ ~

~

> j j T j > i 21 i > tG gj 5 FT ti gi  gj F2T >  G  ti G  Gj 5 ti F g  F > > > > > : GG t  tij Gi  Gj 5 tij FT gi  FT gj 5 FT tij gi  gj F ~

leading to ~

~

~

~

t GG 5 tij Gi  Gj 5 FT tFð 5 t GGT Þ (3.24)

~

~

~

~

t GG , t GG ,

~



G  GT tG 5 ti j Gi  Gj 5 F21 tFð 6¼ t G Þ

~

~

j

T 2T tG ð6¼ t GT  G 5 ti Gi  Gj 5 F tF  G Þ;

~

T t GG 5 tij Gi  Gj 5 F21 tF2T ð 5 t GG Þ;

G t G

GG

and t are called the contravariant, the contravariantwhere covariant, the covariant-contravariant and the covariant pull-back operation, respectively. In the case that the original tensor t is symmetric, the pull-backed tensors in the contravariant and covariant forms are symmetric, that is, T

~

~

~

~

T t GG 5 t GG ; t GG 5 t GG . On the other hand, the pull-backed tensor in the

~

~

~

~

G G G 6¼ t G and t G mixed-variant form is not symmetric, that is, t G  G 6¼ t  G but G GT it holds that t G 5 t  G . Here, the tensor in the current configuration is called the Eulerian tensor and the pulled-back tensor in the reference configuration is called the Lagrangian tensor.

~

~

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

86

3. Tensor operations in convected coordinate system

For the pull-back operation in which only one of the two current base vectors is changed to the reference base vector, it is denoted by adding the hat symbol ð^Þ to the unchanged one as follows: 9 9 8 > > ij ij 21 21 > ij ij = = > t 5t G  g 5t F g  g t gG > Gg^ i ^ 5t gi  Gj 5t gi  F gj j r j > 21 > 5F 5tF2T t; > > g ^ > > > j_ i j_ i g^ 21 ;

> j > T ^ gG i i j = i T ^ G g j i j > t 5tij g  G 5tij g  F g = > t 5tij G  g 5tij F g  g T > > 5tF 5F t; > > G j T j > i i : t G 5tj_Gi  g 5tj_FT gi  g ; ; t 5t g  G 5t g  F g g g^ ^ j j i i j_ i j_ i ~

~

~

~

~

~

~

~

(3.25) Further, for the pull-back operation in which only one of the current base vector is replaced to the reference base vector and the other current base vector is subjected to the rigid-body rotation F 5 R, it follows from Eq. (3.24) that 8 9 9 > G = j T T = i ij > > > t GR 5 t Gi  R gi 5 F21 tR; t R 5 t j R gi G > 5 RT tF > > < t  R 5 ti Gi  RT gj ; j; T i RG t 5 tij R g  G j G 9 9 (3.26) j > = = i T G T > ij > t  R 5 ti G  R gj t RG 5 t R gi  Gj > > 5 FT tR; 5 RT tF2T > > j ; ; : GR T t R G 5 ti R gi  Gj t 5 tij Gi  RT gj ~

~ ~

~

~

~ ~

~

where the subscript R is put instead of G in order to specify the replacement of the current base vector to the reference base vector under F 5 R. Furthermore, when only the rotation is given to both base vectors, the pull-backed tensors are described by the following unique equation in which the distinction of covariant, contravariant, and mixed-variant operations vanishes: ~

 TÞ T  t R 5 RT tRð 5 T

(3.27)

which is derived by setting F 5 R in Eq. (3.24). T is called the corotational Lagrangian tensor. Inversely to the above, the Lagrangian tensor defined by the reference bases fGi g; fGi g is represented in the four types: j

T 5 Tij Gi  Gj 5 T ji Gi  Gj 5 Ti Gi  Gj 5 Tij Gi  Gj

(3.28)

When replacing the reference base vectors to the current base vectors, the following push-forwarded Eulerian tensors are obtained.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

87

3.4 Pull-back and push-forward operations

8 > > > > > >
~ g 5 T  j gi  g 5 T  j F2T Gi  FGj 5 F2T T  j Gi  Gj FT > T > g j i i i > > > : ~ gg 2T i 2T j 2T i j T 5 Tij g  g 5 T ij F G  F G 5 F Tij Gi  Gj F21 leading to     g  gT T ~ T gg ; ~ Tg T gg 5Tij gi  gj 5FTFT 5 ~ T g 5T i j gi  gj 5FTF21 6¼ ~     g gT gg ggT j ~ T  g 5T i gi  gj 5F2T TFT 6¼ ~ Tg ; ~ T T 5T ij gi  gj 5F2T TF21 5 ~ (3.29) g

g ~ T g

gg

where ~ T gg , ~ Tg , and ~ T are called the contravariant, the contravariantcovariant, the covariant-contravariant and the covariant push-forwardoperation, respectively. For the push-forward operation in which only one of the two reference base vectors is changed to the current base vector, it is denoted by adding the hat symbol ð^Þ to the unchanged one as follows: 9 8 ~ T gG^ 5T ij gi Gj 5T ij FGi GJ = > > > > 5FT; > ^ > G ; >

gG^ > j j = 2T i i > ~ 5T g G 5T F G G T > ij ij > > 5F2T T; > : ~g j j ; T  G^ 5Ti gi Gj 5Ti F2T Gi Gj

9 ij ij ~ T Gg ^ 5T Gi gi 5T Gi FGj = ^

G j j ; ~ T  g 5T i Gi gj 5Ti Gi FGj

5TFT ;

9 ^ Gg ~ T 5T ij Gi gi 5T ij Gi F2T Gj = 5TF21 ; g ~ T 5T i G gj 5T i G F2T Gj ; G^

j

i

j

i

(3.30)

Further, for the push-forward operation in which only one of the reference base vector is replaced to the current base vector and the other reference base vector is subjected to the rigid-body rotation F 5 R, it follows from Eq. (3.29) that 8 ) ) g ~ ~ > T gR 5 T ij gi  RGi T R 5 Ti j RGi  gj > T > > 5 RtF21 5 FtR ; R > Rg T ~ g 5 T j gi  RGi T RG 5 Tij RGi  gj > 2T T T R > i > 5 RtF 5 F tR ; R > j gR :~ ~ T 5 T RGi  g t 5 T gi  RGj ij

g

i

j

Furthermore, when only the rotation is given to both base vectors, the push-forwarded tensors are described by the following unique equation in which the distinction of covariant, contravariant, and mixedvariant operations vanishes: R T t  ~ T 5 RTRT ð 5 t Þ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(3.32)

88

3. Tensor operations in convected coordinate system

which is called the corotational Eulerian tensor. As described above, the pull-back and the push-forward operations can be realized originally by exchanging the base vectors between the reference and the current configurations but they are replaced by the multiplication of the deformation gradient F, where there exist four types of operation of F depending on the contravariant or covariant or the mixed-variant forms. The metric tensors in the reference and the current configurations are related in the pull-back and the push-forward operations as follows: 8 < G  Gi ð 5 F21 g  FΤ gi Þ 5 F21 GF 5 g  G i i G G5 : i G  Gi ð 5 FΤ gi  F21 gi Þ 5 FΤ gF2Τ 5 g G G (3.33) 8 g ~ < g  gi ð 5 FGi  F2Τ Gi Þ 5 FGF21 5 G g i g5 : gi  g ð 5 F2Τ Gi  FGi Þ 5 F2Τ GFΤ 5 G ~ g ~

~

i

g

The pull-back and the push-forward operations between the current and the reference configurations are illustrated in Fig. 3.2. The tensors in the current configuration and in the reference configuration are referred to the Eulerian tensor and the Lagrangian tensor, respectively. It is shown that we have the two and the four types of transformations from the Eulerian to the Lagrangian vector and second-order tensor or its inverse transformations, respectively, since the vector and the secondorder tensor possesses one and two base vectors, respectively. In general, there exist n 3 2 types of transformations for the nth order tensor. It should be noticed from the physical point of view that practically the Eulerian tensor is defined first and thereafter the Lagrangian tensors are produced from the Eulerian tensor by the pull-back operation as will be described for stress tensors in Section 5.4.

3.5 Convected time-derivative Rates of tensors used in constitutive equations have to be independent of the rigid-body rotation, possessing the objectivity, which is referred to as the material-frame indifference. The convected derivative of tensor, which is called also the Lie derivative, is explained in this section, which is the rate of tensor observed from the convected coordinate system moving, deforming, and rotating with a material. It is verified that corotational rate of tensor can be regarded as the particular convected derivative observed from the coordinate system rotating with a material. In addition, the objectivities of tensors are examined, and the

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

89

3.5 Convected time-derivative

objectivity of the convected derivative is proved. The convected derivatives of tensors provide the systematic way to derive various objective rate tensors appearing in continuum mechanics. It must be adopted for the variables that are influenced by the rigid-body rotation, for example, the stress, the kinematic and the rotational hardening variables in the current configuration. The convected time-derivative is not required to be adopted in the multiplicative hyperelastic-based plasticity since it is formulated in the configuration, which is independent of the rigid-body rotation as will be explained comprehensively in Chapter 8, Multiplicative decomposition of deformation gradient tensor, and Chapter 9, Multiplicative hyperelastic-based plastic constitutive equations. Therefore the readers who wish to acquire the multiplicative hyperelastic-based plasticity may omit reading this section.

3.5.1 General convected derivative The time-derivative of the vector v in Eq. (2.40) is described as ( r r  ðvr gr Þ 5 v gr 1 vr gr 5 v gr 1 vr lgr v5 (3.34) r  ðvr gr Þ 5 vr gr 1 vr g 5 vr gr 2 vr lT gr

_

_ _

_ _ _ _

noting Eq. (3.12). The first terms of these equations represent the rates of the vector v observed from the embedded coordinate system and thus they are called the convected derivative. In other words, they mean the rate of physical quantity observed from the embedded coordinate system having the base vectors composed of line-elements etched in a material (not material fiber). Also, they are interpreted as the rates observed from material itself and thus they are independent of rigidbody rotation, possessing the objectivity. They are expressed as

_ _ _ _ _ _ _ _ _ _

_ _

r 3 ~ vg  ~ 5 v gr 5 v 2 vr gr 5 v 2 lv 5 FðF21 vÞ 5 F’ vG  V g Vg g r 3g ’  ~g v  ~ 5 vr gr 5 v 2 vr g 5 v 1 lT v 5 F2T ðFT vÞ 5 F2T v G  V V

_

_

(3.35)

~

~

It is known from Eq. (3.35) that the convected derivative is obtained by time-differentiating the pulled-backed quantity F21 v or FT v to the reference configuration and then pushing-forward it to the current configuration. The symbols with the double arrows in the second terms in Eq. (3.35) are introduced in order to stipulate this fact. Similarly to the vector in Eq. (3.34), the material time-derivative of the second-order tensor t [Eq. (2.57)] in the current configuration is described by

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

90

3. Tensor operations in convected coordinate system

8 > > > > > > > >
> > > > > > > :

  ij tij gi  gj 5 t gi  gj 1 tij gi  gj 1 tij gi  gj   i j ti j gi  gj 5 t j gi  gj 1 ti j gi  gj 1 ti j gi  g   j j i j j ti gi  gj 5 ti gi  gj 1 ti g  gj 1 ti gi  gj  i  i j tij g  gj 5 tij gi  gj 1 tij g  gj 1 tij gi  g

_ _ _ _

_ _ _ _

_ _ _ _

(3.36)

from which the convected derivatives, that is, the rate of the tensor t observed from the observer moving with the embedded coordinate system, are given by   8 ij ij > t g  g 5 t g  g 2 tij gi  gj 2 tij gi  gj 5 t 2 tij lgi  gj 2 tij gi  gj lT > i j i j > >    > > < ti g  gj 5 ti g  gj 2 ti g  gj 2 ti g  gj 5 t 2 ti lg  gj 1 ti g  gj l j i j i j i j i j j i i   j j i j j T j i > T i i i i > > > ti g  gj 5 ti g  gj 2 ti g  gj 2 ti g  gj 5 t 1 ti l g  gj 2 t j gi  gj l > >   : i i j  tij g  gj 5 tij gi  gj 2 tij g  gj 2 tij gi  g 5 t 1 tij lT gi  gj 1 tij gi  gj l

_ _ _ _

_ _ _ _

_ _ _ _

_ _ _ _

(3.37)

noting Eqs. (1.137) and (3.12). The following four types of convected derivatives can be defined from Eq. (3.37). 3 ~ 5 tij g  g 5 t 2lt2tlT 5FðF21 tF2T Þ FT 5F’ ~  5t3 T  tgg  ’ t GG FT  T gg i j t gg gg
 g G T 3 g ’ 3 i g ~ 5 t g  gj 5 t 2lt1tl5FðF21 tFÞ F21 5F t F21  ~ 6¼ t tg  ’ j i gg G tg Tg
 G 3g ~ g 5 t j gi  g 5 t 1lT t2tlT 5F2T ðFT tF2T Þ FT 5F2T ’ ~ g 6¼ t3 gT T t F  t g  ’   g j G t g i Tg
 gg gg ggT gg 3 3 ~ 5 t gi  gj 5 t 1lT t1tl5F2T ðFT tFÞ F21 5F2T ’GG ~ 21 t F  5t t ’ ij t T

_ _

_ _

_ _

_

_

_

_

_

_

_

_

_

_

_

_

_

_

(3.38) 3

3 gg

tgg and t are the general forms of the Oldroyd rate and the Cotter 2 Rivlin rate, respectively. The stress rate tensors based on them are shown in Appendix 2. The convected derivatives for only one of the base vectors are given from Eq. (3.38) as follows: 8 3 9 ~ 5F > > ~   t > ^ g g t ^ G g > = Tgg^ > t gg^ > >  > > 5 t 2 lt 5 FðF21 tÞ >  >  > ^  g ^ ~ g^ > 3 g g ^ > ~ > ; > < tg  t g 5 F t G  T g (3.39) 9 3g > > ~g 5 F2T G  ~ g > > t  > > g t  g^ T  g^ = ^ > t  g^ >  > > 5 t 1 lT t 5 F2T ðFT tÞ > > > gg^ > 3 gg^ > > : t  ~gg^ 5 F2T Gg^  ~ ; t T t

_

~

_

~

_

~

~

_

_

_

_

~

~

~

~

_

_

_

_

_

_

_

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

91

3.5 Convected time-derivative

FIGURE 3.3 Convected derivative illustrated in the two-dimensional space.

for the left base vector and 8 9 3 ~ 5 FT  ~ > > > tgg > ^  t ^ > gG = T ^ gg > t gg ^ >  > > 5 t 2 tlT 5 ðtF2T Þ FT > g^ > g^ > 3 > > ~ ~g^ T > ; > < t g  t g 5 F  T  g 9 g > 3  g ~ > > g 5  G F21 ~ > t  > t ^ ^ = g g > T g^ > t g^ >  > > 5 t 1 tl 5 ðtFÞ F21 > > ^ > gg > ^ gg ^ 3 gG > ~ > 21 ^ ; : t  ~gg 5’ t F T t ~

_ _

_

~

~

_

_

~

~

~

_

_

_

_

_

_

_

(3.40)

_

for the right base vector. As described above, the convected derivatives are the rates of vector or tensor observed from the embedded coordinate system moving, deforming, and rotating with material, which possess the objectivity as will be proved in Section 3.5.3. They are attained by 1. first pulling-back the tensor in the current configuration to the reference configuration, 2. next taking its time-differentiating, and 3. finally returning (pushing-forward) it to the current configuration. As shown in Fig. 3.3, while the modes of the pull-back and pushforward operations depend on the variant of tensors. The convected derivative is called often the Lie derivative and denoted by the symbol Lv ð Þ (cf., e.g., Truesdell and Noll, 1965; Marsden and Hughes, 1983;

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

92

3. Tensor operations in convected coordinate system

Simo, 1998; Bonet and Wood, 2008; Belytschko et al., 2014; de Souza Neto et al., 2008). However, the symbol with the double arrows and the indices specifying the variants shown in this section would express the operations and modes of the convected derivatives more clearly and intuitively. It has been introduced by Hashiguchi (2011). Various convective stress rate tensors are shown in Appendix 2.

3.5.2 Corotational rate An irreversible deformation exhibits the loading-path dependency. Therefore a constitutive equation for an irreversible deformation must be described by the rate form. Therein, pertinent rates of stress and internal variables must be used in the constitutive equation, which satisfies the objectivity described in the next subsection. Here, consider the particular cases in which the deformation and its rate are negligible compared with the rotation, that is, UDI or dDO. For the case that the deformation is small compared with the rotation, that is, FDR, all the four kinds of derivatives in Eq. (3.38) reduce to the single equation

_

_ _

_

_ T ~ 3R 3 3 g 3g 3 gg T T _R  t  tgg 5 tg 5 t g 5 t 5 RðRT tRÞ RT 5 RT R  T 5 t 2 RR t 1 tRR

_

(3.41) which is referred to the Green 2 Naghdi rate (Green and Naghdi, 1965). 3R

t is zero when the quantity RT tR, which is observed by the coordinate system moving and rotating with a material line-element, is kept constant. The relation of the base vectors are given from Eq. (3.8) under F 5 R as follows: 8 i < gi 5 RGi ; gi 5 RG (3.42) G 5 RT gi ; Gi 5 RT gi : i R 5 gi  Gi 5 gi  Gi Then, the rate of the current base vector is given as follows:

_ _

_

_gi 5 ΩR gi

(3.43)

noting gi 5 RGi 5 RRT gi , where

_

ΩR  RRT

(3.44)

is referred to the relative spin. It designates the spin of the normalized convected covariant base vector gi =:gi : under the rigid-body rotation in  which ðgi =:gi :Þ 5 ΩR ðgi =:gi :Þ holds because of :gi : 5 const: and :ðgi =:gi :Þ: 5 1. The physical meaning of the relative spin based on the Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

93

3.5 Convected time-derivative

name capped with the adjective “relative” will be given in Section 4.4.1. The Green 2 Naghdi rate in Eq. (3.41) is represented in terms of the relative spin as follows:  3  3 tR 5 t 2 ΩR t 1 tΩR 5 tRT (3.45)

_

Further, in the case that the strain rate d in Eq. (1.375) is infinitesimal compared with the continuum spin w in Eq. (1.376), leading to l 5 w, Eq. (3.38) is reduced to  3  3 tw  t 2 wt 1 tw 5 twT (3.46)

_

which is called the Zaremba 2 Jaumann rate (Zaremba, 1903; Jaumann, 3 1911). Here, tw ignoring the strain rate d is related to the convected derivative in Eq. (3.38) as follows: 3

3

3

3

3

tw 5 tgg 1 td 1 dt 5 tgg 2 td 1 dt 5 tgg 1 td 2 dt 5 tgg 2 td 2 dt resulting in

 1  3  g 3 g  1  3  g  3  g T 3 gg 1 3 tgg 1 t t 1 t g 5 t 1 tg t 5 5 2 2 g 2 g  1 3 g  3 g T t 1 t g 5 2 g

(3.47)

3w

(3.48)

The continuum spin w is induced ceaselessly at a constant rate w12 5 γ =2 leading to the ceaseless rotation of material under the constant shear strain rate γ 5 const: in the simple shear deformation as will be shown in Eq. (5.98) in Section 5.6. Therefore the stress oscillates as shown in Fig. 3.4 when the Jaumann rate is used for the stress rate in the hypoelastic material as was indicated by Dienes (1979), and the stress and the kinematic hardening variable oscillate when the Jaumann rate is used for the rate of the kinematic hardening variable as was revealed for the elastoplastic material by Nagtegaal and de Jong (1982). On the other hand, the relative spin ΩR decreases gradually as Ω R12 5 2γ =ð4 1 γ 2 Þ decreases for the increase of shear strain γ, which will be shown in Eq. (5.103) for the simple shear deformation. Therefore the oscillation is suppressed by adopting the GreenNaghdi rate as was shown for the hypoelastic material by Dienes (1979) and for the kinematic hardening material by Dafalias (1985) (see Fig. 3.4). The corotational rates in Eqs. (3.45) and (3.46) depend only on the geometrical change of material, which can be observed from the outside appearance of material. However, the physically meaningful rotation is the rotation of the substructure of material induced by the substructure

_

_

_

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

94

3. Tensor operations in convected coordinate system

FIGURE 3.4 Descriptions of simple shear behaviors of the hypoelastic material by GreenNaghdi rate, the Jaumann rate, and the modified Jaumann rate with substructure spin for kinematic hardening material with constant size of yield surface, that is, F 5 G.

spin ω, which depends on the plastic deformation and thus cannot be known only from the outside appearance of material. It has been pointed out by Mandel (1971, 1973a), Kratochvil (1971), etc. that the rotation of substructure is suppressed by the plastic deformation such that the substructure spin ω is given by the subtraction of the plastic spin wp (Dafalias, 1984, 1985, 1998) from the continuum spin w, that is, 3

_

t  t 2 ωt 1 tω

(3.49)

ω 5 w 2 wp

(3.50)

with

The explicit equation of the plastic spin wp will be shown in Eqs. (7.75) or (7.119). The oscillations of the stress and the kinematic hardening variable are suppressed by the corotational rate with the modified spin tensor in Eq. (3.50) as illustrated in Fig. 3.4 for the kinematic hardening material with the constant size of the yield surface F 5 G.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

3.5 Convected time-derivative

95

An identical stress-strain behavior is described by any spin tensors, that is, any corotational rates, and thus the plastic spin is negligible up to a quite large deformation. In fact, the same behavior is described by using any corotational rates up to a quite large deformation reaching the engineering shear strain γ up to around 100% in the simple shear deformation as seen in Fig. 3.4. However, a corotational tensor must be used even for an infinitesimal deformation since a rigid-body rotation is induced during an infinitesimal deformation in many cases in engineering practice (suppose gears in machines). Then, it is recommendable to use the Jaumann rate based on the antisymmetric part of the velocity gradient tensor, that is, the continuum spin w, while the rate of deformation is described by the symmetric part of the velocity gradient tensor, that is, the strain rate d in the hypoelastic-based plastic constitutive equation. It should be emphasized that the corotational rate can be used only when the deformation (rate) is negligible compared with the rotation (rate) of material and thus the convected derivatives described in Section 3.5.1 should be used in general.

3.5.3 Objectivity of convected rate Rates of tensors standing for a stress and internal variables are influenced by the rotational rate of material, that is, the spin of rigid-body rotation in general, while rates of tensors standing for a deformation are independent of the rigid-body spin since they are originally excluded the rigid-body spin except for the infinitesimal strain tensor (see Section 4.2). Therefore particular rate tensors of stress and internal variables, which are independent of the rigid-body spin must be adopted in constitutive equations. This fact should be noticed particularly when the description of finite deformation by the hypoelastic-based plastic constitutive equations is required. However, the readers may skip the reading this section, who wish to acquire only the multiplicative hyperplasticbased plastic constitutive equation in which the rates of stress and internal variables based in the configuration excluded originally the influences of the rigid-body spin are used. Constitutive equations describe the mechanical property of material and thus it must be expressed independent of observers. Observers are no more than the coordinate systems by which constitutive equation is described and thus the constitutive equation must be described by the variables obeying a common objective transformation rule. Then, this notion called the objectivity is also called the principle of material-frame indifference by Oldroyd (1950). Here, the common transformation rule is nothing but the rule shown in Eqs. (1.93) or (1.98) [Eqs. (1.95) or (1.99)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

96

3. Tensor operations in convected coordinate system

for vector and second-order tensor], which is called the objective transformation rule in the definition of the tensor described in Section 1.3. All tensor variables obey the abovementioned objective transformation rule between the coordinate systems between which the mutual rotating rate does not exist, satisfying the objectivity. However, some of rate tensors do not obey the objective transformation rule between the coordinate systems between which the mutual rotating velocity exists, violating the objectivity. In other words, some of rate tensors are observed to be different by the coordinate systems rotating at different rotational velocities. Tensors describing the rotational rate of material are not objective tensor in general and some of tensors depending on the rotational rate of material are not objective tensor. The convected derivative described in Section 3.5.1 is based on the variation observed by the material itself. Therefore it would be independent of observers, and thus it would be self-evident for the convected derivative to satisfy the objectivity. In other words, the objective rates are to be based on the convected derivative. In particular, the rates derived from the convected derivative by incorporating only a spin tensor in a velocity gradient tensor are called the corotational rate, while various corotational rate tensors are derived for various spin tensors. The objectivity of the convected derivative tensor will be proved also mathematically in the following. An objective rate variable is required to fulfill the objective transformation rule in Eq. (1.93) even between the coordinate systems with different rates of rotation. Then, consider the fixed base fei g and the other base fei ðtÞg, which coincide with fei g in the reference state at t 5 0, that is, fei ð0Þg 5 fei g with ei ðtÞ 5 QT ðtÞei [Eq. (1.155)]. The following relation between the components observed by these bases holds, while the components observed by the bases fei g and fei ðtÞg are denoted by ð Þ and ð Þ , respectively, for the sake of brevity. (   i Gi 5 Gi Gi r 5 Gr ;   (3.51) gi 5 Qgi gir 5 Qrs gis and

(

  dX 5 dX dXr 5 dXr ;   dx 5 Qdx gr 5 Qrs dxs ;

(3.52)

noting Eq. (1.99)1. All measures describing the deformation/rotation are defined in terms of the deformation gradient tensor. Therefore we must consider the transformation rule of the deformation gradient tensor, which is given by

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

97

3.5 Convected time-derivative

from

  F 5 QF Frs 5 Qrp Fps

(3.53)

  F 5 gi  Gi 5 Qgi  Gi Frs 5 gir Gis 5 Qrp gip Gis

(3.54)

noting Eq. (3.9), or @x @x 5Q F 5 @X @X 


 @xp @xr  Frs 5 5 Qrp @Xs @Xs

(3.55)

The deformation gradient is the second-order tensor obeying the objective transformation rule when it is observed by the coordinate systems between which there does not exist the mutual rotational rate. On the other hand, it is the two-point tensor composed of the initial infinitesimal line-element dX observed by the fixed base fei g and the current infinitesimal line-element dx observed by the current rotated base fei ðtÞg so that it obeys the component transformation rule of one-order tensor (vector) as shown in Eq. (3.53). Then, the following relation holds from Eq. (3.53).

_F 5 Q_F 1 _QF

(3.56)

which is derived also by

_F 5 _gi  Gi 5 ðQ_gi 1 _QgiÞ  Gi 5 Q_gi  Gi 1 _Qgi  Gi

(3.57)

noting

_gi 5 Q_gi 1 _Qgi ; _F 5 _gi  Gi

(3.58)

obtained from Eqs. (3.13)1, and (3.51)2. The transformation rule of the velocity gradient tensor is given by

_

_

l 5 F F21 5 QlQT 1 QQT 5 Qðl 2 ΩÞQT

(3.59)

noting

_F F21 5 ðQFÞ ðQFÞ21 5 ðQ_F 1 _QFÞF21QT 5 Q_FF21QT 1 _QQT 

(3.60)

where

_T

ΩQ Q

(3.61)

which is the spin of the rotating base fei g as known from

_ei 5 ðQT eiÞ 5 _QT ei 5 _QT Qei 5 Ωei 

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(3.62)

98

3. Tensor operations in convected coordinate system

noting Eq. (1.156), while the length of ei does not change but it only rotates. The transformation rule of the continuum spin is given by setting l 5 w in Eq. (3.59) as follows: w 5 Qðw 2 ΩÞQT

(3.63)

Similarly, the transformation rule of the substructure spin in Eq. (3.50) is given by ω 5 Qðω 2 ΩÞQT

(3.64)

The transformation rule of the relative spin is given by setting F 5 R in Eq. (3.59) as follows:

_

ΩR 5 R RT 5 QðΩR 2 ΩÞQT

_

(3.65)

Therefore F; l; w, ΩR , and ω are not objective tensors, and l; w, ΩR , and ω obey the same transformation rule. On the other hand, needless to say, the strain rate d, which is the symmetric part of the velocity gradient excluding the antisymmetric part w describing the rotational velocity from the velocity gradient tensor l is the objective tensor as proved easily by

_

d 5 sym½l  5 sym½QlQT 1 QQT  5 QdQT

(3.66)

The objectivities of the convected derivatives of the vector and the second-order tensor in Eqs. (3.35)1 and (3.38)1 are proved for the contravariant ones as follows: 8    3 3 3 3 > < vg 5 Qvg vgi 5 Qir vgr 3  (3.67) 3 3 3 > : tgg 5 Qtgg QT tggij 5 Qir Qjs tggrs noting

_v 2 l v 5 ðQvÞ 2 ðQlQT 1 _QQT ÞQv 5 Q_v 1 _Qv 2 Qlv 2 _Qv 5 Qð_ v 2 lvÞ  _t 2 l t 2 t lT 5 ðQtQT Þ 2 ðQlQT 1 _ QQT ÞðQtQT Þ 2 ðQtQT ÞðQlQT 1_ QQT ÞT T T 5 Q_ tQT 1 _ QtQT 1 Qt_ Q 2 QltQT 2 _ QtQT 2 QtlT QT 2 Qt_ Q 5 Q_ tQT 2 QltQT 2 QtlT QT 5 Qð_ t 2 lt 2 tlT ÞQT 



with Eqs. (1.99) and (3.59). The objectivities of the vector and the tensor in the other variants can be proved analogously.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

3.5 Convected time-derivative

99

The tensor is calculated for the corotational rate in Eq. (3.49) by the time-integration ð ð  3 t 5 tdt 5 t 1 ωt 2 tω dt (3.68)

_

3

where t is used for the stress and the internal variables in the constitutive relation. The incorporation of the corotational rate is required for the hypoelastic-based constitutive equations since it is formulated in the current configuration, which is influenced by the rigid-body rotation. On the other hand, the multiplicative hyperelastic-based plastic constitutive equation is formulated in the intermediate configuration, which is independent of the rigid-body rotation as will be shown in Section 8.1.2. In addition, the elastic deformation described by the hypoelastic-based plastic constitutive equation is limited to be infinitesimal as will be verified in Section 8.2.3. The hypoelastic-based plastic constitutive equation with the corotational rate has been reluctantly used in order to replace the constitutive equation based on the infinitesimal strain, which is limited to the infinitesimal deformation without the rotation, and thus it should be replaced to the hyperelastic-based constitutive equation in the near future.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

C H A P T E R

4 Deformation/rotation (rate) tensors Any tensors describing a deformation, i.e. strain tensors are represented by the deformation gradient tensor which transforms the reference infinitesimal line-element vector to the current one. However, note that the deformation gradient tensor depends not only on a deformation but also on a rotation of material. Therefore, the positive-definite part of the deformation gradient tensor, which describes a pure deformation, must be adopted for the strain and the strain rate tensors in constitutive equations. One has various strain tensors and its rate tensors satisfying this requirement, while note that this requirement is not satisfied in the so-called infinitesimal strain tensor defined by the symmetric part of the displacement gradient tensor. They are classified into the ones in the reference and the current configuration. The former and the latter are called the Lagrangian and the Eulerian tensor, respectively, while they are transformed to each other by the pull-back and the push-forward operations. Incidentally, one has various rotation tensors and its rate tensors, i.e., spin tensors. These deformation tensors, rotation tensors and their rate tensors are described in this chapter.

4.1 Deformation tensors The covariant pull-back (Eq. (3.24)4) of the metric tensor gðtÞ in the current configuration and the contravariant push-forward (Eq. (3.29)1) of the metric tensor G in the reference configuration lead to the right CauchyGreen tensor C and the left CauchyGreen tensor b, that is, ~

C  U2 5 FT F 5 FT gF 5 gGG 5 gij Gi  Gj ð 5 CT Þ ~ gg 5 Gij g  g ð 5 bT Þ b  V2 5 FFT 5 FGFT 5 G i

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity DOI: https://doi.org/10.1016/B978-0-12-819428-7.00004-3

101

(4.1)

j

© 2020 Elsevier Inc. All rights reserved.

102

4. Deformation/rotation (rate) tensors

as ascertained by

(

FT F 5 Gi  gi gj  Gj 5 gij Gi  Gj FFT 5 gi  Gi Gj  gj 5 Gij gi  gj

so that C and b are related to each other as follows: 8 < G C 5 FT bF2T 5 b G G : b 5 F2T CFT 5 C G ~

(4.2)

~

and their components by bij 5 Gij

Cij 5 gij ;

(4.3)

C and b are referred to as the right and left Cauchy 2 Green deformation tensors, respectively. Their magnitude is identical to each other, that is, jjCjj 5 jjbjj

(4.4)

noting trðCCT Þ 5 trðCCÞ 5 trðFT FFT FÞ 5 trðFFT FFT Þ 5 trðbbÞ 5 trðbbT Þ The inverses of C and b are given by the following equation. ~

C21 5 F21 F2T 5 F21 gF2T 5 g GG 5 gij Gi  Gj ð 5 C2T Þ ~ gg 5 Gij gi  gj ð 5 b2T Þ b21 5 F2T F21 5 F2T GF21 5 G

(4.5)

leading to C21 ij 5 gij ;

b21 ij 5 Gij

(4.6)

where C21 is called sometimes as the Piola deformation tensor and b21 as the Finger deformation tensor. Their magnitude is identical to each other, that is, jjC21 jj 5 jjb21 jj

(4.7)

The application of Eq. (1.286) for the polar decomposition in Section 1.6.6 to the deformation gradient F reads: F 5 RU 5 VR

(4.8)

for which the following relations hold

R 5 FU21=2 5 FðFT FÞ21=2 ;

U 5 C1=2 ðU 5 UT Þ

(4.9)

V 5 b1=2 ðV 5 VT Þ

(4.10)

R 5 V21 F 5 ðFFT Þ21=2 F ðdet R 5 1Þ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(4.11)

103

4.1 Deformation tensors

U 5 RT VR 5 RT F 5 FT R;

V 5 RURT 5 FR 5 RFT

(4.12)

detC 5 detb 5 J 2

(4.13)

detF 5 detU 5 detV 5 J;

where U and V are called the right and the left stretch tensors, respectively. Now, we designate the deformation gradient induced in the duration from the initial time t0 to the current time t by F0 ðtÞ and the one induced in the duration from t to an arbitrary time τ ð . tÞ by Ft ðτÞ. Further, we designate the deformation gradient induced in the duration from t0 to τ by F0 ðτÞ. Then, it follows that F 5 F0 ðtÞ 5

@xðtÞ ; @X

F0 ðτÞ 5

@xðτÞ ; @X

Ft ðτÞ 5

@xðτÞ @xðtÞ

(4.14)

which fulfill F0 ðτÞ 5 Ft ðτÞF0 ðtÞ

(4.15)

because of @xðτÞ @xðτÞ @xðtÞ 5 @X @xðtÞ @X The generalized deformation gradient Ft ðτÞ is referred to as the relative deformation gradient tensor and its polar decomposition is given as follows: Ft ðτÞ 5 Rt ðτÞUt ðτÞ 5 Vt ðτÞRt ðτÞ

(4.16)

The rate of deformation gradient Ft ðτÞ in which τ is chosen as the current time t, called the updated Lagrangian description, is given by @vðτÞ @vðtÞ jτ5t 5 j 5 _Ft ðtÞ 5 @F@τt ðτÞ jτ5t 5 @ð@xðτÞ=@xðtÞÞ @τ @xðtÞ τ5t @xðtÞ

(4.17)

Denoting the principal values and the eigenvectors of U and V by λα ðα 5 1; 2; 3Þ and fNα g and fnα g, respectively, the following relations hold from Eqs. (1.282), (1.283), and (1.288)(1.290). UNα 5 λα Nα ; U5

3 X

λα Nα  Nα ;

α51

Vnα 5 λα nα V5

3 X

λ α nα  n α

(4.18) (4.19)

α51

nα 5 RNα ; R5

3 X

N α 5 R T nα nα  Nα

α51

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(4.20) (4.21)

104

4. Deformation/rotation (rate) tensors

Nα ðtÞ and nα ðtÞ are called the Lagrangian triad and the Eulerian triad, respectively. The deformation gradient F and its inverse are described by Eq. (1.291) as follows: F5

3 X

λ α nα  N α ;

F21 5

α51

3 X 1 N α  nα λ α51 α

(4.22)

Let the mechanical meanings of U; V, and R be examined below. The variation of infinitesimal line-element is given by the polar decomposition F 5 RU as follows: dx 5 FdX 5 RUdX 5 R

3 X

λβ Nβ  Nβ

3 X

dXα Nα 5 R

α51

β51

3 X

λα dXα Nα (4.23)

α51

Eq. (4.23) means that the infinitesimal line-elements dXα Nα ðno sumÞ in the principal directions Nα are first stretched by λα times to λα dXα Nα ðno sumÞ and then undergoes the rotation R. On the other hand, the change in the infinitesimal line-element is described by the polar decomposition VR as dx 5 VRdX 5

3 3 3 3 X X X X λβ nβ  nβ R dXα Nα 5 λ β nβ  nβ dXα nα β51

α51

α51

α51

α51

β51

3 3 X X 5 λα dXα nα 5 λα RdXα Nα

(4.24)

which means that the infinitesimal line-elements dXα Nα ðno sumÞ in the principal directions Nα first becomes dXα nα ðno sumÞ by the rotation R and then it is stretched by λα to λα dXα nα ðno sumÞ. As described above, U and V designates the pure deformation and R the pure rotation. U and V are called the right and left stretch tensors, respectively, and λα is called the principal stretch. The deformation/rotation processes by the polar decomposition of deformation gradient are illustrated in Fig. 4.1. Letting RL and RE designate the rotations of the Lagrangian triad fNα g and the Eulerian triad fnα g, respectively, from the fixed base feα g, we define RL 

3 X

N α  eα ;

α51

RE 

3 X

n α  eα

α51

where the following relations exist.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(4.25)

105

4.1 Deformation tensors

FIGURE 4.1 Deformation/rotation processes by polar decomposition of deformation gradient tensor.

N α 5 R L eα ;

nα 5 R E e α

R 5 RR E

L

(4.26) (4.27)

The right and left Cauchy 2 Green deformation tensors are described, noting Eqs. (4.1), (4.12), and (4.19) as C 5 U2 5 RT V2 R 5 b 5 V 5 RU R 5 2

2

T

3 X α51 3 X α51

λ2α Nα  Nα

λ2α nα

(4.28)  nα

The principal stretches λα are calculated by solving the characteristic equation based on Eq. (1.240): λ3 2 IC λ2 2 IIC λ 1 IIIC 5 0

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(4.29)

106

4. Deformation/rotation (rate) tensors

where IC  trC; IIC 

 1 2 1 1 1 tr C 2 trC2 ; IIIC  tr3 C 2 trCtrC2 1 trC3 2 6 2 3 (4.30)

In order to calculate U; V, and R, it is first required to calculate the eigenvalues λα by Eq. (4.29) and eigenvectors Nα and nα by Eq. (1.237) for the known deformation gradient F and thereafter R can be calculated from Eq. (4.21).

4.2 Strain tensors Various strain measures as the Lagrangian and the Eulerian tensors are described in this section, which often appear in continuum mechanics.

4.2.1 Green and Almansi strain tensors The square of the length of line-element is described by the following equation, noting Eqs. (3.1), (3.4), (3.7), (4.3), and (4.6). ( dx dx 5 dθi gi dθj gj 5 gij dθi dθj 5 gij dΘι dΘj 5 Cij dΘi dΘj (4.31) i j dX dX 5 dΘi Gi dΘj Gj 5 Gij dΘi dΘj 5 Gij dθi dθj 5 b21 ij dθ dθ

 

that is,





 









dx dx 5 FdX FdX 5 dX FT FdX 5 dX CdX dX dX 5 F21 dx F21 dx 5 dx F2T F21 dx 5 dx b21 dx







(4.32)

Then, the change in the square of length of line-element is described from Eq. (4.31) or (4.32) as follows: 8 ðg 2 Gij ÞdΘi dΘj 5 ðCij 2 Gij ÞdΘi dΘj 5 2Eij dΘi dΘj > > > ij > < 5 dΘr Gr ½ð2Eij Gi  Gj ÞðdΘs Gs Þ 5 dX ½ð2Eij Gi  Gj ÞdX dx dx 2 dX dX 5 i j i j > ðgij 2 Gij Þdθi dθj 5 ðgij 2 b21 > ij Þdθ dθ 5 2eij dθ dθ > > : 5 dθr gr ½ð2eij gi  gj Þðdθs gs Þ 5 dx ½ð2eij gi  gj Þdx













(4.33) or





dx dx 2 dX dX 5



 

 

dX ðC 2 GÞdX 5 dX 2EdX dx ðg 2 b21 Þdx 5 dx 2edx

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(4.34)

107

4.2 Strain tensors

where 8 1 1 > > E 5 Eij Gi  Gj 5 ðgij 2 Gij ÞGi  Gj 5 ðCij 2 Gij ÞGi  Gj > < 2 2 1 1 > 21 i i j j i j > > : e 5 eij g  g 5 2 ðgij 2 Gij Þg  g 5 2 ðgij 2 bij Þg  g

(4.35)

that is, 2 0 1 3   1 4 @x T @ @x A 1 1 1 1 E5 2 G5 5 ðFT F 2 GÞ 5 ðC 2 GÞ 5 ðU2 2 GÞ 5 ðgGG 2 GÞ 2 @X @X 2 2 2 2 ~

1 5 ðgij 2 Gij ÞGi  Gj 2 2 0 13  T 14 @X @@XA5 1 1 1 1 ~ gg Þ g2 e5 5 ðg 2 F2T F21 Þ 5 ðg 2 b21 Þ 5 ðg 2 V22 Þ 5 ðg 2 G 2 @x @x 2 2 2 2 1 5 ðgij 2 Gij Þgi  gj 2

(4.36) noting Eqs. (4.1) and (4.5). E and e are called the Green 2 Lagrangian strain tensor and the Almansi 2 Eulerian strain tensor, respectively. It should be noted here that (1) they are expressed as the variation of the metric tensor in the reference and the current configurations and (2) they are described only by the right or left CauchyGreen deformation tensor, excluding the rotation tensor and thus describing the pure deformation exactly. Then, they are often called the finite strain tensor. The physical meanings and necessity of these strains can be captured clearly from the expressions in terms of the metric tensor of the embedded coordinate system as shown in Eq. (4.35) or (4.36). That is, the Green strain tensor E is regarded as the half of the subtraction of the reference metric tensor G from the covariant pull-back of the current metric tensor g, that is, C. On the other hand, the Almansi strain tensor e is regarded as the half of the subtraction of the covariant push-forward of the reference metric tensor G, that is, b21 from the current metric tensor g, noting the relations described in Section 4.1. The strain tensors E and e are related by the covariant pull-back and the push-forward operation in Eq. (3.24)4 and the covariant pushforward operation in Eq. (3.29)4 as follows: ~

E5e

GG

~gg 5 FT eF; e 5 ~ E 5 F2T EF21

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(4.37)

108

4. Deformation/rotation (rate) tensors

Limiting to the fixed coordinate system, the strain tensors E and e in Eq. (4.36) are represented in terms of the position vectors X, x ð 5 X 1 uÞ, and the displacement vector u as follows: 2 3 8  T > 1 @x @x > > > 2 G5 E 5 4 > > 2 @X @X > > > 2 3 2 3 > >  T  T  T > > > 1 @ðX1uÞ @ðX 1 uÞ 1 @u @u @u @u > 5 > 2 G5 5 4 1 5 4 1 > > < 2 @X @X 2 @X @X @X @X 2 3  T > > 1 @X @X > > 5 > e 5 4g 2 > > 2 @x @x > > > 2 3 2 3 > >  T  T  T > > > 1 @ðx2uÞ @ðx 2 uÞ 1 @u @u @u @u > 55 4 1 5 > 5 4g 2 2 > : 2 @x @x 2 @x @x @x @x (4.38) The component descriptions of the strains in terms of the displacement vector require the covariant derivative as explained in the literature (Hashiguchi and Yamakawa, 2012). The current line-element vecor is approximately identical with the reference one leading to dxDdX under the infinitesimal deformation and rotation. The second-order infinitesimal terms in Eq. (4.38) may be ignored in this situation, so that Eq. (4.38) leads to 2 3 2 3  T @u 1 @u @u 5 1 1 5 ðF 1 FT Þ 2 I 5 ðRU 1 URT Þ 2 I ε  sym4 5 5 4 1 @X 2 @X @X 2 2 0 1 @uj A 1 @ui ei  ej 1 5 @ 2 @Xj @Xi (4.39) or

2

3

2

3  T @u 1 @u @u 5 1 1 5I2 ðF21 1F2T Þ5I2 ðRT V21 1V21 RÞ ε  sym4 5 5 4 1 @x 2 @x @x 2 2 0 1 1 @ui @uj 5 @ 1 A e i  ej 2 @xj @xi (4.40)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

109

4.2 Strain tensors

noting

8 @u @ðx 2 XÞ > > 5 5F2I > < @X @X @u @ðx 2 XÞ > > 5 I 2 F21 > : @x 5 @x

(4.41)

The strain ε in Eq. (4.39) or (4.40) describes roughly a deformation, depending not only on U or V but also on the rotation tensor R. Then, ε is called the infinitesimal strain tensor. The infinitesimal strain tensor is reduced to the nominal strain e in the one-dimensional deformation, which satisfies neither additive nor multiplicative superposition, that is,

e0Bn 5

8 l1 2 l0 l2 2 l1 ln 2 ln21 > > 1 1...1 5 e0B1 1 e1B2 1 . . . 1 eðn21ÞBn > l < l1 ln21 0

ln 2 l0 6¼ l1 2 l0 l2 2 l1 ln 2 ln21 > l0 > ... 5 e0B1 e1B2 . . .eðn21ÞBn > : l l l 0

1

n21

(4.42) where l0 is the initial value of the length l of material.

4.2.2 General strain tensors The second-order tensor function fðUÞ of U is coaxial to U, and likewise the function fðVÞ of V is coaxial to V. Therefore it follows noting Eq. (4.19) that 8 3 X > > > fðλα ÞNα  Nα > E  fðUÞ 5 < α51 (4.43) 3 X > > > > fðλα Þnα  nα : e  fðVÞ 5 α51

These functions are related by E 5 RT eR;

e 5 RERT

(4.44)

noting E5

3 X α51

fðλα ÞRT nα  RT nα 5 RT

3 X

fðλα Þnα  nα R

α51

where E and e are referred to as the generalized strain measure or Hill’s strain measure (Hill, 1968, 1978).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

110

4. Deformation/rotation (rate) tensors

Seth (1964) proposed the following particular forms of fðUÞ and fðVÞ in Eq. (4.43): 8 < 1 ðU2m 2 GÞ for m 6¼ 0 (4.45) EðmÞ 5 fðUÞ 5 2m : ln U for m 5 0 and 8 < 1 ðV2m 2 gÞ for m 6¼ 0 ðmÞ e 5 fðVÞ 5 2m : ln V for m 5 0

(4.46)

where 2m is a positive or negative integer. The Green and the Almansi strain tensors are given as E 5 Eð1Þ and e 5 eð21Þ . The strain tensors in Eqs. (4.45) and (4.46) are related as follows:   ’  -  ðmÞ T ðmÞ ðmÞR ðmÞ ðmÞ T ðmÞR E 5R e R 5 e ; e 5 RE R 5 E (4.47) noting Eq. (4.44) with Eqs. (4.45) and (4.46). The principal values of the general strain tensors EðmÞ and eðmÞ in Eqs. (4.45) and (4.46) are identical to each other, and they are given by 8 < 1 ðλ2m 2 1Þ for m 6¼ 0 α (4.48) fðλα Þ 5 2m : ln λα for m 5 0 The function fðλα Þ fulfills fð1Þ 5 0; f 0 ð1Þ 5 1

(4.49)

f 0 ðsÞ . 0

(4.50)

and

for an arbitrary positive scalar s. 0 ^ s rule as (Note) Eq. (4.48)2 for m 5 0 is derived using the l0 Hopital follows:

limm-0

 1  2m @fexpð2mlnλα Þ 2 1g=@m λ 2 1 5 lim m-0 2m α @ð2mÞ=@m expð2mlnλα Þð2lnλα Þ 5 ln λα 5 lim m-0 2

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

111

4.2 Strain tensors

The substitution of Eq. (4.48) into Eq. (4.43), the Seth-Hill’s strains in Eqs. (4.45) and (4.46) are described in the spectral decompositions as follows: 8 3 X 1   > > > λ2m for m 6¼ 0 > α 2 1 Nα  Nα < 2m α51 ðmÞ (4.51) E 5 3 > X > > > ln λα Nα  Nα for m 5 0 : α51

e

ðmÞ

8 3 X 1   > > > λ2m > α 2 1 nα  nα < 2m

for m 6¼ 0

> > > :

for m 5 0

5 α51 3 >X

ln λα nα  nα

(4.52)

α51

The following strain measures are used often in continuum mechanics. 1 1 1 Eð1Þ 5 E5 ðU2 2 GÞ 5 ðC 2 GÞ 5 ðFT F 2 GÞ 2 2 2   E 1=2 5 B 5U 2 G 5 C1=2 2 G 5ðFT FÞ1=2 2G 1 1 Eð0Þ 5 lnU 5 lnC 5 lnðFT FÞ 2 2 1 1 1 Eð21Þ 5 ðG 2 U22 Þ5 ðG 2C21 Þ 5 ðG 2 F21 F2T Þ 2 2 2

materialGreen strain material Biot strain material Hencky strain Piola strain (4.53)

1 1 1 eð1Þ 5 ðV2 2 gÞ5 ðb2 gÞ5 ðFFT 2gÞ 2 2 2 1 1 eð0Þ 5 ln V 5 ln b 5 lnðFFT Þ 2 2   e 21=2 5 g2 V21 5g 2b21=2 5g 2ðFFT Þ21=2 1 1 1 eð21Þ 5 e5 ðg2 V22 Þ5 ðg 2 b21 Þ 5 ðg 2F2T F21 Þ 2 2 2

Finger strain spatial Hencky strain spatial Biot strain Almansi strain (4.54)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

112

4. Deformation/rotation (rate) tensors

The generalized strain tensors in Eqs. (4.45) and (4.46) can be represented in terms of the metric tensors, noting Eqs. (4.1) and (4.5), as follows: 8 1 > > > ððgGG Þm 2 GÞ for m . 0 > > 2m > > > >

> > > > > 1 > > ððg Þ2m 2 GÞ for m , 0 > : 2m GG ~

~

~

and

8 1 ~ m > > ððG gg Þ 2 gÞ > > 2m > > > >

> > > > 1 ~ gg 2m > > > : 2m ððG Þ 2 gÞ

for m . 0 for m 5 0

(4.56)

for m , 0

Then, the following expressions hold for the popular strain tensors: 8 1 1 > > > Eð1Þ 5 ðgGG 2GÞ5 ðgij 2Gij ÞGi  Gj Green strain > > 2 2 > > <   (4.57) E 1=2 5ðgGG Þ1=2 2G material Biot strain > > > > > Eð21Þ 5 1 ðG2 g Þ5 1 ðGij 2gij ÞGi  Gj material Piola strain > > GG : 2 2 ~

~

~

8 1 ~ 1 ij ij > > eð1Þ 5 ðG gg 2 gÞ 5 ðG 2 g Þgi  gj Finger strain > > 2 2 > > <   ~ gg Þ1=2 spatial Biot strain e 21=2 5 g 2 ðG > > > gg 1 1 > eð21Þ 5 ðg 2 G ~ Þ 5 ðg 2 Gij Þgi  gj Almansi strain > > : 2 2 ij

(4.58)

while strains for m 5 0 are shown in Eqs. (4.55) and (4.56) already. Any strain tensor can be described by the variations in the metric tensor in the convected coordinate system. The importance of the incorporation of convected coordinate system for the interpretation of physical meanings of strains could be convinced by this fact.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

113

4.2 Strain tensors

4.2.3 Logarithmic strain tensor In the particular case of m 5 0 in the general strain tensor of Eqs. (4.51) and (4.52), noting λα $ 0, the following strain measures are called the Logarithmic or Hencky strain tensor. right-Hencky strain tensor:

Eð0Þ 5

3 X

ln λα Nα  Nα  ln U 5

α51

left-Hencky strain tensor:

1 lnC 2

3 X 1 eð0Þ 5 ln λα nα  nα  ln V 5 lnb 2 α51

(4.59) where λα ðα 5 1; 2; 3Þ is the eigenvalues U α and V α of U and V, that is, pffiffiffiffiffi pffiffiffiffiffiffi λα  Uα 5 Cα 5 Vα 5 bα (4.60) When the principal directions of C and b are fixed, the following equations hold. @xα ðEð0Þ Þ 5 ðeð0Þ Þα 5 ln λα 5 ln ðno sumÞ (4.61) @Xα   @xα =@Xα @x    ð0Þ ð0Þ   5 α 5 dαα ðno sumÞ (4.62) ½ðE Þα  5 ½ðe Þα  5 ðln λα Þ 5 @xα @xα =@Xα

_

where ln λα in Eq. (4.61) is the logarithmic strain and dαα ðno sumÞ is the normal component in the strain rate tensor d defined in the next section. It holds from Eq. (4.61) that 8 3 3 3 X X > 1X > ð0Þ > > trE 5 ln λ 5 ln C 5 ln Uα 5 ln ðU1 U2 U3 Þ α α > < 2 α51 α51 α51 (4.63) 3 3 3 X X > 1X > ð0Þ > > ln λα 5 ln bα 5 ln Vα 5 ln ðV1 V2 V3 Þ tre 5 > : 2 α51 α51 α51 which is identical to the logarithmic volumetric strain, that is, trEð0Þ 5 treð0Þ 5 lnðλ1 λ2 λ3 Þ 5

3 X α51

ln

@xα dv  εv 5 ln J 5 ln dV @Xα

(4.64)

Obviously, the logarithmic strain satisfies the multiplicative superposition in the one-dimensional deformation, that is, ) ( ð0Þ ð0Þ E0B1 E1B2 . . .Eð0Þ l l l l Eð0Þ n 1 2 n ðn21ÞBn 0Bn 5 ln ln . . .ln 5 5 ln (4.65) ð0Þ ð0Þ ð0Þ l l l l eð0Þ e e . . .e 0 0 1 n21 0Bn 0B1 1B2 ðn21ÞBn

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

114

4. Deformation/rotation (rate) tensors

4.3 Volumetric and isochoric parts of deformation gradient tensor The deformation gradient tensor F is defined as the ratio of the length of the infinitesimal line-element in the current configuration to that of the reference configuration. Therefore it may not be additively decomposed but it is obliged to be multiplicatively decomposed. The multiplicative decomposition of the deformation gradient into and the volume preserving, that is, isochoric (distortional or volume-preserving) part F will be described in the following, where F 5 Fvol F; Fvol  ðdetFÞ1=3 g;

F ðdetFÞ21=3 F

(4.66)

noting F 5 gi  Gi 5 δir gi  Gr 5 gi  gi gr  Gr 5 gF 5 ðdetFÞ1=3 gðdetFÞ21=3 F 5 Fvol F based on Eq. (3.9), where it holds that detFvol 5 detF 5 J; detF 5 1 noting h i h i3 detFvol 5 det ðdetFÞ1=3 g 5 ðdetFÞ1=3 detg

(4.67)

h i h i3 detF 5 det ðdetFÞ21=3 F 5 ðdetFÞ21=3 detF 5 ðdetFÞ21 detF by Eq. (1.189)2 with s 5 ðdetFÞ21=3 . Then, the tensor F for the isochoric part is called a unimodular tensor. Then, let the following decomposition of the right CauchyGreen deformation tensor be defined. C 5 Cvol C Cvol FTvol Fvol 5 ðdetCÞ1=3 g; C FT F 5 ðdetCÞ21=3 C

(4.68)

satisfying detCvol 5 detC 5 J 2 ;

detC 5 1

(4.69)

by Eq. (1.189)2. It holds that detF 5 detC 5 1;

0

trC 5 3; C 5 O for F 5 Fvol

(4.70)

The following partial derivatives hold for the volumetric and the isochoric tensors. Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

4.3 Volumetric and isochoric parts of deformation gradient tensor

@Cvol 1 1 5 ðdetCÞ1=3 g  C21 5 J 2=3 g  C21 3 3 @C C 1 5 ðdetCÞ21=3 ð 2 C  C21 1 SÞ @ @C 3 pffiffiffiffiffiffiffiffiffiffiffi @detC @ detC 1 21 21 @J 5 ðdetCÞC ; 5 5 JC @C @C @C 2 pffiffiffiffiffiffiffiffiffiffiffi @ln detC 1 1 5 ðdetCÞC21 5 C21 @C 2detC 2 2 3 2 3 C 1 1 5 ðdetCÞ21=3 4G 2 ðtrCÞC21 5 5 J22=3 4G 2 ðtrCÞC21 5 @tr @C 3 3

115

(4.71)

noting @Cvol @ðdetCÞ1=3 g 1 @detC 5 g  ðdetCÞ22=3 5 @C 3 @C @C 1 1 5 g  ðdetCÞ22=3 ðdetCÞC21 5 g  ðdetCÞ1=3 C21 3 3 h i 2 3 @ ðdetCÞ21=3 C C 1 5C  42 ðdetCÞ24=3 ðdetCÞC21 5 1 ðdetCÞ21=3 S 5 @ @C @C 3 pffiffiffiffiffiffiffiffiffiffiffi @ detC 1 @detC 1 5 pffiffiffiffiffiffiffiffiffiffiffi 5 pffiffiffiffiffiffiffiffiffiffiffi ðdetCÞC21 @C 2 detC @C 2 detC pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi @ln detC 1 @ detC 5 pffiffiffiffiffiffiffiffiffiffiffi @C detC @C h i 21=3 @ ðdetCÞ trC @trC 1 5 2 ðdetCÞ24=3 ðdetCÞC21 trC 1 ðdetCÞ21=3 G 5 @C @C 3 by virtue of Eqs. (1.301)1 and (1.313) and (1.315). The deviatoric part in the logarithm of the right stretch (right-logarithmic (Hencky) strain) tensor ln U in Eq. (4.59) is described as follows: ðlnUÞ0 5

3  X α51

3 X lnλα 2 ðlnJÞ=3 Nα  Nα 5 ðlnλ α ÞNα  Nα

(4.72)

α51

Where λ α is the isochoric part of the stretch λα , that is, pffiffiffiffiffiffi λ α 5 λα J21=3 5 λα ðU 1 U 2 U 3 Þ21=3 5 Cα ðC1 C2 C3 Þ21=6

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(4.73)

116

4. Deformation/rotation (rate) tensors

leading to



λ 1 λ 2 λ 3 5 λ1 λ2 λ3 J21=3 5 1 lnðλ 1 λ 2 λ 3 Þ 5 1

(4.74)

The second invariant J 2 of deviatoric tensor ðlnUÞ0 is given noting Eq. (1.250) by J2 5

1 1 trðlnUÞ02 5 ðlnλ 1 Þ2 1 ðlnλ 2 Þ2 1 ðlnλ 3 Þ2 2 2

(4.75)

from which one has @J 2 5 lnλ α @lnλ α

(4.76)

It is follows from Eq. (4.73) that hpffiffiffiffiffiffi i 21=6 ðC @ln C C C Þ α 1 2 3 @lnλ α 5 @Cβ @Cβ 2 3 21=6 ffiffiffiffiffiffi p 7 1 ðC C C Þ 1 C C C 2 3 1 2 35 4 1 p ffiffiffiffiffiffi 5 pffiffiffiffiffiffi Cα ðC1 C2 C3 Þ2 6 δαβ 2 6 Cβ 2 Cα Cα ðC1 C2 C3 Þ21=6 leading to @lnλ α 1 1 1 5 δαβ 2 2Cα 6 Cβ @Cβ

(4.77)

It follows by virtue of Eqs. (4.76) and (4.77) that 0 1 3 3 X X @lnλ @J2 @J2 1 1 1A β 5 5 lnλ β @ δαβ 2 2C 6 Cα @Cα β51 @lnλ β @Cα β β51 5

3 1 1 X ðlnλ α ÞC21 2 lnλ β α 2 6Cα β51

from which it follows that  @J2 1 5 lnC1=2 0 C21 2 @C

(4.78)

The volumetric and the isochoric decomposition of deformation gradient described in this section is often adopted in formulations of hyperelastic constitutive equation as will be addressed in Chapter 6.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

117

4.4 Strain rate and spin tensors

4.4 Strain rate and spin tensors Various strain rate and spin tensors adopted in rate (incremental) type constitutive equations will be described in this section.

4.4.1 Strain rate and spin tensors based on velocity gradient tensor The contravariant-covariant pull-back of l in Eq. (1.362) by Eq. (3.24)2 is given as the Lagrangian tensor ~

21 G L  l G 5 F lF

(4.79)

which is the workconjugate pair of the Mandel stress as will be described in Section 5.4. Furthermore, the covariant pull-back of l by Eq. (3.24)4 is given by  T @x @x (4.80) ℒ  l GG 5 ðgi gr ÞGr  Gi 5 FT lF 5 FT FF21 F 5 FT F 5 @X @X

_

_

_

~

which is also the Lagrangian tensor. It follows from Eqs. (4.35) and (4.80) that

_E 5 sym½ℒ 5 12 _gijGi  Gj

(4.81)

while one has the relation ~

_E  FT dFð 5 d

GG

Þ

(4.82)

noting h

_E 5 12 ð_FT F 1 FT_FÞ 5 FT 12 _FF21 1 ð_FF21ÞT

i F

Here, the strain rate tensor d is the symmetric part of l, while the continuum spin w is the antisymmetric part of l, that is, l5d1w where 8 1 > > d 5 sym½l 5 ðv  rx 1 rx  vÞ 5 symðgradvÞ > < 2 1 1 > > > : w 5 ant½l 5 2 ðv  rx 2 rx  vÞ 5 antðgradvÞ 5 2 rotv

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(4.83)

(4.84)

118

4. Deformation/rotation (rate) tensors

which can be rewritten by Eq. (3.14) as follows: 8 h i 1 ~ gg < d 5 sym g  gi 5 1 ðg g 1 i j i j g g Þg  g 5 g g  g 5 i i j i j g 2 h i 2 ij (4.85) : w 5 ant g  gi 5 1 ðg g 1 g g Þgi  gj i i j i j 2

_ _  _ _ 

_

_

~

_ _

noting Eqs. (2.31), (2.32), and j

                                                    !gg  i ðgij G  Gj Þ

~ gg 5g

_

~

_gijg  g 5 i

It follows that trl 5 noting

_vg 5 ðdvÞ vg



(4.86)

dv

_vg 5 ð½g1g2 g3 Þ 5 _g1  ðg2 3 g3Þ 1g1  ð_g2 3 g3 Þ 1 g1  ðg2 3 _g3 Þ 5 vg _ g1  g1 1 _ g2  g2 1 _ g3  g3 5 vg_ gi  gi 5 vg lgi  gi 5 vg trl ð 5 vg l:gÞ 

with the aid of Eqs. (1.47), (2.1), (2.2), and (3.14). Noting that the components of position vector of material particle on i the embedded coordinate system do not change, that is, θ 5 0, the rate of the covariant base vector is given noting Eq. (2.12) by

_

@v _gi 5 @θ@_xi 5 @θ i

(4.87)

_

The relation of the components of the rate of the Green strain tensor E and the strain rate tensor d is given by Eij 5 dij from Eqs. (4.81) and (4.85), and thus it is known that their components have same values, while E is the covariant pull-back of the strain rate tensor d as shown in Eq. (4.82). Substituting Eq. (4.8) into Eq. (3.14), l is described by U; V; and R as follows: 8 < RUR ~ T 1 ΩR (4.88) l5 : ~ 1 VΩR V21 V

_

_

_

_

where

_ _

_ _ _

~  UU21 ; V ~  VV21 U ΩR  RRT noting

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(4.89) (4.90)

119

4.4 Strain rate and spin tensors

 l5

_ _

_ _

ðRUÞ ðRUÞ21 5 ðRU 1 RUÞU21 RT  ðVRÞ ðVRÞ21 5 ðVR 1 V RÞRT V21 

(4.91)

ΩR is the relative spin which appeared already in Section 3.5.2 and the corotational rate based on it was called the Green-Naghdi rate, while the meaning of “relative” will be described later as to Eq. (4.99). Then, d and w are described in terms of U; V; and R from Eq. (4.88) as follows: 8 < d 5 Rsym½U ~ RT (4.92) : w 5 ΩR 1 Rant½ ~ RT U 8 < d 5 sym½V ~  1 sym½VΩR V21  (4.93) : w 5 ant½VΩR V21  1 ant½ ~  V

_

_

_

_

The following strain rate derived from Eqs. (4.1), (4.89), and (4.92) is called the rotationless (or corotational) strain rate (Storen and Rice, 1975). ~

 1 R 1 1 1 ~ 5 ðUU21 1 U21 UÞ 5 C22 CC22 5 RT dR D  d 5 sym U 2 2

_

_

_

_

(4.94)

which depends only on the right stretch tensor U or C. Noting

_RL RL

T

3 X

5



ðNα  eα Þ ðNβ  eβ ÞT 5

α;β51

3 X α;β51

_Nα  eαeβ  Nβ

it follows that

_L

T

ΩL  R RL 5

3 X α51

_Nα  Nα

(4.95)

and thus one has

_Nα 5 ΩL Nα

(4.96)

Therefore ΩL designates the spin of the Lagrangian triad fNα g of the right stretch tensor U and is called the Lagrangian spin tensor. On the other hand, noting

_

E

R RET 5

3 X



ðnα  eα Þ ðnβ  eβ ÞT 5

α;β51

3 X α;β51

_nα  eαeβ  nβ

we have

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

120

4. Deformation/rotation (rate) tensors

_

E

ΩE  R RET 5

3 X α51

and thus one has

_nα  nα

(4.97)

_nα 5 ΩE nα

(4.98)

Therefore ΩE describes the spin of the Eulerian triad fnα g of the left stretch tensor V and is called the Eulerian spin tensor. Further, it follows that

_

ΩR 5 RRT 5 5

3 X



ðnα  Nα Þ ðnβ  Nβ ÞT

α;β51 3 h i X nα  nα 2 nα  Nα ðNγ  Nγ ÞNβ  nβ

α;β;γ51

_

_

and thus the following relations hold. ΩR 5 ΩE 2 RΩL RT ;

ΩE 5 ΩR 1 RΩL RT ;

ΩL 5 RT ðΩE 2 ΩR ÞR

(4.99)

The physical meaning of the relative spin ΩR based on the name capped with the adjective “relative” is captured from Eq. (4.99) because it describes the difference of ΩE , that is, the spin of the Eulerian triad fnα g from ΩL based on the spin of the Lagrangian triad fNα g. ~ 5 V 5V ~ 5 O), it follows from In the rigid-body rotation (U 5 U Eqs. (4.88), (4.92), (4.95) and (4.99) that

_ _ _ _

d 5 O;

l 5 w 5 ΩR 5 ΩE ;

ΩL 5 O

(4.100)

4.4.2 Strain rate tensor based on general strain tensor The rates of the general strain tensors defined in Section 4.2.2 are described below. The rate of the general Lagrangian and Eulerian strain tensors are given by 8 ðmÞ 1 1  > > ðU2m Þ 5 ðUU2m21 1 UUU2m22 1 ? 1 U2m21 UÞ >E 5 > 2m 2m < (4.101) ðmÞ 1 1 > 2m  2m21 2m22 2m21 > > ðV ð e 5 Þ 5 V V 1 V V V 1 ? 1 V V Þ > : 2m 2m

_

_

_

_

_

_

_

_

For example, the rates of the general strain for m 5 6 1 are given by

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

121

4.4 Strain rate and spin tensors

8 GG GG ð1Þ > 1 > < E 5 E 5 C 5 FT dF 5 d 5 g =2 2  > > : ð21Þ E 5 F21 dF2T 5 d GG 5 g GG =2 ~

_

~

_

_

~

_

~

_

8  ð1Þ > 1 1 > ~ gg=2 < e 5 b 5 sym½lb 5 ðlb 1 blT Þ 5 G 2 2  > > : ð21Þ ~ gg=2 e 5 e 5 d 2 el 2 lT e 5 2 G

_

_

_

_

(4.102)

(4.103)

noting 8 ð1Þ T 2  T  T 21 2T T T > < E 5 ðU Þ =2 5 ðF FÞ =2 5 F ðFF 1 F F ÞF=2 5 F ðl 1 l ÞF=2

_ _ ð21Þ 22 21 2T E 5 2 ðU Þ =2 5 2 ðF F Þ =2 > :_ 21 2T 21 21 T 

_



5 2 ½ð 2 F lÞF

1 F ð2F lÞ =2 5 F21 ðl 1 lT ÞF2T =2

8 ð1Þ T T 1 1   > > e 5 ðV2 Þ =2 5 ðFFT Þ =2 5 ðFFT 1 FF Þ 5 ðFF21 FFT 1 FFT F2T F Þ > > 2 2 >
> > > > :





5 ½F F l 1 ðF lÞ F =2 5 ðb21 l 1 lT b21 Þ=2 5 ðg 2 2eÞl 1 lT ðg 2 2eÞ =2

with Eqs. (4.57) and (4.58). The following relation in Eq. (4.82) or (4.102)1 is often used hereinafter.

_E 5 12 _C 5 FT dF;

_

d 5 F2T EF21

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(4.104)

C H A P T E R

5 Conservation laws and stress tensors Conservation laws of mass, linear momentum, angular moment, angular momentum, angular moment and energy which must be satisfied during a deformation are first delineated in this chapter. Concurrently, the Cauchy stress tensor is defined as the Eulerian tensor. However, the material included in the unit current volume element and its mass changes during the deformation. Then, the stress and the strain rate adopted in constitutive equations for the exact description of the finite deformation must be defined for the specific volume element possessing a fixed mass in the reference configuration leading to the Lagrangian tensors. In addition, the work rate involved in the conservation laws must be the work rate done to the reference volume element, while the pair of stress and strain rate tensors inducing this work rate is called the work-conjugate pair. Then, various work-conjugate Lagrangian stress and strain rate tensors are derived from the Cauchy stress tensor and the Eulerian strain rate tensor in the systematic way by means of the pull-back operations.

5.1 Conservation laws Conservation laws in the Eulerian and the Lagrangian descriptions are described in this section, which must be fulfilled throughout all the deformation process.

5.1.1 Conservation law of physical quantity Scalar- or vector- or tensor-valued physical quantity φ must satisfy the following integrated relation in the Eulerian configuration. ð 
ð ð φdv 5 sφ dv 2 vφ nda (5.1) v

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity DOI: https://doi.org/10.1016/B978-0-12-819428-7.00005-5

v

123

a




© 2020 Elsevier Inc. All rights reserved.

124

5. Conservation laws and stress tensors

where vφ is the flow velocity of φ and sφ is the source strength of φ. The left-hand side in Eq. (5.1) is transformed by the Reynolds’ transportation theorem in Eq. (1.342) as follows: ð 
ð   φdv 5 φ_ 1 φdivv dv (5.2) v

v

Substituting Eq. (5.2) and using Eq. (1.339), Eq. (5.1) is rewritten as @φ 1 divðφv 1 vφ Þ 5 sφ @t

(5.3)

which is the conservation law in the Eulerian description in the local form. Denoting the flow velocity of a scalar- or vector- or tensor-valued physical quantity Φ by VΦ and the source strength of Φ by SΦ in the reference configuration, the conservation law in Eq. (5.1) is described in the Lagrangian description as follows: ð 
ð ð ΦdV 1 Vφ NdA 5 Sφ dV (5.4) V

A




V

where V is the volume occupied in the reference configuration. Transforming Eq. (5.4) to the volume integration by use of the Gauss’s divergence theorem in Eq. (1.339), one has ð ð ð _ (5.5) ΦdV 1 DivVφ dV 5 Sφ dV V

V

V

leading to the conservation law in the Lagrangian description in the local form _ 1 DivVΦ 5 SΦ Φ

(5.6)




where DivðÞ  ð@=@XÞ ðÞ.

5.1.2 Conservation law of mass The mass m involved in the material Ð possessing the mass density ρðx; tÞ and the volume v is given as m 5 v ρðx; tÞdv. The following conservation law of mass in the Eulerian description must be satisfied. ð 
_ 50 ρðx; tÞdv ð 5 mÞ (5.7) v

which requires the following relationship for the infinitesimal volume element. ρðx; tÞdv 5 ρ0 ðXÞdV

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(5.8)

125

5.1 Conservation laws

where ρ0 is the mass density in the reference configuration. Applying the Reynolds’ transportation theorem in Eq. (1.342), the following local form of Eq. (5.7) is obtained. ρ_ 1 ρdivv 5 0

(5.9)

which is referred to as the continuity equation. Further, setting Tðx; tÞ  ρφ, where φ is a physical quantity per unit current mass density, it follows from Eq. (5.2) that ð 
ð    ð  @vr @vr
_ ρφdv 5 ðρφÞ 1 ρφ ρφ 1 ρ_ φ 1 ρφ dv 5 dv (5.10) @xr @xr v v v and substituting Eq. (5.9) into Eq. (5.10), there follows ð 
ð _ ρφdv 5 ρφdv v

(5.11)

v

The conservation law of mass in Eq. (5.7) is described in the Lagrangian description as follows: ð 
  _ 50 ρ0 ðXÞdV 5M (5.12) V

where M is the mass density possessed in the reference volume V. The local form of Eq. (5.12) is written as ρ_ 0 ðXÞ 5

@ρ0 ðXÞ 50 @t

(5.13)

which is a trivial equation.

5.1.3 Conservation law of linear momentum The rate of linear momentum is equivalent to the sum of the traction and the body force applied to the body. Therefore, the following Euler’s first law of motion (or conservation law of linear momentum) in the Eulerian description must be satisfied. ð 
ð ð ρvdv 5 tda 1 ρbdv v

or

a

ð

ð

ð

_ 5 ρvdv v

v

tda 1 a

ρbdv

(5.14)

v

by virtue of Eq. (5.11), where t is the traction vector, that is, the force vector applied to per unit area on the surface of the body and b is the body force vector per unit mass.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

126

5. Conservation laws and stress tensors

The linear Ð momentum possessed by the unit reference volume is given by V ρ0 vdV. On the other hand, denoting the traction vector applied to the unit reference area of surface in the region as Τð 5 tda=dAÞ, Ð the total stress vector applied to the surface of the region is given as A TdA, and the body force applied to the region is given by Ð ρ bdV. The conservation law of linear momentum in the Lagrangian V 0 description is given as ð 
ð ð ρ0 vdV 5 ΤdA 1 ρ0 bdV V

or

A

ð V

ð

ð _ ρ0 vdV 5

V

TdA 1 A

V

ρ0 bdV

(5.15)

by virtue of Eqs. (5.8) and (5.11) with ΤdA  tda.

5.1.4 Conservation law of angular momentum Ð The rate of the angular momentum Ðð v ρðx 3 vÞdvÞ
coincides with the

sum of the angular moment of traction a ðx 3 tÞda and the angular moment Ð of body force v ρðx 3 bÞdv. Then, the following Euler’s second law of motion (or conservation law of angular momentum) must be satisfied. ð 
ð ð ρx 3 vdv 5 x 3 tda 1 ρx 3 bdv v

a

which is reduced to ð

ð

v

ð

_ 5 ρx 3 vdv

x 3 tda 1

v

a

ρx 3 bdv

(5.16)

v

_ noting ðx 3 vÞ
5 v 3 v 1 x 3 v_ 5 x 3 v. The conservation law of angular momentum in Eq. (5.16) is rewritten in the reference configuration as follows: ð 
ð ð ρ0 X 3 vdV 5 X 3 ΤdA 1 ρ0 X 3 bdV V

A

which reduces to ð V

ð _ ρ0 X 3 vdV 5

V

ð X 3 ΤdA 1

A

V

ρ0 X 3 bdV

_ 5 0. noting X

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(5.17)

127

5.2 Cauchy stress tensor

5.2 Cauchy stress tensor The Cauchy stress tensor is first defined in this section. Various stress tensors are defined from this tensor through various types of the pullback operations. The explanation in this section is the elaboration of the descriptions in the past literatures (cf., e.g., Flu¨gge, 1972; Hashiguchi, 2017a; etc.).

5.2.1 Definition of Cauchy stress tensor The surface vector da of the triangle ΔABC of the infinitesimal tetrahedron formed by the infinitesimal line vectors dθ1 g1 , dθ2 g2 , and dθ3 g3 parallel to the primary base vectors in the current configuration shown in Fig. 5.1 is given as follows (The expressions in the normalized orthogonal coordinate system are described inside the bracket): da 5 dan 5

3 X

dai

(5.18)

i51

where dai 5 dai gi ðno sumÞ

(5.19)

dai 5 ðdv=dθi Þ=2

(5.20)

dv 5 ½dθ1 g1 dθ2 g2 dθ3 g3 

(5.21)

 i  da 5 dai ei ðno sumÞ; dai 5 ðdv=dxi Þ=2; dv 5 dx1 dx2 dx3

t da − t 1 || d a1 ||

− da1 = − da1g1

C

da

σ g1 21

−σ 22 g 2

− t || d a ||

da 2

B

dθ 2 g 2

σ 23 g 3

− da 2 = − da2 g 2 2

da = da n

n

dθ 3 g 3

dv = [ dθ 1g1 dθ 2 g2 dθ 3 g3] / 2

O

2

dθ 1 g 1 −t 3 || d a3 ||

A

− da 3 = − da3 g 3

FIGURE 5.1 Equilibrium of force applied to tetrahedron formed by edges parallel to primary base vectors in the current configuration.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

128

5. Conservation laws and stress tensors

noting !

!

da 5 dan 5 BA 3 CB 5 ðdθ2 g2 2 dθ1 g1 Þ 3 ðdθ3 g3 2 dθ2 g2 Þ=2 5 ½g1 g2 g3 ðdθ2 dθ3 g1 1 dθ3 dθ1 g2 1 dθ1 dθ2 g3 Þ=2 5 ½dθ1 g1 dθ2 g2 dθ3 g3 ½ð1=dθ1 Þg1 1 ð1=dθ2 Þg2 1 ð1=dθ3 Þg3 =2 5 dv½ð1=dθ1 Þg1 1 ð1=dθ1 Þg2 1 ð1=dθ3 Þg3 Þ=2 noting Eq. (2.1). The fact that the surface vector dai in Eq. (5.19) with Eq. (5.20) designates the infinitesimal triangle composing the tetrahedron is confirmed as shown for the example i 5 1 in the following. da1 5 dθ2 g2 3 dθ3 g2 =2 5 dθ2 dθ3 ½g1 g2 g3 g1 =2 5 ½fðdθ1 dθ2 dθ3 ½g1 g2 g3 Þ=dθ1 g=2g1 5 ½ðdv=dθ1 Þ=2g1

(5.22)

The following expression holds.




dai 5 ðdan gi Þgi ðno sumÞ

(5.23)




ðdai 5 dai ei 5 ðdan ei Þei ðno sumÞÞ leading to




:dai : 5 dan gi :gi : ðno sumÞ   :dai : 5 dan ei

(5.24)




Now, designating the stress (force per unit area) vector applied to the surface ΔABC with the surface vector da by t and the stress vectors applied to the surfaces ΔOAB; ΔOBC, and ΔOCA with the surface vectors dai 5 dai gi (no sum) by ti as shown in Fig. 5.1, the following equilibrium equation of the forces in the tetrahedron must hold. tda 2 ti :dai : 1 bdv 5 0

(5.25)

For the infinitesimal tetrahedron ðdv-0Þ, the substitution of Eq. (5.24) into Eq. (5.25) reduces to   (5.26) t 5 ti :gi : gi n   t 5 ti ðei nÞ







Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

5.2 Cauchy stress tensor

129




(5.27)

Here, we incorporate σij  ti :gi : gj ðno sumÞ   σij  ti ej




from which one has

  ti jjgi jj 5 ti jjgi jj gj gj 5 σij gj ðno sum for iÞ




(5.28)




ðti 5 ðti ej Þej 5 σij ej Þ that is, ti 5 σij gj =:gi : ðno sum for iÞ   ti 5 σij ej

(5.29)

Eq. (5.26) with Eq. (5.27) is rewritten as t 5 σT n

(5.30)

noting







t 5 ti :gi :ðgi nÞ 5 σij gj ðgi nÞ 5 σij gj  gi n   t 5 ti ðei nÞ 5 σij ej ðei nÞ 5 σij ej  ei n




with

(




σT 5 σij gj  gi 5 :gi :ti  gi ; σ 5 :gi :gi  ti










σij 5 gi σgj 5 gi gr  :gr :tr gj 5 :gi :ti gj  T  σ 5 σij ej  ei 5 ti  ei ; σ 5 ei  ti σij 5 ei σej 5 ei er  tr ej 5 ti ej




(5.31)




(5.32)




The objectivity of σT is verified by substituting n 5 Qn and t 5 Qt into Qt 5 QσT QT Qn derived from Eq. (5.30), leading to σT 5 QσT QT . Eq. (5.30) is referred to as the Cauchy’s fundamental theorem or Cauchy’s stress principle and σ to as the Cauchy stress tensor. It insists that the Cauchy stress tensor σ plays the role of the linear mapping of the unit surface vector n to the Cauchy stress vector t. The multiplication to the Cauchy stress vector by the magnitude of the base vector, that is, ti :gi :ðno sumÞ in Eq. (5.28) is decomposed into the Cauchy stress tensor components in the direction of the base vectors, that is, σij gj as illustrated for the relation of t2 :g2 : to σ2j gj in Eq. (5.27) (Fig. 5.1). The multiplication of the magnitude of base vector :gi : is

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

130

5. Conservation laws and stress tensors

required in the curvilinear coordinate system with the base vectors which are not unit vectors. Here, as known from Eq. (5.27), the first index in the components of the tensor σ denotes the direction of the surface to which the stress vector applies and the second index denotes the acting direction. The sign of stress component is defined to be positive when the stress component applies in the positive direction of the coordinate axis for the positive surface, and vice versa for the negative one. Here, the sign of surface is defined to be positive when the outward-normal vector has the positive direction of the coordinate axis. The Cauchy stress tensor σ is represented in general by Eq. (2.57) as σ 5 σij gi  gj 5 σi
j gi  gj 5 σi
gi  gj 5 σij gi  gj j

(5.33)

The relations between the contra-, co-, and mixed-variant components are given by Eq. (2.59) as follows: ( j σij 5 σi
r grj 5 gir σr
5 gir σrs gsj ; σi
j 5 σir grj 5 gir σr
s gsj 5 gir σrj (5.34) j σi
5 gir σrj 5 gir σr
s gsj 5 σir grj ; σij 5 gir σrs gsj 5 gir σr
j 5 σi
r grj All components are illustrated for the two-dimensional state in Fig. 5.2, noting Eq. (5.28).

5.2.2 Symmetry of Cauchy stress tensor Consider the equilibrium of the angular moment applied to the infinitesimal parallelopiped element formed by the infinitesimal sides dθ1 g1 ; dθ2 g2 ; dθ3 g3 parallel to the current primary base fgi g (see Fig. 5.3) in order to verify the symmetry of the Cauchy stress tensor. The load vectors dPi applied to the surface 2dai of the infinitesimal parallelopiped element is given as 

dPi  ti :2dai : 5 σij gj ðdv=dθi Þ ðno sum for iÞ

(5.35)

 dPi  ti jj2dai jj 5 σij ei ðdv=dxi Þ ðno sum for iÞ

noting

   ti :2dai : 5 σij gj =:gi : :2dai gi : 5 σij gj 2dai 5 σij gj dv=dθi ðno sum for iÞ     ti :2dai : 5 σij ej :2dai ei : 5 σij ej 2dai 5 σij ej dv=dxi ðno sum for iÞ

with Eqs. (5.19), (5.20),and (5.29). The angular moment by the load vector dP2 is given by dP2 3 dθ2 g2 for an example as known from Fig. 5.3. The sum of the angular moments must be zero by the equilibrium of the angular moments, noting Eq. (5.35), that is, Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

131

5.2 Cauchy stress tensor 2 2 σ 22 g 2 t || g ||

t 2 || g 2 ||

σ •22 g 2

g2

t 1|| g 1||

σ 21 g1 σ 12 g 2

t 1|| g 1||

σ •21g1

σ •12 g 2

σ 11g1 σ •11 g1

g1 −g

− g1

1

g2

g2 −t || g 1|| 1

− t || g || 1

1

g1 g3 − g2

− t 2 || g 2 ||

(A)

g1

g3

g2

− g2

−t 2 || g 2 || (B)

g2

g1

σ 2•2 g 2

g2

g1

t 2 || g 2||

t 2 || g 2||

σ 22 g 2

σ 2•1 g1 σ 1•2 g 2

σ 21g1

t1 || g1 ||

σ 12 g 2

σ 1•1g 1 g2

−t1 || g1 ||

g3

(C)

g1

−t 2 || g 2|| − g 2

g1

σ 11g1

− g1

− g1 −t 1 || g1 ||

t 1 || g1 ||

g2

g3

(D)

g1

−t 2 || g 2|| − g 2

FIGURE 5.2 Traction vectors and Cauchy stress components acting on parallelopiped formed by base vectors (illustrated in the two-dimensional state). (A) Contravariant components. (B) Contravariantcovariant components. (C) Covariantcontravariant components. (D) Covariant components. 3 X

dPi 3 dθi gi 5 dvσis gs 3 gi

i51

5 dvð2 σ12 g3 1 σ13 g2 2 σ23 g1 1 σ21 g3 2 σ31 g2 1 σ32 g1 Þ

       5 2 dv σ23 2 σ32 g1 1 σ31 2 σ13 g2 1 σ12 2 σ21 g3 5 0

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(5.36)

132

5. Conservation laws and stress tensors

d P 3 = t 3 || 2da 3 ||

2da3 = 2 da3 g 3 dθ 3 g 3

− 2da2 −dθ 2 g 2 /2 dθ 2 g 2 /2

d P 2 = t 2 || 2da 2 ||

0

−d P 2

d θ 1 g1

2da1 = 2 da1g1

dθ 2 g 2

2da2 = 2 da2 g 2

d v = [dθ 1 g 1 dθ 2 g 2 dθ 3 g 3]

d P = t || 2da || 1

1

1

FIGURE 5.3 Load vectors acting on parallelopiped volume element formed by infinitesimal sides parallel to current convected primary base vectors.

0 B B B @

3 X

1 dPi 3 dxi ei 5 dvσis es 3 ei

i51

5 dvð 2σ12 e3 1 σ13 e2 1 σ23 e1 1 σ21 e3 1 σ31 e2 1 σ32 e1 Þ 5 2 dv½ðσ23 2 σ32 Þe1 1 ðσ31 2 σ13 Þe2 1 ðσ12 2 σ21 Þe3  5 0

C C C A

from which it follows that σ12 5 σ21 ; σ23 5 σ32 ; σ31 5 σ13 ; σij 5 σji

(5.37)

leading to the symmetry of the Cauchy stress tensor, that is, σ 5 σT The following relations hold from Eqs. (2.65) and (2.66). ( σij 5 σji ; σi
j 5 σj
i ; σij 5 σji σi
j 6¼ σ
i ; σi
6¼ σj
i j

j

(5.38)

(5.39)

ðσij 5 σji Þ Then, the Cauchy’s fundamental theorem in Eq. (5.30) is rewritten as follows: t 5 σn

(5.40)

5.3 Balance laws in current configuration The balance laws in the current configuration are shown in this section.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

133

5.3 Balance laws in current configuration

5.3.1 Translational equilibrium Substituting the stress tensor in (5.30), the conservation law of linear momentum in Eq. (5.14) is written as ð ð ð _ 5 σT nda 1 ρbdv ρvdv (5.41) v a v leading to ð ð ð _ 5 rx σdv 1 ρbdv ρvdv (5.42) v

v

v

by the Gauss’ divergence theorem in Eq. (1.340), which leads to the local form of the Cauchy’s first law of motion, that is, the translational equilibrium. rx σ 1 ρb 5 ρv_

(5.43)

5.3.2 Rotational equilibrium: symmetry of Cauchy stress tensor The symmetry property of Cauchy stress tensor will be proved by considering the equilibrium of angular moment in this section. Substituting Eq. (5.30) into Eq. (5.16) of the conservation law of angular momentum, one has ð ð ð _ 5 x 3 σT nda 1 ρx 3 bdv ρx 3 vdv (5.44) v

a

v

The first term in the right-hand side of this equation is rewritten as ð ð ð ð x 3 σT nda 5 divðx 3 rx σÞdv 5 ε : σdv 1 ðx 3 rx σÞdv ð

a

ð

εijk xj σrk nr da 5 a

v





ð

v

v

@ εijk xj σrk @xj dv 5 εijk σrk dv 1 @xr @xr a a ð ð @σrk 5 εijk σjk dv 1 εijk xj dv @xr a a

ð εijk xj a

@xrk dv @xr

The substitution of this equation into Eq. (5.44) reads ð ð ε : σdv 1 x 3 ðrx σ 1 ρb 2 ρv_ Þdv 5 0 v

(5.45)

v

from which it follows substituting the equilibrium Eq. (5.43) that ε : σ 5 0; εijk σjk 5 0

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(5.46)

134

5. Conservation laws and stress tensors

resulting in the symmetry of Cauchy stress tensor described in Eq. (5.38) which was derived from the angular moment balance of the parallelopiped formed by the covariant base vectors in Section 5.2.2.

5.3.3 Virtual work principle Giving the scalar product of the arbitrary velocity increment δv to the equilibrium Eq. (5.43), we find










(5.47)




(5.48)

δv rx σ 1 ρδv b 5 ρδv v_ Noting



divðδvσÞ 5 σ : gradδv 1 δv rx σ σ : gradδv 5 σ : δl 5 σ : δd

Eq. (5.47) leads to

 
1 divðδvσÞ 2 σ : δd 1 ρδv b 5 ρδ v v 2




that is,




 
1 σ : δd 5 divðδvσÞ 1 ρδv b 2 ρδ v v 2







(5.49)

which is the local form of the virtual work equation in the current configuration. Now, consider the integration of Eq. (5.49) over the current volume. The integration of the first term in the right-hand side in Eq. (5.49) yields ð ð ð ð divðδvσÞdv 5 ðδvσÞ nda 5 δv ðσnÞda 5 δv tda (5.50) v




a

a




a




by virtue of Eqs. (1.339), (5.30), and (5.38). Further, noting the continuity Eq. (5.11), the integration of Eq. (5.49) leads to 
ð ð ð ð  1 σ : δddv 5 t δvda 1 ρb δvdv 2 ρδ v v dv (5.51) 2 v a v v










Incidentally, applying the velocity v instead of δv, one has the following equation analogously to Eq. (5.51). ð 
ð ð ð 1 v vdv σ : ddv 5 t vda 1 ρb vdv 2 ρ (5.52) v a v v2










Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

135

5.4 Work-conjugacy

where Eq. (5.11) is exploited for the last term. The quantity in the lefthand side means the stress power, and the first, the second, and the third term mean the power done by the work rate by the surface force (traction) and the rate of the kinetic energy. Therefore, σ : d designates the stress power done in the unit current volume.

5.3.4 Conservation law of energy The internal energy E is defined by ð E  ρεdv

(5.53)

v

where ε is the internal energy per unit mass. The heat input caused by the internal heat source r per unit current unit mass and the heat flux vector q per unit area is given by ð ð ð ð Q  ρrdv 2 q nda 5 ρrdv 2 divq dv (5.54) v




a

The stress power is given by Pσ 5

v

v

ð σ : d dv

(5.55)

v

The following energy equilibrium equation must hold. E_ 5 Pσ 1 Q that is,

ð ρεdv




v

ð

(5.56)

ð

5

ð

σ : d dv 1 v

ρrdv 2 divq dv v

(5.57)

v

Noting the continuity Eq. (5.11), Eq. (5.57) leads to ð ðρ_ε 2 σ : d 2 ρr 1 divqÞdv 5 0

(5.58)

v

from which the conservation law of energy in the local form is given as ρ_ε 5 σ : d 1 ρr 2 divq

(5.59)

5.4 Work-conjugacy The stress tensor and the strain rate tensor used in constitutive relations must be the appropriate combination such that their scalar product leads to the physically meaningful stress power (work rate). Such pair is Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

136

5. Conservation laws and stress tensors

called the work-conjugate pair and this notion is called the work-conjugacy. It has been noticed in solid mechanics after Hill (1968, 1978) and Macvean (1968) through Truedell (1952) and Ziegler and McVean (1967). The concept of the work-conjugacy and various work-conjugate pairs will be described in this section.

5.4.1 Kirchhoff stress tensor and work-conjugacy As was described at the end of Section 5.3.3, the stress power w_ done in the unit current volume is given by _ 5 σ : d 5 σ : l ð 5 trðσlÞ 5 trðσdÞÞ w

(5.60)

for the fulfillment of the virtual work principle. This physical interpretation has been described in many literatures (e.g., Ogden, 1984; Asaro and Lubarda, 2006; Bonet and Wood, 2008). On the other hand, it will be directly derived from the deformation of the parallelopiped in the following. The stress power done to the volume element vg 5 ½g1 g2 g3  of the parallelopiped formed by the current primary base vectors fg1 ; g2 ; g3 g is represented as follows (see Fig. 5.4): w_ 5




1 i P g_ i vg

(5.61)

where Pi is the stress vector applied to the area vector of the parallelopiped given by ai 5

1 εijk gj 3 gk 5 vg gi 2

a3 = vg g 3

g3

(5.62)

P3

g3

g3

−a2

P2 0

−P2

g1 g g1

g2

g2 g2

1

a2 = vg g 2

vg = [ g1 g2 g3]

a1 = vg g1

P1

FIGURE 5.4 Stress power done to parallelopiped formed by current primary base vector fg1 ; g2 ; g3 g.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

137

5.4 Work-conjugacy

noting Eqs. (2.1) and (2.2), where a1 is the surface vector normal to the surface formed by a2 and a3. Then, the following relation holds by Cauchy’s fundamental law in Eq. (5.40). Pi 5 σai ð 5 vg σgi Þ

(5.63)

nothing for example of i 5 1: P1 5 t1 jjg2 3 g3 jj; a1 5 g2 3 g3 ; n1 5

g 2 3 g3 jjg2 3 g3 jj

The substitution of Eq. (5.63) into Eq. (5.61) leads to










w_ 5 σgi g_ i 5 σrs gr  gs gi g_ i 5 σri gr g_ i







5 σri ðgr g_ i 1 g_ r gi Þ=2 5 σri g_ ri =2 5 σri dri 5 σ : d

(5.64)

noting Eq. (4.85)1, which is just identical with Eq. (5.60). Note, however, that the stress power w_ done to the current unit volume element does not represent the constitutive property because the mass involved in the current unit volume element changes by the defor_ done to the unit refermation. On the other hand, the stress power W ence volume possessing the fixed mass is physically meaningful for the formulation of constitutive equations, where _  ðσ : dÞdv=dV 5 Jσ : d 5 τ:d W

(5.65)

τ  Jσ

(5.66)

The stress tensor

is referred to as the Kirchhoff stress tensor. The stress tensor and the _ are called the strain rate tensor the scalar product of which leads to W work-conjugate pair.

5.4.2 Work-conjugate pairs Various work-conjugate pairs of the stress tensor and the deformation rate tensor are derived by applying the pull-back operations in Eq. (3.24) to τ and d in Eq. (5.65), noting Eqs. (1.362), (4.79), (4.82), and (4.104), as follows: 8   > tr τF2T ðlFÞT  5 tr PF_ T > > >    >  21  < tr F τ ðlFÞ 5 tr NF_

 21 2T  T      (5.67) trðτdÞ 5 trðτlÞ 5 _ > F dF i5 tr SE_ 5 tr SC=2 > trh F τF >      > > : tr FT τF2T F21 lF T 5 tr MLT

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

138 where

5. Conservation laws and stress tensors

8 8 T > > P  τF2T PF > > < < N  F21 τ FN 21 2T ; i:e: ; τ 5 > > FSFT > S  F τF > : : 2T T 2T M  F τF F MFT

(5.68)

The stress tensors P, N, S, and M are called the first PiolaKirchhoff stress, the nominal stress, the second PiolaKirchhoff stress, and the Mandel stress, respectively. L is defined in Eq. (4.79). Various conjugate pairs of stress tensors and deformation rate tensors are shown in Eq. (5.67). However, note here that the constitutive relations representing the deformation behavior of materials must be formulated primarily by the relations of tensors based in the reference configuration on a fixed material volume element. Therefore, the second Piola-Kirchhoff stress tensor S is used in the pair with the right CauchyGreen deformation tensor C in the hyperelatic constitutive equation which will be described in the next chapter. Further, the Mandel stress M is used in the pair with the velocity gradient tensor L in the plastic constitutive equation, because the plastic deformation/rotation exhibits the loading path-dependence. On the other hand, only the constitutive equation in terms of S is shown in some popular books (Simo, 1998; Simo and Huhges, 1998; Asaro and Lubarda, 2006; Bonet and Wood, 2008; Belytschko et al., 2014) but it would not lead to a pertinent formulation of plastic constitutive equation. The work-conjugate pairs obtained in Eq. (5.67) are rewritten below. τ : d 5 Jσ : d ðEulerianÞ _   W5 (5.69) T _ 5 M : L Lagrangian P : F_ 5 N : F_ 5 S : E_ 5 S : C=2 _ _ 5 trðSLT CÞ 2 trðSC=2Þ 5 tr Incidentally, note that trðMLT Þ 2 trðSC=2Þ T T _ _ ½SðL C 2 C=2Þ 5 0 holds from Eq. (5.67) but L C 2 C=2 5 O does not hold because it is not symmetric tensor in general.

5.4.3 Physical meanings of stress tensors The physical meanings of the Lagrangian(Eulerian) stress tensors derived in Section 5.4.2 will be given in the following. 5.4.3.1 Two-point contravariant pull-back: first PiolaKirchhoff and Nominal stress tensors The first Piola 2 Kirchhoff stress P as the Eulerian 2 Lagrangian twopoint tensor and the nominal stress N as the Lagrangian 2 Eulerian two-point tensor are derived from the Kirchhoff stress tensor by the

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

139

5.4 Work-conjugacy

right and the left contravariant pull-back operations as shown in the following. 8 9 < Pij g  G  τ ij g  G ð 5 τ Þ = ^ j j gG i i 5 τF2T 5 NT ð6¼ PT Þ P5 (5.70) :
j i ; g^ j i
Pi g  Gj  τ i g  Gj ð 5 τ
G Þ 9 8 > = < Nij G  g  τ ij G  g ð 5 τ Þ > i i G^g j j 5 F21 τ 5 PT ð6¼ NT Þ (5.71) N5 > ; : Ni Gi  gj  τ i Gi  gj ð 5 τ
g^ Þ > G
j
j ~

~

~

~

noting Eq. (3.25). The work-conjugate deformation rate tensors for these stress tensors are given from Eq. (5.69) with Eq. (3.13) as ( P3F_ 5 g_ i  Gi 5 lF; (5.72) T N3F_ 5 Gi  g_ i 5 FT lT where the symbol 3 designates the work-conjugate connection. The first PiolaKirchhoff stress tensor plays the important role to transform the equilibrium equation in the current configuration to that in the reference configuration in order to derive the equilibrium equation in the rate form for instance as will be known in Section 5.5.1. 5.4.3.2 Contravariant pull-back: second PiolaKirchhoff stress tensor The following tensor derived by the contravariant pull-back operation in Eq. (3.24)1 from the Kirchhoff stress is referred to as the second Piola 2 Kirchhoff stress S. ( S 5 Sij Gi  Gj  τ ij Gi  Gj 5 F21 τF2T ð 5 τ GG Þð 5 F21 JσF2T Þð 5 ST Þ τ 5 τ ij gi  gj 5 FSFT ~

(5.73) where the components of the second Piola-Kirchhoff stress tensor are identical to those of the Kirchhoff stress tensor, i.e. Sij 5 τ ij ð 5 Jσij Þ. Here, the ratio of the volumes of the parallelepipeds formed by the line-elements θi gi and Θi Gi ð 5 θi Gi Þ are given by J ð 5 dv=dVÞ. Here, τ ij and Sij are the components decomposed into the parallel directions to the surfaces of the parallelepiped formed by the current and the reference primary bases, respectively, as shown in Fig. 5.5. Therefore, Sij possesses the clear physical meaning amongst all various pull-backed stress tensors. On the other hand, the components of the other

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

140

5. Conservation laws and stress tensors

Current configuration at time t

F v σ 22

Reference configuration at time t0 V

S

22

(= τ = J σ ) 22

σ 12

S 21 (=τ 21 = J σ 21)

G2

S 12 (= τ 12 = J σ 12 ) S 11 (= τ 11 = Jσ 11 )

P0

σ 21

g2 22

σ 11 P

g1

G1

FIGURE 5.5 Relation of the contravariant components of the Cauchy and the second PiolaKirchhoff stress tensors illustrated in the two-dimensional space.

pulled-back stress tensors apply to the parallelopiped formed not only by the primary but also by the reciprocal bases, so that these current parallelopipeds are different from those changed from the reference parallelopiped, noting that the reciprocal base vectors are defined momentarily so as to satisfy the relations to the primary base vectors in Eq. (2.1). The work-conjugate strain rate tensor of the second PiolaKirchhoff stress tensor is S3E_ 5

1_ C 2

(5.74)

The Lagrangian tensors, for example, the second PiolaKirchhoff stress tensor and the Mandel stress tensor based in the reference configuration are not influenced by the superposition of the rigid-body rotation and thus they are called the rotation-free tensor, as is proved for the second PiolaKirchhoff stress tensor below, where R denotes the superposed rigid-body rotation tensor. S 5 F21 Jσ F2T 5 ðRFÞ21 JðRσRT ÞðRFÞ2T 5 F21 RJRT σRT R2T F2T 5 F21 JσF21 5 S

(5.75)

The second Piola-Kirchhoff stress tensor S is used in the pair with the right Cauchy-Green deformation tensor C in the hyperelatic constitutive equation as was mentioned in the foregoing and as will be described in Chapter 6 explicitly.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

141

5.4 Work-conjugacy

5.4.3.3 Covariantcontravariant pull-back: Mandel stress tensor The Mandel stress tensor M (Mandel, 1973b) is derived from the Kirchhoff stress tensor by the covariantcontravariant pull-back operation noting Eqs. (3.24)3 and (5.73) as shown in Eq. (5.67):   G j j M 5 Mi
Gi  Gj  τ i
Gi  Gj 5 FT τF2T 5 τ
G ~

    5 FT FF21 τF2T 5 CS 5 gij Gi  Gj ðSrs Gr  Gs Þ 5 gij Srj Gi  Gj 6¼ MT (5.76) which is also derived by multiplying the right CauchyGreen tensor C to the second PiolaKirchhoff stress tensor S. Incidentally, the Mandel stress is the rigorous stress tensor for the expression of constitutive equation for finite deformation as was mentioned in the foregoing. The workconjugate strain rate tensor of the Mandel stress tensor is the velocity gradient tensor in the reference configurationas shown in Eq. (5.69), i.e. M3L 5 F21 lF

(5.77)

Numerous work-conjugate stress tensors have been derived for the generalized strain measure in Eqs. (4.45) and (4.46) (e.g., Guo and Dubey, 1984; Hoger, 1987; Xiao, 1995; Guo and Man, 1992; Farahani and Naghdabadi, 2000; see also Ogden, 1982; Lubarda, 2002). However, the usuful work-conjugate stress tensors are the second Piola-Kirchhoff stress tensor S which is work-conjugate to the right Cauchy-Green deformation tensor C for the hyperelastic-based consitutive equations and the Mandel stress M which is work-conjugate to the velocity gradient tensor L for the hyperelastic-based plastic constitutive equation as will be described in detail in Chapters 6 and 9.

5.4.4 Relations of stress tensors The relations of the above-mentioned various stress tensors are given by τ 5 Jσ 5 FPT 5 FN 5 FSFT 5 F2T MFT 8 21 8 < F ðFΠT Þ 5 PT < ðFPÞT F2T 5 NT FT F2T 5 NT 2T 21 T 2T N 5 F τ 5 F21 ðFSFT Þ 5 SFT P 5 τF 5 ðFSF ÞF 5 FS : 21 2T : 2T T 2T 2T F ðF MFT ÞT 5 MT F21 ðF MF ÞF 5 F M

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

142

5. Conservation laws and stress tensors

8 8 < FT ðFPT ÞF2T 5 CPT F2T < F21 ðFPT ÞT F2T 5 F21 P 21 2T T 2T 21 2T 2T M 5 F τF 5 FT ðFNÞF2T 5 CNF2T S 5 F τF 5 F ðFNÞF 5 NF : T : 21 2T 21 T 2T F ðFSFT ÞF2T 5 CS F ðF MF ÞF 5 C M

where the symmetry of the Kirchhoff stress tensor is exploited. The relations of the above-mentioned stress tensors are shown collectively listed in Table 5.1. Incidentally, relationships of some stress tensors are shown for the simple shear loading in Section 5.6. The details of the other stress tensors, for example, Biot stress tensor, and their strain tensors in the conjugate pairs are referred to Hashiguchi and Yamakawa (2012) and Hashiguchi (2017a).

5.4.5 Relations of stress tensors to traction vectors The relation of the Cauchy stress tensor to the Cauchy stress vector applied to the current surface vector is described by Eq. (5.30). The relations of the above-mentioned stress tensors to the stress vectors applied to the reference surface vectors are formulated below. The stress vector Τ ð 5 tda=dAÞ calculated by supposing that the Cauchy stress vector t applies to the unit reference area is given by Τ 5 τF2T N

(5.78)

noting Τ  tda=dA 5 σnda=dA 5 σJF2T N noting Eqs. (1.359) and (5.40) with Eq. (5.38). TABLE 5.1 Relations of various stress tensors.

Names, Notations

(=

T)

Kirchhoff

2nd Piola -Kirchhoff S Mandel M

(≠

T)

F

FPT

(= J ) 1st Piola -Kirchhoff P Nominal

P (≠ P T )

F −T

T

F −1

PT

F −1 F −T

F −1P

F T F −T

FTP

( ≠ MT )

FSFT

F −T MFT

FS

F −T M

SFT

MT F −1

F −T

C

F −T

M

S (= S T )

C−1M

CS

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

5.4 Work-conjugacy

143

Substituting Eq. (5.68) into Eq. (5.78), the relations between the traction vectors and the stress tensors are represented as follows:

noting

t 5 σn 5 ðτ=JÞn Τ 5 PN 5 NT N F21 Τ 5 SN FT Τ 5 MN

(5.79)

8 > ðPFT ÞF2T N 5 PN > < ðFNÞT F2T N 5 NT FT F2T N Τ5 > ðFSFT ÞF2T N 5 FSN > : 2T F MFT F2T N 5 F2T MN

(5.80)

It is shown in Eq. (5.79) that similar equations to the Cauchy’s fundamental law in Eq. (5.40) hold for these stress tensors, since stress tensors are second-order tensors transforming a vector linearly to other vector. Eq. (5.79) is illustrated in Fig. 5.6. Further, the following relation holds denoting the infinitesimal load vector by df applying to the surface from Eqs. (1.356), (5.40), (5.68), and (5.79). 8 σda > > < ðτ=JÞda (5.81) df 5 PdA > > : T N dA F21 df 5 SdA

(5.82)

FT df 5 MdA

(5.83)

These stress tensors are physically interpreted as follows: 1. The first PiolaKirchhoff stress tensor P is defined by the stress tensor induced by the infinitesimal load vector df applied to the infinitesimal reference surface area dA. 2. The nominal stress tensor N is defined by the transpose of the stress tensor induced by the infinitesimal load vector df applied to the infinitesimal reference surface area dA. 3. The second PiolaKirchhoff stress tensor S is defined by the stress tensor induced by the contravariant pull-backed infinitesimal load vector F21 df applied to the infinitesimal reference surface area dA. The contravariant components of S and τ are identical to each other as described in Section 5.4.3. 4. The Mandel stress tensor M is defined by the stress tensor induced by the covariant pull-backed infinitesimal load vector FT df applied to the infinitesimal reference surface area dA.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

144

5. Conservation laws and stress tensors

FIGURE 5.6 Relations of stress tensors and traction vectors, where tractions per unit area are written for simplicity.

Incidentally, it follows from Eq. (5.81) in the one-dimensional state that df 5 σda 5 NdA

(5.84)

and thus N is called the nominal stress tensor which is nothing but the tensorization of the so-called nominal stress N.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

5.5 Balance laws in reference configuration

145

The stress tensors defined in this section are shown explicitly for the simple shear deformation in Section 5.6.

5.5 Balance laws in reference configuration The balance laws were described in Section 5.3 but they are expressed in the current configuration. They will be represented in the reference configuration by use of the work-conjugate pair of stress and strain in this section.

5.5.1 Translational equilibrium The substitution of σnda 5 ðτ=JÞJF2T NdA 5 PNdA derived from Eqs. (1.356) and (5.70) into Eq. (5.41) reads: ð ð ð ρ0 v_ dV 5 PNdA 1 ρ0 bdV (5.85) v

resulting in

A

ð

ð

ð

V

ρ0 v_ dV5

V

rX PdV 1 V

V

ρ0 bdV

(5.86)

from which it follows that rX P 1 ρ0 b 5 ρ0 v_

(5.87)

which is the transformation of the translational equilibrium Eq. (5.43) into the reference configuration. The time-differentiation of Eq. (5.87) leads to the following rate-(or incremental)-type equilibrium equation in the reference configuration under the state that the acceleration does not change, that is, v̈ 5 0. rX P_ 1 ρ0 b_ 5 0

(5.88)

On the other hand, the time-differentiation of Eq. (5.41) reads: ð 
ð 
ð 
_ ρvdv 5 σnda 1 ρbdv v

that is,

a

ð

ð


ð _ σnda 1 ρbdv

ρ v€ dv 5 v

v

a

v

by virtue of Eq. (5.11). This equation is further rewritten as

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

146

5. Conservation laws and stress tensors

ð

ð

ð

_ _ 0 5 σnda 1 σðndaÞ
1 ρbdv a

a

v

by noting v̈ 5 0. The substitution of Eq. (1.370) with Eq. (1.369) into this equation reads: ð ð ð

 _ 50 _ σnda 1 σ ðtrlÞI 2 lT nda 1 ρbdv a

that is,

a

ð

v

  div σ_ 1 σtrl 2 σlT dv 1 v

ð

_ 50 ρbdv v

noting Eq. (1.340), from which one has the rate-type equilibrium equation in terms of the Cauchy stress as follows: divp σ ˚ T 1 b_ 5 0

(5.89)

where p σ ˚ is the nominal stress rate defined in Eq. (A2.7) in Appendix 2.

5.5.2 Virtual work principle Eq. (5.51) is described in the reference configuration as follows: ð 
ð ð ð 1 _ ρ0 v vdV S : δEdV 5 T δvdA 1 ρ0 b δvdV 2 δ (5.90) V A V V2










_ noting tda 5 TdA and σ : δd dv 5 ðτ=JÞ: δd dv 5 ðS : δE=JÞdv 5 S : δE_ dV in Eq. (5.69).

5.5.3 Conservation law of energy Eq. (5.59) is written in the reference configuration as follows: ρ0 ε_ 5 S : E_ 1 ρ0 r 2 divQ

(5.91)

Q 5 JqF2T

(5.92)

where

noting

 ρ0 _ 5 S : E_ _ 21 Þ 5 trðSEÞ σ : d 5 Jσ : d 5 τ : d 5 trðτdÞ 5 tr FSFT ðF2T EF ρ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

147

5.6 Simple shear

ð

ð divqdv 5

v

ð




a




q JF2T NdA 5

q nda 5 A

ð




Divðq JF2T ÞdV

V

with the aid of Eqs. (1.339), (1.356), (4.82), (4.104), and (5.73) with u Tv 5 uT v.







5.6 Simple shear Various strain (rate) and stress (rate) described in the foregoing are shown explicitly and their relations are described for the simple shear deformations in order to capture their physical meanings in the following. The simple shear deformations are often observed in experiments for measurement of material properties. Homogeneous and isotropic deformation is assumed and the normalized-rectangular coordinate system is adopted for the representation of components in the following. Consider the simple shear behavior in which the material particles move along the x1 -axis as shown in Fig. 5.7. x 5 ðX1 1 γX2 Þe1 1 X2 e2 1 X3 e3

(5.93)

where γ is the engineering shear strain. We introduce the angle φ defined by _ 2φ γ 5 2 tan φ; γ_ 5 2 φsec The following equation holds 2 3 2 1 γ 0 1 F 5 4 0 1 0 5; F21 5 4 0 0 0 1 0

γ 1 0

3 0 0 5; 1

J 5 det F 5 1

(5.94)

(5.95)

where the inverse tensor F21 is derived using (1.186). The components in the third line and the third row are zero except for unity in all tensors γ γ /2 γ /2 1

τ

φ γ X2

X 2 = x2

X

x

e2

0

e1

X1

x1

FIGURE 5.7 Simple shear deformation behavior.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

148

5. Conservation laws and stress tensors

appearing hereinafter for the simple shear deformation. Then, for simplicity, let them be expressed by the matrix with two lines and two rows.     1 2γ 1 γ (5.96) F5 ; F21 5 0 1 0 1 from which we obtain _ 21 5 l 5 FF d5



 γ_ 0 2 1

   γ_ 1 2γ 0 γ_ 0 0 1 0 0    γ_ 0 1 1 ; w5 0 2 21 0

0 0

(5.97) (5.98)

Further, it follows noting Eq. (1.186) that     8 1 γ 1 1 γ2 2 γ > 21 2 T 22 21 2T > Cð 5 U 5 F FÞ 5 ð 5 U 5 F F Þ 5 ; C > < γ 1 1 γ 2 1  2 γ  2 1 1 1 γ 2 γ 2 1 γ > U21 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi > U 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ; > : 2 4 1 γ2 γ 2 1 γ 4 1 γ2 2 γ (5.99)     8 1 2γ 1 1 γ2 γ > 21 2 T 22 2T 21 > b ð 5 V 5 FF Þ 5 ð 5 V 5 F F Þ 5 ; b > 2 < γ  2 γ 1 1 γ   1 2 1 1 2 2γ 21γ γ > > V 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi ; V21 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > 2 2 : 12 2 γ γ γ 2 41γ 41γ 



(5.100)

1 2γ (5.101) 2 γ 1 1 γ2   1 2 γ R 5 FU21 5 V21 F 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi (5.102) 4 1 γ2 2 γ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi   γ_ 2 4p 2ffiffiffiffiffiffiffiffiffiffiffiffi γ 2 1 γffi 4 1 γ 2 2 4 1 γ 2pffiffiffiffiffiffiffiffiffiffiffiffiffi R T _ Ω 5 RR 5 2 γ2 1 γ 4 1 γ2 2 4 ð41γ 2 Þ3=2 2 2 4 1 γ 2 b21 ð 5 V22 5 F2T F21 Þ 5



   1 1 0 γ 1 1 0 γ 21 ðg 2 b E 5 ðC 2 GÞ 5 Þ 5 ; e 5 2 2 γ γ2 2 2 γ 2 γ2   1 0 γ ε5 2 γ 0

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(5.103) (5.104) (5.105)

5.6 Simple shear

149

where γ designating twice the deviatoric component of the strain tensors, noting     1 1 γ 2 1 γ2 2 γ pffiffiffiffiffiffiffiffiffiffiffiffiffi FU21 5 0 1 2 4 1 γ2 2 γ !  
  1 2 γ 2 2γ T 1 _ p ffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi p RR 5 41γ 2 2γ 2 4 1 γ2 γ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi    γ_ 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 γ 1 4 1 γ 2 2 2 γ 5 γ 2 ð41γ 2 Þ3=2 γ 2 4 1 γ 2 2 2 Various stress tensors are described using the notation τ 5 σ12 as follows:   σ11 τ σ5τ5 (5.106) τ σ22      1 0 σ11 2 γτ τ σ11 τ T 2T 5 P 5 N 5 JσF 5 (5.107) 2γ 1 τ σ22 τ 2 γσ22 σ22    1 2 γ σ11 2 γτ τ S 5 JF21 σF2T 5 F21 P 5 0 1 τ 2 γσ22 σ22 (5.108)   2 σ11 2 γ σ22 2 2γτ τ 2 γσ22 5 τ 2 γσ22 σ22

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

C H A P T E R

6 Hyperelastic equations The independent descriptions of the elastic and the plastic deformations are required in elastoplastic constitutive equations of materials which are the assembly of solid particles as will be indicated in Section 7.1. Elastic constitutive equations are classified into hypoelastic, Cauchy elastic, and hyperelastic (or Green elastic) equations. An energy (work) is produced or dissipated during a closed loading (stress or strain) cycle in both the hypoelastic and the Cauchy elastic equations. The one-toone correspondence between stress and strain does not exist in the hypoelastic equation but it exists in the Cauchy elastic equation. On the other hand, the hyperelastic equation is capable of describing a purely elastic deformation without the inexactness contained in the hypoelastic and the Cauchy elastic equations. It exhibits the one-to-one correspondence between the stress and strain and possesses the strain energy function leading to no energy dissipation/production under a cyclic loading. Therefore, the hyperelastic equation must be used in the finite elastoplastic constitutive equation. The general form of hyperelastic constitutive equation will be formulated and various hyperelastic equations for metals, rubbers, and soils will be shown in this chapter.

6.1 Basic hyperelastic equations In the hyperelastic material, the work W done in a unit reference volume during the change from a certain deformation state to the other deformation state is uniquely determined independent of the deformation path between these deformation states. Therefore,
 taking account of the fact that the right Cauchy
Green tensor C 5 U2 in Eq. (4.1) designates the pure deformation independent of the rigid-body rotation R, there must exist the positive scalar function @ψðCÞ and the following relation must hold.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity DOI: https://doi.org/10.1016/B978-0-12-819428-7.00006-7

151

© 2020 Elsevier Inc. All rights reserved.

152

6. Hyperelastic equations

W5

ðC C0

dW 5

 ðC  @ψðCÞ : dC 5 ψðCÞ 2 ψðC0 Þ @C C0

(6.1)

Therefore, it must hold that dW 5

@ψðCÞ @ψðCÞ : dC 5 2 : dC=2 @C @C

(6.2)

On the other hand, the work increment done to the unit reference volume is given by dW 5 S:dC=2 from Eq. (5.69). Therefore, the second Piola
Kirchhoff stress is given by S52

@ψðCÞ @C

(6.3)

Needless to say, the one-to-one correspondence between the stress and the deformation exists. Inversely, an arbitrary function of the deformation gradient cannot be adopted for the stress but the function must satisfy the complete integrability condition @Sij @Skl 5 @Ckl @Cij

(6.4)

imposed by the exchangeability of the order of partial derivatives, that is, @2 ψ @2 ψ 5 @Cij @Ckl @Ckl @Cij

(6.5)

which holds for an arbitrary scalar function @ψðCÞ. The function ψ is called the Helmholtz free energy function, that is, elastic strain energy function which is nonnegative and determined uniquely for the elastic deformation of material by the requirement from the first law of thermodynamics. On the other hand, the constitutive equation, which describes the one-to-one correspondence of stress and strain but does not possess a elastic strain energy function and thus does not fulfill the integrability condition described by the partial derivative of the stress tensor by the strain tensor, is defined as the Cauchy elasticity. Furthermore, a linear relation between a stress rate and a strain rate, which does not lead to the one-to-one correspondence between a stress and a strain and in which the strain energy function does not exist and thus the energy is produced or dissipated during a loading cycle, is called the hypoelasticity (Truesdell, 1955). The fundamentals of these constitutive equations which do not describe the purely elastic deformation behavior are explained in Appendix 3.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

6.1 Basic hyperelastic equations

153

Now, substituting Eq. (6.3) into the stress tensors listed in Table 5.1, we obtain various expressions of the hyperelasticity in terms of the deformation gradient as follows: 8 1 @ψðCÞ T @ψðCÞ T > > σ52 F F ; τ 5 2F F > > J @C @C > > > > < @ψðCÞ @ψðCÞ T ; N52 F P 5 2F (6.6) @C @C > > > > > @ψðCÞ @ψðCÞ > > ; M 5 2C >S52 : @C @C Furthermore, noting @ @ @Crs @ @ðFpr Fps Þ @ @ @ 5 5 5 Fis 1 Fir 5 2Fir @Fij @Crs @Fij @Crs @Fij @Cjs @Crj @Crj

(6.7)

that is, @ @ 5 2F @F @C it holds that

8 @ @ @ > > 5 2F 5F > > @F @C @E > > > > < @ 1 21 @ 1 @ 5 F 5 @C 2 @F 2 @E > > > > > @ @ @ > 21 > > : @E 5 F @F 5 2 @C

(6.8)

(6.9)

It follows from Eq. (6.8) that @C 5 I FT 1 FT I @F by use of the symbols in Eq. (1.221), noting ! @Cij @Cij @Cji 5 Fkr 1 5 Fkr ðδir δjl 1 δjr δil Þ 5 Fki δjl 1 Fkj δil @Fkl @Crl @Crl derived by exploiting Eq. (6.7) with the symmetry Cij 5 Cji or @Cij @ðFri Frj Þ 5 5 δkr δil Frj 1 Fri δkr δjl 5 δil Fkj 1 Fki δjl @Fkl @Fkl derived directly.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(6.10)

154

6. Hyperelastic equations

Then, substituting Eq. (6.9) into Eq. (6.6), the hyperelasticity is expressed in terms of the right Cauchy
Green deformation tensor C and the Green strain tensor E as follows: 1 @ψðCÞ T 1 @ψðEÞ T @ψðCÞ T @ψðEÞ T F 5 F F ; τ 5 2F F 5F F σ52 F J @C J @E @C @E P 5 2F S52

@ψðCÞ @ψðEÞ @ψðCÞ T @ψðEÞ T 5F ; N52 F 5 F @C @E @C @E

(6.11)

@ψðCÞ @ψðEÞ @ψðCÞ @ψðEÞ 5 ; M 5 2C 5 ð2E 1 GÞ @C @E @C @E

The tangent stiffness modulus tensor CS  2

@S @2 ψðCÞ 54 @C @C=@C

(6.12)

in the hyperelastic relation in terms of the second Piola
Kirchhoff stress tensor S satisfies both the minor symmetry and the major symmetry, that is, CS ijkl 5 CS ijikl 5 CS ijlkl ;

CS ijkl 5 CS klij

(6.13)

The Helmholtz free (strain energy) function ψðEÞ and the Gibbs’ free (complementary strain) energy function φðSÞ fulfilling ψðEÞ 1 φðSÞ 5 S:E are schematically shown in Fig. 6.1, noting 8 @ψðEÞ > > : dE 5 S: dE dψðEÞ 5 > < @E (6.14) @φðSÞ > > dφðSÞ 5 : dS 5 E: dS > : @S

dφ (S) = E : d S

dS S

φ (S) ψ (E)

0

dψ (E) = S(E) :d E

E dE

FIGURE 6.1 Helmholtz free (strain) energy function ψðEÞ and Gibbs’ free (complementary strain) energy function φðSÞ.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

6.2 Hyperelastic constitutive equations of metals

155

6.2 Hyperelastic constitutive equations of metals Various hyperelastic constitutive equations are shown in this section, which are used for metals. The partial derivatives described in Section pffiffiffiffiffiffiffiffiffiffiffi1.7.1 are exploited in the following. The equation J 5 detC 5 detU 5 detF is often used below.

6.2.1 St. Venant
Kirchhoff elasticity The St. Venant
Kirchhoff elastic material possesses the following strain energy. ψðEÞ 5

1 λðtrEÞ2 1 μtrE2 ; 2

ψðCÞ 5

1 λftr½ðC2GÞ=2g2 1 μtr½ðC2GÞ=22 2 (6.15)

where the material constants λ and μ correspond to the Lame constants, respectively, in the infinitesimal linear elasticity theory. The substitution of Eq. (6.15) into Eq. (6.11) reads: S 5 λðtrEÞG 1 2μE 5 λftr½ðC 2 GÞ=2gG 1 μðC 2 GÞ

(6.16)

It is followed that @S 5 CSV @E where

(

CSV  λI 1 2μS CSV ijkl  λδij δkl 1 μðδik δjl 1 δil δjk Þ

(6.17)

(6.18)

Further, the Mandel stress in Eq. (5.76) is given for Eq. (6.16) as M 5 λtr½ðC 2 GÞ=2C 1 μCðC 2 GÞ

(6.19)

The Kirchhoff stress is given from Eq. (6.16) as τ 5 λtr½ðb 2 gÞ=2b 1 μbðb 2 gÞ noting

  

 τ 5 FSFT 5 F λ tr ðC 2 GÞ=2 g 1 μðC 2 GÞ FT   1 1 5 λtr ðFT F 2 GÞ FFT 1 2μF ðFT F 2 GÞ FT 2 2

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(6.20)

156

6. Hyperelastic equations

The multiplications of C to S, C, and G from the left lead to M, CC, and CG. On the other hand, the contravariant push-forward operation, that is, the multiplications of F from the left and FT from the right to S,C, and G lead to FSFT 5 τ, FCFT 5 FðFT FÞFT 5 bb, and FGFT 5 b 5 bg. Therefore, the hyperelastic equations in terms of ðM; CC; CGÞ and ðτ; bb; bgÞ possess the identical form. It is unfortunate that the St. Venant-Kirchhoff equation is practically useless except for the small deformation range.

6.2.2 Modified St. Venant
Kirchhoff elasticity The following strain energy function is adopted by Wallin and Ristinmaa (2005), modifying Eq. (6.15) in the St. Venant
Kirchhoff elastic material. ψðCÞ 5

1 KðlnJÞ2 1 2GJ 2 2

(6.21)

with pffiffiffiffiffiffiffiffiffiffiffi 0 1 lnJ 5 ln detCÞ; J 2  trðlnC1=2 Þ 2 2

(6.22)

noting Eq. (4.75), where the material constants K and G correspond to the bulk and the shear moduli, respectively, in the infinitesimal elasticity theory. The differentiation of Eq. (6.21) with respect to C reads: 0

S 5 KðlnJÞC21 1 2GðlnC1=2 Þ C21

(6.23)

noting Eq. (4.78), from which the Mandel stress M is given as M 5 KðlnJÞG 1 2GðlnC1=2 Þ

0

by Eq. (4.78) and the following differential equations pffiffiffiffiffiffiffiffiffiffiffi 8 @J @ detC 1 21 > > > 5 5 JC > < @C @C 2 pffiffiffiffiffiffiffiffiffiffiffi > > > @lnJ 5 @ln detC 5 1 C21 > : @C @C 2 based on Eqs. (1.314) and (1.315).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(6.24)

(6.25)

157

6.2 Hyperelastic constitutive equations of metals

6.2.3 Neo-Hookean elasticity The elastic material possessing the following strain energy is called the neo-Hookean material. ψðCÞ 5

1 1 λðlnJÞ2 2 μlnJ 1 μðtrC 2 3Þ 2 2

(6.26)

where λ and μ are the material constants, from which we have  1 1 1 1 S 5 2 λ2ðlnJÞ C21 2 μ C21 1 μG 2 2 2 2 leading to S 5 λðlnJÞC21 1 μðG 2 C21 Þ

(6.27)

noting Eq. (6.25)2. Then, the Mandel stress M is given by M 5 λðlnJÞG 1 μðC 2 GÞ

(6.28)

Eq. (6.27) is described by the Kirchhoff stress as τ 5 FSFT 5 F½λðlnJÞC21 1 μðG 2 C21 ÞFT 5 F½λðlnJÞF21 F2T 1 μðG 2 F21 F2T ÞFT leading to τ 5 λðlnJÞg 1 μðb 2 gÞ

(6.29)

6.2.4 Modified neo-Hookean elasticity (1) By modifying Eq. (6.26) in the neo-Hookean elasticity, let the following the strain energy function be adopted by Simo and Pister (1984), Ciarlet (1988), etc. ψðCÞ 5

1 2 1 1 λðJ 2 1Þ 2 ð λ 1 μÞlnJ 1 μðtrC 2 3Þ 4 2 2

(6.30)

from which, noting Eqs. (6.44) and (6.49) with Eq. (6.25)1,2, we have 1 2 λðJ 2 1ÞC21 1 μðG 2 C21 Þ 2 1 M 5 λðJ 2 2 1ÞG 1 μðC 2 GÞ 2 1 τ 5 λðJ 2 2 1Þg 1 μðb 2 gÞ 2

S5

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(6.31) (6.32) (6.33)

158

6. Hyperelastic equations

Eq. (6.30) is reduced to the following equation for the incompressible material. ψðCÞ 5 μðtrC 2 3Þ

(6.34)

which is used often for polymers.

6.2.5 Modified neo-Hookean elasticity (2) The following strain energy function is used for the finite elastoplastic constitutive relation by Vladimirov et al. (2008, 2010). ψðCÞ 5

1 μ ΛðJ 2 2 1 2 2lnJÞ 1 ðtrC 2 3 2 2lnJÞ 4 2

(6.35)

where Λ and μ are material constants, from which we have S5

Λ 2 ðJ 2 1ÞC21 1 μðG 2 C21 Þ 2

(6.36)

Λ 2 ðJ 2 1ÞG 1 μðC 2 GÞ 2

(6.37)

Λ 2 ðJ 2 1Þg 1 μðb 2 gÞ 2

(6.38)

M5 τ5

noting Eq. (6.25). Eq. (6.35) will be adopted for the formulation of the finite elastoplastic constitutive equation in Section 9.7 for the multiplicative-hyperelastic-based plasticity.

6.2.6 Modified neo-Hookean elasticity (3) The following strain energy function has been applied to the finite elastoplasticity theory by Brunig (1998) and Helm (2001) and to the finite viscoplasticity theory by Shutov and Kreissing (2008a,b, 2010) and Shutov et al. (2011). ψðCÞ 5

K μ ðlnJÞ2 1 ðtrC 2 3Þ 2 2

(6.39)

where K and μ are material constants. Here, C is the unimodular tensor defined in Eq. (4.68). It follows from Eq. (6.39), noting Eqs. (4.71) and (6.25)1,2, that  1 S 5 KðlnJÞC21 1 μJ22=3 G 2 ðtrCÞC21 (6.40) 3 0

0

M 5 KðlnJÞG 1 μJ22=3 C 5 KðlnJÞG 1 μC

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(6.41)

6.3 Hyperelastic equations of rubbers

τ 5 KðlnJÞg 1 μJ22=3 b

0

159 (6.42)

6.2.7 Modified neo-Hookean elasticity (4) By modifying Eq. (6.26) in the neo-Hookean elasticity, the following the strain energy function is assumed by Simo and Pister (1984), Ciarlet (1988), Simo and Miehe (1992), etc., which has the separated form into two terms related to the volume change J and the isochoric deformation C.   K J2 2 1 μ 2 lnJ 1 ðtrC 2 3Þ ψðCÞ 5 (6.43) 2 2 2 where K and μ are the material constants, from which we have  1 1 S 5 KðJ 2 2 1ÞC21 1 μJ22=3 G 2 ðtrCÞC21 2 3

(6.44)

1 1 0 0 KðJ 2 2 1ÞG 1 μJ22=3 C 5 KðJ 2 2 1ÞG 1 μC (6.45) 2 2 0 1 (6.46) τ 5 KðJ 2 2 1Þg 1 μb 2 Except for the St. Venant-Kirchhoff elastic constitutive equation in Subsection 6.2.1, the hyperelastic equations are described in complex forms in terms of the second Piola-Kirchhoff stress tensor S but it can be described concisely in terms of the Mandel stress tensor M. M5

6.3 Hyperelastic equations of rubbers The strain energy function of the Mooney
Rivlin model (Mooney, 1940; Rivlin, 1948) which is applicable to the elastic deformation of the incompressible rubbers is given as ψ 5 a1 ðIC 2 3Þ 1 a2 ðIIC 2 3Þ ðIIIC 5 1Þ

(6.47)

where a1 and a2 are material parameters and

8 IC  trC 5 J2=3 trC 5 trb 5 λ21 1 λ22 1 λ23 > > > < 1 1 1 IIC  ðtr2 C 2 trC2 Þ 5 J 4=3 ðtr2 C 2 trC 2 Þ 5 ðtr2 b 2 trb2 Þ 5 λ21 λ22 1 λ22 λ23 1 λ23 λ21 2 2 2 > > > : 2 2 2 2 2 IIIC  det C 5 J 5 det b ð 5 J Þ 5 λ1 λ2 λ3

(6.48)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

160 with

6. Hyperelastic equations

8 @IC @trC > > 5G 5 > > @C @C > > > > > 1 > < @ ðtr2 C 2 trC2 Þ @IIC 2 5 IC G 2 C 5 > > @C @C > > > > > @IIIC @detC > 21 > > : @C 5 @C 5 IIIC C

(6.49)

noting Eqs. (1.301), (1.311), and (1.313), where λi are the principal values 3 3 P P of C 5 U2 leading to C 5 λ2i Ni  Ni and C21 5 λ22 i Ni  Ni . Further, i51

i51

the Mooney
Rivlin model is given by the simplification setting a2 5 0 as follows: ψ5

1 νðIC 2 3Þ ðIIIC 5 1Þ 2

(6.50)

where ν is the material parameter. Further, based on the Seth-Hill’s generalized strain in Eq. (4.51), the following strain energy function, called the Ogden model, was proposed by Ogden (1982, 1984), which is applicable to the elastic deformation of the incompressible rubber for a large deformation is given as ψ5

3 X βn

α n51 n

ðλα1 n 1 λα2 n 1 λα3 n 2 3Þ

(6.51)

where αn and β n are material parameters. Eq. (6.51) is reduced to Eq. (6.47) for the Mooney
Rivlin model by choosing the material parameters under the incompressibility λ1λ2λ3 5 1 as follows (cf. Hisada, 1992): 8 α1 5 2 > < β 1 5 2C1 ; > :

β 2 5 2 2C2 ; α2 5 2 2

β 3 5 0;

(6.52)

α3 5 0

6.4 Hyperelastic equations of soils The following strain energy function is assumed by Hashiguchi (2018b).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

6.5 Hyperelasticity in infinitesimal strain

~ M0 1 ϑF0 ÞJ21=κ~ 1 G0 J2n=κ~ ðtrC 2 3Þ ψðlnJ; trCÞ 5 ϑFlnJ 1 κðP

161 (6.53)

noting ~ J2n=κ~ 5 expð2nκlnJÞ where PM0 is the initial value of the pressure defined in terms of the ~ κ, ~ and G0 are Mandel stress M in Eq. (5.76) by PM  2 ð1=3ÞtrM. ϑ, λ, material parameters. The first Piola
Kirchhoff stress and the Mandel stress are given by h i n S 5 ϑF 2 ðPM0 1 ϑF0 ÞJ21=κ~ 2 G0 J2n=κ~ ðtrC 2 3Þ C21 κ~  (6.54) 1 2n=κ~ 22=3 21 1 2G0 J J G 2 ðtrCÞC 3 h i 0 n M 5 CS 5 ϑF 2 ðPM0 1 ϑF0 ÞJ21=κ~ 2 G0 J2n=κ~ ðtrC 2 3Þ G 1 2G0 J2n=κ~ C ~κ (6.55) These formulations will be described in detail in Section 9.7.2.

6.5 Hyperelasticity in infinitesimal strain Apart from the exact formulation based on the deformation gradient tensor, consider the infinitesimal strain ε described in Eq. (4.36). The stress is calculated through the Gibbs’ free energy function ψðεÞ as follows: σ5

@ψðεÞ @ε

(6.56)

In case that the strain energy function is given by the quadratic form 1 ψðεÞ 5 ε:E:ε 2

(6.57)

with the fourth-order tensor E which is the constant tensor independent of ε, the stress and the elastic modulus tensors are given as follows: σ 5 E:ε;

@σ @2 ψðεÞ 5 5E @ε @ε  @ε

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(6.58)

162

6. Hyperelastic equations

where E stands for the elastic modulus which is given in the Hooke’s form as follows: 2 3 8 > E E E E 1 1 0 > > 4 ðδik δjl 1δil δjk Þ2 δij δkl 5 > T1 I ; Eijkl 5 δij δkl 1 E5 > > 3ð122νÞ 11ν 3ð122νÞ 11ν 2 3 < 2 3 > > > 122ν 11ν 122ν 11ν 1 1 0 > E21 5 4 ðδik δjl 1δil δjk Þ2 δij δkl 5 > T1 I ; E21 δij δkl 1 > ijkl 5 : 3E E 3E E 2 3 (6.59)

or 2 3 8 > 1 1 0 > > > E 5 KT 1 2GI ; Eijkl  Kδij δkl 1 2G4 ðδik δjl 1 δil δjk Þ 2 δij δkl 5 > > 2 3 < 2 3 > > > 21 1 1 0 21 1 1 41 1 > 5 > > : E 5 9K T 1 2G I ; Eijkl 5 9K δij δkl 1 2G 2 ðδik δjl 1 δil δjk Þ 2 3 δij δkl (6.60) where E and ν are Yong’s modulus and Poisson’s ratio, respectively, and K and G are called the bulk elastic modulus and the shear elastic modulus, 0 respectively. T is the fourth-order tracing tensor and I is the fourthorder deviatoric projection tensor defined in Eqs. (1.223) and (1.228), respectively. The relations of these material parameters are given by 8 9KG 3K 2 2G > > > E 5 3K 1 G ; ν 5 2ð3K 1 GÞ < (6.61) E E > > ; G  K  > : 3ð1 2 2νÞ 2ð1 1 νÞ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

C H A P T E R

7 Development of elastoplastic and viscoplastic constitutive equations The requirements for the elastoplastic constitutive equations, that is, the decomposition of the deformation into the elastic and the plastic parts and the incorporation of the yield surface and the stress rate versus the strain rate relation due to the stress path dependence of the plastic strain rate, are described first. In addition, the continuity condition and the smoothness condition are defined, which are required for the accurate description of the elastoplastic deformation behavior. Then, the development of elastoplastic constitutive equations is reviewed: The constitutive equations based on the infinitesimal hyperelastic-based plasticity were first studied and then the hypoelastic-based plasticity was incorporated. In the meantime, various cyclic elastoplasticity models describing not only the monotonic but also the cyclic loading behavior, that is, the so-called cyclic plasticity models, have been proposed. Among them, the subloading surface model is capable of describing the monotonic and the cyclic loading behaviors rigorously. The basic concept and the formulation of the subloading surface model in the current configuration are described in detail. Further, the subloading surface model is extended to the subloadingoverstress model to describe the rate-dependent deformation.

7.1 Basis of elastoplastic constitutive equations The fundamental features and the mathematical requirements for elastoplastic constitutive equation are shown below based on the physical observations.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity DOI: https://doi.org/10.1016/B978-0-12-819428-7.00007-9

163

© 2020 Elsevier Inc. All rights reserved.

164

7. Development of elastoplastic and viscoplastic constitutive equations

7.1.1 Fundamental requirements for elastoplasticity Elastoplastic constitutive equations must possess the following fundamental properties. 7.1.1.1 Decomposition of deformation/rotation (rate) into elastic and plastic parts The deformation of solids which are assemblies of sloid particles, e.g. crystal particles in metals and soil particles in sands and clays, are induced by the deformations of material particles themselves and their mutual slips. The deformations of material particles themselves cause the macroscopic elastic deformation of the material as far as a large stress such as material particles deforms irreversibly is not applied. On the other hand, when the macroscopic stress applied to the material increases up to the yield stress, the mutual slips between the material particles are not removed even if the stress is removed (unloaded), so that they cause the plastic deformation of the material. Eventually, the deformation of material is decomposed into the elastic part and the plastic part. Here, note the following facts. 1. The elastic deformation must be described by the hyperelastic constitutive equation described in Chapter 6: Hyperelastic equations. Besides, if the elastic part is described by the Cauchy elastic equation describing the one-to-one correspondence of stress and strain, energy is produced or dissipated by the elastic deformations during cyclic loading of stress. Further, if the elastic part is described by the hypoelastic elastic equation describing the linear relation of stress rate and strain rate, the accumulation of strain in addition to the production or dissipation of energy during the cyclic loading of stress is described by the hypoelastic equation. (Note) The elastic deformation of the material as the assembly of solid particles is caused by the deformation of material particles as described above. It is referred to as the energetic elasticity (cf. e.g. Holzapfel, 2000; Bergstrom, 2015). On the other hand, the elastic deformation of the solids which are composed of the polymer chains, e.g. the rubber and various kinds of polymers is caused by the entropic strengthening of polymer chains leading to the ordered structure. It is referred to as the entropic elasticity. This book is concentrated to the former solids. 2. The variation of the plastic deformation characteristics is not induced by the elastic deformation but induced by the plastic deformation. Then, it is described by incorporating the variables which evolve only by the plastic deformation. Such variables are called the internal

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.1 Basis of elastoplastic constitutive equations

165

(or hidden) variables, since they cannot be known from the current external mechanical state of material but known by pursuing the history of plastic deformation, while the stress, the strain, the temperature, etc. are called the external variables since they are known from the current external mechanical state of material. 3. The mutual slips between material particles are not induced suddenly at once but it is induced gradually from parts in which slips are induced easily. Therefore the plastic deformation develops gradually as the stress approaches the yield stress so that the smooth transition from the elastic state to the elastoplastic state, that is, smooth elasticplastic transition is induced. It leads to a continuous variation of the stiffness modulus tensor. The consideration of this fact is of the crucial importance in the description of cyclic loading behavior. It will be attained rigorously by the concept of the subloading surface leading to the subloading surface model described in later sections. All deformation (rate) measures are defined by the deformation gradient tensor F describing the linear operator that transforms the reference infinitesimal line-element vector to the current infinitesimal lineelement vector. Then, the deformation gradient tensor F is decomposed into the elastic and the plastic parts in order to describe the elastic and the plastic deformations exactly leading to the multiplicative decomposition of the deformation gradient tensor as will be formulated in Section 8.1. However, the strain rate tensor and the spin tensor are additively decomposed into the elastic and the plastic parts. 7.1.1.2 Incorporation of yield surface Yield surface must be incorporated, which specifies the stress state at which the plastic strain rate is induced in the multidimensional stress state. Then, the loading criterion, by which whether or not the plastic strain rate is induced can be judged, is formulated rigorously. Nevertheless, constitutive equations for reversible-irreversible deformation without the abovementioned decomposition of deformation and the yield surface have been studied after the proposition of the hypoelasticity by Truesdell (1955) (see also Truesdell and Noll, 1965), in which the magnitude of the strain rate and/or the stress rate is introduced, resulting in the rate-nonlinearity, called the hypoplasticity by Dafalias (1986). This approach has been fashioned temporarily and the society studying this approach is called the rational mechanics. However, this approach for constitutive equations of metals has been almost deteriorated in the 1980s, recognizing the fundamental inability caused by lack of the decomposition of deformation and the yield surface. Nevertheless, it remains yet in constitutive equations for soils (cf. e.g., Kolymbas and Wu, 1993).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

166

7. Development of elastoplastic and viscoplastic constitutive equations

7.1.1.3 Stress rate versus strain rate relation The elastic deformation is independent of the stress path but the plastic deformation depends on it, exhibiting the so-called stress path dependence. Consequently, there does not exist the stress versus strain relation in the elastoplastic deformation process. Therefore the elastoplastic constitutive equation must be described as the stress rate versus strain rate relation.

7.1.2 Requirements for elastoplastic constitutive equation The fulfillment of the continuity and the smoothness conditions are required to describe the elastoplastic deformation accurately, which will be explained in the following. 7.1.2.1 Continuity condition It is observed in experiments that “Stress rate changes continuously for a continuous change of strain rate.” This fact is called the continuity condition and is expressed mathematically as follows (Hashiguchi, 1993a,b, 1997, 2000): lim ½σðσ; _ _ H; H; ε_ 1 δ_εÞ-σðσ; H; H; ε_ Þ for infinitesimal strain

(7.1)

lim ½σ3 ðσ; H; H; d 1 δdÞ- σ3 ðσ; H; H; dÞ for hypoelasticity

(7.2)

δ_ε-O

δd-O

where H and H are the second-order tensor-valued and the scalar-valued internal variables, respectively, and δ ð Þ stands for an infinitesimal variation. The response of the stress rate to the input of strain rate in the cur3 ðσ; H; H; dÞ rent state of stress and internal variables is designated by σ 3 and σ ðσ; H; H; dÞ. Uniqueness of solution is not guaranteed in constitutive equations violating the continuity condition, predicting different stresses or deformations for identical input loading. Ordinary elastoplastic constitutive equations, in which the plastic strain rate is derived obeying the consistency condition, fulfill the continuity condition. The concept of the continuity condition was first advocated by Prager (1949). However, a mathematical expression of this condition was not given. The condition was defined as the continuity of strain rate to the input of stress rate by Prager (1949) inversely to the definition given above. However, identical stress rate directing inwards the yield surface can induce different strain rates in loading and unloading states for softening materials. Plastic strain rate can be induced indeterminately for the zero stress rate on the yield surface in the perfectly plastic material. For that reason, the Prager’s (1949) notion does not hold in the general loading state including softening and the perfectly plastic states.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.1 Basis of elastoplastic constitutive equations

167

7.1.2.2 Smoothness condition The plastic deformation/sliding is not induced abruptly but develops gradually. In fact, mutual slips of material particles, for example, crystal particles in metals and soil particles in sands and clays are not induced simultaneously but induced gradually in the parts in which mutual slips can be induced easily leading to the smooth transition from the elastic to the plastic transition. Then, it is observed in experiments that “Stress rate induced by identical strain rate changes continuously for continuous change of stress state” leading to the continuous variation of the tangent stiffness modulus tensor. This fact is called the smoothness condition and its mathematical expression was given by Hashiguchi (1993a,b, 1997, 2000) as follows: lim σðσ _ 1 δσ; H; H; ε_ Þ-σðσ; _ H; H; ε_ Þ

δσ-O

(7.3)

for infinitesimal elastoplastic strain theory lim σ3 ðσ 1 δσ; H; H; dÞ- σ3 ðσ; H; H; dÞ for hypoelastic plasticity (7.4)

δσ-O

The rate-linear elastoplastic constitutive relation is described as σ_ 5 Mep ðσ; H; HÞ : ε_ for infinitesimal elastoplastic strain theory 3

σ 5 Mep ðσ; H; HÞ : d for hypoelastic plasticity

(7.5) (7.6)

where the fourth-order tensor Mep is the elastoplastic tangent stiffness modulus tensor, which is a function of the stress and internal variables and described as Mep 5

@σ for infinitesimal elastoplastic strain theory @ε @σ Mep 5 Ð for hypoelastic plasticity @ ddt

(7.7) (7.8)

Consequently, Eqs. (7.3) and (7.4) can be rewritten as lim Mep ðσ 1 δσ; H; HÞ-Mep ðσ; H; HÞ

δσ-O

(7.9)

Elastoplastic constitutive equations assuming the yield surface which encloses a purely elastic domain, that is, all elastoplasticity models other than the subloading surface model violate the smoothness condition, exhibiting the abrupt transition from the elastic to the plastic state resulting in the discontinuous variation of tangent stiffness modulus tensor. Then, they require (1) the judgment on the fulfillment of the yield condition in the loading criterion and (2) the determination of an offset value (plastic strain value in yield point), which is accompanied

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

168

7. Development of elastoplastic and viscoplastic constitutive equations

with an arbitrariness. The smoothness condition must be taken into account in the formulation of the cyclic plasticity model. However, it is violated in the conventional elastoplastic constitutive model and the cyclic kinematic hardening models, that is, the multisurface model (Mroz, 1967; Iwan, 1967), the two-surface model (Dafalais and Popov, 1975; Krieg, 1975; Yoshida and Uemori, 2002), the superposed kinematic hardening models (Chaboche et al., 1979; Ohno and Wang, 1993) because a surface enclosing a purely-elastic domain is assumed in them.

7.2 Historical development of elastoplastic constitutive equations The historical development of elastoplastic constitutive equation is delineated in this section prior to the explanation of the multiplicative hyperelastic-based plastic constitutive equation in the subsequent chapters, which is the main purpose of this book. The multiplicative hyperelastic-based plastic constitutive equation is based on the intermediate configuration pulled back from the current configuration. However, the constitutive relation which one can perceive directly from the measured material behavior and thus is based on the current configuration. Then, one must first formulate the constitutive equation in the current configuration. The infinitesimal hyperelastic-based plasticity and the hypoelastic-based plasticity are formulated in the current configuration. However, the former is irrelevant to the rotation of material, i.e. the spin and the latter is limited to the infinitesimal elastic deformation. Then, both of the concrete formulations for the infinitesimal hyperelastic-based plasticity and for the hypoelastic-based plasticity will be described in this section prior to the formulation of the multiplicative hyperelastic-based plasticity in the subsequent chapters.

7.2.1 Infinitesimal hyperelastic-based plasticity The elastoplastic constitutive equation based on the so-called infinitesimal strain has been studied until the middle of the last century. The infinitesimal strain ε in Eq. (4.39) cannot be described only by the stretch tensor U, depending on the rotation tensor R, so that it describes the deformation roughly. It is expressed from Eq. (4.39) with Eq. (4.41) as follows:   @ ðx 2 XÞ ε 5 sym (7.10) @X

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

169

7.2 Historical development of elastoplastic constitutive equations

The infinitesimal strain ε is reduced to the nominal strain e in a particular direction, that is, e5

l 2 l0 l0

(7.11)

where l and l0 are the current and the initial length, respectively. The nominal strain e is additively decomposed into the nominal elastic strain ee and the nominal plastic strain ep as follows: e 5 ee 1 ep

(7.12)

where ee 5

l2l ; l0

ep 5

l 2 l0 l0

(7.13)

where l is the length unloaded to the stress-free state. The infinitesimal uniaxial strain ε in Eq. (7.10) is regarded as the three-dimensional extension of the nominal strain, replacing l0 -dX; l-dX ; and l-dX, and thus it is additively decomposed into the elastic and the plastic parts, that is, ε 5 εe 1 εp ; ε_ 5 ε_ e 1 ε_ p where ε 5 sym



     @u @x 2 @X @x 2 @X 5 sym ; εe 5 sym ; @X @X @X

(7.14)  εp 5 sym

 @X 2 @X @X (7.15)

X is the position vector of the material particle in the virtually unloaded state to the stress-free state by the hyperelastic constitutive equation. The elastic strain is given in Eq. (7.14) as follows: ð e p ε 5 ε 2 ε 5 ε 2 ε_ p dt (7.16) The defects of the infinitesimal strain theory are as follows: 1. The deformation which can be described by this theory is limited to the infinitesimal deformation owing to the inherent property of the infinitesimal strain tensor. 2. The spin cannot be taken into account in constitutive equations, because the corotational rates of stress and tensor-valued internal variables are not used but their material-time-derivatives in Eq. (1.321) are used, although the spin is described by ant½@v=@X.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

170

7. Development of elastoplastic and viscoplastic constitutive equations

The stress is calculated by the hyperelastic relation in Eq. (6.56) with the replacement of ε to εe as follows: σ5

@ψ ðεe Þ @εe

(7.17)

from which the stress rate is given as follows: σ_ 5

@2 ψðεe Þ : ε_ e @εe  @εe

(7.18)

If the Helmholtz’s free energy function ψðεe Þ is given by the quadratic equation with the constant elastic modulus tensor E independent of εe as ψðεe Þ 5

1 e ε : E : εe 2

(7.19)

it follows that σ 5 E : εe ;

@σ @2 ψðεe Þ 5 5 E; @εe @εe  @εe σ_ 5 E : ε_ e ;

εe 5 E21 : σ

ε_ e 5 E21 : σ_

(7.20) (7.21)

The stress σ can be calculated by substituting the elastic strain εe in Eq. (7.16) into Eq. (7.20) (Eq. 7.17 for the general elastic strain energy function) without the time-integration. Eqs. (7.20) and (7.21) are rewritten by Eq. (7.14) as σ 5 E : ε 2 E : εp ; σ_ 5 E : ε_ 2 E : ε_ p

(7.22)

The second terms in the right-hand sides in these equations are called the plastic (stress) relaxation, which is induced by the plastic deformation caused by the mutual slips between material particles. The following yield condition (surface) with the isotropic and the kinematic hardening is introduced. ^ 5 FðHÞ f ðσÞ

(7.23)

σ^  σ 2 α

(7.24)

where

H is the isotropic hardening variable and α is the kinematic hardening variable, that is, back stress describing the Bauschinger effect. The evolution rule of H is given generally as follows: H_ 5 f Hε ðσ; H; ε_ p Þ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.25)

7.2 Historical development of elastoplastic constitutive equations

171

The isotropic hardening function FðHÞ and the rate of the isotropic hardening variable H_ for metals are given as follows: rffiffiffi    2 p _ FðHÞ 5 F0 1 1 h1 1 2 exp ð2 h2 HÞ ; H 5 (7.26) :_ε : 3 F0 5

dF 5 F0 h1 h2 exp ð2 h2 HÞ dH

(7.27)

where h1 and h2 are the material constants. Here, note that the following Voce-type isotropic hardening function is used widely. h i FðHÞ 5 F0 1 h1 1 2 expð2 h2 HÞ in which the hardening function F saturates at F0 1 h1 , that is, F-F0 1 h1 for H-N, while the material parameters h1 possess the stress dimension. On the other hand, F saturates at ð1 1 h1 ÞF0 , that is, F-ð1 1 h1 ÞF0 for H-N, while h1 is the dimensionless parameter in Eq. (7.27), which would be more reasonable than the Voce-type function. The evolution rule of the kinematic hardening variable is given by modifying the Armstrong and Frederick nonlinear kinematic hardening rule (Armstrong and Frederick, 1966) so as to saturate in the state jjαjj 5 bk F as follows:

1 p p _ 5 ck ε_ 2 jj_ε jjα α (7.28) bk F where ck and bk ð # 1Þ are the material constants. Eq. (7.28) is written in the uniaxial loading state as follows: 8 ! sffiffiffi > 3 α > a > ck 1 2 ε_ pa for ε_ pa $ 0 ! > rffiffiffi > < 2 b F k 3 1 p ! (7.29) α_ a 5 ck ε_ pa 2 j_ε jαa 5 sffiffiffi > 2 bk F a 3 α > a p p > > c 11 ε_ a for ε_ a , 0 > : k 2 bk F which is illustrated in Fig. 7.1. An isotropic hardening variable causes the variation of the size of yield surface and thus its rate is described by scalar-valued variable, that is, invariant(s) of the plastic strain rate, for example, the magnitude of deviatoric plastic strain rate for metals and the plastic volumetric strain rate for soils. It is irrelevant to the loading direction. On the other hand, the rate of anisotropic hardening variable is given by the tensorvalued variables because it depends on the loading direction. The rate

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

172

7. Development of elastoplastic and viscoplastic constitutive equations

(A)

α ck

(

1− 3 α 2 bk F

(

(B)

α

1

(

ck 1 + ck 0

(

1

1

ck (ε p )

ck

ε

1

3 α 2 bk F

εp

1

0

p

FIGURE 7.1 Kinematic hardening rules illustrated in one-dimensional state. (A) Ratenonlinear kinematic hardening rule. (B) Rate-linear kinematic hardening rule.

of anisotropic hardening variable changes abruptly at the moment of inverse loading. This is the inherent property of the anisotropic hardening variable as shown in Eq. (7.29) with Fig. 7.1 for the kinematic hardening variable. A tensor-valued internal variable, denoted by the symbol t, describing the anisotropy is described by the hyperelastic relation.  p @ϕt εts t5 p @εts

(7.30)

 p p where ϕt εts is the potential energy function of the storage part εts of the plastic strain rate. Here, the following relation is incorporated. p

p

εp 5 εts 1 εtd ;

p

p

ε_ p 5 ε_ ts 1 ε_ td

(7.31)

p

where εtd is the dissipative part of the plastic strain, which is caused by purely slips of material particles but does not contribute to the anisotropic hardening. Here, adopt the simplest function  p 1 p p ϕt εts 5 ct εts : εts 2

(7.32)

in the quadratic form, where ct is the material constant, from which one has  p p @2 ψt εts @t @εts p t 5 ct εts ; 5 5 c t p p p p 5 ct I @εts @εts @εts  @εts (7.33)  p p p t_ 5 ct ε_ ts 5 ct ε_ 2 ε_ td

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.2 Historical development of elastoplastic constitutive equations

173

Then, the storage and the dissipative parts of the plastic strain rate are given as follows: p

ε_ ts 5

1_ p 1 p t; ε_ td 5 ε_ p 2 ε_ ts 5 ε_ p 2 _t ct ct

(7.34)

The kinematic hardening variable α and its rate are formulated using p the Helmholtz free energy function ψk ðεks Þ for the kinematic hardening variable in Eq. (7.33) as follows: α5

 p @ψk εks

(7.35)

p

@εks

p

where the decomposition of the plastic strain εp into the storage part εks p and the dissipative part εkd is incorporated for the kinematic hardening following Eq. (7.31) as follows: p

 k

p

p

p

εp 5 εks 1 εkd ; ε_ p 5 ε_ ks 1 ε_ kd

(7.36)

 p 1 p p ψk εks 5 ck εks : εks 2

(7.37)

p If ψ εks is given by the quadratic equation

where ck is the material constant, it follows that p

α 5 ck εks ;

p

_ 5 ck ε_ ks ; α

(7.38)

and p

ε_ ks 5

1 _ α; ck

p

p

ε_ kd 5 ε_ p 2 ε_ ks 5 ε_ p 2

1 _ α ck

(7.39)

noting Eq. (7.34). The rates of the storage and the dissipative parts in the rate of the kinematic hardening variable in Eq. (7.28) are given in Eq. (7.39) as follows: 8 1 p > > jj_εp jjα ε_ 5 ε_ p 2 > < ks bk F 1 p > p > > :ε_ kd 5 bk F jj_ε jjα

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.40)

174

7. Development of elastoplastic and viscoplastic constitutive equations

Consequently, the evolution rule of the kinematic hardening is represented as follows: Storage part

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{

1 _ 5 ck |{z} jj_εp jj α α ε_ p 2 bk F |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

(7.41)

Total

Dissipative part

This equation means that the rate of kinematic hardening is proporp tional to the storage part of plastic strain rate, ε_ ks , causing the prevention of plastic deformation which is subtracted the dissipative part of p plastic strain rate, ε_ kd , due to the mere slips of material particles from p the plastic strain rate ε_ p , while the dissipative part ε_ kd causes the rateirreversibility, which is the fundamental feature of the irreversible phenomena. The variable involved in the dissipative part is not the plastic strain itself but its magnitude that causes the rate-nonlinearity, that is, the abrupt increase of the tangential stiffness modulus at the moment of the inverse loading. The rheological model for isotropic and rate-nonlinear kinematic hardening is shown in Fig. 7.2. The elastic deformation is depicted by the zigzag mark but the storage parts of the plastic strain are depicted by the spiral spring marks. The hardening is given only by the  pisotropic p storage part leading to ε_ is 5 ε_ p ε_ id 5 0 . The following Prager’s linear kinematic hardening rule (Prager, 1956) cannot describe the actual material behavior. _ 5 ck ε_ p α

(7.42)

which lacks the dissipative part leading to the rate-reversibility. In addition, the extension of the material parameter ck to the monotonically

Kinematic hardening

p ·p 1 · 1 p ε·kd = b α | ε· | ε ks = ck α kF α ε kpd ε ksp

p ε· is = ε· p = F· / F ' (ε· ipd = 0 )

Isotropic hardening Sliding between particles

F − F0

εp

σ = F +α ε e = σ /E

F0

ε

εe

FIGURE 7.2 Rheological model for isotropic and rate-nonlinear kinematic hardenings.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.2 Historical development of elastoplastic constitutive equations

175

decreasing function of the plastic strain, that is, ck ðεp Þ (e.g., de Souza Neto et al., 2008) is quite unrealistic such that the ratio of the kinematic hardening rate to the plastic strain rate decreases continuously even in the reverse loading process as shown in Fig. 7.1B. It should be noticed that Eq. (7.41) with the dissipative part is rate-nonlinear relation but Eq. (7.42) without the dissipative part is rate-linear relation, while the linear and the nonlinear relations of α and εp are described in Eq. (7.42), when ck is the material constant and the function of εp , respectively. Now, introduce the associated flow rule (normality rule): ε_ p 5 λ_ n^ ðλ_ . 0Þ

(7.43)

where λ_ is referred to the plastic multiplier or the positive proportionality factor and n^ is the normalized outward-normal of the yield surface, that is, n^ 

^ ^ ^ ^ @fðσÞ @fðσÞ @fðσÞ @fðσÞ ^ 5 1Þ =: :5 =: :ðjjnjj @σ^ @σ^ @σ @σ

(7.44)

noting ^ ^ @ðσ 2 αÞ @fðσÞ ^ @σ^ ^ @fðσÞ @fðσÞ @fðσÞ 5 5 5 @σ^ @σ^ @σ @σ^ @σ @σ Here, it should be noted that the normalized tensor n^ is used in the flow rule of Eq. (7.43) in order to separate the plastic strain rate ε_ p defi^ This definite nitely into the pure magnitude λ_ and the pure direction n. separation is of physical importance for the rigorous formulation of the plastic constitutive equation. The physical interpretations of the associated flow rule have been advocated by Drucker (1950) for positivity of the stress cycle, Ilyushin (1961) for positivity of the strain cycle, etc., as explained in detail in the literature (Hashiguchi, 2017a). The substitution of Eq. (7.43) into Eqs. (7.25) and (7.28) leads to ^ λ_ H_ 5 f Hn^ ðσ; H; nÞ

1 _ ^ _ α α 5 λ fkn ; fkn^  ck n 2 bk F

(7.45) (7.46)

The rate of the isotropic hardening variable H_ for metals is given from Eq. (7.26) as follows: rffiffiffi rffiffiffi! 2 2 λ_ f Hn^ 5 (7.47) H_ 5 3 3 In what follows, the plastic strain rate is formulated in the following.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

176

7. Development of elastoplastic and viscoplastic constitutive equations

The consistency condition for the yield condition in Eq. (7.23) is given by ^ ^ @fðσÞ @fðσÞ _ 5 F0 H_ : σ_ 2 :α @σ @σ

(7.48)

^ in Eq. (7.23) to be the Here, we choose the yield stress function fðσÞ homogeneous function of σ^ in degree-one. Therefore it follows that ^ 5 jsj fðσÞ ^ fðjsjσÞ

(7.49)

^ @fðσÞ ^ 5F : σ^ 5 fðσÞ @σ^

(7.50)

for an arbitrary scalar s and

for the sake of Euler’s theorem for homogeneous function in degree-one (cf. Hashiguchi, 2017a). Then, it follows from Eq. (7.50) that ^  @fðσÞ   : σ^  @fðσÞ   @fðσÞ ^ ^  n^ : σ^ ^ @ σ       5 1   @σ^  5 F ^ fðσÞ @σ^ 

(7.51)

Eq. (7.48) is rewritten noting Eq. (7.51) as 0

_2 n^ : σ_ 2 n^ : α

F _ H n^ : σ^ 5 0 F

(7.52)

The substitution of the associated flow rule in Eq. (7.43) with Eqs. (7.25) and (7.46) into Eq. (7.52) leads to 0

n^ : σ_ 2 n^ : fkn^ λ_ 2

F f λ_ n^ : σ^ 5 0 F Hn^

(7.53)

which is rewritten as n^ : σ_ 2 λ_ Mp 5 0

(7.54)



0 F Mp  n^ : fkn^ 1 f Hn^ σ^ F

(7.55)

where

The plastic multiplier and the plastic strain rate are given by n^ : σ_ n^ : σ_ λ_ 5 n^ ; ε_ p 5 Mp Mp

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.56)

7.2 Historical development of elastoplastic constitutive equations

177

Then, the strain rate is expressed by the stress rate by noting Eqs. (7.14), (7.21), and (7.56) as follows: ε_ 5 E21 : σ_ 1

n^ : σ_ n^ Mp

(7.57)

The substitution of Eqs. (7.14) and (7.43) with Eq. (7.56) into Eq. (7.21) leads to

n^ : σ_ ^ 5 E : ε_ 2 p n^ (7.58) σ_ 5 E : ð_ε 2 ε_ p Þ 5 E : ð_ε 2 λ_ nÞ M and further it follows from Eqs. (7.54) and (7.56) that _ p 5 n^ : σ_ 5 n^ : E : ð_ε 2 λ_ nÞ ^ λM

(7.59)

from which the plastic multiplier in terms of the strain rate is given by _ Λ5

n^ : E : ε_ n^ : E : ε_ ; ε_ p 5 p n^ Mp 1 n^ : E : n^ M 1 n^ : E : n^

(7.60)

Then, the stress rate is expressed by substituting Eq. (7.60) into λ_ in Eq. (7.58) as follows: σ_ 5 E : ε_ 2 E : n^

n^ : E : ε_ Mp 1 n^ : E : n^

The loading criterion is given by  p ^ 5 FðHÞ and Λ_ . 0 ε_ 6¼ O for fðσÞ p ε_ 5 O for others or



^ 5 FðHÞ and n^ : E : ε_ p $ 0 ε_ p 6¼ O for fðσÞ p ε_ 5 O for others

(7.61)

(7.62)

(7.63)

noting that the denominator Mp 1 n^ : E : n^ . 0 in Λ_ because E is the positivedefinite tensor. The elastic modulus tensor E in Eqs. (7.58)(7.63) must be replaced by @2 ψðεe Þ=ð@εe  @εe Þ for the general elastic strain energy function. The physical background of the loading criterion (7.62) or (7.63) is elucidated in Hashiguchi (2000, 2017a). The loading criterion in Eq. (7.62) or (7.63) is applicable not only to the hardening but also to the softening behaviors. The deformation analysis by the abovementioned infinitesimal hyperelastic-based plastic constitutive equation is performed by the following procedure.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

178

7. Development of elastoplastic and viscoplastic constitutive equations

1. The plastic strain, the isotropic hardening variable, and the dissipative part of kinematic hardening variable are calculated by input of ε_ as follows: ð ð ð 1 p _ _ ^ H 5 f Hn^ Λdt; αΛdt (7.64) εkd 5 εp 5 Λ_ ndt; bk F where Λ_ is given in Eq. (7.60). 2. The stress and the kinematic hardening variable are calculated in Eq. (7.20) (Eq. 7.17 for the general elastic strain energy function) and Eq. (7.28) (Eq. 7.35 for the general kinematic hardening storage energy function). These calculation processes are repeated for the further loading. On the other and, the calculation may be performed even by the direct time-integration of the following rate relations. ^ σ_ 5 E : ð_ε 2 Λ_ nÞ

1 _ α _ 5 ck n^ 2 H_ 5 f Hn^ Λ; α Λ_ bk F

(7.65) (7.66)

where Λ_ is given in Eq. (7.60). The abovementioned two kinds of the calculation methods, that is, the former method by the hyperelasticity and the method by the latter fully direct time-integration of rate equations provide almost same accuracy and efficiency in the simple linear elastic material. However, the former method using the hyperelasticity will be inherited to the multiplicative hyperelastic-based plastic constitutive equation described in Chapter 8, Multiplicative decomposition of deformation gradient tensor, and Chapter 9, Subloading-multiplicative hyperelastic-based plastic constitutive equations. The abovementioned infinitesimal hyperelastic-based plastic constative equation is limited to the description of the infinitesimal elastic and plastic deformation without a rotation. Besides, large loading steps can be input in the implicit return-mapping method but only quite small incremental steps are allowed in the explicit-forward method.

7.2.2 Hypoelastic-based plasticity The abovementioned infinitesimal hyperelastic-based plastic constitutive equation has been developed and used widely until the middle of the last century. In that circumstance, the hypoelastic equation was proposed by Truesdell (1955) concurrently with the appearance of the finite element method. Then, the hypoelastic-based plasticity has been developed widely in the leading by Rodney Hill in Cambridge and James

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.2 Historical development of elastoplastic constitutive equations

179

Robert Rice in Harvard and thus it is called often the HillRice manner, while the multiplicative hyperelastic-based plasticity is called the Mandel-Lee manner. The last half-century has been spent for the study of the hypoelastic-based plasticity as known from the fact that most of monographs (Simo, 1998; Simo and Hughes, 1988; Belytschko et al., 2014; de Souza Neto et al., 2008; Hashiguchi, 2017a; etc.) are more or less concerned with the explanation of the hypoelastic-based plasticity. In particular, various corotational rates have been proposed for this purpose as described in Appendix 2. The two defects of the infinitesimal theory described in the beginning of the preceding section are solved by the hypoelastic-based plasticity. However, the elastic deformation is limited to be infinitesimal in the hypoelastic-based plasticity as will be explained in Section 8.2.3. The strain rate d in Eq. (4.84) is additively decomposed into the elastic strain rate de and the plastic strain rate dp , that is, d 5 de 1 dp

(7.67)

Here, note that the additive decomposition of the strain rate in Eq. (7.67) holds on the limitation that the elastic deformation is infinitesimal. The elastic strain rate is linearly related to the corotational rate in the hypoelastic equation (see Appendix 3) as follows: 3

3

σ 5 E : de ; de 5 E21 : σ

(7.68)

where the elastic modulus tensor E is a function of the Cauchy stress in general. Here, the corotational rate with the plastic spin is designated by the symbol ð3 Þ as shown in Eq. (3.49), which must be adopted to the stress, and the kinematic hardening variable. It may be replaced to the Jaumann rate for the small plastic strain rate. The consistency condition is given as follows: ^ ^ @fðσÞ @fðσÞ 0 3 3 :σ2 : α 5 F H_ @σ @σ

(7.69)

that is, 0

3

3

n^ : σ 2 n^ : α 2

F _ H n^ : σ^ 5 0 F

(7.70)

where the evolution rules of the isotropic hardening variable H are given from Eq. (7.25) by H_ 5 f Hd ðσ; H; dp Þ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.71)

180

7. Development of elastoplastic and viscoplastic constitutive equations

in general, which is given from Eq. (7.26) for Mises metals as follows: rffiffiffi 2 p _ (7.72) H5 jjd jj 3 The kinematic hardening variable α is given from Eq. (7.41) by Storage part

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ 1 3 jjdp jjαÞ α 5 ck ð |{z} dp 2 bk F |fflfflfflfflffl{zfflfflfflfflffl}

(7.73)

Total

Dissipative part

Further, introduce the associated flow rule: dp 5 λ_ n^ ðλ_ . 0Þ

(7.74)

The plastic spin wp in Eq. (3.50) is given by Zbib and Aifantis (1998) as follows: 1 wp 5 ηp ðσdp 2 dp σÞ 5 ηp λ_ ant½σn^  2

(7.75)

The substitution of Eq. (7.74) into Eqs. (7.71) and (7.73) leads to ^ λ_ H_ 5 f Hn^ ðσ; H; nÞ

1 3 _ α α 5 fkn^ λ; fkn^  ck n^ 2 bk F

(7.76) (7.77)

Substituting Eqs. (7.76) and (7.77) into Eq. (7.52) leads to 0

3

n^ : σ 2 n^ : fkn^ λ_ 2

F fHn^ λ_ n^ : σ^ 5 0 F

(7.78)

which is rewritten as 3

_ p 50 n^ : σ 2 λM

(7.79)

where

! 0

M  n^ :

F f Hn^ σ^ 1 fkn^ F

p

(7.80)

The plastic multiplier and the plastic strain rate are given by 3

3

n^ : σ n^ : σ n^ λ_ 5 ; ε_ p 5 Mp Mp

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.81)

7.2 Historical development of elastoplastic constitutive equations

181

The strain rate is given from Eqs. (7.67), (7.68), and (7.81) as follows: 3

d 5 E21 : σ 1

3

n^ : σ n^ Mp

(7.82)

from which the proportionality factor described in terms of the strain _ and the plastic strain rate in the flow rule, denoted by Λ_ instead of λ, rate are given as follows: Λ_ 5

n^ : E : d ; 1 n^ : E : n^

Mp

dp 5 Λ_ n^ 5

n^ : E : d n^ 1 n^ : E : n^

Mp

(7.83)

_ The stress rate is given from Eq. (7.82) with the replacement of λ_ to Λ as follows:

n^ : E : d E : n^  n^ : E 3 E : n^ 5 E p σ 5E:d2 p :d (7.84) M 1 n^ : E : n^ M 1 n^ : E : n^ The loading criterion is given as follows (Hashiguchi, 2000, 2017a):  p ^ 5 FðHÞ and Λ_ . 0 d 6¼ O for fðσÞ (7.85) p d 5 O for others that is,



^ 5 FðHÞ and n^ : E : d . 0 dp 6¼ O for fðσÞ dp 5 O for others

(7.86)

It is noticeable in the hypoelastic-based plasticity that the corotational rate is used in order to exclude the influence of the rigid-body rotation from the material-time derivative of stress so that the objectivity (Oldroyd, 1950) is satisfied, while the responses by various corotational stress rate with the plastic spin have been examined by Dafalias (1985), Zbib and Aifantis (1988), Gambirasio et al. (2016), etc. The main features of the hypoelastic-based plasticity are described below. 1. The cumbersome operation is required for the time-integration of the corotational stress rate in numerical calculations (cf. Simo and Hughes, 1988; Fish and Shek, 1999; de Souza Neto et al., 2008; Hashiguchi, 2017a; etc.). 2. The finite deformation with the finite rotation can be described under the restriction of the infinitesimal elastic deformation as will be verified in Section 8.2.3.

7.2.3 Multiplicative hyperelastic-based plasticity The elastoplastic deformation and rotation can be described exactly by the multiplicative decomposition of the deformation gradient tensor

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

182

7. Development of elastoplastic and viscoplastic constitutive equations

into the elastic and the plastic parts, while the deformation gradient tensor is the most basic measure for describing the deformation and the rotation. The multiplicative hyperplastic-based plasticity will be described comprehensively in Chapter 8 and 9.

7.3 Subloading surface model As described in Section 7.1.2, the plastic deformation is not induced abruptly but induced gradually as the stress approaches the yield surface, although it is postulated in the conventional elastoplastic constitutive equation that the plastic deformation is induced abruptly at the moment that the stress reaches the yield stress. Then, the tangent stiffness modulus tensor changes continuously for a continuous variation of stress state. The subloading surface model (Hashiguchi, 1978, 1980, 1989; Hashiguchi and Ueno, 1977) is the natural extension of the conventional elastoplastic constitutive equation taking account of this fact leading to the noticeable abilities, for example, the fulfillments of both the continuity and the smoothness conditions, the unnecessary of the judgment on the fulfillment of the yield condition, the automatic control function attracting the stress to the yield surface. Now, let the following underlying postulate be incorporated in order to describe the plastic strain rate induced by the rate of stress inside the yield surface (Hashiguchi, 1980, 1989, 2009, 2017a). Fundamental postulate of subloading surface concept: The plastic strain rate develops as the stress approaches the yield surface and inversely the stress approaches the yield surface when the plastic strain rate is induced, exhibiting a continuous variation of tangent stiffness modulus. On the contrary, the stress recedes from the yield surface when only an elastic strain rate is induced. Then, it is first required to incorporate the general measure that describes the approaching degree of the stress to the yield surface, renamed the normal-yield surface, because the plastic strain rate is induced even inside it in this model. To this end, the following subloading surface is introduced, which always passes through the current stress and maintains a similar shape and an orientation to the normal-yield surface (see Fig. 7.3). ^ 5 RFðHÞ fðσÞ

(7.87)

where Rð0 # R # 1Þ is the ratio of the size of the subloading surface to that of the normal-yield surface and called the normal-yield ratio designating the approaching degree of the stress to the normal-yield surface. It should be noted that the independent surface is only the

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

183

7.3 Subloading surface model

Normal-yield surface f (σ ) = F ( H )

Subloading surface f (σ ) = RF (H )

σ α

0

σij

FIGURE 7.3 Normal-yield and subloading surfaces.

normal-yield surface but the subloading surface is not independent surface and is determined from the normal-yield surface with the current stress. The rate of the normal-yield ratio must satisfy the following conditions on account of the fundamental concept of the subloading surface. 8 -N for R 5 0 > > < p _ . 0 for R , 1 for d 6¼ O (7.88) R 5 0 for R 5 1 > > : ð, 0 for R . 1Þ Here, the rate of the normal-yield ratio evolves with the plastic strain rate, obeying Eq. (7.88) but it is calculated from the equation of the subloading surface equation in Eq. (7.87), substituting a stress changing by the elastic constitutive relation under fixed internal variables when only the elastic strain rate is induced. Then, let the evolution rule of the normal-yield ratio in the plastic loading process be given as follows: R_ 5 UðRÞjjdp jj for dp 6¼ O

(7.89)

where UðRÞ is the monotonically decreasing function of R fulfilling the conditions (see Fig. 7.4A). 8 - 1 N for R 5 0 ðelastic stateÞ > > < . 0 for R , 1 ðsubyield stateÞ (7.90) UðRÞ 5 0 for R 5 1 ðnormal-yield stateÞ > > : , 0 for R . 1 ðover normal-yield stateÞ The explicit forms of the function UðRÞ fulfilling the conditions in Eq. (7.90) are given by the following equations.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

184

7. Development of elastoplastic and viscoplastic constitutive equations

·

·

p

p

U ( R ) ( = R / || d ||)

U ( R ) ( = R / || d ||)

dp = O, de≠ O

dp = O, de ≠ O

dp ≠ O

dp ≠ O 0

(A)

1

0

dp≠ O R

Re

1

dp ≠ O R

(B)

FIGURE 7.4 Function UðRÞ in rate of normal-yield ratio R. (A) Re 5 0. (B) Re . 0.

UðRÞ 5 u cot

hπ i R 2

UðRÞ 5 2 u ln R

1 21 UðRÞ 5 u R

(7.91) (7.92) (7.93)

where u is the material parameter. Equation (7.91) will be adopted hereinafter. The normal-yield ratio R increases obeying the evolution rule in Eq. (7.89) formulated by the plastic strain rate in the plastic-loading process. On the other hand, it decreases in the elastic loading process, where R is calculated by substituting the stress and the internal variables into the equation of the subloading surface in Eq. (7.87). Here, note that there exist a lot of materials containing usual metals in which the plastic strain rate is hardly induced in a range of lower value of the normal-yield ratio. Then, let the following relation be assumed instead of Eq. (7.90), in which the plastic strain rate is not induced until the normal-yield ratio R reaches a certain value of the material parameter Re ð , 1Þ (see Fig. 7.4B). 8 - 1 N for 0 # R # Re ðelastic stateÞ > > < . 0 for Re , R , 1 ðsubyield stateÞ (7.94) UðRÞ 5 0 for R 5 1 ðnormal-yield stateÞ > > : , 0 for R . 1 ðover normal-yield stateÞ The material parameter Re is interpreted to be the ratio of the (half) stress amplitude σfl at the fatigue (or endurance) limit to the yield stress σy , that is, Re 5 σfl =σy . Fatigue limit is observed in steels, titanium, etc. but it is not observed in other materials involving nonferrous metals.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

185

7.3 Subloading surface model

Eq. (7.91) is extended to satisfy Eq. (7.94) as follows:

π hR 2 Re i UðRÞ 5 u cot 2 1 2 Re

(7.95)

where h i is the Macaulay bracket defined by hsi 5 ðs 1 jsjÞ=2, that is, s , 0:hsi 5 0 and s $ 0:hsi 5 s (s: arbitrary scalar variable). Eq. (7.95) fulfills the smoothness condition since it decreases continuously from infinite value in R 5 Re . In the plastic loading process, the normal-yield ratio R is calculated by Eq. (7.89). However, if u is fixed to be constant, Eq. (7.89) with Eq. (7.95) can be integrated analytically as 9 # "

p

> > 2 π R0 2 Re π εp 2 ε0 21 > R 5 ð1 2 Re Þ cos exp 2u 1 Re> cos > > π 2 1 2 Re 2 1 2 Re > > > >

= π R0 2 Re cos for R0 $Re > 2 1 2 Re > 2 1 2 R p > e >

ln εp 2 ε0 5 > > > π R 2 Re π u > > cos > ; 2 1 2 Re Ð

p

(7.96)

under the initial condition R 5 R0 for εp 5 ε0 , where εp 5 jjdp jjdt. One must set R0 5 Re for R0 , Re . Here, the judgment whether of R , Re or R $ Re is required in Eq. (7.95), although the yield judgment is not required. The analytical integration curve in Eq. (7.96) is shown in Fig. 7.5 for Re 5 0:2 and u 5 50; 100 and 500. On the other hand, the normal-yield ratio R is calculated by ^ R 5 fðσÞ=FðHÞ based on Eq. (7.87) in the elastic deformation process.

1.0

u = 500 100

50

0.8 0.6

R 0.4 0.2 0

0.05

εp

0.10

0.15

FIGURE 7.5 Analytical integration curve of normal-yield ratio.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

186

7. Development of elastoplastic and viscoplastic constitutive equations

The time-differentiation of Eq. (7.87) of the subloading surface leads to ^ ^ @fðσÞ @fðσÞ 0 3 3 _ 50 :σ2 : α 2 RF H_ 2 RF ^ ^ @σ @σ

(7.97)

Substituting the associated flow rule

 

dp 5 λn^ ð λ 5 jjdp jj $ 0Þ into Eq. (7.97), one has



(7.98)





^ ^ @fðσÞ @fðσÞ 0 3 :σ2 : λ fkn^ 2 RF λ f Hn^ 2 U λF 5 0 @σ^ @σ^

(7.99)

The substitution of the relation 8 ^ @fðσÞ > > ^ 5 RF : σ^ 5 fðσÞ > > > ^ @ σ > < ^ @fðσÞ    : σ^  > >     ^ ^  n^ : σ^ @fð σÞ > >  5 @σ^   @fðσÞ > 1 >  @σ^   @σ^  5 RF : ^ fðσÞ into Eq. (7.99) reads:

 n^ : σ 2 n^ : λf 3

from which one has



λ5

! 0 F U ^ : σ^ λf 1 λ 50 kn^ 2 n R F Hn^



n^ : σ3

where

M

p

"

p

M  n^ :

;

dp 5



n^ : σ3 M

p

n^

#

0 F U fHn^ 1 σ^ 1 fkn^ R F

(7.100)

(7.101)

(7.102)

(7.103)

which is reduced to Eq. (7.80) for the conventional plasticity in the normal-yield state (R 5 1: U 5 0). The strain rate is given by Eqs. (7.67), (7.68), and (7.102) as follows: 21

d5E

3

:σ1

n^ : σ3 M

p



n^  n^ 21 n^ 5 E 1 :σ3 p M

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.104)

187

7.3 Subloading surface model



from which the magnitude of plastic strain λ in terms of strain rate,



denoted by the symbol Λ, is derived as follows:



Λ5



n^ : E : d n^ : E : d ; dp 5 Λn^ 5 p n^ M 1 n^ : E : n^ M 1 n^ : E : n^ p

(7.105)

The stress rate is described from Eq. (7.104) with Eq. (7.105) as 3

σ 5E:d2

n^ : E : d p

M 1 n^ : E : n^

E : n^ 5 ðE 2

E : n^  n^ : E p

M 1 n^ : E : n^

Þ:d

The loading criterion is given by ( dp 6¼ O for Λ . 0 dp 5 O for other

(7.106)



or



(7.107)

dp 6¼ O for n^ : E : d . 0 dp 5 O for other

(7.108)

where the judgment of whether or not the stress reaches the yield surface is not required since the plastic strain rate develops continuously as the stress approaches the normal-yield surface. The stress versus strain curve described by the subloading surface model is illustrated in Fig. 7.6 for the simplest case of the nonhardening state in Ð the uniaxial loading, where the uniaxial strain is designated by ε1 ð 5 d1 dtÞ.

σ1

σ1

Normal-yield surface f (σˆ ) = F (H)

Subloading surface model

σ

Subloading surface

R 1

f (σˆ ) = RF (H)

α

0

ε1

0

σ2

σ3

FIGURE 7.6 Smooth stressstrain curve predicted by subloading surface model.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

188

7. Development of elastoplastic and viscoplastic constitutive equations

As known from the abovementioned facts, the distinguished advantages for the descriptions of irreversible mechanical phenomena are gained by the subloading surface model with the simple modification of existing computer program for the conventional elastoplasticity model to add only one material parameter u.

7.4 Cyclic plasticity models The elastoplasticity model for the description of the cyclic loading behavior is called the cyclic plasticity model, which is required to describe the plastic strain rate induced by the rate of stress inside the yield surface. Various cyclic plasticity models have been proposed hitherto. They are classified into two types. One type is based on the concept of the kinematic hardening, that is, the translation of the small loading surface inside the conventional yield surface (Mroz, 1967; Iwan, 1967; Dafalias and Popov, 1975, Krieg, 1975; Yoshida and Uemori, 2002) or the small yield surface excluding the conventional yield surface (Chaboche et al., 1979; Ohno and Wang, 1993) while these surfaces enclose a purely elastic domain. They are called the kinematic cyclic hardening models. The other type is based on the natural concept that the plastic strain rate develops as the stress approaches the yield surface, that is, the extension of the subloading surface model (Hashiguchi, 1978, 1980, 1989) described in the preceding sections in this chapter.

7.4.1 Cyclic kinematic hardening models with yield surface The multi surface model, i.e., Mroz model was first proposed by Mroz (1967) and Iwan (1975), which assume the several encircled loading surface inside the conventional yield surface. Thereafter, the two surface model was proposed by Dafalias and Popov (1975) and Krieg (1975) and later by Yoshida and Umemori (2002), in which a small loading surface enclosing a purely-elastic domain is incorporated inside the conventional yield surface (see Fig. 7.7). These models possess the serious drawbacks as follows: 1. Plastic strain rate cannot be described for the variation of stress inside the small loading surface. It leads to the risky design of the mechanical elements subjected to the cyclic loading with small stress amplitudes at high stress state, although in fact a remarkable strain accumulation exhibiting the mechanical ratchetting is induced. 2. The tangent stiffness modulus lowers suddenly at the moment when the stress reaches the small loading surface enclosing a purely elastic domain so that the smoothness condition in Eq. (7.9) is violated and

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

FIGURE 7.7 Yield and loading surfaces in cyclic plasticity models. .

190

3. 4.

5.

6.

7. Development of elastoplastic and viscoplastic constitutive equations

thus a smooth stress-strain curve cannot be described, resulting in the discontinuous variation of the tangent stiffness modulus. The judgement whether the stress reaches the small loading surface enclosing a purely-elastic domain is required. The determination of the offset value, i.e. the plastic stain at yield point of the small yield surface is required, which is accompanied with an arbitrariness. Quite small loading increments such that a stress does not go out from the small loading surface enclosing a purely-elastic domain must be input, resulting in inefficient calculation in the forwardEuler method in numerical calculations. The extension of the cyclic kinematic hardening model to the finite strain theory based on the multiplicative decomposition of deformation gradient, called the multiplicative finite strain theory, would not be easy or impossible.

In addition to the above-mentioned defects, the multi surface model describes the excessively strong Masing effect, because the plural encircled loading surfaces translate simultaneously with the plastic deformation, so that the mechanical ratchetting phenomenon cannot be not described at all. In contrast, the two surface model cannot describe the plastic deformation in the unloading process, because the plastic modulus depends merely on the distance from the current stress to its conjugate point on the yield surface, so that the Masing effect cannot be described at all and the excessively strong mechanical ratchetting strain is predicted. Therefore, the spring-back phenomenon cannot be described by the two-surface model by which only elastic deformation is described in the unloading process. Nevertheless, the description of the spring-back phenomenon by lowering the Young’s modulus with the plastic deformation is introduced by Yoshida and Uemori model (2002). It causes the serious physical contradiction ignoring the damage phenomenon (mechanics) that the damage develops in accelerating rate with the plastic deformation once the damage proceeds, while the damage is assumed to cease as the plastic deformation proceeds up to several percent of plastic strain in the Yoshida and Uemori model (refer to Hashiguchi (2017a) for the detail). In addition, Yoshida and Amaishi (2020) proposed the reduction of the Yonug’s modulus with a pseudo elastic deformation. However, it again results in the contraiction to describe the hysteresis loop by an elastic equation, while it should be noticed that the hysteresis loop is to be induced by the plastic deformation. All of the above-mentioned defects in the multi surface and the two surface model are entirely dissolved by the subloading surface model described in the subsection 7.4.3.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.4 Cyclic plasticity models

191

7.4.2 Ad hoc Chaboche model and Ohno-Wang model excluding yield surface The Chaboche model (Chaboche et al., 1979) and the Ohno-Wang model (Ohno and Wang, 2003) exclude (ignore) the yield surface which has been introduced in the conventional plasticity theory so that they possess the mechanical structures different basically from the conventional plasticity model as shown in Fig. 7.7. On the other hand, all the other models, i.e. the multi surface model (Mroz, 1967; Iwan, 1967), the two surface model (Dafalias, 1975; Krieg, 1975) and the subloading surface model (Hashiguchi, 1978, 1980, 1989) follow the conventional plasticity model with the yield surface so that they exhibit the plastic behavior identical to the conventional plasticity model when the stress lies on the yield surface. The Chaboche model and the Ohno-Wang model which ignore the historically developed conventional plasticity model possess the following fundamental defects. i. The cylindrical loading surface with a small radius enclosing a purely elastic domain is rapidly translated in parallel by the superposition of many kinematic hardening rules (5 to 8 usually) in the deviatoric tress plane so as to fit to the test curve and thus a lot of material parameters lacking physical meanings are involved. Therefore, these models cannot be regarded as a physical model but they are merely the polynomial approximations of test curves, i.e. the rarely-typical ad hoc models applicable only to the limited 0 metallic materials, i.e. the J2 ð ðtrσ 2 Þ=2Þ material without the influences of the pressure and the third deviatoric invariant. ii. The extension of these models to the description of the anisotropy, e.g. the Hill’s orthotropic anisotropy (Hill, 1948) and the Balat’s anisotropy (Balat et al., 2003) is impossible because the yield surface in the conventional plasticity is abandoned, where the kinematic hardening rule cannot be formulated so as to translate under the limitation due to the anisotropic conventional yield surface, iii. The description of the plastic deformation behavior of the pressuredependent materials is also impossible, because the loading surface independent of the pressure is concerned, iv. The tangent stiffness modulus changes suddenly at the moment when the stress reaches the loading surface enclosing the purelyelastic domain, v. The plastic deformation cannot be described for the small stress variation because the inside of the small loading surface enclosing

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

192

7. Development of elastoplastic and viscoplastic constitutive equations

the purely-elastic domain is adopted. It leads to the risky mechanical designs of solids and structures subjected to the vibrational loading with small stress amplitudes, vi. Quite small loading increments such that a stress does not go out finitely from the small loading surface enclosing a purely-elastic domain must be input, resulting in inefficient calculation by the forward-Euler method in numerical calculations, vii. The extension to the description of the finite deformation based on the multiplicative decomposition is not easy or impossible. although these crucial defects have not been recognized/revealed but the exaggerations of merits have repeated during near the last half century in the reviews (Chaboche, 1989, 2008; Lemaitre and Chaboche, 1990; Ohno-Wang, 1993). Consequently, the Chaboche model and the Ohno-Wang model ignore the historically developed conventional plasticity model but regress to the easy going way by the empirical method to superpose many kinematic hardening rules such that the stress-strain curve fits to test data. Nevertheless, they are diffused widely through the standard installations in the commercial FEM software (Abaqus, Marc, LS-DYNA, ANSIS, etc.), because their formulations can be captured easily even by the beginners who are ignorant of the advanced plasticity theory but know only the kinematic hardening rule. However, it should be recognized that the Chaboche model and the Ohno-Wang model are not applicable to materials other than the isotropic and pressure-independent Mises metal, although the other cyclic plasticity models have been applied to various materials, e.g. soils, rocks, concrete, friction, etc. It is quite pity that the appearances and the diffusion of such ad hoc models are preventing the true development of the plasticity theory, especially resulting in the serious regression of the metal cyclic plasticity. All of the above-mentioned defects in the Chaboche model and the Ohno-Wang model are entirely dissolved by the subloading surface model described in the next subsection. The overall assessments of the cyclic plasticity models are shown in Table 7.1. The cyclic kinematic hardening models possess various basic defects, where the Chaboche model and the Ohno-Wang model are the worst ones possessing the crucial defects amongst all models. However, these defects have never been indicated by the proposers themselves. It is not clear that this fact is caused by the research abilities or the moral senses.

7.4.3 Extended subloading surface model The subloading surface is applicable to the general elastoplastic materials, e.g. soils, rocks and concrete unlimited to the Mises

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

TABLE 7.1

Assessment of cyclic plasticity models.

Cyclic plasticity model

Smoothness condition

Judgment of yielding

Automatic pull-back of stress to yield surface

Strain accumulation during cyclic loading of small stress amplitude

Description Description of of anisotropic pressure hardening dependence

Formulation of multiplicative finite deformation

Multi surface model (Mroz)

Cyclic kinematic Hardening Model: Translation of small surface enclosing elastic domain

Expansion of loading surface

Possible

Two surface model (Dafalias; Yoshida-Uemori) Superposed kinematic hardening model excluding yield surface (Chaboche; Ohno-Wang)

Subloading surface model (Hashiguchi)

Violate

Required Impossible Impossible

Difficult Impossible

Fulfill

Not required

Possible

Formulated already

194

7. Development of elastoplastic and viscoplastic constitutive equations

metals, The subloading surface model formulated in preceding section, called the initial subloading surface model hereinafter, is incapable of describing cyclic loading behavior appropriately, predicting an open hysteresis loop in an unloading-reloading process and thus overestimating a mechanical ratcheting phenomenon. The insufficiency is caused by the fact that the similarity-center of the normalyield and the subloading surfaces is fixed at the origin of stress space and thus a purely elastic deformation is described in the unloading process, resulting in the open hysteresis loop. Here, it should be noted that purely elastic response is induced only in an initiation of reverse loading process in general. Then, the insufficiency was remedied by making the similarity-center of the normal-yield and the subloading surfaces translate with the plastic deformation (Hashiguchi, 1986, 1989). The similarity-center is called the elastic-core and denoted by c, since the most elastic deformation behavior is described when the stress lies on (coindides with) it leading to R 5 0. The uniaxial loading behavior is depicted in Fig. 7.8 for the simple material behavior without a variation of the normal-yield surface. The elastic-core goes up following the stress by the plastic strain rate in the initial loading process. The subloading surface contracts and thus only elastic strain rate is induced until the stress goes down to the elastic-core in the unloading process. However, the plastic strain rate in compression is induced as the subloading surface begins to expand after it once contracts in the unloading-inverse loading process whilst the elastic-core goes down following the stress by the plastic strain rate. Again only the elastic strain rate is induced until the stress goes up to the elastic-core in the reloading process from the complete unloading. After that the subloading surface begins to expand and thus the plastic strain rate is induced whilst the elastic-core goes up following the stress by the plastic strain rate. Consequently, the closed hysteresis loop is depicted realistically as shown in this figure. σ1

σ1

° σ

σ° °c

c° ° α

0

Expands

° α

εp

0 σ2

σ3

FIGURE 7.8 Prediction of uniaxial loading behavior by extended subloading surface model.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

195

7.5 Formulation of (extended) subloading surface model

(

Stress,

Elastic-core,

Center of subloading surface)

σ

σ

εp

0 Initial subloading surface model

0 εp Extended subloading surface model

Elastic-core (similarity-center)

FIGURE 7.9 Modification of subloading surface model to describe cyclic loading behavior.

The extended subloading surface model would describe the cyclic loading behavior realistically as illustratively shown in Fig. 7.9 in which variations of the stress and the elastic-core in the unloading-reloading process are depicted. It does not contain any drawbacks in the cyclic plasticity models based on the kinematic hardening concept, while both of the continuity and the smoothness conditions in Eqs. (7.2) and (7.4) are satisfied only in this model. Then, it has been applied to the descriptions of rate-independent and rate-dependent elastoplastic deformation behavior of not only metals but also geomaterials and further the friction phenomena between solids as will be described in detail in the subsequent chapters.

7.5 Formulation of (extended) subloading surface model The fundamental concept for the extension of the subloading surface model to the description of the cyclic loading behavior was described in the latest section. The explicit formulation will be given in this section.

7.5.1 Normal-yield and subloading surfaces The normal-yield surface with the isotropic and the kinematic hardening is described as ^ 5 FðHÞ fðσÞ

(7.109)

as shown in Eq. (7.23) already. The extended subloading surface for the normal-yield surface in Eq. (7.109) in which the similarity-center of the

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

196

7. Development of elastoplastic and viscoplastic constitutive equations

normal-yield and the subloading surface, i.e. the elastic-core translates is given as follows (see Fig. 7.10). fðσÞ 5 RFðHÞ

(7.110)

where Rð0 # R # 1Þ is the normal-yield ratio and σ  σ2α

(7.111)

α stands for the conjugate (similar) point in the subloading surface to the point α in the normal-yield surface. Here, the following relation holds by virtue of the similarity of the subloading surface to the normal-yield surface (see Figure 7.10).  σ 2 c 5 Rðσy 2 cÞ (7.112) α 2 c 5 Rðα 2 cÞ which yields α 5 c 2 R^c _

σ 5 σ 1 R^c where



c^  c 2 α _ σ  σ2c

(7.113) (7.114)

(7.115)

σy is the conjugate stress on the normal-yield surface. All the relations in Eqs. (7.112)(7.114) hold by virtue of the similarity of the subloading surface to the normal-yield surface. The timedifferentiation of Eq. (7.113) leads to 3 3 3 _c α 5 R α 1 ð1 2 RÞ c 2 R^ Adopt the associated flow rule for the subloading surface:   p dp 5 λ n λ 5 jjd jj . 0



where @fðσÞ n @σ



   @fðσÞ     @σ  ðjjnjj 5 1Þ

(7.116)

(7.117)

(7.118)

The plastic spin in Eq. (3.50) is given as follows:



wp 5 ηp ðσdp 2 dp σÞ 5 λωp ; ωp 5 ηp ðσn 2 nσÞ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.119)

FIGURE 7.10 Normal-yield, subloading, and elastic-core surfaces. (A) General material. (B) Mises material in deviatoric stress plane.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

198

7. Development of elastoplastic and viscoplastic constitutive equations

The rate of the isotropic hardening variable is described as



_ Hðσ; H; dp Þ 5 λ f Hn ðσ; H; nÞ

(7.120)

and the rate of the kinematic hardening variable is described as follows:

1 3 α 5 λ fkn ; fkn 5 ck n 2 α (7.121) bk F



7.5.2 Evolution rule of elastic-core The rigorous evolution rule of the elastic-core is formulated in this section, which plays the central role in the description of the cyclic loading behavior leading to the closed hysteresis loop as shown in Figure 7.9. Now, the following facts should be noticed. 1. In the physical view-point, a smooth elastic-plastic transition is not described if the elas-tic-core lies on the normal-yield surface at which a remarkable plastic deformation is in-duced, 2. In the mathematical view-point, the subloading surface is not determined uniquely if the elastic-core, i.e., the similarity-center lies on the normal-yield surface, noting the fact: If the elastic-core lies on the normal-yield surface and the stress coincides with the elastic-core, R is indeterminate as known from the relation 0 5 R0 which is induced by substituting σ 5 c 5 σy into σ 2 c 5 Rðσy 2 cÞ based on the similarity of the subloading surface to the normal-yield surface. Consequently, the elastic-core is not allowed to approach to the normal-yield surface unlimitedly. Now, let the following elastic-core surface be introduced, which always passes through the elastic-core c and maintains a similarity to the normalyield surface with respect to the kinematic hardening variable α. fð^cÞ 5 ℜc FðHÞ; that is; ℜc 5

fð^cÞ FðHÞ

(7.122)

where ℜc designates the ratio of the size of the elastic-core surface to the normal-yield surface (see Fig. 7.10) so that let it be called the elasticcore yield ratio. Then, let it be postulated that the elastic-core can never reach the normal-yield surface designating the fully plastic stress state so that the elastic-core does not go over the following limit elastic-core surface. fð^cÞ 5 χFðHÞ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.123)

7.5 Formulation of (extended) subloading surface model

199

where χð , 1Þ is material constant or function in general and the following inequality must be satisfied. fð^cÞ # χFðHÞ; that is; ℜc # χ

(7.124)

The material-time derivative of Eq. (7.124) at the limit state that c lies on the limit elastic-core surface in Eq. (1.123) yields: @fð^cÞ 3 3 : ðc 2 αÞ 2 χF_ # 0 @^c

for ℜc 5 χ

(7.125)

Here, noting @fð^cÞ : c^ 5 fð^cÞ 5 χF for ℜc 5 χ (7.126) @^c on account of the Euler’s homogeneous function fð^cÞ in degree-one for the variable c^ , the substitution of Eq. (7.126) to Eq. (7.125) leads to

@fð^cÞ F_ 3 3 : c  α 2 c^ # 0 for ℜc 5 χ (7.127) @^c F The inequality (7.124) is called the enclosing condition of elastic-core and Eq. (7.127) is its rate form. Now, assume the equation (Hashiguchi, 2018e, 2019a) F_ 3 3 c 2 α 2 c^ 5 ce jjdp jjðσχ 2 cÞ F

(7.128)

where ce is the material constant and σχ is the conjugate stress on the limit elastic-core surfaces to the current stress σ on the subloading surface, that is, χ σχ  σ 1 α (7.129) R which is based on the relation  σ 2 α σχ 2 α 5 σy 2 α 5 χ σy 2 α 5 χ R χ

(7.130)

where σy is the conjugate stress on the normal-yield surface (Fig. 7.10). Eq. (7.128) means that the elastic-core translates to approach the conju3 gate stress σχ in the nonhardening state: F_ 5 0 and α 5 O. It follows for Eq. (7.128) that

@fð^cÞ @fð^cÞ F_ 3 3 : c2α2 : ðσχ 2 cÞ 5 ce jjdp jj @^c @^c F (7.131)    cÞ  p  @fð^   n^ c : ðσχ 2 cÞ # 0 for ℜc 5 χ 5 ce jjd jj @^c 

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

200

7. Development of elastoplastic and viscoplastic constitutive equations

FIGURE 7.11 Translation of elastic-core when it lies on the limit surface (ℜc 5 χ) fulfilling the enclosing condition in Eq. (7.127) because of the convexity of the surface.

as far as the normal-yield surface satisfies the convexity condition (cf. Hashiguchi, 2017a), noting that @fð^cÞ=@c which is the outwardnormal of the elastic-core surface at the current elastic-core c and σχ 2 c makes an obtuse angle when c lies on the limit elastic-core surface, while σχ lies on the limit elastic-core surface. n^ c is the normalized outward-normal of the elastic-core surface.   @fð^cÞ  @fð^cÞ  n^ c  (7.132)  @c  ðjjn^ c jj 5 1Þ @c Here, σχ 2 c makes an obtuse angle to n^ c when c lies on the limit elastic-core surface, satisfying ℜc 5 χ, as shown in Fig. 7.11. Therefore Eq. (7.128) satisfies the enclosing condition of the elastic-core so that the elastic-core can never go out from the limit elastic-core surface. Then, the evolution rule of the elastic-core is given from Eq. (7.128) by χ  F_ 3 c3 5 ce jjdp jj σ 2 c^ 1 α 1 c^ R F

(7.133)

where the second and the third terms in the right-hand side designate the influences by the kinematic and the isotropic hardening. Eq. (7.133) is described in the infinitesimal strain theory as follows:

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

201

7.5 Formulation of (extended) subloading surface model

 F_ _ 1 c^ (7.134) σ 2 c^ 1 α R F Eq. (7.133) with Eqs. (7.120) and (7.121) leads to

0 χ  1 Ff 3 p p p σ 2 c^ 1 ck d 2 c 5 ce jjd jj jjd jjα 1 Hn jjdp jj^c 5 fcn λ (7.135) R bk F F c_ 5 ce jj_εp jj

χ



which is described in the infinitesimal theory as follows:

0 χ  1 Ff c_ 5 ce jj_εp jj jj_εp jjα 1 Hn jj_εp jj^c σ 2 c^ 1 ck ε_ p 2 R bk F F 2 3



0 χ 1 F fHn 5 4 α 1 c^ λ 5 fcn λ σ 2 c^ 1 ck n 2 5 ce R bk F F



where fcn  ce



0 1 Ff α 1 Hn c^ σ 2 c^ 1 ck n 2 R bk F F

χ

(7.136)



(7.137)

The elastic-core translates toward the conjugate stress σχ in the simple material without the hardenings H_ 5 0 and α3 5 O, resulting in c_ 5 ce jj_εp jjðσχ 2 cÞ. Considering the uniaxial loading behavior of the further simple with the plastic incompressibility ε_ p 5 ε_ p0 leading pffiffiffiffiffiffiffiffi material p p jj_ε jj 5 3=2j_εa j and designating the components of c and ε_ p in the axial p direction by ca and ε_ a , respectively, it follows noting σχa 5 6 χF p p ðupper: ε_ a . 0; lower: ε_ a , 0Þ that pffiffiffiffiffiffiffiffi c_a 5 ce 3=2ð 6 ε_ pa Þ ð 6 χF 2 ca Þ ðupper: ε_ pa . 0; lower: ε_ pa , 0Þ leading to c_a 5

pffiffiffiffiffiffiffiffi 3=2ce ðχF7ca Þ ε_ pa ðupper:_εpa . 0; lower:_εpa , 0Þ

(7.138)

from which one has

8 pffiffiffiffiffiffiffiffi 3=2ce χF for ca 5 0 > > < ( c_a pffiffiffiffiffiffiffiffi p 3=2ce ðχF7ca Þ 5 pffiffiffi χF and ε_ a , 0 p 5 > ε_ a > 6 c χF for c 5 e a : p 2χF and ε_ a . 0

The time-integration of Eq. (7.138) is given as follows: h pffiffiffiffiffiffiffiffi i χF7ca p 5 exp 7 3=2ce ðεpa 2 εa0 Þ χF7ca0 p

ðupper: ε_ a . 0;

p

lower: ε_ a , 0Þ

i.e. Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.139)

(7.140)

202

7. Development of elastoplastic and viscoplastic constitutive equations

ca 5 6 χF7ðχF7ca0 Þ exp½7

pffiffiffiffiffiffiffiffi p 3=2ce ðεpa 2 εa0 Þ

p

p

where ca0 and εa 0 are the initial values of ca and εa , respectively. The p relation of ca versus εa is shown for the incompressible material without hardening in Fig. 7.12 in which the nonlinear evolution rule is depicted actually. The elastic-core c is formulated using the Helmholtz free energy p function ψc ðεcs Þ as follows: p

c5

@ψc ðεcs Þ p @εcs

(7.141) p

where the decomposition of the plastic strain εp into the storage part εcs p and the dissipative part εcd for the elastic-core is incorporated following Eq. (7.31) as follows: p

p

εp 5 εpcs 1 εcd ; ε_ p 5 ε_ pcs 1 ε_ cd

(7.142)

p

If ψc ðεcs Þ is given by the quadratic equation ψc ðεpcs Þ 5

1 p p ce ε : ε 2 cs cs

(7.143)

where ce is constant, it follows that

ca F

Normal-yield surface

χF

Limit elastic-core surface

3 / 2 ce ( χ F − c a )

3 / 2 ce χ F 1

3 / 2 ce χF

1 0

1

1

3 / 2 ce ( χ F + c a )

1

3 / 2 ce χ F

3 / 2 ce χ F 1

6 ce χ F

1

ε ap

1

6 ce χF

− χF F

FIGURE 7.12 Translation of elastic-core in uniaxial loading for nonhardening Mises material.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

203

7.5 Formulation of (extended) subloading surface model

c 5 ce εpcs ;

c_ 5 ce ε_ pcs ;

(7.144)

and ε_ pcs 5

1 c_ ; ce

p

ε_ cd 5 ε_ p 2 ε_ pcs 5 ε_ p 2

1 c_ ce

(7.145)

noting Eq. (7.34). The storage and the dissipative parts in the rate of the elastic-core in Eq. (7.136) are given noting Eq. (7.145) as follows: 2 3 8 0 χ  1   > > 1 Ff > > jj_εp jjα 1 Hn jj_εp jj^c5 ε_ pcs 5 jj_εp jj σ 2 c^ 1 4ck ε_ p 2 > > R c b F F > e k > > > > 8 9 2 3 > > 0 >

1  F fHn 5= > > 4 > ^ 5 α 1 c^ σ 2 c 1 c n 2 λ k > < : R ; ce bk F F 2 3 > 0 χ  1   > > 1 Ff p > > jj_εp jjα 1 Hn jj_εp jj^c5 σ 2 c^ 2 4ck ε_ p 2 ε_ cd 5 ε_ p 2 jj_εp jj > > R c b F F > e k > > > 8 2 39 > > 0 > < =  χ  1  > > > 4ck n 2 1 α 1 F fHn c^ 5 λ ^ > 5 n 2 σ 2 c 2 > : : ; R ce bk F F





(7.146) that is, Storage part

c_ 5 ce

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{ ! 

 0 χ  1 1 Ff ε_ p 2 ε_ p 2 jj_εp jj 5 λ fcn σ 2 c^ 2 ck ε_ p 2 jj_εp jjα 1 Hn jj_εp jj^c |{z} F R ce bk F |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl } Total



Dissipative part

(7.147) Eq. (7.147) will be used in the formulation of the dissipative part of the plastic strain rate for the rate of the elastic-core in the multiplicative hyperelastic-based plasticity in Section 9.6.3. The rheology model for the subloading surface model is depicted in Fig. 7.13. The following equation different from Eq. (7.128) was adopted in Hashiguchi (2017a) and used by Hashiguchi and Ueno (2017), Iguchi et al. (2017a,b), etc.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

204

7. Development of elastoplastic and viscoplastic constitutive equations

p

ε·cd = ε· p −

χ ( χR σ − cˆ( |ε· | ε· = ( R σ − cˆ( |ε· |

Elastic-core

p

ε cpd p 1 p ε· kd = b α | ε· | kF

Kinematic hardening

Isotropic hardening

ε kpd

p

p

cs

ε csp p 1 ε· ks = ck

α·

α

ε ksp

p ε· is = ε· p = F· / F' (ε· ipd = 0 )

Sliding between particles

σ εe =σ / E

F − F0

εp

F0

ε

εe

FIGURE 7.13 Rheology model for extended subloading surface model.

ℜc F_ _ 2 c^ 5 ce ε_ p 2 c_ 2 α n^ c jj_εp jj F χ

(7.148)

For this equation, the left-hand side in Eq. (7.127) leads to the inequality

@fð^cÞ @fð^cÞ ℜc F_ p p ^ ^ _ _ : ð_c 2 α 2 cÞ 5 : ce ε 2 nc jj_ε jj @^c @^c F χ

@fð^cÞ ℜc ^ ^ jjnc : n 2 5 ce jj nc jj_εp jj (7.149) @^c χ 5 ce jj

@fð^cÞ jjðn^ c : n 2 1Þjj_εp jj # 0 for ℜc 5 χ @^c

Therefore Eq. (7.148) satisfies the enclosing condition of the elastic-core so that the elastic-core can never go out from the limit elastic-core surface. The evolution rule of the elastic-core is given from Eq. (7.148) as follows: c_ 5 ce ð_εp 2

ℜc F_ _ 1 c^ n^ c jj_εp jjÞ 1 α F χ

Eq. (7.150) with Eqs. (7.120) and (7.121) leads to



0 ℜc 1 Ff c_ 5 ce ε_ p 2 n^ c jj_εp jj 1 ck ε_ p 2 jj_εp jjα 1 Hn jj_εp jj^c bk F χ F

(7.150)

(7.151)

The dissipative part of the plastic strain is given simply since the plastic strain is set-offed in this equation. The rate of the elastic-core in this equation depends on the directions of outward-normal on the subloading and the elastic-core surfaces but does not depend on the distance from the elastic-core to the stress.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

205

7.5 Formulation of (extended) subloading surface model 0

The following relation holds in the deviatoric stress plane pffiffiffiffiffiffiffi ffi 0 (σ 5 σ ) for _ 5 O; F_ 5 0; fðσÞ 5 3=2jjσ jjÞ. the nonhardening Mises metals (α



c_ 5 ce λ



pffiffiffiffiffiffiffiffi



3=2jj^cjj=F c^ χ σ RF χ σ 2 c^ 5 ce λ 2 pffiffiffiffiffiffiffiffi jjσjj jj^cjj χ R jjσjj R 3=2jjσjj







 pffiffiffiffiffiffiffiffi ℜc χ ℜc χ ℜc  5 2=3χ ce λ n 2 n^ c jjσjj 5 ce λ n 2 n^ c jjσjj pffiffiffiffiffiffiffiffi n^ c F 5 ce λ n 2 R χ χ χ 3=2jjσjj=F |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}







Eq: ð7:150Þ

(7.152)

Now, we compare the evolution rules of elastic-core in Eqs. (7.133) and (7.150). The translational speed of the elastic-core must depend on the size and the shape of the normal-yield surface. The former depends on them but the latter does not depend on them, concerning only with the outward-normals of the subloading and the elastic-core surfaces as seen in Eq. (7.150). Therefore Eq. (7.150) is applicable only approximately to the nonhardening Mises metals. However, it is inapplicable to the pressure-dependent hardening-softening materials, for example, soils, rocks, and concrete in which the elastic-core translates not only to the deviatoric direction but also to the pressure direction, depending on the distance from the elastic-core to the stress.

7.5.3 Plastic strain rate The material-time derivative of Eq. (7.110) leads to the consistency condition of the subloading surface in the corotational time-derivative: @fðσÞ 3 @fðσÞ 3 _ 50 :σ2 : α 2 RF_ 2 RF @σ @σ

(7.153)

where one has



@fðσÞ @fðσÞ @fðσÞ @ðσÞ @fðσÞ 5 5 : σ 5 fðσÞ 5 RF @σ @σ @σ @σ @σ

(7.154)

based on the homogeneous function fðσÞ of σ in degree-one by the Euler’s theorem. Then, it follows that @fðσÞ :σ RF  ; n : σ 5 @σ  5    @fðσÞ   @fðσÞ   @σ   @σ 

1 n:σ    @fσ  5 RF    @σ 

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.155)

206

7. Development of elastoplastic and viscoplastic constitutive equations

The substitution of Eq. (7.155) into Eq. (7.153) leads to  _ 

F R_ 3 3 1 n: σ 2n: σ 1α 5 0 F R

(7.156)

which is further rewritten by Eq. (7.116) as _  F R_ 3 3 3 n : σ 2 n : σ 1 R α 1 ð1 2 RÞ c 1 ðσ 2 R^cÞ 5 0 F R

(7.157)

Noting the relation _

σ 2 R^c 5 σ 2 α 2 ðc 2 αÞ 5 σ

(7.158)

it follows from Eq. (7.157) that _

F R_ _ 3 3 3 n: σ 2n: σ 1 R α 1 ð1 2 RÞ c 1 σ 5 0 F R

(7.159)

The substitution of Eqs. (7.89), (7.120), (7.121), and (7.135) into Eq. (7.159) leads to  0  F U _ 3 n: σ 2n: λ f σ 1 R λ fkn 1 ð1 2 RÞ λ fcn 1 λ σ 5 0 (7.160) R F Hn











from which the plastic multiplier λ and the plastic strain rate dp are given as follows:



λ5 where p

M 5n:

3

n: σ p

M

3

;

dp 5

n: σ p

M

(7.161)

n

 0  F U_ fHn σ 1 Rfkn 1 ð1 2 RÞfcn 1 σ R F

(7.162)

The plastic modulus in Eq. (7.162) reduces to Eq. (7.80) for the conventional model in the normal-yield state ðR 5 1: U 5 0; α 5 c 5 α; _ ^ σ 5 σ 5 σÞ.

7.5.4 Strain rate versus stress rate relations The strain rate is given by substituting Eqs. (7.68) and (7.161)2 into Eq. (7.67) as follows: 3



3

d 5 E21 : σ 1 λ n 5 E21 : σ 1

n : σ3 M

p

n 5 ðE21 1

n :n M

p

3

Þ: σ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.163)

207

7.5 Formulation of (extended) subloading surface model

from which the magnitude of plastic strain rate described in terms of





the strain rate, denoted by Λ instead of λ, in the flow rule of Eq. (7.117) is given as follows:



Λ5



n:E:d p

M 1n:E:n

; dp 5 Λ n 5

n:E:d p

M 1n:E:n

n

(7.164)

The stress rate is given from Eq. (7.163) with Eq. (7.164) as follows: 3

σ 5E:d2

n:E:d p

M 1n:E:n

E : n 5 ðE 2

E:nn:E p

M 1n:E:n

Þ:d

(7.165)

The loading criterion is given as follows (Hashiguchi, 2000, 2017b): ( dp 6¼ O for Λ . 0 (7.166) dp 5 O for other



or



dp 6¼ O for n : E : d . 0 dp 5 O for other

(7.167)

7.5.5 Calculation of normal-yield ratio Substituting Eq. (7.114) into Eq. (7.110), the subloading surface is described as follows: _

ðfðσÞ 5 Þ fðσ 1 R^cÞ 5 RF ðHÞ

(7.168)

from which the normal-yield ratio R is calculated by substituting the updated values of σ; α; c; F. Eq. (7.168) is expressed for the von Mises material as follows: rffiffiffi

rffiffiffi 3 0 3 _0 (7.169) jjσ jj 5 jj σ 1 R^c0 jj 5 RFðHÞ 2 2 that is, _

tr ðσ0 1R^c0 Þ2 5

2 2 2 R F 3

from which the normal-yield ratio R is given as follows:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 _ _ _ F 2 jj^c0 jj2 jj σ0 jj2 R 5 σ0 : c^ 0 1 ðσ0 : c^ 0 Þ2 1 3 2 2 F 2 jj^c0 jj2 3

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.170)

(7.171)

208

7. Development of elastoplastic and viscoplastic constitutive equations

The normal-yield ratio can be calculated by the following two methods: 1. In the plastic loading process, it is calculated by the time-integration of Eq. (7.89). Here, the analytical time-integration in Eq. (7.96) is beneficial to the enhancement of numerical analysis in the return-mapping projection, provided that the incremental step is chosen to be small such that the change of u is small enough. 2. In the elastic loading process, it is calculated by Eq. (7.168) (Eq. (7.171) for Mises metals) after all the other variables σ; α; c; F were updated. The method (1) and the method (2) are adopted for the plastic loading process and for the elas-tic loading (plastic unloading) process, respectively.

7.5.6 Improvement of inverse and reloading responses p

p

The variation of plastic strain εb 2 εa for a certain variation Rb 2 Ra of the normal-yield ratio induced during the process from the state a to the state b is identical regardless of loading processes, for example, initial loading, reloading and inverse loading, proportional and nonproportional loadings as known from Eq. (7.96), if the parameter u in Eq. (7.95) is a constant. It leads to the impertinent description that the reloading stressstrain curve after a partial unloading returns to the preceding stressstrain curve too gently. It causes the inadequate prediction of cyclic loading behavior, resulting in the unrealistically large plastic strain accumulation during cyclic loading (excessive mechanical ratcheting). The material parameter u is then extended to describe the generalized Masing effect (Masing, 1926) as follows (Hashiguchi, 2013b, 2017a): u 5 u exp ðuc ℜc Cn Þ

ðu exp ð2 uc χÞ # u # u exp ðuc χÞÞ

(7.172)

8 1 ðlargestÞ for ℜc 5 χ and Cn 5 1 > < u expðuc χÞ B C u ðaverageÞ for ℜc 5 0 or Cn 5 0 A @5 > : uexpð2 uc χÞ ðsmallestÞ for ℜc 5 χ and Cn 5 2 1 0

where Cn  n^ c : n ð2 1 # Cn # 1Þ

(7.173)

n^ c is the normalized outward-normal of the elastic-core surface defined in Eq. (7.132). u (average value of u) and uc are the material constants. u

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.5 Formulation of (extended) subloading surface model

209

is the continuous function of the variables ℜc and Cn . The function u for the particular states are shown in the bracket. Cn 5 1, 0, and 21 designate the states that the plastic strain rate is directed outward-normal, tangential and inward-normal, respectively, to the elastic-core surface. ℜc 5 0 and ℜc 5 χ designate the states that the similarity-center lies in the center of the normal-yield surface and on the limit elastic-core surface, respectively. By this modification, the phenomenon that the reloading curve after a partial unloading returns rapidly to the preceding loading curve and the curvature of inverse loading curve decreases can be described realistically. Incidentally, the plastic strain accumulation for the cyclic loading process in the neighborhood of yield surface is suppressed.

7.5.7 Cyclic stagnation of isotropic hardening It is observed through experiments for metals that the isotropic hardening stagnates and only the kinematic hardening proceeds in a certain period of reverse deformation starting from the reverse re-yielding. Ohno (1982) stated “As plastic deformation proceeds, dislocations lose their mobility due to the piling-up of obstacles or the formation of various networks, thus leading to the hardening of metals. When a load reversal occurs, some parts of the immobilized dislocations recover the mobility to result in the temporary softening of metals.” This phenomenon considerably affects the cyclic loading behavior in which the reverse loading is repeated. To describe this phenomenon, the concept of the cyclic stagnation of isotropic hardening, that is, nonhardening region was proposed by Chaboche et al. (1979; see also Chaboche, 1989) and studied also by Ohno (1982). However, their formulations are not reasonable, belonging to the plastic strain space formulation. Thereafter, the stress space formulation based on the translation of the kinematic hardening variable was formulated for the two surface model by Yoshida and Uemori (2002). In what follows, the latter is improved for the subloading surface model. Assuming that the isotropic hardening stagnates when the back stress α lies inside a certain region, let the following surface, called the normal-isotropic hardening surface, be introduced. _

gðαÞ 5 K~

(7.174)

where _

α  α2ρ

(7.175)

K~ and ρ ð 5 ρ0 Þ designate the size and the center, respectively, of the normal-isotropic hardening surface, the evolution rules of which will be formulated later. Furthermore, we introduce the surface, called the

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

210

7. Development of elastoplastic and viscoplastic constitutive equations

subloading-isotropic hardening surface, which always passes through the current back stress α and which has a similar shape and an orientation to the normal-isotropic hardening surface (see Fig. 7.14). It is expressed by the following equation. _

gðαÞ 5 R~ K~

(7.176)

where R~ ð0 # R~ # 1Þ is the ratio of the size of subloading-isotropic hardening surface to that of the normal-isotropic hardening surface. It plays the role as the measure for the approaching degree of the back stress to the normal-isotropic hardening surface. Then, R~ is referred to as the normal-isotropic hardening ratio. It is calculable from the equation _ ~ R~ 5 gðαÞ=K~ in terms of the known values of α, ρ, and K. The consistency condition of the subisotropic hardening surface is given by _

_

 

@gðαÞ 3 @gðαÞ 3 ~ :α2 : ρ 5 R~ K~ 1 R~ K: @α @α

(7.177)

Let the following evolution rules of K~ and ρ be adopted so as to fulfill Eq. (7.177) for R~ 5 1.  _  ς _ 3  @gðαÞ  ~ ~ (7.178) K 5 CR hn : αi @α 



FIGURE 7.14 Normal- and subloading-isotropic hardening surfaces.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.5 Formulation of (extended) subloading surface model

ς _ 3 3 _ ρ 5 ð1 2 CÞR~ hn : αin

211 (7.179)

where 0 # C # 1 and ς ð $ 1Þ are the material constants and _  _  @gðαÞ  @gðαÞ  _ n  @α : @α

(7.180)

Substituting Eqs. (7.178) and (7.179) for the evolution rules of K~ and ρ into Eq. (7.177), the rate of the normal-isotropic hardening ratio is given by  _ !   _ _ 1 @gðαÞ 3 ς @gðαÞ _ ς _ 3 _ 3  @gðαÞ  ~ ~ ~ : α 2 ð1 2 CÞR : hn : αin 2 RCR hn : αi R~ 5 ~ ~ ~ ~  @α @α @α K n  _   o 1 @gðαÞ 3 ~ R~ ς 5 :α 1 2 1 2 Cð1 2 RÞ @α K~



(7.181) which is the monotonically decreasing function of R~ fulfilling   _ 8 1 @gðαÞ 3 > > : α ð . 0Þ for R~ 5 0 5 > > @α > K~ > > >   _ < 1 @gðαÞ 3 R~ , : α ð . 0Þ for R~ , 1 > @α K~ > > > > >5 0 for R~ 5 1 > > : , 0 for R~ . 1



(7.182)

Therefore the normal-isotropic hardening ratio increases when the back stress moves to the outward of the subisotropic hardening surface but it decreases such that the normal-isotropic hardening surface involves the back stress when the back stress goes out from the normal-isotropic hardening surface by virtue of the inequality R~ , 0 for R~ . 1 as shown in Eq. (7.182). Furthermore, needless to say, the judgment of whether the back stress reaches the normal-isotropic hardening surface is not necessary in the present formulation. It is assumed that the isotropic hardening variable H evolves under the following requirements.



3

1. The isotropic hardening is induced when the back stress rate α is induced directing outwards the subisotropic hardening surface, that is,  _ 3 _ . 0 for n H :α.0 (7.183) _ 5 0 for other H 2. The isotropic hardening rate increases as the back stress approaches the normal-isotropic hardening surface, that is, as the normal-

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

212

7. Development of elastoplastic and viscoplastic constitutive equations

isotropic hardening ratio R~ increases. Then, H_ is the monotonically ~ Here, in order that the isotropic hardening increasing function of R. develops continuously, its rate must be zero, that is, H_ 5 0 for R~ 5 0, that is, when the back stress lies just on the center of the normal-isotropic hardening surface because the rate is zero during the process in which the back stress moves inward of the subisotropic stagnation surface. 3. The isotropic hardening rule of Eq. (7.120) in the monotonic loading process holds when the back stress lies on the normal-isotropic hardening surface ðR~ 5 1Þ and the back stress rate is induced in the outward direction of that surface. Then, the evolution rule of isotropic hardening variable in Eq. (7.26) be modified as follows: rffiffiffi 2 ~υ _ _ (7.184) H5 R hn : nkn ijjdp jj 3 where nkn is the normalized direction of increment of the back stress, that is, nkn 

fkn  5 nTkn ; jjnkn jj 5 1 jjfkn jj

(7.185)

υ is the material constant. Employing the extended isotropic hardening rule in Eq. (7.184) instead of Eq. (7.26) into Eq. (7.162), the plastic modulus is modified as follows:  0  F ~υ _ U p M  n: 1 ð1 2 RÞ f (7.186) R hn : nkn iσ 1 Rfkn 1 σ cn R~ F The normal-isotropic hardening surface evolves such that the boundary of the surface always approaches the back stress and moves so as to involve it even if the back stress goes out from the boundary by the inequality R~ , 0 for R~ . 1 as shown in Eq. (7.182). Furthermore, the judgment of whether the back stress reaches the normal-isotropic hardening surface is not necessary in the present formulation. In contrast, the judgment of whether the plastic strain or the back stress reaches the isotropic hardening (stagnation) surface is required in the other models (Chaboche et al., 1979; Chaboche, 1991; Ohno, 1982; Yoshida and Uemori, 2002). In addition, the boundary of the isotropic (stagnation) surface does not approach the plastic strain or the back stress and does not move so as to involve it even if they go out from the surface.



Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.6 Implicit time-integration: return-mapping

213

Therefore these models abandoned the incorporation of isotropic stagnation (Chaboche, 2008; Kobayashi and Ohno, 2002) except for the calculation by the forward-Euler method with infinitesimal loading increments, although the isotropic stagnation formulations have been proposed by the proposers of these cyclic kinematic models themselves. ~ is given in the simplest form as follows: The function gðαÞ  _  _  @gðαÞ  α _ _ _ _  5 jjnjj gðαÞ 5 jjαjj; n 5 _ ;  51 (7.187) jjαjj @α 

7.6 Implicit time-integration: return-mapping The implicit stress integration for the time-independent elastoplastic deformation is called the return-mapping (projection) because the stress is calculated to return to the subloading surface (yield surface for the conventional elastoplastic model). The return-mapping algorithm for the subloading surface model will be shown for the infinitesimal strain theory by the replacement of the strain rate d to the rate of infinitesimal strain ε in this section for the von Mises metals. The subloading surface for the von Mises metals is given by sffiffiffi 8 3 0 > > jjσ jj > fðσÞ 5 > > 2 > > > > > sffiffiffi sffiffiffi > <    2 p 2_ _ jj_ε jj 5 FðHÞ 5 F0 1 1 h1 1 2 expð2 h2 HÞ ; H 5 3 3λ > > > > > > > > dF d2 F > 0 > 5 F 5 h h exp ð2 h HÞ; Fv 5 5 2 F0 h1 h22 exp ð2 h2 HÞ F > 0 1 2 2 : dH dH 2 (7.188) leading to σ0 ; f 5 n5 jjσ0 jj Hn

rffiffiffi 2 3

(7.189)

7.6.1 Return-mapping formulation The implicit stress integration (backward-Euler) method by the returnmapping method will be shown below referring to Hashiguchi and Yamakawa (2012), Anjiki et al. (2016, 2018), and Iguchi et al. (2016, 2019).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

214

7. Development of elastoplastic and viscoplastic constitutive equations p

Suppose the state that the stress σn , the plastic strain εn , and the internal variables αn and cn in the n-th step are already known by performing the calculations in the incremental steps of n times. At the beginning of calculation in the step n 1 1, one calculates first the elastoplastic deformation under a given boundary condition by use of the global tangent stiffness matrix obtained in the previous step n. Then, calculate the strain increment Δε from the deformation in each numerical integration point. Further, calculate the trial stress by inputting the strain increment Δε, postulating that only the elastic deformation is induced. This process and the trial stress calculated in this process are called the elastic trial (or predictor) step and the elastic trial stress, respectively. Designating the trial etrial stress and the elastic strain calculated in this step by σtrial n11 and εn11 , respectively, they are related by the hyperelastic relation as follows: p

etrial σtrial n11 5 E : εn11 5 E : ðεn11 2 εn Þ ptrial

p

trial trial εn11 5 εn ; H trial n11 5 H n ; αn11 5 αn ; cn11 5 cn

(7.190) (7.191)

where p

εn11 5 εn 1 Δεn11 5 εn 1 εetrial n11 p

(7.192)

The strain εn11 is fixed, leading to εn11 5 εen11 1 εn11 5 const: throughout the plastic corrector process until the calculation step n 1 1 is finished, which will be described below. The following simultaneous equation must be fulfilled during every loading increment. 8 Stress variation: > > > > > p > Yσ 5 σn11 2 E : εetrial > n11 1 E : Δεn11 5 O > > > > > > Kinematic hardening rule: > > > > > > > Yα 5 αn11 2 αn 2 Δαn11 5 O > > > > > > < Elastic-core translation rule: (7.193) > > Yc 5 cn11 2 cn 2 Δcn11 5 O > > > > > > > Isotropic hardening rule: > > > > > > > > YH 5 Hn11 2 Hn 2 ΔHn11 5 0 > > > > > Subloading Surface: > > > > : Ys 5 fðσn11 Þ 2 Rn11 FðHn11 Þ 5 0

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.6 Implicit time-integration: return-mapping

215

Eq. (7.193) is explicitly expressed for the Hooke’s law and the von Mises metals the subloading surface of which is given in Eq. (7.110) with Eqs. (7.121) and (7.135) as follows: 8 > Stress variation: > > > > etrial > > > Yσ 5σn11 2E:εn11 12Gnn11 Δλn11 5O > > > Kinematic hardening rule: > > >

> > > > Yα 5αn11 2αn 2ck nn11 2pffiffiffiffiffiffiffiffi1 > α n11 Δλn11 5O > > 3=2bk Fn11 > > > > > > > Elastic-core translation rule: > > > > > Yc 5cn11 2cn > > 3 < 2



pffiffiffiffiffiffiffiffi 0 2=3 F χ 1 n11 > c^ n11 5Δλn11 5O 2 4ce σn11 2 c^ n11 1ck nn11 2 αn11 1 > > Rn11 bk Fn11 Fn11 > > > > > > > Isotropic hardening rule: > > sffiffiffi > > > 2 > > Δλn11 50 YH 5Hn11 2Hn 2 > > > 3 > > > > > > Subloading Surface: > > > sffiffiffi > > > 3 0 > > jjσ jj2Rn11 FðHn11 Þ50 5 Y > : s 2 n11 (7.194)

with 0

nn11 5

σn11 0 jjσn11 jj

(7.195)

noting E : I0 5 ðKI 1 2GI0 Þ : I0 5 2GI0

(7.196)

The five unknown variables σn11 , αn11 , cn11 , Hn11 , and Δλn11 are involved in the simultaneous equation. Various equations may be adopted instead of the ones in Eq. (7.194). For example: 1. The first equation may be replaced by the plastic flow rule p p Yεp 5 εn11 2 εn 2 nn11 Δλn11 5 O. Then, the stress is calculated by p substituting εen11 5 εtrial n11 2 εn11 into the hyperelastic equation. 2. The normal-yield ratio Rn11 is involved as the unknown variable, while the evolution rule is used. However, it is better to calculate it by the analytical time-integration equation. 3. The fourth equation on the isotropic hardening variable is omitted and it is calculated after the calculation of Δλn11 .

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

216

7. Development of elastoplastic and viscoplastic constitutive equations

However, the calculation method shown here would be one of the best calculation methods. Eq. (7.194) is described in the matrix form as follows: YðXÞ 5 O where

9 8 Yσ > > > > > > > > > > > = < Yα > Y  Yc ; > > > > > > > YH > > > > > ; : Ys

(7.197)

9 8 σn11 > > > > > > > > > > > = < αn11 > X  cn11 > > > > > > > Hn11 > > > > > ; : Δλn11

(7.198)

In order to solve Eq. (7.197) numerically, linearizing it by means of Taylor expansion and taking the first-order term, we have



YðXðk11Þ ÞDYðXðkÞ Þ 1 JðXðkÞ Þ dX 5 O where J is the Jacobin 2 @Yσ 6 @σn11 6 6 6 @Yα 6 6 6 @σn11 6 6 6 @Yc 6 J 5 6 @σ 6 n11 6 6 6 @YH 6 6 @σn11 6 6 6 @YS 4 @σn11

matrix given by @Yσ @αn11

@Yσ @cn11

@Yσ @Hn11

@Yα @αn11

@Yα @cn11

@Yα @Hn11

@Yc @αn11

@Yc @cn11

@Yc @Hn11

@YH @αn11

@YH @cn11

@YH @Hn11

@YS @αn11

@YS @cn11

@YS @Hn11

3 @Yσ @Δλn11 7 7 7 @Yα 7 7 7 @Δλn11 7 7 7 @Yc 7 7 @Δλn11 7 7 7 7 @YH 7 7 @Δλn11 7 7 7 @YS 7 5 @Δλn11

(7.199)

(7.200)

Eq. (7.199) is the simultaneous equation for the unknown dX, which is updated by  21   21  dX 5 2 JðXðkÞ Þ Y XðkÞ -Xðk11Þ 5 XðkÞ 1 dX 5 XðkÞ 2 JðXðkÞ Þ Y XðkÞ (7.201) where ðkÞ designates the iteration number of calculation within the step n 1 1. One has the following partial derivatives.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.6 Implicit time-integration: return-mapping

8 @σ @½σ 2 ðc 2 R^cÞ > > 5 S; > @σ 5 > @σ > > > > > @σ @½σ 2 ðc 2 R^cÞ > > > 5 5 2 RS; > > @α @α > > > > < @σ @½σ 2 ðc 2 R^cÞ 5 5 2 ð1 2 RÞS @c @c > > > > > @σ @½σ 2 ðc 2 R^cÞ > > 5 5 c^ > > > @R @R > > > > > @σ @½σ 2 ðc 2 R^cÞ @R > > 5 c^ 5 UðRÞ^c > : @Δλ 5 @Δλ @Δλ N

217

(7.202)



@n @ðσ0 =jjσ0 jjÞ 1 σ0 σ0 1 0  ðI0 2 n  nÞ 5 5 2 I 5 0 0 0 0 0 @σ @σ jjσ0 jj jjσ jj jjσ jj jjσ jj  N:n50 (7.203)

for which Eq. (1.305) is used. Further, one has 8 @n @n @σ0 > > 5 0 : 5 N : I0 5 N σ ; > > @σ @σ @σ > > > > > @n @n @σ0 > 0 > > 5 > @α @σ0 : @α 5 2 N : I 5 2 Nσ > > > > > < @n @n @σ0 5 0 : 5 2 N : ð1 2 RÞI0 5 2 ð1 2 RÞNσ @c @σ @c > > > > > @n @n @σ0 > > 5 0 : 5 N : c^ 0 > > > @R @σ @R > > > > > @n @n @σ0 @R 0 > 0 > 5 > : @Δλ @σ0 : @Δλ 5 N : @Δλ c^ 5 UðRÞ Nσ : c^ 8 @jjσ0 jj @jjσ0 jj @σ0 σ0 > > 5 5 : I0 5 n : I0 5 n : > > @σ @σ0 @σ jjσ0 jj > > > > > @jjσ0 jj @jjσ0 jj @σ0 σ0 > > > 5 5 : I0 5 n : ð2 RI0 Þ 5 2 R n : > < @α @σ0 @α jjσ0 jj @jjσ0 jj @jjσ0 jj @σ0 σ0 > > 5 5 : ½ 2 ð1 2 RÞI0  5 2 ð1 2 RÞ n : > > > @c @σ0 @c jjσ0 jj > > > > > @jjσ0 jj @jjσ0 jj @σ0 > 0 > > : @R 5 @σ0 : @R 5 n : c^

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.204)

(7.205)

218

7. Development of elastoplastic and viscoplastic constitutive equations

The components in Eq. (7.200) are given below, where the subscript n 1 1 added to the unknown variables is omitted hereafter for the sake of simplicity, noting Eqs. (7.194) and (7.202)(7.205). @Yσ @n 5 S 1 2G Δλ 5 S 1 2GNΔλ @σ @σ @Yσ @n Δλ 5 2 RNΔλ 5 2G @α @α @Yσ @n Δλ 5 2 ð1 2 RÞNΔλ 5 @c @c @Yσ 5O @H @Yσ 5 2Gn @Δλ

(7.206) (7.207) (7.208) (7.209) (7.210)

@Yα @n Δλ 5 2 ck NΔλ 5 2 ck (7.211) @σ @σ    @n  @Yα 1 1 2 pffiffiffiffiffiffiffiffi 5 S 2 ck S Δλ 5 S 2 ck 2 RN 2 pffiffiffiffiffiffiffiffi S Δλ @α @α 3=2bk F 3=2bk F 1 5 S 1 ck ðRNσ 1 pffiffiffiffiffiffiffiffi SÞΔλ 3=2bk F (7.212)   @Yα @n 5 2 ck Δλ 5 2 ck 2 ð1 2 RÞN Δλ 5 ck ð1 2 RÞNΔλ (7.213) @c @c 0 @Yα F 5 2 ck pffiffiffiffiffiffiffiffi αΔλ (7.214) @H 3=2bk F2



@Yα @n 1 1 52 ck 2ck n 2 pffiffiffiffiffiffiffiffi α 52 ck UðRÞN^c0 1 ck n 2 pffiffiffiffiffiffiffiffi α @Δλ @Δλ 3=2ζF 3=2ζF @Yc χ 52 ce SΔλ2 ck NΔλ R @σ 2 3

pffiffiffiffiffiffiffiffi 0

2=3 F @Yc χ @σ @^ c @n 1 @α @^ c 5Δλ 2 2 52 4ce 1 ck 1 R @α @α @α bk F @α @α F @α 2 3

pffiffiffiffiffiffiffiffi 0

2=3 F χ 1 ð2RSÞ1 S 1ck 2ℕσ 2 S 2 S 5Δλ 52 4ce R bk F F 82 9 pffiffiffiffiffiffiffiffi 0 3 < = 2=3 F c k 5S 2ck ℕσ  Δλ 52 4ce ð1 2χÞ 2 2 : ; bk F F

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.215) (7.216)

(7.217)

7.6 Implicit time-integration: return-mapping

2 3

pffiffiffiffiffiffiffiffi 0

2=3 F @Yc χ @σ @^ c @n 1 @α @^ c 5Δλ 2 1ck 2 1 52 4ce R @α @α @α bk F @α @α F @α 2 3

pffiffiffiffiffiffiffiffi 0

2=3 F χ 1 ð2RSÞ1S 1ck 2ℕσ 2 S 2 52 4ce S 5Δλ R bk F F 82 9 pffiffiffiffiffiffiffiffi 0 3 < = 2=3 F c k 1 Þ5S1ck ℕ Δλ 5 4 2ce ð12χÞ1 : ; bk F F

pffiffiffiffiffiffiffiffi FvF2ðF0 Þ2 @Yc ck F0 ^ c Δλ 5 α2 2=3 @H bk F2 F2

219

(7.218)

(7.219)

2 3

pffiffiffiffiffiffiffiffi 0

 2=3 F @Yc χ 1 χ @σ @n  52 4ce α 1 1 ck Δλ σ 2 c^ 1ck n2 c^ 5 1 2 ce R bk F R @Δλ @Δλ F @Δλ 2 3

pffiffiffiffiffiffiffiffi 0



2=3 F χ 1 χ α 1 σ 2 c^ 1 ck n 2 524ce c^ 5 1 2 ce UðRÞ^c 1ck UðRÞℕ : c^ 0 Δλ R bk F R F 2 3

pffiffiffiffiffiffiffiffi 0



2=3F 5 χ 1 χ 52 4ce α 1 σ 2 c^ 1 ck n 2 c^ 2 UðRÞ ce c^ 2 ck ℕ : c^ 0 Δλ R bk F R F (7.220)

@YH 5O @σ

(7.221)

@YH 5O @α

(7.222)

@YH 5O @c

(7.223)

@YH 51 @H

(7.224)

pffiffiffiffiffiffiffiffi @YH 52 2=3 @Δλ

(7.225)

@Ys pffiffiffiffiffiffiffiffi @jjσ0 jj pffiffiffiffiffiffiffiffi 5 3=2n 5 3=2 @σ @σ

(7.226)

pffiffiffiffiffiffiffiffi @Ys pffiffiffiffiffiffiffiffi @jjσ0 jj 52 3=2Rn 5 3=2 @α @α

(7.227)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

220

7. Development of elastoplastic and viscoplastic constitutive equations

pffiffiffiffiffiffiffiffi @Ys pffiffiffiffiffiffiffiffi @jjσ0 jj 52 3=2 ð1 2RÞ n 5 3=2 @c @c @Ys 52 RF0 @H @Ys pffiffiffiffiffiffiffiffi @jjσ0 jj @σ0 @R 5 3=2 2 FðHÞ 0 @σ @Δλ @Δλ @Δλ   pffiffiffiffiffiffiffiffi @jjσ0 jj @ σ0 2 ðc2R^cÞ0 @R : 2 FðHÞ 5 3=2 0 @σ @Δλ @Δλ

(7.228) (7.229)

(7.230)

pffiffiffiffiffiffiffiffi  5UðRÞ 3=2 n : c^ 0 2FðHÞ where the parameter u is updated by Eq. (7.172) in every iteration calculation. The normal-yield ratio Rn11 is involved as one more unknown variable in the simultaneous Eq. (7.199). It is calculated more accurately by the following analytically integrated equation than by the numerical time-integration of the original rate equation (7.89), which is given by p setting R0 5 R0n11 and εp 2 ε0 5 Δλ from Eq. (7.96) as follows: 



 2 π hR0n11 2 Re i π Δλn11 21 Rn11 5 ð1 2 Re Þ cos cos exp 2 u 1 Re π 2 1 2 Re 2 1 2 Re (7.231) under the assumption uDconst: during the increment step. Then, @R=@σ 5 O holds, which is used in the partial derivative @Ys =@σ. The iteration calculation in Eq. (7.201) with Eq. (7.231) is continued until the residual becomes sufficiently small: that is, it is judged that the convergence of solution is attained when the following equation in terms of the residual norm is fulfilled. jjYðXðk11Þ Þjj , TOL

(7.232)

The stress σ is adopted as one of the unknown variables in the above. The elastic strain εe may be adopted instead of the stress σ. Let the normal-yield ratio Rn at the end of the step n and the normalyield ratio Rtrial n11 at the end of the elastic trial step be derived for the Mises material below. The following relation must be satisfied at the end of the step n for the Mises material. rffiffiffi 3 0 (7.233) jjσ jj 5 Rn FðHn Þ 2 n

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.6 Implicit time-integration: return-mapping

which is rewritten as

rffiffiffi 0 3 _0 jjσ n 1 Rn c^ n jj 5 Rn FðHn Þ 2

from which Rn is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 _0 0 0 _0 _0 σ n : c^ n 1 ðσ n : c^ n Þ2 1 ½ð2=3ÞðFðHn ÞÞ2 2 jj^cn jj 2 jj σ n jj 2 Rn 5 0 ð2=3ÞðFðHn ÞÞ2 2 jj^cn jj 2

221

(7.234)

(7.235)

Analogously, the subloading surface at the end of the elastic trial step is described by pffiffiffiffiffiffiffiffi trial 0 3=2 jjσn11 jj 5 Rtrial (7.236) n11 FðHn Þ which is rewritten as pffiffiffiffiffiffiffiffi _ trial 0 0 ^ n jj 5 Rtrial 3=2jjσ n11 1 Rtrial n11 c n11 FðHn Þ

(7.237)

from which Rtrial n11 is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 0 _ trial 0 _ _ 0 ^ Þ2 1 ½ð2=3ÞðFðH ÞÞ2 2 jj^ 0 2 σ n11 : c^ n 1 ð σ trial cn jj 2 jj σ trial n n n11 : c n11 jj Rtrial 5 0 n11 ð2=3ÞðFðHn ÞÞ2 2 jj^cn jj 2 (7.238)

7.6.2 Loading criterion The following loading criterion for the return-mapping method was incorporated by Hashiguchi (2013, 2017a) and has been used for the FEM analyses by Hashiguchi (2013a,b, 2017a), Anjiki et al. (2016), Yamakawa et al. (2010), Iguchi (2017a,b), Fincato and Tsutsumi (2017, 2018), and Tsutsumi et al. (2019). (

 p trial trial trial Δεn11 5 O; σFinal n11 5 σn11 for fðσn11 Þ 2 Rn FðH n Þ # 0 or fðσn11 Þ 2 Re F Hn # 0 p

trial Δεn11 6¼ O; σFinal n11 6¼ σn11 for other

(7.239)

which persists that the plastic strain rate is induced only when the subloading surface after the elastic trial step is larger than subloading surface at the beginning of the elastic trial step, that is, at the end of the previous step. Therefore, any elastic trial steps inducing the shrink of the subloading surface are judged as the elastic loading process by the past (incorrect) loading criterion. In fact, however, the plastic strain rate must be induced when once the subloading surface contracts but then it

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

222

7. Development of elastoplastic and viscoplastic constitutive equations

FIGURE 7.15

Defects of past (incorrect) loading criterion in subloading surface model. (A) Elastic trial steps which are judged as elastic deformation process by the past (incorrect) loading criterion. (B) Cyclic loading behavior predicted by past and corrected loading criteria illustrated for α 5 α 5 c.

expands over the elastic domain in the elastic trial step even if the subloading surface after the elastic trial step is smaller than subloading surface in the beginning of the elastic trial step as shown in Fig. 7.15A. Further, the accumulation of plastic strain rate during the cyclic loading for a constant or a decreasing amplitude cannot be described by the past loading criterion as shown in Fig. 7.15B. Eventually, the past loading criterion is inapplicable to the reverse and the cyclic loading behaviors. Therefore the correct loading criterion must be incorporated into the return-mapping method for the analysis of the deformation behavior including the reverse loading by the subloading surface model.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.6 Implicit time-integration: return-mapping

223

The correct loading criterion for the subloading surface model has been proposed by Hashiguchi (2016b, 2017c, 2018a,c) and its validity was verified by Anjiki et al. (2019) for various loading processes, which will be explained in the following. The following facts should be noticed for the formulation of the loading criterion required at the beginning of the plastic corrector step, referring to Fig. 7.16, where the elastic trial step for the initial subloading surface model ðc 5 αÞ and for the extended subloading surface model ðc 6¼ αÞ are shown in Fig. 7.16A and B, respectively (The subscript n and n 1 1 is added to the variables at the end of the step n and the step n 1 1, respectively): p “The plastic strain increment Δεn11 is not induced if the elastic trial stress trial σn11 stays inside the elastic response region, that is,fðσtrial n11 Þ 2 Re FðHn Þ , 0 in the step n or if the stress increment Δσtrial makes an obtuse angle with the n11 outward-normal ntrial of the subloading surface in the elastic trial step n 1 1. n11 Otherwise, it is induced.” Then, the correct loading criterion in the return-mapping method for the subloading surface model is given as follows: Loading criterion in return-mapping for subloading surface model p

trial trial trial trial Δεn11 5 O and σFinal n11 5 σn11 for fðσn11 Þ 2 Re FðH n Þ # 0 or nn11 : Δσn11 # 0 : p trial Δεn11 6¼ O and σFinal n11 6¼ σn11 for others

(7.240)

where trial Δσtrial n11  σn11 2 σn

(7.241)

c^ n  cn 2 αn

(7.242)

_

_ trial σ n11

σ n 5 σn 2 cn ;

5 σtrial n11 2 cn

trial ^n α n 5 cn 2 Rn c^ n ; α en 5 cn 2 Re c^ n ; α trial n11 5 cn 2 Rn11 c _

_

_

(7.243) (7.244)

_

σn 5 σn 2 α n 5 σ n 1 Rn c^ n ; _

_

trial trial trial trial ^n σtrial n11 5 σn11 2 α n11 5 σ n11 1 Rn11 c

(7.245)

_

nn 

@fðσn Þ @σn

    trial trial   @fðσn Þ   ; ntrial  @fðσn11 Þ  @fðσn11 Þ   n11  @σ   @σtrial  @σtrial n n11 n11

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.246)

224

7. Development of elastoplastic and viscoplastic constitutive equations

FIGURE 7.16 Correct loading criterion when elastic trial stress increment is directed inward of subloading surface at step n in return-mapping method for subloading surface model. (A) Initial subloading surface model (α 5 α 5 c). (B) Extended subloading surface model (α 6¼ α 6¼ c).

7.6.3 Initial value of normal-yield ratio in plastic corrector step As known from the correct loading criterion described at the latest section, the initial value of normal-yield ratio must be determined at the beginning of the plastic corrector step. It has been formulated and derived by Hashiguchi (2017c, 2018a,c,d) as explained below.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.6 Implicit time-integration: return-mapping

225

Noting the loading criterion described in the foregoing, the plastic corrector step must be started from: 1. the subloading surface coinciding with the elastic domain surface, when the elastic trial stress increment intersects with the elastic domain surface, for which the normal-yield ratio is given by Re and 2. the subloading surface in the transitional state at which the subloading surface once contracts and then begins to expand in the elastic trial step, while the elastic trial stress increment Δσtrial n11 contacts tangentially to the subloading surface, for which the normal-yield ratio is designated as R0n11 . Then, designating the stress at which the stress increment vector Δσtrial n11 becomes tangential to the subloading surface which changes from the contraction to the expansion by σ0n11 , the following relations must be satisfied at the contact point, that is, the loading-start stress σ0n11 . fðσ0n11 Þ 5 R0n11 FðHn Þ n0n11 : Δσtrial n11 5 0

(7.247)

The first equation is required for the subloading surface to pass through the contact point σ0n11 and the second equation is required for the elastic trial increment Δσtrial n11 to contact tangentially to the subloading surface at the contraction-expansion transition. σ0n11  sΔσtrial n11 1 σn ð0 # s # 1Þ

(7.248)

α0n11 5 cn 2 R0n11 c^ n

(7.249) _

0 ^n σ0n11  σ0n11 2 α0n11 5 sΔσtrial n11 1 σ n 1 Rn11 c     @fðσ0n11 Þ  @fðσ0n11 Þ  n0n11    @σ0 @σ0n11 n11

(7.250) (7.251)

sð0 # s # 1Þ is the unknown scalar parameter which must be determined so as to satisfy Eq. (7.247) at the stress σ0n11 . The two unknown variables s and R0n11 are calculated by solving Eq. (7.247) and then all the variables σ0n11 , α0n11 , σ0n11 in the subloading surface passing through the contact point and the stress increment Δσtrial n11 at the elastic trial step are calculated. Eq. (7.247) is regarded as the simultaneous quadratic equation of the unknown scalar variables R0n11 and s so that the numerical calculation is required for their solutions in general. In what follows, the analytical equation of R0n11 in terms of the known variables will be derived for the Mises material. Eq. (7.247) the Mises material with pffiffiffiffiffiffiffiffi is 0 explicitly described for 0 0 0 fðσ0n11 Þ 5 3=2jjσ0n11 jj leading to n0n11 5 σ0n11 =jjσ0n11 jj as follows:

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

226

7. Development of elastoplastic and viscoplastic constitutive equations

pffiffiffiffiffiffiffiffi 00 3=2jjσn11 jj 5 R0n11 FðHn Þ

(7.252)

0

σ0n11 : Δσtrial n11 5 0 that is,

8 pffiffiffiffiffiffiffiffi 0 _0 0 0 < 3=2jjsΔσtrial ^ n jj 5 R0n11 FðHn Þ n11 1 σn 1 Rn11 c   0 _0 0 0 : sΔσtrial ^ n : Δσtrial n11 1 σn 1 Rn11 c n11 5 0

(7.253)

The upper equation in Eq. (7.253) is expressed as      0 0 _0 _0 0 0 0 0 ^ n : sΔσtrial ^ n 5 ðR0n11 FðHn ÞÞ2 2=3 sΔσtrial n11 1 σn 1 Rn11 c n11 1 σn 1 Rn11 c leading to 0

0

0

_

0

trial trial 0 0 ^nÞ s2 Δσtrial n11 : Δσn11 1 2sΔσn11 : ð σ n 1 Rn11 c  0 0 _ _ 1ðσ 0 n 1 R0n11 c^ n Þ : ð σ n0 1 R0n11 c^ n Þ 2 2=3 ðR0n11 FðHn ÞÞ2 5 0

from which we have h

_  0 0 0 ^n 1 s 5 2 Δσtrial n11 : σ n 1 Rn11 c 0

sffiffiffiffi ffi

h 0 0 _ 0 0 0 trial0 ^ n Þ2 2 Δσtrial Δσtrial n11 :ð σ n 1Rn11 c n11 :Δσn11

h_ ii.   0 0 0 _0 0 2 0 0 0 trial0 :Δσ ð σ n 1 Rn11 c^ n Þ:ð σ n 1 Rn11 c^ n Þ 2 2=3 ðRn11 FðHn ÞÞ Δσtrial n11 n11 (7.254) The substitution of Eq. (7.254) into the second equation in Eq. (7.253)2 leads to 0

0 _0 0 ^ 2Δσtrial nÞ n11 :ðσ n 1Rn11 c rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi _  0  trial _ 0 2 0 2 _0 0 0 ^0 0 1R0 c 0 :Δσtrial 0 σ ^ : 2 2=3 ðR c 1R σ FðH ÞÞ 1 Δσn11 0 : σ n0 1Rn11 c^ n 2Δσtrial n n n n11 n11 n11 n n11 n n11  0 trial 0 _0 0 ^ 1Δσn11 : σ n 1Rn11 cn 50

that is, h i _ 0 0 2 0 0 ^ c Δσtrial : σ 1R n n11 n11 n h i 0 _ 0 0 trial 0 _ 0 2Δσtrial σ n 1 R0n11 c^ n : σ n0 1 R0n11 c^ n 2 ð2=3ÞðR0n11 FðHn ÞÞ2 5 0 n11 : Δσn11 (7.255)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

227

7.6 Implicit time-integration: return-mapping

which is the quadratic equation of R0n11 . Eq. (7.255) is rewritten as h  0 0   i 0 2  0 2 0 0 0 trial0 trial0 ^ n 2 Δσtrial Δσtrial c^ n : c^ n 1 2=3 ðFðHn ÞÞ2 Δσtrial Rn11 n11 : c n11 :Δσn11 n11 :Δσn11 i h  0 0 0 _  0 0 0 _ 0 trial ^ trial trial 12 Δσtrial σ n0 : c^ n R0n11 n11 :σ n Δσn11 : c n 2 Δσn11 :Δσn11    trial0 0 _ 2 0 trial0 _ 0 _ 0 1 Δσtrial σ n :σ n 50 n11 :σ n 2 Δσn11 :Δσn11 The solution of R0n11 in Eq. (7.255) is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 B 6 B2 2 AC 0 Rn11 5 A where

(7.256)

h 0 0  i 8 2 2 > ^ ^ c 2 S : c 2 2=3 ðFðH ÞÞ A  S < ss n n ca n _

0

(7.257)

Sca 2 Sss σ n0 : c^ n > : B  Ssc _ _ C  S2sc 2 Sss σ n0 : σ n0 with 0

0

trial Sss  Δσtrial n11 : Δσn11 ;

0

0

^n; Sca  Δσtrial n11 : c

0

_

0 Ssc  Δσtrial n11 : σ n

(7.258)

One must choose the solution satisfying 0 # R0n11 # 1 in Eq. (7.256). Here, we must set R0n11 5 Re if R0n11 # Re in the calculated result. It is enough to calculate the initial normal-yield ratio R0n11 , while it is not necessary to calculate the scalar number s and the stress σ0n11 for the return-mapping calculation. The correct loading criterion for the return-mapping method in the initial subloading surface model is given by setting c 5 α in the abovementioned formulations, referring to Fig. 7.16A.

7.6.4 Consistent tangent modulus tensor The introduction of the consistent tangent modulus tensor which fulfills the equilibrium equations on the stress (rate) boundary and inside the body is required in order to perform the finite element analysis, while the calculated stresses in the individual finite elements induce the residual stresses in the nodal points. The inverse matrix method will be delineated below referring to de Souza Neto et al. (2008) and particularly to Anjiki et al. (2016, 2019) for the extended subloading surface model, while the complete numerical method by the perturbation

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

228

7. Development of elastoplastic and viscoplastic constitutive equations

method also has been adopted widely (cf. Hashiguchi and Yamakawa, 2012; Iguchi et al., 2016, 2017a). If εetrial n11 is not constant but changes with a plastic deformation, the simultaneous equation in Eq. (7.199) becomes as follows: 3 2 @Yσ @Yσ @Yσ @Yσ @Yσ 6 @σ @α @c @H @ðΔλ Þ 7 n11 7 6 n11 n11 n11 n11 7 6 7 6 7 6 @Yα @Yα @Yα @Yα @Y α 7 6 9 9 8 etrial 9 6 @σ @α @c @H @ðΔλ Þ 78 8 6 n11 n11 n11 n11 n11 7> dσn11 dσn11 dεn11 > > > > > 7> 6 > > > > > > > > > > > > > > > > 7> 6 > > > > > > dα dα O > > > > > > 7 6 @Y @Y @Y @Y n11 n11 =  < = < = @Yc 7< 6 c c c c 7 dcn11 6 5 J 5 dc O n11 6 @σn11 @αn11 @cn11 @H n11 @ðΔλn11 Þ 7> > > > > > > > > 7> 6 > > > > > > 0 > > > > 7> 6 > > > > > dH n11 > > dH n11 > > > > > > > 7> 6 ; ; ; : : : 6 @YH @YH @YH @YH @YH 7 dðΔλn11 Þ dðΔλ Þ 0 n11 7 6 6 @σn11 @αn11 @cn11 @H n11 @ðΔλn11 Þ 7 7 6 7 6 7 6 6 @Ys @Ys @Ys @Ys @Ys 7 5 4 @σn11 @αn11 @cn11 @H n11 @ðΔλn11 Þ (7.259)

Let the inverse relation of Eq. (7.259) be described as follows: 3 2 D11 D12 D13 D14 D15 9 8 etrial 9 8 dσn11 dεn11 > 7> 6 > > > > > > > > 6 D21 D22 D23 D24 D25 7> > > > > > > > > 7> 6 dα O > > > > n11 = < = 7< 6 7 6 5 dcn11 (7.260) O 6 D31 D32 D33 D34 D35 7 > > > 7> 6 > > > > > > > 7> 6 > > > > 0 > > dH n11 > 6 D41 D42 D43 D44 D45 7> > > ; > ; : : 5> 4 dðΔλn11 Þ 0 D51 D52 D53 D54 D55 where the fourth-order tensors, the second-order tensors, and the scalars are denoted by Dij ; Dij ; Dij , respectively. It follows from Eq. (7.260) that dσn11 5 D11 : dεetrial n11

(7.261)

leading to ep; algo

Kn11

5

dσn11 5 D11 dεetrial n11

(7.262)

In this way, the consistent tangent modulus tensor is obtained by calculating the inverse of the matrix J.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.7 Subloading-overstress model

229

7.7 Subloading-overstress model The deformation of solids which are assemblies of material particles, e.g. metals and geomateials consists of the elastic deformation of material particles and the plastic deformation of their mutual slips. Therefore the rate-dependent deformation in a low stress level below the yield stress is induced by the deformations of material particles themselves so that it is described by the viscoelastic constitutive equation. In contrast, the rate-dependent deformation in a high stress level over the yield stress is described by the viscoplastic constitutive equation. This situation is illustrated in Fig. 7.17. The two types of the constitutive models, that is, the overstress model and the creep model have been proposed for the description of the ratedependent plastic (viscoplastic) strain rate. The viscoplastic strain rate is induced by the overstress from the yield surface in the overstress model (Prager, 1961a; Perzyna, 1963, 1966, 1971) based on the Bingham model for the one-dimensional deformation (Bingham, 1922), so that the mechanical response is reduced to that of the plastic constitutive model in the quasistatic deformation process. On the other hand, the

Deformation of solids

{

Deformation of solid particles Mutual slips of solid particles

σ

Yield stress

{ {

Time-independent: Elastic equation Time-dependent: Viscoelastic equation Time-independent: Plastic equation Time-dependent: Viscoplastic equation Elasto-viscoplastic deformation in dynamic loading

Elastoplastic deformation in quasi-static loading

ε Viscoelastic deformation Yield stress

FIGURE 7.17

Classification of rate-dependent and -independent deformations.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

230

7. Development of elastoplastic and viscoplastic constitutive equations

viscoplastic strain rate is induced always depending on the ratio of the stress to the yield stress in the creep model (Norton, 1929; Bodner and Partom, 1975; Peirce et al., 1984; Lemaitre and Chaboche, 1990; Lubarda, 2002, Asaro and Lubarda, 2006; de Souza Neto et. al, 2008; Belytschko et al., 2014). Therefore the mechanical response is not reduced to that of the plastic constitutive model in the quasistatic deformation in the creep model, since the creep model possesses the basically different mechanical structure from that of the ordinary plastic constitutive model as the viscoplastic strain rate is induced depending on the ratio of the stress to the yield stress in any stress level (even below the yield stress). Here, it is noticed that the yield stress decreases with the increase of temperature (cf. Section 7.7.5) so that the creep model is approximately applicable to the description of the deformation behavior at a high temperature. However, it should be recognized that the creep model does not hold but the overstress model holds in general. The subloading surface model described in the preceding sections will be extended to the rate-dependent plastic deformation below based on the concept of the overstress.

7.7.1 Constitutive equation The strain ε is additively decomposed into the elastic strain εe and the viscoplastic strain rate εvp , that is, ε 5 εe 1 εvp ; ε_ 5 ε_ e 1 ε_ vp

(7.263)

The viscoplastic strain rate εvp is further decomposed into the storage vp vp part εks and the dissipative part εkd for the kinematic hardening and vp vp into the storage part εcs and the dissipative part εcd for the elastic-core, that is, vp

vp

vp

vp

(7.264)

vp

(7.265)

εvp 5 εks 1 εkd ; ε_ vp 5 ε_ ks 1 ε_ kd vp

_ vp 5 ε_ vp _ cd εvp 5 εvp cs 1 εcd ; ε cs 1 ε Here, note the following facts.

1. In the sub-yield state ðR , 1Þ in which the stress lies inside the normal-yield surface, the subloading surface is similar to the normalyield surface with respect to the elastic-core c. 2. In the normal-yield state ðR 5 1Þ in which the stress lies on the normal-yield surface, the subloading surface coincides with the normal-yield surface. While α 5 α holds independent of the elasticcore c, leading to the relations σ 5 σ^ and n 5 n^ by virtue of

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.7 Subloading-overstress model

231

Eqs. (7.113), (7.114) and (7.118). This situation is shown by the dashed lines in Figure 7.8 for the uniaxial loading process. 3. In the over-yield state ðR . 1Þ in which the stress lies outside the normal-yield surface, the deformation behavior would depend on the viscous characteristics of material and thus it would be independent of the inside state of the normal-yield surface analogously to the normal-yield state ðR 5 1Þ. This fact has been accepted in the past overstress model as will be shown in Eq. (7.288). Then, let it be assumed that the following relations hold throughout the over normal-yield state. ^ n 5 n^ α 5 α; σ 5 σ;

for R $ 1

(Hashiguchi, 2019b). Then, we introduce the notations  α for R $ 1 ~  α α for R , 1  σ^ for R $ 1 σ~  σ for R , 1  n^ for R $ 1 n~  n for R , 1

(7.266)

(7.267) (7.268) (7.269)

Therefore the subloading surface is similar to the normal-yield surface with respect to the elastic-core c in the subyield state ðR , 1Þ but similar to the normal-yield surface with respect to the back stress α in the overyield state ðR $ 1Þ. Then, the subloading surface is described by ~ 5 RFðHÞ fðσÞ that is,



 ^ 5 RFðHÞ R 5 fðσÞ=FðHÞ ^ fðσÞ for R $ 1 fðσÞ 5 RFðHÞ for R , 1

(7.270)

(7.271)

Then, the normal-yield ratio R is derived from Eq.ffi (7.87) for R $ 1 and pffiffiffiffiffiffiffi 0 Eq. (7.110) for R , 1, and it is given by R 5 3=2jjσ^ jj=F for R $ 1 and Eq. (7.171) for R , 1 in the Mises metals. The normal-yield, the subloading and the elastic-core surfaces are shown in Figure 7.18. Now, let the flow rule of the viscoplastic strain rate be given by the following equation instead of Eqs. (7.43) and (7.117) for the plastic strain rate (Hashiguchi, 2019b), modifying the overstress model (Prager, 1961a; Perzyna, 1963, 1966) and noting Eq. (7.269) as follows: ε_ vp 5 Γ n~

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.272)

232

7. Development of elastoplastic and viscoplastic constitutive equations

FIGURE 7.18 Limit subloading, subloading, normal-yield, static subloading surfaces, limit elastic-core elastic-core, surfaces in subloading-overstress model. (A) Over normalyield state (R $ 1: α 5 α 5 αs 5 c). (B) Below normal-yield state (R , 1: α 6¼ α 6¼ αs 6¼ c).

where Γ

1 hR2Rs in μv 1 2 ½R=ðcm Rs Þm

or

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.273)

233

7.7 Subloading-overstress model

U ( R s ) ( = R· s / ||ε· ||) vp

U ( R s ) ( = R· s / ||ε· ||) vp

vp e ε· = O , ε· ≠ O

vp e ε· = O , ε· ≠ O

ε· ≠ O

ε• ≠ O

vp

0

1

vp

0

ε· ≠ O Rs vp

(A)

Re

1

vp ε· ≠ O

Rs

(B)

FIGURE 7.19 Function UðRs Þ in the evolution rule of static normal-yield ratio. (A) Re 5 0. (B) Re6¼0.

Γ

1 hexp½nðR 2 Rs Þ 2 1i μv 1 2 ½R=ðcm Rs Þm

(7.274)

where μv (viscoplastic coefficient), n, cm ðc1Þ, and m ð $ 1Þ are the material constants. The viscoplastic deformation is induced in the over normal-yield state (R $ 1) in the past overstress model but it is induced for R $ Rs so that the smooth elastic-plastic transition is described in Eq. (7.273) or (7.274), where Rs ( # 1) obeys the evolution rule of the normal-yield ratio in Eq. (7.89) with the replacement of the plastic strain rate to the viscoplastic strain rate as follows (Fig. 7.19): ( R_ s 5 UðRs Þjj_εvp jj for R . Re (7.275) Rs 5 Re for other with



π hRs 2 Re i UðRs Þ 5 u cot 2 1 2 Re

(7.276)

u is given based on Eq. (7.172) as follows: ~ u 5 u exp ðuc ℜc Cn Þ 5 u exp ðuc ℜc n^ c : nÞ

(7.277)

Rs ð # 1Þ is called the static normal-yield ratio because the quasistatic deformation proceeds in the state R 5 Rs . It is analytically expressed in

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

234

7. Development of elastoplastic and viscoplastic constitutive equations

Eq. (7.96) with the replacement of the plastic strain rate to the viscoplastic strain rate for fixing u as follows: 

vp  2 π Rs0 2 Re π εvp 2 ε0 21 Rs 5 ð1 2 Re Þ cos cos exp 2 u π 2 1 2 Re 2 1 2 Re (7.278) 1 Re for Rs0 $ Re Ð vp under the initial condition Rs 5 Rs0 for εvp 5 ε0 , where εvp 5 jj_εvp jj dt. The infinite viscoplastic strain rate proceeds for R-cm Rs leading to Γ -N. Then, cm Rs is called the limit normal-yield ratio. Simulation of test data would be performed more easily by Eq. (7.274) in the exponential type than by Eq. (7.273) in the power type. It is postulated in Eq. (7.272) with Eq. (7.273) or (7.274) that the viscoplastic strain rate is induced by the overstress from the following surface (see Fig. 7.18) given through the relations fðσs Þ 5 Rs FðHÞ

(7.279)

where σs 5 σ 2 αs ;

αs 5 c 2 Rs c^

(7.280)

noting Eq. (7.271)2. Eq. (7.279) designates the subloading surface evolving in the quasistatic deformation process, because Rs evolves in Eq. (7.275). Therefore the surface defined in Eq. (7.279) is called the static subloading surface. The rates of the isotropic hardening variable, the kinematic hardening variable, and the elastic-core are given in Eqs. (7.25), (7.28), and (7.136), respectively, with the replacement of ε_ p to ε_ vp as follows: ~ H_ 5 f Hn ðσ; H; nÞΓ

(7.281) Storage part

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ 1 1 1 vp vp _ 5 ck ðn~ 2 αÞΓ 5 ck ð_ε 2 αjj_ε jjÞ 5 ck ð ε_ vp 2 αjj_εvp jj Þ α |{z} bk F bk F bk F |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} Total Dissipative part



  ~ χ 1 F0 f ðσ; H; nÞ σ~ 2 c^ 1 ck n~ 2 α 1 Hn c_ 5 ce c^ Γ R bk F F



  ~ χ 1 F0 fHn ðσ; H; nÞ σ~ 2 c^ 1 ck n~ 2 α 1 c^ jj_εvp jj 5 ce R bk F F

(7.282)

Storage part

5 ce

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl) ffl{ (

 !  1 χ ~ 1 F0 fHn ðσ; H; nÞ vp vp vp vp vp vp σ~ 2 c^ 2 jj_ε jj^c jj_ε jjα 1 ck ε_ 2 ε_ 2 ε_ 2 jj_ε jj |{z} R ce bk F F |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl {zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} Total Dissipative part

(7.283) Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.7 Subloading-overstress model

235

Needless to say, all the internal variables H, α, c and Rs evolve by the viscoplastic strain rate ε__vp by the common evolution rules in both states under and over the normal-yield state, i.e. R , 1 and R $ 1. The stress is given from Eqs. (7.20), (7.263), and (7.272) by ð ð ~ σ 5 E : ε 2 E : ε_ vp dt 5 E : ε 2 E : Γ ndt (7.284) The second term in the right-hand sides is induced by the viscoplastic deformation caused from the mutual viscoplastic slips of material particles, which are called the viscoplastic (stress) relaxation. In the overstress model, we do not need to calculate the plastic modulus which possesses often a complex form as seen in Eq. (7.162) but instead we have only to perform the update calculation of the viscoplastic internal variables involved in the positive viscoplastic multiplier Γ in ~ Eq. (7.273) or (7.274) and n. The stagnation of isotropic hardening is incorporated by replacing the plastic strain rate ε_ p to the viscoplastic strain rate ε_ vp in the formulation described in Section 7.5.7. The stressstrain curve described by the subloading-overstress model is illustrated in Fig. 7.20. The smooth elastic-plastic transition and the cyclic loading behavior can be described appropriately. The rheological model of the subloading-overstress model is shown in Fig. 7.21, which is the extension of Fig. 7.13 for the subloading surface model, where the overstress is depicted by the dashpot. The resistance of the dashpot reduces to zero and the overstress diminishes and thus Fig. 7.21 reduces to Fig. 7.13 in the quasistatic deformation.

σ

Impact loading || ε·|| → ∞

|| ε·|| increases Quasi static loading || ε·|| ≅ 0

Overstress

0

FIGURE 7.20

ε

Stressstrain curve predicted by subloading-overstress model.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

236

7. Development of elastoplastic and viscoplastic constitutive equations

FIGURE 7.21 Rheological model for subloading-overstress model with decomposition of viscoplastic strain into storage and dissipative parts. (A) Overstress model with kinematic hardening. (B) Overstress model with isotropic and kinematic hardenings. (C) Extended subloading-overstress model.

The following relations hold from Eqs. (7.21), (7.263), and (7.272) in the overstress model.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.7 Subloading-overstress model

(

ε_ 5 E21 : σ_ 1 Γ n~ σ_ 5 E : ε_ 2 Γ E : n~

which is represented in the incremental form as follows:  ~ dε 5 E21 : dσ 1 Γ ndt ~ dσ 5 E : dε 2 Γ E : ndt

237 (7.285)

(7.286)

Then, it follows from Eq. (7.286) with Eq. (7.273) or (7.274) in the quasistatic deformation that Γ 5 0 leading to R 5 Rs for dt-N

(7.287)

so that the stress changes along the static subloading surface given in Eq. (7.279). Consequently, the response of the subloading-overstress model is reduced to that of the original subloading surface model in the rate-independence described in the previous sections. The subloadingoverstress model is no more than the generalization of the subloading surface model to the description of the elastoplastic deformation in the general strain rate. Irreversible deformations in any rate from the static to the impact loading can be described by the subloading-overstress model. Eventually, the elastoplastic constitutive equation with the plastic modulus for the rate-independent elastoplastic deformation which is derived by consistency condition of the yield condition (subloading surface for the subloading surface model) can be disused by adopting only the subloading-overstress model. The creep under the fixed stress and the stress relaxation in the fixed strain under the uniaxial loading process are predicted by the subloading-overstress model as illustrated in Fig. 7.22A and B, respectively. Therein, the nonhardening material is assumed in the stress laxation for the simplicity. In the creep phenomenon under the low stress level illustrated as σ 5 σ1 , the increase of strain increment stops when the stress reaches the yield surface, while it increases ceaselessly in the creep model described in Section 7.7.3. On the other hand, in the stress relaxation, the decrease of stress ceases when the stress lowers to the yield stress, while it decreases ceaselessly in the creep model. Further, the viscoplastic strain rate increases at the infinite deformation rate because of dεvp =dt-N for 1 2 ½R=ðcm Rs Þm -0 as known in Eq. (7.273) or (7.274), although the stress increases infinitely in the creep model analogously to the past overstress model.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

238

7. Development of elastoplastic and viscoplastic constitutive equations

σ

σ2 σ1

σ3

ε

σ3

σ2

σ1 σ1

Stop Creep model

ε

0

0

t

(A)

σ σ0

σy Creep model (B)

0

t

FIGURE 7.22 Creep and stress relaxation predicted by subloading-overstress model. (A) Creep phenomena. (B) Stress relaxation illustrated for constant yield stress.

7.7.2 Defects of past overstress model Eq. (7.273) is the modification of the past overstress model (Prager, 1961a; Perzyna, 1963, 1966) with n 

n 1 fðσÞ 1  21 R21 Γ 5 (7.288) μv FðHÞ μv leading to ε_ 5 E21 : σ_ 1

1 μv

dε 5 E21 : dσ 1

1 μv

that is,





fðσÞ 21 FðHÞ

fðσÞ 21 FðHÞ

n n

(7.289)

n dt

(7.290)

n

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.7 Subloading-overstress model

239

which results in ( dε 5 E21 : dσ for dt=jjdεjjD0 and dt=jjdσjjD0 ðimpact loadingÞ (7.291) dσ 5 E : dε Therefore the elastic deformation behavior is predicted for the impact loading by the past overstress model as shown in Fig. 7.23. The past overstress model is inapplicable to the description of the loading behavior at high rate of deformation, predicting the purely elastic deformation behavior at the impact loading. In addition, it cannot describe the smooth transition from the elastic to the viscoplastic state, exhibiting the discontinuous variation of the tangential stiffness modulus at the yield point. Further, the past overstress model is reduced to the conventional elastoplastic constitutive equation in the quasi-static deformation process and thus it cannot be applicable to the description of the cyclic loading behavior. Lemaitre and Chaboche (1990) proposed the irrational formulation such that the inelastic strain is composed of the plastic strain and the viscoplastic strain. Here, the plastic strain is defined as the inelastic strain induced at the infinite deformation rate (impact loading). This

FIGURE 7.23 Unrealistic stressstrain curve predicted by past overstress model: (1) discontinuous tangent stiffness modulus by abrupt occurrence of viscoplastic strain rate at yield point and (2) elastic response for impact loading.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

240

7. Development of elastoplastic and viscoplastic constitutive equations

idea would be quite irrational, contradicting with the fact that the plastic strain rate in the elastoplasticity theory is determined based on the test data at the quasistatic deformation rate. This would be caused by confusing the viscoplastic deformation to the viscoelastic deformation.

7.7.3 On irrationality of creep model The creep model is used widely for the rate-dependent deformation behavior analyses (e.g., Norton, 1929; Rice, 1970, 1971; Bodner and Partom, 1975; Johnson and Cook, 1983; Peirce et al., 1984; Lemaitre and Chaboche, 1990; Lubarda, 2002, Asaro and Lubarda, 2006; de Souza Neto et. al., 2008; Gurtin et al., 2010; Belytschko et al., 2014). The strain rate is composed of the elastic strain rate and the creep strain rate ε_ c in the creep model, that is, ε_ 5 ε_ e 1 ε_ c 5 E21 : σ_ 1 or ε_ 5 ε_ e 1 ε_ c 5 E21 : σ_ 1

1 μc

1 jjσjjm n μc

fðσÞ FðHÞ

(7.292)

m n

(7.293)

where μc and m are material constants. Eq. (7.292) is the so-called Norton law (Norton, 1929). The rheological model of the creep model is shown in Fig. 7.24, where the dashpot is connected with the spring in series analogously to the Maxwell model in the viscoelasticity. In other words, the creep model is to be the misuse of the viscoelastic model to the rate-dependent elastoplastic deformation. Eq. (7.292) or (7.293) is described in the incremental form as follows: dε 5 E21 : dσ 1

1 jjσjjm n dt μc

(7.294)

σ

εc

= σ dt / μ c

εc

ε

εe

=σ / E

εe

FIGURE 7.24 Rheological model of creep model, which is substantially identical to Maxwell model of viscoelastic model.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

241

7.7 Subloading-overstress model

or 21

dε 5 E

1 : dσ 1 μc



fðσÞ FðHÞ

m n dt

(7.295)

which leads to

8 dt dt > > -N and -N ðquasi-static deformationÞ jjσjj 5 0 or fðσÞ 5 0 for > > < jjdεjj jjdσjj > > dε 5 E21 : dσ > > :

for

dt dt D0 and D0 ðimpact loadingÞ jjdεjj jjdσjj (7.296)

Therefore, the creep model exhibits the unrealistic deformation behavior as shown in Fig. 7.25. It is incapable of describing the deformation behaviors in high and low rates of deformation. In paricular, the infinite stress increase is predicted for the impact loading by the creep model. On the other hand, the creep strain rate is induced even in any low stress level so that it is not reduced to elastoplastic constitutive equation in the quasi-static deformation, exhibiting the substantially different deformation behavior from that of the elastoplastic constitutive equation, while the overstress model is reduced to it in the quasi-static deformation. Then, the creep strain is always induced even in the unloading process below the yield stress. It leads to the excessive strain accumulation with open histeresis loops under the cyclic loading process as shown in Fig. 7.25. Consequently, the creep model is approximately applicable to the moderate rate of deformation but it should be noted that it is inapplicable to the prediction of the deformation at high and low rates as shown in Fig. 7.26. As described at the end of Section 7.7.1 with Fig. 7.22, the creep model leads to the unrealistic descriptions: The unlimited increase of strain in the creep phenomenon even under the fixed low stress and the unlimited stress relaxation to zero stress under the fixed strain are predicted by the creep model. Further, the creep strain is induced even in the unloading process at low stress level. This behavior contradicts with

σ

0

ε

FIGURE 7.25 Cyclic loading behavior described by creep model: Excessive mechanical ratchetting even for stress amplitude below yield stress.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

242

7. Development of elastoplastic and viscoplastic constitutive equations

Impact loading || · ||→ ∞

σ

|| · || increases

Quasi-static loading || · || ≅ 0

0

ε

FIGURE 7.26

Creep model is approximately applicable in limited range of strain rate except for high and low strain rates.

the fact that the mutual slips between material particles are removed by the stress removal. Then, the cyclic loading behavior leads to the excessive mechanical ratcheting as shown in Fig. 7.26. Unfortunately, the creep model is used widely in the crystal plasticity analysis at the room temperature by Peirce et al. (1982, 1983), Havner (1992), Asaro and Lubarda (2006), de Souza Neto et al. (2008), Gurtin (2010), etc., although the original purpose of the crystal plasticity analysis is to clarify the fundamental background of the irreversible deformation of metals. The crystal plasticity analysis based on the creep model is easy because the creep strain rate is induced always and thus any loading criterion is not required. However, the creep model is physically unacceptable as described above. The crystal plasticity analysis would have to be performed rationally by the subloading-crystal plasticity model (Hashiguchi, 2013a,b, 2016a, 2017a, 2018c; Okamura and Hashiguchi, 2015; Hashiguchi and Okamura, 2019). The cyclic plasticity models other than the extended subloading surface model, i.e. the Mroz’s multi surface model (Mroz, 1067), the Dafalias’ two (bounding) surface model (Dafalias and Popov, 1975), the Chaboche’s and Ohno-Wang’s suprposed kinematic hardening models

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.7 Subloading-overstress model

243

(Chaboche et al., 1979; Ohno and Wang, 1993) would not be able to be extended to describe the viscoplatic deformation reasonably. Nevertheless, the creep-type rate-dependent model was formulated by the Ohno’s groupe (Kobayashi et al., 2003) based on the OhnoWang’s superposed kinematic hardening model (Ohno and Wang, 1993). However, the Kobayashi et al. (2003)’s model is not reduced to the Ohno-Wang model in the quasi-static deformation process. It is inapplicable to the rate-independent deformation behavior at the quasistatic deformation process but the Ohno-Wang model (1993) is inapplicable to the rate-dependent deformation behavior, while they possess the basically different physical and the mathematical structures to each other. Therefore, it is impossible to analyze the deformation behavior in the varying rate from quasi-static to high rate and/or in the varying temperature from low to high tenperature by a unified formulation. This fact reveals that these models are the typical ad hoc models without the rationality/generality.

7.7.4 Implicit stress integration The concrete constitutive equation with the plastic modulus is not used but the plastic multiplier as the scalar variable is determined so as to satisfy the yield condition in the implicit numerical method, that is, the return-mapping projection for the rate-independent elastoplasticity as described in Section 7.6. On the other hand, the yield condition is not satisfied except for the quasi-static deformation process and thus the concrete constitutive equations, that is, Eq. (7.272) with Eq. (7.273) or (7.274) and Eq. (7.281), (7.282), and (7.283) for the evolution rules of the internal variables are used in the overstress model. The following simultaneous equation is solved in the implicit stress integration for the subloading-overstress model. 8 Stress variation:  > > vp > > Yσ 5 σn11 2 E : εetrial > n11 2 Δεn11 5 O > > > Kinematic hardening rule: > > > > Yα 5 αn11 2 αn 2 Δαn11 5 O > > < Elastic-core translation rule: (7.297) Yc 5 cn11 2 cn 2 Δcn11 5 O > > > > Isotropic hardening rule: > > > > YH 5 Hn11 2 Hn 2 ΔHn11 5 0 > > > > > Subloading >  Surface:  : Ys 5 f σ~ n11 2 Rn11 F Hn11 5 0

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

244

7. Development of elastoplastic and viscoplastic constitutive equations

Eq. (7.297) is explicitly expressed for the Hooke’s law and the Mises metals, the subloading surface of which is given in Eq. (7.110) with Eqs. (7.26), (7.28), and (7.136) as follows: 8 Stress variation: > > > > Yσ 5 σn11 2 E : εetrial ~ n11 ΔΓ n11 5 O > n11 1 2G n > > > > Kinematic hardening rule:

> > > 1 > > ~ n 5 α 2 α 2 c 2 α Y > α n11 n n11 n11 ΔΓ n11 5 O k > > bk Fn11 > > > > > Elastic-core translation rule: > > > > Yc 52cn11 2 cn > 3 > >



pffiffiffiffiffiffiffiffi 0 < 2=3 F χ 1 n11 σ~ n11 2 c^ n11 1 ck n~ n11 2 2 4ce αn11 1 c^ n11 5ΔΓ n11 5 O > Fn11 Rn11 bk Fn11 > > > > > > Isotropic hardenings rule: > ffiffiffi > > > 2 > > ΔΓ n11 5 0 5 H 2 H 2 Y > H n11 n > > 3 > > > > > Subloading > > sffiffiffi Surface: > > > 3 0 > > Ys 5 jjσ~ jj 2 Rn11 FðHn11 Þ 5 0 > : 2 n11

(7.298)

where n~ 5

σ~ 0 ; jjσ~ 0 jj

f Hn 5

rffiffiffi 2 3

(7.299)

1 hexp ½nðRn11 2 Rs n11 Þ 2 1i Δt (7.300) μv 1 2 ½Rn11 =ðcm Rs n11 Þm adopting Eq. (7.274). The unknown variables σn11 , αn11 , cn11 , Hn11 , and Rn11 are calculated by solving the simultaneous Eq. (7.298), while the variable Rsn11 is updated by 



 2 π Rsn 2 Re π ΔΓ n11 21 Rsn11 5 ð1 2 Re Þ cos cos exp 2 u π 2 1 2 Re 2 1 2 Re (7.301) 1 Re for Rsn $ Re ΔΓ n11 

Eq. (7.298) is expressed as Eq. (7.199) with the following Jacobian matrix 2 @Y σ 6 @σn11 6 6 6 @Yα 6 6 @σ 6 n11 6 6 @Y 6 c J56 6 @σn11 6 6 6 @YH 6 6 @σ 6 n11 6 6 @Y 4 s @σn11

@Yσ @αn11

@Yσ @cn11

@Yσ @Hn11

@Yα @αn11

@Yα @cn11

@Yα @Hn11

@Yc @αn11

@Yc @cn11

@Yc @Hn11

@YH @αn11

@YH @cn11

@YH @Hn11

@Ys @αn11

@Ys @cn11

@Ys @Hn11

@Yσ 3 @Rn11 7 7 7 @Yα 7 7 @Rn11 7 7 7 @Yc 7 7 7 @Rn11 7 7 7 @YH 7 7 @Rn11 7 7 7 @Ys 7 5 @Rn11

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.302)

7.7 Subloading-overstress model

245

Now, the following equations of partial derivatives hold. @σ @R @ΔΓ 5 5 5O @R @σ @σ

(7.303)

because σ and R are the independent variables to each other and Rs are set to be constant during the current increment. Further, the following partial derivatives are derived analogously to Eqs. (7.202) and (7.203), noting Eqs. (7.267)(7.269). 8 @σ~ > > 5 S; > > @σ > > > > > @σ~ > > > 5 ðhR 2 1i 2 RÞS; > > @α > > > > < @σ~ 5 2 h1 2 RiS (7.304) @c > > > > > @σ~ h1 2 Ri > > 5 c^ > > > @R 12R > > > > > @σ~ h1 2 Ri > > > : @ΔΓ 5 1 2 R UðRÞ^c ~  @ n~ 5 1 ðI0 2 n~  nÞ ~ N @ σ~ 0 jjσ~ 0 jj

(7.305)

~ : n~ 5 0Þ ðN Further, analogously to Eqs. (7.204) and (7.205), one has 8 0 @n~ @n~ @σ~ > > ~ 0 5 N; ~ 5 0: 5 N:I > > > @σ @σ ~ @ σ > > > > 0 > > @n~ @n~ @σ~ 0 > ~ ~ > 5 0: 5 N:ðhR 2 1i 2 RÞI 5 ðhR 2 1i 2 RÞN > > @α @α > ~ @ σ > > > 0 > < @n~ @n~ @σ~ ~ ½ 2 h1 2 RiS 5 2h1 2 RiN ~ 5 0: 5 N: @c @σ~ @c > > > > 0 > > @n~ @n~ @σ~ > > ~ h1 2 Ri c^ 0 5 h1 2 Ri N:^ ~ c0 5 5 N: : > 0 > > @R @σ~ @R 12R 12R > > > > 0 > > 0 @n~ @n~ @σ~ h1 2 Ri > ~ 0 ~ h1 2 Ri > > : @ΔΓ 5 @σ~ 0 : @ΔΓ 5 N: 1 2 R UðRÞ^c 5 1 2 R UðRÞN:^c

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.306)

246

7. Development of elastoplastic and viscoplastic constitutive equations

8 0 0 0 0 @jjσ~ jj @jjσ~ jj @σ~ σ~ 0 0 > > > ~ 5 n~ 5 5 :I 5 n:I : 0 0 > > @σ @σ ~ ~ @ σ jj σ jj > > > > 0 0 0 0 > > @jjσ~ jj @jjσ~ jj @σ~ σ~ 0 > > 5 5 :ðhR 2 1i 2 RÞI 5 ðhR 2 1i 2 RÞn~ : > 0 0 > < @α @α @σ~ jjσ~ jj 0

0

0

0

(7.307)

@ΔΓ 5 JR ΔΓ @R

(7.308)

> @jjσ~ jj @jjσ~ jj @σ~ σ~  0 > > 5 5 : 2 h1 2 RiI 5 2h1 2 Rin~ : > 0 0 > @c @c > jjσ~ jj @σ~ > > > 0 0 0 > > @jjσ~ jj @jjσ~ jj σ~ h1 2 Ri 0 h1 2 Ri 0 > > > ~c ~ 5 c^ 5 n:^ 5 n: : 0 > : @R 12R 12R @R @σ~ In addition, one has

where JR  n 1

n m½R=ðcm Rs Þm21 1 hexp½nðR 2 Rs Þ 2 1i cm Rs f1 2 ½R=ðcm Rs Þm g

(7.309)

noting @ΔΓ @ 5 @R @R 5 5

5



1 hexp½nðR 2 Rs Þ 2 1i Δt μv 1 2 ½R=ðcm Rs Þm



1 nexp½nðR 2 Rs Þf1 2 ½R=ðcm Rs Þm g 1 mhexp½nðR 2 Rs Þ 2 1i½R=ðcm Rs Þm21 =ðcm Rs Þ Δt μv f12½R=ðcm Rs Þm g2 1 nfexp½nðR 2 Rs Þ 2 1gf1 2 ½R=ðcm Rs Þm g 1 nf1 2 ½R=ðcm Rs Þm g 1 mhexp½nðR 2 Rs Þ 2 1i½R=ðcm Rs Þm21 =ðcm Rs Þ Δt μv f12½R=ðcm Rs Þm g2 2 3 1 6nfexp½nðR 2 Rs Þ 2 1g n mhexp½nðR 2 Rs Þ 2 1i½R=ðcm Rs Þm21 =ðcm Rs Þ7 1 1 4 5Δt μv 1 2 ½R=ðcm Rs Þm 1 2 ½R=ðcm Rs Þm f12½R=ðcm Rs Þm g2

5 ðn 1

n m½R=ðcm Rs Þm21 1 ÞΔΓ hexp½nðR 2 Rs Þ 2 1i cm Rs f1 2 ½R=ðcm Rs Þm g

ð7:310Þ The components of the Jacobian matrix in Eq. (7.302) are given by @Yσ @n~ ~ 5 S 1 2G ΔΓ 5 S 1 2GNΔΓ @σ @σ @Yσ @n~ ~ ΔΓ 5 2GðhR 2 1i 2 RÞNΔΓ 5 2G @α @α   @Yσ @n~ ~ ΔΓ 5 2 2Gh1 2 RiNΔΓ ~ 5 2G ΔΓ 5 2G 2h1 2 RiN @c @c @Yσ 5O @H

(7.311) (7.312) (7.313) (7.314)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.7 Subloading-overstress model

 

@Yσ @n~ @ΔΓ h1 2 Ri ~ 0 ΔΓ 1 n~ N : c^ 1 J R n~ ΔΓ 5 2G 5 2G @R @R 12R @R

247

(7.315)

@Yα @n~ ~ ΔΓ 5 2 ck NΔΓ 5 2 ck (7.316) @σ @σ



@Yα @n~ 1 ~ 2 1 S ΔΓ 2 5 S 2 ck S ΔΓ 5 S 2 ck ðhR 2 1i 2 RÞN @α bk F bk F @α (7.317)   @Yα @n~ ~ ΔΓ 5 ck h1 2 RiNΔΓ ~ 5 2 ck ΔΓ 5 2 ck 2h1 2 RiN (7.318) @c @c @Yα F0 5 2 ck pffiffiffiffiffiffiffiffi αΔΓ @H 3=2bk F2 2 3

~ @Yα @ n 1 @ΔΓ 5 5 2 ck 4 ΔΓ 1 n~ 2 α @R bk F @R @R 2 3

h1 2 Ri 1 ~ : c^ 0 1 J n~ 2 5 2 ck 4 α 5ΔΓ N R 12R bk F

(7.319)

(7.320)





@Yc χ @σ~ @n~ χ ~ 1 ck 5 2 ce (7.321) ΔΓ 5 2 ce S 1 ck N ΔΓ R @σ @σ R @σ 2 3

pffiffiffiffiffiffiffiffi 0

2=3F @^c 5 @Yc χ @σ~ @^c @n~ 1 @α 2 2 52 4ce 1ck 1 ΔΓ R @α @α @α bk F @α @α F @α 2 3

pffiffiffiffiffiffiffiffi 0

2=3F 5 χ 1 ~ ðhR21i2RÞS1S 1ck ðhR21i2RÞN2 S 2 S ΔΓ 524ce R bk F F 82 9 pffiffiffiffiffiffiffiffi 0 3

< χ = 2=3F 5 ck ~ ΔΓ 5 4ce ðhR21i2RÞ11 1 1 S2ck ðhR21i2RÞN : ; R bk F F 2

3

(7.322)

pffiffiffiffiffiffiffiffi 0

2=3F @^c5 @Yc χ @σ~ @^c @n~ 4 2 ΔΓ 5S2 ce 1ck 1 R @c @c @c @c F @c 2 pffiffiffiffiffiffiffiffi 0 3

  χ ~ 1 2=3F S5ΔΓ (7.323) ð2hR21iÞS2S 1ck ð2hR21iÞN 5S2 4ce R F 82 3

pffiffiffiffiffiffiffiffi 0 < hR21i 2=3F 5 ~ 4 5S1 c χ 11 2 S1ck hR21iNΔΓ : e R F

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

248

7. Development of elastoplastic and viscoplastic constitutive equations



pffiffiffiffiffiffiffiffi FvF2ðF0 Þ2 pffiffiffiffiffiffiffiffi FvF2ðF0 Þ2 @Yc ck F0 ck F 0 c^ ΔΓ 5 c^ ΔΓ 5 α2 2=3 α2 2=3 @H bk F2 F2 bk F 2 F2 2

(7.324)

3

@Yc 4 χ χ @σ~ @n~ ~ 5 ce σ2 1ck 5ΔΓ R2 R @R @R @R 2 3

pffiffiffiffiffiffiffiffi 0

2=3F 5 @ΔΓ χ 1 ~ ~ c^ 1ck n2 2 4ce α 1 c^ σ2 R bk F @R F 8

< χ χ h12Ri h12Ri ~ 0 ^ ~ c 1ck N: c^ 5 ce σ2 2 : R R 12R 12R 2 3 9

pffiffiffiffiffiffiffiffi 0

= 2=3 F χ 1 ~ ~ c^ 1ck n2 α 1 c^ 5J R ΔΓ 24ce σ2 ; R bk F F

(7.325)

@YH 5O @σ

(7.326)

@YH 5O @α

(7.327)

@YH 5O @c

(7.328)

@YH 51 @H rffiffiffi rffiffiffi @YH 2 ΔΓ 2 52 52 J ΔΓ 3 @R 3R @R rffiffiffi rffiffiffi @Ys 3 @jjσ~ 0 jj 3 5 n~ 5 2 @σ 2 @σ rffiffiffi @Ys 2 @jjσ~ 0 jj 5ðhR21i2RÞn~ 5 3 @α @α rffiffiffi rffiffiffi @Ys 2 @jjσ~ 0 jj 2 52 5 h12Rin~ 3 @c 3 @c @Ys 52RF0 @H rffiffiffi rffiffiffi @Ys 2 @jjσ~ 0 jj 2 h12Ri ~ c^ 0 2F 2F5 n: 5 3 @R 3 12R ~ @R

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(7.329) (7.330) (7.331) (7.332) (7.333) (7.334) (7.335)

249

7.8 Fundamental characteristics of subloading surface model

F = F0 {1+ h1 [1− exp(−h2 H )]}exp(−Cθ θ 2 ) F = F0 {1+ h1 [1− exp(−h2 H )]}

1 −2Cθ θ F0 {1+ h1 [1− exp(−h2 H )]}exp(−Cθ θ 2 )

0

FIGURE 7.27

θ

Influence of temperature on isotropic hardening.

The consistent tangent modulus tensor is derived by the way analogous to the one described in Section 7.6.4.

7.7.5 Temperature dependence of isotropic hardening function The isotropic hardening function FðHÞ would decrease with the elevation of temperature. The following equation taken account of the influence of temperature would be able to be postulated (see Fig. 7.27). FðH; θÞ 5 F0 f1 1 h1 ½1 2 expð2 h2 HÞgexpð2 Cθ θ2 Þ

(7.336)

where θ designates the temperature elevation from the ordinary temperature and Cθ is the material constant. Eq. (7.336) is the slight modification of the normal (Gaussian) distribution function. A further study is required for the description of the creep in the tertiary stage which cannot be described by the present formulation of the subloading-overstress model. The overstress-type viscoplastic constitutive equation described in this section is regarded as the generalization of the elastoplastic constitutive equation to the deformation behavior at the general rate of deformation from the static to the impact deformation behavior. Nevertheless, it is much simpler than the original elastoplastic constitutive equations with the complex plastic modulus in general.

7.8 Fundamental characteristics of subloading surface model The basic features of the subloading surface model are described in this section. Incidentally, the basic structure of the bounding surface with radial-mapping is explained briefly.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

250

7. Development of elastoplastic and viscoplastic constitutive equations

7.8.1 Distinguished abilities of subloading surface model The fundamental characteristics of the subloading surface model are summarized in the following. First for the rate-independent elastoplastic subloading surface model: 1. The smoothness condition in Eqs. (7.3) or (7.4) and (7.9) is fulfilled, while it is violated in the other elastoplasticity models because a purely elastic domain is postulated. 2. The smooth transition from elastic to plastic state is described, which is observed in real material behavior. The smaller the value of u, the smoother elastic-plastic transition is depicted as shown in Fig. 7.28. In contrast, the larger the material parameter u, the more rapidly the normal-yield ratio R increases causing the more rapid increase of stress, that is, approaching the behavior of the conventional plasticity. The material parameter u is determined so as to trace the stressstrain curve in test data. 3. The yield judgment of whether or not the stress reaches the yield surface is not required. In contrast, the yield judgment is required in any other elastoplasticity models since they assume a surface enclosing a purely elastic domain. The determination of the yield stress due to the offset value is accompanied with an arbitrariness because the test data are usually smooth. 4. The tangent stiffness modulus changes always continuously, while it changes abruptly at the yield point in the other elastoplasticity model. 5. The plastic strain rate can be described even for low stress level and for cyclic loading process under small stress amplitudes since the surface enclosing a purely elastic domain is not assumed. 6. The automatic stress-controlling function is furnished in the rateindependent elastoplastic subloading surface model such that the σ

U ( R)

u decreases

0 (A)

1

R

0

ε

(B)

FIGURE 7.28 Influence of material parameter u on stressstrain curve. (A) Function U (R). (B) Stressstrain curve.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.8 Fundamental characteristics of subloading surface model

251

stress is always attracted to the normal-yield surface. In particular, it is noticeable that the stress is automatically pulled back to the normal-yield surface when it goes over the surface in numerical calculation because of R_ , 0 for R . 1 based on Eq. (7.89) with Eq. (7.94)4 as seen in Fig. 7.29. In contrast, the cumbersome numerical operation to pull back the stress such as the radial return and mean normal method in the forward-Euler calculation is required in all of the other elastoplastic models. 7. The exact finite elastoplastic constitutive equation, that is, the multiplicative hyperelastic-based plastic constitutive equation can be formulated only by the subloading surface model. It will be explained in the subsequent sections, which is the main subject in this book. The abovementioned advantages of the subloading surface model are furnished in the initial subloading surface model described in Section 7.3 by the simple modification of existing computer program for the conventional elastoplasticity model to add only one material parameter u. Furthermore, 8. The cyclic loading behavior can be described rigorously by incorporating the translation of the elastic-core, that is, the similarity-center of the normal-yield and the subloading surfaces. Therein, the plastic strain rate is described even in the unloading process so that the closed hysteresis loop is depicted realistically. 9. The associated flow rule can be adopted in the subloading surface not only for metals but also for solids in general, including plastically compressible materials, for example, soils and concrete. (cf. Hashiguchi

·

σ ·

R>0 for R 0 for R < 1 ⎪ U ( R) ⎨= 0 for R = 1 ⎪ < 0 for R > 1 ⎩

FIGURE 7.29 Stress is automatically controlled to be attracted to the normal-yield surface in the subloading surface model.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

252

7. Development of elastoplastic and viscoplastic constitutive equations

et al., 2002; Hashiguchi and Mase, 2018), although the nonassociated flow rule with the plastic potential surface different from the yield surface is adopted in the other elastoplasticity models. Further, for subloading-overstress model: 10. The rate-independent and -dependent elastoplastic deformation at the general rate from the quasi-static to the impact loading process can be described rigorously. 11. The cyclic loading behavior at the general rate can be described rigorously, while these deformation behaviors cannot be described appropriately by any other cyclic plasticity model, overstress model and creep model. Furthermore, for the finite deformation: 12. The multiplicative hyperelastic-based plasticity can be formulated only by the subloading surface model even for the description of the finite and cyclic loading behavior as will be described comprehensively ih the subsequent chapters.

7.8.2 Bounding surface model with radial-mapping: Misuse of subloading surface model The bounding surface model with radial-mapping (Dafalias, 1986; Dafalias et al., 2006; Dafalias and Talebat, 2016) is often adopted to the description of deformation behavior of soils as SANICLAY and SANISAND models (Dafalias et al., 2006; Dafalias and Talbat, 2016, etc.), although it is not used for metals so that it is the ad hoc model limited to the monotonic loading behavior of soils. Its basic structure is the inheritance of the initial subloading surface model described in Section 7.3 as is clearly known from the confession “It appears that the first time a radial-mapping formulation was proposed, it was in reference to granular materials by Hashiguchi and Ueno (1977)” which is the original sentence in Dafalias (1986, p. 980). Therefore it remains within the ability of the initial subloading surface model and thus it is incapable of describing the cyclic loading behavior as known also from his continued confection “Because of its simplicity the radial mapping has a relative disadvantage. . . . Hence, upon partial unloading/reloading the stressstrain loops are not closed . . . this feature may not describe realistically the shape of the stressstrain curves.” (sic) by Dafalias (1986, p. 980). Incidentally, the plastic strain rate is not formulated rigorously based on the consistency condition but it is formulated by the interpolation method without any physical logic. Needless to say, the bounding surface model with radial-mapping is incapable of describing realistically

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

7.8 Fundamental characteristics of subloading surface model

253

the cyclic loading behavior because unloading/reloading stressstrain loops are not closed as described in the abovementioned confection. The rotation of the yield surface without the translation of the elasticnucleus is incorporated in order to describe the cyclic mobility in the SANISAND model. However, such an ad hoc primitive method results in the description of the unrealistic cyclic loading behavior with an open hysteresis loop for the stress path along the central axis of the yield surface, including the cyclic isotropic loading. A more detail of the irrationalities in the bounding surface model with radial-mapping can be referred to Hashiguchi (2017a). The above-mentioned crucial defects in the SANICALY and SANISAND models should be recognized in order to avoid the adverse influence to the steady and sound development of elastoplasticity.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

C H A P T E R

8 Multiplicative decomposition of deformation gradient tensor Elastoplastic constitutive equation has been developed first as the infinitesimal elastoplasticity and further as the hypoelastic-based plasticity. However, they are incapable of describing the finite deformation/ rotation exactly. The multiplicative hyperelastic-based plasticity is capable of describing the elastoplastic deformation and rotation accurately, in which the deformation gradient tensor is decomposed definitely into the purely elastic (hyperelastic) part and the purely plastic part by incorporating the multiplicative decomposition of the deformation gradient tensor into the elastic and the plastic parts. Therefore it has been studied widely by numerous workers, for example Lion (2000), Menzel and Steinmann (2003a,b), Wallin et al. (2003), Dettmer and Reese (2004), Menzel et al. (2005), Wallin and Ristinmaa (2005), Vladimirov et al. (2008, 2010), Hashiguchi and Yamakawa (2012), Brepols et al. (2014), Hashiguchi (2013a, 2016a, 2017a,b,c, 2018c), etc. after the advocacy of the multiplicative decomposition by Kroner (1960), Lee and Liu (1967), Lee (1969), Mandel (1971, 1972, 1973a), and Kratochvil (1971). Historically, the elastoplastic constitutive model based on the multiplicative decomposition, that is, the multiplicative hyperelastic-based plastic model is first formulated widely in the current configuration, for which the principal stretch-based elastic constitutive equation is incorporated with the use of the spectral decompositions of stress and deformation tenors, and applied widely to the finite elastoplastic deformation analysis (cf. Simo, 1992, 1998; Simo and Hughes, 1998; Bonet and Wood, 2008; de Sauza Neto et al., 2008; etc.). However, it is restricted to the elastic and plastic isotropy, further requiring the cumbersome operation for the spectral decomposition. Then, it has been replaced recently by the multiplicative hyperelastic-based plastic model formulated in the intermediate configuration, which is not restricted to the isotropy, disusing the spectral decomposition.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity DOI: https://doi.org/10.1016/B978-0-12-819428-7.00008-0

255

© 2020 Elsevier Inc. All rights reserved.

256

8. Multiplicative decomposition of deformation gradient tensor

Incidentally, it should be noted that almost of all hyperelastic-based plastic constitutive equations with elastic and plastic potential functions have been formulated based on the Clausius 2 Duhem inequality. However, it would be impertinent to formulate a plastic flow rule under the thermodynamic restriction as will be interpreted in Section 11.3. Therefore the rigorous formulations independent of this inequality will be given in this chapter. The multiplicative decomposition of the deformation gradient tensor with the isoclinic concept leads to the inclusion of the rigid-body rotation in the elastic deformation gradient tensor and thus leads to the description of constitutive relation in the intermediate configuration unloaded to the stress-free state along the hyperelasticity. Here, it should be emphasized that the multiplicative decomposition with the definite uniqueness holds in the general elastoplastic materials, although the isoclinic concept is often misunderstood to hold only in the crystalline materials. The multiplicative hyperelastic-based plastic constitutive equation will be explained comprehensively in this chapter.

8.1 Elastic-plastic decomposition of deformation measure The deformation measure must be decomposed into the elastic and the plastic parts definitely for the exact description of elastoplastic deformation as was explained in Section 7.1. In addition, any exact deformation (rate) measures are defined by the deformation gradient tensor as described in Chapter 4: Deformation/rotation (rate) coordinate system. These requirements can be realized by the multiplicative decomposition of the deformation gradient tensor. In addition, the multiplicative hyperelastic-based plasticity can be formulated rigorously and concisely in the intermediate configuration based on the isoclinic concept by which the intermediate configuration is not influenced by the rotation of the substructure of material. The physical background and the rigorous formulation of the multiplicative decomposition will be described in this section.

8.1.1 Necessity of multiplicative decomposition of deformation gradient tensor First of all, we must remind 1) the deformation is decomposed into the purely elastic deformation and the purely plastic deformation as was described in Section 7.1.1 and 2) any deformation (strain) measures are defined by the deformation gradient tensor F 5 @x/@X. Therefore, the decomposition of the deformation gradient tensor into the purely elastic part and the purely plastic part is required to formulate the exact elastoplastic constitutive equation. Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

8.1 Elastic-plastic decomposition of deformation measure

257

The infinitesimal strain ε is additively decomposed into the elastic and the plastic parts similarly to the one-dimensional infinitesimal strain e, which is defined by the difference of the current length from the initial length of the line-element in Section 7.2.1. However, the deformation gradient tensor is defined by the ratio of the current infinitesimal line-element vector to the initial one. Therefore, it corresponds to the stretch λ in the one-dimensional state, which is defined by the ratio of the current length l to the initial length l0 of the line-element, that is, λ5

l l0

(8.1)

which is multiplicatively decomposed into the elastic stretch λe and the plastic stretch λp as follows: λ 5 λe λp

(8.2)

where λ5

l l l ; λe 5 ; λp 5 l0 l0 l

(8.3)

l is the length in the unloaded state to the stress-free state. Incidentally the stretch satisfies the multiplicative superposition as follows: λ0Bn 5

ln l1 l2 5 l0 l0 l1

ln 5 λ0B1 λ1B2


λðn21ÞBn


ln21

(8.4)

Here, we need to pay particular attention to the unloading process from the current to the stress-free state. The experimental curve of actual one-dimensional deformation of real solid bar is shown by the solid curve in Fig. 8.1. Suppose simply that the material is the assembly of spherical particles. Supposing the spherical shape of particles for instance; they are elongated to the ellipsoidal shape in the tensional loading process. They return elastically to the spherical shape in the unloading process. It causes the mutual slips of the material particles in the direction opposite to the direction of the mutual slips in the former tensional loading process. Then, the slight compressive plastic deformation is induced in the unloading process as observed in the real material behavior. Here, the purely elastic deformation without a plastic deformation is induced only at the initiation of unloading process. Therefore l is not actual but merely fictitious length of material line-element unloaded fictitiously to the stress-free state along the hyperelastic relation. Consequently, the unloading in the elastic-plastic decomposition of deformation for the exact elastoplastic constitutive equation must be performed conceptually along the hyperelastic constitutive equation as

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

258

8. Multiplicative decomposition of deformation gradient tensor

Stress Actual unloading process

Elastic strain energy Virtual unloading process along hyperelasticity

0

l0

l

l

Length

Initial state Elastic extension

Plastic slip

Current loaded state Elastic unloading Hyperelastic unloading of material particles Inverse plastic slip caused in conjunction with hyper elastic unloading of material particles Actually unloaded state

FIGURE 8.1 Unloading process for decomposition of stretch into elastic and plastic parts in tension of bar.

shown by the dashed line in Fig. 8.1, since the deformation must be decomposed exactly into the purely elastic and the purely plastic parts. Consequently, the exact description of the elastoplastic deformation must be based on the multiplicative decomposition of the deformation gradient tensor. Based on this physical background, the following multiplicative decomposition (or Kro¨ner decomposition or Lee decomposition) of the deformation gradient tensor has been studied by Eckert (1948), Kroner (1960), Lee and Liu (1967), Lee (1969), Mandel (1971, 1972, 1973a), and Kratochvil (1971). F 5 Fe Fp

(8.5)

Here, based on the requirement of the exact decomposition of deformation into the purely elastic and the plastic parts, the elastic deformation gradient tensor Fe must be related to the stress by the hyperelastic relation. Then, the plastic deformation gradient tensor Fp is relevant to the state unloaded to the stress-free state along the hyperelastic relation similarly to λp in Eq. (8.2) for the stretch. The unloaded configuration to the stress-free state is called the intermediate configuration. The elastoplastic deformation process based on this notion is illustrated in Fig. 8.2 where the initial, the intermediate, and the current configurations are specified by the symbols K0 , K, and K, respectively. The tensors in the

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

8.1 Elastic-plastic decomposition of deformation measure

259

FIGURE 8.2 Multiplicative decomposition of deformation gradient.

current configuration are designated by the lowercase letter as t, the ones in the reference configuration by the uppercase letters as T and the ones in the intermediate configuration by the uppercase letters with the upper bar as T. Here, consider the vector dX in the following expression. dX 5 Fe21 dx 5 Fp dX

(8.6)

leading to Fe 5

@x p @X ;F 5 @X @X

(8.7)

where dX and dx are the actual infinitesimal line-element vectors similarly to l0 and l. On the other hand, dX is not actual but merely fictitious infinitesimal line-element vector similarly to l in Eq. (8.3)2,3, which is the line-element vector calculated by unloading to the stress-free state along the hyperelastic relation. Here, note that the micromechanical structure of plastically deformed material is heterogeneous, possessing the statically indeterminate structure, and thus different amounts of distressing are required in order that all material points reach stress-free states. Needless to say, it is impossible to separate the material into infinitesimal pieces and then gather them as ever. Eventually, Eq. (8.7) does not possess any actual physical meaning but it is merely fictitious mathematical expression.

8.1.2 Isoclinic concept The extensive debates as to which an elastic deformation gradient or a plastic deformation gradient must include the rigid-body rotation

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

260

8. Multiplicative decomposition of deformation gradient tensor

have been repeated for a long time after the proposition of the multiplicative decomposition. The inclusion of the rigid-body rotation in the plastic deformation gradient has been insisted by Lee (1969), Green and Naghdi (1971), Fardshisheh and Onate (1974), Casey and Naghdi (1980), Lubarda and Lee (1981), Dafalias (1985), Boyce et al. (1988), Lubarda (1991, 2002, 2004), Khan and Huang (1995), Han et al. (2003), Wu (2004), Simo and Ortiz (1985), Asaro and Lubarda (2006), Harrysson and Ristinmaa (2007), etc. This situation would be caused by worrying the fact that the elastic distortion is known from the current stress but the rigid-body rotation is unknown and thus it is possible to exclude only the elastic distortion but it is impossible to exclude both of the rigidbody rotation and the elastic distortion from the current configuration in order to get to the intermediate configuration. On the other hand, the inclusion of the rigid-body rotation in the elastic deformation gradient has been insisted by Holsapple (1973), White (1975), Van der Giessen (1989), Haupt (2002), Wallin et al. (2003), Dettmer and Reese (2004), Wallin and Ristinmaa (2005), Vladimirov et. al. (2008, 2010), etc. The debates on this issue have been commented repeatedly without a definite conclusion by various authors (Cleja-Tigoiu and Soos, 1970; Clifton, 1972; Dashner, 1986; Lubliner, 1990; Simo, 1992, 1998; Simo and Hughes, 1998; etc.). The debate repeated during the long term has been settled by accepting the isoclinic concept (Mandel, 1971) leading to the inclusion of the rigid-body rotation in the elastic deformation gradient tensor as will be explained in the following. 1. The physically meaningful rotation of material is the rotation of the substructure of material and it is induced by the elastic deformation and the rigid-body rotation, while the plastic deformation induces merely the mutual slips of material particles. 2. Then, let not only the rotation induced by the elastic deformation but also the rigid-body rotation be involved in the elastic deformation gradient tensor. Then, the following polar decompositions hold. Fe 5 Re Ue ; Re 5 Rr Red

(8.8)

F p 5 Rp Up

(8.9)

where Re is the pure rotation of substructure that is composed of the rigid-body rotation Rr and the rotation due to the elastic distortion Red , and Ue is the positive-definite tensor describing the purely elastic deformation. Rp is the rotation induced by the mutual slips between material particles, which is independent of the rotation of substructure and Up is the positive-definite tensor describing the purely plastic deformation.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

8.1 Elastic-plastic decomposition of deformation measure

261

3. Consequently, “the substructure does not rotate in the intermediate configuration.” This idea was named the isoclinic concept by Mandel (1971), while “isoclinic” means “equal inclination,” which is schematically shown in Fig. 8.3. Here, Rp causes the plastic spin (Mandel, 1971, 1972, 1973a; Kratochvil, 1971; Dafalias, 1983, 1985; Loret, 1983; Zbib and Aifantis, 1988), which does not cause the spin of the substructure of material (suppose the typical example: the sliding of the playing cards such that they slide mutually without their rotation). 4. The velocity gradient tensor cannot be decomposed into the elastic and the plastic parts in the current configuration but it can be decomposed exactly in the intermediate configuration as will be shown in

FIGURE 8.3 Multiplicative decomposition of deformation gradient, where the substructure does not rotate in the intermediate configuration: isoclinic concept (Mandel, 1971).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

262

8. Multiplicative decomposition of deformation gradient tensor

Eq. (8.22) with Eq. (8.23) and in Eq. (8.29) with Eq. (8.30) in Section 8.2.2. 5. Consequently, the multiplicative hyperelastic-based plastic constitutive equation is formulated exactly in the intermediate configuration based on the isoclinic concept. However, it should be noted that the current configuration is the actual configuration but the intermediate configuration is the fictitious one. In fact, the actual behavior can be observed and compared with test data in the current configuration. Therefore it is natural to formulate firstly the constitutive relation within the framework of the infinitesimal elastoplasticity in the current configuration and thereafter we may extend it to the multiplicative hyperelastic-based plastic constitutive equation in the intermediate configuration as is done in Chapter 7 to 9 in this book.

8.1.3 Uniqueness of multiplicative decomposition The nonuniqueness of the multiplicative decomposition has been viewed as a problem (Green and Naghdi, 1965; Casey and Naghdi, 1980, 1981; Dashner, 1986; Naghdi, 1990; Simo, 1998; Khan and Huang, 1995; Lubarda, 2002; Haupt, 2002; Han et al., 2003; Nemat-Nasser, 2004; Bonet and Wood, 2008; Henann and Anand, 2009). One of the typical statements is “A much debated issue in finite strain theory is concerned with the apparent lack of uniqueness inherent to the multiplicative factorization, which arises when considering superposed rigid-body motion on the intermediate configuration” (Simo, 1998). In fact, the deformation gradient tensor is not decomposed uniquely as shown formerly in the following. F 5 Fe Fp 5 ðFe RT ÞðR Fp Þ 5 Fe Fp

(8.10)

where R is the proper orthogonal tensor ðdetR 51 1Þ denoting an arbitrary rotation added to the intermediate configuration and Fe 5 Fe RT ; Fp 5 R Fp

(8.11)

Then, the formulation of constitutive equation in the current configuration in terms of the elastic left Cauchy-Green tensor and the plastic right Cauchy-Green tensor which are invariant under rigid-body rotation has been performed by some workers (Simo, 1988a, b, 1992, 1998; Simo and Hughes, 1998; Bonet and Wood, 2008; Bonet et al., 2016; de Souza Neto et al., 2008; Yamakawa et al., 2010) as will be commented in Section 9.12.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

8.1 Elastic-plastic decomposition of deformation measure

263

However, the non-uniqueness of the multiplicative decomposition is the unnecessary anxiety (issue). In facts, it is solved out by formulating the constitutive equation in the intermediate configuration based on the isoclinic concept insisting that the rigid-body rotation is not included in the plastic deformation gradient Fp but included in the elastic deformation gradient Fe. Firstly, the plastic deformation gradient Fp is determined uniquely by the plastic constitutive equation independent of the rigid-body rotation based on the isoclinic concept. Then, the elastic deformation gradient Fe is calculated by excluding Fp from the deformation gradient F. Further, the stress is calculated by substituting Fe into the hyperelastic equation. The multiplicative decomposition based on the isoclinic concept leading to the constitutive formulation in the intermediate configuration holds in the general elastoplastic materials unlimited to the crystalline metals as can be known from the abovementioned facts.

8.1.4 Embedded base vectors in intermediate configuration The relations between the base vectors Gi and Gi in the reference configuration and gi and gi in the current configuration are given in Eqs. (3.8) and (3.11), that is, 8 ~ gÞ ~ g Þ; gi 5 F2T Gi ð5 G < gi 5 FGi ð 5 G (8.12) G : Gi 5 F21 gi ð 5 g G Þ; Gi 5 FT gi ð 5 G Þ ~

~

i

Analogously, the base vectors Gi and G in the intermediate configuration are given by replacing F to Fe or Fp in Eq. (3.11) as follows: 8 8     ~ ~ < Fp Gi 5 p G < Fp2T Gi 5 p G G G i Gi 5 (8.13)  ; G 5 : eT i  e  : e21  e F gi 5 G G F g 5 gG ~

~

The deformation gradient tensor and its elastic and the plastic parts are represented by the base vectors as follows: 8 F 5 gi  Gi ð 5 δi
j gi  Gj 5 Fi
j gi  Gj Þ; F21 5 Gi  gi > > > < i j j Fe 5 gi  G ð 5 δi
j gi  G 5 Fei
j gi  G Þ; Fe21 5 Gi  gi (8.14) > > > : Fp 5 G  Gi ð 5 δi G  Gj 5 Fpi G  Gj Þ; Fp21 5 G  Gi i i
j i
j i The metric tensors are expressed as

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

264

8. Multiplicative decomposition of deformation gradient tensor

8 i i T > < G  Gi  G 5 G  Gi ð 5 G Þ i i T G  Gi  G 5 G  Gi ð 5 G Þ > : g  gi  gi 5 gi  gi ð 5 gT Þ

(8.15)

Other than the multiplicative decomposition in Eq. (8.5), the multiplicative decomposition in the inverse order, that is, F 5 Fp Fe (Clifton, 1972; Nemat-Nasser, 1979, 2004; Davidson, 1995; Lubarda,1999, 2002; Sansour et al., 2006) and the additive decomposition F 5 Fp 1 Fe 2 I e p (Nemat-Nasser, 1979, 1982, 2004) leading to F_ 5 F_ 1 F_ (Clifton, 1972; Labara, 1999, 2002) have been proposed. However, they have been disused because of their respective irrationalities.

8.2 Deformation tensors The multiplicative hyperelastic-based plasticity can be formulated essentially in the intermediate configuration as will be explained in Sections 9.3B9.6. However, various deformation (rate) measures in the current, the intermediate, and the reference configurations appear in the background of the formulation. They are explained in this section.

8.2.1 Elastic and plastic right Cauchy 2 Green deformation tensor Let the following elastic and plastic Cauchy 2 Green deformation tensors be defined noting Eqs. (3.24), (3.29), (4.1), and (4.5). 9 8   > > = < FeT gFe 5 e g GG 5 g Gi  Gj e i e j ij eT e 5 Cij G  G (8.16) C F F 5   > > GG i j ; : pT p21 ~ 5 pC 5 Cij G  G F CF 8 9 < Fe21 gFe2T  5 e g  5 gij G  G =   21 e21ij e21 i j GG 5C C  FeT Fe 5 Gi  Gj : p 21 pT  p ~21 ; F C F 5 C GG Þ5 C21ij Gi  Gj ~

~

(8.17)  ’  p Cp  FpT Fp 5FpT GFp 5 p GGG 5Gij Gi  Gj 5Cij Gi  Gj

(8.18)

 21  ’  ij Cp1  FpT Fp 5 Fp21 GFp2T 5 p GGG 5 G Gi  Gj 5 Cp21ij Gi  Gj (8.19) because of

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

8.2 Deformation tensors

(

265

e

C 5 FeT Fe 5 ðFFp21 ÞT FFp21 5 Fp2T FT FFp21 5 Fp2T CFp21 e21

C

5 ðFeT Fe Þ21 5 ðFp2T CFp21 Þ21 5 Fp C21 FpT

(8.20)

The following relations of the components hold from Eqs. (8.16) and (8.17). e

Cij 5 gij 5 Cij ; C

e21ij

5 gij 5 C21ij

(8.21)

8.2.2 Strain rate and spin tensors The velocity gradient tensors in the current and the intermediate configurations are introduced in this subsection. They are additively decomposed into the elastic and the plastic parts. Here, the decomposition into the purely elastic and the purely plastic parts is not realized in the current configuration but it can be realized in the intermediate configuration. Therefore the velocity gradient tensor in the intermediate configuration is adopted in the multiplicative hyperelastic-based plastic constitutive equation, which is derived by the elastic contravariant-covariant pull-back of the velocity gradient tensor in the current configuration. The rates of elastic and the plastic deformations and rotations are defined by the additive decomposition into the symmetric and the antisymmetric parts, respectively, of the elastic and the plastic parts of the velocity gradient tensor in the intermediate configuration. 8.2.2.1 Strain rate and spin tensors in current configuration Substituting Eq. (8.5) into Eq. (1.362), the velocity gradient is additively decomposed into the elastic and the plastic parts based in the current configuration, as follows:

where

noting

l 5 le 1 lp

(8.22)

8 _ 21 l  FF > > > < le  F_ e Fe21 ; > lp  Fe F_ p Fp21 Fe21 5 Fe Lp Fe21 > > : p p L  F_ Fp21

(8.23)

  _ 21 5 ðFe Fp Þ
ðFe Fp Þ21 5 F_ e Fp 1 Fe F_ p Fp21 Fe21 FF 5 F_ Fe21 1 Fe F_ Fp21 Fe21 e

p

Incidentally, we have the expressions in terms of the base vectors: Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

266

8. Multiplicative decomposition of deformation gradient tensor

8 l 5 g_ i  gi >   > > _ g  gj > < le 5 g_ i  gi 2 Gi G j i i p _ > L 5Gi  G  > > > _ g  gj : lp 5 Gi G







j

i

noting _ 21 5 ðg  Gi Þ
Gj  gj 5 g_  Gi Gj  gj 5 g_  δi gj 5 g_  gi FF j i i i i    i e e21 i
i _ G  gj F_ F 5 gi  G Gj  gj 5 g_ i  G 1 gi  G j  i   i  _ G g  gj 5 g_  gi 2 G G _ g  gj 5 g_ i  gi 1 G j j i i i     p _  Gi G  Gj 5 G _  Gi F_ Fp21 5 G i j i  p       i _  i r  _ j Fe F_ Fp21 Fe21 5 gi  G Gr  G Gj  gj 5 G G j gi  g










  i _ 5 0. _ i G 1 Gi G with ðδij Þ
5 G Gj
5 G j j p Here, L in Eq. (8.23) is apparently referred to as the plastic velocity gradient tensor based in the intermediate configuration because of










ðdXÞ
5 L dX p

(8.24)

p p noting ðdXÞ
5 ðFp dXÞ
5 F_ dX 5 F_ Fp21 dX with Eq. (8.6). The strain rates and spins and their elastic and the plastic parts in the current configuration are defined based on Eq. (8.22) with Eq. (8.23) as  d 5 de 1 dp (8.25) w 5 we 1 wp (  21    _ 21 RT _ 5 Rsym UU d 5 sym½l 5 sym FF (8.26)  21    _ T 1 Rant UU _ _ 21 RT 5 RR w 5 ant½l 5 ant FF

8 h e i h e i   > _ Ue21 ReT < de 5 sym le 5 sym F_ Fe21 5 Re sym U h i h e i   > _ Ue21 ReT _ e ReT 1 Re ant U : we 5 ant le 5 ant F_ e Fe21 5 R 8 h i   p > < dp 5 sym lp 5 sym Fe L Fe21 h i   > : wp 5 ant lp 5 ant Fe Lp Fe21

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(8.27)

(8.28)

267

8.2 Deformation tensors

8.2.2.2 Strain rate and spin tensors in intermediate configuration The following velocity gradient tensor based on the contravariant 2 covariant pull-back of the velocity gradient tensor l by Fe to the intermediate configuration is additively decomposed into the purely elastic and plastic parts, noting Eq. (8.22) with Eq. (8.23). e

L5L 1L

p

(8.29)

where 8   e   j > > > L  Fe21 lFe 5 l G_ G 5 gi g_ j Gi  G > > > <    e e  j i e e _ e21 e e i _ G 5 Fe21 F _ g L  F l F l 5 g 5 _ j Gi  G 2 Gi  G > G > > > >   > p p _  Gi : L  Fe21 lp Fe 5 e l p G_G 5 F_ Fp21 5 G i ~




~




(8.30)

~

noting Eq. (3.14) and

    j j Fe21 lFe 5 Gi  gi g_ r  gr gj  G 5 gi g_ j Gi  G n  r  o _ g  gs g  Gj Fe21 le Fe 5 Gi  gi g_ r  gr 2 G G s r j  r   i  j j _ 5 g g_ r Gi  G 2 G Gs Gi  G    i  j _ G  Gj 5 gi g_ j Gi  G 2 G G j i   j _  Gi 5 gi g_ j Gi  G 2 G i











e




p

Therefore L and L ; L can be pertinently adopted in the formulation of elastoplastic constitutive equation. Let them be decomposed additively into the symmetric and the antisymmetric parts, that is, ( L5D1W (8.31) e e e p p p L 5D 1W ; L 5D 1W with e

p

e

D5D 1D ;W5W 1W where

(

e

e

p

(8.32)

p

p

D  sym ½L; D  sym ½L ; D  sym ½L  e

e

p

p

W  ant ½L; W  ant ½L ; W  ant ½L 

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(8.33)

268

8. Multiplicative decomposition of deformation gradient tensor

Noting Eq. (2.31), it follows from Eq. (8.30) that 8 h  i 1 h    j i j j > i i i _ > _ _ g g g g D 5sym g G  G G  G 1 g G  G 5 > i i i j j j > 2 > > > > h   i >   1 > i j > > gr g_ j Gir 1 gr g_ I Gjr G  G 5 > > 2 > > > >    j i < e   i 1 h i j j p g g_ j Gi  G 1 gi g_ j G  Gi 2 Gi Gj
G  G D 5D 2D 5 2 > > > h   i >   > 1 > _ Gi  Gj > gr g_ j Gir 1 gr g_ i Gjr 2 G 5 > ij > 2 > > > > h i > > p _ Gi  Gj _  Gi 5 1 G G 
Gi  Gj 5 1 G > > D 5 sym G i i j ij > : 2 2


































(8.34) because of




j

2sym½ðgi g_ j ÞGi  G  j

j r j j r 5 ðgi
g_ j ÞðGi
Gr ÞG  G 1 ðgi
g_ j ÞG  ðGi
Gr ÞG i j i j 5 ðgr
g_ j ÞðGr
Gi ÞG  G 1 ðgr
g_ i ÞG  ðGr
Gj ÞG

5 ðgi g_ j ÞGi  G 1 ðgi g_ j ÞG  Gi

_ G 2sym½G i _  Gi 1 Gj  G _ 5 ðG _ G ÞGr  Gi 1 Gi  ðG _ G ÞGr 5G i i i r i r i j i j _ G ÞG  G 1 G  ðG _ G ÞG 5 ðG G Þ
Gi  Gj 5 ðG i

j







i

i




j

i







j

Incidentally, it follows from Eqs. (2.32) and (8.30)3 that h i p _ 2G _  Gi 5 1 ðG G _ G ÞGi  Gj W 5 ant G i i j i j 2







(8.35)

p

The plastic strain rate D is related to the rate of plastic right CauchyGreen deformation tensor as follows: ’

_ p 5 2FpT Dp Fp 5 2p D pGG C

(8.36)

noting ðFpT Fp Þ
5 F_ Fp 1 FpT F_ 5 FpT ðFp2T F_ pT

p

pT

1 F_ Fp21 ÞFp p

e

The rate of C is given from Eqs. (8.16) and (8.29) with Eq. (8.30)2 as

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

8.3 On limitation of hypoelastic-based plasticity

269

h i h  i _ e 5 2sym Ce Le 5 2sym Ce L 2 Lp C

(8.37)

noting

      _ e 5 FeT Fe
5 FeT F_ e 1 F_ eT Fe 5 FeT Fe Fe21 F_ e 1 F_ eT Fe2T FeT Fe C e e

eT

e

5C L 1L C

8.2.2.3 Substructure spin The substructure of material is its skeleton. The plastic deformation is induced by the mutual slips between material particles along the substructure independently of the rotation of substructure. On the other hand, the elastic distortion induces a rotation of substructure as has been suggested by Mandel (1971), Kratochvil (1971), Dafalias (1983, 1984, 1985), Loret (1983), etc. Eventually, the rotation of substructure is independent of the plastic deformation but induced by the elastic distortion and the rigid-body rotation, both of which are included in the elastic deformation s gradient Fe . Then, the spin of substructure W is given by the elastic spin, that is, s

e

W 5W 5W2W

p

(8.38)

The continuum spin W is known from the outside appearance of s material. In contrast, the spin of substructure W consisting of the elastic distortion spin and the rigid-body rotation is unknown from the outside appearance of material as far as the plastic spin is unknown, which depends on the constitutive property due to the substructure of material. Thus it is obliged to formulate the plastic spin as the constitutive equation. Then, the substructure spin can be calculated by subtracting the plastic spin from the continuum spin.

8.3 On limitation of hypoelastic-based plasticity The plastic strain rate and the plastic spin in the intermediate configuration depend only on the plastic deformation gradient as shown in Eqs. (8.30) and (8.33). Then, let their flow rules be assumed as follows: 8   h p i p > _ p ONp O 5 1 < D 5 sym F_ Fp21 5 λN   h p i λ_ . 0 (8.39) p > _ : wp 5 ant F_ Fp21 5 λΩ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

270

8. Multiplicative decomposition of deformation gradient tensor

where λ_ is the positive plastic multiplier, and N and Ω are the functions of stress and internal variables in the intermediate configuration, respectively. Now, we assume that the elastic deformation is infinitesimal so that p

p

Ue DI; Ve DI

8 e e F DR > h e i h ei > < _ Ue21 Dsym U _ 5U _e sym U h e i h ei > > : _ Ue21 Dant U _ 5O ant U

(8.40)

(8.41)

The elastic and the plastic strain rates and spins in Eqs. (8.27) and (8.28) in the current configuration are reduced for Eq. (8.41) as h ei 8 e e _ ReT 5 Re U _ e ReT > DR sym U d > > > > h ei > > > _ e ReT 1 Re ant U _ ReT 5 R _ e ReT < w e DR h i (8.42) p > p e eT e p eT > d DR sym L 5 R D R R > > > > h i > p > p : w DRe ant L ReT 5 Re wp ReT The plastic flow rules for Eq. (8.42) are given from Eq. (8.39) as follows: ( p _ p ðOnp O 5 1Þ d 5 λn (8.43) _ p wp 5 λω where

(

p

np 5 Re N ReT p

ωp 5 Re Ω ReT

(8.44)

On the other hand, the elastic strain rate in the current configuration 3 is given by de 5 E21 :σ in Eq. (7.68) as the hypoelasticity. Consequently, the following hypoelastic-based plastic constitutive equation is given from Eq. (8.42)2 and (8.43) as follows:  3 _ p d 5 de 1 dp 5 E21 :σ 1 λn (8.45) e eT e p _ p _ R 1 λω w5w 1w 5R As shown above, the hypoelastic-based plastic constitutive equation can hold under the infinitesimal elastic deformation. In addition, the cumbersome time-integrations of cororational rates are required for the numerical calculation of the stress and internal variales.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

8.4 Multiplicative decomposition for kinematic hardening

271

8.4 Multiplicative decomposition for kinematic hardening The plastic strain is decomposed into the storage and the dissipative parts for the kinematic hardening as was shown in Eq. (7.31) for the infinitesimal hyperelastic plasticity in Section 7.2.1. Likewise, the plastic p deformation gradient Fp is decomposed into the plastic storage part Fks p causing the kinematic hardening and its plastic dissipative part Fkd multiplicatively after Lion (2000) as follows (see Fig. 8.4): p

p

F 5 Fe FP ; Fp 5 Fks Fkd

(8.46)

Based on the right Cauchy 2 Green deformation tensor C  FT F, the e _p following tensors of the storage parts C ; C ks and the dissipative parts _ p Cp ; C kd are defined. 8     > < Ce  FeT Fe 5 e g G G 5 Re Ue T Re Ue 5 Ue2 (8.47) ’ ’__ > _ p2 pT p  p GG  p pT p  p _ GG  : _p 5 U ks ; Ckd  Fkd Fkd 5 kd G C ks  Fks Fks 5 ks G ~

where one has p

pG G p _~

Cks  ks C ks

p2T _ p

p21

5 Fks C ks Fks 5 G

(8.48)

FIGURE 8.4 Multiplicative decomposition of deformation gradient for nonlinear kinematic hardening.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

272

8. Multiplicative decomposition of deformation gradient tensor

where G is the metric tensor in the intermediate configuration. The hat symbols (_) are added to the variables based in the kinematic hardening _ intermediate configuration K. p Further, the plastic velocity gradient L is additively decomposed for the kinematic hardening as follows: p

(

p

p

L 5 Lks 1 Lkd p

p

p

p

(8.49)

p

D 5 Dks 1 Dkd

(8.50)

p

W 5 Wks 1 Wkd where

8 p p p p < Lks  F_ ks Fp21 ks 5 Dks 1 wks   p p p21 p _ ~
G 5 Dp 1 wp : Lp  Fp _ L F 5 L kd kd kd kd ks ks ks kd G 8 h i h i p p p p > < Dks  sym Lks ; Wks  ant Lks h i h i > : Dpkd  sym Lpkd ; Wpkd  ant Lpkd _p L kd

p p21 5 F_ kd Fkd



(8.51)

(8.52)

_




’p G  p21 p p  p Fks Lkd Fks 5 ks L kdG_

(8.53)

noting  p p   p p 21 p p21 p p p p21 p21 L 5 Fks Fkd
Fks Fkd 5 F_ ks Fks 1 Fks F_ kd Fkd Fks _p

The time-derivative of C ks in Eq. (8.47) is given by __

__ p C ks

 ’ pGG  p pT p p  p pT 5 2Fks Dks Fks 5 2ks D ks 5 2Fks D

noting __ p C ks

 p p 2 Dkd Fks

  pT p pT p
pT p 5 Fks Fks 5 Fks F_ ks 1 F_ ks Fks pT p

p

pT pT p

pT

p

p

5 Fks Lks Fks 1 Fks Lks Fks 5 2Fks Dks Fks

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(8.54)

C H A P T E R

9 Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations The multiplicative decomposition of the deformation gradient, the strain rate and the spin in the intermediate configuration were explained in the preceding chapter. The multiplicative hyperelastic plastic constitutive equations based on them are explained comprehensively in this chapter. The constitutive equation within the framework of the conventional elastoplasticity assuming a yield surface enclosing the purely elastic domain is first shown, which is limited to the description of the monotonic loading behavior. Then, incorporating the subloading surface model, the extended multiplicative hyperelastic
based plastic constitutive equation for the description of the monotonic and the cyclic loading behaviors are formulated, while only the subloading surface model would be able to be extended to the multiplicative hyperelasticity among various plasticity models. The rigorous flow rules of plastic strain rate and the plastic spin are introduced in them. Furthermore, the extension to the rate-dependence is formulated as the subloading-multiplicative overstress model by incorporating the concept of overstress. Incidentally, the constitutive equations of metals and soils are given based on these formulations.

9.1 Stress measures Four types of stress tensors are introduced by pulling-back the Cauchy stress tensor in the current configuration. Now, introduce the second Piola
Kirchhoff stress tensor S in the intermediate configuration, which is the elastic contravariant pull-back of the Kirchhoff stress Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity DOI: https://doi.org/10.1016/B978-0-12-819428-7.00009-2

273

© 2020 Elsevier Inc. All rights reserved.

274

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

tensor τ defined through multiplying the Cauchy stress by the elastic volume ratio, i.e., 8

   < T S  Fp SFpT 5 p~ S G G 5 Fe21 FSFT Fe2T  Fe21 τFe2T 5 e τ G G 5 S   : τ 5 Jσ 5 FSFT 5 ~ S gg J  dv=dV ~

(9.1) and the Mandel stress (Mandel, 1973b), which is the redefinition of the Mandel stress M in Eq. (5.76) into the intermediate configuration, ~

noting

 e G  T e eT e2T 5 τ  G ð6¼ M Þ M  C S 5 F τF

(9.2)

   e C S 5 FeT Fe Fe21 τFe2T 5 FeT τFe2T

(9.3)

The following work-conjugacies hold for the Mandel stress tensor in the intermediate configuration. 8

< τ : l 5 tr½ðFe2T MFeT ÞðFe LFe21 Þ  5 trðFe2T MFeT Fe2T L FeT Þ 5 trðM L Þ e eT eT τ : le 5 tr½ðFe2T MFeT ÞðFe L Fe21 ÞT  5 trðFe2T MFeT Fe2T L FeT Þ 5 trðM L Þ > : p pT pT τ : lp 5 tr½ðFe2T MFeT ÞðFe L Fe21 ÞT  5 trðFe2T MFeT Fe2T L FeT Þ 5 trðM L Þ by virtue of Eq. (8.30) with Eq. (9.2). On the other hand, Simo (1998) and Belytschko et al. (2014) adopted e p p S and D  sym½C L  instead of M and L as the work conjugate pair, noting T

e

e

T

e

e

e

M : L 5 trðM LÞ 5 tr½ðC SÞT L 5 trðSC LÞ 5 trðS C LÞÞ 5 S : C L ð 5 S : sym½C LÞ p

Then, the associated flow rule is applied to D by Simo (1998) but it is p irigorous because D is influenced by the elastic deformation. On the p other hand, the associated flow is applied to L by Belytschko et al. (2014) but the plastic spin cannot be formulated rigorously in the formulation. Consequently, it is inevitable to incorporate the Mandel stress M for the rigorous formulation of the multiplicative hyperelastic-based p plasticity. The associated flow rule should be applied to D as will be shown later.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

275

9.2 Hyperelastic constitutive equations

Further, the contravariant push-forward of the second Piola
Kirchhoff _ _ stress-like variable for the kinematic hardening variable S k from K to K is given by ~

_ _T T p _ pT p _ p21 p2T p Sk  Fks Sk Fks ð5ks ~ SkG G Þð 5 Sk Þ; S k  Fks Sk Fks ð5ks SkG_G_ Þ ð 5 Sk Þ

(9.5)

Further, the Mandel-like variable Mk for the kinematic hardening variable is given by p

T

Mk 5 Cks Sk 5 G Sk ; that is; Mk 5 Sk ð5 Mk Þ (9.6) noting Eq. (8.48). Note here that the Mandel stress M is the asymmetric tensor in general but the Mandel-like kinematic hardening variable Mk is the symmetric tensor. The material-time derivative of the kinematic hardening variable Sk in the intermediate configuration is given by

_

_

p _

p

pT

Sk 5 Fks S k Fks 1 2sym½Lks Sk 

(9.7)

from Eqs. (8.51)1 and (9.5), noting

_

_

_p _ pT p _ _pT pT p2T pT p p21 p2T p _ _pksFp21 _pT 5 Fks _ S k Fks 1 F ks Sk Fks Fks 1 Fks Fks Sk Fks Fks p _

pT

Sk 5 Fks S k Fks 1 Fks S k Fks 1 Fks S k Fks

p p2T _pT _ pT _ p21 p _ FpT 1 Lp S 1 ðLp S ÞT 5 Fks _ S k ks ks k ks k p _ 5 Fks S k Fks 1 Fks Fks Sk 1 Sk Fks Fks

Further, the material-time derivative of Mk is given from Eq. (9.7) with Eqs. (8.49) and (9.6) by

_

_

_

p _

pT

p

_

p _

pT

p

Mk 5 Sk 5 Fks S k Fks 1 2sym½Lks Mk  5 Fks S k Fks 1 2sym½ðL 2 Lkd ÞMk 

(9.8)

9.2 Hyperelastic constitutive equations The second Piola
Kirchhoff stress push-forwarded to the intermediate configuration, S, is given by incorporating the strain energy function e e e2 ψðC Þ, where C ð 5 U Þ stands for the purely elastic deformation, as e

S52

@ψe ðC Þ e

@C

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.9)

276

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

and the Mandel stress is given as e

M 5 2C

e

@ψe ðC Þ

(9.10)

e

@C

T

The tensor M satisfies the symmetry M 5 M for the particular case that e ψ is the function of invariants of only C leading to the elastic isotropy. The material-time derivative of the Mandel stress is given noting Eq. (8.37) as e

_

e

e

p

M 5 2L : sym½C ðL 2 L Þ

(9.11)

noting e

@ðC SÞ

e

ðC SÞ 5

@C

e

e e
e e  @S  @ψe ðC Þ  : C 5 I ~ S 1 Ce e : C 5 IS ~ 1 2Ce e e : C @C @C  @C

_e

_

_

e

e

_

p

e 5 L : C 5 2L : sym½C ðL 2 L Þ e

where L is the fourth-order hyperelastic tangent stiffness modulus tensor given by e

L 

@M e

@C

~ 1 5 IS

1 e e C C 2

(9.12)

with e

C 2 _

e

@S e

@C

54

@2 ψe ðC Þ e

@C  @C

(9.13)

e

Further, let S k be formulated incorporating the strain energy function as

_p ψk ðCks Þ

_

_p

Sk 5 2

@ψk ð C ks Þ

(9.14)

_p

@C ks

from which Sk and Mk are given by p _ pT Sk 5 Fks S k Fks

p _

_p

p 5 2Fks

@ψk ðC ks Þ _p @C ks

pT

(9.15)

Fks

_p

pT

p

Mk 5 Sk 5 Fks S k Fks 5 2Fks

@ψk ð C ks Þ _p

@C ks

pT

Fks

noting Eqs. (8.48), (9.5), and (9.6).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.16)

277

9.3 Conventional elastoplastic model _p

e

The strain energy functions ψðC Þ and ψk ðCks Þ are given by St. Venant
Kirchhoff equation, the neo-Hookean equation and their various modifications for metals (cf. Section 6.2 and Hashiguchi and Yamakawa, 2012). _ The material-time derivative of S k is given from Eq. (9.14) as

__

_k

Sk 5C :

_

1 _p C 2 ks

(9.17)

where _k

_

C 2

@Sk _p

@C ks

_p

54

@2 ψk ð C ks Þ _p

(9.18)

_p

@C ks  @C ks

_

Substituting Eq. (9.17) with Eq. (8.54) into Eq. (9.8), Mk is given as follows:

_

p _k

pT

p

p

p

pT

p

p

Mk 5 FksC : Fks ðD 2 Dkd ÞFks Fks 1 2sym½ðL 2 Lkd ÞMk 

(9.19)

9.3 Conventional elastoplastic model The multiplicative hyperelastic-based plastic constitutive equation (Hashiguchi, 2017b) will be shown within the framework of the conventional elastoplasticity assuming the yield surface enclosing the purely elastic domain (Drucker, 1988) in this section.

9.3.1 Flow rules for plastic strain rate and plastic spin The yield surface with the isotropic and the kinematic hardenings is described by the following equation (see Fig. 9.1). ^ 5 FðH Þ fðMÞ

(9.20)

in the intermediate configuration, where H is the isotropic hardening variable and ^ TÞ ^  M 2 M ð6¼ M M k

(9.21)

^ be chosen to be homogeneously degreeHere, let the function fðMÞ ^ one of M.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

278

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

ˆ) ˆ (≠ N N

M ˆ M

Mk

0

M ij

Yield surface ˆ ) = F (H ) f (M

FIGURE 9.1 Conventional yield surface in the intermediate configuration.

Let the plastic strain rate be given by the symmetrized associated flow rule (Hashiguchi, 2017b, 2018c):

_ _

p ^ ðλ^ $ 0Þ D 5 λ^ N

_

(9.22)

^ is the normalized and where λ^ is the positive plastic multiplier and N symmetrized outward-normal tensor of the yield surface, that is,   ^  ^  @fðMÞ @fðMÞ ^ ^ 5 1Þ N  sym (9.23) =jjsym jjðjjNjj @M @M The elastic isotropy caused by the isotropic strain energy function ψe of e T only C leading to the symmetry M 5 M holds as mentioned as to Eq. (9.10). The plastic dissipative part of the plastic strain rate for the kinematic hardening variable in Eq. (8.52) is given referring to the second term in Eq. (7.40) or (7.73) as follows (Hashiguchi, 2016a, 2017b, 2018c): p

Dkd 5

_

1 1 ^ p jjD jjMk 5 λMk bk F bk F

p

(9.24)

Let the plastic spin W in Eq. (8.33) and the kinematic hardening spin in Eq. (8.52), which are induced by the plastic strain rate and the dissipative part of kinematic hardening rate, respectively, be given extending the equation in the hypoelastic-based plasticity by Zbib and Aifantis (1988) as follows (Hashiguchi, 2016a, 2017b, 2018c): p Wkd

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

9.3 Conventional elastoplastic model

_

p p ^ ^ W 5 ηp ant½M D  5 ηp λant½M N

_

^ Wkd 5 ηk ant½M Dkd  5 ½ηk =ðbk FÞλant½M Mk  p

p

p

p

279

(9.25)

p

where ηp and ηk are the material parameters, while the flow rules in Eqs. (9.22) and (9.24) are exploited. The elastic isotropy caused by the elase T tic strain energy function ψe of only C leads to the symmetry M 5 M holds as mentioned as to Eq. (9.10), and further the isotropic yield function of only symmetric tensor M due to the plastic isotropy results in no p plastic spin W 5 O. Likewise, the plastic isotropy due to Mk 5 O in addiT tion to M 5 M caused by the elastic isotropy results in no kinematic spin p p p Wkd 5 O. The plastic spin equation W 5 ηp ant½M L  was adopted by Wallin et al. (2003) and Wallin and Ristinmaa (2005), which results in the p p p peculiar relation W 5 ηp ðant½M D  1 ant½M W Þ. Incidentally, the neglect of plastic spin tensor is called the spinless (Dafalias, 1998), the inelastically spin-free (Gurtin and Anand, 2005), and the irrotational (Henann and Anand, 2009). The plastic velocity gradients are given by substituting Eqs. (9.22), (9.24), and (9.25) into Eqs. (8.31) and (8.51) as follows:

_ _

p ^ ^ 1 ηp ant½M NÞ ^ N L 5 λð p ^ k 1 ηp ant½M Mk Þ=ðbk FÞ L 5 λðM kd

(9.26)

k

The substitutions of Eqs. (9.22), (9.24) and (9.26) into Eqs. (9.11) and (9.19) yield:

_

_

e e ^ ^ 1 ηp ant½M NÞg ^ N M 5 2L : sym½C fL 2 λð

 p _k pT ^ p pT Mk 5 λ^ Fks C : Fks ðN 2 ð1=bk ÞMk ÞFks Fks

_

_

^ 1 ηp ant½MN ^ 2 ðMk 1 ηp ant½M Mk Þ=ðbk FÞgMk  1 2sym½fN k

(9.27)

 (9.28)

The following various plastic flow rules other than Eq. (9.22) with Eq. (9.26) have been postulated so far. p

_

D 5 λ@f=@M

(9.29)

adopted by Lubarda and Lee (1981), Miehe (1994), Svendsen et al. (1998), Eidel and Gruttmann (2003), and Han et al. (2003) is limited to e the elastic and plastic isotropy with the strain energy function ψe ðC Þ of e T C only leading to the symmetry M 5 M .

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

280

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

_

p

D 5 λ@f=@M=jj@f=@Mjj

(9.30)

adopted by Anand et al. (2012) is also limited to the elastic and plastic isotropy.

_

p ^ ^ D 5 λ@f=@ M

(9.31)

adopted by Wallin et al. (2003), Wallin and Ristinmaa (2005), Dettmer and Reese (2004), Gurtin and Anand (2005), and Vladimirov et al. (2008, 2010) is limited to the elastic isotropy.

_

p ^ ^ ^ D 5 λ@f=@ M=jj@f=@ Mjj

(9.32)

adopted by Henann and Anand (2009) and Iguchi et al. (2017a) is limited to the elastic isotropy.

_

p

D 5 λsym½@f=@M

(9.33)

adopted by Svendsen (2001) is limited to the plastic isotropy.

_

p ^ ^ D 5 λsym½@f=@ M

(9.34)

adopted by Harrysson and Ristinmaa (2007), Hashiguchi and Yamakawa (2012), and Brepols et al. (2014) is irrelevant to the plastic spin.

_

p ^ ^ ^ D 5 λsym½@f=@ M=jj@f=@ Mjj

(9.35)

adopted by Tsakmakis (2004) and Hausler et al. (2004) is also irrelevant to the plastic spin. p

_

L 5 λ@f=@M

(9.36)

adopted by Lubliner (1986), Svendsen (2001), Eidel and Gruttmann (2003), and Balieu and Kringos (2015) is limited to the plastic isotropy. p

_

D 5 λ@f=@S

(9.37)

is adopted by Aravas (1994), Haupt (1985, 2002), and Gurtin and Anand p (2005), while the variables S and D are not work-conjugate pair. p

_

L 5 λ@f=@S

(9.38) p

is adopted by Belytschko et al. (2014), while the variables S and L are not work-conjugate pair.

_p _

Lp  Fp21 F 5 λ@f=@S

(9.39)

in the reference configuration is adopted by Sansour et al. (2006, 2007), while the variables S and Lp are not work-conjugate pair.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

9.3 Conventional elastoplastic model

_p _

p ^ M=jj ^ MjjÞC ^ C 5 λð

281 (9.40)

in the reference configuration adopted by Vladimirov et al. (2008, 2010) and Brepols et al. (2014) is limited to the Mises material and irrelevant to the plastic spin. p p The plastic spin is assumed to be W 5 R Rp (constitutive spin: p ^ ^ ^ M=jj@f=@ Mjj by Dafalias, 1998) by Han et al. (2003), W 5 λant½@f=@ h i p p p Tsakmakis (2004), W 5 η ant ML by Wallin et al. (2003) and Wallin and

_ _

p

_

Ristinmaa (2005) and W 5 λant½@f=@M by Svendsen (2001) as was described just after Eq. (9.25), which would involve each problem. All the above-mentioned flow rules are limited to the description of the monotonic loading behavior. The plastic flow rule of the plastic strain rate in Eq. (9.22) with Eq. (9.25) as the modification of Eqs. (9.34) and (9.35) and that of the plastic spin in Eq. (9.25) are rigorous possessing the generality for the monotonic/cyclic loading.

9.3.2 Confirmation for uniqueness of multiplicative decomposition The issue on the nonuniqueness of the multiplicative decomposition was p delineated in Section 8.1.3. However, the velocity gradient tensor L in the intermediate configuration is determined uniquely in Eq. (9.26). Then, the plastic deformation gradient tensor Fp is also updated uniquely and further the elastic deformation gradient tensor Fe is determined uniquely by Fe 5 FFp21 . The calculation procedure will be described in detail in Section 9.8. The uniqueness of the multiplicative decomposition is the distinctive advantage caused by the formulation in the intermediate configuration based on the isoclinic concept insisting the independence of the plastic deformation gradient tensor from the rigid-body rotation.

9.3.3 Plastic strain rate The time-differentiation of Eq. (9.20) leads to the consistency condition as follows: ^ @fðMÞ : ð M 2 Mk Þ 2 F 5 0 @M

_ _

_

(9.41)

Here, it holds from Eq. (9.20) that ^ @fðMÞ ^ 5F ^ 5 fðMÞ : M @M

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.42)

282

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

^ of M ^ in by the Euler’s theorem for the homogenous function fðMÞ degree-one, and then it follows that    ^ ^ ^ ^ @fðMÞ ^ @fðMÞ 5 fðMÞ ^ ^ 5 @fðMÞ : M ^ :M 5 F @fðMÞ N ^ ^ ^ ^ @M @M @M @M which leads to  ^ N ^ ^ @fðMÞ 5 : M 1 ^ F @M

(9.43)

^  @fðMÞ ^ @fðMÞ ^ ^ T ; jjNjj ^ 5 1Þ ð6¼ N N ^ ^ @M @M

(9.44)

where

The substitution of Eq. (9.43) into Eq. (9.41) leads to

_

_ _

^ 50 ^ : ðM 2 M Þ 2 F N ^ :M N k F

(9.45)

resulting in

_

^ : ^ :M2N N



 F0 H ^ M 1 Mk 5 0 F

_

_

(9.46)

0

where F  dF=dH and

_H 5 fHd ðM; H; Dp

. ^ λ^ Dp Þ Dp 5 fHn^ ðM; H; NÞ

_

_

(9.47) p

notingpEq. ffiffiffiffiffiffiffiffi (9.22) and the homogeneity of H in degree-one of D , while fHn^ 5 2=3 holds for the equivalent plastic strain hardening. The substitutions of Eqs. (9.27), (9.28), and (9.47) into Eq. (9.46) lead to

_

_

^ : M 2 Mp λ^ 5 0 N

(9.48)

from which it follows that

_

_

^ :M ^ :M N N p ^ ; D 5 N λ^ 5 Mp Mp

_

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.49)

283

9.4 Continuity and smoothness conditions

where ^
0 _k pT ^ p p pT ^ : F fHn^ ðM; F; NÞ M ^ 1 Fp C Mp  N : Fks ðN 2 ½ηk =ðbk FÞMk ÞFks Fks ks F 

^ ^ 2 fM 1 ηp ant M M g=ðb FÞÞM  1 2sym ðN 1 ηp ant½M N k k k k k (9.50) The substitution of Eq. (9.27) into Eq. (9.48) leads to the consistency condition: ^ 1 ηp ant½MNÞ ^ 1 Mp gΛ ^ : Le : sym Ce L 2 fN ^ : Le : sym½Ce ðN ^ 50 (9.51) N

_

_

^ for the plastic multiplier in terms of the strain rate using the symbol Λ instead of λ^ in terms of the stress rate. The plastic multiplier is expressed from Eq. (9.51) as follows:

_

_

^5 Λ

^ : Le : sym½Ce L N ^ 1 ηp ant½MNÞ ^ ^ : Le : sym½Ce ðN Mp 1 N

(9.52)

The loading criterion is given as follows (Hashiguchi, 2000, 2017a): ( p ^ 5 FðHÞ and Λ ^ .0 D 6¼ O for fðMÞ (9.53) p D 5 O for others

_

which can be given actually as ( p ^ 5 FðHÞ and N ^ : Le : sym½Ce L . 0 D 6¼ O for fðMÞ p D 5 O for others

(9.54)

The stress rate versus strain rate relation is not used in the hyperelastic-based plasticity because the stress is calculated by substituting the elastic deformation gradient into the hyperelastic equation. Here, the elastic deformation gradient is calculated by excluding the plastic deformation gradient from the deformation gradient, while the plastic deformation gradient is updated by the plastic velocity gradient in Eq. (9.26) after the calculation of the plastic multiplier. The detailed calculation procedure will be described in Section 9.8.

9.4 Continuity and smoothness conditions The continuity condition expressed in the current configuration in Eq. (7.2) is extended for the intermediate configuration as follows:

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

284

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

_

_

lim MðM; H; H; D 1 δDÞ-MðM; H; H; DÞ

δD-O

(9.55)

where H designates collectively the tensor-valued internal variables based in the intermediate configuration. The smoothness condition expressed in the current configuration in Eq. (7.4) is extended for the intermediate configuration as follows:

_

_

lim MðM 1 δM; H; H; DÞ-MðM; H; H; DÞ

δM-O

(9.56)

The elastoplastic modulus tensor M

ep 

 @M M; H; H 5 Ð @ Ddt

(9.57)

in the constitutive equation fulfilling the smoothness condition satisfies lim Mep ðM 1 δM; H; HÞ-Mep ðM; H; HÞ

δM-O

(9.58)

Both of the continuity and the smoothness conditions are satisfied only in the subloading surface model, while the smoothness condition is violated in the other elastoplasticity models as described in Section 7.1.2.

9.5 Initial subloading surface model The initial subloading surface model formulated in the current configuration in Subsection 7.3 will be extended to the multiplicative hyperelastic-based plasticity in this section. The subloading surface is represented by the following equation (see Fig. 9.2). ^ 5 RFðHÞ fðMÞ

(9.59)

The evolution rule of normal-yield ratio is given based on Eq. (7.89) as follows:

_R 5 UðRÞjjDp jj for Dp 6¼ O

 e p 5 0 for D 5 O R for D 5 O e , 0 for D 6¼ O

_

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.60) (9.61)

285

9.5 Initial subloading surface model

R

1

ˆ) ˆ (≠ N N

ˆ M

M

ˆ ( ≠ Nˆ ) N

Mk

0

M ij

Sunloading surface ˆ ) R F (H ) f (M = Normal - yield surface ˆ ) = F (H ) f (M

FIGURE 9.2 Normal-yield and subloading surfaces.

The time-differentiation of the subloading surface in Eq. (9.59) reads  ^ @fðMÞ ^ 2 RF 2 RF 5 0 :M ^ @M

_

_

(9.62)

which can be described as

!  F R ^ ^ ^ ^ 1 N: M 2 N:M50 F R

_ _

(9.63)

where ^  ^ ^ 5 1Þ ^  @fðMÞ @fðMÞ ðjjNjj N @M @M

(9.64)

The following relation based on the Euler’s homogenous function is ^ in the derivation of Eq. (9.63) from Eq. (9.62), noting used for M Eq. (9.59). ^ @fðMÞ ^ :M ^ RF ^ ^ @M 5 ; N : M 5 ^ ^ @fðMÞ @fð MÞ @M^ ^ @M

^ ^ :M N 5 ^ RF @fðMÞ @M^ 1

(9.65)

The plastic modulus is given by the following equation instead of Eq. (9.50).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

286

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

2 ^ F0 fHn ðM; F; NÞ UðRÞ  ^ ^ 4 p 1 : M M N F R


p   p pT p _k pT ^ 1 FksC : Fks N 2 ηk =ðbk FÞ Mk Fks Fks h
 i o. ^ 1 ηp ant M N ^  2 M 1 ηp ant M M  ðb FÞ M 1 2sym N k k k k k

#

(9.66) The loading criterion is given as follows (Hashiguchi, 2000, 2017a): ( p ^ .0 D 6¼ O for Λ (9.67) p D 5 O for other

_

which can be given actually as ( p ^ e :sym Ce L . 0 D 6¼ O for N:L p D 5 O for other

(9.68)

in which the yield judgment is not required.

9.6 Multiplicative extended subloading surface model Multiplicative hyperelastic-based plastic constitutive equation will be given below based on the extended subloading surface model in the current configuration described in Section 7.5. On the other hand, the cyclic plasticity models other than the subloading surface model have not been hitherto and would not be able to be extended to the multiplicative hyperelastic-based plasticity.

9.6.1 Multiplicative decomposition of plastic deformation gradient for elastic-core The plastic strain is decomposed into the storage and the dissipative parts for the elastic-core in Eq. (7.147). Analogously to the multiplicative decomposition of the plastic deformation gradient for the kinematic p hardening in Eq. (8.46), decompose Fp into the plastic storage part Fcs causing the translation of the elastic-core and its plastic dissipative part p Fcd multiplicatively as follows (Hashiguchi, 2016a, 2018c): p

p

p

F 5 Fe Fp ; Fp 5 Fks Fkd ; Fp 5 Fpcs Fcd

(9.69)

The configurations based on these decompositions are illustrated in Fig. 9.3.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

9.6 Multiplicative extended subloading surface model

287

FIGURE 9.3 Multiplicative decompositions of deformation gradient tensor for material with translations of kinematic hardening variable and elastic-core.

 p and the dissipative part The following tensors of the storage part C cs are defined.

p C cd

 p  FpT Fp 5 U  p2 ; Cp  FpT Fp C cs cs cs cs cd cd cd

(9.70)

 p21 Ccs  Fp
T cs C cs Fcs 5 G

(9.71)

where one has p

p

Tensor variables in the elastic-core intermediate configuration are specified by adding the hat symbol ð Þ Further, the following additive decomposition of the velocity gradient holds for the elastic-core analogously to Eqs. (8.49)
(8.53) for the kinematic hardening variable. p

p

p

L 5 Lcs 1 Lcd where

8 < Lp  Fp Fp21 5 Dp 1 Wp ; cs cs cs cs cs : Lp  Fp L p Fp21  5 p L ~p G  5 Dp 1 Wp cs cd cs cs cdG cd cd cd ( p

p p

p Dcs  sym Lcs ; Wcs  ant Lcs

p p

p p Dcd  sym Lcd ; Wcd  ant Lcd

_

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.72)

(9.73)

(9.74)

288

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

’ p  p p p p21 p p  G L cd 5 Fcd Fcd  Fp21 cs Lcd Fcs 5 cs Lcd G

_

(9.75)

noting p p  p p   p 21 p21 p21 p L 5 Fpcs Fcd Fpcs Fcd 5 Fcs Fp21 cs 1 Fcs Fcd Fcd Fcs

_

_

 p in Eq. (9.70) is given by The material-time derivative of C ks p    5 2FpT Dp Fp 5 2FpT Dp 2 Dp Fp C cs cs cs cd cs cs cs

_

(9.76)

The contravariant push-forward of the second Piola
Kirchhoff  to K is given by stress-like variable for the elastic-core S c from K   T T p21 p2T  Sc  Fpcs S c FpT 5 S c (9.77) cs 5 Sc ; S c  Fcs Sc Fcs Further, the Mandel-like variable Mc for the elastic-core is given by  p T Mc 5 Ccs Sc 5 G Sc 5 Sc 5 Mc (9.78) noting Eq. (9.71). Note here that the Mandel stress M is the asymmetric tensor in general but the Mandel-like elastic-core Mc is the symmetric tensor. The material-time derivative of Mc is given analogously to Eq. (9.8) for that of the kinematic hardening as follows: 

 p  (9.79) Mc 5 Sc 5 Fpcs S c FpT cs 1 2sym L 2 Lcd Mc

_

_

_

Let S c be formulated as the hyperelastic relation by incorporating the  p Þ as strain energy function ψc ðC cs   c p S c 5 2 @ψ Cp cs (9.80)  @C cs from which Sc and Mc are given by 

 p Mc 5 Ccs Sc 5 Sc

5 Fpcs S c FpT cs

5 2Fpcs

 p  @ψc C cs FpT cs p @C

(9.81)

cs

noting Eqs. (9.71), (9.77), and (9.80). The material-time derivative of S c is given from Eq. (9.80) with Eq. (9.76) as

_

_

p  c : 1C  c : FpT ðDp 2 Dp ÞFp  5C S c 5 C cd cs cs 2 cs

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.82)

9.6 Multiplicative extended subloading surface model

where

 p  @2 ψ c C @S c c cs  C  2 p 54 p  p   @C @C @ C cs cs cs

289

(9.83)

_

Substituting Eq. (9.82) into Eq. (9.79), Mc is given as follows:

      c : FpT Dp 2 Dp Fp FpT 1 2sym Lp 2 Lp Mc Mc 5 Fpcs C cd cd cs cs cs

_

(9.84)

9.6.2 Normal-yield, subloading, and elastic-core surfaces The subloading surface in Eq. (9.59) for the initial subloading surface model is extended as follows:   f M 5 RFðHÞ (9.85) which is depicted in Figure 9.4. The symbols for the stress-dependent variables in the infinitesimal and the hypoelastic-based plasticity in the current configuration correspond to those in the intermediate configuration for the multiplicative hyperelastic-based plasticity as follows: 8 T > > σ-M ð6¼ M Þ; > > T > > > > α-Mk ð 5 Mk Þ; > > > ^ 5 M 2 M ð6¼ M ^ T Þ; > ^ M σ> k > > T > > c-Mc ð 5 Mc Þ; > > > > > ^ 5M 2M ð5M ^ T Þ; > c^ 5 c 2 α-M > c c k c > > _ _ _T > > > σ 5 σ 2 cM 2 M ð6 ¼ Þ; 5 M M c > > > > ^ ð 5 M T Þ; > α 5 c 2 R^c ðc 2 α 5 Rðc 2 αÞÞ-M k 5 Mc 2 RM > c k > > T > _ _

σχ 5 σ 1 α-Mχ 5 M 1 Mk 6¼ Mχ ; > > R R > > > > > > T > @fðσÞ @fðσÞ @fðMÞ @fðMÞ > >n5 = -N 5 = ð6¼ N Þ > > @σ @σ > @M @M > > 2 3 2 3 > > 

> > T > @fðMÞ @fðMÞ > 4 5 4 5 > N 6¼ N 5 sym =:sym :ð 5 N Þ > > > @M @M > > > > > ^ Þ @fðM ^ Þ > > @fð^cÞ @fð^cÞ ^ @fðM c c > ^ TÞ > = = -Nc 5 ð5N > n^ c 5 c > @c @c @c : @Mc

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

290

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

FIGURE 9.4 Normal-yield, subloading, and elastic-core surfaces in the intermediate configuration. (A) General material. (B) Mises metals.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

291

9.6 Multiplicative extended subloading surface model

for elastoplasticity 8 ( ( ^ > > ^ $ 1Þ σðR ~ > > σ~ 5 -M 5 MðR $ 1Þ > > σðR , 1Þ > MðR , 1Þ > >   < Mk ðR $ 1Þ αðR $ 1Þ ~ ~5 -Mk 5 α αðR , 1Þ > > ( ( M k ðR , 1Þ > > > ^ $ 1Þ > ^ $ 1Þ nðR > ~ 5 NðR > ~ n 5 N > : nðR , 1Þ NðR , 1Þ for viscoplasticity 8 > ρ-Θ > > _ _ >

> =jj jj- N  n =jj jjð 5 N Þ > > : @α @α @Mk @Mk

(9.86)2

(9.86)3

for isotropic stagnation. The subloading surface is given from Eq. (9.85) with Eq. (9.86)1.8 as follows: _ ^  5 RFðHÞ f M 1 RM (9.87) c from which the normal-yield ratio R is calculated. The material-time derivative of the conjugate kinematic hardening variable Mk is given by

_

_

_

_

^ Mk 5 RMk 1 ð1 2 RÞMc 2 RM c

(9.88)

leading to 

_ _

_

_

_

_

^ M 5 M 2 Mk 5 M 2 RMk 2 ð1 2 RÞMc 1 RM c

(9.89)

The elastic-core surface which passes through the elastic-core Mc and is similar to the normal-yield surface with respect to the back-stress Mk is given noting Eq. (7.122) as follows: ^  ^  f M c 5 Rc FðH Þ; that is; Rc 5 f Mc =FðH Þ

(9.90)

9.6.3 Plastic flow rules The plastic strain rate is given in the following associated flow rule proposed by Hashiguchi (2016b, 2018c).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

292

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations p

_

D 5λN



5D

pT 

_

ðλ $ 0Þ

(9.91)

_

where λ is the positive plastic multiplier and "  # "  #

  T
@f M @f M N  sym N = sym 5 N 5 1 @M @M

(9.92)

which is the normalized and symmetrized tensor. If the strain energy e functions ψe are given by invariants of C leading to the elastic isotropy, T the symmetry of the Mandel stress M 5 M holds resulting in the sym   T   metry @f M @M 5 @f M @M . The dissipative parts of the plastic velocity gradient for the kinematic hardening variable and the elastic-core are given by referring to Eqs. (7.40) and (7.146) as follows: p

Dkd 5

_

(9.93)

 ^  1   F0 fHn ^  1 p sym M 2 M ck N 2 Mk 1 M c gjjD jj c 2 R ce bk F F  pT  5M λ 6¼ Dcd

Dcd 5 fN 2 p

 1 1 p pT  jjD jjMk 5 λ Mk 5 Dkd bk F bk F

χ

_

(9.94) where M N2

T

 ^  1   F0 fHn ^  χ 1 sym M 2 M ck N 2 Mk 1 Mc 5 M c 2 R ce bk F F (9.95)

The variables Rc and Cn in the material parameter u in Eq. (7.95) involved in Eq. (7.89) for the evolution rule of the normal-yield ratio R are given by Eqs. (7.122) and (7.173) for the infinitesimal strain theory. Further, they are given for the multiplicative hyperelastic-based plasticity as follows (Hashiguchi, 2016a):
 ^ =F; C  N ^ : N ð2 1 # C # 1Þ Rc  f M (9.96) c n c n where  ^   ^  @f Mc  ^ T   ^  @f M c ^ Nc 5 1 Nc  ^ 5 Nc ^ @M c @M c

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.97)

293

9.6 Multiplicative extended subloading surface model

which is the normalized outward-normal to the elastic-core surface in Eq. (9.90). p p Let the plastic spin W , the kinematic hardening spin Wkd , and the p  be given extending the equation in the hypoelasticelastic-core spin W cd based plasticity by Zbib and Aifantis (1988) and incorporating Eqs. (9.91), (9.93), and (9.94) as follows:

p p W 5 ηp ant M D 5 ηp ant½M Nλ 

p p  p p Wkd 5 ηk ant M Dkd 5 ½ηk =ðbk FÞant M Mk λ

p p  p p Wcd 5 ηc ant M Dcd 5 ηc A λ

_

_

_

(9.98)

p

where ηc is the material parameter and A  ant½M M

(9.99)

p

The plastic spin tensor W diminishes if the symmetry of the Mandel T stress, that is, M 5 M due to the elastic isotropy and the plastic isotp p ropy due to Mk 5 Mc 5 O hold. Further, the spin tensors Wkd and Wcd diminish for the plastic isotropy due to Mk 5 Mc 5 O. The velocity gradients are given by substituting Eqs. (9.91), (9.93), (9.94), and (9.98) into Eqs. (8.31)3, (8.51)2, and (9.73)2 as follows: 

 p L 5 λ N 1 ηp ant M N    

p p Lkd 5 λ Mk 1 ηk ant M Mk = bk F
 p p Lcd 5 λ M 1 ηc A

_ _ _

(9.100)

The substitution of Eq. (9.100) into Eqs. (9.11), (9.19), and (9.84) yield:

e

  e M 5 L : sym C fL 2 λðN 1 ηp ant M N Þg (9.101)

_

_

h n _k

  p pT

 p pT Mk 5 λ FksC : Fks N 2 1=ðbk FÞ Mk Fks Fks 1 2sym N 1 ηp ant M N

_

_



 io

p 2 1=ðbk FÞ Mk 1 ηk ant M Mk Mk

(9.102)


hn  

  o i  c : FpT N 2 M Fp FpT 1 2sym Mc 5 λ Fpcs C N 1 ηp ant M N 2 M 1 ηpc A Mc cs cs cs

_ _

(9.103)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

294

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

9.6.4 Plastic strain rate The elastic constitutive equation is given in Eqs. (9.9) and (9.10). The plastic strain rate will be formulated in this section. The formulation of the plastic modulus given in this section is not necessary in the implicit numerical calculation by the return-mapping based on the closet-point projection. Incidentally, the plastic modulus is not required also in the overstress model by which an elastoplastic deformation can be analyzed by the calculation as the quasi-static deformation as will be described in Section 11.2. The time-differentiation of Eq. (9.85) leads to the consistency condition of the subloading surface as follows:    @f M (9.104) : M 2 RF 2 RF 5 0 @M

_

It holds from Eq. (9.85) that   @f M @M

_

   : M 5 f M 5 RF

(9.105)

by the Euler’s theorem for the homogenous function fðMÞ of M in degree-one, and then it follows that       @f M @f M   @f M 5 f 5 RF= M = = : M N:M5 @M @M @M @M   @f M

which leads to    @f M N : M 5 1 RF @M

(9.106)

     T @f M @f M  N 6¼ N ; jjNjj 5 1 @M @M

(9.107)

where

The substitution of Eq. (9.106) into Eq. (9.104) leads to 

N: M 2

!

_F 1 _R F

R

N : M50

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.108)

295

9.6 Multiplicative extended subloading surface model

The further substitution of Eq. (9.89) into Eq. (9.108) leads to " #  F R ^ N:M2N: M1 M 2 RMc 1 RMk 1 ð1 2 RÞMc 5 0 F R

_

_

_

_

_

(9.109)

^ 5 M 2 M 2 M 2 M  5 _ M 2 RM M c c k k

(9.110)

Furthermore, substituting the relation

Eq. (9.109) is rewritten as " # F0 H R_ N :M2N : M 1 M 1 RMk 1 ð1 2 RÞMc 5 0 F R

_

_

_

_

_

(9.111)

where    p p  p H 5 fHd M; H; D =jjD jj jjD jj 5 fHn M; H; N λ p p R 5 UðRÞ D 5 UðRÞλ for D 6¼ O

_

_

_

_

(9.112) (9.113)

based on Eqs. (9.47) and (9.60) with Eq. (9.91). The substitutions of Eqs. (9.102), (9.103), (9.112), and (9.113) into Eq. (9.111) lead to the consistency condition:

_

p

_

N : M2M λ50

(9.114)

from which it follows that

_

λ5 where p

M N:

2   0 4F fHn M; F; N F

_;

N:M

M1

M

p

p

D 5

_

N:M M

p

N

(9.115)

UðRÞ _ R M

n h

  p pT

 p _k pT 1R FksC : Fks N 2 1=ðbk FÞ Mk Fks Fks 12sym N 1ηp ant MN n



  io  p  c : FpT N 2M Fp FpT 2 1=ðbk FÞ Mk 1ηk ant MMk Mk 1ð12RÞ Fpcs C cs cs cs 12sym

hn

#   o i o p N 1η ant MN 2 M 1ηc A Mc p

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.116)

296

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

The substitution of Eq. (9.101) into Eq. (9.114) leads to the consistency condition: o

e  n

e

 p e e N : L : sym C L 2 N : L : sym C N 1 ηp ant M N 1 M Λ 5 0

_

(9.117)

_

using the symbol Λ for the plastic multiplier in terms of the strain rate

_

instead of λ in terms of the stress rate. The plastic multiplier is given from Eq. (9.117) as follows:

e  e N : L : sym C L (9.118) Λ5 p

e

 e M 1 N : L : sym C N 1 ηp ant M N

_

Here, the following holds.

e  e C L 5 FeT lFe ; sym C L 5 FeT dFe

The loading criterion is given by ( p D 6¼ O for Λ . 0 p D 5 O for other

_

which can be given actually as (

e  p e D 6¼ O for N : L : sym C L . 0 p D 5 O for other

(9.119)

(9.120)

(9.121)

9.7 Material functions of metals and soils Metals exhibit the pressure-independence of the plastic deformation leading to the plastic incompressibility and thus its yield surface has cylindrical shape in the stress space. On the other hand, soils exhibit the pressure-dependence of plastic deformation leading to the plastic compressibility and thus it possesses ellipsoidal shape, conical shape, hexagonal pyramid shape, etc. Material functions contained in the constitutive equations formulated in the preceding sections are given for metals and soils in this section.

9.7.1 Metals The hyperelastic equation and the yield function for metals are shown below in the intermediate configuration.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

297

9.7 Material functions of metals and soils

9.7.1.1 Hyperelastic equation The following strain energy function of elastic deformation in Eq. (6.35) for the modified neo-Hookean elasticity (2) may be adopted (Vladimirov et al., 2008, 2010) in the multiplicative hyperelastic-based plastic constitutive equation. e

ψe ðC Þ 5

pffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffie  1  e 1 e e Λ detC 2 1 2 2 ln 1 μ trC 2 3 2 2 ln detC detC 4 2 (9.122)

where μ and Λ are the material constants. The substitution of Eq. (9.122) into Eq. (9.9) leads: S5 noting

 e21  1  e e21  Λ detC 2 1 C 1 μ G 2 C 2

(9.123)

8 e  @ detC > e  e21 > > 5 detC C > e > > @C > > pffiffiffiffiffiffiffiffiffiffiffiffi > > > < @ detCe 1 pffiffiffiffiffiffiffiffiffiffiffiffie e21 detC C 5 e 2 @C > > > > pffiffiffiffiffiffiffiffiffiffiffiffie > > > @ ln detC 1 e21 > > > 5 C e > : 2 @C

(9.124)

by virtute of Eqs. (1.313)
(1.315). It is follows from Eq. (9.123) that 1 e e ΛðdetC 2 1ÞG 1 μðC 2 GÞ 2   e e Mij 5 ΛðdetC 2 1ÞGij =2 1 μðCij 2 Gij Þ e

M5C S5

(9.125) (9.126)

τ 5 Fe SFeT 5 Λðdetbe 2 1Þg 1 μbe ðbe 2 gÞ e

The hyperelastic tangent stiffness modulus tensor C in Eq. (9.13) is given for Eq. (9.123) by

   e e e e  e21 e21 e C 5 Λ detC C  C 1 detC 2 1 ℭ 2 2μℭ e

where ℭ is the fourth-order tensor defined as e

ℭ 

e21

@C

e

@C

e21

52C

e21

IC

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.127)

298

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

0

e21

@ ℭe ijkl



@Cij

5

e

Ckl

e21 e21 2 Cik Cjl

52



e21 e21 Cik Cjl

e21 e21  1 Cil Cjk

e

e

.

1 2A

(9.128)

e

which possesses the minor symmetry ℭijkl 5 ℭjikl 5 ℭijlk and the major e e symmetry ℭijkl 5 ℭklij . Further, it follows from Eq. (9.125) that @M @C

e

e

5 ΛðdetC ÞG  C

e21

1 μI

(9.129)

noting @Mij e @Ckl

5

@ e @Ckl

 e e e e21 ΛðdetC 2 1ÞGij 1 μðCij 2 Gij Þ 5 ΛdetC Ckl Gij 1 μδik δjl

Further, it follows for Eq. (9.129) with the aid of Eq. (6.10) that @M e e21 e21 ~ ~ e 1 Fe IÞ 5 ΛðdetC ÞGðC FeT 1 FeT C Þ 1 2μðIF @Fe

(9.130)

noting e

e

e

@Mij @Mij @Crs 5 e e e @Fkl @Crs @Fkl

5

@½ΛðdetC 2 1ÞGij 1 μðCij 2 Gij Þ @ðFerp Feps Þ e

@Fekl

@Crs e

e21

5 ½ΛðdetC ÞCrs Gij 1 μδir δjs ðδrk δpl Feps 1 Ferp δpk δsl Þ e

e21

e21

5 ΛðdetC ÞGij ðCks Fels 1 Crl Ferk Þ 1 μðδrk Felj 1 Feik δjl Þ 9.7.1.2 Hyperelastic equation for kinematic hardening variable Assume the following strain energy function for the kinematic hardening, which possesses the identical form to the shear part in Eq. (9.122).  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  _ p   1 _p _p tr C ks 2 3 2 ln det C ks ψk C ks 5 Ck (9.131) 2 where Ck is material constant. It is derived from Eq. (9.14) that _ p  _ @ψk C ks _ _ p21  Sk 5 2 5 Ck G 2 C ks _p @C ks

(9.132)

Then, the Mandel-like kinematic hardening variable is given from Eqs. (9.16) and (9.132) as follows:  p pT  Mk 5 Ck Fks Fks 2 G (9.133)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

299

9.7 Material functions of metals and soils

noting  _ _ p21  pT

_  pT p 21  pT p p Fks Mk 5 Fks Ck G 2 C ks Fks 5 Fks Ck G 2 Fks Fks _k

C in Eq. (9.18) is given for Eq. (9.132) as _k

_ p21 _ p21

C 5 Ck C ks I C ks

(9.134)

9.7.1.3 Hyperelastic equation for elastic-core Assume the following strain energy function for elastic-core analogously to the kinematic hardening variable in Eq. (9.131).  qffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1  p p  p ψ ðC cs Þ 5 Cc trC cs 2 3 2 ln detC cs 2 c

(9.135)

where Cc is material constant. The following relations hold.   c p   S c 5 2 @ψ Cp cs 5 Cc G  2C  p21 cs  @C cs   Mc 5 Cc Fpcs FpT cs 2 G

(9.136) (9.137)

 c in Eq. (9.83) is given for Eq. (9.135) as C  c 5 Cc C  p21  p21 IC C cs cs

(9.138)

9.7.1.4 Yield function Assume the von Mises yield function and the plastic equivalent hardening, that is,  ^ 0 5 f M

rffiffiffi 3 ^ 0 M 2

(9.139)

ð rffiffiffi 2 p FðHÞ 5 F0 1 1 h1 1 2 expð2 h2 HÞ ; H 5 D dt 3 



(9.140)

for which we have

rffiffiffi rffiffiffi 2 p 2 F 5 F H ; F  F0 h1 h2 expð2 h2 HÞ; H 5 D 5 λ 3 3

_

0

_

0

_

_

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.141)

300

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

F-ð1 1 h1 ÞF0 holds for H-N in Eq. (9.140). It follows for Eq. (9.139) that

^ 0 ^0 ^ 5 sym M ^ 5 M ; N N (9.142) ^

^  0 sym M M0 The subloading surface for the normal-yield surface in Eq. (9.139) is given noting Eq. (9.85), that is, (9.87) by the following equation. rffiffiffi 3 0 M 5 RFðHÞ (9.143) 2 that is,

rffiffiffi 0 3 _ 0 ^ M 1 RMc 5 RFðHÞ 2

from which the normal-yield yield ratio is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 u _ u ^ 2 1  2 F2 2 M ^ 2  _ 0 2 0 0 : M _0 u M M c c ^ 1t 3 R 5M : M 0 c 2 2 2 ^ 3 F 2 Mc

(9.144)

(9.145)

9.7.2 Soils The hyperelastic equation and the yield functions for soils are shown below in the intermediate configuration. 9.7.2.1 Hyperelastic equation The hyperelastic equation of soils was delineated in Section 6.4. It will be described below in detail. The hyperelastic constitutive equation of soils was proposed first by Houlsby (1985) (see also Houlsby et al., 2005; Houlsby and Puzrin, 2010) and it has been extended to various multiplicative hyperelastic equations by Borja and Tamagnini (1998), Callari et al. (1998), etc. using the Hencky (logarithmic) strain in the current configuration and by Yamakawa et al. (2010) using the second Piola
Kirchhoff stress in the intermediate configuration, while they include various impertinences or inexactness. The hyperelastic equation within the framework of the multiplicative finite strain theory for soils was shown by Hashiguchi (2017a) but it includes some impertinences. The exact equation (Hashiguchi, 2018b) will be shown in the following. The isotropic hyperelastic equation is given by introducing the funce e tion of the variables ln J e and trC (ð1=2ÞðtrC 2 3Þ in detailed expression)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

9.7 Material functions of metals and soils

301

which stand for the volumetric strain and deviatoric strain in the infinitesimal strain theory, respectively, as follows:   e e e e @ψ ln J e ; trC @ψ ln J e ; trC @ ln J e @ψðln J e ; trC Þ @trC S52 5 2 1 2 e e e e @ ln J e @C @C @C @trC (9.146) where

8 e e e J 5 detFe ; C 5 FeT Fe ; detC 5 J e2 > > > > e > F 5 Fevol Fe ; Fevol  J e1=3 g; Fe J e21=3 Fe > > < e e e e C  FeT Fe 5 J e22=3 C ; trC 5 J e22=3 trC > ! > > detFe 5 detCe 5 1 > > > > : e e e0 C 5 G; trC 5 3; C 5 O for Fe 5 Fevol

(9.147)

noting Eqs. (4.66)
(4.71). Fevol is the elastic volumetric part and Fe is the so-called unimodular tensor designating the isochoric (constant volume, e i.e., deviatoric) part of Fe . In addition, trC 5 3 ðFe 5 Fevol Þ and e e e ln J 5 0 ðF 5 F Þ are required in the purely volumetric deformation and the purely deviatoric deformation, respectively. The necessity of ð1=2Þ in e ð1=2ÞtrC for the deviatoric infinitesimal strain was suggested by Prof. Yuki Yamakawa, Tohoku university. The following partial derivatives hold. 8 @ ln J e 1 1 e21 e e21 > > pffiffiffiffiffiffiffiffiffiffiffiffie ðdetC ÞC 5 C > e 5 > > 2 e @C < 2J detC (9.148)   e > @trC 1  e  e21 > e22=3 > 5J G 2 trC C > > : @Ce 3 noting

8 pffiffiffiffiffiffiffiffiffiffiffiffie e e e e > > @ln J @ln J @J 1 @ detC 1 1 @detC > > 5 e 5 e pffiffiffiffiffiffiffiffiffiffiffiffie > e 5 e > J J 2 detC @Ce @J e @Ce > @C @C > > > > > >  > 1 e  e21 > > pffiffiffiffiffiffiffiffiffiffiffiffie detC C 5 > > < 2J e detC



 > e 21=3 e  e 21=3 e e > > @tr detC C @ detC trC @trC > > 5 > e 5 e e > > @C @C @C > > > > > >  1  e  e21  > e 21=3 > > 5 detC G 2 trC C > : 3

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

302

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

The substitution of Eq. (9.148) into Eq. (9.146) reads:     e  e21 e @ψ ln J e ; trC C 1 @ψ ln J e ; trC e22=3 1  e  e21 S5 J G 2 C trC (9.149) e 3 @ ln J e @trC Let the following strain energy function be assumed.   e    e ψ ln J e ; trC 5 ϑF ln J e 1 κ~ PM0 1 ϑF0 J e21=κ~ 1 ð1=2ÞG0 J e2n=κ~ trC 2 3 (9.150) noting

  ~ Je J e2n=κ~ 5 exp 2nκln

where PM0 is the initial value of the pressure defined in terms of the Mandel stress M, that is, PM  2 ð1=3ÞtrM. λ~ and κ~ are the material constants standing for the inclinations of the isotropic and the swelling lines in both the logarithmic linear relation of pressure and the volume (cf. Hashiguchi, 2018a). ϑð , 1=2Þ is material constant, while the volume becomes infinite as the pressure approaches ϑF, while the hardening function F coincides to the preconsolidation pressure in the isotropic consolidation process. G0 is the initial value of elastic shear modulus, n standing for the pressure-dependence of the shear modulus. The following partial derivatives hold for Eq. (9.150). 8  e e   e   @ψ ln J ; trC > n > 5 ϑF 2 PM0 1 ϑF0 J e21=κ~ 2 G0 J2n=κ~ trC 2 3 > < ~ κ @ ln J (9.151)  e e > > @ψ ln J ; trC > : 5 G0 J e2n=κ~ e @ trC ~ e2n=κ~ . noting @J e2n=κ~ =@lnJ e 5 2 ðn=κÞJ Eq. (9.149) with Eq. (9.151) reads:   e

  e21 n S 5 ϑF 2 PM0 1 ϑF0 J e21=κ~ 2 G0 J e2n=κ~ trC 2 3 C κ~ 

1 e  e21  1 G0 J e2n=κ~ J e22=3 G 2 trC C 3 from which the Mandel stress is given as follows:   e

  e n M 5 C S 5 ϑF 2 PM0 1 ϑF0 J e21=κ~ 2 G0 J e2n=κ~ trC 2 3 G κ~ e0

1 G0 J e2n=κ~ C e0

e0

noting J e22=3 C 5 C by virtue of Eq. (9.147)7.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.152)

(9.153)

303

9.7 Material functions of metals and soils

It follows from Eq. (9.153) that

 PM 1 ϑF 1 e21=κ~ e 5J 5 exp 2 ln J for Fe 5 Fevol κ~ PM0 1 ϑF0

(9.154)

that is, PM 1 ϑF for Fe 5 Fevol PM0 1 ϑF0

εev 5 ln J e 5 2 κ~ ln

(9.155)

and 0

M 5 G0 J

e2n=κ~



e0

C 5 G0

PM 1ϑF PM0 1ϑF0

n C

e0

(9.156)

which would describe appropriately the basic characteristics in the volumetric and the deviatoric deformations. Eq. (9.153) is rewritten in terms of the Kirchhoff stress τ and the elastic unimodular left Cauchy
Green deformation tensor be in the current configuration as follows: h  i   0 n τ 5 ϑF 2 pτ0 1 ϑF0 J e21=κ~ 2 G0 J e2n=κ~ trbe 2 3 g 1 G0 J e2n=κ~ be κ~ (9.157) where

  τ 5 Fe2T MFeT 5 Fe SFeT

(9.158)

be Fe FeT

(9.159)

and pτ0 is the initial value of pτ  2 ð1=3Þtrτ, noting ( 0 e e0 Fe2T C FeT 5 b e ; Fe2T C FeT 5 be e

trC 5 trb e

(9.160)

It follows from Eq. (9.157) that ln J e 5 2 κ~ ln

pτ 1 ϑF for Fe 5 Fevol pτ0 1 ϑF0

and 0

τ 5 G0



pτ 1ϑF pτ0 1ϑF0

n be

0

(9.161)

(9.162)

The above-mentioned elastic equation is reduced to the infinitesimal strain theory by adopting the strain energy function

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

304

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

e  e      ε ε ψ εev ; εed 5 ϑFεev 1 κ~ p0 1 ϑF0 exp 2 v 1 G0 exp n 2 v εe2 (9.163) d κ~ κ~ as follows: 

e  e    e  εv n εv ε 0 e2 σ5 ϑF2 p0 1ϑF0 exp 2 2 G0 exp n 2 εd g1G0 exp n 2 v εe κ~ κ~ κ~ κ~ 



(9.164)

resulting in

e   p 1 ϑF ε 0 5 exp 2 v for εe 5 O εed 5 0 p0 1 ϑF0 κ~

that is,



εev and

p 1 ϑF 5 2 κ~ ln p0 1 ϑF0



  0 for εe 5 O εed 5 0

  e  ε p1ϑF n e0 0 σ0 5 2G0 exp n 2 v εe 5 2G0 ε p0 1ϑF0 κ~

(9.165)

(9.166)

where p is the pressure, that is, p  2 ðtrσÞ=3 and its initial value is denoted by p0 . εe is the infinitesimal elastic strain tensor and εev 5 trεe ; 0 εed 5 jjεe jj. The elastic balk modulus K and the elastic shear modulus G adopted in the hypoelasticity for the above-mentioned elastic equations are given as follows: ! 0 0 !

 σ σ σm 2p p 1 ϑF p1ϑF n D e0 5 G0 K5 e D e 5 ; G5 0 p0 1ϑF0 dv κ~ εv 2 ε 2 de

_

_ _

_

_ _

(9.167) dev

e

e

e

where  trl ; d  sym½l , while nD0:5 can be chosen in most of soils (Tatsuoka et al., 1978). The elastic constitutive equations of soils formulated above possess the following physical validities. 1. It is applicable up to the finite deformation/rotation. 2. It is applicable up to the negative pressure range which depends on the preconsolidation pressure. 3. The shear modulus increases depending on the pressure. 4. They are consistent for the multiplicative hyperelasticity, the infinitesimal hyperelasticity and the hypoelasticity.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

305

9.7 Material functions of metals and soils

|| M' || MF /2

(1 – x)x MF

1 M

M (= M /(1 – 2 x))

1

– ∂F

0

(1 / 2 – x )F

(– ∂ F for ξ h 5 0 PM 1 1 X=PM > <     f PM ; X 5 1 P 2 ξPM for ξh 6¼ 0 > > : ξ~ Mχ ξ~  2ð1 2 ξÞξ; ξ  1 2 2ξ; PMχ  0 M X M

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ~ 2 PM 1 2ξX

(9.172)

(9.173) (9.174)

and the isotropic hardening function is given by (Hashiguchi, 2017a).

 2 ln J p FðHÞ 5 F0 exp (9.175) λ~ 2 κ~ where J p 5 detFp and F0 is the initial value of F. The Drucker
Prager’s conical yield surface (Drucker and Prager, 1952) has been used widely for soils. However, it requires to incorporate the another conical potential surface different from the yield surface in order to suppress the plastic volumetric strain rate, while the plastic volumetric strain is described realistically by the subloading surface model with the associated flow rule (cf. Hashiguchi et al., 2002).

9.8 Calculation procedure The calculation procedure for the above-mentioned formulations is described in this section. The deformation gradient tensor is updated by Fn11 5 f½n;n11 Fn

(9.176)

f½n;n11  I 1 Δu  rxn

(9.177)

where

with the displacement vector u, designating rxn ð Þ  @ð Þ=@xn and noting f½n;n11  Fn11 F21 n 5

@xn11 @X @xn11 @ðxn 1 ΔuÞ 5 5 5 I 1 Δu  rxn @xn @X @xn @xn (9.178)

The rates of the plastic gradient and its dissipative parts are given from Eqs. (8.30)3, (8.53), and (9.75) as follows:

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

307

9.8 Calculation procedure

8 p p p > > < Fp 5 L F  p21 p p  p _p p Fkd 5 L kd Fkd 5 Fks Lkd Fks Fkd > >  p21 p p  p p p : p F 5 L F 5 Fcs L Fcs F

_ _ _cd

p Lkd ,

cd cd

cd

(9.179)

cd

p Lcd

p

where L, and are given in Eq. (9.100). The storage parts Fe ; Fks , p and Fcs of the deformation gradient are given by substituting the results of the time-integrations of Eq. (9.179) into Eq. (9.69) as follows: p

p21

p21

Fe 5 FFp21 ; Fks 5 Fp Fkd ; Fpcs 5 Fp Fcd e _p C ; C ks ,

(9.180)

p C cs

Further, and are calculated by substituting Eq. (9.180) into Eqs. (8.47) and (9.70). Further, the stress S, the kinematic hardening _ e _p variable S k , and the elastic-core S c are calculated by substituting C ; C ks , p  into Eqs. (9.9), (9.14), and (9.80). The isotropic hardening variand C cs able and the normal-yield ratio are calculated by the time-integration of Eqs. (9.112) and (9.113). The stress tensors in the current configuration are calculated in Eq. (9.1)1, that is, τ 5 Fe SFeT and σ 5 ðdetFÞτ. p p The time-integrations of Eq. (9.179) for the tensors Fp , Fkd , and Fcd can be executed in high efficiency by the tensor exponential method (Miehe, 1996; Weber and Anand, 1990; Hashiguchi and Yamakawa, 2012) which is delineated below. The equations in Eq. (9.179) are collectively described in terms of arbitrary second-order tensors T and Z as follows:

_TðtÞ 5 ZTðtÞ

(9.181)

Consider the following candidate as the numerical solution of Eq. (9.181) for the time-interval Δt 5 ½tn ; tn11 . Tn11 5 expðZΔtÞTn

(9.182)

provided that Z and Tn are constant during the time-interval. The timedifferentiation of Eq. (9.182) leads noting Eq. (1.264) to

_

Tn11 5

dTn11 d½expðZΔtÞTn  dexpðZΔtÞ Tn 5 Tn 5 dt dt dt

5

 d 1 1 I 1 ZΔt 1 ðZΔtÞ2 1 ðZΔtÞ3 1 ? Tn dt 2! 3!

 1 5 Z 1 ðZΔtÞZ 1 ðZΔtÞ2 Z 1 ? Tn 2!

 1 5 Z I 1 ðZΔtÞ 1 ðZΔtÞ2 1 ? Tn 5 ZexpðZΔtÞTn 5 ZTn11 2! (9.183)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

308

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

coinciding to Eq. (9.181) and thus the rightness of the candidate in Eq. (9.182) is proven. Eq. (9.182) is given by the generalized mid-point rule as follows: Tn11 5 expðZn1θ ΔtÞTn ð0 # θ # 1Þ

(9.184)

In what follows, one adopts the backward implicit scheme θ 5 1, that is, Tn11 5 expðZn11 ΔtÞTn

(9.185)

_p p p p The tensor Z is given by L, L kd , and L cd for Fp , Fkd , and Fcd , respectively, as shown in Eq. (9.179), so that they are updated by substituting Eq. (9.100) as follows: 8 n
o   p  p

p p p > > 5 exp L Δt F 5 exp N 1 η ant M N F Fn Δλ n11 n11 n11 n11 n n11 n11 > > > >  p21 p  p > p p > > F 5 exp F L ΔtF Fkdn > kdn11 kdn11 ksn11 ksn11 > > n o >



 > p21 p p p > > 5 exp Fksn11 1=ðbk FÞ Mkn11 1 ηk ant Mn11 Mkn11 Δλn11 Fksn11 Fkdn > > > >  p21 p  p < p p Fcdn11 5 exp Fcsn11 Lcdn11 ΔtFcsn11 Fcdn h i >   p > p21 p p > > 5 exp Fcsn11 Δλn11 M n11 1 ηc A n11 Fcsn11 Fcdn > > > > > > > Hn11 5 Hn 1 fHnn11 Δλn11 > > > > >

 π hRn 2 Re i   2 π Δλn11  > > > exp 2u 1 Re Rn11 5 ð1 2 Re Þcos21 cos > : π 2 1 2 Re 2 1 2 Re

(9.186) in which the plastic multiplier Δλn11 is calculated by Eq. (9.118) or by the return-mapping projection which will be described in the next section. Then, all the internal variables can be calculated uniquely. Here, it is required to input their initial values at the start of the numerical p p p calculation. One may input F0 5 Fkd0 5 Fcd0 5 G for initial isotropic materials under a null initial stress state, while Fe , Feks , and Fecs are calp p culated uniquely by substituting Fp ; Fkd and Fcd calculated from Eq. p

p21

p

p21

(9.186) into Eq. (9.69), i.e. Fe 5 FFp21 ; Fks 5 Fp Fkd ; Fcs 5 Fp Fcd . Further, the Mandel stress M and the internal state variables Mk and Mc are p p calculated by substituting Fe ; Fks and Fcs into the hyperplastic equations (9.10), (9.16) and (9.81). Therein, the rigid-body rotation is not included in the plastic deformation gradient but included in the elastic deformation gradient tensor, obeying the isoclinic concept, that is, the independence of the intermediate configuration on the direction of the rotation of the substructure. Consequently, the fundamental issues described in Sections 8.1.2 and 8.1.3, that is,

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

9.9 Implicit calculation by return-mapping

309

1. Which the elastic or the plastic deformation must include the rigidbody rotation. 2. The nonuniqueness of the multiplicative decomposition are solved out completely. p

p

The calculation by use of Cp , Cks , Ckd , etc. in the reference configuration are adopted widely (cf. e.g., Dettmer and Reese, 2004; Vladimirov et al., 2008, 2010; Hashiguchi and Yamakawa, 2012). However, it would be difficult to take the plastic spin into account by that method.

9.9 Implicit calculation by return-mapping The return-mapping equations for the subloading surface model will be formulated in this section. The calculation by the plastic strain rate by the plastic constitutive equation with the complicated equation of the plastic modulus in Eq. (9.115) is not needed in the return-mapping method.

9.9.1 Return-mapping The deformation gradient tensors in the elastic trial step are updated by

8 > Ftrial 5 f½n;n11 Fn ; > < n11 trial p21 Fetrial n11 5 Fn11 Fn > > p ptrial p ptrial p : ptrial Fn11 5 Fn ; Fksn11 5 Fksn ; Fcsn11 5 Fcsn

(9.187)

while the kinematic hardening variable and the elastic-core are fixed to those at the previous step n. The following simultaneous equation must be fulfilled in every loading increment process during the plastic corrector process. 8 p e > Deformation : YF 5 Ftrial > n11 2 Fn11 Fn11 5 O > > p p p > > Kinematic hardening : Yk 5 Fn11 2 Fksn11 Fkdn11 5 O > > < p p p Elastic-core : Yc 5 Fn11 2 Fcsn11 Fcdn11 5 O (9.188) > > > > Isotropic hardening : YH 5 Hn11 2 Hn 1 fHnn11 Δλn11 5 0 > > >   > : Subloading surface : Ys 5 f M 2 RFðHÞ 5 0 Eq. (9.188) is described explicitly for the Mises metals by exploiting Eq. (9.186) with Eqs. (9.139) and (9.141) as follows:

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

310

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

8 i h


 p e > YF 5Ftrial N n11 1ηp ant Mn11 N n11 Δλn11 Fn 5O > n11 2Fn11 exp > > > ih h . ii h > > p p p > Y 5Fp 2Fp expfFp21 1 b F M > > k k kn11 1ηk ant Mn11 Mkn11 Δλn11 Fksn11 gFkdn 5O n11 ksn11 ksn11 > < h   p i p p p p21 p Yc 5Fn11 2Fcsn11 exp Fcsn11 Δλn11 M n11 1ηc A n11 Fcsn11 Fcdn 5O > > > > pffiffiffiffiffiffiffiffi > > > YH 5Hn11 2Hn 1 2=3Δλn11 50 > > > > pffiffiffiffiffiffiffiffi : Ys 5 3=2 M 0 2RFðHÞ50

ð9:189Þ p

p

while M, Mk , Mc , N, and A can be described by Fe , Fks , and Fcs as shown in Eqs. (9.125), (9.133), and (9.137) with Eqs. (9.92) and (9.99). p p The five unknown variables Fen11 , Fksn11 , Fcsn11 , Hn11 , and Δλn11 are involved in Eqs. (9.188) and (9.189). Eq. (9.189) is described in the matrix form as follows:

where

YðXÞ 5 O

(9.190)

9 8 e 9 8 Fn11 > YF > > > > > > > > > > > > > p > > > > > > > > F Y > > = = < ksn11 > < k> p Y  Yc ; X  Fcsn11 > > > > > > > > > > > > > > YH > Hn11 > > > > > > > > > > ; > : ; : Ys Δλn11

(9.191)

In order to solve Eq. (9.190) numerically, linearizing it by means of the Taylor expansion and taking the first-order infinitesimal term, we have       Y Xðk11Þ DY XðkÞ 1 J XðkÞ dX 5 O (9.192)



where

2

@YF 6 @Fe 6 n11 6 6 @Yk 6 6 @Fe 6 n11 6 6 @Yc 6 J 5 6 @Fe 6 n11 6 6 @YH 6 e 6 @F 6 n11 6 6 @YS 4 e @Fn11

3 @YF @YF @YF @YF p p @Fksn11 @Fcsn11 @Hn11 @Δλn11 7 7 7 @Yk @Yk @Yk @Yk 7 7 p p @Fksn11 @Fcsn11 @Hn11 @Δλn11 7 7 7 @Yc @Yc @Yc @Yc 7 7 p p @Fksn11 @Fcsn11 @Hn11 @Δλn11 7 7 7 @YH @YH @YH @YH 7 7 p p @Fksn11 @Fcsn11 @Hn11 @Δλn11 7 7 7 @YS @YS @YS @YS 7 5 p p @Fksn11 @Fcsn11 @Hn11 @Δλn11

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.193)

9.9 Implicit calculation by return-mapping

311

Here, the partial-derivatives of the components in this matrix can be derived by exploiting Eq. (6.10) analytically or by the numerical method. Eq. (9.192) is the simultaneous equation for the unknown dX, and thus, solving this equation, it is updated by

 21  ðkÞ 

 21  ðkÞ  dX 5 2 J XðkÞ Y X -Xðk11Þ 5 XðkÞ 1 dX 5 XðkÞ 2 J XðkÞ Y X (9.194) where ðkÞ designates the number of iterations. Therein, the normal-yield ratio Rn11 is updated by the analytical integration in Eq. (7.231). The iteration calculation in Eq. (9.194) is continued until the residual becomes sufficiently small: that is, it is judged that the convergence of solution is attained when the following equation in terms of the residual norm is fulfilled.  ðk11Þ  Y X , TOL (9.195) The normal-yield ratio Rn at the end of the step n and the normalyield ratio Rtrial n11 at the end of the elastic trial step are given for the Mises material below. The following equation holds at the end of the step n for the Mises material. pffiffiffiffiffiffiffiffi 0 3=2 M n 5 Rn FðHn Þ (9.196) which is rewritten as 0 pffiffiffiffiffiffiffiffi 0 ^ 5 R FðH Þ 1 R M 3=2 M n n n cn ~n

(9.197)

from which Rn is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 u _ 0 0 u ^ 2 1 ½ð2=3ÞðFðH ÞÞ2 2 M ^ 2  _ 0 2 : M _ 0 u n Mn cn cn ^ 1 t Mn Rn 5 Mn : M cn ^ 0 2 2 ð2=3ÞðFðHn ÞÞ 2 Mcn (9.198) Analogously, the following equation holds at the end of the elastic trial step for the Mises material. pffiffiffiffiffiffiffiffi trial0 3=2 Mn11 5 Rtrial (9.199) n11 FðHn Þ which is rewritten as 0 pffiffiffiffiffiffiffiffi _ trial0 ^ trial 3=2 Mn11 1 Rtrial n11 Mcn 5 Rn11 FðHn Þ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.200)

312

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

from which Rtrial n11 is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i  _ trial0 ^ 2 h _ trial0 ^ 0 2 ^ 0 2 _ trial0 2 0 Mn11 : Mcn 1 Mn11 : Mcn 1 ð2=3ÞðFðHn ÞÞ 2 Mcn Mn11 Rtrial n11 5 ^ trial0 2 ð2=3ÞðFðHn ÞÞ2 2 M n11 (9.201)

9.9.2 Loading criterion The loading criterion in the current configuration in Section 7.6.2 has been extended to that for the multiplicative hyperelastic-based plasticity in Hashiguchi (2018a). Consider the loading criterion at the initial stage of the plastic corrector step. The following facts should be noticed, referring to Fig. 9.6 where the elastic trial steps for the initial subloading surface model ðMc 5 Mk Þ and the extended subloading surface model ðMc 6¼ Mk Þ are shown in Fig. 9.6A and B, respectively. p trial “The plastic strain rate Dn11 is not induced if the elastic trial stress Mn11  trial trial  stays inside the elastic response region, i.e. f Mn11 2 Mkn11 2 Re FðHn Þ # 0 trial

trial

in the step n or if the stress increment ΔMn11  Mn11 2 Mn makes an obtuse angle with the outward-normal of the subloading surface in the elastic trial step n 1 1. Otherwise, it is induced.” Then, the loading criterion in the return-mapping method for the subloading surface model is given as follows: Loading criterion in return
mapping for subloading surface model trial   trial p Final trial trial Dn11 5O and Mn11 5Mn11 for f Mn11 2Mkn11 2Re FðHn Þ#0 or N n : ΔMn11 #0 p

Final

trial

Dn11 6¼ O and Mn11 6¼ Mn11 for others (9.202)

where trial

_

trial

ΔMn11  Mn11 2 Mn

(9.203)

^  M 2M M c cn kn

(9.204)

Mn 5 Mn 2 Mcn ;

_ trial Mn11

trial

5 Mn11 2 Mcn

(9.205)

^ ; Me 5 M 2 R M ^ ; Mtrial 5 M 2 Rtrial M ^ Mkn 5 Mcn 2 Rn M cn cn e cn cn kn kn11 n11 cn (9.206)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

9.9 Implicit calculation by return-mapping

313

FIGURE 9.6 Loading criterion when the elastic trial stress is directed inward of the subloading surface at the step n in return-mapping method for subloading surface model. (A) Initial subloading surface model ðMc 5 Mk Þ. (B) Extended subloading surface model ðMc 6¼ Mk Þ.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

314

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

_ _ trial ^ ; M trial 5 Mtrial 2 Mtrial 5 M trial ^ M n 5 Mn 2 Mkn 5 Mn 1 Rn M cn n11 kn11 n11 1 Rn11 Mcn n11

(9.207)  trial    trial       @f Mn @f Mn11 trial @f Mn11 @f Mn (9.208) ; Nn11  Nn  trial trial @Mn @Mn @Mn11 @Mn11

9.9.3 Initial value of normal-yield ratio in plastic corrector step The initial value R0n11 of the normal-yield ratio in the plastic corrector step is required in the return-mapping calculation. The equation of R0n11 will be given below in the analogous formulation to the one given in Section 7.6.3. The subloading surface once shrinks inducing only elastic strain increment and turns to expand inducing the plastic strain incretrial ment after it contacts tangentially to the stress increment ΔMn11 in the process of the stress variation from Mn as shown in Figure 9.6. Then, we need to calculate the normal-yield ratio R0n11 in the state that the subloading surface contacts tangentially to the lineelement. Here, the following relations must be satisfied at the tangential contact point, where the stress at the tangential contact 0 point, that is, the plastic-loading initiation stress is denoted by Mn11 from which the subloading surface changes from the contraction to the expansion.   f M 0n11 5 R0n11 FðHn Þ (9.209) 0 trial N n11 : ΔMn11 5 0 where 0

trial

Mn11  sΔMn11 1 Mn ð0 # s # 1Þ

(9.210)

0 ^ Mkn11 5 Mcn 2 R0n11 M cn

(9.211)

0 0 _ 0 trial ^ M n11  Mn11 2 Mkn11 5 sΔMn11 1 Mn 1 R0n11 M cn 0

Nn11

 0    0  @f Mn11 @f Mn11  0 0 @Mn11 @Mn11

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.212) (9.213)

315

9.9 Implicit calculation by return-mapping

sð0 # s # 1Þ is the unknown scalar parameter which must be determined 0 so as to satisfy Eq. (9.209) at the stress Mn11 . The two unknown vari0 ables s and Rn11 are included in Eq. (9.209). In what follows, the equation of R0n11 in terms of the known variables will be derived for the Mises material. Eq. (9.209) is explicitly described for the Mises material with pffiffiffiffiffiffiffiffi 00 00 . 00 00 0 fðMn11 Þ 5 3=2 Mn11 leading to Nn11 5 Mn11 Mn11 as follows: 8 pffiffiffiffiffiffiffiffi 0 < 3=2jjM 0 jj 5 R0 FðH Þ n n11 n11 0 0 : trial M n11 : ΔMn11 5 0 that is,

( pffiffiffiffiffiffiffiffi 0 trial0 ^ 0 5 R0 FðH Þ 1 R0n11 M 3=2 sΔMn11 1 M n n11 ~ n cn 0   _0 trial0 trial ^ 0 sΔMn11 1 Mn 1 Rn11 Mcn : ΔMn11 5 0

(9.214)

(9.215)

The upper equation in Eq. (9.215) is expressed as

trial0 _0 ^ 0  : sΔMtrial0 1 _ 0 1R0 M ^ 0  5 R0 FðH Þ2 ð3=2Þ sΔMn11 1Mn 1R0n11 M n Mn n11 n11 cn n11 cn leading to trial0 trial0 trial0  _ 0 ^0  s2 ΔMn11 : ΔMn11 1 2sΔMn11 : Mn 1 R0n11 M cn 0 _0 2  0 ^ 0  :  _ 1 R0 M ^0  1 Mn 1 R0n11 M M cn n n11 cn 2 ð2=3Þ Rn11 FðHn Þ 5 0

from which we have

n h trial0  _ 0 trial0 trial0 ^ 0  1O ΔMtrial0 :  _ 0 1R0 M ^ 0 2 s5 2ΔMn11 : M 1R0n11 M Mn n11 cn 2ΔMn11 :ΔMn11 cn n11 n h 0 0    0 2 ioi _ ^0   0 0 0 _ Mn 1Rn11 Mcn : σ~ n 1Rn11 Mn 2 2=3 Rn11 FðHn 

trial0

trial0 

ΔMn11 : ΔMn11

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.216)

316

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

The substitution of Eq. (9.216) into Eq. (9.215)2 leads to trial0  _ 0 ^0  2 ΔMn11 : Mn 1 R0n11 M cn ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

trial0  _ 0 ^ 0 2 2 ΔMtrial0 : ΔMtrial0  _ 0 1 R0 M ^0 : 0 1 ΔMn11 : Mn 1Rn11 M Mn n11 n11 n11 cn cn  0 2  _0 ^0   0 Mn 1 Rn11 Mcn 2 2=3 Rn11 FðHn trial0  _ 0 ^ 0 50 1 ΔMn11 : Mn 1 R0n11 M cn

resulting in i h trial0  _ 0 ^0  2 0 ΔMn11 : M n1Rn11 Mcn h trial0 trial0  _ 0 ^ 0  :  _ 0 1R0 M ^ 0  2 2=3R0 FH 2 50 2ΔMn11 :ΔMn11 Mn 1R0n11 M n cn cn n11 n11 Mn (9.217)

which is the quadratic equation of R0n11 . Eq. (9.217) is rewritten as h trial0 ^ 0 2  trial0 trial0  ^ 0 ^0  Mcn : M ΔMn11 : M cn 2 ΔMn11 :ΔMn11 cn     2  trial0 trial0  1 2=3 F Hn ΔMn11 :ΔMn11 R02 n11 h i trial0 _ 0  trial0 ^ 0   trial0 trial0  _ 0 ^ 0  0 12 ΔMn11 : Mn ΔMn11 : M cn 2 ΔMn11 :ΔMn11 Mn : Mcn Rn11 0   trial0 _ 0 2  trial0 trial0  _ 0 1 ΔMn11 : Mn 2 ΔMn11 :ΔMn11 Mn : M ~ n 50

Then, the solution of R0n11 in Eq. (9.217) is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 B 1 B2 2 AC 0 Rn11 5 A where

(9.218)

8 h 0 i ^ 0 2 2=3ðFðH ÞÞ2 ^ :M > A  S2ca 2 Sss M > n cn cn > > < _0 ^0 B  Ssc Sca 2 Sss Mn : M cn > > > > _0 _0 : 2 C  Ssc 2 Sss Mn : Mn

(9.219)

with 0

trial0 trial0 trial0 ^ 0 ; S  ΔMtrial0 : _ Sss  ΔMn11 : ΔMn11 ; Sca  ΔMn11 : M sc cn n11 Mn (9.220)

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

9.10 Cyclic stagnation of isotropic hardening

317

One must choose the solution satisfying 0 # R0n11 # 1 in Eq. (9.218). Here, we must set R0n11 5 Re if R0n11 # Re in the calculated result. It is enough to calculate the initial normal-yield ratio R0n11 , while it is not 0 necessary to calculate the scalar number s and the stress Mn11 for the return-mapping calculation. The loading criterion for the return-mapping method in the initial subloading surface model is given by setting Mc 5 Mk in the abovementioned formulations, referring to Fig. 9.6A.

9.10 Cyclic stagnation of isotropic hardening The cyclic stagnation of isotropic hardening described in Section 7.5.7 for the hypoelastic-based plasticity will be extended to the multiplicative hyperelastic-based plasticity in this section. The normal-isotropic hardening surface in the intermediate configuration is given by _  g Mk 5 K~ (9.221) where  _ ^ T 6 M Mk  Mk 2 Θ ¼ k

(9.222)

The scalar variable K~ and the second-order tensor variable T Θ ð6¼ Θ ; trΘ 5 0Þ designate the size and the center, respectively, of the normal-isotropic hardening surface, the evolution rules of which will be formulated later. Further, the subisotropic hardening surface, which always passes through the back-stress Mk in the intermediate configuration and has a similar shape and a same orientation to the normal-isotropic hardening surface is expressed by the following equation (see Fig. 9.7). _  g Mk 5 R~ K~ (9.223) where R~ is called the normal-isotropic hardening ratio and is calculable _ from the equation R~ 5 gðMk Þ=K~ in terms of the known values Mk , Θ, ~ and K. The consistency condition of the subisotropic hardening surface is given by _  _  @g Mk @g Mk : Mk 2 : Θ 5 R~ K~ 1 R~ K~ (9.224) _ _ @ Mk @ Mk

_

_

_ _

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

318

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

FIGURE 9.7 Normal- and sub-isotropic hardening surfaces in multiplicative elastoplasticity.

The rates of K~ and Θ are given by the following equations (cf. Hashiguchi, 2017a).  _  @g Mk ς _ ~ ~ (9.225) K 5 CR hN : Mk i @Mk

_

_

_

_

_

_

ς Θ 5 ð1 2 CÞR~ hN : Mk i Nk

where C ð0 # C # 1Þ and ς are the material constants and  _   _  _ @g Mk @g Mk  _ T  6¼ Nk N = @Mk @Mk

(9.226)

(9.227)

Substituting Eqs. (9.225) and (9.226) for the evolution rules of K~ and Θ into Eq. (9.224), the rate of the normal-isotropic hardening ratio is given by 0*  _  *  _  +1 + _  @g _ ς _ @g @g M M k M 1 ς k k ~ R~ : Mk A R~ 5 @ : Mk 2 : ð1 2 CÞR~ h N : Mk iN 2 RC @Mk @Mk @Mk K~

_

_

_

* _  + n

  ζ o 1 @g Mk 1 2 1 2 C 1 2 R~ R~ : Mk 5 ~ @Mk K

_

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

_

(9.228)

319

9.10 Cyclic stagnation of isotropic hardening

FIGURE 9.8 Evolution of normal-isotropic hardening ratio.

which is the monotonically decreasing function of R~ fulfilling 8 + * _  > > 1 @g Mk > > : Mk ð . 0Þ for R~ 5 0 5 > > > @gMk K~ > > > * _  + < R~ > , 1 @g Mk : M ð . 0Þ for R~ , 1 k > > @gMk > K~ > > > > > 5 0 for R~ 5 1 > : , 0 for R~ . 1

_

_

_

(9.229)

Therefore, Mk is attracted automatically to the normal-isotropic hardening surface even if it goes out from that surface by virtue of the inequality R~ , 0 for R~ . 1 as shown in Eq. (9.229) and Fig. 9.8. Furthermore, the

_

judgment of whether Mk lies on the normal-isotropic hardening surface is not required in the present formulation. The evolution rule of isotropic hardening is given as follows (cf. Hashiguchi, 2017a):  υ _ H 5 R~ N : A= A f Hn λ 5 f Hsn λ

_

_

_

where υ is the material constant and D E υ _ f Hsn  R~ N : A= A f Hn

(9.230)

(9.231)


h
 

 p pT

 p_ k pT p A  Mk =λ 5 Fks C : Fks N 2 1=bk sym Mk Fks Fks 1 2sym N 1 η ant M N  



 i p Mk (9.232) 2 1=bk sym Mk 1 ηk ant Mk sym Mk

_ _

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

320

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

which denotes the direction of the kinematic hardening rate in Eq. (9.102). The plastic modulus is given by replacing f Hn in Eq. (9.112) to f Hsn in Eq. (9.231) in Eq. (9.116). _ The function gð MÞ is given in the simplest form as follows:  _  _ (9.233) g Mk 5 Mk It follows from Eqs. (9.227) and (9.233) that _  _  _ T  _  @g Mk _ M 5 _ k 5 N N 5 1 N5 M k @Mk

(9.234)

9.11 Multiplicative subloading-overstress model The subloading-overstress model described in Section 7.7 will be extended to the multiplicative hyperelastic-based viscoplasticity in this section.

9.11.1 Constitutive equation The limit subloading, the subloading, the normal-yield, the static subloading surfaces, the limit elastic-core, and the elastic-core surface in the intermediate configuration are shown in Fig. 9.9. The deformation gradient F is multiplicatively decomposed into the elastic deformation gradient Fe and the viscoplastic deformation gradient Fvp instead of the plastic deformation gradient Fp in the multiplicative elastoplasticity described in the preceding sections. Then, we first adopt the following equation instead of Eq. (8.5). F 5 Fe Fvp

(9.235)

Further, the viscoplastic deformation gradient Fvp is multiplicatively vp decomposed into the viscoplastic storage part Fks causing the kinematic vp hardening and its dissipative part Fkd and into the viscoplastic storage vp vp part Fcs causing the elastic-core and its dissipative part Fcd as follows: ( vp vp Fks Fkd p F 5 (9.236) vp vp Fcs Fcd

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

9.11 Multiplicative subloading-overstress model

321

FIGURE 9.9 Limit subloading, subloading, normal-yield, static subloading surfaces, limit elastic-core, elastic-core, and in subloading-overstress model. (A) Over normal-yield state ðR $ 0: Mk 5 Mk 5 Mks 5 Me Þ. (B) Below normal-yield state ðR , 0: Mk 6¼ Mk 6¼ Mks 6¼ Mc Þ.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

322

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

Then, the following right Cauchy
Green deformation tensors for the viscoplastic deformation are introduced analogously to Eqs. (8.18) and (9.70). 8 vp C  FvpT Fvp ; > > < _ vp vpT vp vp vpT vp C ks  Fks Fks ; Ckd  Fkd Fkd (9.237) > > :  vp vpT vp vp vpT vp C cs  Fcs Fcs ; Ccd  Fcd Fcd The velocity gradient l in the current configuration is additively decomposed into the elastic and the viscoplastic parts: l 5 le 1 lvp

(9.238)

8 l  FF21 ; > > < e vp vp le  F Fe21 ; lp  Fe F Fvp21 Fe21 5 Fe L Fe21 > > vp : vp L  F Fvp21

(9.239)

where

_ _

_

_

Further, the velocity gradient L in the intermediate configuration is additively decomposed into the elastic and the viscoplastic parts as follows: e

L5L 1L where

(

L  Fe21 lFe

_e

e

L  Fe21 le Fe 5 Fe21 F ;

L

vp

vp

(9.240)

_vp

 Fe21 lvp Fe 5 F Fvp21

from which it follows that ( L5D1W e

e

e

e

vp

vp

vp

L 5D 1W ; L 5D 1W e

D5D 1D ;W5W 1W where

vp

vp

8 > D 5 sym½L; W 5 ant½L > < e e e e D 5 sym½L ; W 5 ant½L  > > vp vp vp : vp D 5 sym½L ; W 5 ant½L 

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.241)

(9.242) (9.243)

(9.244)

323

9.11 Multiplicative subloading-overstress model vp

The viscoplastic velocity gradient L is additively decomposed for the kinematic hardening and the elastic-core into the storage and the dissipative parts as follows: vp

vp

vp

vp

vp

vp

L 5 Lks 1 Lkd ; L 5 Lcs 1 Lcd

(9.245)

8 vp vp vp vp21 > < Lvp 5 Dks 1 Wks ks  Fks Fks - > _ vp vp21  vp vp vp _ vp G  : Lvp  Fvp L 5 5 Dkd 1 Wkd F L kd ks kd ks ks kd G

(9.246)

where

_

_ vp

vp  vp Dks  sym Lks ;

vp  vp Dkd  sym Lkd ;

_vp

vp  vp Wks  ant Lks

vp  vp Wkd  ant Lkd

~

(

vp21

L kd 5 Fkd Fkd

vp21 vp vp  vp vp G  Lkd Fks 5 ks L _ kd G

 Fks



_

8 vp vp vp < Lvp  F Fvp21 5 Dcs 1 Wcs ; cs cs cs vp  G  : vp vp vp vp vp vp21  vp ~ 5 cs L cd G 5 Dcd 1 Wcd Lcd  Fcs L cd Fcs

_

vp  vp Dcs  sym Lcs ;

vp  vp Dcd  sym Lcd ;

vp  vp Wcs  ant Lcs

vp  vp Wcd  ant Lcd

(9.248)

(9.249)

(9.250)

~

(

(9.247)

vp  vp vp  G  vp vp vp21 Lcd Fvp L cd 5 Fcd Fcd  Fvp21 cs cs 5 cs Lcd G

_

(9.251)

Now, the following tensors are defined based on Eqs. (7.267)
(7.269). ( Mk for R $ 1 ~ M  (9.252) k

( ~  M

Mk for R , 1 ^ for R $ 1 M

M for R , 1 " # " # ~  ~ @fðMÞ @fðMÞ ~ N  sym sym @M @M

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.253)

(9.254)

324

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

The subloading surface is expressed based on Eq. (7.270) in the current configuration as follows: ~ 5 RFðHÞ fðMÞ i.e.,

(

^ 5 RFðHÞ ðR 5 fðMÞ=FðHÞÞ ^ fðMÞ for R $ 1Þ fðMÞ 5 RFðHÞ for R , 1

(9.255)

(9.256)

The viscoplastic strain rate is given by extending Eq. (7.272) as follows: vp ~ D 5 ΓN

(9.257)

where Γ is given by Eqs. (7.273) and (7.274), that is, hR2Rs in 1

 μv 1 2 R=ðcm Rs Þ m

(9.258)

1 hexp½nðR 2 Rs Þ 2 1i μv 1 2 ½R=ðcm Rs Þm

(9.259)

Γ or Γ

The evolution rule of the static normal-yield ratio Rs is given by ( vp Rs 5 UðRs ÞjjD jj for R . Re (9.260) Rs 5 Re for other

_

with UðRs Þ 5 ucot by replacing ε_ vp to D Eq. (7.277) as follows:

vp

 π hRs 2 Re i 2 1 2 Re

(9.261)

in Eq. (7.275), where u is given analogously to

 ~ ^ :N (9.262) u 5 u exp ðuc Rc Cn Þ 5 uexp uc Rc N c Ð vp Rs is analytically expressed by Eq. (7.278) with εvp 5 jjD jjdt for fixing u. The static subloading surface (Fig. 9.9) over which the viscoplastic strain rate is induced is given by   (9.263) f Ms 5 Rs FðHÞ analogously to Eq. (7.279), where M s 5 M 2 Mks

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.264)

9.11 Multiplicative subloading-overstress model

^ Mks 5 Mc 2 Rs M c

325 (9.265)

noting Eq. (7.280). The dissipative parts in the viscoplastic strain rates for the kinematic hardening and the elastic-core are given extending Eqs. (9.93) and (9.94) as follows: 1 vp D Mk bk F vp ~ Dvp D 5M

vp

Dkd 5

(9.266)

cd

~ is defined based on Eq. (9.95) as follows: where M 
h i 0 ~ 2 χ sym M ~ N ~ 2M ~ 2 1 M  1 F fHn M ^ ^ 2 1 c N M c c k k R ce bk F F

(9.267)

p

The viscoplastic spin W , the kinematic hardening viscoplastic spin p and the elastic-core viscoplastic spin Wcd are given extending Eq. (9.98) by Eqs. (9.257) and (9.266) as follows: p Wkd ,

vp vp ~ W 5 ηvp ant½M D  5 ηvp ant½MNΓ vp

vp

vp

cd

c

cd

vp

Wkd 5 ηk ant½M Dkd  5 ½ηk =ðbk FÞant½MMk Γ vp vp vp vp ~ W 5 η ant½M D  5 η AΓ vp

(9.268)

c

vp

where ηvp , ηk , and ηc are the material parameters. The velocity gradients are given by substituting Eqs. (9.257), (9.266), and (9.268) into Eq. (9.242)3, (9.246)2, and (9.249)2, as follows: ~

~  vp L 5Γ N 1 ηvp ant M N .  

vp vp bk F Lkd 5 Γ Mk 1 ηk ant M Mk ~ vp vp ~  Lcd 5 Γ M 1 ηc A ~ is defined based on Eq. (9.99) by where A 8 ^  for R $ 1 < ant M M ~ A : ant M M for R , 1

(9.269)

(9.270)

leading to ~  ant½M M ~ A

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.271)

326

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

The rate of the viscoplastic gradient is given from Eq. (9.239)3, (9.248), and (9.251) as follows: 8 vp vp vp > > > < Fvp 5 L vpF  vp21 vp vp  vp _ vp (9.272) Fkd 5 L kd Fkd 5 Fks Lkd Fks Fkd > > vp >  vp21 vp vp  vp vp vp : Fcd 5 L cd Fcd 5 Fcs Lcd Fcs Fcd

_ _ _

The numerical time-integration for these variables can be performed effectively by the tensor exponential method which are shown for the plastic deformation in Eq. (9.186). vp vp The storage parts Fe ; Fks , and Fcs of the deformation gradient are given by substituting the results of the time-integrations of Eq. (9.272) into Eqs. (9.235) and (9.236) as follows: vp

vp21

vp21

vp F 5 FFvp21 ; Fks 5 Fvp Fkd ; Fvp cs 5 F Fcd e _ vp C ; C ks ,

(9.273)

 vp C cs

and are calculated by substituting Eq. (9.273) Further, into Eq. (9.237). Furthermore, the stress S, the kinematic hardening vari_ e _ vp able S k , and the elastic-core S c are calculated by substituting C ; C ks ,  vp into the hyperelastic Eqs. (9.9), (9.14), and (9.80) with the replaand C cs

_p _ vp  p into C  vp cements of C ks and C ks and C cs . The isotropic hardening varics able and the normal-yield ratio are calculated by the time-integration of Eqs. (9.112) and (9.113). Furthermore, M, Mk , and Mc are calculated in Eqs. (9.10), (9.16), and (9.81) with the replacement of the plastic strain rate to the viscoplastic one. The rate of isotropic hardening variable is generally described as follows:
~ H 5 fHn M; H; NÞΓ (9.274)

_

The stagnation of isotropic hardening is incorporated by replacing

_p

the plastic strain rate λ to hte viscoplastic strain rate Γ in the formulation described in Section 9.10. The stress-strain behavior predicted by the multiplicative subloadingoverstress model is illustrated in Fig. 9.10. As described in Section 7.7, the subloading-overstress model is no more than the generalization of the subloading surface model to the description of the elastoplastic deformation in the general strain rate and thus the original subloading surface model in the rateindependence can be disused by using the subloading-overstress model.

9.11.2 Calculation procedure The numerical calculation is performed by the procedure described in the following.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

9.11 Multiplicative subloading-overstress model

FIGURE 9.10

327

Stress-strain curve predicted by multiplicative subloading-overstress

model.

vp

vp

Fvp , Fkd , Fcd , H, and Rs are updated analogously to Eq. (9.186) as follows: n
o
 8 

vp vp ~ ~ vp vp > Fvp ΔΓ F 5 exp L Δt F 5 exp N 1 η ant M N > n11 n11 n11 n11 n11 n n n11 > > > >
 > > vp vp21 vp vp vp > > F 5 exp Fksn11 Lkdn11 ΔtFksn11 Fkdn > > > kdn11 > > n o >

  > vp21 vp vp vp > > ΔΓ 5 exp F 1= ð b F Þ M 1 η ant M M F n11 n11 k kn11 kn11 > ksn11 k ksn11 Fkdn > > > >
 < vp vp21 vp vp vp Fcdn11 5 exp Fcsn11 Lcdn11 ΔtFcsn11 Fcdn > > > h > ~  vp i vp > vp21 vp ~ > 5 exp Fcsn11 ΔΓ n11 M > n11 1 ηc An11 Fcsn11 Fcdn > > > > > > > Hn11 5 Hn 1 fHnn11 ΔΓ n11 > > > > > >

 π hRsn 2 Re i   > 2 π ΔΓ n11  > 21 > > : Rsn11 5 π ð1 2 Re Þcos cos 2 1 2 Re exp 2μ 2 1 2 Re 1 Re (9.275)

in which the viscoplastic multiplier ΔΓ n11 is calculated in Eq. (9.258) or (9.259) or by the return-mapping projection which will be described in the next section. The stress, the kinematic hardening variable and the elastic-core are calculated by the procedure described at the end of the preceding subsection.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

328

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

It should be noticed that the calculation by the elastoplastic constitutive equation with the quite complex plastic modulus in Eq. (9.116) is not necessary even for the time-independent elastoplastic deformation analysis with the forward-Euler method by calculating it as a quasistatic deformation using the subloading-overstress model.

9.11.3 Implicit calculation by return-mapping The deformation gradient tensors in the elastic trial step are updated by

8 trial F 5 f½n;n11 Fn ; > > < n11 vp21 etrial Fn11 5 Ftial n11 Fn > > : vptrial vp vptrial vp vptrial vp Fn11 5 Fn ; Fksn11 5 Fksn ; Fcsn11 5 Fcsn

(9.276)

The following simultaneous equation must be fulfilled in every loading increment process during the plastic corrector process. 8 vp e Deformation : YF 5 Ftrial > n11 2 Fn11 Fn11 5 O > > > vp vp vp > Kinematic hardening : Yk 5 Fn11 2 Fksn11 Fkdn11 5 O > > < vp vp vp Elastic-core : Yc 5 Fn11 2 Fcsn11 Fcdn11 5 O (9.277) > > > > Isotropic hardening : YH 5 Hn11 2 Hn 1 fHnn11 ΔΓn11 5 0 > > > ~  : Subloading surface : Ys 5 f M n11 2 Rn11 FðH n11 Þ 5 0 which is described explicitly for the Mises metals as follows:

8 i nh o

vp ~  ~ vp > YF 5Ftrial 2Fen11 exp N > n11 1η ant Mn11 Nn11 ΔΓ n11 Fn 5O n11 > > > n o > 

 > vp vp21 vp vp vp > > Yk 5Fvp > n11 2Fksn11 exp Fksn11 1=ðbk FÞ Mkn11 1ηk ant Mn11 Mkn11 ΔΓ n11 Fksn11 Fkdn 5O > > < h ~  vp i vp vp vp vp21 vp ~ Yc 5Fn11 2Fcsn11 exp Fcsn11 ΔΓ n11 M n11 1ηc An11 Fcsn11 Fcdn 5O > > > pffiffiffiffiffiffiffiffi > > > YH 5Hn11 2Hn 1 2=3ΔΓ n11 50 > > > > pffiffiffiffiffiffiffiffi ~ 0 > > : Ys 5 3=2 M n11 2Rn11 FðHn11 Þ50

ð9:278Þ where ΔΓ n11 

1 hexp½nðRn11 2 Rsn11 Þ 2 1i Δt μv 1 2 ½Rn11 =ðcm Rsn11 Þm

while the variable Rsn11 is updated in Eq. (7.301).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(9.279)

329

9.11 Multiplicative subloading-overstress model vp

vp

The five unknown variables Fen11 , Fksn11 , Fcsn11 , Hn11 , and Rn11 are ~ can be described by Fe , ~ and A involved in Eq. (9.278), while M, M , M , N, vp

k

vp

c

Fks , and Fcs as shown in Eqs. (9.125), (9.133), (9.137), (9.254), and (9.270). Eq. (9.278) is described in the matrix form as follows: YðXÞ 5 O where 9 8 YF > > > > > > > > > > > = < Yk > Y  Yc ; > > > > > > > YH > > > > > ; : Ys

(9.280)

8 e 9 F > > > > > > n11 > vp > > > > Fksn11 > > > = < vp X  Fcsn11 > > > > > > > > > Hn11 > > > > > ; : Rn11

(9.281)

In order to solve Eq. (9.280) numerically, linearizing it by means of the Taylor expansion and adopting its the first-order term, we have



YðXðk11Þ ÞDYðXðkÞ Þ 1 JðXðkÞ Þ dX 5 O

(9.282)

where 2 @Y F 6 @Fen11 6 6 @Y 6 k 6 e 6 @Fn11 6 6 @Y c 6 J56 6 @Fen11 6 6 @Y H 6 6 e 6 @Fn11 6 6 @Y S 4 @Fen11

@YF p @Fksn11 @Yk p @Fksn11 @Yc p @Fksn11 @YH p @Fksn11 @YS p @Fksn11

@YF p @Fcsn11 @Yk p @Fcsn11 @Yc p @Fcsn11 @YH p @Fcsn11 @YS p @Fcsn11

@YF @Hn11 @Yk @Hn11 @Yc @Hn11 @YH @Hn11 @YS @Hn11

@YF 3 @Rn11 7 7 @Yk 7 7 7 @Rn11 7 7 @Yc 7 7 7 @Rn11 7 7 @YH 7 7 7 @Rn11 7 7 @YS 7 5 @Rn11

(9.283)

Here, the partial-derivatives of the components in this matrix can be derived by exploiting Eq. (6.10) or by the numerical method. Eq. (9.280) is the simultaneous equation for the unknown dX, and thus, solving this equation, it is updated by

 21  ðkÞ 

 21  ðkÞ  dX 5 2 J XðkÞ Y X -Xðk11Þ 5 XðkÞ 1 dX 5 XðkÞ 2 J XðkÞ Y X (9.284) where ðkÞ designates the iteration number of iterations.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

330

9. Subloading-multiplicative hyperelastic-based plastic and viscoplastic constitutive equations

The iteration calculation in Eqs. (9.280) and (9.284) is continued until the residual becomes sufficiently small: that is, it is judged that the convergence of solution is attained when the following equation in terms of the residual norm is fulfilled. YðXðk11Þ Þ , TOL (9.285)

9.12 On multiplicative hyperelastic-based plastic equation in current configuration The multiplicative hyperelastic-based plastic equation in the intermediate configuration has been explained in detail in the preceding chapters. Prior to it, the multiplicative hyperelastic-based plastic equation in the current configuration has been studied by adopting the left Cauchy
Green elastic deformation tensor be , the Kirchhoff stress tensor τ and the plastic velocity gradient tensor lp in the current configuration e p instead of C , M, and L in the intermediate configuration by many researchers in the central contribution of Simo (1988a,b, 1992, 1998; Simo and Hughes, 1998) (see also Bonet and Wood, 2008; Bonet et al., 2016; de Souza Neto et al., 2008; Yamakawa et al., 2010). This approach results in the following defects. 1. The left elastic Cauchy
Green deformation tensor be is not influenced by the rigid-body rotation R added to the intermediate configuration, because of be 5 ðFe RT ÞðFe RT ÞT 5 be , while the right elastic Cauchy
Green deformation tensor Ce does not possess the objectivity, which is influenced by the rotation R , because of Ce 5 ðFe RT ÞT ðFe RT Þ 5 R Ce RT . Then, the hyperelastic equation is formulated in terms of the Kirchhoff stress tensor τ to be , resulting in the coaxiality of τ and be . Then, this formulation is limited to the elastic isotropy. 2. The Eulerian tensor lp in Eq. (8.23) is influenced by the elastic deformation so that a plastic deformation cannot be described exactly. 3. lp is formulated by the associated flow rule in terms of the symmetric tensor τ. However, this formulation leads to the neglect of the plastic spin and the plastic anisotropy. 4. All the variables τ, be , and lp become to be coaxial, possessing the same principal directions, so that they can be described in the spectral decompositions. Then, the calculation is performed in the principal directions, and thus it is greatly simplified as explained in detail

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

9.12 On multiplicative hyperelastic-based plastic equation in current configuration

331

in Bonet and Wood (2008). However, it is limited to the description of the elastically and plastically isotropic deformation. Eventually, the exact description of the finite elastoplastic deformation/rotation cannot be formulated by the multiplicative elastoplasticity in the current configuration.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

C H A P T E R

10 Subloading-friction model: finite sliding theory The reduction of the friction coefficient from the static to kinetic friction coefficient and the recovery of the friction coefficient due to the reduction of sliding velocity are the fundamental characteristics in friction phenomena, which are observed in the dry friction. Therein, the friction resistance decreases with the sliding velocity. On the other hand, the friction resistance increases with the sliding velocity in the fluid friction observed in the sliding between the lubricated contact surfaces. The subloading-friction model is explained first, which is capable of describing the dry friction rigorously. Subsequently, it is extended to the subloading-overstress friction model which is capable of describing the general friction behavior ranging from the dry to and the fluid frictions uniquely. The finite sliding behavior under the finite rotation of the contact surface can be described exactly by the subloading-friction and the subloading-overstress friction model. Therein, the hyperelastic-based plastic subloading-friction model is formulated, although the hypoelastic-based subloading-friction model, in which the elastic sliding is limited to be infinitesimal and the cumbersome time-integration of the contact traction rate is required, has been given in the past (Hashiguchi et al., 2005; Hashiguchi and Ozaki, 2008; Hashiguchi, 2017a).

10.1 History of friction models The sliding with the friction between solids is the irreversible phenomenon and thus it must be formulated within the framework of the elastoplasticity. Constitutive equations of friction within the framework of elastoplasticity were first formulated as the rigid-plasticity (Seguchi et al., 1974; Fredriksson, 1976). Subsequently, they were

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity DOI: https://doi.org/10.1016/B978-0-12-819428-7.00010-9

333

© 2020 Elsevier Inc. All rights reserved.

334

10. Subloading-friction model: finite sliding theory

extended to the elasto-perfect-plasticity (Michalowski and Mroz, 1978; Oden and Pires, 1983a,b; Curnier, 1984; Cheng and Kikuchi, 1985; Kikuchi and Oden, 1988; Wriggers et al., 1990; Peric and Owen, 1992; Anand, 1993; Mroz and Stupkiewicz, 1994; Wriggers, 2003). Further, they were extended incorporating the isotropic hardening (Oden and Martines, 1986; Gearing et al., 2001). However, the interior of the sliding-yield surface was assumed as an elastic domain, falling within the framework of the conventional elastoplasticity. Therefore, they possess the following defects. 1. The plastic sliding cannot be described for the rate of contact stress inside the friction-yield surface. The accumulation of the plastic sliding under the cyclic loading of contact stress inside the friction-yield surface cannot be described. 2. The plastic sliding rate is induced suddenly when the contact stress reaches the friction-yield surface, leading to the abrupt change of the tangent friction stiffness modulus at the friction-yield point. The judgment whether or not the contact stress reaches the friction-yield surface is required. 3. The input of sliding increment must be infinitesimal in the numerical calculation, since the pull-back operation of the contact stress to the friction-yield surface is not furnished. Incidentally, the friction models other than the subloading-friction model are incapable of describing 1. the reduction of friction resistance from the static to the kinetic friction and the recovery of friction resistance to the static friction under the stop of sliding; 2. the fluid (lubricated) friction behavior with the positive rate-sensitivity. The formulation of the friction behavior without the abovementioned limitations in the past models has been attained by the subloading-friction model (Hashiguchi et al., 2005; Hashiguchi and Ozaki, 2008; Hashiguchi, 2017a). However, it falls within the framework of the hypoelastic-based plasticity and thus it is limited to be the infinitesimal elastic deformation and obliged to perform the cumbersome time-integration of the contact traction rate. The hyperelastic-based subloading-friction model will be described in this chapter, which is capable of describing the finite sliding behavior under the finite rotation of the contact surface without the timeintegration of the contact stress rate.

10.2 Sliding displacement and contact traction vectors The sliding displacement vector u, which is defined as the sliding displacement of the counter (slave) body to the main (master) body, is Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

335

10.2 Sliding displacement and contact traction vectors

orthogonally decomposed into the normal sliding displacement vector un and the tangential sliding displacement vector ut to the contact surface as follows: u 5 un 1 ut where




(10.1)



un 5 ðu nÞn 5 ðn  nÞu 5 2 un n ut 5 u 2 un 5 ðI 2 n  nÞu

(10.2)

n being the unit outward-normal vector of the surface of main body and





u n  2 n un 5 2 n u

(10.3)

The minus sign is added for un to be positive when the counter body approaches the main body. The sliding displacement vector u can be exactly decomposed into the elastic sliding displacement ue and the plastic sliding displacement up in the additive form even for the finite sliding displacement, that is, u 5 ue 1 up

(10.4)

u 5 uen 1 uet p p up 5 un 1 ut

(10.5)

uen 5 ðue nÞn 5 ðn  nÞue 5 2 uen n uet 5 ue 2 uen 5 ðI 2 n  nÞue



(10.6)



(10.7)




where





p

e

un 5 ðup nÞn 5 ðn  nÞup 5 2 upn n p p ut 5 up 2 un 5 ðI 2 n  nÞup

setting





uen  2 n uen 5 2 n ue ;



p



upn  2 n un 5 2 n up

(10.8)

The elastic sliding displacement vector ue is formulated by the hyperelastic relation to the current contact stress vector f, where ue is calculated by subtracting the plastic sliding displacement up from the total sliding displacement vector u. Note here that the additive decomposition of the sliding displacement vector holds, although the deformation gradient tensor defined by the ratio of the current to the initial infinitesimal line-element vectors is obliged to be decomposed multiplicatively into the elastic and the plastic parts as studied in the preceding chapters. The finite elastoplastic sliding behavior can be described exactly by

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

336

10. Subloading-friction model: finite sliding theory

fn f

Counter (slave) body ˆe3 = n

·

ft

eˆ 2

·

un

eˆ1

ut

u·

Main (master) body

FIGURE 10.1 Contact traction and sliding velocity.

the hyperelastic-based plastic sliding formulation (Hashiguchi, 2018c). On the other hand, the hypoelatic-based plastic sliding formulation e relating the elastic sliding rate u_ to the contact stress rate f_ (Hashiguchi et al., 2005; Hashiguchi and Ozaki, 2008; Hashiguchi, 2017a) is limited to the description of infinitesimal elastic sliding and requires the cumbersome time-integration of the corotational rate of contact stress vector. The contact traction vector f acting on the main body is additively decomposed into the normal traction vector fn and the tangential traction vector ft as follows (see Fig. 10.1): f 5 fn 1 ft 5 2 f n n 1 f t t f where

(



fn  ðn fÞn 5 ðn  nÞf 5 2 f n n ft  f 2 fn 5 ðI 2 n  nÞf 5 f t tf

8 < fn  2 n f ðI 2 n  nÞf : f t  tf f 5 jjft jj; tf  jjðI 2 n  nÞfjj 5 2 ðn tf 5 0; jjtf jj 5 1Þ







(10.9)

(10.10)

(10.11)

The minus sign is added for f n to be positive when the compression is applied to the main body by the counter body. The contact traction vector f, fn , and ft can be calculated from the Cauchy stress σ applied in the contact bodies by virtue of the Cauchy’s fundamental theorem in Eq. (5.30) as follows: f 5 σn



fn 5 ðn σnÞn 5 ðn  nÞσn ft 5 ðI 2 n  nÞσn

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(10.12)

10.3 Hyperelastic sliding displacement

337

10.3 Hyperelastic sliding displacement Let the contact traction vector f be given by the hyperelastic relation with the elastic sliding displacement energy function ϕðue Þ as follows: f5

@ϕðue Þ @ue

(10.13)

The simplest function @ϕðue Þ is given by the quadratic form:



@ϕðue Þ 5 ue Eue =2

(10.14)

where the second-order symmetric tensor E is the elastic contact tangent T stiffness modulus fulfilling the symmetry E 5 E . The substitution of Eq. (10.14) into Eq. (10.13) leads to f 5 Eue

(10.15)

The inverse relation of Eq. (10.15) is given by 21

ue 5 E f

(10.16)

Assuming the isotropy on the contact surface and introducing the normalized rectangular coordinate system ð^e1 ; e^ 2 ; e^ 3 Þ 5 ð^e1 ; e^ 2 ; nÞ fixed to the contact surface, the elastic contact tangent stiffness modulus tensor E (second-order tensor) is given as follows: 8 > E 5 αt ðI 2 n  nÞ 1 αn n  n 5 αt ð^e1  e^ 1 1 e^ 2  e^ 2 Þ 1 αn n  n > < 1 1 1 1 21 > E 5 ðI 2 n  nÞ 1 n  n 5 ð^e1  e^ 1 1 e^ 2  e^ 2 Þ 1 nn > : αt αn αt αn (10.17) where αn and αt are the normal and tangential contact elastic moduli, respectively. Their values are quite large usually as 102 2 105 GPa=mm3 for metals because the elastic sliding is caused by elastic deformations of the surface asperities. Eqs. (10.15) and (10.16) with Eq. (10.17) leads to 8 > e e > > < f 5 α t ut 1 α n un (10.18) 1 1 e > > u 5 f 1 f t n > : α α t

n

Further, the contact stress and the elastic sliding displacement are described in the rectangular coordinate system as follows: 8 < f 5 f1 e^ 1 1 f2 e^ 2 1 fn n (10.19) : e u 5 ue1 e^ 1 1 ue2 e^ 2 1 uen n

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

338

10. Subloading-friction model: finite sliding theory

The substitution of Eqs. (10.17) and (10.19) into Eq. (10.15) leads to the matrix representation: 8 9 2 38 9 8 9 2 38 9 αt 0 0 < ue1 = < ue1 = 1=αt 0 0 < f1 = < f1 = ue2 5 4 0 f2 5 4 0 αt 0 5 ue2 ; 1=αt 0 5 f2 : ; : e; : e; : ; fn un un fn 0 0 αn 0 0 1=αn (10.20) Here, note that one does not need to adopt a corotational rate but one has only to use the time derivative for the contact traction vector f by the fact: The contact traction f is calculated from the hyperelastic equation with the substitution of the elastic displacement ue which is obtained by subtracting the plastic displacement vector up from the displacement vector u as will be explained in Section 10.8.

10.4 Normal-sliding and subloading-sliding surfaces Assume the following sliding-yield surface with the isotropic hardening/softening, which describes the sliding-yield condition. fðfÞ 5 μ

(10.21)

μ is the sliding hardening/softening function denoting the variation of the size of the sliding-yield surface. The friction-yield stress function fðfÞ for the Coulomb friction law is given by fðfÞ 5 f t =f n

(10.22)

for which μ specifies the coefficient of friction. The normal-contact stress dependence of the friction behavior incorporating the normal-contact stress dependent sliding-yield surface was formulated by Hashiguchi et al. (2005) with the application (Ozaki et al., 2007) and the anisotropic friction behaviour incorporating the orthotropic sliding-yield surface was formulated by Hashiguchi (2009) and verified by Ozaki et al. (2012). Now, incorporate the subloading surface concept: The platic sliding velocity denelops as the contact stress approaches the normal-sliding surface, renamed the sliding normal-yield surface. Then, introduce the sliding-subloading surface which is similar to the sliding normal-yield surface and passess through the current contact stress. Further, introduce the sliding normal-yield ratio defined by the ratio of the size of the sliding-subloading surface to that of the sliding normal-yield surface, which play the role to designate the approaching degree of the contact stress to the sliding-normal yield surface. The sliding-subloading surface is represented by the following equation (see Fig. 10.2).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

339

10.5 Evolution rule of friction coefficient

fn

f

eˆ 3 =n 1

r eˆ 1

∂f (f ) ∂f

eˆ 2

tf

(n · ∂f∂(ff ) ) n

ft nt

( ∂f∂(f ) )t ≡ ∂f∂(ff ) − (n · ∂f∂(ff )) n f

Main body

Sliding subloading surface Sliding normal-yield surface

ft / fn = rμ

ft / fn = μ

FIGURE 10.2

Coulomb-type sliding normal-yield and sliding subloading surfaces.

fðfÞ 5 rμ

(10.23)

where r ð0 # r # 1Þ is the sliding normal-yield ratio. The time differentitation of Eq. (10.23) leads to the consistency condition for the sliding subloading surface: @fðfÞ _ : f 5 rμ_ 1 r_μ @f

(10.24)

Here, it is required to formulate the evolution rules of the friction coefficient μ and the sliding normal-yield ratio r as will be done in the subsequent sections.

10.5 Evolution rule of friction coefficient Consider the dry friction without any medium between contact surfaces. The friction resistance is caused by the destruction of the adhesion between surface asperities in a metal friction as has been pointed out by Boden and Tabor (1950). On the other hand, it is caused by pushing out the surface asperities of counter body in most materials (e.g., rubbers, woods, plastics, stones) other than metals. The following facts can be recognized commonly in the dry friction. 1. First, the friction resistance reaches the highest friction coefficient, that is, the so-called static friction just after the initiation of sliding.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

340

10. Subloading-friction model: finite sliding theory

2. Thereafter, the friction coefficient decreases with the sliding displacement from the static to the kinetic friction by the decrease of the asperity engaging. 3. However, the friction coefficient recovers by the reconstruction of asperity engaging under the stop of sliding. 4. The faster is the sliding, the fewer the reconstruction of the asperity engaging, resulting in the negative rate-sensitivity, that is, the decrease of friction resistance with the increase of sliding velocity. In contrast, the positive rate-sensitivity is induced in the fluid friction as will be described in Section 10.10. Based on the above-mentioned physical interpretation, let the evolution rule of the friction coefficient μ be given as follows (Hashiguchi, 2005, 2017a, 2018c; Hashiguchi and Ozaki, 2008):   μ μ p 2 1 jju_ jj 1 ξð1 2 Þ μ_ 5 2 k μk μ |fflfflfflfflfflffl{zfflfflfflfflfflffls} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Negative

(10.25)

Positive

  where μs and μk μs $ μ $ μk are material constants designating the maximum and minimum values of μ for the static friction and the kinetic friction, respectively. κ is a material constant specifying the rate of decrease of μ in the plastic sliding process, and ξ is a material constant specifying the rate of recovery of μ as time elapses. Eq. (10.25) is rewritten in the incremental form as follows: μ μ dμ 5 2 κð 2 1Þjjdup jj 1 ξð1 2 Þdt μk μs |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflffl ffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Negative

(10.26)

Positive

The first and second terms in Eq. (10.25) or (10.26) are relevant to the decrease and the increase of the friction coefficient by the progress and the cease, respectively, of the sliding.

10.6 Evolution rule of sliding normal-yield ratio The evolution rule of the sliding normal-yield ratio is given as follows (Hashiguchi, 2017a): p p r_ 5 UðrÞ:u_ :for u_ 6¼ 0

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(10.27)

10.6 Evolution rule of sliding normal-yield ratio

341

U (r )

·

·

p e u = 0, u ≠ 0

·

p u ≠0

0

FIGURE 10.3

1

·

p u ≠0

r

Function UðrÞ for the evolution rule of the sliding normal-yield ratio r.


e 5 0 for u_ 5 0 r_ e , 0 for u_ 6¼ 0

p for u_ 5 0

(10.28)

where UðrÞ is the monotonically increasing function of r fulfilling the conditions (see Fig. 10.3). 8 - 1 N for r 5 0 ðquasi-elastic sliding stateÞ > > < . 0 for 0 , r , 1 ðsliding subyield stateÞ (10.29) UðrÞ 5 0 for r 5 1 ðsliding normal-yield stateÞ > > : ð , 0 for r . 1 : over sliding normal-yield stateÞ The explicit example of UðrÞ is ~ UðrÞ 5 ucot

π r ; 2

(10.30)

where u~ is a material constant. Eq. (10.28) with Eq. (10.30) can be analytically integrated for the monotonic sliding process as follows: n h π io 2 π 2 1 cosðπ2 r0 Þ ~ p 2 up0 Þ ; up 2 up0 5 In ; r 5 cos21 cosð r0 Þexp 2 uðu cosðπ2 rÞ π 2 2 π u~ Ð

(10.31)

where up ð 5 jju_ jjdtÞ, and r0 and are the initial values of r and up , respectively. The adoption of the analytical integration in Eq. (10.31) would be beneficial for the return-mapping in numerical calculations. The general trend of the effect of u~ on UðrÞ is shown schematically in Fig. 10.4. The contact stress is automatically attracted to the sliding normalyield surface in the plastic sliding process and it is pulled back to that surface even when it goes over the surface in numerical calculation because of r_ , 0 for r . 1 from Eq. (10.28) with Eq. (10.29)4 as seen in Fig. 10.5. p

p u0

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

342

10. Subloading-friction model: finite sliding theory

FIGURE 10.4 Influence of the material parameter u~ on contact stress versus sliding distance curve.

ft / f n

· · 0 r

0

·

up

r· = U ( r) || u· p || for u· p ≠ 0 U (r ) > 0 for r < 1 ⎫ ⎪ U (r ) = 0 for r = 1 ⎬ U (r ) < 0 for r > 1 ⎪⎭

FIGURE 10.5 Contact stress controlling function in the subloading-friction model: contact stress is automatically attracted to sliding normal-yield surface in plastic sliding process.

10.7 Plastic sliding velocity The substitution of Eqs. (10.25) and (10.27) into Eq. (10.24) leads to

 @fðfÞ _ μ μ p p f 5 r 2 κð 2 1Þjju_ jj 1 ξð1 2 Þ 1 UðrÞjju_ jjμ (10.32) @f μk μs



Now, assume that the direction of plastic sliding velocity is tangential to the contact plane and outward-normal to the curve generated by the intersection of the sliding subloading surface and the constant normal traction plane fn 5 const:, leading to the tangential associated flow rule (see Fig. 10.2):



p p p _ _ _ u_ 5 λnt ðλ $ 0Þ ðjju_ jj 5 λ; n u_ 5 0Þ

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(10.33)

343

10.7 Plastic sliding velocity

where

with

 nt 

@fðfÞ @f





@fðfÞ @f



 

@fðfÞ





@f t t



ðjjnt jj 5 1; n nt 5 0Þ

(10.34)

  @fðfÞ @fðfÞ @fðfÞ 2 n  n 5 ðI 2 n  nÞ @f @f @f

(10.35)



t

where λ_ and nt are the magnitude and the direction, respectively, of the plastic sliding velocity. The substitution of Eq. (10.33) into Eq. (10.32) reads:

 @fðfÞ _ μ μ _ _ f 5 r 2 κð 2 1Þλ 1 ξð1 2 Þ 1 UðrÞλμ (10.36) @f μk μs



that is, @fðfÞ _ _ p f 5 λm 1 mc @f



(10.37)

where mp  2 κð

μ μ 2 1Þr 1 UðrÞμ; mc  ξð1 2 Þr ð $ 0Þ μk μs

(10.38)

are relevant to the plastic and the creep sliding velocity, respectively. It is obtained from Eqs. (10.33) and (10.37) that @fðfÞ _ @fðfÞ _ f 2 mc f 2 mc p _λ 5 @f @f _ ; u 5 nt mp mp





(10.39)

Substituting the rate form of Eq. (10.16) and Eq. (10.39) into the rate form of Eq. (10.4), the sliding velocity is given by @fðfÞ _ f 2 mc _u 5 E21 f_ 1 @f nt mp



(10.40)

The plastic multiplier in terms of the sliding velocity, denoted by the _ is given from Eq. (10.40) as symbol Λ, _5 Λ

@fðfÞ @fðfÞ E u_ 2 mc E u_ 2 mc @f _up 5 @f nt @fðfÞ @fðfÞ mp 1 E nt mp 1 E nt @f @f









Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(10.41)

344

10. Subloading-friction model: finite sliding theory

The inverse relation of Eq. (10.40) is given by substituting the rate form of Eq. (10.4) with Eq. (10.41) into the rate form of Eq. (10.15) as follows: 0 1 @fðfÞ Ent  E mc B C_ @f f_ 5 @E 2 Ent (10.42) Au 1 @fðfÞ @fðfÞ mp 1 E nt mp 1 E nt @f @f





noting 0

1 @fðfÞ E u_ 2 mc B C f_ 5 E@u_ 2 @f nt A @fðfÞ mp 1 E nt @f





The loading criterion is given as follows:
p _ .0 u_ 6¼ 0 for Λ p u_ 5 0 for other or

8 > < u_ p 6¼ 0 for @fðfÞ E u_ 2 mc . 0 @f > : _p u 5 0 for other



(10.43)

(10.44)

The following relations hold for the Coulomb friction condition in Eq. (10.22). f @fðfÞ @fðfÞ 1 5 2 2t ; 5 @f n fn f n @f t   f @fðfÞ 1 5 tf 1 t n @f fn fn   @fðfÞ 1 5 tf @f t f n nt 5 t f 5

ft ðI 2 n  nÞf 5 jjft jj jjðI 2 n  nÞfjj

(10.45) (10.46) (10.47) (10.48)

The rheological model of the subloading-friction model is shown in Fig. 10.6. Some numerical experiments (Ozaki and Hashiguchi, 2010) for the linear sliding process for subloading-friction model formulated above are shown in the following. The seven material constants of μs , μk , κ, ξ,

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

345

10.7 Plastic sliding velocity

ξ (1 −

μ μs ) dt

Hardening

μ

− κ (μ − 1) |d u vp |

μ − μ0

k

f

Softening

μ0

Plastic sliding

u e = f /E

u vp

ue u

FIGURE 10.6

Rheological model for subloading-friction model.

•

u t = 0.0001mm/s

0.4

•

u t = 0.001mm/s •

ft /fn

0.3

u t = 0.01mm/s •

u t = 0.1mm/s

0.2

0.1 0.0

FIGURE 10.7

0

0.02

0.04 0.06 ut [mm]

0.08

0.1

Influence of sliding velocity on friction (Ozaki and Hashiguchi, 2010).

~ αn , and αt and the initial value μ0 of the friction coefficient are chosen u, as follows: μ0 5 μs 5 0:4; μk 5 0:2 κ 5 10 mm21 ; ξ 5 0:01=s u~ 5 1000 mm21 αn 5 αt 5 1000 kN=mm3 The influence of the sliding velocity on the relation of the traction ratio ft =fn versus the tangential sliding displacement ut is shown in Fig. 10.7 (Ozaki and Hashiguchi, 2010), where u_ t  jju_ t jj. Smooth transitions from the static friction to the kinetic friction and the decreases of the friction coefficient are depicted. Faster the decrease of friction coefficient is shown for the higher sliding velocity.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

346

10. Subloading-friction model: finite sliding theory

FIGURE 10.8

Influence of stationary time on recovery of friction (Ozaki and

Hashiguchi, 2010).

The recovery of the static friction coefficient from the kinetic friction with the elapsed time t after the stop of sliding (Ozaki and Hashiguchi, 2010) is shown in Fig. 10.8. In the calculation, the constant sliding velocity u_ t 5 0:1 mm=s is given in the first stage reaching the kinetic friction and then the tangential contact traction is unloaded to zero. After the cessation of sliding for several elapsed times, the same sliding velocity in the first stage is given again. The recovery is larger for a longer stationary time. The comparisons with test data for the monotonic sliding and the unloadingreloading sliding processes are described in the literature (Hashiguchi, 2017a). The numerical experiments of the cyclic sliding processes are also shown in this literature.

10.8 Calculation procedure The exact calculation of the contact stress can be performed through the hyperelastic relation, while the cumbersome calculation procedure for the time-integration of the corotational contact stress rate in the hypoelastic relation is not required. It will be shown for the sliding processes under the rotation/deformation of the contact surface in the following. The sliding displacement is calculated by Ð the time-integration of the _ input value of the sliding rate, that is, u 5 udt. The plastic sliding displacement is calculated by the time-integration of the plastic sliding Ð p rate, that is, up 5 u_ dt based on Eq. (10.41). The friction coefficient μ is p updated by substituting the plastic sliding rate u_ into Eq. (10.25).

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

347

10.9 Return-mapping

Further, the contact traction f is calculated by the hyperelastic relation in Eq. (10.13) as f5

@ϕðu 2 up Þ @u

(10.49)

the simplest form of which is given for Eq. (10.15) as follows: f 5 Eðu 2 up Þ

(10.50)

Therefore, the cumbersome operation for the time-integration of the corotational rate of contact traction is not required. Eq. (10.50) is specialized for the elastic modulus tensor in Eq. (10.17) as follows: f 5 αt ðI 2 n  nÞðu 2 up Þ 1 αn ðn  nÞðu 2 up Þ      p p 5 αt u1 2 u1 e^ 1  e^ 1 1 u2 2 u2 e^ 2  e^ 2 1 αn un 2 upn n  n (10.51) noting Eue 5 αn uen 1 αt uet 5 αn ðn  nÞue 1 αt ðI 2 n  nÞue Here, note that the rate form in Eq. (10.42) is not necessary in the calculation procedure by the hyperelastic equation described in this section.

10.9 Return-mapping The return-mapping equations for the subloading-friction model will be formulated in this section.

10.9.1 Return-mapping formulation p

Suppose that the contact stress fn , the plastic displacement un , and the friction coefficient μn in the n-th step are already known. Firstly, the trial contact stress ftrial n11 is calculated by inputting the displacement increment Δu, postulating that only the elastic displacement is induced. This process and the trial contact stress calculated in this process are called the elastic trial (or predictor) step and the elastic trial contact stress, respectively. Designating the trial stress and the elastic displacement calcuetrial lated in this step by ftrial n11 and un11 , respectively, they are related by the hyperelastic relation as follows: p

etrial ftrial n11 5 Eun11 5 Eðun11 2 un Þ ptrial

p

un11 5 un ;

μtrial n11 5 μn

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(10.52) (10.53)

348

10. Subloading-friction model: finite sliding theory

where the sliding displacement un11 is fixed leading to p

un11 5 un 1 Δun11 5 un 1 uetrial n11 5 const:

(10.54)

throughout the plastic corrector process until the calculation step n 1 1 is finished, which will be described below. The following simultaneous equation must be fulfilled in each loading step. 8 Contact stress variation: > > > p > trial > Yf 5 fn11 2 Euen11 1 EΔun11 5 0 > > > > < Isotropic hardening rule: (10.55) > Yμ 5 μn11 2 μn 1 Δμn11 5 0 > > > > > > Subloading-friction surface: > > : Ys 5 fðfÞ 2 rn11 μn11 5 0 which is explicitly described for the Coulomb friction surface given in Eq. (10.22) with Eqs. (10.25) and (10.33) as follows: 8 Contact stress variation: > > > > trial > Yf 5 fn11 2 Euen11 1 αt nt n11 Δλn11 5 0 > > > > > Isotropic hardening rule: > > > > < μ μ Yμ 5 μn11 2 μn 2 κð n11 2 1ÞΔλn11 1 ξð1 2 n11 ÞΔtn11 5 0 (10.56) μk μs > > > > > > Subloading friction surface: > > > > > f t n11 > > 2 rn11 μn11 5 0 > : Ys 5 f n n11 The three unknown variables fn11 , μn11 , and Δλn11 are involved in the simultaneous equation in Eq. (10.56). Eq. (10.56) is described in the matrix form as follows:

where

YðXÞ 5 0

(10.57)

8 9 8 9 < Yf =

> 5n  n 5 > < @f @f @ft @½ðI 2 n  nÞf > > 5 ðI 2 n  nÞI 5 I 2 n  n > : @f 5 @f 8 @f n 5 @ð2 f nÞ > > 5 2 nI 5 2 n > @f < @f



@ft @jjft jj @jjft jj @ft ft @½ðI-n  nÞf ft > > 5 ðI 2 n  nÞ  tf > : @f 5 @f 5 @f @f 5 jjf jj @f jjft jj t t @ðf t =f n Þ @ðf t =fÞf n 2 @ðf n =fÞf t f 1 5 5 ðtf 1 t nÞ fn @f fn2 fn   @ðf t =f n Þ 1 5 tf f @f t n nt 5

tf ft =jjft jj ft 5 5 5 tf jjtf jj jjft =jjft jjjj jjft jj

1 @jjvjj @ðv vÞ1=2 1 v 21=2 5 5 ðv vÞ I v325 C B 2 jjvjj @v C B @v C B   B @ v ð@v=@vÞjjvjj 2 v  ð@jjvjj=@vÞ Ijjvjj 2 v  ðv=jjvjjÞ C C B 5 5 C B @v jjvjj jjvjj2 jjvjj2 C B C B C B A @ 5 I 2 ðv=jjvjjÞ  ðv=jjvjjÞ jjvjj 0



@nt @ 5 @ft @f





ft jjft jj





@ft I 2 ðft =jjft jjÞ  ðft =jjft jjÞ 5 ðI 2 n  nÞ jjft jj @f I 2 n  n 2 ðft =jjft jjÞ  ðft =jjft jjÞ 5 jjft jj

The normal frictionyield ratio r is updated by the analytical integrap tion in Eq. (10.31) setting r0 5 r0n11 and up 2 u0 5 Δλn11 as follows:

  π  2 π 21 ~ n11 rn11 5 cos cos r0 exp 2 uΔλ (10.71) π 2 2 The iteration calculation in Eq. (10.61) is continued until the residual becomes sufficiently small: that is, it is judged that the convergence of solution is attained when the following equation in terms of the residual norm is fulfilled. jjYðXðk11Þ Þjj , TOL

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(10.72)

10.9 Return-mapping

351

10.9.2 Loading criterion Analogously to the subloading surface model for the elastoplastic deformation described in Section 7.6.2, the plastic sliding increment is induced if the sliding subloading surface once shrinks and then expands again even if the trail contact stress increment in the elastic trial step is directed toward the inside of the current sliding subloading surface as shown in Fig. 10.9. The pertinent loading criterion for the subloadingfriction model will be shown below (Hashiguchi, 2018d). The following facts for the loading criterion should be noticed for the formulation of loading criterion, referring to Fig. 10.10. The subscript n is added to the variables at the end of the step n and the subscript n 1 1 is added to the variables at the step n 1 1.

FIGURE 10.9 Defects of past loading criterion in subloading-friction surface model. (A) Elastic trial steps which are judged as elastic deformation process by the past loading criterion. (B) Cyclic loading behavior predicted by past and corrected loading criterions.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

352

10. Subloading-friction model: finite sliding theory

FIGURE 10.10 Loading criterion when elastic trial contact stress increment is directed inward of subloading-friction surface at step n in return-mapping method for subloadingfriction model shown in the hydrostatic plane. p

“The plastic sliding increment Δun11 is not induced if the contact trial stress increment Δftrial n11  fn11 2 fn makes an obtuse angle with the outward-normal of the sliding subloading surface in the elastic trial step n 1 1. Otherwise, it is induced.” Then, the loading criterion in the return-mapping method for the subloading-friction model is given as follows: Loading criterion in return 2 mapping for subloading 2 friction model 8 p trial trial trial

> > >     > > μn11 μn11 > > > μn11 5 μn 2 κ 2 1 ΔΓ n11 1 ξ 1 2 Δt > > μk μs > > < 1 hexp½nðrn11 2 rs n11 Þ 2 1i > > Δt ΔΓ n11  > > rn11 > μv > 1 2 > > cm rs n11 > > > > > : fðfn11 Þ 2 rn11 μn11 5 0

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

(10.97)

10.12 On crucially important applications of subloading-friction model

361

The four unknown variables fn11 , μn11 , rn11 , and ΔΓ n11 can be determined by solving the simultaneous Eq. (10.97), where rsn11 is calculated by rsn11 5

i 2 21 h π π cos cosð r0sn11 Þexpð2 u ΔΓ n11 Þ π 2 2

(10.98)

In the quasistatic sliding behavior, the equality rn11 5 rsn11 holds and thus the number of equations is reduced to three from four. The implicit stress integration for the rate-independent subloadingfriction model is performed by the method: The increment of the plastic multiplier ΔΛn11 is calculated such that the contact stress goes down to the subloading-sliding surface. Therefore, it is not necessary to calculate the concrete equation of the plastic multiplier in Eq. (10.41) involving the plastic modulus. On the other hand, the calculation of the viscoplastic multiplier ΔΓ n11 must be performed in its concrete form in Eq. (10.88) or (10.89), because the contact stress is not settled down to the subloading surface with rs which evolves by the viscoplastic sliding in Eq. (10.91) but it must be settled down to the sliding subloading surface with r. The subloading-overstress friction model is regarded as the generalization of the elastoplastic sliding constitutive equation to the deformation behavior at the general rate of sliding from the static to the impact sliding behavior. Nevertheless, it is much simpler than the original elastoplastic sliding constitutive equations with the complex plastic modulus.

10.12 On crucially important applications of subloadingfriction model The friction phenomena can be analyzed by the subloading-friction model accurately. There are various friction phenomena which cannot be analyzed without using the subloading-overstress friction model. The typical examples will be delineated in the following.

10.12.1 Loosening of screw The loosening of the screw from the nut is observed often even if the friction between the screw and the nut is lower than the frictionyield condition. This phenomenon cannot be analyzed by the

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

362

10. Subloading-friction model: finite sliding theory

conventional friction model based on the assumption that the sliding does not occur until the contact stress reaches the friction-yield surface. The gradual loosening of the screw can be predicted by the subloading-friction model which is capable of describing the plastic sliding rate by the change of the contact stress inside the normal friction-yield surface.

10.12.2 Deterministic prediction of earthquake occurrence The rate-and-state friction model (cf., e.g., Dieterich, 1978; Ruina, 1980, 1983; Rice and Ruina, 1983; Scholz, 1998; Rice et al., 2001) the basic idea of which is the dependence of the contact shear stress or friction coefficient on the rate of sliding and some state variables has been widely used for the prediction of earthquake phenomena. An earthquake is a typical irreversible phenomenon which can be described appropriately by elastoplasticity but the rate-and-state friction model is not based on elastoplasticity which is premised on (1) decomposition of the rate of deformation or sliding into the reversible, that is, elastic part and the irreversible, that is, plastic part, (2) incorporation of the yield condition, and (3) the potential flow rule of plastic strain rate or plastic sliding rate. Therefore, the rate-and-state friction model possesses various defects as follows: 1. It is limited to the description of one-dimensional sliding behavior and inapplicable to the two-dimensional sliding behavior in which the sliding direction changes, because whether the rate of plastic sliding is induced cannot be judged rigorously as far as the yield surface is not incorporated. 2. The hardening/softening of friction coefficient cannot be described appropriately so that the transition from the static to the kinetic friction and the recovery of the static friction cannot be described appropriately. The rate-and-state model would be easily understood and felt familiar by the workers who are unfamiliar to the elastoplasticity. It should be noted that the rate-and-state model is not a physical model but just an empirical law or an ad hoc model. The earthquake is regarded as the stick-slip phenomenon, that is, the intermittent sliding phenomenon. The subloading-friction model is capable of describing the stick-slip phenomenon rigorously (cf. Ozaki and Hashiguchi, 2010; Hashiguchi, 2017a). To establish the deterministic prediction of earthquake occurrence, the deformation/sliding behaviors of the continent plates must be analyzed as the elastoplastic

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

10.12 On crucially important applications of subloading-friction model

363

(viscoplastic) deformation/sliding by the numerical method, for example, the finite element method, the boundary element method, etc., with the subloading(-overstress) surface model and the subloading-friction model, utilizing the supercomputers.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

C H A P T E R

11 Comments on formulations for irreversible mechanical phenomena Some comments on the formulation/analyses of irreversible mechanical phenomena of solids are given for the sound development of the irreversible mechanics. First, the present state on the utilization of the subloading surface model is described and further utilization is recommended for the further development of the mechanical designs in engineering practice. Incidentally, the disuse of the rate-independent elastoplastic constitutive model and instead the utilization of the subloading-overstress model are recommended, because the former is involved in the latter. Besides, it is described that the formulations of plastic flow rules based on the second law of thermodynamics is not physically meaningful and thus it should be discontinued towards the true development of the irreversible mechanics.

11.1 Utilization of subloading surface model The applications and the usages of the subloading surface model are briefly explained in this section.

11.1.1 Mechanical phenomena described by subloading surface model Among various cyclic plasticity models, only the subloading surface model is free from the existence of the loading surface enclosing a purely elastic domain but instead incorporates the subloading surface, which is determined from the normal-yield surface (conventional yield surface) by the geometrical similarity. Further, the normal-yield ratio R (r

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity DOI: https://doi.org/10.1016/B978-0-12-819428-7.00011-0

365

© 2020 Elsevier Inc. All rights reserved.

366

11. Comments on formulations for irreversible mechanical phenomena

for sliding behavior), which plays the fundamental role to describe the approaching-degree to and/or the protruding-degree from the normalyield surface, is incorporated in this model. It is inevitable for the rigorous formulations of irreversible deformation of solids, including the elastoplastic and the viscoplastic deformations. The subloading surface model is capable of describing the smooth elastic-plastic transition and the continuous variation of the tangent stiffness modulus tensor. Further, it does not require the yield judgment and possesses the controlling function to attract the stress to the yield surface. The mechanical phenomena which have been described rigorously by the subloading surface model are enumerated below. 1. Monotonic/cyclic elastoplastic deformation behaviors of solids, for example, metals (Hashiguchi, 1980, 1989, 1993a,b, 1997, 2017a; Hashiguchi et al., 2012; Hashiguchi and Yoshimaru, 1995; Hashiguchi and Yamakawa, 2012; Hashiguchi and Ueno, 2017; Fincato and Tsutsumi, 2017, 2018; MSC Soft. Corp., 2017), soils (Hashiguchi, 1978, 1980, 2018b; Topolnicki, 1990; Hashiguchi and Chen, 1998; Hashiguchi et al., 2002; Zhao, 2005; Hashiguchi and Mase, 2007; Wongsaroj et al., 2007; Farias et al., 2009; Yuanming et al., 2009; Yamakawa et al., 2010; Yadav et al., 2019; Pedroso, 2014; Noda et al., 2013; Lin et al., 2015; Sakai and Nakano, 2015; Yamada et al., 2015; Zhou and Zhang, 2015; Zhang and Zhou, 2016; Ghasemzadeh et al., 2017; Zhang et al., 2018; Fuente, 2019; Gang, 2019; Rojas, 2019; Xiong et al., 2019; Yadav et al., 2019; etc.), rocks (Fu et al., 2012; Zhu et al., 2013; Xiong et al., 2014, 2017; Zhou et al., 2019; 2020; etc.), coal (Liu et al., 2019), and concrete (Hashiguchi and Mase, 2018). Unfortunately, however, the application to the description of metal behavior is stagnating by the appearance and the diffuse of the easy-going ad hoc Chaboche model and the Ohno-Wang model as was described in Section 7.4.2; 2. Monotonic/cyclic viscoplastic deformation of solids (Hashiguchi, 2017a); 3. Damage of solids (Hashiguchi, 2015, 2017a; Yuanming et al., 2009; Hashiguchi and Mase, 2018), which requires the description of the smooth elastic-plastic transition because it exhibits the remarkable softening behavior. Incidentally, the Gurson model (Gurson, 1977) is extended to the subloading-Gurson model (Hashiguchi, 2017a); 4. Fatigue of metals cannot be predicted without incorporating the subloading surface model as reviewed in Toyosada (2015), in which cyclic loading with small stress amplitude is applied. However, the formulation of constitutive equation for high cycle fatigue phenomenon is quite difficult or impossible at present because when and where a crack starts cannot be captured by the present level of natural science;

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

11.1 Utilization of subloading surface model

367

5. Phase transformation of metals (Hashiguchi, 2017a), which requires the accurate description of the plastic strain rate with the smooth elastic-plastic transition; 6. Dry and wet friction behaviors of solids (Hashiguchi, 2009, 2013b, 2017a, 2018c; Hashiguchi et al., 2005; Hashiguchi and Ozaki, 2008; Hashiguchi et al., 2016); 7. Crystal plasticity of metals, which requires the analyses of slips between numerous number of crystal systems (2013a,b, 2016a, 2017a, 2018c; Okamura and Hashiguchi, 2015; Hashiguchi and Okamura, 2019), although the creep model which cannot be accepted mechanically has been used widely after Peirce et al. (1982, 1983); 8. Multiplicative hyperelastic-based plastic constitutive equations (Hashiguchi and Yamakawa, 2012; Hashiguchi, 2013a,b, 2016a, 2017a, 2018c). It has been attained by incorporating the subloading surface model for the cyclic loading behavior. However, it has never been formulated and will never been attained by incorporating the other cyclic plasticity models which adopt the plural independent yield surfaces (multisurface model: Mroz, 1967; Iwan, 1967; two-surface model: Dafalias, 1975; Krieg, 1975) or the plural kinematic hardening rules (superposed-kinematic hardening model: Chaboche et al., 1979; Ohno and Wang, 1993) leading to the complexity and/or the limitation to the Mises metals even in the current configuration. Consequently, the subloading surface model is regarded as the governing law of irreversible mechanical phenomena of solids, which is capable of describing the global mechanical behaviors expanding from the micro to the macro levels.

11.1.2 Standard installation to commercial software The subloading surface model for metals has been incorporated to the commercial software “Marc” in MSC Software Corporation as the standard installation by the name “Hashiguchi model”, which can be used by all Marc users (contractors) since October, 2017, which can be referred to the Marc user manual (MSC Software Corporation, 2017). Further, the subloading-friction model also will be incorporated to Marc as the standard installation until the end of 2020. The subloading surface model is concerned with the general elastoplastic deformation/sliding behaviors of solids unlimited to a particular material in contrast to the superposed kinematic hardening rules (Chaboche et al., 1979; Ohon-Wang, 1993). Therefore, the deterimination of the material parameters requires the learning on the expilicit formulation of the subloading surface model. However, the new function for the automatic determinations of the material parameters in the

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

368

11. Comments on formulations for irreversible mechanical phenomena

subloading surface model by an input of stress-strain curve is added to the Hashiguchi model in Marc from June, 2019. Now, the highly accurate and efficient function of the subloading surface model can be utilized by the Marc users even if they are unaware of the detailed formulation of the subloading surface model.

11.2 Disuses of rate-independent elastoplastic constitutive equations The rate-independent elastoplastic constitutive equations for deformation and sliding have been used hitherto. They can be described as the particular case of the quasistatic deformation/sliding phenomena by the overstress model. In other words, the overstress constitutive equations are regarded as the generalizations of the elastoplastic constitutive equation to the deformation/sliding phenomena at the general rate of deformation/sliding from the quasistatic to the impact deformation/ sliding. Nevertheless, they are much simpler than the original plastic constitutive equations with the complex plastic moduli, which are used for the explicit forward-Euler numerical calculations. Consequently, in the deformation analyses of the rate-independent elastoplastic deformation/sliding analyses, the elastoplastic constitutive equations for the elastoplastic deformation/sliding with the plastic moduli, which are used for the forward-explicit stress integration methods in numerical calculations, that is, dp 5

M 1 n:E:n

n:Eq: ð7:164Þ

e

e

N:L :sym½C L

p

D 5

n:E:d p

p

e

e

M 1 N:L :sym½C ðN 1 ηp ant½M NÞ

:Eq: ð9:91Þ with Eq: ð9:115Þ

and p u_ 5

@fðfÞ E u_ 2 mc @f nt :Eq: ð10:41Þ @fðfÞ mp 1 E nt @f





should be disused. Then, they will be replaced by the subloadingoverstress models which are capable of describing the elastoplastic deformation/sliding behavior in the general rate from the quasistatic to the impact loadings. Consequently, the calculation by the elastoplastic constitutive equation with the quite complex plastic modulus (see

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

11.3 Impertinence of formulation of plastic flow rule based on second law of thermodynamics

369

Eq. (9.116) in particular) is not necessary even for the time-independent elastoplastic deformation analysis with the forward-Euler method by calculating it as a quasi-static deformation using the subloading-overstress model.

11.3 Impertinence of formulation of plastic flow rule based on second law of thermodynamics The hyperelasticity based on the elastic potential energy is adopted in the formulations of constitutive equations for the damage, the phase transformation phenomena, and the finite elastic plasticity, since the large elastic deformation is concerned in them. It is based on the first law of thermodynamics, that is, the energy conservation law. On the other hand, the derivations of the flow rules for the plastic strain rate and the dissipative rates of internal variables in the potential types from the second law of thermodynamics, that is, the ClausiusDuhem inequality are fashioned widely (cf., e.g., Lemaitre and Chaboche, 1990; Lemaitre, 1996; Collins and Houlsby, 1997; Holzapfel, 2000; Haupt, 2002; Wallin et al., 2003; Han et al., 2003; Dettmer and Reese, 2004; Tsakmakis, 2004; Houlsby et al., 2005; Voyiadjis and Kattan, 2005; Voyiadjis et al., 2008; Wallin and Ristinmaa, 2005; de Souza Neto et al., 2008; Vladimirov et al., 2008, 2010; Gurtin et al., 2010; Murakami, 2012; Belytschko et al., 2014). Not a few spaces are wasted for the thermodynamic explanations without a substantial significance in these literatures. However, it would be reckless to expect that a valuable information can be extracted in order to formulate the explicit constitutive equation for the irreversible deformation of solids, which exhibit complex deformation behaviors contrastive to the simple behaviors, e.g., the volumetric change in the ideal gas and the entropy-elastic deformation (NeoHookean elasticity) in rubbers from the second law of thermodynamics, that is, the ClausiusDuhem inequality or the principle of increase of entropy, which is to be the universal law governing all the phenomena in the natural world (cf., Hashiguchi, 2001, 2017a). Incidentally, the principle of maximum dissipation is a postulate so that the derivation of the associated flow rule from this principle is merely to be one of the interpretations of the particular flow rule. It is comparable to the positivity of the work done by the stress cycle (Drucker, 1950) and the positivity of the work done by the strain cycle (Ilyushin, 1961) which are the postulates possessing the particular physical meanings. Only the stress for the elastic strain and internal variables for storage parts of plastic strain are required the formulations by the potential

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

370

11. Comments on formulations for irreversible mechanical phenomena

functions, which are based on the first law of thermodynamics independent of the second law. In contrast, the potential theory would not hold exactly for the irreversible phenomena in general, although only the potential theory has been formulated by quoting the second law of thermodynamics. Note that only a rule with the generality has to be formulated based on the universal law: the second law of thermodynamics. Then, the derivation of the potential theory for the irreversible phenomena would be misuse of the second law of thermodynamics. In fact, any flow rule which can be proved to violate the second law of thermodynamics has never been proposed hitherto. In order to avoid the misunderstanding, one should not say that the plastic potential flow rule is derived from the thermodynamics but one should say merely that the plastic potential flow rule does not violate the thermodynamic laws. One should make effort to formulate constitutive equations by clarifying the definite physical backgrounds, without falling into the thermodynamic formalism. Incidentally, it would not be beneficial for the education of mechanics to describe as if the plastic flow rules are derived based on the second law of thermodynamics or as if the second law of thermodynamics is inevitable for the formulation of plastic constitutive relations. In fact, it makes difficult for the readers to understand the constitutive formulations by adding the unnecessary explanations by the second law of thermodynamics. In addition, it makes the readers to waste not a short time for reading explanations without a substantial-physical significance. Consequently, the people who use the second law of thermodynamics for constitutive formulations of irreversible (plastic, damage, fatigue, phase transformation, etc.) deformations would have to arise from the thermodynamic religion.

Nonlinear Continuum Mechanics for Finite Elasticity-Plasticity

A P P E N D I X

1 Proofs for formula of scalar triple products with invariants The proof of the scalar triple products with invariants is given below referring to Chadwick (1976), Hisada (1992), Kyoya (2008), Hashiguchi (2017a), etc. Proof of Eq. (1.262)1: We have the relation ½Tabc 1 ½a Tb c 1 ½a b Tc 5 ai bj ck ½Tei ej ek  1 ½ei Tej ek  1 ½ei ej Tek 





5 εijk ai bj ck ½Te1 e2 e3  1 ½e1 Te2 e3  1 ½e1 e2 Te3 
5 ½abc ½Te1 e2 e3  1 ½e1 Te2 e3  1 ½e1 e2 Te3 

(A1.1)

making use of Eq. (1.52) and noting that ½Tei ej ek  1 ½ei Tej ek  1 ½ei ej Tek  5 εijk ½Te1 e2 e3  1 ½e1 Te2 e3  1 ½e1 e2 Te3 




(A1.2) due to Eq. (1.50). Further, the inside of bracket in the last equation in Eq. (A1.1) is described as follows: ½Te1 e2 e3  1 ½e1 Te2 e3  1 ½e1 e2 Te3  5 ½T r1 er e2 e3  1 ½e1 Tr2 er e3  1 ½e1 e2 T r3 er  5 Tr1 ½er e2 e3  1 Tr2 ½e1 er e3  1 T r3 ½e1 e2 er  5 T 11 ½e1 e2 e3  (A1.3) 1 T22 ½e1 e2 e3  1 T 33 ½e1 e2 e3  5 T11 1 T 22 1 T 33 5 trT Eq. (1.262)1 is obtained by substituting Eq. (A1.3) into Eq. (A1.1).

371

372

Appendix 1

Proof of Eq. (1.262)2: In the similar way as Eq. (A1.1), we have first ½a Tb Tc 1 ½Ta b Tc 1 ½Ta Tb c


5 εijk ai bj ck ½e1 Te2 Te3  1 ½Te1 e2 Te3  1 ½Te1 Te2 e3 
5 ½abc ½e1 Te2 Te3  1 ½Te1 e2 Te3  1 ½Te1 Te2 e3 

(A1.4)

Then, applying Eqs. (1.50) and (A1.2), the inside of bracket in this equation is described as follows: ½e1 Te2 Te3  1 ½Te1 e2 Te3  1 ½Te1 Te2 e3  5 Tr2 Ts3 ½e1 er es  1 Tr1 Ts3 ½er e2 es  1 Tr1 Ts2 ½er es e3  5 Tr2 Ts3 ε1rs 1 Tr1 Ts3 εr2s 1 Tr1 Ts2 εrs3 5 ðT22 T33 2 T23 T32 Þ 1 ðT11 T33 2 T13 T31 Þ 1 ðT11 T22 2 T12 T12 Þ 5 T11 T22 1 T22 T33 1 T33 T11 2 ðT12 T12 1 T23 T32 1 T31 T13 Þ   5 T211 1 T222 1 T 233 1 2ðT 11 T 22 1 T 22 T33 1 T33 T11 Þ 2  2½T 211 1 T 222 1 T 233 1 2ðT12 T12 1 T23 T 32 1 T 31 T 13 Þ 2  5 ðTrr Tss 2 Trs Tsr Þ 2

(A1.5)

leading to

  ½e1 Te2 Te3  1 ½Te1 e2 Te3  1 ½Te1 Te2 e3  5 ðtrTÞ2 2 tr2 T 2

(A1.6)

Eq. (1.262)2 is obtained by substituting Eq. (A1.6) into Eq. (A1.4) with Eq. (1.242). Proof of Eq. (1.262)3: Changing the representations of the tree vectors to the componentbased representations and then applying Eqs. (1.50) and (A1.2), we have ½Ta Tb Tc 5 εijk ai bj ck ½Te1 Te2 Te3  5 ½abc½Te1 Te2 Te3 

(A1.7)

Eq. (1.262)3 is obtained by substituting Eq. (1.206) into Eq. (A1.7).

A P P E N D I X

2 Convective stress rate tensors Various convective (convected) stress rate tensors are shown below referring to Hashiguchi (2017a). As was mentioned in the beginning of Section 3.5, the convective time-derivative of stress tensor is not used in the multiplicative hyperelastic-based plasticity and thus the readers who wish to know only it may omit reading this appendix. A variety of stress rate tensors are obtained from the Cauchy stress σ and/or the Kirchhoff stress τð 5 JσÞ by the convected derivatives in Section 3.5, while the convected derivative tensors possess the objectivity as described in Section 3.5.3. Here, denoting their convected deriva~ ~ ’ ’ _ _ tives in four types by σ and τ collectively, the following relations hold ~ ’ _

~ ’ _ τ 5 Jð σ 1 σtrdÞ;

~ ’ _

σ5

~ 1’ _ τ 2 σtrd J

(A2.1)

_ 1 σJ_ 5 Jðσ_ 1 σtrdÞ. because of the relation τ_ 5 σJ The contravariant convected rate of the Cauchy stress σ is given from Eq. (3.38)1 by



  ~
_ 3 Ol ’ 3 OlT
T _ 5σ σ  σ gg 5F F21 σF2T  FT 5F S=J FT 5 S=J gg 5 σ2lσ2σl (A2.2) which is termed the Oldroyd rate of Cauchy stress. Likewise, for the Kirchhoff stress we have 3 Ol

τ

~
’ _ 3 OlT
_ T 5~  τ gg 5 F F21 τF2T  FT ð 5 FSF S_ gg Þ 5 τ_ 2 lτ 2 τlT 5 τ (A2.3)

which is termed the Oldroyd rate of Kirchhoff stress. Further,

373

374

Appendix 2


~ ’ _

_ T 5 J21~ S_ gg  J21 Jσ gg 5 J21 FðF21 JσÞF2T  FT 5 J21 FSF 3 Ol 3 TrT
5 σ 1 σtrd 5 σ_ 2 lσ 2 σlT 1 σtrd 5 σ

3 Tr

σ

(A2.4)

is termed the Truesdell rate of Cauchy stress. The covariantcontravariant convected rate of the Kirchhoff stress τ is given from Eq. (3.38)3 as g

g
~
3c 3 cT
’ _ ~ T T _ _ T 5M t  τ  g 5 F2T FT τF2T  FT 5 F2T MF  g 5 τ_ 1 l τ 2 τl 6¼ t (A2.5)

Based on Eq. (3.40), one has g^

~
’ _ ~_ g^
3 3 T
T 2T  T _ T 5L 5 LF τ  τ 5 τF F P g  g 5 τ_ 2 τl 6¼ P τ

(A2.6)

which is termed the relative first PiolaKirchhoff stress rate. The following stress rate is termed the nominal stress rate. 3 Pσ



1 31 _ T 3
τ LF 5 σ_ 2 σlT 1 σtrl 6¼ P σT JP J

(A2.7)

which is used in Eq. (5.89) for the equilibrium equation of rate-form in the current configuration. The covariant convected rate of Cauchy stress is given from Eq. (3.38)4 as gg ~
’ _ 3 CRT
 σ 5 F2T FT σF  F21 5 σ_ 1 lT σ 1 σl 5 σ

3 CR

σ

(A2.8)

which is termed the CotterRivlin rate of Cauchy stress. Likewise, the covariant convected rate of Kirchhoff stress is given by 3 CR

τ

~ ’ _ τ

gg


3 CRT
5 F2T FT τF  F21 5 τ_ 1 lT τ 1 τl 5 τ

(A2.9)

The following corotational convected derivative based on Eq. (3.41) is termed the GreenNaghdi rate of Cauchy stress. 3 GN

σ

R ~ ’ _ 3 GNT
 σ 5 RðRT σRÞ R 5 σ_ 2 ΩR σ 1 σΩR 5 σ

(A2.10)

Similarly, the GreenNaghdi rate of Kirchhoff stress is given by 3 GN

τ

~ ’ _

R

3 GNT


 τ 5 RðRT τRÞ  RT 5 τ_ 2 ΩR τ 1 τΩR 5 τ

The corotational rate based on Eq. (3.46) is given by

(A2.11)

375

Appendix 2 w ~ ’ _ 3w 3 wT
σ  σ 5 σ_ 2 wσ 1 σw 5 σ

(A2.12)

which is termed the ZarembaJaumann rate of Cauchy stress. Likewise, it follows for the Kirchhoff stress that w ~ ’ _ 3w τ  τ 5 τ_ 2 wτ 1 τw

3 wT


5τ

(A2.13)

The convected stress rates shown above are used in the hypoelastic materials and fluid mechanics in which an elastic deformation is not induced or induced only infinitesimally. They cannot be adopted in the exact constitutive equation of materials exhibiting finite elastic deformation as will be examined in Section 8.3 for the hyperelastic-based plastic constitutive equation.

A P P E N D I X

3 Cauchy elastic and hypoelastic equations There are elastic equations other than the hyperelastic equation, that is, the Cauchy elastic and the hypoelastic equations. These elastic equations will be addressed below.

A3.1 Cauchy elastic equation Consider the elastic constitutive equations in terms of the second Piola
Kirchhoff stress tensor S and the Green tensor E as an example. The hyperelastic equation is given by Eq. (6.11), that is, S5

@ψðEÞ ; @E

Sij 5

@ψðEÞ @Eij

(A3.1)

which leads to the complete differential equation dS 5

@ψðEÞ dE; @E  @E

Sij 5

@ψðEÞ dEkl @Eij @Ekl

(A3.2)

Here, the complete integrability condition @Sij @Skl 5 @Ekl @Eij

(A3.3)

is satisfied by virtue of the reciprocality (commutativity) of the orders in partial derivatives   2   @Sij @ ψ @2 ψ @Skl 5 5 5 (A3.4) @Eij @Ekl @Ekl @Eij @Ekl @Eij

377

378

Appendix 3

On the other hand, the relation between S and E S 5 SðEÞ;

Sij 5 Sij ðEÞ

(A3.5)

leading to dS 5

@S dE; @E

Sij 5

@Sij dEkl @Ekl

(A3.6)

where the complete integrability condition, that is the exact differentiability condition @Sij @Skl 6¼ @Ekl @Eij

(A3.7)

is not satisfied in general. The elastic equation in Eq. (A3.5) with Eq. (A3.7) is called the Cauchy elastic equation. The Cauchy elastic equation possesses the one-to-one correspondence between the stress and the strain, but it causes the energy production or dissipation even during the stress or the strain cycle.

A3.2 Hypoelastic equation Truesdell (1955) proposed the linear relation between the objective 3 stress rate σ and the strain rate d, that is 3

σ 5 Eðσ; H; HÞ : d

(A3.8)

where E is the fourth-order elastic modulus tensor, which is the function of the stress σ, the second-order tensor H and the scalar H standing for the internal variables in general. Eq. (A3.8) is called the hypoelastic equation. The hypoelastic equation does not possess the one-to-one correspondence between the stress and strain, and further it causes the energy production or dissipation even during the stress or the strain cycle. It can take account of the material rotation. However, the hypoelasticbased plastic constitutive equation is limited to the infinitesimal elastic deformation as described in Section 8.2.3.

Bibliography

Anand, L., 1993. A constitutive model for interface friction. Comput. Mech. 12, 197 213. Anand, L., 2016. A theory for non-Newtonian viscoelastic polymeric liquids. Int. J. Plasticity 83, 273 301. Anand, L., 2017. A large deformation poroplasticity theory for microporous polymeric materials. J. Mech. Phys. Soilds 98, 126 155. Anand, L., Gurtin, M.E., 2003. A theory of amorphous solids undergoing large deformations, with application to polymer glasses. Int. J. Solids Struct. 40, 1465 1487. Anjiki, T., Oka, M., Hashiguchi, K., 2016a. Elastoplastic analysis by complete implicit stress-update algorithm based on the extended subloading surface model. Trans. Jpn. Soc. Mech. Eng. 82 (839), No.16-00029 (in Japanese). Available from: https://doi.org/ 10.1299/transjsme.16-00029. Anjiki, T., Oka, M., Hashiguchi, K., 2016b. Elastoplastic analysis by return-mapping method with rigorous loading criterion for extended subloading surface model. Trans. Jpn. Soc. Mech. Eng. 85 (870), No.18-00327 (in Japanese). Available from: https://doi. org/10.1299/transjsme.18-00327. Anjiki, T., Oka, M., Hashiguchi, K., 2020. Complete implicit stress integration algorithm with extended subloading surface model for elastoplastic deformation analysis. Int. J. Numer. Methods Eng 121, 945 966. Aravas, N., 1994. Finite-strain anisotropic plasticity and plastic spin. Model. Simul. Mater. Sci. Eng. 2, 483 504. Armstrong, P.J., Frederick, C.O., 1966. A mathematical representation of the multiaxial Bauschinger effect. CEGB Report RD/B/N 731 (or in Materials at High Temperature, 24, 1 26, 2007). Asaro, R., Lubarda, V.A., 2006. Mechanics of Solids and Materials. Cambridge University Press. Balieu, R., Kringos, N., 2015. A new thermodynamic framework for finite strain multiplicative elastoplasticity coupled to anisotropic damage. Int. J. Plasticity 70, 126 150. Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourgoghrat, F., Choi, S. H., Chu, E., 2003. Plane stress yield function for aluminum alloy sheets - part 1: theory. Int. J. Plasticity 19, 1297 1319. Bazar, Y., Weichert, D., 2000. Nonlinear Continuum Mechanics: Fundamental Mathematical and Physical Concept. Springer. Belytschko, T., Liu, W.K., Moran, B., Elkhodary, K.I., 2014. Nonlinear Finite Elements for Continua and Structures, second ed. John-Wiley. Bergstrom, J., 2015. Mechanics of solids polymers: theory and computational modelling. Elsevier. Bingham, E.C., 1922. Fluidity and Plasticity. McGraw-Hill. Biot, M.A., 1965. Mechanics of Incremental Deformations. John Wiley & Sons, New York. Boden, F.P., Tabor, D., 1950. The Friction and Lubrication of Solids. Oxford University Press.

379

380

Bibliography

Bodner, S.R., Partom, Y., 1975. Constitutive equations for viscoplastic strain hardening. J. Appl. Mech 42, 385 389. Bonet, J., Gil, A.J., Wood, R.D., 2016. Nonlinear continuum mechanics for finite element analysis, Fourth edition Cambridge Univ. Press. Bonet, J., Wood, R.D., 2008. Nonlinear Continuum Mechanics for Finite Element Analysis, second ed. Cambridge University Press. Borja, R.I., Tamagnini, C., 1998. Cam-Clay plasticity, part III: extension of the infinitesimal model to include finite strains. Comput. Methods Appl. Mech. Eng. 155, 73 95. Bowen, R.M., Wang, C.-C., 1976. Introduction to Vector and Tensor Linear Multilinear Algebra. Plenum Press. Boyce, M.C., Park, D.M., Argon, Ali, S., 1988. Large inelastic deformation of glassy polymers Part I: Rate dependent constitutive model. Mech. Mater. 7, 15 33. Brepols, T., Vladimirov, I.N., Reese, S., 2014. Numerical comparison of isotropic hypo- and hyperelastic-based plasticity models with application to industrial forming processes. Int. J. Plasticity 63, 18 48. Brunig, M., 1998. Nonlinear finite element analysis based on a large strain deformation theory of plasticity. Comput. Struct. 69, 117 128. Burland, J.B., 1965. The yielding and dilatation of clay, correspondence. Ge´otechnique 15, 211 214. Callari, C., Auricchio, F., Sacco, E., 1998. A finite-strain Cam-Clay model in the framework of multiplicative elasto-plasticity. Int. J. Plasticity 14, 1155 1187. Casey, J., Naghdi, P.M., 1980. Remarks on the use of the decomposition F 5 FeFp in plasticity. J. Appl. Mech. (ASME) 47, 672 675. Casey, J., Naghdi, P.M., 1981. Discussion of “a correct definition of elastic and plastic deformation and its computational significance. J. Appl. Mech. (ASME) 48, 983 984. Chaboche, J.L., 1989. Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int. J. Plasticity 5, 247 302. Chaboche, J.L., 1991. On some modifications of kinematic hardening to improve the description of ratcheting effects. Int. J. Plasticity 7, 661 678. Chaboche, J.L., Dang-Van, K., Cordier, G., 1979. Modelization of the strain memory effect on the cyclic hardening of 316 stainless steel. In: Proceedings of 5th International Conference on SMiRT, Berlin, Division L., Paper No. L. 11/3. Chadwick, P., 1976. Continuum Mechanics. George Allen & Unwin Ltd. Cheng, J.-H., Kikuchi, N., 1985. An incremental constitutive relation of uniaxial contact friction for large deformation analysis. J. Appl. Mech. (ASME) 52, 639 648. Ciarlet, P.G., 1988. Mathematical Elasticity I: Three-Dimensional Elasticity. North-Holland. Cleja-Tigoiu, S., Soos, E., 1990. Elastoviscoplastic models with relaxed configurations and internal state variables. Appl. Mech. Rev. (ASME) 43, 131 151. Clifton, R.J., 1972. On the equivalence of F 5 FeFp and F 5 F̅pF̅e. J. Appl. Mech. (ASMS) 14, 703 717. Collins, I.F., Houlsby, G.T., 1997. Application of the thermomechanical principles to the modeling of geomechanical materials. Proc. R.l Soc. Lond. A Math. Phys. Eng. Sci. 453, 1975 2001. Curnier, A., 1984. A theory of friction. Int. J. Solids Struct. 20, 637 647. Dafalias, Y.F., 1983. Corotational rates for kinematic hardening at large plastic deformations. J. Appl. Mech. (ASME) 50, 561 565. Dafalias, Y.F., 1984. The plastic spin concept and a simple illustration of its role in finite plastic transformation. Mech. Mater. 3, 223 233. Dafalias, Y.F., 1985. The plastic spin. J. Appl. Mech. (ASME) 52, 865 871. Dafalias, Y.F., 1986. Bounding surface plasticity. I: mathematical foundation and hypoplasticity. J. Eng. Mech. (ASCE) 112, 966 987.

Bibliography

381

Dafalias, Y.F., 1987. Issues on the constitutive formulation at large elastoplastic deformations, part 1: kinematics. Acta Mech. 69, 119 138. Dafalias, Y.F., 1998. Plastic spin: necessity or redundancy? Int. J. Plasticity 14, 909 931. Dafalias, Y.F., Popov, E.P., 1975. A model of nonlinearly hardening materials for complex loading. Acta Mech. 23, 173 192. Dafalias, Y.F., Manzari, M.T., Papadimitriou, A.G., 2006. SANICLAY: simple anisotropic clay plasticity model. Int. J. Numer. Anal. Methods Geomech. 30, 1231 1257. Dafalias, Y.F., Talebat, M., 2016. SANISAND-Z: zero elastic range plasticity model. Ge´otechnique 66, 999 1013. Dashner, P.A., 1986. Invariance considerations in large strain elasto-plasticity. J. Appl. Mech. (ASME) 53, 55 60. Davidson, L., 1995. Kinematics of finite elastoplastic deformation. Mech. Mater. 21, 73 88. de Souza Neto, E.N., Peri´c, D., Owen, D.J.R., 2008. Computational Methods for Plasticity: Theory and Applications. John Wiley & Sons. del Peiro, G., 1979. Some properties of the set of fourth-order tensors, with application to elasticity. J. Elasticity 9, 245 261. Dettmer, W., Reese, S., 2004. On the theoretical and numerical modelling of ArmstrongFrederic kinematic hardening in the finite strain regime. Comput. Methods Appl. Mech. Eng. 193, 87 116. Dienes, J.K., 1979. On the analysis of rotation and stress rate in deforming bodies. Acta Mech. 32, 217 232. Dieterich, J.H., 1978. Time-dependent friction and the mechanism of stick-slip. Pure Appl. Geophys. 116, 790 806. Drucker, D.C., 1950. A more fundamental approach to plastic stress-strain relations, Proceedings of the 1st US National Congress on Applied Mechanics (ASME), 1. pp. 487 491. Drucker, D.C., 1988. Conventional and unconventional plastic response and representation. Appl. Mech. Rev. (ASME) 41, 151 167. Drucker, D.C., Prager, W., 1952. Soil mechanics and plastic analysis or limit design. Q. Appl. Math. 10, 157 165. Eckart, C., 1948. The thermodynamics of irreversible processes, IV: the theory of elasticity and anelasticity. Phys. Rev. 73, 373 380. Eidel, B., Gruttmann, F., 2003. Elastoplastic orthotropy at finite strains: multiplicative formulation and numerical implementation. Comput. Mater. Sci. 28, 732 742. Eringen, A.C., 1962. Nonlinear Theory of Continuous Media. McGraw-Hill. Eringen, A.C., 1975. Continuum Physics, Part 2. Academic Press. Farahani, K., Naghdabadi, R., 2000. Conjugate stresses of the Seth-Hill strain tensors. Int. J. Solids Struct. 37, 5247 5255. Fardshisheh, F., Onate, E.T., 1974. Representations of elastoplastic behavior by means of state variables. In: Sawczuk, A. (Ed.), Problems of Plasticity, pp. 89 115. Farias, M., Pedroso, D., Nakai, T., 2009. Automatic substepping integration of the subloading tij model with stress path dependent hardening. Comput. Geotech. 36, 537 548. Fincato, R., Tsutsumi, S., 2017. Closest-point projection method for the extended subloading surface model. Acta Mech. 228, 4213 4233. Available from: https://doi.org/ 10.1007/s00707-017-1926-0. Fincato, R., Tsutsumi, S., 2018. A return mapping algorithm for elastoplastic and ductile damage constitutive equations using the subloading surface method. Int. J. Numer. Methods Eng. Available from: https://doi.org/10.1002/nme.5718. Fish, J., Shek, K., 1999. Computational aspects of incrementally objective algorithms for large deformation plasticity. Int. J. Numer. Methods Eng. 44, 839 851. Flu¨gge, W., 1972. Tensor Analysis and Continuum Mechanics. Springer-Verlag.

382

Bibliography

Fredriksson, B., 1976. Finite element solution of surface nonlinearities in structural mechanics with special emphasis to contact and fracture mechanics problems. Comput. Struct. 6, 281 290. Fu, Y., Iwatab, M., Dinga, W., Zhang, F., Yashima, A., 2012. An elastoplastic model for soft sedimentary rock considering inherent anisotropy and confining-stress dependency. Soils Found. 52, 575 589. Fuente, M.D.L., Vaunat, J., Marin-Moreno, H., 2019. A densification mechanism to model the mechanical effect of methane hydrates in sandy sediments. Int. J. Numer. Anal. Meth. Geomech. 27, 1 21. Gambirasio, L., Chiantoni, G., Rizzi, E., 2016. On the consequences of the adoption of the Zaremba Jaumann objective stress rate in FEM codes. Arch. Comput. Methods Eng. 23, 39 67. Gang, W., Horikoshi, K., Akiyoshi, A., 2019. Effects of internal erosion on parameters of subloading Cam-Clay model. Geotech. Geol. Eng. 878. Available from: https://doi. org/10.1007/s10706-019-01093-8. Gearing, B.P., Moon, H.S., Anand, L., 2001. A plasticity model for interface friction: application to sheet metal forming. Int. J. Plasticity 17, 237 271. Ghasemzadeh, H., Sojoudi, M.H., Hhoreishian, G., Karami, M.H., 2017. Soils Found. 57, 371 383. Green, A.E., Naghdi, P.M., 1965. A general theory of an elastic-plastic continuum. Arch. Ration. Mech. Anal. 18, 251 281. Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth: part I—yield criteria and flow rules for porous media. J. Eng. Mater. Technol 99, 2 15. Gurtin, M.E., Anand, L., 2005. The decomposition F 5 FeFp, materials symmetry, and plastic irrotationality for solids that are isotropic-viscoplastic or amorphous. Int. J. Plasticity 21, 1686 1719. Guo, Z.H., Dubey, R.N., 1984. Basic aspects of Hill’s method in solid mechanics. Solid Mech.s Arch. 9, 353 380. Guo, Z.H., Man, C.S., 1992. Conjugate stress and tensor equation. Int. J. Solids Struct. 29, 2063 2076. Gurtin, M.E., Fried, E., Anand, L., 2010. The Mechanics and Thermodynamics of Continua. Cambridge University Press. Han, C.-S., Chung, K., Wagoner, R.H., Oh, S.-I., 2003. A multiplicative finite elasto-plastic formulation with anisotropic yield functions. Int. J. Plasticity 19, 197 211. Harrysson, M., Ristinmaa, M., 2007. Description of evolving anisotropy at large strains. Mech. Mater. 39, 267 282. Hashiguchi, K., 1978. Plastic constitutive equations of granular materials. In: Cowin, S.C., Satake, M. (Eds.), Proceedings of US-Japan Seminar on Continuum Mechanical and Statistical Approaches In the Mechanics of Granular Materials, Sendai, pp. 321 329. Hashiguchi, K., 1980. Constitutive equations of elastoplastic materials with elastic-plastic transition. J. Appl. Mech. (ASME) 47, 266 272. Hashiguchi, K., 1986. Elastoplastic constitutive model with a subloading surface. Proceedings of International Conference on Computational Mechanics. pp. IV65 IV70. Hashiguchi, K., 1989. Subloading surface model in unconventional plasticity. Int. J. Solids Struct. 25, 917 945. Hashiguchi, K., 1993a. Fundamental requirements and formulation of elastoplastic constitutive equations with tangential plasticity. Int. J. Plasticity 9, 525 549. Hashiguchi, K., 1993b. Mechanical requirements and structures of cyclic plasticity models. Int. J. Plasticity 9, 721 748. Hashiguchi, K., 1997. The extended flow rule in plasticity. Int. J. Plasticity 13, 37 58. Hashiguchi, K., 2000. Fundamentals in constitutive equation: continuity and smoothness conditions and loading criterion. Soils Found. 40 (3), 155 161.

Bibliography

383

Hashiguchi, K., 2001. On the thermomechanical approach to the formulation of plastic constitutive equations. Soils Found. 41 (4), 89 94. Hashiguchi, K., 2002. A proposal of the simplest convex-conical surface for soils. Soils Found. 42 (3), 107 113. Hashiguchi, K., 2009. Elastoplasticity Theory, first ed. Springer. Hashiguchi, K., 2011. General interpretations and tensor symbols for pull-back, pushforward and convected derivative. Proc. 24th Comp. Mech. Div. JSME. pp. 669 671. Hashiguchi, K., 2013a. General description of elastoplastic deformation/sliding phenomena of solids in high accuracy and numerical efficiency: subloading surface concept. Arch. Comput. Methods Eng. 20, 361 417. Hashiguchi, K., 2013b. Elastoplasticity Theory, second ed. Springer. Hashiguchi, K., 2015a. Subloading-damage constitutive equation. Proc. Comput. Eng. Conf. JSCES 20, D2-4. Hashiguchi, K., 2015b. Subloading-Gurson model. Proc. Comput. Mech. Div., JSME OS02-33. Hashiguchi, K., 2016a. Exact formulation of subloading surface model: unified constitutive law for irreversible mechanical phenomena in solids. Arch. Comput. Methods Eng. 23, 417 447. Hashiguchi, K., 2016b. Loading criterion in return-mapping for subloading surface model. Proc. 29th Comput. Mech. Div. JSME 036. Hashiguchi, K., 2017a. Foundations of Elastoplasticity: Subloading Surface Model. Springer. Hashiguchi, K., 2017b. Basic constitutive equation in multiplicative hyperelastic-based plasticity. Bull. JSME Mech. Eng. Lett. 3, Paper No.17-00402. Available from: https:// doi.org/10.1299/mel.17-00402. Hashiguchi, K., 2017c. Loading criterion in return-mapping method for subloading surface model. Proc. Tech. Symp., West Branch, JSCE. pp. 19 24. Hashiguchi, K., 2018a. Extension of loading criterion in return-mapping from infinitesimal to multiplicative hyperelastic-based plasticity for subloading surface model. Proc. Conf. Jpn. Comput. Eng., JSCES 23, A-08-01. Hashiguchi, K., 2018b. Hypo-elastic and hyper-elastic equations of soils. Int. J. Numer. Methods Geomech 42, 1554 1564. Hashiguchi, K., 2018c. Multiplicative hyperplastic-based plasticity for finite elastoplastic deformation/sliding: a comprehensive review. Arch. Comput. Methods Eng. 26, 597 637. Available from: https://doi.org/10.1007/s11831-018-9256-5. Hashiguchi, K., 2018d. Return-mapping formulation for subloading-friction model. Proc. 31th Comput. Mech. Div., JSME No. 18-8, 044. Hashiguchi, K., 2018e. Evolution rule of elastic-core in subloading surface model in current and multiplicative hyperelastic-based plasticity. Proc. Conf. Jpn. Comput. Eng., JSCES 23, A-09-04. Hashiguchi, K., 2019a. Formulation for evolution rule of elastic-core in subloading surface model. Proc. Materials & Mech. Divis., JSME GS04. Hashiguchi, K., 2019b. Subloading-overstress model for general rate of deformation. Proc. JSME Okinawa Conference 424. Hashiguchi, K., Chen, Z.-P., 1998. Elastoplastic constitutive equation of soils with the subloading surface and rotational hardening. Int. J. Numer. Anal. Methods Geomech. 22, 197 227. Hashiguchi, K., Mase, T., 2007. Extended yield condition of soils with tensile strength and rotational hardening. Int. J. Plasticity 23, 1939 1956. Hashiguchi, K., Mase, T., 2018. Subloading-unilateral-damage model of concrete. Proc. 31th Comput. Mech. Div. JSME No. 18-8, 070.

384

Bibliography

Hashiguchi, K., Okamura, K., 2014. Subloading phase-transformation model. Proc. 27th Comput. Mech. Div., JSME 1707, pp. 409 411. Hashiguchi, K., Okamura, K., 2019. Subloading-crystal plasticity model. Proc. Comput. Mech. Div. JSME 286. Hashiguchi, K., Ozaki, S., 2008. Constitutive equation for friction with transition from static to kinetic friction and recovery of static friction. Int. J. Plasticity 24, 2102 2124. Hashiguchi, K., Ozaki, S., Okayasu, T., 2005. Unconventional friction theory based on the subloading surface concept. Int. J. Solids Struct. 42, 1705 1727. Hashiguchi, K., Protasov, A., 2004. Localized necking analysis by the subloading surface model with tangential-strain rate and anisotropy. Int. J. Plasticity 20, 1909 1930. Hashiguchi, K., Saitoh, K., Okayasu, T., Tsutsumi, S., 2002. Evaluation of typical conventional and unconventional plasticity models for prediction of softening behavior of soils. Ge´otechnique 52, 561 573. Hashiguchi, K., Tsutsumi, S., 2001. Elastoplastic constitutive equation with tangential stress rate effect. Int. J. Plasticity 17, 117 145. Hashiguchi, K., Tsutsumi, S., 2003. Shear band formation analysis in soils by the subloading surface model with tangential stress rate effect. Int. J. Plasticity 19, 1651 1677. Hashiguchi, K., Ueno, M., 1977. Elastoplastic constitutive laws of granular materials, Constitutive Equations of Soils. In: Murayama S., Schofield, A.N. (Eds.), Proceedings of the 9th International Conference on Soil Mechanics and Foundation Engineering, Spec. Ses. 9, Tokyo, JSSMFE, pp. 73 82. Hashiguchi, K., Ueno, M., 2017. Elastoplastic constitutive equation of metals under cyclic loading. Int. J. Eng. Sci. 111, 86 112. Hashiguchi, K., Ueno, M., Ozaki, T., 2012. Elastoplastic model of metals with smooth elastic-plastic transition. Acta Mech. 223, 985 1013. Hashiguchi, K., Ueno, M., Kuwayama, T., Suzuki, N., Yonemura, S., Yoshikawa, N., 2016. Constitutive equation of friction based on the subloading-surface concept. Proc. R.l Soc. Lond. A Math. Phys. Eng. Sci. 472. Available from: https://doi.org/10.1098/rspa.2016.0212. Hashiguchi, K., Yamakawa, Y., 2012. Introduction to Finite Strain Theory for Continuum Elasto-Plasticity, Wiley Series in Computational Mechanics. Wiley. Hashiguchi, K., Yoshimaru, T., 1995. A generalized formulation of the concept of nonhardening region. Int. J. Plasticity 11, 347 365. Haupt, P., 1985. On the concept pf an intermediate configuration and its application to representation of viscoelastic-plastic material behavior. Int. J. Plasticity 1, 303 316. Haupt, P., 2002. Continuum Mechanics and Theory of Materials. Springer. Hausler, O., Schick, D., Tsakmakis, Ch, 2004. Description of plastic anisotropic effects at large deformations. Part II: the case of traverse isotropy. Int. J. Plasticity 20, 199 223. Havner, K.S., 1992. Finite Plastic Deformation of Crystalline Solids. Cambridge University Press. Helm, D., 2001. Formgedachtnislegierungen, experimentelle Untersuchung, phanomenologische Modellierung und numerische Simulation der thermomechanischen Materialeigenschaften, Universitatsbibliothek Kassel. Henann, D.L., Anand, L., 2009. A large deformation theory for rate-dependent elastic-plastic materials with combined isotropic and kinematic hardening. Int. J. Plasticity 25, 1833 1878. Hill, R., 1948. Theory of yielding and plastic flow of anisotropic metals. Proc. Royal Soc., London A193, 281 297. Hill, R., 1968. On constitutive inequality for simple materials 2 1. J. Mech. Phys. Solids 16, 229 242. Hill, R., 1978. Aspects of invariance in solid mechanics. Adv. Appl. Mech. 18, 1 75. Hisada, T., 1992. Tensor Analysis for Nonlinear Finite Element Method. Maruzen Publishing, Inc (in Japanese).

Bibliography

385

Hoger, A., 1987. The stress conjugate to logarithmic strain. Int. J. Solids Struct. 23, 1645 1656. Holsapple, K.A., 1973. A finite elastic-plastic theory and invariance requirements. Acta Mech. 17, 277 290. Holzapfel, G.A., 2000. Nonlinear Solid Mechanics: A Continuum Approach for Engineering. John Wiley & Sons. Houlsby, G.T., 1985. The use of a variable shear modulus in elastic-plastic models for clays. Comput. Geotech. 1, 3 13. Houlsby, G.T., Amorosi, A., Rojas, E., 2005. Elastic moduli of soils dependent on pressure: a hyperelastic formulation. Ge´otechnique 55 (5), 383 392. Houlsby, G.T., Puzrin, A.M., 2010. Principles of Hypoelasticity. Springer. Iguchi, T., Yamakawa, Y., Ikeda, K., 2016. A re-formulation of extended subloading surface model for cyclic plasticity within small strain framework: hyperelastic-based formulation and fully implicit return-mapping scheme. Trans. Jpn. Soc. Mech. Eng. 82, 841, No.16-00197 (in Japanese). Available from: https://doi.org/10.1299/transjsme.16-00197. Iguchi, T., Yamakawa, K., Hashiguchi, K., Ikeda, K., 2017a. Extended subloading surface model based on multiplicative finite strain elastoplasticity framework: constitutive formulation and fully implicit return-mapping scheme. Trans. Jpn. Soc. Mech. Eng. 843 (848), (in Japanese). Available from: https://doi.org/10.1299/transjsme.17-00008. Iguchi, T., Fukuda, T., Yamakawa, Y., Ikeda, K., Hashiguchi, K., 2017b. An improvement of loading criterion for stress calculation based on elastic predictor and returnmapping scheme for extended subloading surface plasticity model. J. Appl. Mech. (JSCE) 20, I_363-I_375 (in Japanese). Ilyushin, A.A., 1961. On the postulate of plasticity. Appl. Math. Mech. 25, 746 752 (Translation of O postulate plastichnosti, Prikladnaya Mathematika i Mekkanika, 25, 503 507). Iwan, W.D., 1967. On a class of models for the yielding behavior of continuous and composite systems. J. Appl. Mech. (ASME) 34, 612 617. Jaumann, G., 1911. Geschlossenes System physicalisher und chemischer Differentialgesetze. Sitzber. Akad. Wiss. Wien (IIa) 120, 385 530. Johnson, G.R., Cook, W.H., 1983. A constitutive model and data for metals subjected to large strain rates, and high temperatures. In: Proceedings of the 7th International Symposium on Ballistics, SDPA, The Hague, pp. 1 7. Kay, D.C., 1988. Tensor Calculus. McGraw-Hill. Khan, A.S., Huang, S., 1995. Continuum Theory of Plasticity. John Wiley & Sons. Kikuchi, N., Oden, J.T., 1988. Contact Problem in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia. Kintzel, O., Bazar, Y., 2006. Fourth-order tensors tensor differentiation with applications to continuum mechanics, part I: classical tensor analysis. Z.A.M.M. 86, 291 311. Kobayashi, M., Mukai, M., Takahashi, H., Ohno, N., Kawakami, T., Ishikawa, T., 2003. Implicit integration and consistent tangent modulus of a time-dependent non-unified constitutive model. Int. J. Numer. Meth. Eng. 58, 1523 1543. Kobayashi, M., Ohno, N., 2002. Implementation of cyclic plasticity models based on a general form of kinematic hardening. Int. J. Numer. Methods Eng. 53, 2217 2238. Kolymbas, D., Wu, W., 1993. Introduction to plasticity. Modern Approaches to Plasticity. Elsevier, pp. 213 224. Kratochvil, J., 1971. Finite-strain theory of crystalline elastic-inelastic materials. J. Appl. Phys. 42, 1104 1108. Krieg, R.D., 1975. A practical two surface plasticity theory. J. Appl. Mech. (ASME), 42, 641 646. Kroner, E., 1960. Allgemeine Kontinuumstheoreie der Versetzungen und Eigenspannnungen. Arch. Ration. Mech. Anal. 4, 273 334. Kyoya, T., 2008. Continuum Mechanics Note. Morikita Publishing, Inc (in Japanese).

386

Bibliography

Lamaitre, J., Chaboche, J.-L., 1990. Mechanics of Solid Materials. Cambridge University Press. Lee, E.H., 1969. Elastic-plastic deformation at finite strain. J. Appl. Mech. (ASME) 36, 1 6. Lee, E.H., Liu, D.T., 1967. Finite-strain elastic-plastic theory with application to plane-wave analysis. J. Appl. Phys. 38, 19 27. Leigh, D.C., 1968. Nonlinear Continuum Mechanics: An Introduction to the Continuum Physics and Mechanical Theory of the Nonlinear Mechanical Behavior of Materials. McGraw-Hill, New York. Lemaitre, J.A., 1996. A Course on Damage Mechanics. Springer-Verlag. Lemaitre, J.A., Chaboche, J.-L., 1990. Mechanics of Solid Materials. Cambridge University Press. Lin, J.-S., Seol, Y., Choi, J.H., 2015. An SMP critical state model for methane hydrate-bearing sands. Int. J. Numer. Anal. Meth. Geomech. 39, 969 987. Lion, A., 2000. Constitutive modeling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. Int. J. Plasticity 16, 469 494. Liu, S.-f, Wan, Z.-j, Wang, J.-c, Lu, S.-f, Li, T.-h, 2019. Subloading surface model and experimental study of coal failure under cyclic loading. Advan. Mater. Sci. Eng. Available from: https://doi.org/10.1155/2019/9841297. Loret, B., 1983. On the effects of plastic rotation in the finite deformation of anisotropic elastoplastic materials. Mech. Mater. 2, 287 304. Lubarda, V.A., 1991. Constitutive analysis of large elasto-plastic deformation on the multiplicative decomposition gradient. Int. J. Solids Struct. 27, 885 895. Lubarda, V.A., 1999. Duality in constitutive formulation of finite-strain elastoplasticity based on F 5 FeFp and F 5 FpFe decompositions. Int. J. Plasticity 15, 1277 1290. Lubarda, V.A., 2002. Elastoplasticity Theory. CRC Press. Lubarda, V.A., 2004. Constitutive theories based on the multiplicative decomposition of deformation gradient: thermoplasticity, elastoplasticity, and biomechanics. Appl. Mech. Rev. 57, 95 108. Lubarda, V.A., Lee, E.H., 1981. A correct definition of elastic and plastic deformation and its computational significance. J. Appl. Mech. (ASME) 48, 35 40. Lubliner, J., 1986. Normality rules in large-deformation plasticity. Mech. Mater. 5, 29 34. Lubliner, J., 1990. Plasticity Theory. Dover Publications. Macvean, D.B., 1968. Die Elementararbeit in einem Kontinuum und die Zuordnung von Spannungs- und Verzerrungstensoren. Z. Angew. Math. Phys. 19, 157 185. Mandel, J., 1971. Plastidite classique et viscoplasticity. Course & Lectures, No. 97, Int. Center Mech. Sci. Udine. Springer-Verlag. Mandel, J. (1972): Director vectors and constitutive equations for plastic and viscoplastic media. In: Sawczuk, A. (Ed.), Problems of Plasticity (Proc. Int. Symp. Foundation of Plasticity), Noordhoff, pp. 135 141. Mandel, J., 1973a. Equations constitutives directeurs dans les milieux plastiques at viscoplastiques. Int. J. Solids Struct. 9, 725 740. Mandel, J., 1973b. Thermodynamics and plasticity. Proceedings of International Symposium on the Foundations of Continuum Thermodynamics. Halsted Press, pp. 238 304. Maranha, J.R., Pereira, C., Viera, A., 2016. A viscoplastic subloading soil model for ratedependent cyclic anisotropic structured behavior. Int. J. Numer. Anal. Meth. Geomech. 40, 1531 1555. Marsden, J.E., Hughes, T.J.R., 1983. Mathematical Foundation of Elasticity. Prentice-Hall, Englewood Cliffs. Masing, G., 1926. Eigenspannungen und verfestigung beim messing. Proc. 2nd Int. Congr. Appl. Mech. pp. 332 335. Zurich.

Bibliography

387

Menzel, A., Steinmann, P., 2003a. Geometrically non-linear anisotropic inelasticity based on fictitious configurations: application to the coupling of continuum damage and multiplicative elasto-plasticity. Int. J. Numer. Methods Eng. 56, 2233 2266. Menzel, A., Steinmann, P., 2003b. On the spatial formulation of anisotropic multiplicative elasto-plasticity. Comput. Methods Appl. Mech. Eng. 192, 3431 3470. Menzel, A., Ekh, M., Runesson, K., Steinmann, P., 2005. A framework for multiplicative elastoplasticity with kinematic hardening coupled to anisotropic damage. Int. J. Plasticity 21, 397 434. Michalowski, R., Mroz, Z., 1978. Associated and non-associated sliding rules in contact friction problems. Arch. Mech 30, 259 276. Miehe, C., 1994. On the representation of Prandtl-Reuss tensors within the framework of multiplicative elastoplasticity. Int. J. plasticity 6, 609 621. Miehe, C., 1996. Exponential map algorithm for stress updates in anisotropic multiplicative elastoplasticity for single crystals. Int. J. Numer. Methods Eng. 39, 3367 3390. Mooney, M., 1940. A theory of large elastic deformation. J. Appl. Phys. 11 (9), 582 592. Mroz, Z., 1967. On the description of anisotropic workhardening. J. Mech. Phys. Solids 15, 163 175. Mroz, Z., Stupkiewicz, S., 1994. An anisotropic friction and wear model. Int. J. Solids Struct. 31, 1113 1131. MSC Software Corporation, 2017. User manual for Hashiguchi model, Marc and Mentat Release Guide 2017.1, Material Behavior. Murakami, S., 2012. Continuum Damage Mechanics: A Continuum Mechanics Approach to the Analysis of Damage and Fracture. Springer-Verlag. Naghdi, P.M., 1990. A critical review of the state of finite plasticity. Z. Angew. Math. Phys 41, 315 394. Nagtegaal, J.C., De Jong, J.E., 1982. Some aspects of non-isotropic workhardening in finite strain plasticity. In: Lee, E.H., Mallett, R.L. (Eds.), Plasticity of Metals and Finite Strain: Theory, Experiment and Computation. Div. Appl. Mech., Stanford Univ. and Dept. Mech. Eng., Rensselaer Poly. Inst., pp. 65 102. Nemat-Nasser, S., 1979. Decomposition of strain measures and their rates in finite deformation elastoplasticity. Int. J. Solids Struct. 15, 155 166. Nemat-Nasser, S., 1982. On finite deformation elasto-plasticity. Int. J. Solids Struct. 18, 857 872. Nemat-Nasser, S., 2004. Plasticity, A Treatise on Finite Deformation of Homogeneous Inelastic Materials. Cambridge University Press. Noda, T., Xu, B., Asaoka, A., 2013. Acceleration generation due to strain localization of saturated clay specimen based on dynamic soil water coupled finite deformation analysis. Soils Found. 53 (5), 625 670. Norton, F.H., 1929. Creep of Steel at High Temperature. McGraw-Hill. Oden, J.T., Martines, J.A.C., 1986. Models and computational methods for dynamic friction phenomena. Comput. Methods Appl. Mech. Eng. 52, 527 634. Oden, J.T., Pires, E.B., 1983a. Algorithms and numerical results for finite element approximations of contact problems with non-classical friction laws. Comput. Struct. 19, 137 147. Oden, J.T., Pires, E.B., 1983b. Nonlocal and nonlinear friction laws and variational principles for contact problems in elasticity. J. Appl. Mech. (ASME) 50, 67 76. Odqvist, F.K., 1966. Mathematical Theory of Creep and Creep Ruptures. Oxford University Press. Ogden, R.W., 1982. Elastic deformations of rubberlike solids. In: Hopkins, H.G., Sewell, M. J. (Eds.), Mechanics of Solids: Rodney Hill 60th Anniversary Volume. Pergamon Press, pp. 499 537. Ogden, R.W., 1984. Non-Linear Elastic Deformations. Dover Publications Inc.

388

Bibliography

Ohno, N., 1982. A constitutive model of cyclic plasticity with a non-hardening strain region. J. Appl. Mech. (ASME) 49, 721 727. Ohno, N., Wang, J.D., 1993. Kinematic hardening rules with critical state of dynamic recovery, part I: formulation and basic features for ratcheting behavior. part II: application to experiments of ratcheting behavior. Int. J. Plasticity 9, 375 403. Okamura, K., Hashiguchi, K., 2015. Crystal plasticity analysis with subloading surface model. In: Proc. 64th Annual Meeting, Society of Materials Science, Japan, pp. 268 269 (in Japanese). Oldroyd, J.G., 1950. On the formulation of rheological equations of state. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. Ser. A. 200, 523 541. Ozaki, S., Hashiguchi, K., 2010. Numerical analysis of stick-slip instability by a ratedependent elastoplastic formulation for friction. Tribol. Int. 43, 2120 2133. Ozaki, T., Hashiguchi, K., 2019. Unified description of dry and fluid frictions by subloading-overstress friction model. In: Proceedings of International Conference on Coupled Problems in Science and Engineering, 3 5 June 2019, Barcelona. Ozaki, S., Hashiguchi, K., Okayasu, T., Chan, D.H., 2007. Finite element analysis of particle assembly-water coupled frictional contact problem. Comput. Meth. Eng. Sci., 18, 101 119. Ozaki, S., Hikida, K., Hashiguchi, K., 2012. Elastoplastic formulation for friction with orthotropic anisotropy and rotational hardening. Int. J. Solids Struct. 49, 648 657. Pedroso, D.M., 2014. The subloading isotropic plasticity as a variable modulus model. Comput. Geotech. 61, 230 240. Peirce, D., Asaro, J.R., Needleman, A., 1982. Overview 21: an analysis of nonuniform and localized deformation in ductile single crystals. Acta. Metall 30, 1087 1119. Peirce, D., Asaro, J.R., Needleman, A., 1983. Overview 32: material rate dependence and localized deformation in crystal solids. Acta Metall. 31, 1951 1976. Peirce, D., Shih, C.F., Needleman, A., 1984. A tangent modulus method for rate dependent solids. Comput. Struct. 18, 875 887. Peri´c, D., Owen, R.J., 1992. Computational model for 3-D contact problems with friction based on the penalty method. Int. J. Numer. Methods Eng. 35, 1289 1309. Perzyna, P., 1963. The constitutive equations for rate sensitive plastic materials. Q. Appl. Math. 20, 321 332. 1963. Perzyna, P., 1966. Fundamental problems in viscoplasticity. Adv. Appl. Mech. 9, 243 377. Perzyna, P., 1971. Thermodynamic theory of viscoplasticity. Adv. Appl. Mech. 11, 313 354. Prager, W., 1949. Recent development in the mathematical theory of plasticity. J. Appl. Mech. (ASME) 20, 235 241. Prager, W., 1956. A new methods of analyzing stresses and strains in work hardening plastic solids. J. Appl. Mech. (ASME) 23, 493 496. Prager, W., 1961a. Linearization in visco-plasticity. Ing. Arch 15, 152 157. Prager, W., 1961b. Introduction to Mechanics of Continua. Ginn & Company. Rice, J.R., 1970. On the structure of stress-strain relations for time dependent plastic deformation in metals. J. Appl. Mech. (ASME) 37, 728 737. Rice, J.R., 1971. Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433 455. Rice, J.R., Lapusta, N., Ranjith, K., 2001. Rate and state dependent friction and the stability of sliding between elastically deformable solids. J. Mech. Phys. Solids 49, 1865 1898. Rice, J.R., Ruina, A.L., 1983. Stability of steady friction slipping. J. Appl. Mech 50, 343 349. Rivlin, R.S., 1948. Large elastic deformations of isotropic materials. IV. Further developments of the general theory. Phil. Trans. R. Soc. Lond. A Math. Phys. Sci. 241 (835), 379 397.

Bibliography

389

Rojas, E., Horta, J., Perez-Rea, M., Hernandez, C.E., 2019. A fully coupled simple model for unsaturated soils. Int. J. Nemer. Anal. Meth. Geomech. 43, 1143 1161. Roscoe, K.H., Burland, J.B., 1968. On the generalized stress-strain behaviour of ‘wet’ clay. Engineering Plasticity. Cambridge University Press, pp. 535 608. Ruina, A.L., 1980. Friction Laws and Instabilities: Quasistatic Analysis of some Dry Frictional Behavior. Ph.D. Thesis, Brown University, Providence. Ruina, A.L., 1983. Slip instability and state variable friction laws. J. Geophys. Res. 88, 10359 10370. Sakai, T., Nakako, M., 2015. Interpretation of the mechanical behavior of embankments having various compaction properties based on the soil skeleton structure. Soils Found. 55 (5), 1069 1085. Sansour, C., Karsaj, I., Soric, J., 2006. A formulation of anisotropic continuum elastoplasticity at finite strains. Part I: modelling. Int. J. Plasticity 22, 2346 2365. Sansour, C., Karsaj, I., Soric, J., 2007. On anisotropic flow rules in multiplicative elastoplasticity at finite strains. Comput. Methods Appl. Mech. Eng. 196, 1294 1309. Scholz, C.H., 1998. Rate-and state-variable friction law. Nature 391, 37 41. Sedov, L.I., 1966. Foundations of the Non-Linear Mechanics of Continua. Pergamon Press. Seguchi, Y., Shindo, A., Tomita, Y., Sunohara, M., 1974. Sliding rule of friction in plastic forming of metal. Comput. Methods Nonlinear Mech. University of Texas at Austin, pp. 683 692, 1974. Seth, B.R., 1964. Generalized strain measure with applications to physical problems. Second-order Effects Inelasticity, Plasticity, and Fluid Dynamics. Pergamon Press. Shutov, A.V., Kreissig, R., 2008a. Finite strain viscoplasticity with nonlinear kinematic hardening: phenomenological modeling and time integration. Comput. Methods. Appl. Mech. Eng. 197, 2015 2029. Shutov, A.V., Kreissig, R., 2008b. Application of a coordinate-free tensor formalism to the numerical implementation of a material model. Z.A.M.M 88, 888 909. Shutov, A.V., Kreissig, R., 2010. Geometric integrators for multiplicative viscoplasticity: analysis of error accumulation. Comput. Methods Appl. Mech. Eng. 199, 700 7011. Shutov, A.V., Panhans, S., Kreissig, R., 2011. A phenomenological model of finite strain viscoplasticity with distortional hardening. Z.A.M.M 91, 653 680. Simo, J.C., 1985. On the computational significance of the intermediate configuration and hyperelastic relations in finite deformation elastoplasticity. Mech. Mater. 4, 439 451. Simo, J.C., 1988a. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: part I: continuum formulation. Comput. Methods Appl. Mech. Eng. 66, 199 219. Simo, J.C., 1988b. A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: part II: computational aspects. Comput. Methods Appl. Mech. Eng. 68, 1 31. Simo, J.C., 1992. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Comput. Methods Appl. Mech. Eng. 99, 61 112. Simo, J.C., 1998. Numerical analysis and simulation of plasticity. In: Ciarlet, P.G., Lions, J. L. (Eds.), Handbook of Numerical Analysis, vol. IV. Elsevier, pp. 183 499. Simo, J.C., Hughes, T.J.R., 1998. Computational Inelasticity. Springer. Simo, J.C., Miehe, C., 1992. Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Comput. Methods Appl. Mech. Eng. 98, 41 104. Simo, J.C., Ortiz, M., 1985. A unified approaches to finite deformation elastoplasticity based on the use of hyperelastic constitutive equations. Comput. Methods Appl. Mech. Eng. 49, 221 245.

390

Bibliography

Simo, J.C., Pister, K.S., 1984. Remarks on rate constitutive equations for finite deformation problems: computational implications. Comput. Methods Appl. Mech. Eng. 46, 201 215. Steinmann, P., Larsson, R., Runesson, K., 1997. On the localization properties of multiplicative hyperelasto-plastic continua with strong discontinuities. Int. J. Solids Struct. 34, 969 990. Storen, S., Rice, J.R., 1975. Localized necking in thin sheet. J. Mech. Phys. Solids 23, 421 441. Svendsen, B., 2001. On the modelling of anisotropic elastic and inelastic material behavior at large deformation. Int. J. Solids Struct. 38, 9579 9599. Svendsen, B., Arndt, S., Klingbeil, D., Sievert, R., 1998. Hyperelastic models for elastoplasticity with non-linear isotropic and kinematic hardening at large deformation. Int. J. Solids Struct. 35, 3363 3389. Tanahashi, T., 1988. Mechanics of Continua: (5) Vector Analysis and Physical Component. Rikoh-Tosho Publ., Inc (in Japanese). Tanahashi, T., 2004. Foundations of Deformation and Flow. Sankeisha Publ. Inc (in Japanese). Tatsuoka, F., Iwasaki, T., Takagi, Y., 1978. Hysteretic damping of sands under cyclic loading and its relation to shear modulus. Soils Found. 18 (2), 25 40. Topolnicki, M., 1990. An elasto-plastic subloading surface model for clay with isotropic and kinematic mixed hardening parameters. Soils Found. 30 (2), 103 113. Toyosada, M., 2015. Applied Nonlinear Fracture Mechanics. Fatigue-Free Structure Solutions, Ltd (in Japanese). Truedell, C., 1952. The mechanical foundations of elasticity and fluid mechanics. J. Ration. Mech. Anal 1, 125 300. Truesdell, C., 1955. Hypo-elasticity. J. Ration. Mech. Anal 4, 83 133. Truesdell, C., Noll, W., 1965. The nonlinear field theories of mechanics. In: Flugge, S. (Ed.), Encyclopedia of Physics, vol. III. Springer. Tsakmakis, Ch, 2004. Description pf plastic anisotropy effects at large deformations—part I: restrictions imposed by the second law and the postulate of Il’iushin. Int. J. plasticity 20, 167 198. Van der Giessen, E., 1989. Micromechanical and thermodynamic aspects of the plastic spin. Int. J. Plasticity 7, 365 386. Vladimirov, I.N., Pietryga, M.P., Reese, S., 2008. On the modeling of nonlinear kinematic hardening at finite strains with application to springback—comparison of time integration algorithm. Int. J. Numer. Methods Eng. 75, 1 28. Vladimirov, I.N., Pietryga, M.P., Reese, S., 2010. Anisotropic finite elastoplasticity with nonlinear kinematic and isotropic hardening and application to shear metal forming. Int. J. Plasticity 26, 659 687. Voyiadjis, G.Z., Kattan, P.I., 2005. Damage Mechanics (Mechanical Engineering). CRC Press, New York. Voyiadjis, G.Z., Taqieddin, Z.N., Kattan, P.I., 2008. Anisotropic damage-plasticity model for concretes. Int. J. Plasticity 24, 1946 1965. Wachtman Jr., J.B., Tefft, W.E., Lam, D.G., Apstein, C.S., 1961. Exponential temperature dependence of Young’s modulus for several oxides. Physical Review 122, 1754 1759. Wallin, M., Ristinmaa, M., 2005. Deformation gradient based kinematic hardening model. Int. J. Plasticity 21, 2025 2050. Wallin, M., Ristinmaa, M., Ottesen, N.S., 2003. Kinematic hardening in large strain plasticity. Eur. J. Mech. A/Solids 22, 341 356. Wang, Z.-Q., Dui, G.-S., 2008. Two-point constitutive equations and integration algorithms for isotropic-hardening rate-independence elastoplastic materials in large deformation. Int. J. Numer. Methods Eng. 75, 1435 1456.

Bibliography

391

Watanabe, O., 1990. Objective constitutive equation and numerical algorithm in finite strain. Trans. Jpn. Soc. Mech. Eng. 56 524, 893 902. (in Japanese). Weber, G., Anand, L., 1990. Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids. Comput. Methods Appl. Mech. Eng. 79, 173 202. White, P.S., 1975. Elastic-plastic solids as simple materials. Q. J. Mech. Appl. Math. 28, 483 496. Wongsaroj, J., Soga, K., Mair, R.J., 2007. Modeling of long-term ground response to tunneling under St James’ Park, London. Ge´otechnique 57, 75 90. Wriggers, P., 2003. Computational Contact Mechanics. Wiley. Wriggers, P., Vu Van, T., Stein, E., 1990. Finite element formulation of large deformation impact-contact problems with friction. Comput. Struct. 37, 319 331. Wu, H.-C., 2004. Continuum Mechanics and Plasticity. Chapman & Hall/CRC. Xiao, H., 1995. Unified explicit basis-free expressions for time rate and conjugate stress of an arbitrsary Hill’s strain. Int. J. Solds Struct. 22, 3327 3340. Xiong, Y., Zhang, S., Ye, G., Zhang, F., 2014. Modification of thermo-elasto-viscoplastic model for soft rock and its application to THM analysis of heating tests. Soils Found. 54 (2), 176 179. Xiong, Y., Ye, G., Zhu, H., Zhang, S., Zhang, F., 2017. A unified thero-elasto-viscoplastic model for soft rock. Int. J. Rock Mech. Mining Sci. 93, 1 12. Xiong, Y., Ye, G., Xie, Y., Zhang, S., Zhang, F., 2019. A unified constitutive model for unsaturated soil under monotonic and cyclic loading. Acta Geotech. 14, 313 328. Yadav, S.K., Ye, G.-l, Khalid, U., Fuluda, M., 2019. Numerical and centrifugal physical modelling on soft clay improved with floating and fixed sand compaction piles. Comput. Geotech. 115, 1 16. Yamada, S., Noda, T., Tashiro, M., Nguyen, H.-S., 2015. Macro-element method with water absorption and discharge functions for vertical drains. Soils Found. 55 (5), 113 1128. Yamakawa, Y., Hashiguchi, K., Ikeda, K., 2010. Implicit stress-update algorithm for isotropic Cam-clay model based on the subloading surface concept at finite strains. Int. J. Plasticity 26, 634 658. Yoshida, F., Uemori, T., 2002. Elastic-plastic behavior of steel sheets under in-plane cyclic tension-compression at large strain. Int. J. Plasticity 18, 633 659. Yoshida, F., Amaishi, T., 2020. Model for description of nonlinear unloading-reloading stress-strain responses with special reference to plastic-strain dependent chord modulus. Int. J. Plasticity. Available from: https://doi.org/10.1016/j.ijplas.2020.102708. Yuanming, L., Long, J., Xiaoxiao, C., 2009. Yield criterion and elasto-plastic damage constitutive model for frozen sandy soil. Int. J. Plasticity 25, 1177 1205. Zaremba, S., 1903. Su une forme perfectionnee de la theorie de la relaxation. Bull. Int. Acad. Sci, Crac. 594 614 (in French). Zbib, H.M., Aifantis, E.C., 1988. On the concept of relative and plastic spins and its implications to large deformation theories. Part I: hypoelasticity and vertex-type plasticity. Acta Mech. 75, 15 33. Zhang, S., Leng, W., Zhang, F., Xiong, Y., 2012. A simple thermo-elastoplastic model for geomaterials. Int. J. Plasticity 34, 93 113. Zhang, S., Ye, G., Wang, J., 2018. Elastoplastic model for overconsolidated clays with focus on volume change under general loading conditions. Int. J. Geomech. 18. Available from: https://doi.org/10.1061/(ASCE)GM.1943-5622.0001101. Zhang, Y., Zhou, A., 2016. Explicit integration of a porosity-dependent hydromechanicalmodel for unsaturated soils. Int. J. Numer. Anal. Meth. Geomech. 40, 2353 2238. Zhoa, J., Sheng, D., Rouainia, M., Sloan, S.W., 2005. Explicit stress integration of complex soil models. Int. J. Numer. Anal. Meth. Geomech. 29, 1209 1229.

392

Bibliography

Zhou, Y., Sheng, Q., Li, N., Fu, X., 2019. Numerical investigation of the deformation properties of rock materials subjected to cyclic compression by the finite element method. Soil Dyn. Earthq. Eng 126, 1 13. Zhou, Y., Sheng, Q., Li, N., Fu, X., Zhang, Z., Gao, L., 2020. A constitutive model for rock materials subjected to triaxial cyclic compression. Mech. Mater. Available from: https:// doi.org/10.1016/j.mechmat.2020.103341. Zhou, A., Zhang, Y., 2015. Explicit integration scheme for a non-isothermal elastoplastic model with convex and nonconvex subloading surfaces. Comput. Mech. 55, 924 961. Zhu, H., Ye, B., Cai, Y., Zhang, F., 2013. An elasto-viscoplastic model for soft rock around tunnels considering overconsolidation and structure effects. Comput. Geotech. 50, 6 16. Ziegler, H., 1986. An Introduction to Thermomechanics. North-Holland. Ziegler, H., McVean, D., 1967. Recent Progress in Applied Mechanics. Wiley.

Index Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively.

A Additive decomposition, 4 5, 265 strain rate, 201 202 Almansi Eulerian strain tensor, 106 107 Alternating symbol, 2, 5 6 Angular moment, 130 132 Angular momentum, conservation laws of, 126 Anisotropy kinematic hardening. See Kinematic hardening Anti(skew)-symmetric tensor. See Tensor Anti-symmetric tensor. See Tensor, skewsymmetric (anti-symmetric) Anti-symmetrizing. See Tensor, skewsymmetrizing (anti-symmetrizing) Armstrong Frederick kinematic hardening rule. See Kinematic hardening Armstrong Frederick nonlinear kinematic hardening, 171 Associated flow rule (associativity), 175 176, 274 Clausius Duhem inequality, 256, 369 tangential, 342 343 Associative law of vector. See Vector Axial vector. See Vector

B Back stress. See Kinematic hardening Balance laws in current configuration, 132 135 conservation law of energy, 135 rotational equilibrium, 133 134 translational equilibrium, 133 virtual work principle, 134 135, 136f in reference configuration, 145 147 conservation law of energy, 146 147 translational equilibrium, 145 146 virtual work principle, 146 Base vector algebra, 65 68 Bauschinger effect, 170 Bingham model, 229 230

Body force, 125 126 Boundary friction, 358 Bounding surface model with radialmapping, 252 253 Bulk elastic modulus, 161 162

C Cam-clay model (modified), 305 Canonical expression for orthogonal tensor, 24 Cartesian decomposition. See Tensor summation convention, 1 2 Cauchy’s elastic equation, 377 378 elasticity, 152 first law of motion. See Translational equilibrium fundamental theorem. See Cauchy’s, stress tensor stress. See Stress stress principle. See Cauchy’s, stress tensor stress tensor, 127 132, 132f definition, 127 130 symmetry, 130 134 stress vector, 129 Cauchy Green deformation tensor, 140 left and right, 101 102, 105, 114 elastic and plastic, 264 265 Cayley Hamilton theorem, 38 39 Chaboche model. See Superposed kinematic hardening model Characteristic equation, 35 value. See Eigen (principal), values vector. See Eigen (principal), vectors Clausius Duhem inequality, 256, 369 Coaxiality, 38 Cofactor, 3 4 Communicative law of vector. See Vector

393

394 Complementary strain energy function, 154 155, 154f Complete integrability condition, 152, 377 378 Component of tensor. See Tensor of vector. See Vector Configuration current, 132 135, 265 266, 330 331 conservation law of energy, 135 rotational equilibrium, 133 134 translational equilibrium, 133 virtual work principle, 134 135, 136f intermediate, 258 259, 267 269 reference (initial, Lagrangian), 145 147 conservation law of energy, 146 147 translational equilibrium, 145 146 virtual work principle, 146 Conservation laws, 123 126 angular momentum, 126 energy, 135, 146 147 Eulerian description, 124 Lagrangian description, 124 linear momentum, 125 126, 127f mass, 124 125, 131f physical quantity, 123 124 Consistency condition conventional plasticity, 179 180 initial subloading surface model, 210 extended subloading surface model, 205 Consistency condition for friction, 339 Consistent tangent modulus tensor, 227 228 Constitutive equation, 230 237 multiplicative subloading-overstress model, 320 326, 327f rate-independent elastoplastic, 368 369 subloading-overstress model, 213 228 Contact elastic moduli for friction, 337 elastic modulus tensor for friction, 347 stress controlling function, 342f traction for friction, 334 336, 336f Continuity condition, 124 125, 163 elastoplastic constitutive equations, 166, 179 180 subloading-multiplicative hyperelasticbased plastic and viscoplastic constitutive equations, 283 284 Continuity equation, 124 125 Continuum spin. See Spin tensor Contraction of tensor. See Tensor

Index

Contravariant base vector, 70, 74 metric tensor, 71 tensor, 85, 87 88 vector, 83 84 Convected base vectors, 79 80, 79f, 83f coordinate system, 77 stress rate tensors, 373 376 contravariant, 373 374 corotational, 374 covariant, 374 covariant contravariant, 374 tensor, 77 78 term. See Steady term time-derivative, 88 99 convected rate, objectivity of, 95 99 corotational rate, 92 95, 94f general, 89 92, 91f vector, 79 Convective. See Convected Conventional plasticity model, 250 Convexity condition, 199 200 Convexity of yield surface, 199 200 Coordinate system convected, 77 curvilinear, 61 Coordinate transformation tensor, 22 Core surfaces normal-yield, subloading, and elasticcore surfaces, 289 291 metals, 299 300 Corotational Eulerian, 87 88 Green Naghdi rate, 374 Lagrangian, 86 rate, 92 96, 94f Zaremba Jaumann rate, 374 375 Cotter Rivlin rate, 89 90, 374 Coulomb friction condition, 344 Covariant base vector, 70, 74 75 metric tensor, 71 stress, 117 vector, 83 Creep model irrationality of, 240 243 rheological model, 240f Cross product. See Vector Curl of tensor field. See Tensor, field Current configuration. See Configuration

Index

Cyclic kinematic hardening models, 188 190 Cyclic loading, 163 elastoplastic deformation, 188 friction, 351f Cyclic plasticity models, 188 195 Ad hoc Chaboche model, 191 192 cyclic kinematic hardening models, 188 190 Dafalias model, 188 190 extended subloading surface model, 192 213 initial subloading surface model, 192 194 inverse and reloading responses, improvement of, 208 209 isotropic hardening, cyclic stagnation of, 209 213, 210f multi surface model (Mroz model), 188 190 normal-yield ratio, calculation of, 207 208 Ohno-Wang model, 191 192, 243 plastic strain rate, 205 206 strain rate versus stress rate relations, 206 207 superposed-kinematic hardening, 188 two surface model (bounding surface model), 171 Yoshida Uemori model, 190 Cyclic stagnation of isotropic hardening, 209 213, 317 320

D Deformation gradient elastic, 256, 258 260, 263, 269, 281, 283, 308, 320 plastic, 258 260, 263, 269 271, 281, 283, 286 289 polar decomposition, 53 54 relative, 103 tensor, 53 56, 56f, 80 83, 101 106, 105f isochoric part, 114 116 multiplicative decomposition of, 255, 259f volumetric part, 114 116 Del operator. See Hamilton operator Derivatives of tensor field, 49 51 Description Eulerian, 48 49, 124 125 Lagrangian, 48 49, 103, 124 126 material. See Description, Lagrangian

395

relative, 48 49 spatial. See Description, Eulerian Determinant, 27 30 definition, 3 derivative product law, 5 Deterministic prediction of earthquake occurrence, 362 363 Deviatoric part, 25 principal invariant, 10 11 projection tensor (fourth-order), 33 Diagonal component. See Tensor Differential formulae, 43 53 derivatives of tensor field, 49 51 Gauss’ divergence theorem, 51 52 material-time derivative of volume integration, 52 53 partial derivatives of tensor functions, 43 47 time derivatives in Lagrangian and Eulerian descriptions, 48 49 Direction cosine, 7 8 Direct notation. See Tensor, notations Dissipation energy, 164 Dissipative part of plastic strain elastic-core, 202 203 kinematic hardening, 172 175, 178 Distributive law of vector. See Vector Divergence of tensor field, 50 Divergence theorem Gauss’ divergence theorem Dry friction, 358 359, 359f, 359t Dummy index, 2 Dyad. See Vector

E Earthquake occurrence, deterministic prediction of, 362 363 Eddington’s epsilon. See Alternating symbol Eigen (principal) direction. See Principal, direction values, 35 43 second-order tensors vectors, 35 43 second-order tensors, 35 37 Einstein’s summation convention, 2 Elastic bulk elastic modulus, 161 162 Cauchy elasticity, 152

396

Index

Elastic (Continued) constitutive equation, 151, 255, 294, 304 305, 377 deformation gradient, 256, 258 260, 263, 269, 281, 283, 308, 320 Green elasticity, 151 hyperelasticity infinitesimal strain, 161 162 hyperelastic model of soils, 160 161 hypoelasticity, 152 infinitesimal strain, 161 162 modified neo-Hookean elasticity, 157 159 modified St. Venant Kirchhoff elasticity, 156 modulus, 161 162, 170, 177, 179, 347 Mooney Rivlin model, 159 160 neo-Hookean elasticity, 157 Ogden model, 160 Poisson’s ratio, 161 162 predictor step. See Elastic, trial step shear elastic modulus, 161 162 sliding displacement energy function, 337 spin, 269 St. Venant Kirchhoff elasticity, 155 156 strain energy function, 137, 151 152, 154 162 elastic-core, 170 kinematic hardening, 167 168 strain rate, 179, 182 183, 194, 240, 270 stress rate, 93 tangent modulus, 227 228 trial step, 220 223, 222f, 225, 309, 311 312, 328, 351 354, 351f trial contact stress, 347 348 Yong’s modulus, 161 162 Elastic-core enclosing condition, 199, 200f evolution rule of, 198 205 limit elastic-core surface, 198 199 nonhardening Mises material, uniaxial loading for, 202f yield ratio, 198 199 Elastic-plastic transition, 165, 198, 233, 250, 365 367 Elastoplastic constitutive equations, 163 continuity condition, 166 cyclic plasticity models, 188 195 Ad hoc Chaboche model, 191 192 cyclic kinematic hardening models, 188 190

extended subloading surface model, 192 213 inverse and reloading responses, improvement of, 208 209 isotropic hardening, cyclic stagnation of, 209 213, 210f normal-yield ratio, calculation of, 207 208 Ohno-Wang model, 191 192 plastic strain rate, 205 206 strain rate versus stress rate relations, 206 207 fundamental requirements, 164 166 deformation/rotation (rate) into elastic and plastic parts, decomposition of, 164 165 stress rate versus strain rate relation, 166 yield surface, incorporation of, 165 historical development, 168 182 hypoelastic-based plasticity, 178 181 infinitesimal hyperelastic-based plasticity, 168 178 multiplicative hyperelastic-based plasticity, 181 182 rate-independent, 368 369 smoothness condition, 167 168 subloading-overstress model, 229 249, 229f constitutive equation, 230 237, 232f, 233f, 235f creep and stress relaxation, 238f creep model, irrationality of, 240 243, 240f, 241f, 242f defects, 238 240, 239f implicit stress integration, 243 249 isotropic hardening function, temperature dependence of, 249, 249f rheological model, 236f subloading surface model. See Subloading surface model Elastoplastic stiffness modulus tensor, 182 Embedded. See Convected Energy, conservation law of, 135, 146 147 Equilibrium equation, 135, 374 Cauchy stress, 145 146 first Piola Kirchhoff stress, 139 rate type, 145 Equivalent plastic strain, 133 Eulerian

Index

configuration. See Configuration description. See Description (spatial) description time derivatives, 48 49 conservation law in, 124 spin tensor, 120 strain. See Strain tensor. See Tensor triad, 104 Eulerian Lagrangian two-point tensor, 80 82 Euler’s first law of motion. See Linear momentum, conservation law of formula, 56 second law of motion. See Angular momentum, conservation laws of theorem for homogeneous function, 176 Extended subloading surface model. See Subloading surface model

F Fatigue, 184, 366 Fatigue limit, 184 Finger deformation tensor, 102 Finger strain. See Strain Finite sliding theory, 333 Finite strain theory, 190, 262, 300 First Piola Kirchhoff stress. See Stress Flow rule associated, 175 176, 274 for friction, 342 343 plastic. See Plastic, flow rules Fluid friction, 358 359, 359f, 359t Formula of scalar triple products with invariants, 17 Forward-Euler method, 190, 208, 212 213 Fourth-order identity tensor, 32 33 Fourth-order tracing tensor, 33 Fourth-order transposing tensor, 32 33 Friction coefficient evolution rule of, 339 340 kinetic, 333 static, 339 negative rate-sensitivity, 340 positive rate-sensitivity, 334, 340, 358

G Gauss’ divergence theorem, 51 52 conservation laws, 124 Generalized strain measure, 109

397

Gibbs’ free (complementary strain) energy function. See Complementary strain energy function Gradient of tensor field, 49 Green elastic equation. See Hyperelastic equations strain. See Strain Green and Almansi strain tensors, 106 109 Green Lagrangian strain tensor, 106 107 Green Naghdi rate, 119 Cauchy stress, 374 Kirchhoff stress, 374 stress rate, 92 93 Gurson model, 366

H Hamilton operator, 49 Hardening anisotropic, 171 172 isotropic cyclic stagnation of, 317 320 cyclic stagnation, 209 213 evolution rule, 179 180 rheological model, 174, 174f variable, 170 172, 180 Voce-type function, 171 linear kinematic, 171, 172f, 174 175 nonlinear kinematic, 172f, 174, 174f, 184f Hashiguchi model, 367 368 Helmholtz’ free energy function, 152, 154 155, 154f, 170 Hencky strain tensor. See Logarithmic strain tensor Hill Rice manner, 178 179 Hill’s strain measure. See Generalized strain measure Hooke’s law, 215 Hyperelastic-based plasticity infinitesimal, 163, 168 178 multiplicative, 181 182 Hyperelastic equations, 151 154 complete integrability condition, 152 constitutive subloading-multiplicative, 275 277, 278f current configuration, 330 331 elastic-core, 195 elastic sliding, 337 338 kinematic hardening, 170 metals, 155 159, 297 298 rubbers, 159 160

398

Index

Hyperelastic equations (Continued) soils, 160 161 Hyperelasticity infinitesimal strain, 161 162 Hyperelastic sliding displacement, 337 338 Hypoelastic-based plasticity, 163, 178 181 limitation of, 269 270 Hypoelastic equation, 378 Hypoelasticity, 152 Hypoplasticity, 165 Hysteresis loop, 192 194, 198, 251

I Identity tensor, 13 14 fourth-order, 32 33 second-order, 81 82 Impact load deformation, 239 240 friction, 358 359 Implicit stress integration subloading-overstress friction model, 360 361 subloading-overstress model, 243 249 return-mapping subloading surface model, 213 228 Infinitesimal hyperelastic-based plasticity, 163, 168 178 Infinitesimal strain, 161 162 tensor, 109 theory, 200 201 Infinitesimal volume element, 55 56, 124 125 Influence of temperature on isotropic hardening, 249, 249f Initial configuration. See Configuration Initial subloading surface model. See Subloading surface model Inner product. See Vector Intermediate configuration. See Configuration Internal energy, 135 heat source, 135 variable, 92, 95, 164 167, 172, 179, 183 184, 214, 243 244, 269 270, 308, 369 370 Intrinsic. See Convective Invariant. See Principal, invariants Inverse loading, 208 209 Inverse tensor, 21 22 Irreversible mechanical phenomena formulation of, 365

plastic flow rule based on second law of thermodynamics, 369 370 rate-independent elastoplastic constitutive equations, 368 369 subloading surface model, utilization of, 365 368 commercial software, standard installation to, 367 368 mechanical phenomena, 365 367 Isochoric (distortional) part of deformation gradient tensor, 114 116 Isoclinic concept, 256, 259 262, 261f Isotropic hardening (variable) cyclic stagnation, 209 213, 317 320 evolution rule, 179 180 rheological model, 174, 174f variable, 170 172, 180 hardening function, 171, 305 306 temperature dependence, 249 tensors, 34 35 definition, 34 fourth-order, 34 35 second-order, 34 third-order, 34 Voce-type function, 171

J Jacobian, 55 Jacobian matrix, 244 249 Jaumann (Zaremba-Jaumann) rate. See Corotational rate

K Kinematic hardening Armstrong Frederick nonlinear, 171 dissipative part of plastic strain, 172 175, 178 evolution rule, 174 linear, 172f multiplicative decomposition, 271 272, 271f nonlinear, 172f, 174 Prager’s linear kinematic hardening rule, 174 175 rheological model, 174, 174f storage part of plastic strain, 172 variable (back stress), 173 174, 298 299 Kinetic friction. See Friction Kirchhoff stress tensor, 273 274, 373 covariant contravariant convected rate, 374

Index

Green Naghdi rate, 374 Oldroyd rate, 373 374 and work-conjugacy, 136 137 Kronecker’s delta, 2, 78, 81 82 Kro¨ner decomposition, 258

L Lagrangian configuration. See Configuration description. See Description (material) description (representation) time derivatives, 48 49 tensor, 77 78, 80, 85 86, 88 corotational, 86 spin, 119 120 strain. See Strain triad, 104 105 updated description, 103 Lame´ constants, 43 Laplacian (Laplace operator), 50 51 Lee decomposition. See Multiplicative decomposition, of deformation gradient tensor l’Hoˆpital’s rule, 110 111 Lie derivative, 78, 88 89, 91 92 Limit sliding normal-yield ratio subloading-overstress friction model, 355 356 Linear elastic equation, 43 Linear kinematic hardening. See Kinematic hardening Linear momentum, conservation law of, 125 126, 127f Linear transformation, 17 Loading criterion, 221 223 plastic sliding velocity, 344 plastic strain rate, 283 return-mapping for subloading surface model, 221 223, 222f, 224f, 312 314 subloading-friction model, 351 354, 351f, 352f Local form, 124 125, 133 135 Localization of deformation, 48 Local-time derivative. See Spatial-time derivative Local-time derivative term. See Nonsteady term Lode’s angle, 45 Logarithmic strain tensor, 113, 115 116 left and right, 113 volumetric, 113 Loosening of the screw, 361 362 Lubricated friction, 358

399

M Macaulay bracket, 185 Maclaurin expansion, 39 40 Magnitude of tensor, 19, 23, 73 of vector, 7 8, 22 23 Mandel Lee manner, 178 179 Mandel stress tensor, 137 138, 141, 144f, 159, 273 275 Masing rule, 208 209 Mass, conservation law of, 124 125, 131f Material Biot strain. See Strain description. See Lagrangian description frame-indifference. See Objectivity Hencky strain. See Strain Piola strain. See Strain -time derivative, 48 volume integration, 52 53 Matrix, transpose of, 3 Matrix algebra, 1 6 Kronecker’s delta and alternating symbol, 2 matrix notation and determinant, 2 6 summation convention, 1 2 Mean part of tensor. See Tensor Mechanical ratcheting effect, 192 194, 208 209, 241 242 Metals hyperelastic constitutive equations of, 155 159 modified neo-Hookean elasticity, 157 159 modified St. Venant Kirchhoff elasticity, 156 neo-Hookean elasticity, 157 St. Venant Kirchhoff elasticity, 155 156 material functions of, 296 300 elastic core, 299 kinematic hardening variable, 298 299 stress, 297 298 yield function, 299 300 Metric tensors, 65 68 covariant, 71 contravariant, 71 Mises yield condition, 299 300 Mixed friction, 358 Modified Cam-clay model, 305 Modified Neo-Hookean elasticity. See Elastic Modified St. Venant Kirchhoff elasticity. See Elastic

400

Index

Moments, 36 Momentum angular, 126 linear, 125 126 Mooney Rivlin model, 160 Mroz model. See Multi-surface model Multiplicative decomposition, of deformation gradient tensor, 255, 259f elastic and plastic right Cauchy Green deformation tensor, 264 265 elastic-plastic decomposition of deformation measure, 256 264 embedded base vectors, 263 264 isoclinic concept, 259 262, 261f kinematic hardening, 271 272, 271f necessity of, 256 259, 258f strain rate and spin tensors, 265 269 current configuration, 265 266 intermediate configuration, 267 269 substructure spin, 269 hypoelastic-based plasticity, limitation of, 269 270 uniqueness of, 262 263 Multiplicative extended subloading surface model, 286 296 normal-yield, subloading, and elasticcore surfaces, 289 291 plastic deformation gradient for elasticcore, multiplicative decomposition of, 286 289, 290f plastic flow rules, 291 293 plastic strain rate, 294 296 Multiplicative finite strain theory, 190 Multiplicative hyperelastic-based plasticity, 181 182, 255 Multi-surface model. See Cyclic plasticity models

N Nabla operator. See Hamilton operator Nanson’s formula, 56 Negative rate-sensitivity, 340 Neo-Hookean elasticity, 157 modified, 157 159 Nominal strain. See Strain stress. See Stress Nonhardening region. See Cyclic stagnation of isotropic hardening Nonlinear kinematic hardening. See Kinematic hardening

Nonsingular tensor, 21 22 Nonsteady term. See Local-time derivative Normal component, 35, 113 Normal (Gaussian) distribution curve for temperature, 249 Normal-isotropic hardening ratio, 209 210, 213 228, 317 Normal-isotropic hardening surface, 209 210, 210f, 212 213 Normality rule. See Associated flow rule (associativity) Normalized orthogonal coordinate system, 7 Normal sliding for friction evolution rule, 339 340 ratio, 338 339 surfaces, 338 339, 339f Normal traction vector, 336 Normal-yield evolution rule, 340 341 ratio, 182 184, 183f, 184f, 185f, 340 341 calculation, 207 208 in plastic corrector step, initial value of, 224 227, 314 317, 319f and subloading surfaces, 195 198, 197f surface, 182 183 Norton law, 240

O Objective rate of tensor, 88 89 rate of vector, 16 tensor, 16 17 time-derivative of scalar-valued tensor function time-integration of rate tensor, 98 stress rate tensor, 373 transformation, 16 17 Objectivity, 137 of convected rate, 89, 95 99 Ogden model, 160 Oldroyd rate Cauchy stress, 373 374 Kirchhoff stress, 373 374 stress rate, 89 90 Operational tensors, 32 34 Orthogonal coordinate system, 7 tensor, 22 24 canonical expression, 24 Overstress model

Index

friction model. See Subloading-overstress friction model implicit stress integration, 360 361 subloading. See Subloading-overstress model

P Partial derivative, 5 of tensor functions, 43 47 Partial differential calculi, 43 Permutation symbol. See Alternating symbol Permutation tensor, 13 14 Phase transformation of metals, 367 Physical components, 74 75 Physical quantity, conservation law of, 123 124 Piola deformation tensor, 102 Piola Kirchhoff stress tensors, 140f first, 137 139 second, 137 140 Plastic corrector step, 214, 347 348 normal-yield ratio in, initial value of, 224 227 deformation gradient. See Deformation gradient flow rules, 277 281 based on second law of thermodynamics, impertinence of formulation of, 369 370 multiplicative extended subloading surface model, 291 293 -loading initiation stress, 314 315 material spin, 93 modulus, 206, 212, 237, 249, 285 286, 309, 320, 361 multiplier, 175 177 potential, 251 252, 256, 370 sliding velocity, 342 346, 345f, 346f spin, 93 95, 179 181, 196 198, 261, 269 270, 293 strain rate, 205 206, 281 283, 285f flow rules for, 277 281 multiplicative extended subloading surface model, 294 296 stress relaxation, 170 volumetric strain, 171 172, 306 Poisson’s ratio, 161 162 Polar decomposition, 40 42 spin. See Spin

401

Positive-definite tensor, 40 42 Positive proportionality factor, 175 Positive rate-sensitivity, 358 Prager’s continuity condition, 166 linear kinematic hardening rule, 174 175 overstress model, 231 233 Primary base vectors, 61 65, 63f vector, 11 14 Principal direction, 37 38, 104, 113, 330 331 invariants, 36 stretch, 104 value. See Eigen (principal), values vector. See Eigen (principal), vectors Principle of material-frame indifference. See Objectivity of convected rate maximum plastic work, 170 of objectivity. See Objectivity Product law of determinant, 5 Projection of tensor deviatoric, 33 deviatoric-tangential Proper value. See Eigen (principal), values Proper vector. See Eigen (principal), vectors Pull-back operation, 77, 83 88, 83f first Piola Kirchhoff and Nominal stress tensors, 138 139 second Piola Kirchhoff stress tensor, 139 140 strain tensors, 107 108 Push-forward operation, 77, 83 88, 83f strain tensors, 107 108

Q Quasistatic deformation, 229 230, 233 235, 237, 239 241, 355 356, 358 361, 368

R Ratcheting effect. See Mechanical ratcheting effect Rate of elongation, 257 normal vector of surface, 56 shear strain, 93 surface area, 56 Rate-and-state friction model, 362 Rate-independent elastoplastic constitutive equations, 368 369

402

Index

Rate-(or incremental)-type equilibrium equation, 145 Rate-sensitivity negative, 340 positive, 334, 340, 358 Rational mechanics, 165 Reciprocal base vectors, 61 65, 64f vector, 11 14 Reference configuration. See Configuration Relative deformation gradient tensor, 103 first Piola Kirchhoff stress rate, 374 left and right Cauchy-Green deformation tensor, 102 spin, 92 93, 119 Reloading behavior, in subloading surface model, 208 209 Representation theorem of isotropic tensorvalued tensor function, 42 43 Return-mapping, 213 228, 347 354 bounding surface model with, 252 253 consistent tangent modulus tensor, 227 228 elastic trial (predictor) step, 214, 220 223, 222f, 224f formulation, 213 221 loading criterion, 221 223 multiplicative extended subloading surface model, 309 317, 313f, 318f multiplicative subloading-overstress model, 328 330 normal-yield ratio in plastic corrector step, initial value of, 224 227 subloading-friction model formulation, 347 350 loading criterion, 351 354, 351f, 352f Reynolds’ transportation theorem, 52 53, 123 124 Rheology model creep model, 240f isotropic hardening, 174, 174f kinematic hardening, 174, 174f subloading-friction model, 345f subloading-overstress model, 236f subloading-overstress friction model, 357f subloading surface model, 204f Rigid-body rotation, 16 17, 48 49, 88 89, 92 93, 95, 99, 120, 140, 151 152, 181, 256, 259 260, 263, 269, 281, 308, 330 Rigid plastic material, 48 49

Rotation (curl) of tensor field, 50 51 -free (insensitive) tensor, 140 rate of Lagrangian or Eulerian triad, 104 105 rate tensor of material, 101 of triad, 104 105 Rotation-free tensor, 140 Rotational equilibrium, 133 134 Rotationless (or corotational) strain rate, 119 Rubbers, hyperelastic equations of, 159 160

S SANICLAY model, 252 253 SANISAND model, 252 253 Scalar product. See Vector triple product. See Vector triple product with invariants, 39, 371 372 Second law of thermodynamics plastic flow rule based on, impertinence of formulation of, 369 370 Second-order tensor functions, 39 40 properties of, 18 19 Second Piola Kirchhoff stress tensor, 159, 377 Shear deformation behavior, 147f Shear elastic modulus, 161 162 Shifter, 65 66, 69 Similarity-center enclosing condition, 199 surface, 198 translation rule, 252 253 yield ratio, 192 196 Similar tensor, 195 196 Simple shear, 147 149 Simultaneous equation for vector components, 30 31 Skew-symmetric (anti-symmetric). See Tensor, skew-symmetric (antisymmetric) Skew-symmetrizing (anti-symmetrizing). See Tensor, skew-symmetrizing (anti-symmetrizing) Sliding displacement, 334 336, 336f elastic and plastic, 166 hyperelastic, 337 338

Index

normal and tangential, 167 normal-yield ratio, evolution rule of, 340 341, 341f, 342f subloading surface, 333 yield condition, 338 Smoothness condition, 163, 167 168 subloading-multiplicative hyperelasticbased plastic and viscoplastic constitutive equations, 283 284 Soils, hyperelastic equations of, 160 161 Spatial Biot. See Strain Spatial description. See Eulerian, description Spatial-time derivative, 48 Spectral decomposition (representation), 37 38 polar, 41 42 Spherical part of tensor. See Tensor Spin base vector, 27 constitutive, 95 continuum, 58, 117 118 elastic, 269 plastic, 93 95, 179 181, 196 198, 261, 269 270, 293, 309 flow rules, 262 263 substructure, 93 94, 94f, 98, 269 hypoelastic-based plasticity, limitation of, 269 270 Spin tensor, 117 121 based on velocity gradient tensors, 117 120 continuum, 58, 117 118 Eulerian, 120 Lagrangian, 119 120 multiplicative decomposition, 265 269 current configuration, 265 266 intermediate configuration, 267 269 relative (polar), 119 Spring-back phenomenon, 190 Stagnation of isotropic hardening. See Cyclic stagnation of isotropic hardening Static friction. See Friction Steady term, 48 49 Stick-slip phenomenon, 362 363 St. Venant Kirchhoff elasticity, 155 156 modified, 156 Storage part of plastic strain elastic-core, 202, 230, 320 322 kinematic hardening, 172 174, 230, 271, 286, 320 322

403

Strain Almansi Eulerian, 106 107 Biot, 142 Finger, 102 Green Lagrangian, 106 107 Hencky, 113, 115 116 infinitesimal, 109 logarithmic (natural), 113 logarithmic volumetric strain, 113 nominal, 109, 169 Strain rate cyclic plasticity models, 206 207 elastoplastic constitutive equations, 166 tensor, 58, 117 121 based on general strain tensor, 120 121 based on velocity gradient tensors, 117 120 current configuration, 265 266 intermediate configuration, 267 269 multiplicative decomposition, 265 269 rotationless (or corotational), 119 Strain tensors, 106 113 Almansi Eulerian, 106 107 Eulerian, 120 general, 109 112 Green and Almansi, 106 109 Green Lagrangian, 106 107 infinitesimal, 109 logarithmic, 113 Stress Biot, 142 first Piola Kirchhoff (nominal), 137 139, 160 161 Kirchhoff, 373 path dependence, 166 rate contravariant convected rate, 373 374 covariant convected rate, 374 Cotter Rivlin rate, 374 cyclic plasticity models, 206 207 elastoplastic constitutive equations, 166 Green Naghdi rate, 374 Oldroyd rate, 373 374 Truesdell rate, 374 Zaremba Jaumann rate, 374 375 second Piola Kirchhoff (nominal), 137 141, 140f, 143, 152, 154, 273 276, 288, 300, 377 tensors Cauchy, 127 134, 132f, 373 Kirchhoff stress tensor, 136 137

404

Index

Stress (Continued) Mandel, 137 138, 141, 144f nominal, 137 139 physical meanings, 138 141 relations, 141 142, 142t traction vectors, 142 145 Stretch principal, 104 tensor, left and right, 104 Stretching. See Strain rate Subloading-friction model, 333 calculation procedure, 346 347 friction coefficient, evolution rule of, 339 340 history of, 333 334 hyperelastic sliding displacement, 337 338 normal-sliding and subloading-sliding surfaces, 338 339, 339f plastic sliding velocity, 342 346, 345f, 346f return-mapping, 347 354 formulation, 347 350 loading criterion, 351 354, 351f, 352f rheological model, 345f sliding displacement and contact traction vectors, 334 336, 336f sliding normal-yield ratio, evolution rule of, 340 341, 341f, 342f subloading-overstress friction model, 354 360, 357f applications, 361 363 earthquake occurrence, deterministic prediction of, 362 363 implicit stress integration, 360 361 loosening of the screw, 361 362 Subloading-isotropic hardening surface, 209 210, 210f Subloading-multiplicative hyperelasticbased plastic and viscoplastic constitutive equations, 273, 321f calculation procedure, 306 309 continuity and smoothness conditions, 283 284 conventional elastoplastic model, 277 283 plastic strain rate and plastic spin, flow rules for, 277 281 plastic strain rate, 281 283, 285f uniqueness, confirmation of, 281 hyperelastic constitutive equations, 275 277, 278f

current configuration, 330 331 initial subloading surface model, 284 286, 287f isotropic hardening, cyclic stagnation of, 317 320 metals, material functions of, 296 300 elastic core, 299 kinematic hardening variable, 298 299 stress, 297 298 yield function, 299 300 multiplicative extended subloading surface model, 286 296 normal-yield, subloading, and elasticcore surfaces, 289 291 plastic deformation gradient for elastic-core, multiplicative decomposition of, 286 289, 290f plastic flow rules, 291 293 plastic strain rate, 294 296 multiplicative subloading-overstress model, 320 330 calculation procedure, 326 328 constitutive equation, 320 326, 327f return-mapping, 328 330 return-mapping, implicit calculation by, 309 317, 313f, 318f loading criterion, 312 314 normal-yield ratio in plastic corrector step, initial value of, 314 317, 319f soils, material functions of, 300 306 hyperelastic equation, 300 304, 305f yield function, 305 306 stress measures, 273 275 Subloading-overstress friction model, 354 360, 357f boundary friction, 358 dry friction, 358 359, 359f, 359t fluid friction, 358 359, 359f, 359t implicit stress integration, 360 361 limit sliding normal-yield ratio, 355 356 lubricated friction, 358 mixed friction, 358 positive rate-sensitivity, 358 rheological model, 357f sliding normal-yield ratio, 355 356 static sliding normal-yield ratio, 355 356 Subloading-overstress model, 229 249, 229f constitutive equation, 230 237, 232f, 233f, 235f creep and stress relaxation, 238f creep model, irrationality of, 240 243, 240f, 241f, 242f

Index

defects, 238 240, 239f elastic-core surface, 230 implicit stress integration, 243 249 isotropic hardening function, temperature dependence of, 249, 249f limit elastic-core surface, 232f limit normal-yield ratio, 234 limit subloading surface, 232f multiplicative, 320 330 calculation procedure, 326 328 constitutive equation, 320 326, 327f return-mapping, 328 330 normal-yield surface, 195 198 rheological model, 236f static normal-yield ratio, 233 234 static subloading surface, 234 subloading surface, 230 viscoplastic (stress) relaxation, 235 viscoplastic strain rate, 229 230 Subloading-sliding surfaces, 338 339, 339f Subloading surface model, 163 164, 252 253, 367 368 concept, 182 consistency condition, 210 extended, 192 195, 195f elastic-core, evolution rule of, 198 205 formation, 195 213 normal-yield and subloading surfaces, 195 198, 197f rheology model, 204f fatigue (or endurance) limit, 184 fundamental characteristics, 249 253 abilities, 250 252, 250f, 251f bounding surface model with radialmapping, 252 253 initial, 192 194, 284 286, 287f multiplicative extended, 286 296 normal-yield, subloading, and elasticcore surfaces, 289 291 plastic deformation gradient for elastic-core, multiplicative decomposition of, 286 289, 290f plastic flow rules, 291 293 plastic strain rate, 294 296 return-mapping, 213 228 consistent tangent modulus tensor, 227 228 formulation, 213 221 loading criterion, 221 223, 222f, 224f normal-yield ratio in plastic corrector step, initial value of, 224 227

405

uniaxial loading behavior, 194 utilization of, 365 368 commercial software, standard installation to, 367 368 mechanical phenomena, 365 367 Substructure spin, 269 hypoelastic-based plasticity, limitation of, 269 270 Summation convention, 1 2 Cartesian, 1 2 Einstein’s, 2 Superposed kinematic hardening model, 188 Surface element vector, 71, 72f Surface strain rate tensor, 57 58 Symbolic notation. See Tensor, notations Symmetric tensor, 24 25 Symmetrizing tensor, 33

T Tangential for friction associated flow rule, 342 343 contact traction, 336 Temperature dependence of isotropic hardening function, 249, 249f Tensor anti-symmetric. See Tensor, skewsymmetric (anti-symmetric) anti-symmetrizing. See Tensor, skewsymmetrizing (anti-symmetrizing) Cartesian decomposition, 24 25 characteristic equation, 35 coaxial (coaxiality), 38 components, 19 20 continuum spin tensor, 58 contraction, 17 contravariant metric tensor, 71 coordinate transformation tensor, 22 covariant metric tensor, 71 decompositions, 24 25 definition, 15 18 deformation gradient tensor, 53 56, 56f deformation tensors, 101 106, 264 269 derivatives of tensor field, 49 51 determinant, 27 30 deviatoric part, 25 deviatoric projection tensor, 33 eigenvalues, 35 43 eigenvectors, 35 43 Eulerian tensor, 77 78, 80, 85 88 Eulerian Lagrangian two-point tensor, 80 82

406

Index

Tensor (Continued) exponentiation, 19 field divergence, 50 gradient, 49 rotation (curl), 50 51 Finger deformation tensor, 102 fourth-order, 31 32 identity tensor, 32 33 tracing tensor, 33 identity, 13 14 inverse tensor, 21 22 isotropic tensors, 34 35 Lagrangian tensor, 77 78, 80, 85 86, 88 linear transformation, 17 mean part. See Tensor, spherical part metric tensor, 65 68 moments, 36 nonsingular tensor, 21 22 notations, 31 32 objective tensor, 16 17 objective transformation, 16 17 operational, 32 34 operations, 18 31 orthogonal tensor, 22 24 partial derivatives of tensor functions, 43 47 Piola deformation tensor, 102 polar decomposition, 40 42 positive-definite tensor, 40 42 principal invariants. See Principal, invariants product, 14 15 relative deformation gradient tensor, 103 representations in coordinate system, 68 75, 69f representation theorem of isotropic tensor-valued tensor function, 42 43 scalar triple products with invariants, 39 second-order, 31 32 tensor functions, 39 40 tensor, properties of, 18 19 skew-symmetric (anti-symmetric), 24 25 skew-symmetrizing (anti-symmetrizing), 33 spectral decomposition (representation), 37 38 spherical part, 25 spin tensor, 117 121 strain rate tensor, 58, 117 121 surface element vector, 71, 72f surface strain rate tensor, 57 58

symmetric, 24 25 symmetrizing tensor, 33 third-order, 31 32 time derivative convective term, 48 49 Eulerian description, 48 49 Lagrangian description, 48 49 material-time, 48, 52 53 nonsteady term, 48 spatial-time, 48 volume integration, 52 53 transposed tensor, 20 21 transposing, 32 33 two-point tensor, 80 82 unimodular tensor, 114 115 velocity gradient tensor, 56 59 zero tensor, 19 Traction vectors relations of stress tensors, 142 145 Translational equilibrium current configuration, 133 reference configuration, 145 146 Transposed tensor, 20 21 Transpose of matrix, 3 Truesdell rate of Cauchy stress, 374

U Unimodular tensor, 114 115, 300 301 Unit direction vector, 8 vector, 7 8, 27, 35, 58, 74 75, 129 130 Updated Lagrangian description, 103

V Vector, 6 15 associative law, 7 axial, 25 27 base vector algebra, 65 68 Cauchy stress vector, 129 communicative law, 7 components, simultaneous equation for, 30 31 contact traction vectors, 334 336 cross product. See Tensor product definition, 7 distributive law, 7 dyad. See Tensor product normal traction vector, 336 operations, 7 15 primary base vector, 61 65, 63f primary vector, 11 14 product, 8 9

Index

reciprocal base vector, 61 65, 64f reciprocal vector, 11 14 representations in coordinate system, 69f scalar product, 7 8, 9f tangential traction vector, 336 triple product, 9 11 unit, 7 direction vector, 8 zero, 7 Velocity gradient tensor, 56 59 strain rate and spin tensors based on, 117 120 Virtual work principle current configuration, 134 135, 136f reference configuration, 146 Viscoplastic coefficient, 231 233 constitutive equation, 229, 249 deformation, 233, 235, 236f, 239 240, 322 deformation gradient, 320 322 storage part, 320 322 strain rate, 229 234, 239 240, 239f, 324 325 stress relaxation, 235 velocity gradient, 323 Volume element, 9f, 53 56 Volumetric part of deformation gradient tensor, 114 116

407

W Work-conjugacy, 135 145 Kirchhoff stress tensor and, 136 137 stress tensors, physical meanings of, 138 141 work-conjugate pairs, 137 138, 140f Work-conjugate pairs, 135 138, 140f Work rate (stress power), 135 136

Y Yield condition (surface) Cam-clay, 305 metals, 299 300 Mises, 299 300 soils, 305 306 Yield surface elastoplastic constitutive equations, 165 Yong’s modulus, 161 162 Yoshida Uemori model, 190 Young’s modulus. See Elastic

Z Zaremba Jaumann rate, 93 Cauchy stress, 374 375 Kirchhoff stress, 374 375 Zero tensor, 19 Zero vector, 7