computational methods for platicity-souza

COMPUTATIONAL METHODS FOR PLASTICITYTHEORY AND APPLICATIONSEA de Souza Neto D Peri c DRJ OwenCivil and Computational Engineering Centre, Swansea University

A John Wiley and Sons, Ltd, Publication

COMPUTATIONAL METHODS FOR PLASTICITY

COMPUTATIONAL METHODS FOR PLASTICITYTHEORY AND APPLICATIONSEA de Souza Neto D Peri c DRJ OwenCivil and Computational Engineering Centre, Swansea University

A John Wiley and Sons, Ltd, Publication

This edition rst published 2008 c 2008 John Wiley & Sons Ltd Registered ofce John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial ofces, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identied as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Neto, Eduardo de Souza. Computational methods for plasticity : theory and applications / Eduardo de Souza Neto, Djordje Peric, David Owens. p. cm. Includes bibliographical references and index. ISBN 978-0-470-69452-7 (cloth) 1. PlasticityMathematical models. I. Peric, Djordje. II. Owens, David, 1948- III. Title. TA418.14.N48 2008 531.385dc22 2008033260 A catalogue record for this book is available from the British Library. ISBN 978-0-470-69452-7 Set in 10/12pt Times by Sunrise Setting Ltd, Torquay, UK. Printed in Singapore by Markono.

This book is lovingly dedicated to

Deise, Patricia and Andr; Mira, Nikola and Nina; Janet, Kathryn and Lisa.

CONTENTS

Preface

xx

Part One1

Basic concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 . 3 . 4 . 4 . 7 . 7 . 8 . 9 . 13 . 14 17 17 17 18 19 19 20 21 22 23 23 24 25 28 28 29 30 30 30 30

Introduction 1.1 Aims and scope . . . . . . . . . . . . . . . . . . . . . 1.1.1 Readership . . . . . . . . . . . . . . . . . . 1.2 Layout . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The use of boxes . . . . . . . . . . . . . . . 1.3 General scheme of notation . . . . . . . . . . . . . . 1.3.1 Character fonts. General convention . . . . . 1.3.2 Some important characters . . . . . . . . . . 1.3.3 Indicial notation, subscripts and superscripts 1.3.4 Other important symbols and operations . . .

2

Elements of tensor analysis 2.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Inner product, norm and orthogonality . . . . . . . 2.1.2 Orthogonal bases and Cartesian coordinate frames 2.2 Second-order tensors . . . . . . . . . . . . . . . . . . . . . 2.2.1 The transpose. Symmetric and skew tensors . . . . 2.2.2 Tensor products . . . . . . . . . . . . . . . . . . . 2.2.3 Cartesian components and matrix representation . 2.2.4 Trace, inner product and Euclidean norm . . . . . 2.2.5 Inverse tensor. Determinant . . . . . . . . . . . . 2.2.6 Orthogonal tensors. Rotations . . . . . . . . . . . 2.2.7 Cross product . . . . . . . . . . . . . . . . . . . . 2.2.8 Spectral decomposition . . . . . . . . . . . . . . . 2.2.9 Polar decomposition . . . . . . . . . . . . . . . . 2.3 Higher-order tensors . . . . . . . . . . . . . . . . . . . . . 2.3.1 Fourth-order tensors . . . . . . . . . . . . . . . . 2.3.2 Generic-order tensors . . . . . . . . . . . . . . . . 2.4 Isotropic tensors . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Isotropic second-order tensors . . . . . . . . . . . 2.4.2 Isotropic fourth-order tensors . . . . . . . . . . .

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viii 2.5 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 The derivative map. Directional derivative . . . . . . . . . 2.5.2 Linearisation of a nonlinear function . . . . . . . . . . . . 2.5.3 The gradient . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Derivatives of functions of vector and tensor arguments . . 2.5.5 The chain rule . . . . . . . . . . . . . . . . . . . . . . . 2.5.6 The product rule . . . . . . . . . . . . . . . . . . . . . . 2.5.7 The divergence . . . . . . . . . . . . . . . . . . . . . . . 2.5.8 Useful relations involving the gradient and the divergence Linearisation of nonlinear problems . . . . . . . . . . . . . . . . . 2.6.1 The nonlinear problem and its linearised form . . . . . . . 2.6.2 Linearisation in innite-dimensional functional spaces . .

CONTENTS

2.6

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32 32 32 32 33 36 37 37 38 38 38 39 41 41 44 46 46 47 49 49 52 55 56 57 57 58 58 60 61 61 62 64 66 67 67 67 67 68 68 69 69 69 71 74

3

Elements of continuum mechanics and thermodynamics 3.1 Kinematics of deformation . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Material and spatial elds . . . . . . . . . . . . . . . . . . . . 3.1.2 Material and spatial gradients, divergences and time derivatives 3.1.3 The deformation gradient . . . . . . . . . . . . . . . . . . . . . 3.1.4 Volume changes. The determinant of the deformation gradient . 3.1.5 Isochoric/volumetric split of the deformation gradient . . . . . 3.1.6 Polar decomposition. Stretches and rotation . . . . . . . . . . . 3.1.7 Strain measures . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.8 The velocity gradient. Rate of deformation and spin . . . . . . . 3.1.9 Rate of volume change . . . . . . . . . . . . . . . . . . . . . . 3.2 Innitesimal deformations . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The innitesimal strain tensor . . . . . . . . . . . . . . . . . . 3.2.2 Innitesimal rigid deformations . . . . . . . . . . . . . . . . . 3.2.3 Innitesimal isochoric and volumetric deformations . . . . . . 3.3 Forces. Stress Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Cauchys axiom. The Cauchy stress vector . . . . . . . . . . . 3.3.2 The axiom of momentum balance . . . . . . . . . . . . . . . . 3.3.3 The Cauchy stress tensor . . . . . . . . . . . . . . . . . . . . . 3.3.4 The First PiolaKirchhoff stress . . . . . . . . . . . . . . . . . 3.3.5 The Second PiolaKirchhoff stress . . . . . . . . . . . . . . . . 3.3.6 The Kirchhoff stress . . . . . . . . . . . . . . . . . . . . . . . 3.4 Fundamental laws of thermodynamics . . . . . . . . . . . . . . . . . . . 3.4.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Momentum balance . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 The rst principle . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 The second principle . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 The ClausiusDuhem inequality . . . . . . . . . . . . . . . . . 3.5 Constitutive theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Constitutive axioms . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Thermodynamics with internal variables . . . . . . . . . . . . . 3.5.3 Phenomenological and micromechanical approaches . . . . . .

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CONTENTS

ix 3.5.4 The purely mechanical theory . . . . . 3.5.5 The constitutive initial value problem . Weak equilibrium. The principle of virtual work 3.6.1 The spatial version . . . . . . . . . . . 3.6.2 The material version . . . . . . . . . . 3.6.3 The innitesimal case . . . . . . . . . The quasi-static initial boundary value problem . 3.7.1 Finite deformations . . . . . . . . . . . 3.7.2 The innitesimal problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 76 77 77 78 78 79 79 81 83 84 85 86 90 93 94 95 95 96 96 98 99 101 102 102 103 103 103 106 106 107 107 108 111 115 115 115 116 116 117 117 117 119

3.6

3.7

4

The finite element method in quasi-static nonlinear solid mechanics 4.1 Displacement-based nite elements . . . . . . . . . . . . . . . . . 4.1.1 Finite element interpolation . . . . . . . . . . . . . . . . 4.1.2 The discretised virtual work . . . . . . . . . . . . . . . . 4.1.3 Some typical isoparametric elements . . . . . . . . . . . . 4.1.4 Example. Linear elasticity . . . . . . . . . . . . . . . . . 4.2 Path-dependent materials. The incremental nite element procedure 4.2.1 The incremental constitutive function . . . . . . . . . . . 4.2.2 The incremental boundary value problem . . . . . . . . . 4.2.3 The nonlinear incremental nite element equation . . . . . 4.2.4 Nonlinear solution. The NewtonRaphson scheme . . . . 4.2.5 The consistent tangent modulus . . . . . . . . . . . . . . 4.2.6 Alternative nonlinear solution schemes . . . . . . . . . . 4.2.7 Non-incremental procedures for path-dependent materials 4.3 Large strain formulation . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The incremental constitutive function . . . . . . . . . . . 4.3.2 The incremental boundary value problem . . . . . . . . . 4.3.3 The nite element equilibrium equation . . . . . . . . . . 4.3.4 Linearisation. The consistent spatial tangent modulus . . . 4.3.5 Material and geometric stiffnesses . . . . . . . . . . . . . 4.3.6 Conguration-dependent loads. The load-stiffness matrix . 4.4 Unstable equilibrium. The arc-length method . . . . . . . . . . . . 4.4.1 The arc-length method . . . . . . . . . . . . . . . . . . . 4.4.2 The combined NewtonRaphson/arc-length procedure . . 4.4.3 The predictor solution . . . . . . . . . . . . . . . . . . . Overview of the program structure 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Objectives . . . . . . . . . . . . . . . . . . . . . 5.1.2 Remarks on program structure . . . . . . . . . . 5.1.3 Portability . . . . . . . . . . . . . . . . . . . . . 5.2 The main program . . . . . . . . . . . . . . . . . . . . . 5.3 Data input and initialisation . . . . . . . . . . . . . . . . 5.3.1 The global database . . . . . . . . . . . . . . . . 5.3.2 Main problem-dening data. Subroutine INDATA

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CONTENTS

5.4

5.5 5.6

5.7

5.3.3 External loading. Subroutine INLOAD . . . . . . . . . . . . . 5.3.4 Initialisation of variable data. Subroutine INITIA . . . . . . . The load incrementation loop. Overview . . . . . . . . . . . . . . . . 5.4.1 Fixed increments option . . . . . . . . . . . . . . . . . . . . 5.4.2 Arc-length control option . . . . . . . . . . . . . . . . . . . . 5.4.3 Automatic increment cutting . . . . . . . . . . . . . . . . . . 5.4.4 The linear solver. Subroutine FRONT . . . . . . . . . . . . . . 5.4.5 Internal force calculation. Subroutine INTFOR . . . . . . . . . 5.4.6 Switching data. Subroutine SWITCH . . . . . . . . . . . . . . 5.4.7 Output of converged results. Subroutines OUTPUT and RSTART Material and element modularity . . . . . . . . . . . . . . . . . . . . . 5.5.1 Example. Modularisation of internal force computation . . . . Elements. Implementation and management . . . . . . . . . . . . . . . 5.6.1 Element properties. Element routines for data input . . . . . . 5.6.2 Element interfaces. Internal force and stiffness computation . 5.6.3 Implementing a new nite element . . . . . . . . . . . . . . . Material models: implementation and management . . . . . . . . . . . 5.7.1 Material properties. Material-specic data input . . . . . . . . 5.7.2 State variables and other Gauss point quantities. Material-specic state updating routines . . . . . . . . . . . . 5.7.3 Material-specic switching/initialising routines . . . . . . . . 5.7.4 Material-specic tangent computation routines . . . . . . . . 5.7.5 Material-specic results output routines . . . . . . . . . . . . 5.7.6 Implementing a new material model . . . . . . . . . . . . . .

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119 120 120 120 120 123 124 124 124 125 125 125 128 128 129 129 131 131 132 133 134 134 135

Part Two6

Small strains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137139 140 141 142 143 143 144 145 145 146 147 148 148 148 150 150 151 152

The mathematical theory of plasticity 6.1 Phenomenological aspects . . . . . . . . . . . . . . . . . . . 6.2 One-dimensional constitutive model . . . . . . . . . . . . . . 6.2.1 Elastoplastic decomposition of the axial strain . . . . 6.2.2 The elastic uniaxial constitutive law . . . . . . . . . 6.2.3 The yield function and the yield criterion . . . . . . 6.2.4 The plastic ow rule. Loading/unloading conditions 6.2.5 The hardening law . . . . . . . . . . . . . . . . . . 6.2.6 Summary of the model . . . . . . . . . . . . . . . . 6.2.7 Determination of the plastic multiplier . . . . . . . . 6.2.8 The elastoplastic tangent modulus . . . . . . . . . . 6.3 General elastoplastic constitutive model . . . . . . . . . . . . 6.3.1 Additive decomposition of the strain tensor . . . . . 6.3.2 The free energy potential and the elastic law . . . . . 6.3.3 The yield criterion and the yield surface . . . . . . . 6.3.4 Plastic ow rule and hardening law . . . . . . . . . 6.3.5 Flow rules derived from a ow potential . . . . . . . 6.3.6 The plastic multiplier . . . . . . . . . . . . . . . . .

CONTENTS

xi 6.3.7 Relation to the general continuum constitutive theory . . . . . . . 6.3.8 Rate form and the elastoplastic tangent operator . . . . . . . . . . 6.3.9 Non-smooth potentials and the subdifferential . . . . . . . . . . . Classical yield criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 The Tresca yield criterion . . . . . . . . . . . . . . . . . . . . . 6.4.2 The von Mises yield criterion . . . . . . . . . . . . . . . . . . . 6.4.3 The MohrCoulomb yield criterion . . . . . . . . . . . . . . . . 6.4.4 The DruckerPrager yield criterion . . . . . . . . . . . . . . . . Plastic ow rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Associative and non-associative plasticity . . . . . . . . . . . . . 6.5.2 Associative laws and the principle of maximum plastic dissipation 6.5.3 Classical ow rules . . . . . . . . . . . . . . . . . . . . . . . . . Hardening laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Perfect plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Isotropic hardening . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Thermodynamical aspects. Associative isotropic hardening . . . . 6.6.4 Kinematic hardening. The Bauschinger effect . . . . . . . . . . . 6.6.5 Mixed isotropic/kinematic hardening . . . . . . . . . . . . . . . 153 153 153 157 157 162 163 166 168 168 170 171 177 177 178 182 185 189

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7

Finite elements in small-strain plasticity problems 7.1 Preliminary implementation aspects . . . . . . . . . . . . . . . . . . 7.2 General numerical integration algorithm for elastoplastic constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The elastoplastic constitutive initial value problem . . . . . 7.2.2 Euler discretisation: the incremental constitutive problem . . 7.2.3 The elastic predictor/plastic corrector algorithm . . . . . . . 7.2.4 Solution of the return-mapping equations . . . . . . . . . . 7.2.5 Closest point projection interpretation . . . . . . . . . . . . 7.2.6 Alternative justication: operator split method . . . . . . . 7.2.7 Other elastic predictor/return-mapping schemes . . . . . . . 7.2.8 Plasticity and differential-algebraic equations . . . . . . . . 7.2.9 Alternative mathematical programming-based algorithms . . 7.2.10 Accuracy and stability considerations . . . . . . . . . . . . 7.3 Application: integration algorithm for the isotropically hardening von Mises model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 The implemented model . . . . . . . . . . . . . . . . . . . 7.3.2 The implicit elastic predictor/return-mapping scheme . . . . 7.3.3 The incremental constitutive function for the stress . . . . . 7.3.4 Linear isotropic hardening and perfect plasticity: the closed-form return mapping . . . . . . . . . . . . . . . . . 7.3.5 Subroutine SUVM . . . . . . . . . . . . . . . . . . . . . . . 7.4 The consistent tangent modulus . . . . . . . . . . . . . . . . . . . . 7.4.1 Consistent tangent operators in elastoplasticity . . . . . . . 7.4.2 The elastoplastic consistent tangent for the von Mises model with isotropic hardening . . . . . . . . . . . . . . . . . . . 7.4.3 Subroutine CTVM . . . . . . . . . . . . . . . . . . . . . . .

191 . . . 192 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 193 194 196 198 200 201 201 209 210 210 215 216 217 220 223 224 228 229

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xii 7.4.4

CONTENTS

7.5

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The general elastoplastic consistent tangent operator for implicit return mappings . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.5 Illustration: the von Mises model with isotropic hardening . . . 7.4.6 Tangent operator symmetry: incremental potentials . . . . . . . Numerical examples with the von Mises model . . . . . . . . . . . . . . 7.5.1 Internally pressurised cylinder . . . . . . . . . . . . . . . . . . 7.5.2 Internally pressurised spherical shell . . . . . . . . . . . . . . . 7.5.3 Uniformly loaded circular plate . . . . . . . . . . . . . . . . . 7.5.4 Strip-footing collapse . . . . . . . . . . . . . . . . . . . . . . . 7.5.5 Double-notched tensile specimen . . . . . . . . . . . . . . . . Further application: the von Mises model with nonlinear mixed hardening 7.6.1 The mixed hardening model: summary . . . . . . . . . . . . . 7.6.2 The implicit return-mapping scheme . . . . . . . . . . . . . . . 7.6.3 The incremental constitutive function . . . . . . . . . . . . . . 7.6.4 Linear hardening: closed-form return mapping . . . . . . . . . 7.6.5 Computational implementation aspects . . . . . . . . . . . . . 7.6.6 The elastoplastic consistent tangent . . . . . . . . . . . . . . .

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238 240 243 244 244 247 250 252 255 257 257 258 260 261 261 262 265 266 268 279 283 286 291 295 297 310 315 316 319 324 325 334 337 337 340 343 343 344 346 350 351

8

Computations with other basic plasticity models 8.1 The Tresca model . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 The implicit integration algorithm in principal stresses . . 8.1.2 Subroutine SUTR . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Finite step accuracy: iso-error maps . . . . . . . . . . . . 8.1.4 The consistent tangent operator for the Tresca model . . . 8.1.5 Subroutine CTTR . . . . . . . . . . . . . . . . . . . . . . 8.2 The MohrCoulomb model . . . . . . . . . . . . . . . . . . . . . 8.2.1 Integration algorithm for the MohrCoulomb model . . . 8.2.2 Subroutine SUMC . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Accuracy: iso-error maps . . . . . . . . . . . . . . . . . . 8.2.4 Consistent tangent operator for the MohrCoulomb model 8.2.5 Subroutine CTMC . . . . . . . . . . . . . . . . . . . . . . 8.3 The DruckerPrager model . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Integration algorithm for the DruckerPrager model . . . 8.3.2 Subroutine SUDP . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Iso-error map . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Consistent tangent operator for the DruckerPrager model 8.3.5 Subroutine CTDP . . . . . . . . . . . . . . . . . . . . . . 8.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Bending of a V-notched Tresca bar . . . . . . . . . . . . . 8.4.2 End-loaded tapered cantilever . . . . . . . . . . . . . . . 8.4.3 Strip-footing collapse . . . . . . . . . . . . . . . . . . . . 8.4.4 Circular-footing collapse . . . . . . . . . . . . . . . . . . 8.4.5 Slope stability . . . . . . . . . . . . . . . . . . . . . . .

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xiii 357 357 358 359 360 361 362 362 364 366 366 367 367 368 370 371 373 378 382 383 386 387 387 390 391 396 396 400 403 403 404 406 409 410 412 413 414 414 420 423 427 431

9

Plane stress plasticity 9.1 The basic plane stress plasticity problem . . . . . . . . . . . . . 9.1.1 Plane stress linear elasticity . . . . . . . . . . . . . . . 9.1.2 The constrained elastoplastic initial value problem . . . 9.1.3 Procedures for plane stress plasticity . . . . . . . . . . . 9.2 Plane stress constraint at the Gauss point level . . . . . . . . . . 9.2.1 Implementation aspects . . . . . . . . . . . . . . . . . . 9.2.2 Plane stress enforcement with nested iterations . . . . . 9.2.3 Plane stress von Mises with nested iterations . . . . . . 9.2.4 The consistent tangent for the nested iteration procedure 9.2.5 Consistent tangent computation for the von Mises model 9.3 Plane stress constraint at the structural level . . . . . . . . . . . . 9.3.1 The method . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 The implementation . . . . . . . . . . . . . . . . . . . 9.4 Plane stress-projected plasticity models . . . . . . . . . . . . . . 9.4.1 The plane stress-projected von Mises model . . . . . . . 9.4.2 The plane stress-projected integration algorithm . . . . . 9.4.3 Subroutine SUVMPS . . . . . . . . . . . . . . . . . . . . 9.4.4 The elastoplastic consistent tangent operator . . . . . . 9.4.5 Subroutine CTVMPS . . . . . . . . . . . . . . . . . . . . 9.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Collapse of an end-loaded cantilever . . . . . . . . . . . 9.5.2 Innite plate with a circular hole . . . . . . . . . . . . . 9.5.3 Stretching of a perforated rectangular plate . . . . . . . 9.5.4 Uniform loading of a concrete shear wall . . . . . . . . 9.6 Other stress-constrained states . . . . . . . . . . . . . . . . . . . 9.6.1 A three-dimensional von Mises Timoshenko beam . . . 9.6.2 The beam state-projected integration algorithm . . . . .

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10 Advanced plasticity models 10.1 A modied Cam-Clay model for soils . . . . . . . . . . . . . . . 10.1.1 The model . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Computational implementation . . . . . . . . . . . . . . 10.2 A capped DruckerPrager model for geomaterials . . . . . . . . . 10.2.1 Capped DruckerPrager model . . . . . . . . . . . . . . 10.2.2 The implicit integration algorithm . . . . . . . . . . . . 10.2.3 The elastoplastic consistent tangent operator . . . . . . 10.3 Anisotropic plasticity: the Hill, Hoffman and BarlatLian models 10.3.1 The Hill orthotropic model . . . . . . . . . . . . . . . . 10.3.2 Tensioncompression distinction: the Hoffman model . 10.3.3 Implementation of the Hoffman model . . . . . . . . . . 10.3.4 The BarlatLian model for sheet metals . . . . . . . . . 10.3.5 Implementation of the BarlatLian model . . . . . . . .

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xiv

CONTENTS

11 Viscoplasticity 11.1 Viscoplasticity: phenomenological aspects . . . . . . . . . . . . . . . . 11.2 One-dimensional viscoplasticity model . . . . . . . . . . . . . . . . . . 11.2.1 Elastoplastic decomposition of the axial strain . . . . . . . . . . 11.2.2 The elastic law . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 The yield function and the elastic domain . . . . . . . . . . . . 11.2.4 Viscoplastic ow rule . . . . . . . . . . . . . . . . . . . . . . . 11.2.5 Hardening law . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.6 Summary of the model . . . . . . . . . . . . . . . . . . . . . . 11.2.7 Some simple analytical solutions . . . . . . . . . . . . . . . . . 11.3 A von Mises-based multidimensional model . . . . . . . . . . . . . . . 11.3.1 A von Mises-type viscoplastic model with isotropic strain hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Alternative plastic strain rate denitions . . . . . . . . . . . . . 11.3.3 Other isotropic and kinematic hardening laws . . . . . . . . . . 11.3.4 Viscoplastic models without a yield surface . . . . . . . . . . . 11.4 General viscoplastic constitutive model . . . . . . . . . . . . . . . . . . 11.4.1 Relation to the general continuum constitutive theory . . . . . . 11.4.2 Potential structure and dissipation inequality . . . . . . . . . . 11.4.3 Rate-independent plasticity as a limit case . . . . . . . . . . . . 11.5 General numerical framework . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 A general implicit integration algorithm . . . . . . . . . . . . . 11.5.2 Alternative Euler-based algorithms . . . . . . . . . . . . . . . . 11.5.3 General consistent tangent operator . . . . . . . . . . . . . . . 11.6 Application: computational implementation of a von Mises-based model 11.6.1 Integration algorithm . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 Iso-error maps . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.3 Consistent tangent operator . . . . . . . . . . . . . . . . . . . . 11.6.4 Perzyna-type model implementation . . . . . . . . . . . . . . . 11.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.1 Double-notched tensile specimen . . . . . . . . . . . . . . . . 11.7.2 Plane stress: stretching of a perforated plate . . . . . . . . . . . 12 Damage mechanics 12.1 Physical aspects of internal damage in solids . . . . . 12.1.1 Metals . . . . . . . . . . . . . . . . . . . . . 12.1.2 Rubbery polymers . . . . . . . . . . . . . . 12.2 Continuum damage mechanics . . . . . . . . . . . . . 12.2.1 Original development: creep-damage . . . . 12.2.2 Other theories . . . . . . . . . . . . . . . . . 12.2.3 Remarks on the nature of the damage variable 12.3 Lemaitres elastoplastic damage theory . . . . . . . . 12.3.1 The model . . . . . . . . . . . . . . . . . . 12.3.2 Integration algorithm . . . . . . . . . . . . . 12.3.3 The tangent operators . . . . . . . . . . . . .

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435 436 437 437 438 438 438 439 439 439 445 445 447 448 448 450 450 451 452 454 454 457 458 460 460 463 464 466 467 467 469 471 472 472 473 473 474 475 476 478 478 482 485

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xv A simplied version of Lemaitres model . . . . . . . . . . . . 12.4.1 The single-equation integration algorithm . . . . . . . 12.4.2 The tangent operator . . . . . . . . . . . . . . . . . . 12.4.3 Example. Fracturing of a cylindrical notched specimen Gursons void growth model . . . . . . . . . . . . . . . . . . . 12.5.1 The model . . . . . . . . . . . . . . . . . . . . . . . 12.5.2 Integration algorithm . . . . . . . . . . . . . . . . . . 12.5.3 The tangent operator . . . . . . . . . . . . . . . . . . Further issues in damage modelling . . . . . . . . . . . . . . . 12.6.1 Crack closure effects in damaged elastic materials . . 12.6.2 Crack closure effects in damage evolution . . . . . . . 12.6.3 Anisotropic ductile damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 486 490 493 496 497 501 502 504 504 510 512

12.4

12.5

12.6

Part Three

Large strains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

517519 520 520 521 524 525 525 525 527 528 530 530 531 532 533 534 535 535 537 537 538 538 542 546 547 547 550 551 552

13 Finite strain hyperelasticity 13.1 Hyperelasticity: basic concepts . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Material objectivity: reduced form of the free-energy function 13.1.2 Isotropic hyperelasticity . . . . . . . . . . . . . . . . . . . . 13.1.3 Incompressible hyperelasticity . . . . . . . . . . . . . . . . . 13.1.4 Compressible regularisation . . . . . . . . . . . . . . . . . . 13.2 Some particular models . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 The MooneyRivlin and the neo-Hookean models . . . . . . 13.2.2 The Ogden material model . . . . . . . . . . . . . . . . . . . 13.2.3 The Hencky material . . . . . . . . . . . . . . . . . . . . . . 13.2.4 The BlatzKo material . . . . . . . . . . . . . . . . . . . . . 13.3 Isotropic nite hyperelasticity in plane stress . . . . . . . . . . . . . . 13.3.1 The plane stress incompressible Ogden model . . . . . . . . . 13.3.2 The plane stress Hencky model . . . . . . . . . . . . . . . . 13.3.3 Plane stress with nested iterations . . . . . . . . . . . . . . . 13.4 Tangent moduli: the elasticity tensors . . . . . . . . . . . . . . . . . . 13.4.1 Regularised neo-Hookean model . . . . . . . . . . . . . . . . 13.4.2 Principal stretches representation: Ogden model . . . . . . . . 13.4.3 Hencky model . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.4 BlatzKo material . . . . . . . . . . . . . . . . . . . . . . . 13.5 Application: Ogden material implementation . . . . . . . . . . . . . . 13.5.1 Subroutine SUOGD . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Subroutine CSTOGD . . . . . . . . . . . . . . . . . . . . . . . 13.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.1 Axisymmetric extension of an annular plate . . . . . . . . . . 13.6.2 Stretching of a square perforated rubber sheet . . . . . . . . . 13.6.3 Ination of a spherical rubber balloon . . . . . . . . . . . . . 13.6.4 Rugby ball . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6.5 Ination of initially at membranes . . . . . . . . . . . . . .

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CONTENTS

13.7

13.6.6 Rubber cylinder pressed between two plates . . . . . . . . . 13.6.7 Elastomeric bead compression . . . . . . . . . . . . . . . . Hyperelasticity with damage: the Mullins effect . . . . . . . . . . . . 13.7.1 The GurtinFrancis uniaxial model . . . . . . . . . . . . . 13.7.2 Three-dimensional modelling. A brief review . . . . . . . . 13.7.3 A simple rate-independent three-dimensional model . . . . 13.7.4 Example: the model problem . . . . . . . . . . . . . . . . . 13.7.5 Computational implementation . . . . . . . . . . . . . . . . 13.7.6 Example: ination/deation of a damageable rubber balloon

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555 556 557 560 562 562 565 565 569 573 574 575 575 576 576 576 577 577 578 578 582 583 586 588 590 590 591 595 597 598 599 599 600 601 601 601 604 605 606 606 606 607 611 613 614

14 Finite strain elastoplasticity 14.1 Finite strain elastoplasticity: a brief review . . . . . . . . . . . . 14.2 One-dimensional nite plasticity model . . . . . . . . . . . . . . 14.2.1 The multiplicative split of the axial stretch . . . . . . . . 14.2.2 Logarithmic stretches and the Hencky hyperelastic law . 14.2.3 The yield function . . . . . . . . . . . . . . . . . . . . 14.2.4 The plastic ow rule . . . . . . . . . . . . . . . . . . . 14.2.5 The hardening law . . . . . . . . . . . . . . . . . . . . 14.2.6 The plastic multiplier . . . . . . . . . . . . . . . . . . . 14.3 General hyperelastic-based multiplicative plasticity model . . . . 14.3.1 Multiplicative elastoplasticity kinematics . . . . . . . . 14.3.2 The logarithmic elastic strain measure . . . . . . . . . . 14.3.3 A general isotropic large-strain plasticity model . . . . . 14.3.4 The dissipation inequality . . . . . . . . . . . . . . . . 14.3.5 Finite strain extension to innitesimal theories . . . . . 14.4 The general elastic predictor/return-mapping algorithm . . . . . . 14.4.1 The basic constitutive initial value problem . . . . . . . 14.4.2 Exponential map backward discretisation . . . . . . . . 14.4.3 Computational implementation of the general algorithm 14.5 The consistent spatial tangent modulus . . . . . . . . . . . . . . 14.5.1 Derivation of the spatial tangent modulus . . . . . . . . 14.5.2 Computational implementation . . . . . . . . . . . . . . 14.6 Principal stress space-based implementation . . . . . . . . . . . . 14.6.1 Stress-updating algorithm . . . . . . . . . . . . . . . . 14.6.2 Tangent modulus computation . . . . . . . . . . . . . . 14.7 Finite plasticity in plane stress . . . . . . . . . . . . . . . . . . . 14.7.1 The plane stress-projected nite von Mises model . . . . 14.7.2 Nested iteration for plane stress enforcement . . . . . . 14.8 Finite viscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . 14.8.1 Numerical treatment . . . . . . . . . . . . . . . . . . . 14.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9.1 Finite strain bending of a V-notched Tresca bar . . . . . 14.9.2 Necking of a cylindrical bar . . . . . . . . . . . . . . . 14.9.3 Plane strain localisation . . . . . . . . . . . . . . . . . 14.9.4 Stretching of a perforated plate . . . . . . . . . . . . . . 14.9.5 Thin sheet metal-forming application . . . . . . . . . .

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CONTENTS

xvii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 619 621 622 624 625 628 632 633 633 637 642 644 644 647 648 649 651 652 655 656 656 663 663 665 669 669 671 676 678 678 683 683 685 687 689 691 692 692 693 694 694 695 696

14.10 Rate forms: hypoelastic-based plasticity models . . . . . . . . 14.10.1 Objective stress rates . . . . . . . . . . . . . . . . . 14.10.2 Hypoelastic-based plasticity models . . . . . . . . . 14.10.3 The Jaumann rate-based model . . . . . . . . . . . . 14.10.4 Hyperelastic-based models and equivalent rate forms 14.10.5 Integration algorithms and incremental objectivity . 14.10.6 Objective algorithm for Jaumann rate-based models . 14.10.7 Integration of GreenNaghdi rate-based models . . . 14.11 Finite plasticity with kinematic hardening . . . . . . . . . . . 14.11.1 A model of nite strain kinematic hardening . . . . 14.11.2 Integration algorithm . . . . . . . . . . . . . . . . . 14.11.3 Spatial tangent operator . . . . . . . . . . . . . . . 14.11.4 Remarks on predictive capability . . . . . . . . . . . 14.11.5 Alternative descriptions . . . . . . . . . . . . . . . 15 Finite elements for large-strain incompressibility 15.1 The F-bar methodology . . . . . . . . . . . . . . . . . . . 15.1.1 Stress computation: the F-bar deformation gradient 15.1.2 The internal force vector . . . . . . . . . . . . . . 15.1.3 Consistent linearisation: the tangent stiffness . . . 15.1.4 Plane strain implementation . . . . . . . . . . . . 15.1.5 Computational implementation aspects . . . . . . 15.1.6 Numerical tests . . . . . . . . . . . . . . . . . . . 15.1.7 Other centroid sampling-based F-bar elements . . 15.1.8 A more general F-bar methodology . . . . . . . . 15.1.9 The F-bar-Patch Method for simplex elements . . 15.2 Enhanced assumed strain methods . . . . . . . . . . . . . . 15.2.1 Enhanced three-eld variational principle . . . . . 15.2.2 EAS nite elements . . . . . . . . . . . . . . . . 15.2.3 Finite element equations: static condensation . . . 15.2.4 Implementation aspects . . . . . . . . . . . . . . . 15.2.5 The stability of EAS elements . . . . . . . . . . . 15.3 Mixed u/p formulations . . . . . . . . . . . . . . . . . . . 15.3.1 The two-eld variational principle . . . . . . . . . 15.3.2 Finite element equations . . . . . . . . . . . . . . 15.3.3 Solution: static condensation . . . . . . . . . . . . 15.3.4 Implementation aspects . . . . . . . . . . . . . . . 16 Anisotropic finite plasticity: Single crystals 16.1 Physical aspects . . . . . . . . . . . . . . . . . . 16.1.1 Plastic deformation by slip: slip-systems . 16.2 Plastic slip and the Schmid resolved shear stress . 16.3 Single crystal simulation: a brief review . . . . . . 16.4 A general continuum model of single crystals . . . 16.4.1 The plastic ow equation . . . . . . . . . 16.4.2 The resolved Schmid shear stress . . . .

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xviii 16.4.3 Multisurface formulation of the ow rule . . . . . . . . 16.4.4 Isotropic Taylor hardening . . . . . . . . . . . . . . . . 16.4.5 The hyperelastic law . . . . . . . . . . . . . . . . . . . A general integration algorithm . . . . . . . . . . . . . . . . . . 16.5.1 The search for an active set of slip systems . . . . . . . An algorithm for a planar double-slip model . . . . . . . . . . . 16.6.1 A planar double-slip model . . . . . . . . . . . . . . . . 16.6.2 The integration algorithm . . . . . . . . . . . . . . . . 16.6.3 Example: the model problem . . . . . . . . . . . . . . . The consistent spatial tangent modulus . . . . . . . . . . . . . . 16.7.1 The elastic modulus: compressible neo-Hookean model . 16.7.2 The elastoplastic consistent tangent modulus . . . . . . Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . 16.8.1 Symmetric strain localisation on a rectangular strip . . . 16.8.2 Unsymmetric localisation . . . . . . . . . . . . . . . . Viscoplastic single crystals . . . . . . . . . . . . . . . . . . . . . 16.9.1 Rate-dependent formulation . . . . . . . . . . . . . . . 16.9.2 The exponential map-based integration algorithm . . . . 16.9.3 The spatial tangent modulus: neo-Hookean-based model 16.9.4 Rate-dependent crystal: model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CONTENTS

16.5 16.6

16.7

16.8

16.9

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696 698 698 699 703 705 705 707 710 713 713 714 716 717 720 721 723 724 725 726

AppendicesA Isotropic functions of a symmetric tensor A.1 Isotropic scalar-valued functions . . . . . . . . . . . . A.1.1 Representation . . . . . . . . . . . . . . . . A.1.2 The derivative of an isotropic scalar function A.2 Isotropic tensor-valued functions . . . . . . . . . . . . A.2.1 Representation . . . . . . . . . . . . . . . . A.2.2 The derivative of an isotropic tensor function A.3 The two-dimensional case . . . . . . . . . . . . . . . A.3.1 Tensor function derivative . . . . . . . . . . A.3.2 Plane strain and axisymmetric problems . . . A.4 The three-dimensional case . . . . . . . . . . . . . . A.4.1 Function computation . . . . . . . . . . . . A.4.2 Computation of the function derivative . . . A.5 A particular class of isotropic tensor functions . . . . A.5.1 Two dimensions . . . . . . . . . . . . . . . A.5.2 Three dimensions . . . . . . . . . . . . . . . A.6 Alternative procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

729731 731 732 732 733 733 734 735 736 738 739 739 740 740 742 743 744

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xix 747 747 748 749 750 751 751 752 753 753 755 755 756

B The tensor exponential B.1 The tensor exponential function . . . . . . . . . . . . . . . B.1.1 Some properties of the tensor exponential function B.1.2 Computation of the tensor exponential function . . B.2 The tensor exponential derivative . . . . . . . . . . . . . . B.2.1 Computer implementation . . . . . . . . . . . . . B.3 Exponential map integrators . . . . . . . . . . . . . . . . . B.3.1 The generalised exponential map midpoint rule . . C Linearisation of the virtual work C.1 Innitesimal deformations . . . C.2 Finite strains and deformations . C.2.1 Material description . C.2.2 Spatial description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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D Array notation for computations with tensors 759 D.1 Second-order tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759 D.2 Fourth-order tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 D.2.1 Operations with non-symmetric tensors . . . . . . . . . . . . . . 763 References Index 765 783

PREFACE

T

HE purpose of this text is to describe in detail numerical techniques used in small and large strain nite element analysis of elastic and inelastic solids. Attention is focused on the derivation and description of various constitutive models based on phenomenological hyperelasticity, elastoplasticity and elasto-viscoplasticity together with the relevant numerical procedures and the practical issues arising in their computer implementation within a quasi-static nite element scheme. Many of the techniques discussed in the text are incorporated in the FORTRAN program, named HYPLAS, which accompanies this book and can be found at www.wiley.com/go/desouzaneto. This computer program has been specially written to illustrate the practical implementation of such techniques. We make no pretence that the text provides a complete account of the topics considered but rather, we see it as an attempt to present a reasonable balance of theory and numerical procedures used in the nite element simulation of the nonlinear mechanical behaviour of solids. When we embarked on the project of writing this text, our initial idea was to produce a rather concise book based primarily on our own research experience whose bulk would consist of the description of numerical algorithms required for the nite element implementation of small and large strain plasticity models. As the manuscript began to take shape, it soon became clear that a book designed as such would be most appropriate to those already involved in research on computational plasticity or closely related areas, being of little use to those willing to learn computational methods in plasticity from a fundamental level. A substantial amount of background reading from other sources would be required for readers unfamiliar with topics such as basic elastoplasticity theory, tensor analysis, nonlinear continuum mechanics particularly nonlinear kinematics nite hyperelasticity and general dissipative constitutive theory of solids. Our initial plan was then gradually abandoned as we chose to make the text more self-contained by incorporating a considerable amount of basic theory. Also, while writing the manuscript, we decided to add more advanced (and very exciting) topics such as damage mechanics, anisotropic plasticity and the treatment of nite strain single crystal plasticity. Following this route, our task took at least three times as long to complete and the book grew to about twice the size as originally planned. There remains plenty of interesting material we would like to have included but cannot due to constraints of time and space. We are certainly far more satised with the text now than with its early versions, but we do not believe our nal product to be optimal in any sense. We merely offer it to ll a gap in the existing literature, hoping that the reader will benet from it in some way. The text is arranged in three main parts. Part One presents some basic material of relevance to the subject matter of the book. It includes an overview of elementary tensor analysis, continuum mechanics and thermodynamics, the nite element method in quasi-static nonlinear solid mechanics and a brief description of the computer program HYPLAS. Part Two

xxii

PREFACE

deals with small strain problems. It introduces the mathematical theory of innitesimal plasticity as well as the relevant numerical procedures for the implementation of plasticity models within a nite element environment. Both rate-independent (elastoplastic) and ratedependent (elasto-viscoplastic) theories are addressed and some advanced models, including anisotropic plasticity and ductile damage are also covered. Finally, in Part Three we focus on large strain problems. The theory of nite hyperelasticity is reviewed rst together with details of its nite element implementation. This is followed by an introduction to large strain plasticity. Hyperelastic-based theories with multiplicative elastoplastic kinematics as well as hypoelastic-based models are discussed, together with relevant numerical procedures for their treatment. The discussion on nite plasticity and its nite element implementation culminates with a description of techniques for single crystal plasticity. Finite element techniques for large-strain near-incompressibility are also addressed. We are indebted to many people for their direct or indirect contribution to this text. This preface would not be complete without the due acknowledgement of this fact and a record of our sincere gratitude to the following: to J.M.P. Macedo for the numerous valuable suggestions during the design of the program HYPLAS at the very early stages of this project; to R. Billardon for the many enlightening discussions on damage modelling; to R.A. Feijo and E. Taroco for the fruitful discussions held on many occasions over a long period of time; to M. Dutko for producing some of the numerical results reported; to Y.T. Feng for helpful discussions on the arc-length method; to F.M. Andade Pires for his key contribution to the development of F-bar-Patch elements, for producing the related gures presented and for thoroughly reviewing early versions of the manuscript; to P.H. Saksono for his involvement in the production of isoerror maps; to A. Orlando for literally scanning through key parts of the text to nd inconsistencies of any kind; to L. Driemeier, W. Dettmer, M. Vaz Jr, M.C. Lobo, M. Partovi, D.C.D. Speirs, D.D. Somer, E. Saavedra, A.J.C. Molina, S. Giusti and P.J. Blanco for carefully reviewing various parts of the manuscript, spotting hard-to-nd mistakes and making several important suggestions for improvement. Last, but not least, to our late colleague and friend Mike Criseld, for the numerous illuminating and passionate discussions (often held on the beach or late in the bar) on many topics addressed in the book. EA de Souza Neto D Peri c DRJ Owen Swansea

Part OneBasic concepts

1 INTRODUCTION

O

VER the last four decades, the use of computational techniques based on the Finite Element Method has become a rmly established practice in the numerical solution of nonlinear solid mechanics problems both in academia and industry. In their early days, these techniques were largely limited to innitesimal deformation and strain problems with the main complexity arising from the nonlinear constitutive characterization of the underlying material by means of basic elastoplastic or elasto-viscoplastic theories. Applications were mostly conned to the modelling of the behaviour of solids in conventional areas of engineering and analyses were carried out on crude, user-unfriendly software that typically required highly specialized users. Since those days, this area of solid mechanics generally known as computational plasticity has experienced dramatic developments. Fuelled by the steady increase in computing power at decreasing costs together with the continuous industrial demand for accurate models of solids, the evolution of computational plasticity techniques have made possible the development of rened software packages with a considerable degree of automation that are today routinely employed by an everincreasing number of engineers and scientists. The variety of practical problems of interest to which such techniques are currently applied with acceptable levels of predictive capability is very wide. They range from traditional engineering applications, such as stress analysis in structures, soil and rock mechanics, to the simulation of manufacturing processes such as metal forming. Also included are much less conventional applications, such as the simulation of food processing, mining operations and biological tissue behaviour. Many such problems are characterised by extremely large straining and material behaviour often described by means of rather complex constitutive equations.

1.1. Aims and scopeThe main objective of this text is to describe in detail numerical techniques used in the small and large strain analysis of elastic and inelastic solids by the Finite Element Method. Particular emphasis is placed on the derivation and description of various constitutive models based on phenomenological hyperelasticity, elastoplasticity and elasto-viscoplasticity as well as on the relevant numerical procedures and the practical issues arising in their computer implementation. The range covered goes from basic innitesimal isotropic to more sophisticated nite strain theories, including anisotropy. Many of the techniques discussed in the text are implemented in the FORTRAN computer program, named HYPLAS, which accompanies this book. Parts of its source code are included in the text and should help readers correlate the relevant numerical methods with their computer implementation in practice. Another important aspect to emphasise is that the performance of many ofComputational Methods for Plasticity: Theory and Applications EA de Souza Neto, D Peri and DRJ Owen c c 2008 John Wiley & Sons, Ltd

4

COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS

the models/techniques described in the text is documented in numerical examples. These should be of particular relevance to those involved in software research and development in computational plasticity. In order to make the book more self-contained, we have chosen to incorporate a considerable amount of basic theory within the text. This includes some material on elementary tensor analysis, an introduction to the nonlinear mechanics and thermodynamics of continuous media and an overview of small and large strain elastoplasticity and viscoplasticity theory, nite hyperelasticity and nite element techniques in nonlinear solid mechanics. Having a sound knowledge of such topics is essential, we believe, to the clear understanding of the very problems the numerical techniques discussed in this book are meant to simulate. We reiterate, however, that our main focus here is computational. Thus, the volume of theory and the depth at which it is presented is kept to the minimum necessary for the above task. For example, in the presentation of tensor analysis and continuum mechanics and thermodynamics, we omit most proofs for standard relations. In plasticity, viscoplasticity and hyperelasticity, we limit ourselves mainly to presenting the constitutive models together with their most relevant properties and the essential relations needed in their formulation. Issues such as material stability and the existence and uniqueness of solutions to initial boundary value problems are generally not addressed. 1.1.1. READERSHIP This book is intended for graduate students, research engineers and scientists working in the eld of computational continuum mechanics. The text requires a basic knowledge of solid mechanics especially the theory of linear elasticity as well as the Finite Element Method and numerical procedures for the approximate solution of ordinary differential equations. An elementary understanding of vector and tensor calculus is also very helpful. Readers wishing to follow the computer implementation of the procedures described in the text should, in addition, be familiar at a fairly basic level with the FORTRAN computer programming language. It is worth remarking here that the choice of the FORTRAN language is motivated mainly by the following: (a) its widespread acceptance in engineering computing in general and, in particular, within the nite element community; (b) the suitability of procedural languages for codes with relatively low level of complexity, such as HYPLAS. In the present case, the use of more advanced programming concepts (e.g. object-oriented programming) could add a further difculty in the learning of the essential concepts the HYPLAS code is meant to convey; (c) its relative clarity in the coding of short algorithmic procedures such as those arising typically in the implementation of elastic and inelastic material models the main subject of this book.

1.2. LayoutIn line with the above aims, the book has been divided into three parts as follows.

INTRODUCTION

5

Part One: Basic concepts. In this part we introduce concepts of fundamental relevance to the applications presented in Parts Two and Three. The following material is covered: elementary tensor analysis; introductory continuum mechanics and thermodynamics; nite elements in quasi-static nonlinear solid mechanics; a concise description of the computer program HYPLAS. Part Two: Small strains. Here, the theory of innitesimal plasticity is introduced together with the relevant numerical procedures used for its implementation into a nite element environment. A relatively wide range of models is presented, including both rate-independent (elastoplastic) and rate-dependent (elasto-viscoplastic) theories. The following main topics are considered: the theory of innitesimal plasticity; nite elements in innitesimal plasticity; advanced plasticity models, including anisotropy; viscoplasticity; elastoplasticity with damage. Part Three: Large strains. This part focuses on nite strain hyperelasticity and elastoplasticity problems. The models discussed here, as well as their computational implementation, are obviously more complex than those of Part Two. Their complexity stems partly from the nite strain kinematics. Thus, to follow Part Three, a sound knowledge of the kinematics of nite deformations discussed in Chapter 3 (in Part One) is essential. The following topics are addressed: large strain isotropic hyperelasticity; large strain plasticity; nite element techniques for large strain incompressibility; single crystal (anisotropic) nite plasticity. The material has been organised into sixteen chapters and four appendices. These will now be briey described. The remainder of Chapter 1 discusses the general scheme of notation adopted in the book. Chapter 2 contains an introduction to elementary tensor analysis. In particular, the material is presented mainly in intrinsic (or compact) tensor notation which is heavily relied upon thoughout the book. Chapter 3 provides an introdution to the mechanics and thermodynamics of continuous media. The material presented here covers the kinematics of deformation, balance laws and constitutive theory. These topics are essential for an in-depth understanding of the theories discussed in later chapters. Chapter 4 shows the application of the Finite Element Method to the solution of problems in quasi-static nonlinear solid mechanics. A generic dissipative constitutive model, initially presented in Chapter 3, is used as the underlying material model.

6


Chapter 5 describes the general structure of the program HYPLAS, where many of the techniques discussed in the book are implemented. We remark that the program description is rather concise. Further familiarisation with the program will require the reader to follow the comments in the FORTRAN source code together with the cross-referencing of the main procedures with their description in the book. This is probably more relevant to those wishing to use the HYPLAS program for research and development purposes. Chapter 6 is devoted to the mathematical theory of innitesimal plasticity. The main concepts associated with phenomenological time-independent plasticity are introduced here. The basic yield criteria of Tresca, von Mises, MohrCoulomb and DruckerPrager are reviewed, together with the most popular plastic ow rules and hardening laws. In Chapter 7, we introduce the essential numerical methods required in the nite element solution of initial boundary value problems with elastoplastic underlying material models. Applications of the von Mises model with both isotropic and mixed isotropic/kinematic hardening are described in detail. The most relevant subroutines of the program HYPLAS are also listed and explained in detail. Chapter 8 focuses on the detailed description of the implementation of the basic plasticity models based on the Tresca, MohrCoulomb and DruckerPrager yield criteria. Again, the relevant subroutines of HYPLAS are listed and explained in some detail. In Chapter 9 we describe the numerical treatment of plasticity models under plane stress conditions. Different options are considered and their relative merits and limitations are discussed. Parts of source code are also included to illustrate some of the most important programming aspects. The application of the concepts introduced here to other stressconstrained states is briey outlined at the end of the chapter. In Chapter 10 advanced elastoplasticity models are considered. Here we describe the computational implementation of a modied Cam-Clay model for soils, a capped Drucker Prager model for geomaterials and the Hill, Hoffman and BarlatLian anisotropic models for metals. The numerical techniques required for the implementation of such models are mere specialisations of the procedures already discussed in Chapters 7 and 8. However, due to the inherent complexity of the models treated in this chapter, their actual implementation is generally more intricate than those of the basic models. Chapter 11 begins with an introduction to elasto-viscoplasticity theory within the constitutive framework for dissipative materials described in Chapter 3. The (rate-independent) plasticity theory is then obtained as a limiting case of viscoplasticity. The numerical methods for a generic viscoplastic model are described, following closely the procedures applied earlier in elastoplasticity. Application of the methodology to von Mises criterion-based viscoplastic models is described in detail. In Chapter 12 we discuss continuum damage mechanics the branch of Continuum Solid Mechanics devoted to the modelling of the progressive material deterioration that precedes the onset of macroscopic fracturing. Some elastoplastic damage models are reviewed and their implementation, with the relevant computational issues, is addressed in detail. Chapter 13 introduces nite strain hyperelasticity. The basic theory is reviewed and some of the most popular isotropic models are presented. The nite element implementation of the Odgen model is discussed in detail with relevant excerpts of HYPLAS source code included. In addition, the modelling of the so-called Mullins dissipative effect by means of a hyperelasticdamage theory is addressed at the end of the chapter. This concept is closely related to those already discussed in Chapter 12 for ductile elastoplastic damage.

INTRODUCTION

7

In Chapter 14 we introduce nite strain elastoplasticity together with the numerical procedures relevant to the nite element implementation of nite plasticity models. The main discussion is focused on hyperelastic-based nite plasticity theories with multiplicative kinematics. The nite plasticity models actually implemented in the program HYPLAS belong to this class of theories. However, for completeness, a discussion on the so-called hypoelasticbased theories is also included. The material presented is conned mostly to isotropic elastoplasticity, with anisotropy in the form of kinematic hardening added only at the end of the chapter. Chapter 15 is concerned with the treatment of large strain incompressibility within the Finite Element Method. This issue becomes crucial in large-scale nite strain simulations where the use of low-order elements (which, without any added specic techniques, are generally inappropriate near the incompressible limit) is highly desirable. Three different approaches to tackle the problem are considered: the so-called F -bar method (including its more recent F -bar-Patch variant for simplex elements); the Enhanced Assumed Strain (EAS) technique, and the mixed u/p formulation. Finally, in Chapter 16 we describe a general model of large-strain single-crystal plasticity together with the relevant numerical procedures for its use within a nite element environment. The implementation of a specialisation of the general model based on a planar double-slip system is described in detail. This implementation is incorporated in the program HYPLAS. In addition to the above, four appendices are included. Appendix A is concerned with isotropic scalar- and tensor-valued functions of a symmetric tensor that are widely exploited throughout the text. It presents some important basic properties as well as formulae that can be used in practice for the computation of function values and function derivatives. Appendix B addresses the tensor exponential function. The tensor exponential is of relevance for the treatment of nite plasticity presented in Chapters 14 and 16. In Appendix C we derive the linearisation of the virtual work equation both under small and large deformations. The expressions derived here provide the basic formulae for the tangent operators required in the assembly of the tangent stiffness matrix in the nite element context. Finally, in Appendix D we describe the handling including array storage and product operations of second and fourth-order tensors in nite element computer programs. 1.2.1. THE USE OF BOXES Extensive use of boxes has been made to summarise constitutive models and numerical algorithms in general (refer, for example, to pages 146 and 199). Boxes should be of particular use to readers interested mostly in computer implementation aspects and who wish to skip the details of derivation of the models and numerical procedures. Numerical algorithms listed in boxes are presented in the so-called pseudo-code format a format that resembles the actual computer code of the procedure. For boxes describing key procedures implemented in the program HYPLAS, the name of the corresponding FORTRAN subprogram is often indicated at the top of the box (see page 221, for instance).

1.3. General scheme of notationThroughout the text, an attempt has been made to maintain the notation as uniform as possible by assigning specic letter styles to the representation of each type of mathematical entity.

8


At the same time, we have tried to keep the notation in line with what is generally adopted in the present subject areas. Unfortunately, these two goals often conict so that, in many cases, we choose to adopt the notation that is more widely accepted instead of adhering to the use of specic fonts for specic mathematical entities. Whenever such exceptions occur, their meaning should either be clear from the context or will be explicitly mentioned the rst time they appear. 1.3.1. CHARACTER FONTS. GENERAL CONVENTION The mathematical meaning associated with specic font styles is given below. Some important exceptions are also highlighted. We emphasise, however, that other exceptions, not mentioned here, may occur in the text. Italic light-face letters A, a, . . .: scalars and scalar-valued functions. Italic bold-face majuscules B, C, . . .: second-order tensors or tensor-valued functions. Light-face with indices Bij , B , . . .: components of the corresponding tensors. Important exceptions: A (set of thermodynamical forces), H (generalised elastoplastic hardening modulus), J (generalised viscoplastic hardening constitutive function). Italic bold-face minuscules p, v, . . .: points, vectors and vector-valued functions. Light-face with indices pi , p , . . .: coordinates (components) of the corresponding points (vectors). Important exception: s (stress tensor deviator). Sans-serif (upright) bold-face letters A, a, . . .: fourth-order tensors. Light-face with indices Aijkl , A , . . .: the corresponding Cartesian components. Greek light-face letters , , . . ., , , . . .: scalars and scalar-valued functions. Important exception: (region of Euclidean space occupied by a generic body). Greek bold-face minuscules , , , . . .: second-order tensors. Light-face with indices ij , , . . .: the corresponding components. Important exceptions: (generic set of internal state variables), (deformation map) and (when meaning virtual displacement elds). Upright bold-face majuscules, minuscules and greek letters A, a, , . . .: nite element arrays (vectors and matrices) representing second or fourth-order tensors and general nite element operators. Script majuscules A, B, . . .: spaces, sets, groups, bodies. German majuscules F, G, . . .: constitutive response functionals. Calligraphic majuscules X, Y, . . .: generic mathematical entity (scalar, vector, tensor, eld, etc.) Typewriter style letters HYPLAS, SUVM, . . .: used exclusively to denote FORTRAN procedures and variable names, instructions, etc.

INTRODUCTION

9

1.3.2. SOME IMPORTANT CHARACTERS The specic meaning of some important characters is listed below. We remark that some of these symbols may occasionally be used with a different connotation (which should be clear from the context). A Generic set of thermodynamical forces Finite element assembly operator (note the large font) First elasticity tensor Spatial elasticity tensor Left CauchyGreen strain tensore

AA a B B B B b b C c D D D D D D De ep e p

Elastic left CauchyGreen strain tensor Discrete (nite element) symmetric gradient operator (strain-displacement matrix) Generic body Body force Reference body force Right CauchyGreen strain tensor Cohesion Damage internal variable Stretching tensor Elastic stretching Plastic stretching Innitesimal consistent tangent operator Innitesimal elasticity tensor Innitesimal elastoplastic consistent tangent operator Consistent tangent matrix (array representation of D) Elasticity matrix (array representation of De ) Elastoplastic consistent tangent matrix (array representation of Dep ) Youngs modulus Eigenprojection of a symmetric tensor associated with the ith eigenvalue Three-dimensional Euclidean space; elastic domain Set of plastically admissible stresses Generic base vector; unit eigenvector of a symmetric tensor associated with the ith eigenvalue

D De Dep E Ei E E ei

10 F F F f fe p


Deformation gradient Elastic deformation gradient Plastic deformation gradient Global (nite element) external force vector External force vector of element e Global (nite element) internal force vector Internal force vector of element e Virtual work functional; shear modulus Discrete (nite element) full gradient operator Hardening modulus Generalised hardening modulus Principal invariants of a tensor Fourth-order identity tensor: Iijkl = ik jl Fourth-order symmetric identity tensor: Iijkl = 1 (ik jl + il jk ) 2 Deviatoric projection tensor: Id IS 1 I I 3 Second-order identity tensor Array representation of IS Array representation of I Jacobian of the deformation map: J det F Stress deviator invariants Generalised viscoplastic hardening constitutive function Bulk modulus Global tangent stiffness matrix Tangent stiffness matrix of element e Set of kinematically admissible displacements Velocity gradiente p

ext

ext f(e) int int f(e)

G G H H I1 , I2 , I3 I IS Id I IS i J J2 , J3 J K KT(e) KT

K L L L

Elastic velocity gradient Plastic velocity gradient Unit vector normal to the slip plane of a single crystal Plastic ow vector Unit plastic ow vector: N N/ N The orthogonal group The rotation (proper orthogonal) group

m N N O O

+

INTRODUCTION

11

0 o P p p Q q R R R r s s s t t U U U u u V V V V v W W W x p e p e p e e

Zero tensor; zero array; zero generic entity Zero vector First PiolaKirchhoff stress tensor Generic material point Cauchy or Kirchhoff hydrostatic pressure Generic orthogonal or rotation (proper orthogonal) tensor von Mises (Cauchy or Kirchhoff) effective stress Rotation tensor obtained from the polar decomposition of F Elastic rotation tensor Real set Global nite element residual (out-of-balance) force vector Entropy Cauchy or Kirchhoff stress tensor deviator Unit vector in the slip direction of slip system of a single crystal Surface traction Reference surface traction Right stretch tensor Elastic right stretch tensor Plastic right stretch tensor Space of vectors in E Generic displacement vector eld Global nite element nodal displacement vector Left stretch tensor Elastic left stretch tensor Plastic left stretch tensor Space of virtual displacements Generic velocity eld Spin tensor Elastic spin tensor Plastic spin tensor Generic point in space Ogden hyperelastic constants (p = 1, . . . , N ) for a model with N terms in the Ogden strain-energy function series Generic set of internal state variables

Up

12 ij e p , , e p


Back-stress tensor Plastic multiplier Krnecker delta Strain tensor; also Eulerian logarithmic strain when under large strains Elastic strain tensor; also elastic Eulerian logarithmic strain when under large strains Plastic strain tensor Array representation of , e and p , respectively Axial total, elastic and plastic strain in one-dimensional models Effective (or accumulated) plastic strain Virtual displacement eld; relative stress tensor in kinematic hardening plasticity models Isotropic hardening thermodynamical force One of the Lam constants of linear elasticity; axial stretch; load factor in proportional loading Elastic and plastic axial stretch p i Total, elastic and plastic principal stretches One of the Lam constants of linear elasticity Ogden hyperelastic constants (p = 1, . . . , N ) for a model with N terms in the Ogden strain-energy function series Poisson ratio Dissipation potential Isoparametric coordinates of a nite element Mass density Reference mass density Cauchy stress tensor Axial stress in one-dimensional models Principal Cauchy stress Yield stress (uniaxial yield stress for the conventional von Mises and Tresca models) Initial yield stress Array representation of Kirchhoff stress tensor Principal Kirchhoff stress

, e , p p e , p i , e , i p i y y 0 i

INTRODUCTION

13

(e)

Resolved Schmid shear stress on slip system Array representation of Yield function; damage function Deformation map; motion Plastic ow potential Helmholtz free-energy per unit mass; strain-energy per unit mass Domain of a body in the reference conguration Domain of nite element e in the reference conguration

1.3.3. INDICIAL NOTATION, SUBSCRIPTS AND SUPERSCRIPTS When indicial notation is used, the following convention is adopted for subscripts: Italic subscripts i, j, k, l, . . ., as in the Cartesian components ui , Bij , aijkl , or for the basis vectors ei , normally range over 1, 2 and 3. In a more general context (in an n-dimensional space), their range may be 1, 2, . . . , n. Greek subscripts , , , , . . .: range over 1 and 2. When an index appears twice in the same product, summation over the repeated index (Einstein notation) is implied unless otherwise stated. For example,3

u i ei =i=1

u i ei .

We remark that subscripts are not employed exclusively in connection with indicial notation. Different connotations are assigned to subscripts throughout the text and the actual meaning of a particular subscript should be clear from the context. For example, in the context of incremental numerical procedures, subscripts may indicate the relevant increment number. In the expression = n+1 n , the subscripts n and n + 1 refer to the values of , respectively, at the end of increments n and n + 1. Superscripts Superscripts are also used extensively throughout the text. The meaning of a particular superscript will be stated the rst time it appears in the text and should be clear from the context thereafter.

14


1.3.4. OTHER IMPORTANT SYMBOLS AND OPERATIONS The meanings of other important symbols and operations are listed below. det() dev() divp () divx () exp() ln() o() sign() skew() sym() tr() () () () p () x () , s x () ()s s p ,

Determinant of () Deviator of () Material divergence of () Spatial divergence of () Exponential (including tensor exponential) of () Natural logarithm (including tensor logarithm) of () A term that vanishes faster than () The signum function: sign() ()/|()| Skew-symmetric part of () Symmetric part of () Trace of () Increment of (). Typically, () = ()n+1 ()n Iterative increment of () Gradient of () Material gradient of () Spatial gradient of () Corresponding symmetric gradients of () Boundary of the domain () Subdifferential of () with respect to a Derivative of () with respect to a Material time derivative of () The transpose of () Means a is dened as b. The symbol is often used to emphasise that the expression in question is a denition. Assignment operation. The value of the right-hand side of the expression is assigned to its left-hand side. The symbol := is often used to emphasise that a given expression is an assignment operation performed by a computational algorithm. Double contraction of tensors (internal product of second-order tensors)

a () () a () ()T

ab a := b, a := a + b

S : T, S : T, S:T

INTRODUCTION

15 Single contraction of vectors and tensors. The single contraction symbol (the single dot) is usually omitted in single contractions between a tensor and a vector or between tensors; that is, T u and S T are normally represented simply as Tu and ST. Vector product Tensor product of tensors or vectors The appropriate product between two generic entities, X and Y, in a given context. Norm (absolute value) of the scalar () Euclidean norm of tensors and vectors: ||T || T : T, ||v|| v v Used in the description of arguments of FORTRAN subprograms listed in the text. An arrow pointing to the right followed by an argument name A means that the value of A at entry is used by the relevant subprogram and is not changed during its execution. Analogously to above, an arrow pointing to the left followed by an argument name A means that the value of A is calculated and returned by the relevant subprogram and its value at entry is ignored. Analogously to and above, a double arrow followed by an argument name A means that the value of A at entry is used by the relevant subprogram and is changed during its execution.

u v, T u, ST

uv S T, uv XY |()| ||T ||, ||v|| A

A

A

2 ELEMENTS OF TENSOR ANALYSIS

T

HIS chapter introduces the notation and reviews some fundamentals of vector and tensor calculus which are extensively employed in this book. Throughout this text, preference is given to the use of intrinsic (or compact) tensor notation where no indices are used to represent mathematical entities. However, in many of the denitions introduced in this chapter, indicial notation is also used. This will allow readers not yet familiar with compact notation to associate compactly written entities and operations with their indicial forms, which will be expressed exclusively in terms of Cartesian coordinate systems. We note that the use of Cartesian, rather than curvilinear, coordinates for indicial representation is sufciently general for the applications considered in this book. In the subsequent chapters, the use of indicial notation will be much less frequent. Readers who are familiar with tensor analysis and, in particular, the use of compact notation, may comfortably skip this chapter. We remark that no proofs are given to most relations presented in this chapter. Readers interested in such proofs and a more in-depth treatment of the subject are referred to other textbooks such as Gurtin (1981).

2.1. VectorsLet E be an n-dimensional Euclidean space and let U be the space of n-dimensional vectors associated with E. Points of E and vectors U satisfy the basic rules of vector algebra, with which we assume the reader to be familiar. 2.1.1. INNER PRODUCT, NORM AND ORTHOGONALITY Let uv denote the inner product (or scalar product) between two arbitrary vectors of U. The Euclidean norm (or, simply, norm) of a vector u is dened as (2.1) u = u u, and u is said to be a unit vector if u = 1. The zero vector, here denoted o, is the element of U that satises o = 0.Computational Methods for Plasticity: Theory and Applications EA de Souza Neto, D Peri and DRJ Owen c c 2008 John Wiley & Sons, Ltd

(2.2)

(2.3)

18


A vector u is said to be orthogonal (or perpendicular) to a vector v if u v = 0. 2.1.2. ORTHOGONAL BASES AND CARTESIAN COORDINATE FRAMES A set {ei } {e1 , e2 , . . . , en } of n mutually orthogonal vectors satisfying ei ej = ij , where ij = 1 0 if i = j if i = j (2.6) (2.5) (2.4)

is the Krnecker delta, denes an orthonormal basis for U . Any vector u U can be represented as u = u 1 e1 + u 2 e2 + + u n en = u i ei , where u i = u ei , i = 1, 2, . . . , n (2.8) are the Cartesian components of u relative to the basis {ei }. Any vector of U is uniquely dened by its components relative to a given basis. This allows us to represent any vector u as a single column matrix, denoted [u], of components u1 u2 [u] = . . . . un An orthonormal basis, {ei }, together with an origin point, x0 E, denes a Cartesian coordinate frame. Analogously to the representation of vectors, any point x of E can be represented by an array x1 x2 [x] = . , (2.10) . . xn of Cartesian coordinates of x. The Cartesian coordinates {xi } of x are the Cartesian components of the position vector r = x x0 , (2.11) of x relative to the origin x0 . That is, xi = (x x0 ) ei . (2.12) (2.7)

(2.9)

ELEMENTS OF TENSOR ANALYSIS

19

2.2. Second-order tensorsSecond-order tensors are linear transformations from U into U, i.e. a second-order tensor T:UU maps each vector u into a vector v = T u. (2.13) Any linear vector-valued function of a vector is a tensor. The operations of sum and scalar multiplication of tensors are dened by (S + T )u = S u + T u ( S)u = (S u), (2.14)

where R. In addition, the zero tensor, 0, and the identity tensor, I, are, respectively, the tensors that satisfy 0u = o (2.15) Iu=u u U. The product of two tensors S and T is the tensor ST dened by ST u = S (T u). In general, ST = TS. If ST = TS, then S and T are said to commute. 2.2.1. THE TRANSPOSE. SYMMETRIC AND SKEW TENSORS The transpose, T T , of a tensor T is the unique tensor that satises T u v = u T T v, If T = TT, then T is said to be symmetric. If T = T T , then T is said to be skew symmetric (or, simply, skew). Any tensor T can be decomposed as the sum T = sym(T ) + skew(T ) of its symmetric part sym(T ) 1 (T + T T ) 2 and its skew part skew(T ) 1 (T T T ). 2 (2.23) (2.22) (2.21) (2.20) (2.19) u, v U. (2.18) (2.17) (2.16)

20


Basic properties The following basic properties involving the transpose, skew and symmetric parts of a tensor hold: (i) (S + T )T = ST + T T . (ii) (S T )T = T T ST . (iii) (T T ) = T. (iv) If T is symmetric, then skew(T ) = 0, (v) If T is skew, then skew(T ) = T, 2.2.2. TENSOR PRODUCTS The tensor product of two vectors u and v, denoted u v, is the tensor that maps each vector w into the vector (v w)u: (u v) w = (v w) u. The tensor product is sometimes referred to as the dyadic product. Some properties of the tensor product The following relations hold for any vectors s, t, u, v, w and tensor S : (i) u (v + w) = u v + u w. (ii) (u v)T = v u. (iii) (u v)(s t) = (v s)u t. (iv) ei ei = I. (v) S (u v) = (S u) v. (vi) (u v)S = (u ST v). (2.24) sym(T ) = 0. sym(T ) = T.T


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2.2.3. CARTESIAN COMPONENTS AND MATRIX REPRESENTATION Any second-order tensor T can be represented as T = T11 e1 e1 + T12 e1 e2 + + Tnn en en = Tij ei ej where Tij = ei T ej (2.26) are the Cartesian components of T. Note that in (2.25) no summation is implied over the index n. Any second tensor is uniquely dened by its Cartesian components. Thus, by arranging the components Tij in a matrix, we may have the following matrix representation for T : T21 [T ] = . . . Tn1 T11 T12 T22 . . . Tn2 .. . T1n (2.27) T2n . . . . (2.25)

Tnn

For instance, the Cartesian components of the identity tensor, I, read Iij = ij , so that its matrix representation is given by 1 0 0 1 [I ] = . . . . . . 0 0 0 0 . . .. . . . 1 (2.28)

(2.29)

The Cartesian components of the vector v = T u are given by vi = [Tjk (ej ek ) ul el ] ei = Tij uj . . (2.31) (2.30)

Thus, the array [v] of Cartesian components of v is obtained from the matrix-vector product v2 [v ] . . . vn v1 T21 = . . . Tn1 T11 T12 T22 . . . Tn2 .. . T1n T2n u2 . . . . . . un u1

Tnn

T It can be easily proved that the Cartesian components Tij of the transpose T T of a tensor T are given by T Tij = Tji . (2.32)

22


Thus, T T has the following Cartesian matrix representation T11 T21 Tn1 T12 T22 Tn2 [T T ] = . . . . .. . . . . . . . T1n T2n Tnn

(2.33)

2.2.4. TRACE, INNER PRODUCT AND EUCLIDEAN NORM For any u, v U, the trace of the tensor (u v) is the linear map dened as tr(u v) = u v. For a generic tensor, T = Tij ei ej , it then follows that tr T = Tij tr(ei ej ) = Tij ij = Tii , (2.35) (2.34)

that is, the trace of T is the sum of the diagonal terms of the Cartesian matrix representation [T ]. The inner product, S : T, between two tensors S and T is dened as S : T = tr(ST T ), or, in Cartesian component form, S : T = Sij Tij . The Euclidean norm (or simply norm) of a tensor T is dened as 2 2 2 T T : T = T11 + T12 + + Tnn . Basic properties The following basic properties involving the internal product of tensors hold for any tensors R, S, T and vectors s, t, u, v: (i) I : T = tr T. (ii) R : (S T ) = (ST R) : T = (R T T ) : S. (iii) u S v = S : (u v). (iv) (s t) : (u v) = (s u)(t v). (v) Tij = T : (ei ej ). (vi) (u v)ij = (u v) : (ei ej ) = ui vj . (vii) If S is symmetric, then S : T = S : T T = S : sym(T ). (viii) If S is skew, then S : T = S : T T = S : skew(T ). (ix) If S is symmetric and T is skew, then S : T = 0. (2.37) (2.36)

(2.38)


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2.2.5. INVERSE TENSOR. DETERMINANT A tensor T is said to be invertible if its inverse, denoted T 1 , satisfying T 1 T = T T 1 = I exists. The determinant of a tensor T, denoted det T, is the determinant of the matrix [T ]. A tensor T is invertible if and only if det T = 0. A tensor T is said to be positive denite if T u u > 0, Any positive denite tensor is invertible. Basic relations involving the determinant and the inverse tensor Relation (i) below holds for any tensors S and T and relations (ii)(iv) hold for any invertible tensors S and T : (i) det(ST ) = det S det T. (ii) det T 1 = (det T ) (iii) (ST )1 T 1

(2.39)

(2.40)

u = o.

(2.41)

.

= T 1 S1 .1

(iv) (T 1 ) = (T T )

.

2.2.6. ORTHOGONAL TENSORS. ROTATIONS A tensor Q is said to be orthogonal if QT = Q1 . (2.42)

The set of all orthogonal tensors will be denoted O. The determinant of any orthogonal tensor equals either +1 or 1. An orthogonal tensor Q with det Q = 1 (2.43)

is called a proper orthogonal tensor (or a rotation). The set of all proper orthogonal (or rotation) tensors is the proper orthogonal group. It will be denoted O + . The product Q1 Q2 of any two orthogonal tensors Q1 and Q2 is an orthogonal tensor. If Q1 and Q2 are rotations, then the product Q1 Q2 is also a rotation. For all vectors u and v, an orthogonal tensor Q satises Qu Qv = u v. (2.44)

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Rotations and changes of basis Let {ei } and {e } be two orthonormal bases of U. Such bases are related by i e = R ei , i for i = 1, 2, . . . , n, (2.45)

where R is a rotation. Let T and u be, respectively, a tensor and a vector with matrix representations [T ] and [u] with respect to the basis {ei }. The matrix representations [T ] and [u ] of T and u relative to the basis {e } are given by the following products of matrices: i [T ] = [R]T [T ] [R ]; Equivalently, in component form, we have Tij = Rki Tkl Rlj ,

[u ] = [R]T [u ]. u = Rji uj . i .. . e1 e n

(2.46)

(2.47)

The matrix [R] is given by

e2 e 1 [R] = . . . en e 1 or, in component form,

e1 e 1

e1 e 2 e2 e 2 . . . en e 2

e2 e n . , . .

(2.48)

en e n (2.49)

Rij = ei e . j

Example. A rotation in two dimensions In two-dimensional space, the rotation tensor has a simple Cartesian representation. Let the tensor R be a transformation that rotates all vectors of the two-dimensional space by an (anti-clockwise positive) angle . The matrix representation of R reads [R] = cos sin sin cos . (2.50)

2.2.7. CROSS PRODUCT Let us now restrict ourselves to the three-dimensional vector space. In this space, we dene the cross product (o

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