1. 1. COMPUTATIONAL METHODS FOR PLASTICITY THEORY AND APPLICATIONS EA de Souza Neto D Peric DRJ Owen Civil and Computational Engineering Centre, Swansea University A John Wiley and Sons, Ltd, Publication
2. 2. COMPUTATIONAL METHODS FOR PLASTICITY
3. 3. COMPUTATIONAL METHODS FOR PLASTICITY THEORY AND APPLICATIONS EA de Souza Neto D Peric DRJ Owen Civil and Computational Engineering Centre, Swansea University A John Wiley and Sons, Ltd, Publication
4. 4. 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.
5. 5. This book is lovingly dedicated to Deise, Patricia and Andr; Mira, Nikola and Nina; Janet, Kathryn and Lisa.
6. 6. CONTENTS Preface xx Part One Basic concepts 1 1 Introduction 3 1.1 Aims and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Readership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 The use of boxes . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 General scheme of notation . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Character fonts. General convention . . . . . . . . . . . . . . . . 8 1.3.2 Some important characters . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 Indicial notation, subscripts and superscripts . . . . . . . . . . . 13 1.3.4 Other important symbols and operations . . . . . . . . . . . . . . 14 2 Elements of tensor analysis 17 2.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.1 Inner product, norm and orthogonality . . . . . . . . . . . . . . . 17 2.1.2 Orthogonal bases and Cartesian coordinate frames . . . . . . . . 18 2.2 Second-order tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.1 The transpose. Symmetric and skew tensors . . . . . . . . . . . . 19 2.2.2 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.3 Cartesian components and matrix representation . . . . . . . . . 21 2.2.4 Trace, inner product and Euclidean norm . . . . . . . . . . . . . 22 2.2.5 Inverse tensor. Determinant . . . . . . . . . . . . . . . . . . . . 23 2.2.6 Orthogonal tensors. Rotations . . . . . . . . . . . . . . . . . . . 23 2.2.7 Cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.8 Spectral decomposition . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.9 Polar decomposition . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Higher-order tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.1 Fourth-order tensors . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.2 Generic-order tensors . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4 Isotropic tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.1 Isotropic second-order tensors . . . . . . . . . . . . . . . . . . . 30 2.4.2 Isotropic fourth-order tensors . . . . . . . . . . . . . . . . . . . 30
7. 7. viii CONTENTS 2.5 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.5.1 The derivative map. Directional derivative . . . . . . . . . . . . . 32 2.5.2 Linearisation of a nonlinear function . . . . . . . . . . . . . . . . 32 2.5.3 The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.5.4 Derivatives of functions of vector and tensor arguments . . . . . . 33 2.5.5 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.5.6 The product rule . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.5.7 The divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.5.8 Useful relations involving the gradient and the divergence . . . . 38 2.6 Linearisation of nonlinear problems . . . . . . . . . . . . . . . . . . . . . 38 2.6.1 The nonlinear problem and its linearised form . . . . . . . . . . . 38 2.6.2 Linearisation in innite-dimensional functional spaces . . . . . . 39 3 Elements of continuum mechanics and thermodynamics 41 3.1 Kinematics of deformation . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.1 Material and spatial elds . . . . . . . . . . . . . . . . . . . . . 44 3.1.2 Material and spatial gradients, divergences and time derivatives . 46 3.1.3 The deformation gradient . . . . . . . . . . . . . . . . . . . . . . 46 3.1.4 Volume changes. The determinant of the deformation gradient . . 47 3.1.5 Isochoric/volumetric split of the deformation gradient . . . . . . 49 3.1.6 Polar decomposition. Stretches and rotation . . . . . . . . . . . . 49 3.1.7 Strain measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1.8 The velocity gradient. Rate of deformation and spin . . . . . . . . 55 3.1.9 Rate of volume change . . . . . . . . . . . . . . . . . . . . . . . 56 3.2 Innitesimal deformations . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.1 The innitesimal strain tensor . . . . . . . . . . . . . . . . . . . 57 3.2.2 Innitesimal rigid deformations . . . . . . . . . . . . . . . . . . 58 3.2.3 Innitesimal isochoric and volumetric deformations . . . . . . . 58 3.3 Forces. Stress Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3.1 Cauchys axiom. The Cauchy stress vector . . . . . . . . . . . . 61 3.3.2 The axiom of momentum balance . . . . . . . . . . . . . . . . . 61 3.3.3 The Cauchy stress tensor . . . . . . . . . . . . . . . . . . . . . . 62 3.3.4 The First PiolaKirchhoff stress . . . . . . . . . . . . . . . . . . 64 3.3.5 The Second PiolaKirchhoff stress . . . . . . . . . . . . . . . . . 66 3.3.6 The Kirchhoff stress . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4 Fundamental laws of thermodynamics . . . . . . . . . . . . . . . . . . . . 67 3.4.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.2 Momentum balance . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4.3 The rst principle . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4.4 The second principle . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4.5 The ClausiusDuhem inequality . . . . . . . . . . . . . . . . . . 69 3.5 Constitutive theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.5.1 Constitutive axioms . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.5.2 Thermodynamics with internal variables . . . . . . . . . . . . . . 71 3.5.3 Phenomenological and micromechanical approaches . . . . . . . 74
8. 8. CONTENTS ix 3.5.4 The purely mechanical theory . . . . . . . . . . . . . . . . . . . 75 3.5.5 The constitutive initial value problem . . . . . . . . . . . . . . . 76 3.6 Weak equilibrium. The principle of virtual work . . . . . . . . . . . . . . 77 3.6.1 The spatial version . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.6.2 The material version . . . . . . . . . . . . . . . . . . . . . . . . 78 3.6.3 The innitesimal case . . . . . . . . . . . . . . . . . . . . . . . 78 3.7 The quasi-static initial boundary value problem . . . . . . . . . . . . . . . 79 3.7.1 Finite deformations . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.7.2 The innitesimal problem . . . . . . . . . . . . . . . . . . . . . 81 4 The finite element method in quasi-static nonlinear solid mechanics 83 4.1 Displacement-based nite elements . . . . . . . . . . . . . . . . . . . . . 84 4.1.1 Finite element interpolation . . . . . . . . . . . . . . . . . . . . 85 4.1.2 The discretised virtual work . . . . . . . . . . . . . . . . . . . . 86 4.1.3 Some typical isoparametric elements . . . . . . . . . . . . . . . . 90 4.1.4 Example. Linear elasticity . . . . . . . . . . . . . . . . . . . . . 93 4.2 Path-dependent materials. The incremental nite element procedure . . . . 94 4.2.1 The incremental constitutive function . . . . . . . . . . . . . . . 95 4.2.2 The incremental boundary value problem . . . . . . . . . . . . . 95 4.2.3 The nonlinear incremental nite element equation . . . . . . . . . 96 4.2.4 Nonlinear solution. The NewtonRaphson scheme . . . . . . . . 96 4.2.5 The consistent tangent modulus . . . . . . . . . . . . . . . . . . 98 4.2.6 Alternative nonlinear solution schemes . . . . . . . . . . . . . . 99 4.2.7 Non-incremental procedures for path-dependent materials . . . . 101 4.3 Large strain formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.3.1 The incremental constitutive function . . . . . . . . . . . . . . . 102 4.3.2 The incremental boundary value problem . . . . . . . . . . . . . 103 4.3.3 The nite element equilibrium equation . . . . . . . . . . . . . . 103 4.3.4 Linearisation. The consistent spatial tangent modulus . . . . . . . 103 4.3.5 Material and geometric stiffnesses . . . . . . . . . . . . . . . . . 106 4.3.6 Conguration-dependent loads. The load-stiffness matrix . . . . . 106 4.4 Unstable equilibrium. The arc-length method . . . . . . . . . . . . . . . . 107 4.4.1 The arc-length method . . . . . . . . . . . . . . . . . . . . . . . 107 4.4.2 The combined NewtonRaphson/arc-length procedure . . . . . . 108 4.4.3 The predictor solution . . . . . . . . . . . . . . . . . . . . . . . 111 5 Overview of the program structure 115 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.1.2 Remarks on program structure . . . . . . . . . . . . . . . . . . . 116 5.1.3 Portability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.2 The main program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3 Data input and initialisation . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3.1 The global database . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3.2 Main problem-dening data. Subroutine INDATA . . . . . . . . . 119
9. 9. x CONTENTS 5.3.3 External loading. Subroutine INLOAD . . . . . . . . . . . . . . . 119 5.3.4 Initialisation of variable data. Subroutine INITIA . . . . . . . . . 120 5.4 The load incrementation loop. Overview . . . . . . . . . . . . . . . . . . 120 5.4.1 Fixed increments option . . . . . . . . . . . . . . . . . . . . . . 120 5.4.2 Arc-length control option . . . . . . . . . . . . . . . . . . . . . . 120 5.4.3 Automatic increment cutting . . . . . . . . . . . . . . . . . . . . 123 5.4.4 The linear solver. Subroutine FRONT . . . . . . . . . . . . . . . . 124 5.4.5 Internal force calculation. Subroutine INTFOR . . . . . . . . . . . 124 5.4.6 Switching data. Subroutine SWITCH . . . . . . . . . . . . . . . . 124 5.4.7 Output of converged results. Subroutines OUTPUT and RSTART . . 125 5.5 Material and element modularity . . . . . . . . . . . . . . . . . . . . . . . 125 5.5.1 Example. Modularisation of internal force computation . . . . . . 125 5.6 Elements. Implementation and management . . . . . . . . . . . . . . . . . 128 5.6.1 Element properties. Element routines for data input . . . . . . . . 128 5.6.2 Element interfaces. Internal force and stiffness computation . . . 129 5.6.3 Implementing a new nite element . . . . . . . . . . . . . . . . . 129 5.7 Material models: implementation and management . . . . . . . . . . . . . 131 5.7.1 Material properties. Material-specic data input . . . . . . . . . . 131 5.7.2 State variables and other Gauss point quantities. Material-specic state updating routines . . . . . . . . . . . . . . 132 5.7.3 Material-specic switching/initialising routines . . . . . . . . . . 133 5.7.4 Material-specic tangent computation routines . . . . . . . . . . 134 5.7.5 Material-specic results output routines . . . . . . . . . . . . . . 134 5.7.6 Implementing a new material model . . . . . . . . . . . . . . . . 135 Part Two Small strains 137 6 The mathematical theory of plasticity 139 6.1 Phenomenological aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.2 One-dimensional constitutive model . . . . . . . . . . . . . . . . . . . . . 141 6.2.1 Elastoplastic decomposition of the axial strain . . . . . . . . . . . 142 6.2.2 The elastic uniaxial constitutive law . . . . . . . . . . . . . . . . 143 6.2.3 The yield function and the yield criterion . . . . . . . . . . . . . 143 6.2.4 The plastic ow rule. Loading/unloading conditions . . . . . . . 144 6.2.5 The hardening law . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.2.6 Summary of the model . . . . . . . . . . . . . . . . . . . . . . . 145 6.2.7 Determination of the plastic multiplier . . . . . . . . . . . . . . . 146 6.2.8 The elastoplastic tangent modulus . . . . . . . . . . . . . . . . . 147 6.3 General elastoplastic constitutive model . . . . . . . . . . . . . . . . . . . 148 6.3.1 Additive decomposition of the strain tensor . . . . . . . . . . . . 148 6.3.2 The free energy potential and the elastic law . . . . . . . . . . . . 148 6.3.3 The yield criterion and the yield surface . . . . . . . . . . . . . . 150 6.3.4 Plastic ow rule and hardening law . . . . . . . . . . . . . . . . 150 6.3.5 Flow rules derived from a ow potential . . . . . . . . . . . . . . 151 6.3.6 The plastic multiplier . . . . . . . . . . . . . . . . . . . . . . . . 152
10. 10. CONTENTS xi 6.3.7 Relation to the general continuum constitutive theory . . . . . . . 153 6.3.8 Rate form and the elastoplastic tangent operator . . . . . . . . . . 153 6.3.9 Non-smooth potentials and the subdifferential . . . . . . . . . . . 153 6.4 Classical yield criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.4.1 The Tresca yield criterion . . . . . . . . . . . . . . . . . . . . . 157 6.4.2 The von Mises yield criterion . . . . . . . . . . . . . . . . . . . 162 6.4.3 The MohrCoulomb yield criterion . . . . . . . . . . . . . . . . 163 6.4.4 The DruckerPrager yield criterion . . . . . . . . . . . . . . . . 166 6.5 Plastic ow rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6.5.1 Associative and non-associative plasticity . . . . . . . . . . . . . 168 6.5.2 Associative laws and the principle of maximum plastic dissipation 170 6.5.3 Classical ow rules . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.6 Hardening laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 6.6.1 Perfect plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 177 6.6.2 Isotropic hardening . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.6.3 Thermodynamical aspects. Associative isotropic hardening . . . . 182 6.6.4 Kinematic hardening. The Bauschinger effect . . . . . . . . . . . 185 6.6.5 Mixed isotropic/kinematic hardening . . . . . . . . . . . . . . . 189 7 Finite elements in small-strain plasticity problems 191 7.1 Preliminary implementation aspects . . . . . . . . . . . . . . . . . . . . . 192 7.2 General numerical integration algorithm for elastoplastic constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.2.1 The elastoplastic constitutive initial value problem . . . . . . . . 193 7.2.2 Euler discretisation: the incremental constitutive problem . . . . . 194 7.2.3 The elastic predictor/plastic corrector algorithm . . . . . . . . . . 196 7.2.4 Solution of the return-mapping equations . . . . . . . . . . . . . 198 7.2.5 Closest point projection interpretation . . . . . . . . . . . . . . . 200 7.2.6 Alternative justication: operator split method . . . . . . . . . . 201 7.2.7 Other elastic predictor/return-mapping schemes . . . . . . . . . . 201 7.2.8 Plasticity and differential-algebraic equations . . . . . . . . . . . 209 7.2.9 Alternative mathematical programming-based algorithms . . . . . 210 7.2.10 Accuracy and stability considerations . . . . . . . . . . . . . . . 210 7.3 Application: integration algorithm for the isotropically hardening von Mises model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 7.3.1 The implemented model . . . . . . . . . . . . . . . . . . . . . . 216 7.3.2 The implicit elastic predictor/return-mapping scheme . . . . . . . 217 7.3.3 The incremental constitutive function for the stress . . . . . . . . 220 7.3.4 Linear isotropic hardening and perfect plasticity: the closed-form return mapping . . . . . . . . . . . . . . . . . . . . 223 7.3.5 Subroutine SUVM . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.4 The consistent tangent modulus . . . . . . . . . . . . . . . . . . . . . . . 228 7.4.1 Consistent tangent operators in elastoplasticity . . . . . . . . . . 229 7.4.2 The elastoplastic consistent tangent for the von Mises model with isotropic hardening . . . . . . . . . . . . . . . . . . . . . . 232 7.4.3 Subroutine CTVM . . . . . . . . . . . . . . . . . . . . . . . . . . 235
11. 11. xii CONTENTS 7.4.4 The general elastoplastic consistent tangent operator for implicit return mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 238 7.4.5 Illustration: the von Mises model with isotropic hardening . . . . 240 7.4.6 Tangent operator symmetry: incremental potentials . . . . . . . . 243 7.5 Numerical examples with the von Mises model . . . . . . . . . . . . . . . 244 7.5.1 Internally pressurised cylinder . . . . . . . . . . . . . . . . . . . 244 7.5.2 Internally pressurised spherical shell . . . . . . . . . . . . . . . . 247 7.5.3 Uniformly loaded circular plate . . . . . . . . . . . . . . . . . . 250 7.5.4 Strip-footing collapse . . . . . . . . . . . . . . . . . . . . . . . . 252 7.5.5 Double-notched tensile specimen . . . . . . . . . . . . . . . . . 255 7.6 Further application: the von Mises model with nonlinear mixed hardening . 257 7.6.1 The mixed hardening model: summary . . . . . . . . . . . . . . 257 7.6.2 The implicit return-mapping scheme . . . . . . . . . . . . . . . . 258 7.6.3 The incremental constitutive function . . . . . . . . . . . . . . . 260 7.6.4 Linear hardening: closed-form return mapping . . . . . . . . . . 261 7.6.5 Computational implementation aspects . . . . . . . . . . . . . . 261 7.6.6 The elastoplastic consistent tangent . . . . . . . . . . . . . . . . 262 8 Computations with other basic plasticity models 265 8.1 The Tresca model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 8.1.1 The implicit integration algorithm in principal stresses . . . . . . 268 8.1.2 Subroutine SUTR . . . . . . . . . . . . . . . . . . . . . . . . . . 279 8.1.3 Finite step accuracy: iso-error maps . . . . . . . . . . . . . . . . 283 8.1.4 The consistent tangent operator for the Tresca model . . . . . . . 286 8.1.5 Subroutine CTTR . . . . . . . . . . . . . . . . . . . . . . . . . . 291 8.2 The MohrCoulomb model . . . . . . . . . . . . . . . . . . . . . . . . . 295 8.2.1 Integration algorithm for the MohrCoulomb model . . . . . . . 297 8.2.2 Subroutine SUMC . . . . . . . . . . . . . . . . . . . . . . . . . . 310 8.2.3 Accuracy: iso-error maps . . . . . . . . . . . . . . . . . . . . . . 315 8.2.4 Consistent tangent operator for the MohrCoulomb model . . . . 316 8.2.5 Subroutine CTMC . . . . . . . . . . . . . . . . . . . . . . . . . . 319 8.3 The DruckerPrager model . . . . . . . . . . . . . . . . . . . . . . . . . . 324 8.3.1 Integration algorithm for the DruckerPrager model . . . . . . . 325 8.3.2 Subroutine SUDP . . . . . . . . . . . . . . . . . . . . . . . . . . 334 8.3.3 Iso-error map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 8.3.4 Consistent tangent operator for the DruckerPrager model . . . . 337 8.3.5 Subroutine CTDP . . . . . . . . . . . . . . . . . . . . . . . . . . 340 8.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 8.4.1 Bending of a V-notched Tresca bar . . . . . . . . . . . . . . . . . 343 8.4.2 End-loaded tapered cantilever . . . . . . . . . . . . . . . . . . . 344 8.4.3 Strip-footing collapse . . . . . . . . . . . . . . . . . . . . . . . . 346 8.4.4 Circular-footing collapse . . . . . . . . . . . . . . . . . . . . . . 350 8.4.5 Slope stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
12. 12. CONTENTS xiii 9 Plane stress plasticity 357 9.1 The basic plane stress plasticity problem . . . . . . . . . . . . . . . . . . 357 9.1.1 Plane stress linear elasticity . . . . . . . . . . . . . . . . . . . . 358 9.1.2 The constrained elastoplastic initial value problem . . . . . . . . 359 9.1.3 Procedures for plane stress plasticity . . . . . . . . . . . . . . . . 360 9.2 Plane stress constraint at the Gauss point level . . . . . . . . . . . . . . . 361 9.2.1 Implementation aspects . . . . . . . . . . . . . . . . . . . . . . . 362 9.2.2 Plane stress enforcement with nested iterations . . . . . . . . . . 362 9.2.3 Plane stress von Mises with nested iterations . . . . . . . . . . . 364 9.2.4 The consistent tangent for the nested iteration procedure . . . . . 366 9.2.5 Consistent tangent computation for the von Mises model . . . . . 366 9.3 Plane stress constraint at the structural level . . . . . . . . . . . . . . . . . 367 9.3.1 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 9.3.2 The implementation . . . . . . . . . . . . . . . . . . . . . . . . 368 9.4 Plane stress-projected plasticity models . . . . . . . . . . . . . . . . . . . 370 9.4.1 The plane stress-projected von Mises model . . . . . . . . . . . . 371 9.4.2 The plane stress-projected integration algorithm . . . . . . . . . . 373 9.4.3 Subroutine SUVMPS . . . . . . . . . . . . . . . . . . . . . . . . . 378 9.4.4 The elastoplastic consistent tangent operator . . . . . . . . . . . 382 9.4.5 Subroutine CTVMPS . . . . . . . . . . . . . . . . . . . . . . . . . 383 9.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 9.5.1 Collapse of an end-loaded cantilever . . . . . . . . . . . . . . . . 387 9.5.2 Innite plate with a circular hole . . . . . . . . . . . . . . . . . . 387 9.5.3 Stretching of a perforated rectangular plate . . . . . . . . . . . . 390 9.5.4 Uniform loading of a concrete shear wall . . . . . . . . . . . . . 391 9.6 Other stress-constrained states . . . . . . . . . . . . . . . . . . . . . . . . 396 9.6.1 A three-dimensional von Mises Timoshenko beam . . . . . . . . 396 9.6.2 The beam state-projected integration algorithm . . . . . . . . . . 400 10 Advanced plasticity models 403 10.1 A modied Cam-Clay model for soils . . . . . . . . . . . . . . . . . . . . 403 10.1.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 10.1.2 Computational implementation . . . . . . . . . . . . . . . . . . . 406 10.2 A capped DruckerPrager model for geomaterials . . . . . . . . . . . . . . 409 10.2.1 Capped DruckerPrager model . . . . . . . . . . . . . . . . . . . 410 10.2.2 The implicit integration algorithm . . . . . . . . . . . . . . . . . 412 10.2.3 The elastoplastic consistent tangent operator . . . . . . . . . . . 413 10.3 Anisotropic plasticity: the Hill, Hoffman and BarlatLian models . . . . . 414 10.3.1 The Hill orthotropic model . . . . . . . . . . . . . . . . . . . . . 414 10.3.2 Tensioncompression distinction: the Hoffman model . . . . . . 420 10.3.3 Implementation of the Hoffman model . . . . . . . . . . . . . . . 423 10.3.4 The BarlatLian model for sheet metals . . . . . . . . . . . . . . 427 10.3.5 Implementation of the BarlatLian model . . . . . . . . . . . . . 431
13. 13. xiv CONTENTS 11 Viscoplasticity 435 11.1 Viscoplasticity: phenomenological aspects . . . . . . . . . . . . . . . . . 436 11.2 One-dimensional viscoplasticity model . . . . . . . . . . . . . . . . . . . 437 11.2.1 Elastoplastic decomposition of the axial strain . . . . . . . . . . . 437 11.2.2 The elastic law . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 11.2.3 The yield function and the elastic domain . . . . . . . . . . . . . 438 11.2.4 Viscoplastic ow rule . . . . . . . . . . . . . . . . . . . . . . . . 438 11.2.5 Hardening law . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 11.2.6 Summary of the model . . . . . . . . . . . . . . . . . . . . . . . 439 11.2.7 Some simple analytical solutions . . . . . . . . . . . . . . . . . . 439 11.3 A von Mises-based multidimensional model . . . . . . . . . . . . . . . . 445 11.3.1 A von Mises-type viscoplastic model with isotropic strain hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 11.3.2 Alternative plastic strain rate denitions . . . . . . . . . . . . . . 447 11.3.3 Other isotropic and kinematic hardening laws . . . . . . . . . . . 448 11.3.4 Viscoplastic models without a yield surface . . . . . . . . . . . . 448 11.4 General viscoplastic constitutive model . . . . . . . . . . . . . . . . . . . 450 11.4.1 Relation to the general continuum constitutive theory . . . . . . . 450 11.4.2 Potential structure and dissipation inequality . . . . . . . . . . . 451 11.4.3 Rate-independent plasticity as a limit case . . . . . . . . . . . . . 452 11.5 General numerical framework . . . . . . . . . . . . . . . . . . . . . . . . 454 11.5.1 A general implicit integration algorithm . . . . . . . . . . . . . . 454 11.5.2 Alternative Euler-based algorithms . . . . . . . . . . . . . . . . . 457 11.5.3 General consistent tangent operator . . . . . . . . . . . . . . . . 458 11.6 Application: computational implementation of a von Mises-based model . 460 11.6.1 Integration algorithm . . . . . . . . . . . . . . . . . . . . . . . . 460 11.6.2 Iso-error maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 11.6.3 Consistent tangent operator . . . . . . . . . . . . . . . . . . . . . 464 11.6.4 Perzyna-type model implementation . . . . . . . . . . . . . . . . 466 11.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 11.7.1 Double-notched tensile specimen . . . . . . . . . . . . . . . . . 467 11.7.2 Plane stress: stretching of a perforated plate . . . . . . . . . . . . 469 12 Damage mechanics 471 12.1 Physical aspects of internal damage in solids . . . . . . . . . . . . . . . . 472 12.1.1 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 12.1.2 Rubbery polymers . . . . . . . . . . . . . . . . . . . . . . . . . 473 12.2 Continuum damage mechanics . . . . . . . . . . . . . . . . . . . . . . . . 473 12.2.1 Original development: creep-damage . . . . . . . . . . . . . . . 474 12.2.2 Other theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 12.2.3 Remarks on the nature of the damage variable . . . . . . . . . . . 476 12.3 Lemaitres elastoplastic damage theory . . . . . . . . . . . . . . . . . . . 478 12.3.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 12.3.2 Integration algorithm . . . . . . . . . . . . . . . . . . . . . . . . 482 12.3.3 The tangent operators . . . . . . . . . . . . . . . . . . . . . . . . 485
14. 14. CONTENTS xv 12.4 A simplied version of Lemaitres model . . . . . . . . . . . . . . . . . . 486 12.4.1 The single-equation integration algorithm . . . . . . . . . . . . . 486 12.4.2 The tangent operator . . . . . . . . . . . . . . . . . . . . . . . . 490 12.4.3 Example. Fracturing of a cylindrical notched specimen . . . . . . 493 12.5 Gursons void growth model . . . . . . . . . . . . . . . . . . . . . . . . . 496 12.5.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 12.5.2 Integration algorithm . . . . . . . . . . . . . . . . . . . . . . . . 501 12.5.3 The tangent operator . . . . . . . . . . . . . . . . . . . . . . . . 502 12.6 Further issues in damage modelling . . . . . . . . . . . . . . . . . . . . . 504 12.6.1 Crack closure effects in damaged elastic materials . . . . . . . . 504 12.6.2 Crack closure effects in damage evolution . . . . . . . . . . . . . 510 12.6.3 Anisotropic ductile damage . . . . . . . . . . . . . . . . . . . . 512 Part Three Large strains 517 13 Finite strain hyperelasticity 519 13.1 Hyperelasticity: basic concepts . . . . . . . . . . . . . . . . . . . . . . . . 520 13.1.1 Material objectivity: reduced form of the free-energy function . . 520 13.1.2 Isotropic hyperelasticity . . . . . . . . . . . . . . . . . . . . . . 521 13.1.3 Incompressible hyperelasticity . . . . . . . . . . . . . . . . . . . 524 13.1.4 Compressible regularisation . . . . . . . . . . . . . . . . . . . . 525 13.2 Some particular models . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 13.2.1 The MooneyRivlin and the neo-Hookean models . . . . . . . . 525 13.2.2 The Ogden material model . . . . . . . . . . . . . . . . . . . . . 527 13.2.3 The Hencky material . . . . . . . . . . . . . . . . . . . . . . . . 528 13.2.4 The BlatzKo material . . . . . . . . . . . . . . . . . . . . . . . 530 13.3 Isotropic nite hyperelasticity in plane stress . . . . . . . . . . . . . . . . 530 13.3.1 The plane stress incompressible Ogden model . . . . . . . . . . . 531 13.3.2 The plane stress Hencky model . . . . . . . . . . . . . . . . . . 532 13.3.3 Plane stress with nested iterations . . . . . . . . . . . . . . . . . 533 13.4 Tangent moduli: the elasticity tensors . . . . . . . . . . . . . . . . . . . . 534 13.4.1 Regularised neo-Hookean model . . . . . . . . . . . . . . . . . . 535 13.4.2 Principal stretches representation: Ogden model . . . . . . . . . . 535 13.4.3 Hencky model . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 13.4.4 BlatzKo material . . . . . . . . . . . . . . . . . . . . . . . . . 537 13.5 Application: Ogden material implementation . . . . . . . . . . . . . . . . 538 13.5.1 Subroutine SUOGD . . . . . . . . . . . . . . . . . . . . . . . . . 538 13.5.2 Subroutine CSTOGD . . . . . . . . . . . . . . . . . . . . . . . . . 542 13.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 13.6.1 Axisymmetric extension of an annular plate . . . . . . . . . . . . 547 13.6.2 Stretching of a square perforated rubber sheet . . . . . . . . . . . 547 13.6.3 Ination of a spherical rubber balloon . . . . . . . . . . . . . . . 550 13.6.4 Rugby ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 13.6.5 Ination of initially at membranes . . . . . . . . . . . . . . . . 552
15. 15. xvi CONTENTS 13.6.6 Rubber cylinder pressed between two plates . . . . . . . . . . . . 555 13.6.7 Elastomeric bead compression . . . . . . . . . . . . . . . . . . . 556 13.7 Hyperelasticity with damage: the Mullins effect . . . . . . . . . . . . . . . 557 13.7.1 The GurtinFrancis uniaxial model . . . . . . . . . . . . . . . . 560 13.7.2 Three-dimensional modelling. A brief review . . . . . . . . . . . 562 13.7.3 A simple rate-independent three-dimensional model . . . . . . . 562 13.7.4 Example: the model problem . . . . . . . . . . . . . . . . . . . . 565 13.7.5 Computational implementation . . . . . . . . . . . . . . . . . . . 565 13.7.6 Example: ination/deation of a damageable rubber balloon . . . 569 14 Finite strain elastoplasticity 573 14.1 Finite strain elastoplasticity: a brief review . . . . . . . . . . . . . . . . . 574 14.2 One-dimensional nite plasticity model . . . . . . . . . . . . . . . . . . . 575 14.2.1 The multiplicative split of the axial stretch . . . . . . . . . . . . . 575 14.2.2 Logarithmic stretches and the Hencky hyperelastic law . . . . . . 576 14.2.3 The yield function . . . . . . . . . . . . . . . . . . . . . . . . . 576 14.2.4 The plastic ow rule . . . . . . . . . . . . . . . . . . . . . . . . 576 14.2.5 The hardening law . . . . . . . . . . . . . . . . . . . . . . . . . 577 14.2.6 The plastic multiplier . . . . . . . . . . . . . . . . . . . . . . . . 577 14.3 General hyperelastic-based multiplicative plasticity model . . . . . . . . . 578 14.3.1 Multiplicative elastoplasticity kinematics . . . . . . . . . . . . . 578 14.3.2 The logarithmic elastic strain measure . . . . . . . . . . . . . . . 582 14.3.3 A general isotropic large-strain plasticity model . . . . . . . . . . 583 14.3.4 The dissipation inequality . . . . . . . . . . . . . . . . . . . . . 586 14.3.5 Finite strain extension to innitesimal theories . . . . . . . . . . 588 14.4 The general elastic predictor/return-mapping algorithm . . . . . . . . . . . 590 14.4.1 The basic constitutive initial value problem . . . . . . . . . . . . 590 14.4.2 Exponential map backward discretisation . . . . . . . . . . . . . 591 14.4.3 Computational implementation of the general algorithm . . . . . 595 14.5 The consistent spatial tangent modulus . . . . . . . . . . . . . . . . . . . 597 14.5.1 Derivation of the spatial tangent modulus . . . . . . . . . . . . . 598 14.5.2 Computational implementation . . . . . . . . . . . . . . . . . . . 599 14.6 Principal stress space-based implementation . . . . . . . . . . . . . . . . . 599 14.6.1 Stress-updating algorithm . . . . . . . . . . . . . . . . . . . . . 600 14.6.2 Tangent modulus computation . . . . . . . . . . . . . . . . . . . 601 14.7 Finite plasticity in plane stress . . . . . . . . . . . . . . . . . . . . . . . . 601 14.7.1 The plane stress-projected nite von Mises model . . . . . . . . . 601 14.7.2 Nested iteration for plane stress enforcement . . . . . . . . . . . 604 14.8 Finite viscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 14.8.1 Numerical treatment . . . . . . . . . . . . . . . . . . . . . . . . 606 14.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 14.9.1 Finite strain bending of a V-notched Tresca bar . . . . . . . . . . 606 14.9.2 Necking of a cylindrical bar . . . . . . . . . . . . . . . . . . . . 607 14.9.3 Plane strain localisation . . . . . . . . . . . . . . . . . . . . . . 611 14.9.4 Stretching of a perforated plate . . . . . . . . . . . . . . . . . . . 613 14.9.5 Thin sheet metal-forming application . . . . . . . . . . . . . . . 614
16. 16. CONTENTS xvii 14.10 Rate forms: hypoelastic-based plasticity models . . . . . . . . . . . . . . . 615 14.10.1 Objective stress rates . . . . . . . . . . . . . . . . . . . . . . . . 619 14.10.2 Hypoelastic-based plasticity models . . . . . . . . . . . . . . . . 621 14.10.3 The Jaumann rate-based model . . . . . . . . . . . . . . . . . . . 622 14.10.4 Hyperelastic-based models and equivalent rate forms . . . . . . . 624 14.10.5 Integration algorithms and incremental objectivity . . . . . . . . 625 14.10.6 Objective algorithm for Jaumann rate-based models . . . . . . . . 628 14.10.7 Integration of GreenNaghdi rate-based models . . . . . . . . . . 632 14.11 Finite plasticity with kinematic hardening . . . . . . . . . . . . . . . . . . 633 14.11.1 A model of nite strain kinematic hardening . . . . . . . . . . . 633 14.11.2 Integration algorithm . . . . . . . . . . . . . . . . . . . . . . . . 637 14.11.3 Spatial tangent operator . . . . . . . . . . . . . . . . . . . . . . 642 14.11.4 Remarks on predictive capability . . . . . . . . . . . . . . . . . . 644 14.11.5 Alternative descriptions . . . . . . . . . . . . . . . . . . . . . . 644 15 Finite elements for large-strain incompressibility 647 15.1 The F-bar methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 648 15.1.1 Stress computation: the F-bar deformation gradient . . . . . . . . 649 15.1.2 The internal force vector . . . . . . . . . . . . . . . . . . . . . . 651 15.1.3 Consistent linearisation: the tangent stiffness . . . . . . . . . . . 652 15.1.4 Plane strain implementation . . . . . . . . . . . . . . . . . . . . 655 15.1.5 Computational implementation aspects . . . . . . . . . . . . . . 656 15.1.6 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 15.1.7 Other centroid sampling-based F-bar elements . . . . . . . . . . 663 15.1.8 A more general F-bar methodology . . . . . . . . . . . . . . . . 663 15.1.9 The F-bar-Patch Method for simplex elements . . . . . . . . . . 665 15.2 Enhanced assumed strain methods . . . . . . . . . . . . . . . . . . . . . . 669 15.2.1 Enhanced three-eld variational principle . . . . . . . . . . . . . 669 15.2.2 EAS nite elements . . . . . . . . . . . . . . . . . . . . . . . . 671 15.2.3 Finite element equations: static condensation . . . . . . . . . . . 676 15.2.4 Implementation aspects . . . . . . . . . . . . . . . . . . . . . . . 678 15.2.5 The stability of EAS elements . . . . . . . . . . . . . . . . . . . 678 15.3 Mixed u/p formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 15.3.1 The two-eld variational principle . . . . . . . . . . . . . . . . . 683 15.3.2 Finite element equations . . . . . . . . . . . . . . . . . . . . . . 685 15.3.3 Solution: static condensation . . . . . . . . . . . . . . . . . . . . 687 15.3.4 Implementation aspects . . . . . . . . . . . . . . . . . . . . . . . 689 16 Anisotropic finite plasticity: Single crystals 691 16.1 Physical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 16.1.1 Plastic deformation by slip: slip-systems . . . . . . . . . . . . . . 692 16.2 Plastic slip and the Schmid resolved shear stress . . . . . . . . . . . . . . 693 16.3 Single crystal simulation: a brief review . . . . . . . . . . . . . . . . . . . 694 16.4 A general continuum model of single crystals . . . . . . . . . . . . . . . . 694 16.4.1 The plastic ow equation . . . . . . . . . . . . . . . . . . . . . . 695 16.4.2 The resolved Schmid shear stress . . . . . . . . . . . . . . . . . 696
17. 17. xviii CONTENTS 16.4.3 Multisurface formulation of the ow rule . . . . . . . . . . . . . 696 16.4.4 Isotropic Taylor hardening . . . . . . . . . . . . . . . . . . . . . 698 16.4.5 The hyperelastic law . . . . . . . . . . . . . . . . . . . . . . . . 698 16.5 A general integration algorithm . . . . . . . . . . . . . . . . . . . . . . . 699 16.5.1 The search for an active set of slip systems . . . . . . . . . . . . 703 16.6 An algorithm for a planar double-slip model . . . . . . . . . . . . . . . . 705 16.6.1 A planar double-slip model . . . . . . . . . . . . . . . . . . . . . 705 16.6.2 The integration algorithm . . . . . . . . . . . . . . . . . . . . . 707 16.6.3 Example: the model problem . . . . . . . . . . . . . . . . . . . . 710 16.7 The consistent spatial tangent modulus . . . . . . . . . . . . . . . . . . . 713 16.7.1 The elastic modulus: compressible neo-Hookean model . . . . . . 713 16.7.2 The elastoplastic consistent tangent modulus . . . . . . . . . . . 714 16.8 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716 16.8.1 Symmetric strain localisation on a rectangular strip . . . . . . . . 717 16.8.2 Unsymmetric localisation . . . . . . . . . . . . . . . . . . . . . 720 16.9 Viscoplastic single crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 721 16.9.1 Rate-dependent formulation . . . . . . . . . . . . . . . . . . . . 723 16.9.2 The exponential map-based integration algorithm . . . . . . . . . 724 16.9.3 The spatial tangent modulus: neo-Hookean-based model . . . . . 725 16.9.4 Rate-dependent crystal: model problem . . . . . . . . . . . . . . 726 Appendices 729 A Isotropic functions of a symmetric tensor 731 A.1 Isotropic scalar-valued functions . . . . . . . . . . . . . . . . . . . . . . . 731 A.1.1 Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 732 A.1.2 The derivative of an isotropic scalar function . . . . . . . . . . . 732 A.2 Isotropic tensor-valued functions . . . . . . . . . . . . . . . . . . . . . . . 733 A.2.1 Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 A.2.2 The derivative of an isotropic tensor function . . . . . . . . . . . 734 A.3 The two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . 735 A.3.1 Tensor function derivative . . . . . . . . . . . . . . . . . . . . . 736 A.3.2 Plane strain and axisymmetric problems . . . . . . . . . . . . . . 738 A.4 The three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . 739 A.4.1 Function computation . . . . . . . . . . . . . . . . . . . . . . . 739 A.4.2 Computation of the function derivative . . . . . . . . . . . . . . 740 A.5 A particular class of isotropic tensor functions . . . . . . . . . . . . . . . 740 A.5.1 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 742 A.5.2 Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 743 A.6 Alternative procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744
18. 18. CONTENTS xix B The tensor exponential 747 B.1 The tensor exponential function . . . . . . . . . . . . . . . . . . . . . . . 747 B.1.1 Some properties of the tensor exponential function . . . . . . . . 748 B.1.2 Computation of the tensor exponential function . . . . . . . . . . 749 B.2 The tensor exponential derivative . . . . . . . . . . . . . . . . . . . . . . 750 B.2.1 Computer implementation . . . . . . . . . . . . . . . . . . . . . 751 B.3 Exponential map integrators . . . . . . . . . . . . . . . . . . . . . . . . . 751 B.3.1 The generalised exponential map midpoint rule . . . . . . . . . . 752 C Linearisation of the virtual work 753 C.1 Innitesimal deformations . . . . . . . . . . . . . . . . . . . . . . . . . . 753 C.2 Finite strains and deformations . . . . . . . . . . . . . . . . . . . . . . . . 755 C.2.1 Material description . . . . . . . . . . . . . . . . . . . . . . . . 755 C.2.2 Spatial description . . . . . . . . . . . . . . . . . . . . . . . . . 756 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 765 Index 783
19. 19. PREFACE THE 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 phenomenolog- ical 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 non- linear solid mechanics and a brief description of the computer program HYPLAS. Part Two
20. 20. 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 rate- dependent (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 Peric DRJ Owen Swansea
21. 21. Part One Basic concepts
22. 22. 1 INTRODUCTION OVER 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 characteriza- tion 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 ever- increasing 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 scope The 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 of Computational Methods for Plasticity: Theory and Applications EA de Souza Neto, D Peric and DRJ Owen c 2008 John Wiley & Sons, Ltd
23. 23. 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 consider- able 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. Layout In line with the above aims, the book has been divided into three parts as follows.
24. 24. 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.
25. 25. 6 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS 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 stress- constrained 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 con- stitutive 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 hyperelastic- damage 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.
26. 26. 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 hypoelastic- based 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 plas- ticity 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 notation Throughout 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.
27. 27. 8 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS 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.
28. 28. 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 A Finite element assembly operator (note the large font) A First elasticity tensor a Spatial elasticity tensor B Left CauchyGreen strain tensor Be Elastic left CauchyGreen strain tensor B Discrete (nite element) symmetric gradient operator (strain-displacement matrix) B Generic body b Body force b Reference body force C Right CauchyGreen strain tensor c Cohesion D Damage internal variable D Stretching tensor De Elastic stretching Dp Plastic stretching D Innitesimal consistent tangent operator De Innitesimal elasticity tensor Dep Innitesimal elastoplastic consistent tangent operator D Consistent tangent matrix (array representation of D) De Elasticity matrix (array representation of De ) Dep Elastoplastic consistent tangent matrix (array representation of Dep ) E Youngs modulus Ei Eigenprojection of a symmetric tensor associated with the ith eigenvalue E Three-dimensional Euclidean space; elastic domain E Set of plastically admissible stresses ei Generic base vector; unit eigenvector of a symmetric tensor associated with the ith eigenvalue
29. 29. 10 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS F Deformation gradient F e Elastic deformation gradient F p Plastic deformation gradient fext Global (nite element) external force vector fext (e) External force vector of element e fint Global (nite element) internal force vector fint (e) Internal force vector of element e G Virtual work functional; shear modulus G Discrete (nite element) full gradient operator H Hardening modulus H Generalised hardening modulus I1, I2, I3 Principal invariants of a tensor I Fourth-order identity tensor: Iijkl = ikjl IS Fourth-order symmetric identity tensor: Iijkl = 1 2 (ikjl + iljk) Id Deviatoric projection tensor: Id IS 1 3 I I I Second-order identity tensor IS Array representation of IS i Array representation of I J Jacobian of the deformation map: J det F J2, J3 Stress deviator invariants J Generalised viscoplastic hardening constitutive function K Bulk modulus KT Global tangent stiffness matrix K (e) T Tangent stiffness matrix of element e K Set of kinematically admissible displacements L Velocity gradient Le Elastic velocity gradient Lp Plastic velocity gradient m Unit vector normal to the slip plane of a single crystal N Plastic ow vector N Unit plastic ow vector: N N/ N O The orthogonal group O + The rotation (proper orthogonal) group
30. 30. INTRODUCTION 11 0 Zero tensor; zero array; zero generic entity o Zero vector P First PiolaKirchhoff stress tensor p Generic material point p Cauchy or Kirchhoff hydrostatic pressure Q Generic orthogonal or rotation (proper orthogonal) tensor q von Mises (Cauchy or Kirchhoff) effective stress R Rotation tensor obtained from the polar decomposition of F Re Elastic rotation tensor R Real set r Global nite element residual (out-of-balance) force vector s Entropy s Cauchy or Kirchhoff stress tensor deviator s Unit vector in the slip direction of slip system of a single crystal t Surface traction t Reference surface traction U Right stretch tensor U e Elastic right stretch tensor U p Plastic right stretch tensor U Space of vectors in E u Generic displacement vector eld u Global nite element nodal displacement vector V Left stretch tensor V e Elastic left stretch tensor V p Plastic left stretch tensor V Space of virtual displacements v Generic velocity eld W Spin tensor W e Elastic spin tensor W p Plastic spin tensor x Generic point in space p 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
31. 31. 12 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS Back-stress tensor Plastic multiplier ij Krnecker delta Strain tensor; also Eulerian logarithmic strain when under large strains e Elastic strain tensor; also elastic Eulerian logarithmic strain when under large strains p Plastic strain tensor , e , p Array representation of , e and p , respectively , e , p Axial total, elastic and plastic strain in one-dimensional models p 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 e , p Elastic and plastic axial stretch i, e i , p i Total, elastic and plastic principal stretches One of the Lam constants of linear elasticity p 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 i Principal Cauchy stress y Yield stress (uniaxial yield stress for the conventional von Mises and Tresca models) y0 Initial yield stress Array representation of Kirchhoff stress tensor i Principal Kirchhoff stress
32. 32. INTRODUCTION 13 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 (e) 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, uiei = 3 i=1 uiei. 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.
33. 33. 14 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS 1.3.4. OTHER IMPORTANT SYMBOLS AND OPERATIONS The meanings of other important symbols and operations are listed below. det() Determinant of () dev() Deviator of () divp() Material divergence of () divx() Spatial divergence of () exp() Exponential (including tensor exponential) of () ln() Natural logarithm (including tensor logarithm) of () o() A term that vanishes faster than () sign() The signum function: sign() ()/|()| skew() Skew-symmetric part of () sym() Symmetric part of () tr() Trace of () () Increment of (). Typically, () = ()n+1 ()n () Iterative increment of () () Gradient of () p() Material gradient of () x() Spatial gradient of () s , s p , s x () Corresponding symmetric gradients of () () Boundary of the domain () a() Subdifferential of () with respect to a a () Derivative of () with respect to a () Material time derivative of () ()T The transpose of () a b Means a is dened as b. The symbol is often used to emphasise that the expression in question is a denition. a := b, a := a + b 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. S : T, S : T, S : T Double contraction of tensors (internal product of second-order tensors)
34. 34. INTRODUCTION 15 u v, T u, S T 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. u v Vector product S T, u v Tensor product of tensors or vectors X Y The appropriate product between two generic entities, X and Y, in a given context. |()| Norm (absolute value) of the scalar () ||T ||, ||v|| Euclidean norm of tensors and vectors: ||T || T : T, ||v|| v v A 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. A 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. A 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.
35. 35. 2 ELEMENTS OF TENSOR ANALYSIS THIS 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. Vectors Let 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 u v 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 u = u u, (2.1) and u is said to be a unit vector if u = 1. (2.2) The zero vector, here denoted o, is the element of U that satises o = 0. (2.3) Computational Methods for Plasticity: Theory and Applications EA de Souza Neto, D Peric and DRJ Owen c 2008 John Wiley & Sons, Ltd
36. 36. 18 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS A vector u is said to be orthogonal (or perpendicular) to a vector v if u v = 0. (2.4) 2.1.2. ORTHOGONAL BASES AND CARTESIAN COORDINATE FRAMES A set {ei} {e1, e2, . . . , en} of n mutually orthogonal vectors satisfying ei ej = ij, (2.5) where ij = 1 if i = j 0 if i = j (2.6) is the Krnecker delta, denes an orthonormal basis for U . Any vector u U can be represented as u = u1 e1 + u2 e2 + + un en = ui ei, (2.7) where ui = 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 [u] = u1 u2 ... un . (2.9) 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 [x] = x1 x2 ... xn , (2.10) 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)
37. 37. ELEMENTS OF TENSOR ANALYSIS 19 2.2. Second-order tensors Second-order tensors are linear transformations from U into U, i.e. a second-order tensor T : U U 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 I u = u (2.15) u U. The product of two tensors S and T is the tensor ST dened by ST u = S (T u). (2.16) In general, ST = TS. (2.17) 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, u, v U. (2.18) If T = T T , (2.19) then T is said to be symmetric. If T = T T , (2.20) 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 ) (2.21) of its symmetric part sym(T ) 1 2 (T + T T ) (2.22) and its skew part skew(T ) 1 2 (T T T ). (2.23)
38. 38. 20 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS 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 = T. (iv) If T is symmetric, then skew(T ) = 0, sym(T ) = T. (v) If T is skew, then skew(T ) = T, sym(T ) = 0. 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. (2.24) 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).
39. 39. ELEMENTS OF TENSOR ANALYSIS 21 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 (2.25) 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 : [T ] = T11 T12 T1n T21 T22 T2n ... ... ... ... Tn1 Tn2 Tnn . (2.27) For instance, the Cartesian components of the identity tensor, I, read Iij = ij, (2.28) so that its matrix representation is given by [I ] = 1 0 0 0 1 0 ... ... ... ... 0 0 1 . (2.29) The Cartesian components of the vector v = T u are given by vi = [Tjk(ej ek) ulel] ei = Tij uj. (2.30) Thus, the array [v] of Cartesian components of v is obtained from the matrix-vector product [v ] v1 v2 ... vn = T11 T12 T1n T21 T22 T2n ... ... ... ... Tn1 Tn2 Tnn u1 u2 ... un . (2.31) It can be easily proved that the Cartesian components T T ij of the transpose T T of a tensor T are given by T T ij = Tji. (2.32)
40. 40. 22 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS Thus, T T has the following Cartesian matrix representation [T T ] = T11 T21 Tn1 T12 T22 Tn2 ... ... ... ... 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. (2.34) For a generic tensor, T = Tij ei ej, it then follows that tr T = Tij tr(ei ej) = Tij ij = Tii, (2.35) that is, the trace of T is the sum of the diagonal terms of the Cartesian matrix representa- tion [T ]. The inner product, S : T, between two tensors S and T is dened as S : T = tr(ST T ), (2.36) or, in Cartesian component form, S : T = Sij Tij. (2.37) The Euclidean norm (or simply norm) of a tensor T is dened as T T : T = T 2 11 + T 2 12 + + T2 nn. (2.38) 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.
41. 41. ELEMENTS OF TENSOR ANALYSIS 23 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 (2.39) 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. (2.40) A tensor T is said to be positive denite if T u u > 0, u = o. (2.41) 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 ) 1 . (iii) (ST ) 1 = T 1 S1 . (iv) (T 1 ) T = (T T ) 1 . 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 Q1Q2 of any two orthogonal tensors Q1 and Q2 is an orthogonal tensor. If Q1 and Q2 are rotations, then the product Q1Q2 is also a rotation. For all vectors u and v, an orthogonal tensor Q satises Qu Qv = u v. (2.44)
42. 42. 24 COMPUTATIONAL METHODS FOR PLASTICITY: THEORY AND APPLICATIONS Rotations and changes of basis Let {ei} and {e i } be two orthonormal bases of U. Such bases are related by e i = R ei, 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 i } are given by the following products of matrices: [T ] = [R]T [T ] [R ]; [u ] = [R]T [u ]. (2.46) Equivalently, in component form, we have T ij = Rki Tkl Rlj, u i = Rji uj. (2.47) The matrix [R] is given by [R] = e1 e 1 e1 e 2 e1 e n e2 e 1 e2 e 2 e2 e n ... ... ... ... en e 1 en e 2 en e n , (2.48) or, in component form, Rij = ei e j . (2.49) 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 (or vector product) between two vectors u and v as the vector w = u v, (2.51) whose components are given by wi = ijk uj vk, (2.52) where ijk denotes the alternating tensor ijk = +1 if {i, j, k} is an even permutation of {1, 2, 3} 1 if {i, j, k} is an odd permutation of {1, 2, 3} 0 if at least two indices coincide. (2.53)
43. 43. ELEMENTS OF TENSOR ANALYSIS 25 Basic relations with the cross product Some useful relations involving the cross product are listed in the following: (i) u v = v u. (ii) (u v) w = (v w) u = (w u) v. (iii) u u = o. (iv) For any tensor T and a set {u, v, w} of linearly independent vectors, det T = (T u T v) T w (u v) w . (2.54) (v) For any skew tensor W , there is a unique vector w, called the axial vector of W , such that W u = w u. (2.55) In terms of the Cartesian components of W , the axial vector is expressed as w = W32 e1 + W13 e2 + W21 e3. (2.56) The matrix representation of W , reads [W ] = 0 w3 w2 w3 0 w1 w2 w1 0 , (2.57) where {wi} are the components of w. 2.2.8. SPECTRAL DECOMPOSITION Given a tensor T, a non-zero vector u is said to be an eigenvector of T associated with the eigenvalue (or principal value) if T u = u. (2.58) The space of all vectors u satisfying the above equation is called the characteristic space of T correspondin