This page intentionally left blank Non-linear Modeling and Analysis of Solids and Structures Steen Krenk CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521830546 © Cambridge University Press 2009 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2009 ISBN-13 978-0-511-60413-3 eBook (EBL) ISBN-13 978-0-521-83054-6 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate To Jette Contents Preface 1.1 page ix 1.3 1.4 Introduction A simple non-linear problem 1.1.1 Equilibrium 1.1.2 Virtual work and potential energy Simple non-linear solution methods 1.2.1 Explicit incremental method 1.2.2 Newton–Raphson method 1.2.3 Modified Newton–Raphson method Summary and outlook Exercises 13 14 15 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Non-linear bar elements Deformation and strain Equilibrium and virtual work Tangent stiffness matrix Use of shape functions Assembly of global stiffness and forces Total or updated Lagrangian formulation Summing up the principles Exercises 17 18 20 24 26 31 36 39 43 3.1 3.2 3.3 3.4 3.5 Finite rotations The rotation tensor Rotation of a vector into a specified direction The increment of the rotation variation Parameter representation of an incremental rotation Quaternion parameter representation 3.5.1 Representation of the rotation tensor 47 49 53 55 60 63 64 1.2 v vi 3.6 3.7 3.8 4.1 4.2 4.3 4.4 4.5 4.6 5.1 5.2 5.3 5.4 6.1 6.2 Contents 3.5.2 Addition of two rotations 3.5.3 Incremental rotation from quaternion parameters 3.5.4 Mean and difference of two rotations Alternative representation of the rotation tensor Summary of rotations and their virtual work Exercises 65 67 68 69 72 73 Finite rotation beam theory Equilibrium equations Virtual work, strain and curvature Increment of the virtual work equation 4.3.1 Constitutive stiffness 4.3.2 Geometric stiffness 4.3.3 The load increments Finite element implementation 4.4.1 Element stiffness matrix 4.4.2 Loads and internal forces 4.4.3 Shear locking Summary of ‘elastica’ beam theory Exercises 76 77 78 81 82 83 85 86 87 89 91 98 99 Co-rotating beam elements Co-rotating beams in two dimensions 5.1.1 Co-rotation form of the tangent stiffness 5.1.2 Element deformation stiffness 5.1.3 Total tangent stiffness 5.1.4 Finite element implementation Co-rotating beams in three dimensions 5.2.1 Co-rotation form of the tangent stiffness 5.2.2 Element deformation stiffness 5.2.3 Total tangent stiffness 5.2.4 Finite element implementation Summary and extensions Exercises 100 101 104 107 110 112 117 120 127 130 133 139 141 Deformation and equilibrium of solids Deformation and strain 6.1.1 Non-linear strain 6.1.2 Decomposition into deformation and rigid body motion Virtual work and stresses 6.2.1 Piola–Kirchhoff stress 6.2.2 Cauchy and Kirchhoff stresses 145 146 148 151 154 155 158 Contents 6.3 6.4 6.5 6.6 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 8.1 vii 6.2.3 Stress rates Total Lagrangian formulation 6.3.1 Equilibrium and residual forces 6.3.2 Tangent stiffness 6.3.3 Finite element implementation Updated Lagrangian formulation 6.4.1 Transformation from total to updated format 6.4.2 Virtual work in the current configuration 6.4.3 Finite element implementation Summary of non-linear motion of solids Exercises 160 165 166 167 170 174 174 176 180 185 186 Elasto-plastic solids Elastic solids 7.1.1 Stress invariants 7.1.2 Strain invariants and small strain elasticity 7.1.3 Isotropic elasticity at finite strain General plasticity theory 7.2.1 Reversible deformation 7.2.2 Maximum plastic dissipation rate 7.2.3 Evolution equations 7.2.4 Isotropic and kinematic hardening Von Mises plasticity models 7.3.1 Yield surface and flow potential 7.3.2 Explicit integration 7.3.3 Radial return algorithm General aspects of plasticity models 7.4.1 Combined isotropic and kinematic hardening 7.4.2 Internal variables and non-associated flow 7.4.3 General computational procedure Models for granular materials 7.5.1 Flow potential and yield surface 7.5.2 Elasticity and hardening Finite strain plasticity Summary Exercises 189 190 192 198 200 203 204 207 212 216 218 219 222 225 229 230 234 237 241 242 247 249 252 253 Numerical solution techniques Iterative solution of equilibrium equations 8.1.1 Non-linear iteration strategies 256 257 259 viii 8.2 8.3 8.4 8.5 8.6 9.1 9.2 9.3 9.4 9.5 9.6 Contents 8.1.2 Direction and step-size control Orthogonal residual method Arc-length methods 8.3.1 General constraint formulation 8.3.2 Hyperplane constraints 8.3.3 Hypersphere constraint Quasi-Newton methods Summary Exercises 260 263 270 272 274 278 283 287 288 Dynamic effects and time integration Newmark algorithm for linear systems 9.1.1 Energy balance and stability 9.1.2 Numerical accuracy and damping Non-linear Newmark algorithm Energy-conserving integration 9.3.1 State-space formulation 9.3.2 Non-linear kinematics for Green strain 9.3.3 Energy-conserving algorithm Algorithmic energy dissipation 9.4.1 Spectral analysis of linear systems 9.4.2 Linear algorithm with energy dissipation 9.4.3 Non-linear algorithm with energy dissipation Summary and outlook Exercises 290 292 295 300 304 309 310 311 315 323 323 325 327 331 333 References Index 336 345 ... reader on a concentrated tour of some of the central issues of non- linear modeling and analysis of structures and solids Traditionally, the non- linear theories of solids have been treated in books... the analysis of nonlinear problems, such as geometrical and material non- linear behavior of solids and structures The solution of non- linear problems by the finite element method involves modeling, ... Non- linear Modeling and Analysis of Solids and Structures Steen Krenk CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge