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Non-linear Finite Element Analysis of Solids and Structures ~ ~ ~~ ~~ VOLUME 1: ESSENTIALS Non-linear Finite Element Analysis of Solids and Structures VOLUME 1: ESSENTIALS M. A. Crisfield FEA Professor of Computational Mechanics Department of Aeronautics Imperial College of Science, Technology and Medicine London, UK JOHN WILEY & SONS . - - . Chichester New York Brisbane Toronto Singapore Copyright $3 1991 by John Wiley & Sons Ltd. Bafins Lane, Chichester West Sussex PO19 IUD, England Reprinted April 2000 All rights reserved. No part of this book may be reproduced by any means. or transmitted, or translated into a machine language without the written permission of the publisher. Other Wiley Editorial Offices John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, USA Jacaranda Wiley Ltd, G.P.O. Box 859, Brisbane, Queensland 4001, Australia John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario M9W 1 LI, Canada John Wiley & Sons (SEA) Pte Ltd, 37 Jalan Pemimpin 05-04, Block B, Union Industrial Building, Singapore 2057 Library of Congress Cataloging-in-Publication Data: Crisfield, M. A. Non-linear finite element analysis of solids and structures / M. A. Crisfield. p. cm. Includes bibliographical references and index. Contents: v. 1. Essentials. ISBN 0 471 92956 5 (v. I); 0 471 92996 4 (disk) 1. Structural analysis (Engineering)-Data processing. 2. Finite element method-Data processing. I. Title. TA647.C75 199 1 624.1 '7 1 -d c20 90-278 15 CI P A catalogue record for this book is available from the British Library Typeset by Thomson Press (India) Ltd., New Delhi, India Printed in Great Britain by Courier International, East Killbride Contents Preface xi Notation xiii 1 General introduction, brief history and introduction to geometric non-linearity 1 1 1 General introduction and a brief history 1 1 1 1 A brief history 1 12 A simple example for geometric non-linearity with one degree of freedom 2 1 2 1 An incremental solution 6 1 2 2 An iterative solution (the Newton-Raphson method) a 1 2 3 Combined tncremental/iterative solutions (full or modified Newton-Raphson or the initial-stress method) 10 13 A simple example with two variables 13 1 3 1 ‘Exact solutions 16 132 The use of virtual work 18 133 An energy basis 19 14 Special notation 19 15 List of books on (or related to) non-linear finite elements 20 16 References to early work on non-linear finite elements 20 2 A shallow truss element with Fortran computer program 23 2 1 A shallow truss element 23 22 A set of Fortran subroutines 26 2 2 1 Subroutine ELEMENT 27 2 2 2 Subroutine INPUT 29 2 2 3 Subroutine FORCE 30 2 2 4 Subroutine ELSTRUC 31 2 2 5 Subroutine BCON and details on displacement control 32 2 2 6 Subroutine CROUT 34 2 2 7 Subroutine SOLVCR 35 2 3 A flowchart and computer program for an incremental (Euler) solution 36 2 3 1 Program NONLTA 37 24 A flowchart and computer program for an iterative solution using the Newton-Raphson method 39 24 1 Program NONLTB 39 2 4 2 Flowchart and computer listing for subroutine ITER 41 2 5 A flowchart and computer program for an incrementaViterative solution procedure using full or modified Newton-Raphson iterations 44 2 5 1 Program NONLTC 45 V vi CONTENTS 2 6 Problems for analysis 48 2 6 1 Single variable with spring 48 26 1 1 Incremental solution using program NONLTA 49 2 6 1 2 Iterative solution using program NONLTB 49 2 6 1 3 Incremental/iterative solution using program NONLTC 49 262 Single variable no spring 49 2 6 3 Perfect buckling with two variables 50 2 6 4 Imperfect 'buckling with two variades 51 2 6 4 1 Pure incremental solution using program NONLTA 51 2 6 4 2 An incremental/\terative solution using program NONLTC with small increments 52 2 6 4 3 An incremental/iterative solution using program NONLTC with large increments 54 2 6 4 4 An incremental/iterative solution using program NONLTC with displacement control 55 27 Special notation 56 2 8 References 56 3 Truss elements and solutions for different strain measures 57 3.1 A simple example with one degree of freedom 57 3.1.1 A rotated engineering strain 58 3.1.2 Green's strain 59 3.1.3 A rotated log-strain 59 3.1.4 A rotated log-strain formulation allowing for volume change 60 3.1.5 Comparing the solutions 61 3.2 Solutions for a bar under uniaxial tension or compression 62 3.2.1 Almansi's strain 63 3.3 A truss element based on Green's strain 65 3.3.1 Geometry and the strain-displacement relationships 65 3.3.2 Equilibrium and the internal force vector 68 3.3.3 The tangent stiffness matrix 69 3.3.4 Using shape functions 70 3.3.5 Alternative expressions involving updated coordinates 72 3.3.6 An updated Lagrangian formulation 73 3.4 An alternative formulation using a rotated engineering strain 75 3.5 An alternative formulation using a rotated log-strain 76 3.6 An alternative corotational formulation using engineering strain 77 3.7 Space truss elements 80 3.8 Mid-point incremental strain updates 82 3.9 Fortran subroutines for general truss elements 85 3.9.1 Subroutine ELEMENT a5 3.9.2 Subroutine INPUT a7 3.9.3 Subroutine FORCE 88 3.10 Problems for analysis 90 3.10.1 Bar under uniaxial load (large strain) 90 3.10.2 Rotating bar 90 3.10.2.1 Deep truss (large-strains) (Example 2.1) 90 3.10.2.2 Shallow truss (small-strains) (Example 2.2) 91 3.10.3 Hardening problem with one variable (Example 3) 93 3.10.4 Bifurcation problem (Example 4) 94 3.10.5 Limit point with two variables (Example 5) 96 3.10.6 Hardening solution with two variables (Example 6) 98 3.10.7 Snap-back (Example 7) 100 CONTENTS vii 3.1 1 Special notation 102 3.12 References 103 4 Basic continuum mechanics 104 4.1 Stress and strain 105 4.2 St ress-st ra i n relationsh i ps 107 42 1 Plane strain axial symmetry and plane stress 107 4 2 2 Decomposition into vo,umetric and deviatoric components 108 4 2 3 An alternative expression using the Lame constants 109 4.3 Transformations and rotations 110 43 1 Transformations to a new set of axes 110 4 3 2 A rigid-body rotation 113 4.4 Green’s strain 116 4 4 1 Virtual work expressions using Green s strain 118 4 4 2 Work expressions using von Karman s non-linear strain-displacement relqtionships for a plate 119 4.5 Almansi’s strain 120 4.6 The true or Cauchy stress 121 4.7 Summarising the different stress and strain measures 124 4.8 The polar-decomposition theorem 126 4 8 1 Ari example 129 4.9 Green and Almansi strains in terms of the principal stretches 130 4.10 A simple description of the second Piola-Kirchhoff stress 131 4.1 1 Corotational stresses and strains 131 4.12 More on constitutive laws 132 4.13 Special notation 134 4.1 4 References 135 5 Basic finite element analysis of continua 136 5 1 Introduction and the total Lagrangian formulation 136 5 1 1 Element formulation 137 5 1 2 The tangent stiffness matrix 139 5 1 3 Extension to three dimensions 140 5 1 4 An axisymmetric membrane 142 5 2 Implenientation of the total Lagrangian method 144 5 2 1 With dn elasto-plastic or hypoelastic material 144 5 3 The updated Lagrangian formulation 146 5 4 Implementation of the updated Lagrangian method 147 5 4 1 Incremental formulation involving updating after convergence 147 5 4 2 A total formulation for an elastic response 140 5 4 3 An approximate incremental formulation 149 55 Special notation 150 5 6 References 151 6 Basic plasticity 152 6 1 Introduction 152 6 2 Stress updating incremental or iterative strains? 154 6 3 The standard elasto-plastic modular matrix for an elastic/perfectly plastic von Mises material under plane stress 156 6 3 1 Non-associative plasticity 158 6 4 Introducing hardening 159 viii CONTENTS 6 4 1 Isotropic strain hardening 159 6 4 2 Isotropic work hardening 160 6 4 3 Kinematic hardening 161 65 Von Mises plasticity in three dimensions 162 6 5 1 Splitting the update into volumetric and deviatoric parts 164 6 5 2 Using tensor notation 165 66 Integrating the rate equations 166 661 Crossing the yield surface 168 6 6 2 Two alternative predictors 170 6 6 3 Returning to the yield surface 171 6 6 4 Sub-incrementation 172 6 6 5 Generalised trapezoidal or mid-point algorithms 173 6 6 6 A backward-Euler return 176 6 6 7 The radial return algorithm a special form of backward-Euler procedure 177 67 The consistent tangent modular matrix 178 6 7 1 Splitting the deviatoric from the volumetric components 178 6 7 2 A combined formulation 180 6 8 Special two-dimensional situations 181 6 8 1 Plane strain and axial symmetry 181 682 Plane stress 181 6 8 2 1 A consistent tangent modular matrix for plane stress 184 6 9 Numerical examples 185 6 9 1 Intersection point 185 6 9 2 A forward-Euler integration 185 6 9 3 Sub-increments 188 6 9 4 Correction or return to the yield surface 189 6 9 5 Backward-Euler return 189 6 9 5 1 General method 189 6 9 5 2 Specific plane-stress method 190 6 9 6 Consistent and inconsistent tangents 191 6 9 6 1 Solution using the general method 191 6 9 6 2 Solution using the specific plane-stress method 192 6 10 Plasticity and mathematical programming 193 6 10 1 A backward-Euler or implicit formulation 195 6 11 Special notation 196 6 12 References 197 7 Two-dimensional formulations for beams and rods 201 7 1 A shallow-arch formulation 20 1 7 1 1 The tangent stiffness matrix 205 7 1 2 Introduction of material non-linearity or eccentricity 205 7 13 Numerical integration and specific shape functions 206 7 1 4 Introducing shear deformation 208 7 15 Specific shape fur,ctions, order of integration and shear-locking 210 7 2 A simple corotational element using Kirchhoff theory 21 1 7 2 1 Stretching 'stresses and 'strains 21 3 7 2 2 Bending 'stresses' and 'strains 21 3 7 2 3 The virtual local displacements 214 7 2 4 The virtual work 21 5 7 2 5 The tangent stiffness matrix 216 7 2 6 llsing shape functions 21 7 7 2 7 Including higher-order axial terms 21 7 7 2 8 Some observations 21 9 73 A simple corotational element using Timoshenko beam theory 21 9 74 An alternative element using Reissner's beam theory 22 1 CONTENTS ix 7 4 1 The introduction of shape functions and extension to a general isoparametric element 223 7.5 An isoparametric degenerate-continuum approach using the total Lagrangian formulation 225 7.6 Special notation 229 7.7 References 23 1 8 Shells 234 8 1 A range of shallow shells 236 8 1 1 Strain-displacement relationships 236 8 1 2 Stress-strain relatiomhips 238 8 1 3 Shape functions 239 8 1 4 Virtual work and the internal force vector 240 8 1 5 The tangent stiffness matrix 24 1 8 1 6 Numerical integration matching shape functions and 'locking 242 8 1 7 Extensions to the shallow-shell formulation 242 8 2 A degenerate-continuum element using a total Lagrangian formulation 243 82 1 The tangent stiffness matrix 246 83 Special notation 247 8 4 References 249 9 More advanced solution procedures 252 9 1 The total potential energy 253 92 Line searches 254 921 Theory 254 9 2 2 Flowchart and Fortran subroutine to find the new step length 258 9 2 2 1 Fortran subroutine SEARCH 259 9 2 3 Implementation within a finite element computer program 261 9231 Input 26 1 9 2 3 2 Changes to the iterative subroutine ITER 263 9 2 3 3 Flowchart for Iine-search loop at the structural level 264 93 The arc-length and related methods 266 931 The need for arc-length or similar techniques and examples of their use 266 9 3 2 Various forms of generalised displacement control 271 9 3 2 1 The 'spherical arc-length method 273 9 3 2 2 Linearised arc-length methods 274 9 3 2 3 Generalised displacement control at a specific variable 275 9 4 Detailed formulation for ttre 'cylindrical arc-length' method 276 9 4 1 Flowchart and Fortran subroutines for the application of the arc-length constraint 276 94 1 1 Fortran subroutines ARCLl and QSOLV 278 9 4 2 Flowchart and Fortran subroutine for the main structural iterative loop (ITER) 280 9 4 2 1 Fortran subroutine ITER 282 943 The predictor solution 285 9 5 Automatic increments, non-proportional loading and convegence criteria 286 9 5 1 Automatic increment cutting 288 9 5 2 The current stiffness parameter and automatic switching to the arc-length method 288 9 5 3 Non-proportional loading 289 9 5 4 Convergence criteria 289 9 5 5 Restart facilities and the computation of the lowest eigenmode of K, 290 96 The updated computer prcgram 29 1 9 6 1 rcjrtran subroutine LSLOOP 292 9 6 2 Input for incremental/iterative control 294 9 6 2 1 Subroutine INPUT2 296 9 6 3 flowchart and Fortran subroutine for the main program module NONLTD 298 963 1 Fortran for main program module NONLTD 299 X CONTENTS 9 6 4 Flowchart and Fortran subroutine. for routine SCALUP 303 9 6 4 1 Fortran for routine SCALUP 303 965 Flowchart and Fortran for subroutine NEXINC 305 9 6 5 1 Fortran for subroutine NEXINC 305 9 7 Quasi-Newton methods 307 9 8 Secant-related acceleration tecriniques 31 0 98 1 Cut-outS 31 1 9 8 2 Flowchart and Fortran for subroutine ACCEL 31 2 9 8 2 1 Fortran for subroutine ACCEL 313 99 Problems for analysts 31 4 99 1 The problems 314 9 9 2 Small-strain limit-point cxample with one variable (Example 2 2) 31 4 9 9 3 Hardening problem with one variable (Example 3) 316 9 9 4 Bifurcation problem (Example 4) 31 7 9 9 5 Limit point with two variables (Example 5) 31 9 9 9 6 Hardening solution with two variable (Example 6) 322 9 9 7 Snap-back (Example 7) 323 9 10 Further work on solution procedures 324 9 11 Special notation 326 9 12 References 327 Appendix Lobatto rules for numerical integration 334 Subject index 336 Author index 341