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FINITE ELEMENTS IN
PLASTICITY:
Theory and Practice
D. R. J. OWEN
E. HINTON
Department of Civil Engineering
University College of Swansea, U.K.
First published 1980 by
Pineridge Press Limited
91 West Cross Lane, West Cross, Swansea U.K.
ISBN 0-906674-05-2
Copyright © 1980 by
Pineridge Press Limited.
OWEN, D. R. J.
Finite Elements in Plasticity
-
Elasticity
-
Plasticity
-
Finite Element Method
I. Title II. HINTON, E.
620.1'123 TA418
ISBN 0-906674-05-2
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, without the prior written permission of the publishers.
Contents
PART I
1 Introduction 3
1.1 Introductory remarks 3
1.2 Aims and layout 3
1.3 Program structure 8
1.4 References 11
2 One-dimensional nonlinear problems 13
2.1 Introduction 13
2.2 Basic numerical solution processes for nonlinear problems 13
2.3 Systems governed by a quasi-harmonic equation 22
2.4 Nonlinear elastic problems 25
2.5 Elasto-plastic problems in one dimension 26
2.6 Problems 29
2.7 References 31
3 Structure of computer programs for the solution of nonlinear problems 33
3.1 Introduction 33
3.2 Input data subroutine, DATA 35
3.3 Subroutine NONAL 40
3.4 Subroutines for equation assembly and solution 42
3.5 Output of results 58
3.6 Subroutine INITIAL 59
3.7 Load increment subroutine, INCLOD 60
3.8 The master or controlling segment 61
3.9 Program for the solution of one-dimensional quasi-harmonic problems by direct iteration 63
3.10 Program for the solution of one-dimensional quasi-harmonic problems by the Newton–Raphson method 68
CONTENTS
3.11 Program for the solution of nonlinear elastic problems 74
3.12 Program for the solution of elasto-plastic problems 78
3.13 Problems 90
3.14 References 94
4 Viscoplastic problems in one dimension 95
4.1 Introduction 95
4.2 Basic theory 95
4.3 Numerical solution process 99
4.4 Limiting time-step length 102
4.5 Computational procedure 103
4.6 Program structure 104
4.7 Element stiffness subroutine STUNVP 106
4.8 Subroutine INCVP for the evaluation of end of time-step quantities and equilibrium correction terms 107
4.9 Convergence monitoring subroutine, CONVP 109
4.10 Subroutine INCLOD 110
4.11 The main, master or controlling segment 111
4.12 Numerical examples 113
4.13 Problems 117
4.14 References 119
5 Elasto-plastic Timoshenko beam analysis 121
5.1 Introduction 121
5.2 The basic assumptions of Timoshenko beam theory 122
5.3 Finite element idealisation for linear elastic Timoshenko beams 125
5.4 Elasto-plastic nonlayered Timoshenko beams 129
5.5 Elasto-plastic layered Timoshenko beams 141
5.6 Problems 148
5.7 References 152
PART II
6 Preliminary theory and standard subroutines for two-dimensional elasto-plastic applications 157
6.1 Introduction 157
6.2 Virtual work expression for various solid mechanics applications 162
6.3 Isoparametric finite element representation 169
FINITE ELEMENTS IN PLASTICITY
6.4 Standard subroutines for linear elastic finite element analysis 174
6.5 Standard subroutines for elasto-plastic finite element analysis 205
6.6 Problems 214
6.7 References 214
7 Elasto-plastic problems in two dimensions 215
7.1 Introduction 215
7.2 The mathematical theory of plasticity 215
7.3 Matrix formulation 227
7.4 Alternative form of the yield criteria for numerical computation 229
7.5 Basic expressions for two-dimensional problems 232
7.6 Singular points on the yield surface 234
7.7 Finite element expressions and program structure 235
7.8 Additional program subroutines 237
7.9 Numerical examples 262
7.10 Problems 265
7.11 References 268
8 Elasto-viscoplastic problems in two dimensions 271
8.1 Introduction 271
8.2 Theory of elasto-viscoplastic solids 272
8.3 Selection of the time-step length 276
8.4 Computational procedure 278
8.5 Evaluation of matrix H 279
8.6 Program structure 281
8.7 Formulation of the tangential stiffness matrix 283
8.8 Subroutine STEPVP for the evaluation of end of time step quantities and equilibrium correction terms 289
8.9 Subroutine FLOWVP 294
8.10 Subroutine STRESS 295
8.11 Subroutine ZERO 297
8.12 Subroutine STEADY for monitoring steady state convergence 297
8.13 The main, master or controlling segment 299
8.14 General comparison of implicit and explicit time integration schemes 302
8.15 The overlay method for improved material response 304
8.16 Numerical examples 310
8.17 Problems 315
8.18 References 317
9 Elasto-plastic Mindlin plate bending analysis 319
9.1 Introduction 319
9.2 Equilibrium equations 321
9.3 Discretisation 324
9.4 Solution of nonlinear equations 326
9.5 Software for the nonlayered approach 331
9.6 Software for the layered approach 355
9.7 Examples 370
9.8 Problems 372
9.9 References 373
PART III
10 Explicit transient dynamic analysis 377
10.1 Introduction 377
10.2 Dynamic equilibrium equations 378
10.3 Modelling of nonlinearities 381
10.4 Explicit time integration scheme 388
10.5 Critical time step 391
10.6 Program DYNPAK 392
10.7 Examples 420
10.8 Problems 428
10.9 References 429
11 Implicit-explicit transient dynamic analysis 431
11.1 Introduction 431
11.2 Implicit time integration 432
11.3 Implicit-explicit algorithm 434
11.4 Evaluation of the tangential stiffness matrix 439
11.5 Program MIXDYN 440
11.6 Examples 458
11.7 Problems 462
11.8 References 462
12 Alternative formulations and further applications 465
12.1 Introduction 465
12.2 List of subroutines 466
12.3 Alternative material models 476
12.4 Further applications 480
12.5 Equation solving techniques 490
12.6 Other enhancements in elasto-plastic analysis 493
12.7 Concluding remarks 495
12.8. References 496
FINITE ELEMENTS IN PLASTICITY
ix
Appendix I Instructions for preparing input data for one-dimensional problems 503
Appendix II Instructions for preparing input data for plane, axisymmetric and plate bending problems 511
Appendix III Instructions for preparing input data for dynamic transient problems 521
Appendix IV Sample input data and line printer output for one- and two-dimensional applications 529
Author Index 585
Subject Index 589
Preface
The purpose of this text is to present and demonstrate the use of finite element based methods for the solution of problems involving plasticity. As well as the conventional quasi-static incremental theory of plasticity, attention is given to the slow transient phenomenon of elasto-viscoplastic behaviour and also to dynamic transient problems. We make no pretence that the text provides a complete treatment of any of these topics but rather we see it as an attempt to present numerical solution techniques, which have been well tried and tested, for selected important areas of application.
In our earlier books on finite elements we have concentrated on linear applications. Here we attempt the much more daunting task of introducing, in detail, the use of finite elements for solving problems in which plasticity effects are present. To our knowledge it is the first such book. Our main idea is to present the theory and detailed algorithms in the form of modular routines written in FORTRAN which can be linked together to form 13 finite element plasticity programs.
Writing this book has been in itself, rather like solving a nonlinear finite element problem. We have gone through many iterations and we hope that we have now converged to a reasonable ‘solution’. As in many real engineering situations our convergence criterion has been influenced by a deadline. In our case the deadline was largely self-imposed as we have already been engaged on this project for more than three years. We do not believe our solution to be unique or in any sense optimal. We merely offer it to fill a gap in the existing literature.
The text is arranged in three main parts. Part I is devoted to one-dimensional problems. These relatively simple applications are possibly the most important in the book; since all the essential features of nonlinear finite element analysis are immediately recognisable without the distractions and complications that are present in general continuum problems. Part II deals with the two-dimensional applications of plane stress/strain and axisymmetric continua and plate bending problems. Finally in Part III we present some dynamic transient applications and briefly describe some further developments.
All of the programs presented in this text have been specially written by the authors. In the development of the subroutines for the solution algorithms described, a conflict inevitably arose between computational efficiency
and clarity of coding. Whatever sacrifices have been made have been biased towards satisfying the latter condition. However, we believe that the codes presented are both reasonably efficient and flexible and have potential usage in commercial as well as teaching and research environments. A total of 132 subroutines are presented which amount to more than 8,000 statements. The 13 assembled programs comprise approximately 20,000 statements. To aid readers wishing to implement the programs a magnetic tape of the computer codes together with the test input data listed in Appendix IV is available from the publishers. Although every attempt has been made to verify the programs, no responsibility can be accepted for their performance in practice.
A further feature of the book is that each chapter contains several exercises for further study.
We are indebted to many people for their direct or indirect assistance in the preparation of this text. This preface would not be complete without an acknowledgment of this debt and a record of our gratitude to the following: To Professor O. C. Zienkiewicz for his pioneering work and stimulating influence. To Professor G. C. Nayak whose work on numerical analysis of plasticity problems has significantly influenced the present text. To Dr. I. C. Cormeau whose thesis on viscoplasticity has been an invaluable source of information. To Professor K. J. Bathe for permission to use the profile equation solver included in Chapter 11. To N. Bicanic, D. K. Paul, H. H. Abdel Rahman and M. M. Huq for their generous assistance in the preparation of several chapters. To our colleagues and former research workers in the Department of Civil Engineering, University College of Swansea for helpful discussions and suggestions. To E. S. Caldis for his care in preparing annotated computer listings and, finally, to Mrs. M. J. Davies for her skill and patience in typing the manuscript.
D. R. J. OWEN E. HINTON
Swansea, May 1980