8.5 SOLVER DETAILS 200

8.6 CONSTITUTIVE EQUATION SUMMARY 201

8.7 PROGRAM STRUCTURE 206

8.8 MAIN ROUTINE flagshyp 206

8.9 ROUTINE elemtk 214

8.10 ROUTINE ksigma 220

8.11 ROUTINE bpress 221

8.12 EXAMPLES 223

8.12.1 Simple Patch Test 223

8.12.2 Nonlinear Truss 224

8.12.3 Strip With a Hole 225

8.12.4 Plane Strain Nearly Incompressible Strip 225

8.13 APPENDIX : Dictionary of Main Variables 227

APPENDIX LARGE INELASTIC DEFORMATIONS 231

A.1 INTRODUCTION 231

A.2 THE MULTIPLICATIVE DECOMPOSITION 232

xii

A.3 PRINCIPAL DIRECTIONS 234

A.4 INCREMENTAL KINEMATICS 236

A.5 VON MISES PLASTICITY 238

A.5.1 Stress Evaluation 238

A.5.2 The Radial Return Mapping 239

A.5.3 Tangent Modulus 240

243

Bibliography

245

Index

PREFACE

A fundamental aspect of engineering is the desire to design artifacts that

exploit materials to a maximum in terms of performance under working con-

ditions and e¬ciency of manufacture. Such an activity demands an increas-

ing understanding of the behavior of the artifact in its working environment

together with an understanding of the mechanical processes occuring during

manufacture.

To be able to achieve these goals it is likely that the engineer will need

to consider the nonlinear characteristics associated possibly with the manu-

facturing process but certainly with the response to working load. Currently

analysis is most likely to involve a computer simulation of the behavior. Be-

cause of the availability of commercial ¬nite element computer software, the

opportunity for such nonlinear analysis is becoming increasingly realized.

Such a situation has an immediate educational implication because, for

computer programs to be used sensibly and for the results to be interpreted

wisely, it is essential that the users have some familiarity with the funda-

mentals of nonlinear continuum mechanics, nonlinear ¬nite element formu-

lations, and the solution techniques employed by the software. This book

seeks to address this problem by providing a uni¬ed introduction to these

three topics.

The style and content of the book obviously re¬‚ect the attributes and

abilities of the authors. Both authors have lectured on this material for a

number of years to postgraduate classes, and the book has emerged from

these courses. We hope that our complementary approaches to the topic

will be in tune with the variety of backgrounds expected of our readers

and, ultimately, that the book will provide a measure of enjoyment brought

about by a greater understanding of what we regard as a fascinating subject.

xiii

xiv PREFACE

Although the content has been jointly written, the ¬rst author was the

primary contributor to Chapter 5, on hyperelasticity, and the Appendix, on

inelastic deformation. In addition, he had the stamina to write the computer

program.

READERSHIP

This book is most suited to a postgraduate level of study by those either

in higher education or in industry who have graduated with an engineering

or applied mathematics degree. However the material is equally applica-

ble to ¬rst-degree students in the ¬nal year of an applied maths course or

an engineering course containing some additional emphasis on maths and

numerical analysis. A familiarity with statics and elementary stress anal-

ysis is assumed, as is some exposure to the principles of the ¬nite element

method. However, a primary objective of the book is that it be reason-

ably self-contained, particularly with respect to the nonlinear continuum

mechanics chapters, which comprise a large portion of the content.

When dealing with such a complex set of topics it is unreasonable to

expect all readers to become familiar with all aspects of the text. If the

reader is prepared not to get too hung up on details, it is possible to use the

book to obtain a reasonable overview of the subject. Such an approach may

be suitable for someone starting to use a nonlinear computer program. Al-

ternatively, the requirements of a research project may necessitate a deeper

understanding of the concepts discussed. To assist in this latter endeavour

the book includes a computer program for the nonlinear ¬nite deformation

¬nite element analysis of two- and three-dimensional solids. Such a pro-

gram provides the basis for a contemporary approach to ¬nite deformation

elastoplastic analysis.

LAYOUT

Chapter 1 “ Introduction

Here the nature of nonlinear computational mechanics is discussed followed

by a series of very simple examples that demonstrate various aspects of

nonlinear structural behavior. These examples are intended, to an extent,

to upset the reader™s preconceived ideas inherited from an overexposure to

linear analysis and, we hope, provide a motivation for reading the rest of

xv

PREFACE

the book! Nonlinear strain measures are introduced and illustrated using

a simple one-degree-of-freedom truss analysis. The concepts of lineariza-

tion and the directional derivative are of su¬cient importance to merit a

gentle introduction in this chapter. Linearization naturally leads on to the

Newton“Raphson iterative solution, which is the fundamental way of solv-

ing the nonlinear equilibrium equations occurring in ¬nite element analysis.

Consequently, by way of an example, the simple truss is solved and a short

FORTRAN program is presented that, in essence, is the prototype for the

main ¬nite element program discussed later in the book.

Chapter 2 “ Mathematical Preliminaries

Vector and tensor manipulations occur throughout the text, and these are

introduced in this chapter. Although vector algebra is a well-known topic,

tensor algebra is less familiar, certainly, to many approaching the subject

with an engineering educational background. Consequently, tensor algebra

is considered in enough detail to cover the needs of the subsequent chapters,

and in particular, it is hoped that readers will understand the physical inter-

pretation of a second-order tensor. Crucial to the development of the ¬nite

element solution scheme is the concept of linearization and the directional

derivative. The introduction provided in Chapter 1 is now thoroughly de-

veloped. Finally, for completeness, some standard analysis topics are brie¬‚y

presented.

Chapter 3 “ Kinematics

This chapter deals with the kinematics of ¬nite deformation, that is, the

study of motion without reference to the cause. Central to this concept is

the deformation gradient tensor, which describes the relationship between

elemental vectors de¬ning neighboring particles in the undeformed and de-

formed con¬gurations of the body whose motion is under consideration.

The deformation gradient permeates most of the development of ¬nite de-

formation kinematics because, among other things, it enables a variety of

de¬nitions of strain to be established. Material (initial) and spatial (cur-

rent) descriptions of various items are discussed, as is the linearization of

kinematic quantities. Although dynamics is not the subject of this text, it

is nevertheless necessary to consider velocity and the rate of deformation.

The chapter concludes with a brief discussion of rigid body motion and

objectivity.

xvi PREFACE

Chapter 4 “ Stress and Equilibrium

The de¬nition of the true or Cauchy stress is followed by the development

of standard di¬erential equilibrium equations. As a prelude to the ¬nite

element development the equilibrium equations are recast in the weak inte-

gral virtual work form. Although initially in the spatial or current deformed

con¬guration, these equations are reformulated in terms of the material or

undeformed con¬guration, and as a consequence alternative stress measures

emerge. Finally, stress rates are discussed in preparation for the following

chapter on hyperelasticity.

Chapter 5 “ Hyperelasticity

Hyperelasticity, whereby the stress is found as a derivative of some potential

energy function, encompasses many types of nonlinear material behavior and

provides the basis for the ¬nite element treatment of elastoplastic behavior

(see Appendix A). Isotropic hyperelasticity is considered both in a material

and in a spatial description for compressible and incompressible behavior.

The topic is extended to a general description in principle directions that is

specialized for the cases of plane strain, plane stress and uniaxial behavior.

Chapter 6 “ Linearized Equilibrium Equations

To establish the Newton“Raphson solution procedure the virtual work ex-

pression of equilibrium may be linearized either before or after discretiza-

tion. Here the former approach is adopted. Linearization of the equilibrium

equations includes consideration of deformation-dependant surface pressure

loading. A large proportion of this chapter is devoted to incompressibility

and to the development, via the Hu-Washizu principle, of the mean dilata-

tion technique.

Chapter 7 “ Discretization and Solution

All previous chapters have provided the foundation for the development of

the discretized equilibrium and linearized equilibrium equations considered

in this chapter. Linearization of the virtual work equation leads to the fa-

miliar ¬nite element expression of equilibrium involving B T σdv, whereas

discretization of the linearized equilibrium equations leads to the tangent

matrix, which comprises constitutive and initial stress components. Dis-

cretization of the mean dilatation technique is presented in detail. The

tangent matrix forms the basis of the Newton“Raphson solution procedure,

xvii

PREFACE

which is presented as the fundamental solution technique enshrined in the

computer program discussed in the following chapter. The chapter con-

cludes with a discussion of line search and arc length enhancement to the

Newton“Raphson procedure.

Chapter 8 “ Computer Implementation

Here information is presented on a nonlinear ¬nite element computer pro-

gram for the solution of ¬nite deformation ¬nite element problems employing

the neo-Hookean hyperelastic compressible and incompressible constitutive

equations developed in Chapter 5. The usage and layout of the FORTRAN

program is discussed together with the function of the various key subrou-

tines. The actual program is available free on the Internet.

The source program can be accessed via the ftp site: ftp.swansea.ac.uk

use “anonymous” user name and go to directory pub/civeng

and get flagshyp.f. Alternatively access the WWW site address:

http://www. swansea.ac.uk/civeng/Research/Software/flagshyp/ to

obtain the program and updates.

Appendix “ Introduction to Large Inelastic Deformations

This appendix is provided for those who wish to gain an insight into the

topic of ¬nite deformation elastoplastic behavior. Because the topic is ex-

tensive and still the subject of current research, only a short introduction

is provided. The Appendix brie¬‚y extends the basic nonlinear kinematics

of Chapter 3 to cater for the elastoplastic multiplicative decomposition of

the deformation gradient required for the treatment of elastoplastic ¬nite

deformation. In particular, Von-Mises behavior is considered together with

the associated radial return and tangent modulus expressions.

Finally, a bibliography is provided that enables the reader to access the

background to the more standard aspects of ¬nite element analysis, also

listed are texts and papers that have been of use in the preparation of this

book and that cover material in greater depth.

xviii

CHAPTER ONE

INTRODUCTION

1.1 NONLINEAR COMPUTATIONAL MECHANICS

Two sources of nonlinearity exist in the analysis of solid continua, namely,

material and geometric nonlinearity. The former occurs when, for whatever

reason, the stress strain behavior given by the constitutive relation is nonlin-

ear, whereas the latter is important when changes in geometry, however large

or small, have a signi¬cant e¬ect on the load deformation behavior. Material

nonlinearity can be considered to encompass contact friction, whereas ge-

ometric nonlinearity includes deformation-dependent boundary conditions

and loading.

Despite the obvious success of the assumption of linearity in engineering

analysis it is equally obvious that many situations demand consideration

of nonlinear behavior. For example, ultimate load analysis of structures

involves material nonlinearity and perhaps geometric nonlinearity, and any

metal-forming analysis such as forging or crash-worthiness must include both

aspects of nonlinearity. Structural instability is inherently a geometric non-

linear phenomenon, as is the behavior of tension structures. Indeed the

mechanical behavior of the human body itself, say in impact analysis, in-

volves both types of nonlinearity. Nonlinear and linear continuum mechanics

deal with the same subjects such as kinematics, stress and equilibrium, and

constitutive behavior. But in the linear case an assumption is made that