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Non-linear Finite Element Analysis

of Solids and Structures

~ ~ ~~ ~~

VOLUME 1: ESSENTIALS

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.

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John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, USA

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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.

Crisfield.

Non-linear finite element analysis of solids and structures / M. A.

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 1991 624.1'7 1 - dc20 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

Notation

1 General introduction, brief history and introduction to geometric

non-linearity

1 1 General introduction and a brief history 1 1 1 A brief history 1 2 A simple example for geometric non-linearity with one degree of freedom 1 2 1 An incremental solution 1 2 2 An iterative solution (the Newton-Raphson method) 1 2 3 Combined tncremental/iterative solutions (full or modified Newton-Raphson or the initial-stress method) 1 3 A simple example with two variables 1 3 1 ‘Exact solutions 1 3 2 1 3 3 An energy basis

List of books on (or related to) non-linear finite elements References to early work on non-linear finite elements

The use of virtual work

1 4 Special notation 1 5 1 6

2 A shallow truss element with Fortran computer program

2 1 A shallow truss element 2 2 A set of Fortran subroutines 2 2 1 Subroutine ELEMENT 2 2 2 Subroutine INPUT 2 2 3 Subroutine FORCE 2 2 4 Subroutine ELSTRUC 2 2 5 2 2 6 Subroutine CROUT 2 2 7 Subroutine SOLVCR

2 3 A flowchart and computer program for an incremental (Euler) solution

2 3 1 Program NONLTA 2 4 A flowchart and computer program for an iterative solution using the Newton-Raphson method 2 4 1 Program NONLTB 2 4 2 A flowchart and computer program for an incrementaViterative solution procedure using full or modified Newton-Raphson iterations 2 5 1 Program NONLTC

Subroutine BCON and details o n displacement control

Flowchart and computer listing for subroutine ITER 2 5

x i

xiii

1 1 1 2 6 a

10 13 16 18 19

V

CONTENTS vii

3.1 1 Special notation

3.12 References

4 Basic continuum mechanics

Stress and strain St ress-st ra i n relationsh i ps 4 2 1 Plane strain axial symmetry and plane stress 4 2 2 Decomposition into vo,umetric and deviatoric components 4 2 3 An alternative expression using the Lame constants Transformations and rotations 4 3 1 Transformations to a new set of axes 4 3 2 A rigid-body rotation Green’s strain 4 4 1 Virtual work expressions using Green s strain 4 4 2 Work expressions using von Karman s non-linear strain-displacement relqtionships for a plate Almansi’s strain The true or Cauchy stress Summarising the different stress and strain measures The polar-decomposition theorem 4 8 1 Ari example Green and Almansi strains in terms of the principal stretches

4.10 A simple description of the second Piola-Kirchhoff stress

4.1 1 Corotational stresses and strains

4.12 More on constitutive laws

4.13 Special notation

4.1 4 References

5 Basic finite element analysis of continua

5 1 Introduction and the total Lagrangian formulation

5 1 1 Element formulation 5 1 2 The tangent stiffness matrix 5 1 3 Extension to three dimensions 5 1 4 An axisymmetric membrane

5 2 Implenientation of the total Lagrangian method

5 2 1 With dn elasto-plastic or hypoelastic material

5 3 The updated Lagrangian formulation

5 4 Implementation of the updated Lagrangian method

Incremental formulation involving updating after convergence A total formulation for an elastic response An approximate incremental formulation

5 5 Special notation

5 6 References

6 Basic plasticity

6 1 Introduction

6 2 Stress updating incremental or iterative strains?

6 3 The standard elasto-plastic modular matrix for an elastic/perfectly plastic

von Mises material under plane stress 6 3 1 Non-associative plasticity

6 4 Introducing hardening

viii CONTENTS

6 4 1 6 4 2 6 4 3 Kinematic hardening Von Mises plasticity in three dimensions 6 5 1 Splitting the update into volumetric and deviatoric parts 6 5 2 Using tensor notation

6 6 Integrating the rate equations

6 6 1 Crossing the yield surface 6 6 2 Two alternative predictors 6 6 3 Returning to the yield surface 6 6 4 Sub-incrementation 6 6 5 Generalised trapezoidal or mid-point algorithms 6 6 6 A backward-Euler return 6 6 7 The radial return algorithm a special form of backward-Euler procedure The consistent tangent modular matrix 6 7 1 Splitting the deviatoric from the volumetric components 6 7 2 A combined formulation

6 8 Special two-dimensional situations

6 8 1 Plane strain and axial symmetry 6 8 2 Plane stress 6 8 2 1 A consistent tangent modular matrix for plane stress

Isotropic strain hardening Isotropic work hardening

6 9 Numerical examples

6 9 1 Intersection point 6 9 2 6 9 3 Sub-increments 6 9 4 6 9 5 Backward-Euler return

A forward-Euler integration

Correction or return to the yield surface

6 9 5 1 General method 6 9 5 2 Specific plane-stress method 6 9 6 Consistent and inconsistent tangents 6 9 6 1 Solution using the general method 6 9 6 2 Solution using the specific plane-stress method

6 10 Plasticity and mathematical programming

6 1 1 Special notation

6 12 References

A backward-Euler or implicit formulation

7 Two-dimensional formulations for beams and rods

7 1 A shallow-arch formulation

A simple corotational element using Kirchhoff theory 7 2 1 Stretching 'stresses and 'strains 7 2 2 7 2 3 7 2 4 7 2 5 7 2 6 7 2 7 7 2 8 Some observations

7 3 A simple corotational element using Timoshenko beam theory

7 4 An alternative element using Reissner's beam theory

The tangent stiffness matrix Introduction of material non-linearity or eccentricity Numerical integration and specific shape functions Introducing shear deformation Specific shape fur,ctions, order of integration and shear-locking

Bending 'stresses' and 'strains The virtual local displacements The virtual work The tangent stiffness matrix llsing shape functions Including higher-order axial terms

X CONTENTS

Flowchart and Fortran subroutine. for routine SCALUP 9 6 4 1 Fortran for routine SCALUP Flowchart and Fortran for subroutine NEXINC 9 6 5 1 Fortran for subroutine NEXINC

9 7 Quasi-Newton methods

9 8 Secant-related acceleration tecriniques

9 8 1 Cut-outS 9 8 2 Flowchart and Fortran for subroutine ACCEL 9 8 2 1 Fortran for subroutine ACCEL

9 9 Problems for analysts

9 9 1 The problems 9 9 2 9 9 3 9 9 4 9 9 5

Small-strain limit-point cxample with one variable (Example 2 2) Hardening problem with one variable (Example 3) Bifurcation problem (Example 4) Limit point with two variables (Example 5) Hardening solution with two variable (Example 6) Snap-back (Example 7) 9 10 Further work o n solution procedures

9 11 Special notation

9 12 References

Appendix Lobatto rules for numerical integration

Subject index

Author index

Preface

This book was originally intended as a sequal to my book Finite Elements and Solution

Proc.t.dures,fhr Structural Anufysis, Vol 1 -Linear Analysis, Pineridge Press, Swansea,

1986. However, as the writing progressed, it became clear that the range of contents was becoming much wider and that it would be more appropriate to start a totally new book. Indeed, in the later stages of writing, it became clear that this book should itself be divided into two volumes; the present one on ‘essentials’ and a future one on

‘advanced topics’. The latter is now largely drafted so there should be no further

changes in plan!

Some years back, I discussed the idea of writing a book on non-linear finite elements with a colleague who was much better qualified than I to write such a book. He

argued that it was too formidable a task and asked relevant but esoteric questions

such as ‘What framework would one use for non-conservative systems?’ Perhaps

foolishly, I ignored his warnings, but 1 am, nonetheless, very aware of the daunting

task of writing a ‘definitive work’ on non-linear analysis and have not even attempted

such a project.

Instead, the books are attempts to bring together some concepts behind the various

strands of work on non-linear finite elements with which I have been involved. This

involvement has been on both the engineering and research sides with an emphasis on the production of practical solutions. Consequently, the book has an engineering rather than a mathematical bias and the developments are closely wedded to computer

applications. Indeed, many of the ideas are illustrated with a simple non-linear finite

element computer program for which Fortran listings, data and solutions are included

(floppy disks with the Fortran source and data files are obtainable from the publisher

by use of the enclosed card). Because some readers will not wish to get actively involved in computer programming, these computer programs and subroutines are

also represented by flowcharts so that the logic can be followed without the finer detail.

Before describing the contents of the books, one should ask ‘Why further books

on non-linear finite elements and for whom are they aimed?’ An answer to the first

question is that, although there are many good books on linear finite elements, there

are relatively few which concentrate on non-linear analysis (other books are discussed

in Section 1. I ). A further reason is provided by the rapidly increasing computer power

and increasingly user-friendly computer packages that have brought the potential

advantages of non-linear analysis to many engineers. One such advantage is the

ability to make important savings in comparison with linear elastic analysis by

allowing, for example, for plastic redistribution. Another is the ability to directly

x i

PREFACE xiii

then applied to a range of problems using truss elements to illustrate such responses

as limit points, bifurcations, ‘snap-throughs’ and ‘snap-backs’.

It is intended that Volume 2 should continue straight on from Volume 1 with, for

example, Chapter 10 being devoted to ‘more continuum mechanics’. Among the

subjects to be covered in this volume are the following: hyper-elasticity, rubber, large

strains with and without plasticity, kinematic hardening, yield criteria with volume

effects, large rotations, three-dimensional beams and rods, more on shells, stability

theory and more on solution procedures.

REFERENCES

At the end of each chapter, we will include a section giving the references for that

chapter. Within the text, the reference will be cited using, for example, [B3] which

refers to the third reference with the first author having a name starting with the

letter ‘B’. If, in a subsequent chapter, the same paper is referred to again, it would

be referred to using, for example, CB3.41 which means that it can be found in the References at the end of Chapter 4.

NOTATION

We will here give the main notation used in the book. Near the end of each chapter

(just prior to the References) we will give the notation specific to that particular

chapter.

General note on matrix/vector and/or tensor notation

For much of the work in this book, we will adopt basic matrix and vector notation

where a matrix or vector will be written in bold. I t should be obvious, from the

context, which is a matrix and which is a vector.

In Chapters 4-6 and 8, tensor notation will also be used sometimes although, throughout the book, all work will be referred to rectangular cartesian coordinate systems (so that there are no differences between the CO- and contravariant compo- nents of a tensor). Chapter 4 gives references to basic work on tensors. A vector is a first-order tensor and a matrix is a second-order tensor. If we use the direct tensor (or dyadic) notation, we can use the same convention as for matrices

and vectors and use bold symbols. In some instances, we will adopt the suffix notation

whereby we use suffixes to refer to the components of the tensor (or matrix or vector). For clarity, we will sometimes use a suffix on the (bold) tensor to indicate its order.

These concepts are explained in more detail in Chapter 4, with the aid of examples.

Scalars

E = Young’s modulus

e =error

xiv PREFACE

,f = yield function

g

G =shear modulus

I

J =det(F)

k = bulk modulus

K , =tangent stiffness

t =thickness

U, U , w = displacements corresponding to coordinates x, y, z

V

V =virtual work

Vi =internal virtual work

V , =external virtual work

x, y , z = rectangular coordinates

c =strain

p = shear modulus

i. = load-level parameter

v = Poisson's ratio

<,7, i; = non-dimensional (natural) coordinates

fl =stress

z =shear stress

4 =total potential energy

= out-of-balance force or gradient of potential energy

= 2nd moment of area

= volume: dV = increment of volume; also

i^1 = shear strain

=curvature

Subscripts

2 =second order

4 =fourth order

cr = 'critical' (in relation to buckling)

e =external

ef = external (fixed)

i =internal

n =new

o =old

t =tangential

v =virtual

g =global

Superscripts and special symbols

. = rate or time-derivative

T =transpose

: = contraction (see equation (4.6))

0 =tensor product (see equation (4.31))

tr = trace ( = sum of diagonal elements)

General introduction, brief

history and introduction

to geometric non-linearity

1.1 GENERAL INTRODUCTION AND A BRIEF HISTORY

At the end of the present chapter (Section 1.5), we include a list of books either fully devoted to non-linear finite elements or else containing significant sections on the subject. Of these books, probably the only one intended as an introduction is the book edited by Hinton and commissioned by the Non-linear Working Group of NAFEMS (The National Agency of Finite Elements). The present book is aimed to start as an introduction but to move on to provide the level of detail that will generally not be found in the latter book. Later in this section, we will give a brief history of the early work on non-linear

finite elements with a selection of early references being provided at the end of the

chapter. References to more recent work will be given at the end of the appropriate chapters. Following the brief history, we introduce the basic concepts of non-linear finite

element analysis. One could introduce these concepts either via material non-linearity

(say, using springs with non-linear properties) or via geometric non-linearity. I have

decided to opt for the latter. Hence, in this chapter, we will move from a simple truss system with one degree of freedom to a system with two degrees of freedom. To simplify the equations, the ‘shallowness assumption’ is adopted. These two simple systems allow the introduction of the basic concepts such as the out-of-balance force vector and the tangent stiffness matrix. They also allow the introduction of the basic solution procedures such as the incremental approach and iterative techniques based on the Newton-Raphson method. These procedures are introduced firstly via the

equations of equilibrium and compatibility and later via virtual work. The latter

will provide the basis for most of the work on non-lineat finite elements.

1.1.1 A brief history

The earliest paper on non-linear finite elements appears to be that by Turner et ul. [T2] which dates from 1960 and, significantly, stems from the aircraft industry. The

1

2 INTRODUCTION TO GEOMETRIC NON-LINEARITY

present review will cover material published within the next twelve years (up to and

including 1972).

Most of the other early work on geometric non-linearity related primarily to the linear buckling problem and was undertaken by amongst others [H3, K I], Gallagher

et al. [G I , G21. For genuine geometric non-linearity, ‘incremental’ procedures were

originally adopted (by Turner et al. [T2] and Argyris [A2, A31) using the ‘geometric

stiffness matrix’ in conjunction with an updating of coordinates and, possibly, an

initial displacement matrix [Dl, M1, M31. A similar approach was adopted with

material non-linearity [Z2, M61. In particular, for plasticity, the structural tangent

stiffness matrix (relating increment of load to increments of displacement)incorporated

a tangential modular matrix [PI, M4, Y I , Z 1,221 which related the increments of

stress to the increments of strain.

Unfortunately, the incremental (or forward-Euler) approach can lead to an unquantifiable build-up of error and, to counter this problem, Newton-Raphson iteration was used by, amongst others, Mallet and Marcal [M I ] and Oden [Ol]. Direct energy search [S2,M2] methods were also adopted. A modified Newton-Raphson procedure was also recommended by Oden [02], Haisler c t al. [HI] and Zienkiewicz [Z2]. In contrast to the full Newton-Raphson method, the

stiffness matrix would not be continuously updated. A special form using the very

initial, elastic stiffness matrix was referred to as the ‘initial stress’ method [Zl] and

much used with material non-linearity. Acceleration procedures were also considered

“21. The concept of combining incremental (predictor) and iterative (corrector)

methods was introduced by Brebbia and Connor [B2] and Murray and Wilson

[M8, M9] who thereby adopted a form of ‘continuation method’. Early work on non-linear material analysis of plates and shells used simplified methods with sudden plastification [AI,BI]. Armen p t al. [A41 traced the

elasto-plastic interface while layered or numerically integrated procedures were

adopted by, amongst others, Marcal c’t al. [M5, M7] and Whang [Wl] combined

material and geometric non-linearity for plates initially involved ‘perfect elasto-plastic buckling’ [Tl, H21. One of the earliest fully combinations employed an approximate approach and was due to Murray and Wilson [MlO]. A more rigorous ‘layered approach’ was applied to plates and shells by Marcal [M3, M51, Gerdeen et ul. [G3] and Striklin et nl. [S4]. Various procedures were used for integrating through the

depth from a ‘centroidal approach’ with fixed thickness layers [P2] to trapezoidal

[M7] and Simpson’s rule [S4]. To increase accuracy, ‘sub-increments’were introduced for plasticity by Nayak and Zienkiewicz [NI]. Early work involving ‘limit points‘ and ‘snap-through’ was due to Sharifi and Popov [S3] and Sabir and Lock [Sl].

1.2 A SIMPLE EXAMPLE FOR GEOMETRIC

NON-LINEARITY WITH ONE DEGREE OF FREEDOM

Figure l.l(a) shows a bar of area A and Young’s modulus E that is subject to a load W so that it moves a distance U’. From vertical equilibrium,

N(z + M’) (^) - N(z + NI) W = N sin 0 = -

I” 1

4 INTRODUCTION TO GEOMETRIC NON-LINEARITY

and, from ( 1 .l), the relationship between the load Wand the displacement, w is given by

E A W = ~ - (z2w + gzw2 + iw”. 1 3

This relationship is plotted in Figure 1.2(a). If the bar is loaded with increasing - W,

at point A (Figure 1.2(a)), it will suddenly snap to the new equilibrium state at point

C. Dynamic effects would be involved so that there would be some oscillation about

the latter point. Standard finite element procedures would allow the non-linear equilibrium path to be traced until a point A’ just before point A, but at this stage the iterations would probably fail (although in some cases it may be possible to move directly to point

C-see Chapter 9). Methods for overcoming this problem will be discussed in

Chapter9. For the present, we will consider the basic techniques that can be used for the equilibrium curve, OA’. For non-linear analysis, the tangent stiffness matrix takes over the role of the

stiffness matrix in linear analysis but now relates small changes in load to small

changes in displacement. For the present example, this matrix degenerates to a scalar

dW/dw and, from ( l. l ) ,^ this quantity is given by

d W ( z + w ) d N N

dw I dw I

K , = E - -

1

(1.10)

Equation (1.6) can be substituted into (1.10)so that K , becomes a direct function

of the initial geometry and the displacement w. However, there are advantages in

maintaining the form of (1.10) (or (1.9)), which is consistent with standard finite

element formulations. If we forget that there is only one variable and refer to the

constituent terms in (1.10) as ‘matrices’, then conventional finite element terminology

would describe the first term as the linear stiffness matrix because it is only a function

of the initial geometry. The second term would be called the ‘initial-displacement’ or

‘initial-slope matrix’ while the last term would be called the ‘geometric’ or ‘initial-stress

matrix’. The ‘initial-displacement’ terms may be removed from the tangent stiffness

matrix by introducing an ‘updated coordinate system’ so that z ’ = z + w. In these

circumstances, equation (1.9) will only contain a ‘linear’ term involving z’ as well as

the ‘initial stress’ term. The most obvious solution strategy for obtaining the load-deflection response

OA’ of Figure 1.2(a)is to adopt ‘displacement control’ and, with the aid of (1.7) (or (1.6)

and (1.1))’ directly obtain W for a given w. Clearly this strategy will have no difficulty with the ‘local limit point’ at A (Figure 1.2(a)) and would trace the complete

equilibrium path OABCD. For systems with many degrees of freedom, displacement

control is not so trivial. The method will be discussed further in Section 2.2.5. For

the present we will consider load control so that the problem involves the computation

of w for a given W.

- w

E A

" 1 0.2 0.4 0.6 0.8 1.0, 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.

0.2 t

~-

\ / - - - w -

(b)

Figure 1.2 Load/deflection relationships for simple one-dimensional problem

(a) Response for bar alone. (b) Set of responses for bar-spring system.