Extended Finite Element Method for Crack Propagation

271 Pages·2011·5.88 MB·English

by  Sylvie Pommier

Extended Finite Element Method for Crack Propagation
Extended Finite Element
Method for Crack Propagation

Sylvie Pommier
Anthony Gravouil
Alain Combescure
Nicolas Moës
First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Adapted and updated from La simulation numérique de la propagation des fissures : milieux
tridimensionnels, fonctions de niveau, éléments finis étendus et critères énergétiques published 2009 in
France by Hermes Science/Lavoisier © LAVOISIER 2009
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,
stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers,
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CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the
undermentioned address:
ISTE Ltd John Wiley & Sons, Inc.
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UK USA
www.iste.co.uk www.wiley.com
© ISTE Ltd 2011

The rights of Sylvie Pommier, Anthony Gravouil, Alain Combescure, Nicolas Moës to be identified as
the authors of this work have been asserted by them in accordance with the Copyright, Designs and
Patents Act 1988.
____________________________________________________________________________________
Library of Congress Cataloging-in-Publication Data

Extended finite element method for crack propagation / Sylvie Pommier ... [et al.].
p. cm.
Includes bibliographical references and index.
ISBN 978-1-84821-209-1
1. Fracture mechanics--Mathematics. 2. Finite element method. I. Pommier, Sylvie.
TA409.E98 2011
620.1'1260151825--dc22
2010048620

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-84821-209-1

Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.
Table of Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Chapter 1. Elementary Concepts of
Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Superposition principle . . . . . . . . . . . . . . . . . 3
1.3. Modes of crack straining . . . . . . . . . . . . . . . . 4
1.4. Singular fields at cracking point . . . . . . . . . . . 5
1.4.1. Asymptotic solutions in Mode I . . . . . . . . 8
1.4.2. Asymptotic solutions in Mode II . . . . . . . . 9
1.4.3. Asymptotic solutions in Mode III . . . . . . . 9
1.4.4. Conclusions . . . . . . . . . . . . . . . . . . . . . . 10
1.5. Crack propagation criteria . . . . . . . . . . . . . . . 10
1.5.1. Local criterion . . . . . . . . . . . . . . . . . . . . 10
1.5.2. Energy criterion . . . . . . . . . . . . . . . . . . . 13
1.5.2.1. Energy release rate G. . . . . . . . . . . . 13
1.5.2.2. Relationship between G and
stress intensity factors . . . . . . . . . . . 15
v
vi X-FEMforCrackPropagation
1.5.2.3. How the crack is propagated . . . . . . . 16
1.5.2.4. Propagation velocity. . . . . . . . . . . . . 16
1.5.2.5. Direction of crack propagation . . . . . . 17
Chapter 2. Representation of Fixed and
Moving Discontinuities . . . . . . . . . . . . . . . . . . . . 21
2.1. Geometric representation of a crack: a
scale problem . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.1. Link between the geometric
representation of the crack and the crack
model . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.2. Link between the geometric representation
of the crack and the numerical method used
for crack growth simulation . . . . . . . . . . . 25
2.2. Crack representation by level sets . . . . . . . . . 29
2.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . 29
2.2.2. Definition of level sets . . . . . . . . . . . . . . . 32
2.2.3. Level sets discretization . . . . . . . . . . . . . 38
2.2.4. Initialization of level sets . . . . . . . . . . . . 49
2.3. Simulation of the geometric propagation
of a crack . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.1. Some examples of strategies for crack
propagation simulation . . . . . . . . . . . . . . 53
2.3.2. Crack propagation modeled by
level sets . . . . . . . . . . . . . . . . . . . . . . . . 55
2.3.3. Numerical methods dedicated to
level set propagation . . . . . . . . . . . . . . . . 61
2.4. Prospects of the geometric representation
of cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Chapter 3. Extended Finite Element Method
X-FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2. Going back to discretization methods . . . . . . . 70
3.2.1. Formulation of the problem and
notations . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2.2. The Rayleigh-Ritz approximation . . . . . . . 72
TableofContents vii
3.2.3. Finite element method . . . . . . . . . . . . . . 73
3.2.4. Meshless methods . . . . . . . . . . . . . . . . . 75
3.2.5. The partition of unity . . . . . . . . . . . . . . . 78
3.3. X-FEM discontinuity modeling . . . . . . . . . . . . 79
3.3.1. Introduction, case of a cracked bar . . . . . . 80
3.3.1.1. Case a: crack positioned on a node . . . 80
3.3.1.2. Case b: crack between two nodes . . . . 81
3.3.2. Variants . . . . . . . . . . . . . . . . . . . . . . . . 83
3.3.3. Extension to two-dimensional and
three-dimensional cases . . . . . . . . . . . . . 85
3.3.4. Level sets within the framework of the
eXtended finite element method . . . . . . . . 93
3.4. Technical and mathematical aspects . . . . . . . . 94
3.4.1. Integration . . . . . . . . . . . . . . . . . . . . . . 94
3.4.2. Conditioning . . . . . . . . . . . . . . . . . . . . . 96
3.5. Evaluation of the stress intensity factors . . . . . 98
3.5.1. The Eshelby tensor and the J integral. . . . 99
3.5.2. Interaction integrals . . . . . . . . . . . . . . . . 103
3.5.3. Considering volumic forces . . . . . . . . . . . 106
3.5.4. Considering thermal loading . . . . . . . . . . 107
Chapter 4. Non-linear Problems, Crack
Growth by Fatigue . . . . . . . . . . . . . . . . . . . . . . . 109
4.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 109
4.2. Fatigue and non-linear fracture
mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.2.1. Mechanisms of crack growth by fatigue . . . 114
4.2.1.1. Crack growth mechanism at
low ∆K . . . . . . . . . . . . . . . . . . . . . 115
I
4.2.1.2. Crack growth mechanisms at
average or high ∆K . . . . . . . . . . . . 116
I
4.2.1.3. Macroscopic crack growth rate and
striation formation. . . . . . . . . . . . . . 119
4.2.1.4. Fatigue crack growth rate of long
cracks, Paris law . . . . . . . . . . . . . . . 121
4.2.1.5. Brief conclusions . . . . . . . . . . . . . . . 122
viii X-FEMforCrackPropagation
4.2.2. Confined plasticity and consequences for
crack growth . . . . . . . . . . . . . . . . . . . . . 122
4.2.2.1. Irwin’s plastic zones . . . . . . . . . . . . . 122
4.2.2.2. Role of the T stress . . . . . . . . . . . . . 125
4.2.2.3. Role of material hardening . . . . . . . . 125
4.2.2.4. Cyclic plasticity . . . . . . . . . . . . . . . . 129
4.2.2.5. Effect of residual stress on crack
propagation. . . . . . . . . . . . . . . . . . . 134
4.3. eXtended constitutive law . . . . . . . . . . . . . . . 137
4.3.1. Scale-up method for fatigue crack
growth . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.3.1.1. Procedure . . . . . . . . . . . . . . . . . . . . 137
4.3.1.2. Scaling . . . . . . . . . . . . . . . . . . . . . . 139
4.3.1.3. Assessment . . . . . . . . . . . . . . . . . . . 149
4.3.2. eXtended constitutive law . . . . . . . . . . . . 150
4.3.2.1. Damage law . . . . . . . . . . . . . . . . . . 151
4.3.2.2. Plasticity threshold . . . . . . . . . . . . . 152
4.3.2.3. Plastic flow rule. . . . . . . . . . . . . . . . 157
4.3.2.4. Evolution law of the center of the
elastic domain . . . . . . . . . . . . . . . . . 159
4.3.2.5. Model parameters . . . . . . . . . . . . . . 160
4.3.2.6. Comparisons . . . . . . . . . . . . . . . . . . 160
4.4. Applications . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.4.1. Mode I crack growth under variable
loading . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.4.2. Effect of the T stress . . . . . . . . . . . . . . . . 166
Chapter 5. Applications: Numerical Simulation of
Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.1. Energy conservation: an essential ingredient . . 173
5.1.1. Proof of energy conservation . . . . . . . . . . 174
5.1.1.1. X-FEM approach . . . . . . . . . . . . . . . 174
5.1.1.2. Cohesive zone models . . . . . . . . . . . . 178
5.1.1.3. Energy conservation for adaptive
cohesive zones . . . . . . . . . . . . . . . . . 178
5.1.2. Case where the material behavior
depends on history . . . . . . . . . . . . . . . . . 180
TableofContents ix
5.2. Examples of crack growth by fatigue
simulations . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.2.1. Calculation of linear fatigue crack
growth simulation . . . . . . . . . . . . . . . . . 182
5.2.2. Two-dimensional fatigue tests . . . . . . . . . 183
5.2.2.1. Test-piece CTS: crack growth in
mode 1 . . . . . . . . . . . . . . . . . . . . . . 183
5.2.2.2. Arcan test piece: crack growth in
mixed mode. . . . . . . . . . . . . . . . . . . 184
5.2.3. Three-dimensional fatigue cracks.
Propavanfiss project . . . . . . . . . . . . . . . . 187
5.2.3.1. Internal crack growth rate . . . . . . . . 187
5.2.4. Propagation of corner cracks . . . . . . . . . . 192
5.3. Dynamic fracture simulation . . . . . . . . . . . . . 192
5.3.1. Effects of crack speed a˙ and crack growth
criteria . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.3.2. Analytical solution: rectilinear crack
propagation on a reference problem . . . . . 195
5.3.3. Kalthoff experiment . . . . . . . . . . . . . . . . 197
5.3.4. Tests on test pieces CCS of Maigre-Rittel. . 200
5.3.5. Réthoré, Gregoire and Maigre tests . . . . . 202
5.3.6. X-FEM method in explicit dynamics . . . . . 206
5.4. Simulation of ductile fracture. . . . . . . . . . . . . 207
5.4.1. Characteristics of material 16MND5. . . . . 208
5.4.1.1. Dynamic characterization of the
material . . . . . . . . . . . . . . . . . . . . . 208
5.4.1.2. Fracture tests . . . . . . . . . . . . . . . . . 209
5.4.1.3. Crack advancement measurement
device. . . . . . . . . . . . . . . . . . . . . . . 209
5.4.1.4. Description of tests on CT test
pieces . . . . . . . . . . . . . . . . . . . . . . . 211
5.4.1.5. Numerical simulation. . . . . . . . . . . . 212
5.4.2. Ring test and interpretation . . . . . . . . . . 219
5.4.2.1. Geometry, mesh, and loading. . . . . . . 220
5.4.2.2. Interpretation of the test in Mode I . . 220
5.4.2.3. Interpretation of the test in mixed
mode . . . . . . . . . . . . . . . . . . . . . . . 222
x X-FEMforCrackPropagation
Conclusions and Open Problems . . . . . . . . . . . . 227
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Foreword
This book, offered to us by Sylvie Pommier, Anthony
Gravouil, Nicolas Moës, and Alain Combescure, is undoubt-
edly the most awaited book today in the solid mechanics
community. It certainly fills a gap in the very important
subjectofthenew“eXtended”FiniteElementMethod,known
only very recently and available only through specialized
articles.Thefact,thatoneoftheauthorsofthisnew“X-FEM”
method is also the joint author of this book, makes it all the
more valuable.
The raison d’être of this book is to provide a method,
certainly the only truly efficient one, to model real crack
propagation, from the initiation stage to final fracture.
We know that the conventional Finite Element Methods
using fixed meshes can only deal with this type of problem,
either if the crack path travels through mesh nodes, or if
we remove mesh elements. This is an extremely important
limitation in industrial applications.
The new “X-FEM” method no longer requires the mesh to
be constantly stuck to discontinuous surfaces or propagating
cracks.Itintroducesdiscontinuousfiniteelements.Ituses,as
inotherknownmethods,thepartitionofunityandintroduces
levelsetstomodelthecracksurfaceandthecrackfront,while
maintaining the same mesh.
xi