Extended Finite Element Method for Crack Propagation

by Pommier, Sylvie; Gravouil, Anthony; Moes, Nicolas; Combescure, Alain

• ISBN13: 9781848212091
• ISBN10: 1848212097
• Edition: 1st
• Format: Hardcover
• Copyright: 2011-02-21
• Publisher: Wiley-ISTE
• More Book Details

Summary

Novel techniques for modeling 3D cracks and their evolution in solids are presented. Cracks are modeled in terms of signed distance functions (level sets). Stress, strain and displacement field are determined using the extended finite elements method (X-FEM). Non-linear constitutive behavior for the crack tip region are developed within this framework to account for non-linear effect in crack propagation. Applications for static or dynamics case are provided.

Author Biography

Sylvie Pommier is Professor at Ecole Normale Suprieure de Cachan and the LMT-Cachan Laboratory in France. Her main research topics include the development of fatigue crack growth rate predictions and methods accounting for load history effects under complex loading conditions (mixed mode loading, non-isothermal fatigue, variable amplitude, corrosion assisted fatigue). Anthony Gravouil is Professor at INSA and the LaMCoS Laboratory in Lyon, France. His main research topics include the development of efficient and robust numerical methods (X-FEM) for the simulation of crack growth without remeshing, local multi-grid strategy coupled with X-FEM with a 3D representation of "real" cracks by level sets from 3D imaging and the development of space-time multi-scale methods for transient nonlinear dynamics (simulation of crash and impact phenomena). Alain Combescure is Professor at INSA in Lyon, France. His specialties include buckling, fracture mechanics, dynamics (mainly computational mechanics). Nicolas Mos is Professor at Ecole Centrale de Nantes in France. His research interests include computational methods in engineering for fracture and impact, and X-FEM

Table of Contents

Foreword	p. xi
Acknowledgements	p. xiii
List of Symbols	p. xv
Introduction	p. xvii
Elementary Concepts of Fracture Mechanics	p. 1
Introduction	p. 1
Superposition principle	p. 3
Modes of crack straining	p. 4
Singular fields at cracking point	p. 5
Asymptotic solutions in Mode I	p. 8
Asymptotic solutions in Mode II	p. 9
Asymptotic solutions in Mode III	p. 9
Conclusions	p. 10
Crack propagation criteria	p. 10
Local criterion	p. 10
Energy criterion	p. 13
Energy release rate G	p. 13
Relationship between G and stress intensity factors	p. 15
How the crack is propagated	p. 16
Propagation velocity	p. 16
Direction of crack propagation	p. 17
Representation of Fixed and Moving Discontinuities	p. 21
Geometric representation of a crack: a scale problem	p. 22
Link between the geometric representation of the crack and the crack model	p. 22
Link between the geometric representation of the crack and the numerical method used for crack growth simulation	p. 25
Crack representation by level sets	p. 29
Introduction	p. 29
Definition of level sets	p. 32
Level sets discretization	p. 38
Initialization of level sets	p. 49
Simulation of the geometric propagation of a crack	p. 52
Some examples of strategies for crack propagation simulation	p. 53
Crack propagation modeled by level sets	p. 55
Numerical methods dedicated to level set propagation	p. 61
Prospects of the geometric representation of cracks	p. 66
Extended Finite Element Method X-FEM	p. 69
Introduction	p. 69
Going back to discretization methods	p. 70
Formulation of the problem and notations	p. 70
The Rayleigh-Ritz approximation	p. 72
Finite element method	p. 73
Meshless methods	p. 75
The partition of unity	p. 78
X-FEM discontinuity modeling	p. 79
Introduction, case of a cracked bar	p. 80
Case a: crack positioned on a node	p. 80
Case b: crack between two nodes	p. 81
Variants	p. 83
Extension to two-dimensional and three-dimensional cases	p. 85
Level sets within the framework of the eXtended finite element method	p. 93
Technical and mathematical aspects	p. 94
Integration	p. 94
Conditioning	p. 96
Evaluation of the stress intensity factors	p. 98
The Eshelby tensor and the J integral	p. 99
Interaction integrals	p. 103
Considering volumic forces	p. 106
Considering thermal loading	p. 107
Non-linear Problems, Crack Growth by Fatigue	p. 109
Introduction	p. 109
Fatigue and non-linear fracture mechanics	p. 114
Mechanisms of crack growth by fatigue	p. 114
Crack growth mechanism at low ¿KI	p. 115
Crack growth mechanisms at average or high ¿KI	p. 116
Macroscopic crack growth rate and striation formation	p. 119
Fatigue crack growth rate of long cracks, Paris law	p. 121
Brief conclusions	p. 122
Confined plasticity and consequences for crack growth	p. 122
Irwin's plastic zones	p. 122
Role of the T stress	p. 125
Role of material hardening	p. 125
Cyclic plasticity	p. 129
Effect of residual stress on crack propagation	p. 134
eXtended constitutive law	p. 137
Scale-up method for fatigue crack growth	p. 137
Procedure	p. 137
Scaling	p. 139
Assessment	p. 149
eXtended constitutive law	p. 150
Damage law	p. 151
Plasticity threshold	p. 152
Plastic flow rule	p. 157
Evolution law of the center of the elastic domain	p. 159
Model parameters	p. 160
Comparisons	p. 160
Applications	p. 164
Mode I crack growth under variable loading	p. 164
Effect of the T stress	p. 166
Applications: Numerical Simulation of Crack Growth	p. 173
Energy conservation: an essential ingredient	p. 173
Proof of energy conservation	p. 174
X-FEM approach	p. 174
Cohesive zone models	p. 178
Energy conservation for adaptive cohesive zones	p. 178
Case where the material behavior depends on history	p. 180
Examples of crack growth by fatigue simulations	p. 182
Calculation of linear fatigue crack growth simulation	p. 182
Two-dimensional fatigue tests	p. 183
Test-piece CTS: crack growth in mode 1	p. 183
Arcan test piece: crack growth in mixed mode	p. 184
Three-dimensional fatigue cracks Propavanfiss project	p. 187
Internal crack growth rate	p. 187
Propagation of corner cracks	p. 192
Dynamic fracture simulation	p. 192
Effects of crack speed a and crack growth criteria	p. 193
Analytical solution: rectilinear crack propagation on a reference problem	p. 195
Kalthoff experiment	p. 197
Tests on test pieces CCS of Maigre-Rittel	p. 200
Réthoré, Gregoire and Maigre tests	p. 202
X-FEM method in explicit dynamics	p. 206
Simulation of ductile fracture	p. 207
Characteristics of material 16MND5	p. 208
Dynamic characterization of the material	p. 208
Fracture tests	p. 209
Crack advancement measurement device	p. 209
Description of tests on CT test pieces	p. 211
Numerical simulation	p. 212
Ring test and interpretation	p. 219
Geometry, mesh, and loading	p. 220
Interpretation of the test in Mode I	p. 220
Interpretation of the test in mixed mode	p. 222
Conclusions and Open Problems	p. 227
Summary	p. 233
Bibliography	p. 235
Index	p. 253
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