IUTAM Symposium on discretization methods for evolving discontinuities

1 Introduction......Page 10
2 Element-free Galerkin shape functions......Page 13
3 Nonlocal continuum damage theory......Page 15
4.1 Wave propagation in a one-dimensional bar......Page 18
4.2 Notched specimen tension test......Page 19
5.1 Tapered bar in tension......Page 22
6 Conclusions......Page 24
References......Page 26
1 Introduction......Page 28
2 Element Free Galerkin (EFG) Approximation......Page 30
3 Discontinuous enrichment of the displacement Field......Page 31
3.1 Closing the crack along the front......Page 32
4.3 Tracking the crack path......Page 33
5.2 Cohesive cracks......Page 34
6 Discretized Equations......Page 35
7.1 Pull-out test......Page 37
7.2 Taylor bar impact......Page 39
8 Conclusions......Page 40
References......Page 41
1 Introduction......Page 44
2 The meshfree natural element method......Page 45
2.1 Natural neighbor interpolation......Page 46
2.3 The alpha-shape based natural element method – alpha-NEM –......Page 48
3.1 Tracking versus capturing......Page 50
Interpolation enrichment based on MLS-NEM......Page 51
Explicit interface representation......Page 55
4.1 Fundamentals: Karhunen-Lo`eve decomposition and reduced basis enrichment......Page 56
Enriching the approximation basis......Page 58
4.2 Accounting for weak discontinuities......Page 59
References......Page 61
1.1 Numerical Representations of Evolving Discontinuities......Page 64
Discontinuity/Shock Capturing with h-Refinement......Page 65
Cohesive Damage Models for Fracture......Page 66
1.2 Spacetime Discontinuous Galerkin Methods......Page 67
1.3 Organization of This Paper......Page 68
3 Formulations......Page 69
3.1 Elastodynamics......Page 70
3.2 Euler Equations......Page 72
4 Causal Spacetime Meshing and Patch-wise Solution Procedure......Page 73
Error Indicator for Element-wise Numerical Dissipation......Page 75
Numerical Example: Crack-Tip Wave Scattering......Page 76
Incorporation of Cohesive Damage Model in SDG Framework......Page 79
Adaptive Enforcement of the Traction–Separation Relation......Page 80
Numerical Example: Cohesive Fracture Under Shock Loading Conditions......Page 81
5.3 Front Tracking and Combined Tracking/Capturing Methods......Page 83
New Adaptive Meshing Operations......Page 85
Numerical Example: Combined Interface Tracking and Shock Capturing......Page 86
6 Conclusions......Page 89
Acknowledgements......Page 90
References......Page 91
1 Introduction......Page 93
2 Cahn-Hilliard equation......Page 94
3.1 Definitions......Page 95
3.2 Semi-discrete formulation......Page 96
4.1 Uniqueness......Page 97
4.2 Stability......Page 100
4.3 Accuracy......Page 101
5 Numerical example of phase separation......Page 103
References......Page 105
1 Introduction......Page 108
2.1 Problem definition......Page 110
2.2 The finite element approximation......Page 111
3 Finite elements with linear jumps......Page 114
4 Continuous interpolations of the discontinuity jumps......Page 117
5.1 Element bending test......Page 119
5.3 Three-point bending test......Page 121
5.4 Continuous interpolations......Page 122
References......Page 124
1 Introduction......Page 126
2 Strong Discontinuity Kinematics......Page 128
3 Continuum Constitutive Models and Traction-Separation Laws......Page 129
3.1 Damage and Plasticity Continuum Models......Page 130
Strong discontinuity activation criteria......Page 131
4 Variational Formulation and Finite Element Approach......Page 132
4.1 Strong Discontinuity Tracking Algorithm......Page 133
5 Constitutive Equation Integration. IMPL-EX Scheme......Page 135
6.1 Notched Cylindrical Bar in Tensile State......Page 136
6.2 Embankment Stability Analysis......Page 138
7 Conclusions......Page 139
References......Page 140
1 Introduction......Page 142
2 Cohesive theory of fracture......Page 143
3.1 Experimental set-up and numerical treatment......Page 145
3.2 The macro crack patterns......Page 147
3.3 The micro branch profile......Page 149
3.4 The energy dissipation mechanism......Page 150
3.5 Another example of branching instability......Page 152
References......Page 153
On Applications of XFEM to Dynamic Fracture and Dislocations......Page 156
1 Dynamic Crack Propagation......Page 158
2 Benchmark Problem......Page 160
3 Dislocation Modeling by XFEM......Page 163
Acknowledgments......Page 170
References......Page 171
1 Introduction......Page 172
2 Model Problem......Page 173
3 The classical Xfem......Page 174
4 Representation of the discontinuity......Page 175
5 Asymptotic displacement at the crack tip......Page 176
5.1 Adapted cubature formula for the asymptotic displacement......Page 177
5.2 Fixed enriched area......Page 178
6.1 The dof gathering technique......Page 180
6.2 Enrichment with a cut-off function......Page 181
6.3 Enrichment with a bonding condition......Page 183
Concluding remarks......Page 184
References......Page 185
1 Introduction......Page 186
2.1 Theoretical background......Page 187
2.3 Remarks on extension to other cases like different brittlefailure mechanism or ductile dynamic fracture......Page 190
3.1 X-FEM strategy: description and energy conservationproperties......Page 191
3.2 Time integration......Page 192
4.1 Experimental set up......Page 193
4.2 Material data......Page 194
4.3 One hole specimen......Page 195
5 Conclusion......Page 197
References......Page 198
1 Introduction......Page 200
2 Modeling assumptions......Page 201
Strong discontinuity kinematics......Page 202
Variational formulation......Page 203
Constitutive formulations......Page 204
Partition of unity finite element method (PUFEM)......Page 205
Crack tracking algorithm......Page 206
3 Numerical example......Page 207
4 Summary and Conclusion......Page 209
References......Page 211
1 Introduction......Page 213
2.1 Experimental Observations......Page 214
2.2 Implicit Representation of the Crack Using the Level SetMethod......Page 216
2.3 Strategy of Enrichment and SIFS Calculation......Page 217
3 Development of a 3D Local Paris Law......Page 220
4.1 Strategy of Enrichment......Page 223
4.2 Application to Fatigue Crack Growth with Confined Plasticity......Page 226
5 Conclusions......Page 229
References......Page 230
1 Introduction......Page 232
2 Polynomial cohesive law for quasi-brittle materials......Page 234
3 Crack tip asymptotic fields of frictionless andfrictional cohesive cracks in quasi-brittle materials......Page 237
4 Simulation of cohesive crack growth in quasi-brittlematerials using XFEM......Page 241
5 Illustration of the mode I cohesive crack tip fields......Page 244
6 Numerical example......Page 246
7 Discussion and conclusions......Page 251
References......Page 252
1 Motivation......Page 254
2 Governing Equations of Strong Discontinuities......Page 256
3 Discretization of Strong Discontinuities......Page 257
4 Simulations of Strong Discontinuities – 3D Examples......Page 260
5 Discussion......Page 261
Acknowledgements......Page 262
References......Page 263
1 Introduction......Page 265
The shape functions......Page 266
The critical time step......Page 268
Kinetic energy conservation......Page 269
2.2 2D element: Triangular element......Page 272
2.3 3D element: Tetrahedral element......Page 275
2.4 Results analysis......Page 276
3 Numerical applications......Page 277
4 Conclusion......Page 278
References......Page 279
1 Introduction......Page 280
2 Variational Formulation of Cracked Bodies......Page 281
3.1 Enhanced Displacement Approximation......Page 283
3.2 Traction-Separation Law for Mixed-Mode Conditions......Page 285
3.3 Finite Element Formulation......Page 286
4 Influence of Integration......Page 287
5.1 Crack Propagation in a L-Shaped Panel......Page 290
5.2 Crack Propagation in a Tension-Shear Specimen......Page 292
5.3 Crack Propagation in a Slab Subjected to Biaxial Compression......Page 294
6 Concluding Remarks......Page 296
References......Page 297
1 Introduction......Page 299
2 Finite Element Formulation......Page 300
3 Solution Procedure......Page 302
4 Benchmark Problem......Page 304
5 Stability of the Time-Integration Algorithm......Page 310
6 A Propagating Discontinuity......Page 313
7 Conclusions......Page 317
References......Page 318
1 Introduction......Page 319
2 Balance equations......Page 320
3 Discontinuities in a two-phase medium......Page 323
4.1 Models for the bulk......Page 324
4.2 Interface behaviour......Page 325
5.1 Weak forms......Page 327
5.2 Discretisations......Page 328
6 Example calculations......Page 331
6.1 Tresca-like initation criterion......Page 332
6.2 Coulomb initiation criterion......Page 334
7 Concluding remarks......Page 336
References......Page 337
1 Introduction......Page 339
2 Analysis of the Problem......Page 340
3 Step Function Enrichment......Page 343
4 Material Discontinuity Problems......Page 344
5 Regularized Heaviside Function......Page 347
6 Gaussian Quadrature for the Equivalent Polynomials......Page 349
6.1 Quadrature errors......Page 350
7 Conclusions......Page 353
Appendix......Page 355
References......Page 356
1 Introduction......Page 359
2 Simple gradient elasticity theory and integral formulation......Page 361
3 Discontinuous line element of variable order singularity......Page 363
4 BEM procedure......Page 365
6 Numerical examples......Page 368
6.1 Mixed mode I & II crack problem......Page 371
7 Conclusions......Page 372
References......Page 373
Introduction......Page 375
1 Francfort and Marigo’s model for quasi-static brittle fracture......Page 376
2 Numerical implementation......Page 377
2.1 Approximation of the Francfort-Marigo energy......Page 378
2.2 Minimization methods......Page 379
3 Numerical Experiments......Page 380
4 Extensions to thermal......Page 382
5 Conclusion and extensions......Page 385
1 Introduction......Page 388
2.2 General Framework......Page 390
2.3 First Case: Q4-DIC......Page 391
2.4 Second Case: I-DIC......Page 393
3.2 Second Method: Interaction Integral......Page 394
3.4 Comparison between the different approaches......Page 397
4.1 Experimental Configuration and Kinematic Measurements......Page 399
4.2 Analysis of the Maximum Load Level......Page 401
5 Summary......Page 403
References......Page 405
1 Introduction......Page 406
2.1 Problem Setting......Page 407
2.3 Solid Subsystem......Page 408
2.4 Interface Conditions......Page 410
2.5 Conservation Properties......Page 411
2.6 Incompatibility and Conservation......Page 412
3.1 Finite-Element Setting......Page 413
3.3 Boundary Conditions......Page 414
4 Conservation under Incompatibility: the NPCL Method......Page 415
4.2 The NPCL Method......Page 416
5 Example......Page 419
6 Conclusion......Page 423
References......Page 424