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 |  موضوع: كتاب Fracture Mechanics - Inverse Problems and Solutions السبت 23 فبراير 2019, 10:40 pm | |
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أحضرت لكم كتاب
Fracture Mechanics - Inverse Problems and Solutions
H. D. BUI
Ecole Polytechnique
Department of Mechanics
Palaiseau, France
and Research & Development Division,
Electricité de France,
Clamart, France
ويتناول الموضوعات الأتية :
Contents
Preface xiii
Notations xvii
Color plates xxi
1 Deformation and Fracture 1
1.1 Deformation 2
1.1.1 Geometric transforms 2
1.1.2 Small strain 3
1.1.3 Compatibility conditions 4
1.1.4 Stress 5
1.2 Elasticity 6
1.2.1 Constitutive laws 6
1.2.2 Tonti’s diagram in elasticity 7
1.3 Plasticity 10
1.3.1 Experimental yield surfaces 10
1.3.2 Prandtl-Reuss equations 13
1.3.3 Generalized standard materials 17
1.4 Fracture 18
1.4.1 Introduction to fracture mechanics
1.4.2 Stress intensity factors
1.4.3 On the physics of separation
1.4.4 Different types of fracture
1.4.5 Brittle fracture criterion
2 Energetic Aspects of Fracture 29
2.1 Griffith’s theory of fracture 29
2.2 Some expressions of G in quasi-statics 31
2.3 Irwin’s formula 32
2.4 Barenblatt’s cohesive force model 34
2.5 Berry’s interpretation of energies 35
2.6 Stability analysis of multiple cracks 37
2.7 An inverse energetic problem 40
2.8 Path-independent integrals in quasi-statics 43
2.8.1 The path-independent J-integral 44
2.8.2 Associated J-integrals for separating mixed modes 45
2.8.3 The T-integral in linear thermo-elasticity 47
2.8.4 Lagrangian derivative of energy and the G-? integral 50viii Contents
2.9 Generalization of Griffith’s model in three dimensions 52
2.9.1 A local model of viscous fracture 54
2.9.2 A non local model of fracture 55
2.9.3 A dissipation rate model for non local brittle fracture 56
2.9.4 Convex analysis of three-dimensional brittle fracture 57
3 Solutions of Cracks Problems 61
3.1 Mathematical problems in plane elasticity 61
3.1.1 Plane strain and antiplane strain 61
3.1.2 Plane stress condition revisited 62
3.1.3 Complex variables in elasticity 63
3.1.4 The Hilbert problem 65
3.2 The finite crack in an infinite medium 68
3.2.1 The auxiliary problem 69
3.2.2 Dugdale-Barenblatt’s model 72
3.3 The kinked crack in mixed mode 73
3.3.1 An integral equation of the kinked crack problem 74
3.3.2 The asymptotic equation 76
3.4 Crack problems in elastoplasticity 78
3.4.1 Matching asymptotic solutions 79
3.4.2 A complete solution in plasticity and damage 80
3.4.3 A review of non linear asymptotic solutions 85
3.5 Inverse geometric problem with Coulomb’s friction 87
3.5.1 Non-uniqueness of solutions 88
3.5.2 Frictional crack problem without opening 89
3.5.3 The energy release rate of a frictional interface crack 92
3.5.4 The frictional interface crack with an opening zone 93
4 Thermodynamics of Crack Propagation 97
4.1 An elementary example 97
4.2 Dissipation analysis 99
4.3 Thermal aspects of crack propagation 101
4.4 Singularity of the temperature in thermo-elasticity 106
4.5 Asymptotic solution of the coupled equations 107
5 Dynamic Fracture Mechanics 113
5.1 Experimental aspects of crack propagation 113
5.2 Fundamental equations 115
5.3 Steady state solutions 116
5.4 Transient crack problems 118
5.4.1 Symmetric extension of a crack 118
5.4.2 Semi-infinite crack with arbitrary propagation 121
speedContents ix
5.5 Diffraction of time harmonic plane wave by a crack 122
5.6 Path-independent integrals 126
5.6.1 Path-independent integrals for moving cracks 126
5.6.2 A path-independent integral for crack initiation 128
5.6.3 Inverse problems in dynamic fracture 129
5.7 An experimental method for dynamic toughness 130
5.8 Identification of energies in dynamic fracture
6 Three-Dimensional Crack Problems 135
6.1 Fundamental tensors in elastostatics 135
6.1.1 The Kelvin-Somigliana tensor 135
6.1.2 The Kupradze-Bashelishvili tensor 136
6.2 Fundamental theorems in elastostatics 137
6.2.1 Solution of the Neumann boundary value problem 138
6.2.2 Solution of the Dirichlet boundary value problem
6.2.3 Direct methods using Kelvin-Somigliana’s tensor
6.3 A planar crack in an infinite elastic medium
6.3.1 The symmetric opening mode I 143
6.3.2 The shear modes 144
6.4 A planar crack in a bounded elastic medium 145
6.4.1 Singularity analysis 147
6.4.2 Solutions to some crack problems 149
6.5 The angular crack in an infinite elastic medium 153
6.6 The edge crack in an elastic half-space 155
6.7 On some mathematical methods for BIE in 3D
6.7.1 The Kupradze elastic potential theory 160
6.7.2 On the regularization of hypersingular integrals 162
6.7.3 Other regularization methods 165
6.8 An integral equation in elasto-plasticity 169
7 Non Linear Fracture Mechanics 173
7.1 Introduction 173
7.2 Ductile fracture 175
7.2.1 Rousselier’s model 176
7.2.2 The micromechanics of plasticity 182
7.2.3 Gurson’s model 186
7.2.4 Extension of porous plasticity models to aggregates 187
7.3 Bifurcation problems in plasticity 189
7.4 A finite strain theory of cavitation in solids 193
7.4.1 Abeyaratne and Hou’s solution in finite elasticity 196
7.4.2 Solution for a creeping material 197
160x Contents
8 The Fluid-Filled Crack 199
8.1 Introduction 199
8.2 The leak before break inverse problem 200
8.2.1 Empirical models of fluid flow in a crack breach 201
8.2.2 Variable breach area 205
8.3 Wear mechanics 210
8.3.1 Wear criterion and wear rate 212
8.3.2 Conservation of mass 214
8.3.3 Rheology of the third body 214
8.3.4 The W-equation for a sliding punch on a half-plane 216
8.3.5 Identification of constants 218
8.4 Hydraulic fracturing of rocks 218
8.4.1 The physical problems 218
8.4.2 Equations in hydraulic fracturing of rocks 220
8.5 Capillary phenomenon in fracture mechanics 222
8.5.1 The equilibrium crack partially filled with a fluid 223
8.5.2 Capillary stress intensity factor
8.6 Viscous fluid flow solution near a moving crack tip 229
8.6.1 Equations of the fluid-filled moving crack 229
8.6.2 Numerical results 230
9 Crack Detection by Scattering of Waves 233
9.1 Introduction 233
9.2 Scattering of acoustic waves 234
9.2.1 Detection of a rigid inclusion 235
9.2.2 Detection of a flat cavity 238
9.2.3 Finite spectrum and finite number of incident waves 240
9.3 Diffraction of elastic waves 243
9.4 Non destructive testing of materials. A case study 246
9.5 Time reversal mirror (TRM) 248
9.5.1 Experimental validation of TRM 250
9.5.2 The mathematics of time reversal mirror 251
10 Tomographic Evaluation of Materials 253
10.1 Introduction 253
10.2 X-rays tomography 254
10.2.1 Inverse Radon’s transform 255
10.2.2 Example of crack detection 257
10.3 Attenuated Radon transform 258
228Contents xi
10.3.1 Functional tomography by SPECT and PET 258
10.3.2 Novikov’s inversion formula 260
10.4 Conical Radon’s transform in Compton scattering 260
10.4.1 The conical Radon transform 261
10.4.2 Nguyen and Truong’s inversion formula 263
11 The Reciprocity Gap Functional for Crack Detection 267
11.1 Distributed defect and crack 267
11.2 Planar crack identification in quasi-static elasticity 270
11.2.1 Determination of the normal to the crack plane 271
11.2.2 Determination of the crack plane 272
11.2.3 Determination of the crack shape 272
11.3 The instantaneous RG functional 273
11.4 Inverse problem for the heat diffusion equation 277
11.4.1 Solution for the crack plane location
11.4.2 Solution for the crack shape 280
11.5 Inverse acoustic scattering of a crack in time domain 281
11.6 Elastodynamic scattering of a crack in time domain 283
11.6.1 The observation equation in elastodynamics 284
11.6.2 Solution 286
11.7 The earthquake inverse problem and solution 287
12 Methods of Solution to Inverse Problems 293
12.1 The ill-posedness of the inverse problem 293
12.2 General considerations on inverse problems 295
12.3.1 Optimal choice of the regularization parameter 298
12.3.2 Error estimate 299
12.4 Optimal control theory 300
12.4.1 State equation and optimal control 301
12.4.2 Adjoint system of equations 302
12.5 The dynamic system of quasi-static elasticity 305
12.5.1 Smoothing operators 309
12.5.2 Transfer matrix operator in elasticity 310
12.6 Quasi-reversibility method 313
12.7 Control of partial derivative equations 314
12.7.1 Inverse problems in parabolic systems 314
12.7.2 Identification of materials 317
12.8 Stochastic inversion methods 318
12.8.1 Tarentola’s inversion method 318
12.8.2 Kalman’s filter 319
12.9 Duality in solid mechanics and inverse problems 320
12.3 Tikhonov’s regularization 296xii Contents
A Residual Stresses in Fracture Mechanics 323
A.1 An approximate theory of residual stress 323
A.1.1 A theory of fracture with initial stresses 324
A.1.2 Energy release rate 325
A.2 Residual stress identification 326
A.2.1 Origin of the residual stresses 326
A.2.2 Determination of the residual stresses 327
B Weak Interface Singularities 329
B.1 The interface crack between dissimilar materials 329
B.2 Discontinuous Neumann boundary conditions 332
B.3 Thermo-plastic discontinuity on a composite tube 333
C Problems and Solutions 337
References
Index
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