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This book deals with the mechanics and physics of fractures at various scales. Based on advanced continuum mechanics of heterogeneous media, it develops a rigorous mathematical framework for single macrocrack problems as well as for the effective properties of microcracked materials. In both cases, two geometrical models of cracks are examined and discussed: the idealized representation of the crack as two parallel faces (the Griffith crack model), and the representation of a crack as a flat elliptic or ellipsoidal cavity (the Eshelby inhomogeneity problem).
The book is composed of two parts:
Luc Dormieux is Professor at Ecole Nationale des Ponts et Chaussées (Laboratoire NAVIER) in Marne-la-Vallée, France.
Djimédo Kondo is Professor at Sorbonne University (UPMC, Institut D'Alembert) in Paris, France.
Notations xiii
Preface xv
Part 1. Elastic Solutions to Single Crack Problems 1
Chapter 1. Fundamentals of Plane Elasticity 3
1.1. Complex representation of Airy's biharmonic stress function 3
1.2. Force acting on a curve or an element of arc 7
1.3. Derivation of stresses 9
1.4. Derivation of displacements 11
1.5. General form of the potentials ¿ and ¿ 12
1.6. Examples 15
1.6.1. Circular cavity under pressure 15
1.6.2. Circular cavity in a plane subjected to uniaxial traction at infinity 16
1.7. Conformal mapping 18
1.7.1. Application of conformal mapping to plane elasticity problems 18
1.7.2. The domain ¿ is the unit disc |¿| ¿ 1 20
1.7.3. The domain ¿ is the complement ¿¿ of the unit disc 23
1.8. The anisotropic case 26
1.8.1. General features 26
1.8.2. Stresses, displacements and boundary conditions 28
1.9. Appendix: mathematical tools 29
1.9.1. Theorem 1 30
1.9.2. Theorem 2 31
1.9.3. Theorem 3 31
Chapter 2. Fundamentals of Elasticity in View of Homogenization Theory 33
2.1. Green's function concept 33
2.2. Green's function in two-dimensional conditions 34
2.2.1. The general anisotropic case 34
2.2.2. The isotropic case 35
2.3. Green's function in three-dimensional conditions 38
2.3.1. The general anisotropic case 38
2.3.2. The isotropic case 39
2.4. Eshelby's problems in linear microelasticity 41
2.4.1. The (elastic) inclusion problem 41
2.4.2. The Green operator of the infinite space 44
2.4.3. The Green operator of a finite domain 48
2.4.4. The inhomogeneity problem 50
2.4.5. The inhomogeneity problem with stress boundary conditions 51
2.4.6. The infinite heterogeneous elastic medium 52
2.5. Hill tensor for the elliptic inclusion 54
2.5.1. Properties of the logarithmic potential 54
2.5.2. Integration of the r,ir,l term 57
2.5.3. Components of the Hill tensor 59
2.6. Hill's tensor for the spheroidal inclusion 60
2.6.1. Components of the Hill tensor 63
2.6.2. Series expansions of the components of the Hill tensor for flat spheroids 64
2.7. Appendix 65
2.8. Appendix: derivation of the ¿ij 67
Chapter 3. Two-dimensional Griffith Crack 71
3.1. Stress singularity at crack tip 72
3.1.1. Stress singularity in plane elasticity: modes I and II 73
3.1.2. Stress singularity in antiplane problems in elasticity: mode III 78
3.2. Solution to mode I problem 80
3.2.1. Solution of PI 82
3.2.2. Solution of PI 90
3.2.3. Displacement jump across the crack surfaces 91
3.3. Solution to mode II problem 92
3.3.1. Solution of PII 93
3.3.2. Solution of PII 96
3.3.3. Displacement jump across the crack surfaces 97
3.4. Appendix: Abel's integral equation 98
3.5. Appendix: Neuber-Papkovitch displacement potentials 101
Chapter 4. The Elliptic Crack Model in Plane Strains 103
4.1. The infinite plane with elliptic hole 103
4.1.3. Elliptic cavity in a plane subjected to a remote stress state at infinity 107
4.1.4. Stress intensity factors 108
4.1.5. Some remarks on unilateral contact 111
4.2. Infinite plane with elliptic hole: the anisotropic case 112
4.2.1. General properties 112
4.2.2. Complex potentials for an elliptic cavity in the presence of traction at infinity 115
4.2.3. Complex potentials for an elliptic cavity in the case of shear at infinity 116
4.2.5. Displacement discontinuities 121
4.2.6. Closed cracks 123
4.3. Eshelby approach 130
4.3.1. Mode I 130
4.3.2. Mode II 133
Chapter 5. Griffith Crack in 3D 137
5.1. Griffith circular (penny-shaped) crack in mode I 138
5.1.1. Solution of PI 139
5.1.2. Solution of PI 143
5.2. Griffith circular (penny-shaped) crack under shear loading 144
5.2.1. Solution of PII 146
5.2.2. Solution of PII 151
Chapter 6. Ellipsoidal Crack Model: the Eshelby Approach 155
6.1. Mode I 156
6.2. Mode II 159
Chapter 7. Energy Release Rate and Conditions for Crack Propagation 163
7.1. Driving force of crack propagation 163
7.2. Stress intensity factor and energy release rate 167
Part 2. Homogenization of Microcracked Materials 173
Chapter 8. Fundamentals of Continuum Micromechanics 175
8.1. Scale separation 175
8.2. Inhomogeneity model for cracks 177
8.2.1. Uniform strain boundary conditions 177
8.2.2. Uniform stress boundary conditions 181
8.2.3. Linear elasticity with uniform strain boundary conditions 182
8.2.4. Linear elasticity with uniform stress boundary conditions 185
8.3. General results on homogenization with Griffith cracks 187
8.3.1. Hill's lemma with Griffith cracks 187
8.3.2. Uniform strain boundary conditions 188
8.3.3. Uniform stress boundary conditions 190
8.3.4. Derivation of effective properties in linear elasticity: principle of the approach 190
8.3.5. Appendix 194
Chapter 9. Homogenization of Materials Containing Griffith Cracks 197
9.1. Dilute estimates in isotropic conditions 197
9.1.1. Stress-based dilute estimate of stiffness 199
9.1.2. Stress-based dilute estimate of stiffness with closed cracks 202
9.1.3. Strain-based dilute estimate of stiffness with opened cracks 204
9.1.4. Strain-based dilute estimate of stiffness with closed cracks 205
9.2. A refined strain-based scheme 206
9.3. Homogenization in plane strain conditions for anisotropic materials 208
9.3.1. Opened cracks 208
9.3.2. Closed cracks 211
Chapter 10. Eshelby-based Estimates of Strain Concentration and Stiffness 213
10.1. Dilute estimate of the strain concentration tensor: general features 213
10.1.1. The general case 213
10.2. The particular case of opened cracks 215
10.2.1. Spheroidal crack 215
10.2.2. Elliptic crack 216
10.2.3. Crack opening change 218
10.3. Dilute estimates of the effective stiffness for opened cracks 220
10.3.1. Opened parallel cracks 222
10.3.2. Opened randomly oriented cracks 224
10.4. Dilute estimates of the effective stiffness for closed cracks 226
10.4.1. Closed parallel cracks 228
10.4.2. Closed randomly oriented cracks 228
10.5. Mori-Tanaka estimate of the effective stiffness 229
10.5.1. Opened cracks 231
10.5.2. Closed cracks 233
Chapter 11. Stress-based Estimates of Stress Concentration and Compliance 235
11.1. Dilute estimate of the stress concentration tensor 235
11.2. Dilute estimates of the effective compliance for opened cracks 236
11.2.1. Opened parallel cracks 237
11.2.2. Opened randomly oriented cracks 239
11.2.3. Discussion 239
11.3. Dilute estimate of the effective compliance for closed cracks 240
11.3.1. 3D case 241
11.3.2. 2D case 242
11.3.3. Stress concentration tensor 243
11.3.4. Comparison with other estimates 244
11.4. Mori-Tanaka estimates of effective compliance 244
11.4.1. Opened cracks 246
11.4.2. Closed cracks 246
11.5. Appendix: algebra for transverse isotropy and applications 246
Chapter 12. Bounds 251
12.1. The energy definition of the homogenized stiffness 252
12.2. Hashin-Shtrikman's bound 255
12.2.1. Hashin-Shtrikman variational principle 255
12.2.2. Piecewise constant polarization field 259
12.2.3. Random microstructures 261
12.2.4. Application of the Ponte-Castaneda and Willis (PCW) bound to microcracked media 270
Chapter 13. Micromechanics-based Damage Constitutive Law and Application 273
13.1. Formulation of damage constitutive law 273
13.1.1. Description of damage level by a single scalar variable 274
13.1.2. Extension to multiple cracks 276
13.2. Some remarks concerning the loss of uniqueness of the mechanical response in relation to damage 277
13.3. Mechanical fields and damage in a hollow sphere subjected to traction 280
13.3.1. General features 280
13.3.2. Case of damage model based on the dilute estimate 284
13.3.3. Complete solution in the case of the damage model based on PCW estimate 285
13.4. Stability of the solution to damage evolution in a hollow sphere 296
13.4.1. The MT damage model 298
13.4.2. The general damage model [13.44] 300
Bibliography 305
Index 309
