1. Introduction: Shape derivatives in smooth domains ............ 7
1.1. The speed method ........................................ 7
1.2. The first order shape derivative ........................ 9
1.3. The second order shape derivative ....................... 9
2. Shape sensitivity analysis for cracks ....................... 10
2.1. The structure theorem .................................. 11
2.1.1. Introduction .................................... 11
2.1.2. The structure theorem in 3D ..................... 11
2.1.3. The structure theorem in 2D ..................... 15
2.2. Semi-derivatives of the eigenvalues .................... 17
2.2.1. Introduction, notations and main result ......... 18
2.2.2. Outline of the proof of Theorem 2.9 ............. 22
2.2.3. Proof of Theorem 2.9 ............................ 23
2.3. The energy functional for elastic bodies with cracks ... 33
2.3.1. Introduction .................................... 33
2.3.2. Problem formulation ............................. 33
2.3.3. Convergence of solutions ........................ 36
2.3.4. Main result ..................................... 38
3. Frechet differentiability in domains with cracks ............ 41
3.1. The structure theorem in dimension d ≥ 3 ............... 42
3.1.1. The structure theorem ........................... 42
3.1.2. Normal and tangential perturbations ............. 43
3.1.3. Proof of the structure theorem .................. 48
3.2. The structure theorem in dimension 2 ................... 50
3.2.1. The structure theorem ........................... 50
3.2.2. Normal and tangential perturbations ............. 51
3.2.3. Proof of the structure theorem .................. 53
4. Tangent sets in Banach spaces ............................... 55
4.1. Introduction ........................................... 56
4.2. Notation and preliminaries ............................. 56
4.2.1. Duality mapping ................................. 57
4.2.2. Examples covered by our setup ................... 58
4.3. Non-linear potential theory ............................ 59
4.4. Tangent cones .......................................... 59
4.5. Conical differentiability for evolution variational
inequalities ........................................... 63
4.5.1. Tangent sets and measures of finite energy ...... 63
4.5.2. Conical differentiability ....................... 65
4.6. Applications ........................................... 69
4.6.1. An abstract result .............................. 69
4.6.2. Unilateral conditions on the crack .............. 70
5. Non-penetration conditions on crack faces in elastic
bodies ...................................................... 74
5.1. Introduction ........................................... 74
5.2. Existence of solution .................................. 76
5.2.1. Mixed formulation of the problem ................ 78
5.2.2. Smooth domain formulation ....................... 78
5.3. Fictitious domain method ............................... 79
5.4. Crack on the boundary of rigid inclusion ............... 81
5.5. Shape derivatives of energy functionals ................ 84
5.6. Evolution of a kinking crack ........................... 90
5.7. 3D problems and open questions ......................... 92
6. Smooth domain method for crack problems ..................... 93
6.1. Introduction ........................................... 94
6.1.1. Main results .................................... 96
6.2. Two-dimensional elasticity ............................. 98
6.2.1. Variational formulation ......................... 98
6.2.2. Mixed formulation ............................... 99
6.2.3. Smooth domain method ........................... 103
6.3. Kirchhoff plate with a crack .......................... 106
6.4. Three-dimensional case ................................ 112
6.4.1. Preliminaries .................................. 112
6.4.2. Existence of solutions ......................... 115
6.4.3. Smooth domain method ........................... 119
6.5. Elastoplastic problems for plates with cracks ......... 119
6.5.1. Existence of solutions—Smooth domain method .... 119
7. Bridged crack models and singular integral equations ....... 125
7.1. Introduction and derivation of the model .............. 125
7.2. Mathematical problems ................................. 128
7.2.1. Existence and uniqueness using semigroup
methods ........................................ 128
7.2.2. Numerical example .............................. 132
7.3. Pseudo-differential operators ......................... 134
7.3.1. Introduction and statement of the result ....... 134
7.3.2. Notations and statement of the main results .... 135
7.3.3. Proof of Proposition 7.3 ....................... 136
7.3.4. Concluding remarks ............................. 141
References .................................................... 142
Index ......................................................... 149
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