Geodesic
Differentiation of energy functionals in the three-dimensional theory of elasticity for bodies with surface cracks
Sibirskij žurnal industrialʹnoj matematiki, Tome 8 (2005) no. 1, pp. 106-116

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A three-dimensional elastic body with a surface crack is considered. The boundary nonpenetration conditions in the form of inequalities (the Signorini type conditions) are given at the faces of the crack. The convergence is proved of a sequence of equilibrium problems in perturbed domains to the solution of an equilibrium problem in the unperturbed domain in a suitable Sobolev function space. The derivative is calculated of the energy functional with respect to the perturbation parameter of the surface crack. 
@article{SJIM_2005_8_1_a11,
     author = {E. M. Rudoy},
     title = {Differentiation of energy functionals in the three-dimensional theory of elasticity for bodies with surface cracks},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {106--116},
     publisher = {mathdoc},
     volume = {8},
     number = {1},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2005_8_1_a11/}
}
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E. M. Rudoy. Differentiation of energy functionals in the three-dimensional theory of elasticity for bodies with surface cracks. Sibirskij žurnal industrialʹnoj matematiki, Tome 8 (2005) no. 1, pp. 106-116. http://geodesic.mathdoc.fr/item/SJIM_2005_8_1_a11/