On equilibrium of the elastic bodies with cracks crossing thin inclusions

Under study is the equilibrium problem of a two-dimensional elastic body with a crack crossing a thin rigid inclusion at some point. Nonpenetration conditions in the form of inequalities are put on the crack faces and at the intersection point of the crack with the rigid inclusion. The equilibrium problem of an elastic body with a crack crossing a thin elastic inclusion is also considered. The theorems of unique solvability of these problems are proved, and some complete systems of boundary conditions are obtained. The equivalence of the two formulations, variational and differential, is examined. We establish that the limit transition with respect to the rigidity parameter in the problems on the equilibrium of an elastic body with an elastic inclusion leads to the equilibrium problem of an elastic body with a rigid inclusion.