We consider an equilibrium problem for a 2D elastic body with a thin elastic T-shape inclusion. A part of the inclusion is delaminated from the elastic body forming a crack between the inclusion and the surrounding elastic body. Inequality type boundary conditions are imposed at the crack faces preventing interpenetration between the crack faces. The model is characterized by a damage parameter. This parameter is responsible for connection at the junction points between different parts of the considered structure. Dependence of solutions on the damage parameter is investigated, in particular, a passage to infinity and to zero is analyzed. Inverse problems are considered provided that the damage parameter and Lamé parameters of the elastic body are unknown. In this case, a displacement of the tip point of the inclusion is assumed to be known. A solution existence of the inverse problems is proved.
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