Thin inclusion in elastic body: identification of damage parameter

In the paper, we consider an equilibrium problem for a 2D elastic body with a thin elastic inclusion crossing an external boundary. The elastic body has a defect which is characterized by a positive damage parameter. The presence of a defect means that the problem is formulated in a non-smooth domain. Non-linear boundary conditions at the defect faces are imposed to prevent the mutual penetration between the faces. Both variational and differential problem formulations are proposed, and existence of solutions is established. We study an asymptotics of solutions with respect to the damage parameter as well as with respect to a rigidity parameter of the inclusion. Identification problems for finding the damage parameter are investigated. To this end, existence of solutions of optimal control problems is proven.