Crack on the boundary of a thin elastic inclusion inside an elastic body

We propose a model for a 2D elastic body with a thin elastic inclusion in which delamination of the inclusion may take place, thus forming a crack. Non-linear boundary conditions at the crack faces are imposed to prevent mutual penetration. We prove existence and uniqueness of the equilibrium configuration, considering both the variational and the differential formulations. Moreover, we study the dependence of solutions on the rigidity of the beam and we prove that in the limit corresponding to infinite and zero rigidity, we recover the case of a semi-rigid inclusion and the case of a crack with non-penetration conditions, respectively. The convergence of solutions is proved both using variational inequalities and G-convergence.