Under study are the boundary value problems describing the equilibrium of twodimensional elastic bodies with thin anisotropic weakly curved inclusions in presence of separations. The latter implies the existence of a crack between the inclusion and the matrix. Nonlinear boundary conditions in the form of inequalities are imposed on the crack faces that exclude mutual penetration of the crack faces. This leads to the formulation of the problems with unknown contact area. The passage to limits with respect to the rigidity parameters of the thin inclusions is inspected. In particular, we construct the models as the rigidity parameters go to infinity and analyze their properties.