The conjunction problem for thin elastic and rigid inclusions in an elastic body

Under consideration is the conjunction problem for a thin elastic and a thin rigid inclusions that are in contact at one point and placed in an elastic body. Depending on what kind of conjunction conditions are set at the contact point of inclusions, we consider the two cases: the case of no fracture, where, as the conjunction conditions, we take the matching of displacements at the contact point and preservation of the angle between the inclusions, and the case with a fracture in which only the matching of displacements is assumed. At the point of conjunction, we obtain the boundary conditions for the differential formulation of the problem. On the positive face of the rigid inclusion, there is delamination. On the crack faces, some nonlinear boundary conditions of the type of inequalities are set, that prevent mutual penetration of the faces. The existence and uniqueness theorems for the solution of the equilibrium problem are proved in both cases.