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

We consider the problem of the conjugation of a thin elastic inclusion and a thin rigid inclusion that are in contact at one point and are placed in an elastic body. Depending on what kind of conjugation conditions are given at the contact point of the inclusions, we consider the two cases: the case of no fracture, where as the conjugation conditions we take the coincidence of the displacements at the contact point and the preservation of the angle between the inclusions, and the case with a fraction, where only the coincidence of the displacements is given. At the conjugation point, we obtain boundary conditions for a differential statement of the problem. Delamination happens at the positive face of the rigid inclusion. On the crack faces, inequality-type nonlinear boundary conditions are given to prevent the mutual penetration of the crack faces. Existence and uniqueness theorems for a solution to the equilibrium problem are proved for each of the cases.