The equilibrium problem of the structure, which consists of two elastic plates, is considered. It is assumed that the plates are flatly deformed, and the layers are modeled as elastic bodies. Plates are glued along a given line. In addition there is a defect along the gluing line in one of the layers. On the defect faces, nonlinear boundary conditions containing the damage parameter are established. Using the variational approach, the solvability of this problem is proved. In the problem, the passage to the limit is carried out when the damage parameter tends to zero and to infinity. Differential formulations for the corresponding limit problems are obtained. The case of the rigidity of one of the layers tends to infinity is considered; the obtained limit problem is analyzed.

The equilibrium problem of a two-layer elastic structure is investigated. In the lower layer there is a rectilinear defect. The upper layer covers one of the defect tips and is glued to the lower layer along its edge. Nonlinear boundary conditions are used to model the defect. Using the variational approach, the existence of a solution of the problem is established. Passages to the limit in the problem with respect to a parameter characterizing the elasticity of the upper layer, as well as to the defect damage parameter are carried out. The optimal control problem is considered, in which the cost functional is the derivative of the energy functional with respect to the defect length, and two parameters mentioned above act as control functions. The solvability of the optimal control problem is proved.