On the equilibrium problem for a two-layer structure with the upper layer covering a defect tip

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.