A contact problem for two plates of the same shape glued along one edge of a crack

Under study is the equilibrium problem for two plates with possible contact between them. It is assumed that the plates of the same shape and size are located in parallel without a gap. The clamped edge condition is stated on their lateral boundaries. The deflections of the plates satisfy the nonpenetration condition. There is a vertical crack in the lower layer. Along one edge of the crack, the plates are rigidly glued with each other. The three cases are studied in the paper: In the first case, the both layers are elastic, whereas in the second and third cases, the lower or upper layer respectively is rigid. To describe the displacement of the points of elastic plates, the Kirchhoff–Love model is used. Variational and differential formulations of the problems are derived and the unique solvability of the problems is established.