On control problem for two-layers elastic body with a crack

A two layer elastic body equilibrium problem is considered in this paper. One of the layers contains a crack. The other one is glued to the first one by its edge to cover one of the crack tips. The unique solvability of the problem with non-penetration condition on the crack faces is proved. A problem in the case when the second layer is rigid is studied. It is shown that a “rigid patch” problem could be obtained as a limit problem for a family of “elastic patch” problems when the rigidity parameter of the second layer tends to infinity. An optimal control problem is considered. The vector of exterior forces acting on two layers is chosen as a control function. Two aim functions are used. The first of them is a functional that characterizes potential energy change with respect to the crack length. The other one is a functional equal to an average opening of the crack along its length. The existence of solutions that minimize each of the functionals is proved.