Variational Equilibrium Problem for a Plate with a Vertical Crack with a Geometrically Nonlinear Nonpenetration Condition

We consider a variational problem of minimizing energy functional of an elastic isotropic plate with a crack. On the curve describing the crack we impose boundary conditions prohibiting mutual penetration of the crack edges. Boundary conditions introduced are inequalities which define a non-convex set of admissible displacement functions. We prove the existence of the solution for this variational problem and establish that sufficiently smooth solutions satisfy boundary condition on the crack curve. For the problem with a given angle of curving of one of edges, the equivalence of variational and differential forms of problem is established.