Extreme Crack Shapes in a Plate Timoshenko Model

We consider equilibrium problem for Timoshenko-type plate. The problem is formulated as variational one. The plate is assumed to have a vertical crack with a variable shape. The nonpenetration condition imposed on crack faces is formulated in the form of inequality. An extreme shape of the crack is sought among all admissible cracks with fixed ends. We prove that functional, depending on crack and describing deformation has extreme cracks. We establish weakly convergence of solutions depending on parameter, which describing the crack shape. Furthermore, if functions of external forces is Lipschitz continuous, we prove prove that the solutions converge strongly.