Existence of an extremal crack shape in the equilibrium problem for the timoshenko plate

We consider the variational problem of equilibrium of an elastic plate (the Timoshenko model) containing a crack. On the curve describing the crack, we impose boundary conditions in the form of inequalities. We study the dependence of the solutions on the curve shape. We prove the existence of an extremal curve shape and show that the solutions weakly converge in the Sobolev space as the parameter describing the crack shape tends to zero. The convergence is strong if the external forces satisfy the Lipschitz condition. © 2013 Springer Science+Business Media New York.