Derivatives of the energy functional for 2D-problems with a crack under Signorini and friction conditions

We consider the two-dimensional elasticity problem for an elastic body with a crack under unilateral constraints imposed at the crack. We assume that both the Signorini condition for non-penetration of the crack faces and the condition of given friction between them are fulfilled. The problem is non-linear and can be described by a variational inequality. Varying the shape of the crack by a local coordinate transformation of the domain, the first derivative of the energy functional to the problem with respect to the crack length is obtained, which gives the criterion for the crack growing. The regularity of the solution is discussed and the singular solution is performed.