On crack propagations in elastic bodies with thin inclusions

The paper concerns an analysis of a crack propagation phenomena for an elastic body with thin inclusions and cracks. In the frame of free boundary approach, we investigate a dependence of the solutions on a rigidity parameter of the inclusion. A passage to the limit is justified as the parameter goes to infinity. Derivatives of the energy functionals are found with respect to the crack length for the models considered with different rigidity parameters. The Griffith criterion is used to describe a crack propagation. In so doing, an optimal control problem is investigated with a rigidity parameter being a control function. A cost functional coincides with a derivative of the energy functional with respect to the crack length. A solution existence is proved.