Optimal size control of a rigid inclusion in equilibrium problems for inhomogeneous three-dimensional bodies with a crack

We consider equilibrium problems for an inhomogeneous three-dimensional body with a crack at the inclusion-matrix interface. The matrix of the plate is assumed to be elastic. The boundary condition on the crack curve is given in the form of inequality and describes mutual nonpenetration of the crack faces. We analyze the dependence of solutions on the size of the rigid inclusion. It is shown that as the size of the rigid inclusion's volume tends to zero the solutions of the corresponding equilibrium problems converge to the solution of the equilibrium problem for a body containing a thin rigid delaminated inclusion. The existence of the solution to the optimal control problem is proved. For that problem, the size parameter of the rigid inclusion is chosen as the control function, while the cost functional is an arbitrary continuous functional.