Optimal location of a thin rigid inclusion for a problem describing equilibrium of a composite Timoshenko plate with a crack

Abstract

Abstract We consider equilibrium problems for a cracked composite plate with a thin cylindrical rigid inclusion. Deformation of an elastic matrix is described by the Timoshenko model. The plate is assumed to have a through crack that does not touch the rigid inclusion. In order to describe mutual nonpenetration of the crack faces we impose a boundary condition in the form of inequality on the crack curve. For a family of appropriate variational problems, we analyze the dependence of their solutions on the location of the rigid inclusion. We formulate an optimal control problem with a cost functional defined by an arbitrary continuous functional on the solution space, while the location parameter of inclusion is chosen as the control parameter. The existence of a solution to the optimal control problem and a continuous dependence of the solutions in a suitable Sobolev space with respect to the location parameter are proved.

Keywords

• Variational inequality
• Optimal control problem
• Nonpenetration
• Nonlinear boundary conditions
• Crack
• Rigid inclusion