Problem of the optimal amount of rigid thin sections for an equilibrium model of a Timoshenko plate with a crack

Abstract

Variational problems for composite plates with a system of joined rigid inclusions are considered. It is supposed that plate consist of an elastic matrix and cylindrical rigid inclusions. Inclusions are joined at common fibers so that displacements of these joined inclusions coincide at the corresponding points of fibers. Variational approach is used in order to deal with the Signorini-type boundary condition that describes mutual nonpenetration of opposite crack faces. We study two types of equilibrium models that correspond to different types of rigid inclusions. For the first model, we suppose that the plate has a volume rigid inclusion which is described by a corresponding 3D domain, and the second one describes the body containing a set of fastened thin rigid inclusions, each of which corresponds to a cylindrical surface. The crack is defined by the same curve in both models so that crack curve lies on the part of the boundary of the volume inclusion. An optimal control problem is formulated in the framework of the both model, such that a control is specified by the number of thin rigid thin plane inclusions and by the limiting case which as it turned out, fits to the first model. A quality functional is defined by an arbitrary continuous functional in a suitable Sobolev space. The solvability of the optimal control problem is proved.