In the talk we propose a new model to describe a state of equilibrium for an elastic plate with a thin rigid inclusion. It is assumed that a delamination of the inclusion takes place thus forming a crack between the elastic and rigid parts of the plate. We consider the free boundary approach and do not fix a contact set at the crack faces. Therefore, non-linear boundary conditions of inequality type are considered at the crack faces to prevent a mutual penetration. This approach is much more favorable from mechanical standpoint as compared to classical linear boundary conditions. Moreover, it is turned out that a new type of non-local boundary conditions are appeared at the inclusion describing an equilibrium of the inclusion. Correct mathematical formulation of the problem is proposed. We prove a solution existence for the suitable free boundary problems and analyze other properties of the solution for different location of the thin inclusion with respect to the external boundary. In particular, in the case of zero angle between the inclusion and the external boundary we propose a new fictitious domain method providing a solution existence for the problem. In this case we introduce an extended domain and consider a family of free boundary problems with a positive parameter in the extended domain. It is proved that for any fix parameter the problem is solvable, and, moreover, the solutions ( in fact, their restrictions to the original domain) are converging to the solution of the original boundary value problem.
展开▼
