Abstract:

In the paper, we analyze an equilibrium problem for a three-dimensional elastic body with incorporated thin two-dimensional elastic inclusion. The inclusion is delaminated from the surrounding elastic body forming therefore an interfacial crack. To avoid a mutual penetration between the opposite crack faces, the inequality type boundary conditions at the faces are considered. In addition to this, the Neumann type condition is imposed at the external boundary of the elastic body, which implies a non-coercivity of the problem. Assuming that the elasticity tensor depends on a positive parameter in a given subdomain we prove an existence of solutions and investigate the asymptotic behavior of the solutions as this parameter tends to infinity. It is proved that the limit model describes an equilibrium state of the elastic body with the thin and volume  inclusions. Necessary and sufficient conditions imposed on the external forces are found, and solution existence for all cases analyzed is proved.