On Equilibrium Problem for Elastic Body with Neumann Boundary Condition and Delaminated Timoshenko Inclusion

This paper addresses the investigation of an equilibrium problem describing an elastic body with an incorporated thin elastic Timoshenko inclusion, provided that the Neumann boundary condition is imposed on the external boundary of the body. The inclusion is delaminated from the elastic body, thus forming an interfacial crack. A restriction of the inequality type considered at the crack faces does not allow the opposite faces to penetrate each other. Moreover, the crack faces’ boundary conditions depend on a positive parameter characterizing the adhesion. We find the condition imposed on the external forces, proving the existence of a solution to the boundary value problem analyzed, and prove the solution’s existence. The asymptotic analysis of the solutions with respect to the adhesion parameter and the stiffness parameter of the inclusion is investigated. Limit models are analyzed. Both variational and differential statements of the considered problems are obtained, and a connection between them is established.