The derivative of the energy functional in an equilibrium problem for a Timoshenko plate with a crack on the boundary of an elastic inclusion

We consider an equilibrium of a composite plate containing a through vertical crack of variable length on the separation boundary between the matrix and the elastic inclusion. The deformation of the matrix is described by the Timoshenko model, and the deformation of the elastic inclusion is described by the Kirchhoff–Love model. We obtain a formula for the derivative of the energy functional with respect to the crack length.