The Griffith formula and J integral for elastic bodies with Timoshenko inclusions
The paper contains an analysis of a two dimensional equilibrium problem for an elastic body with a thin elastic inclusion. The thin elastic inclusion is modeled within the framework of Timoshenko beam theory. There is a crack on the interface between two media, displacements of the opposite crack faces are constrained with nonpenetration conditions. We derive the Griffith formula, which gives the first derivative of the energy functional with respect to the crack length. It is proved that the formula for the derivative can be represented as a path independent integral along a smooth curve surrounding the crack tip. The invariant integral consists of a regular part and a singular part and is an analogue of the classical Eshelby Cherepanov Rice J integral.