Singular path-independent energy integrals for elastic bodies with Euler-Bernoulli inclusions

This paper is devoted to a rigorous analysis of an equilibrium problem for a two-dimensional homogeneous anisotropic elastic body containing a thin isotropic elastic inclusion. The thin inclusion is modelled within the framework of Euler-Bernoulli beam theory. Partial delamination of the inclusion from the elastic body results in the appearance of an interfacial crack. We deal with nonlinear conditions that do not allow the opposing crack faces to penetrate each other. We derive a formula for the first derivative of the energy functional with respect to the regular crack perturbation along the interface, which is related to energy release rates. It is proved that the energy release rates associated with crack translation and self-similar expansion are represented as path-independent integrals along smooth curves surrounding one or both crack tips. The path-independent integrals consist of regular and singular terms and are analogues of the well-known Eshelby-Cherepanov-Rice J-integral and Knowles-Sternberg M-integral.