Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions

In this paper, we present some rigorous results for an equilibrium problem arising from the study of fiber-reinforced composites. We consider a two-dimensional homogeneous anisotropic linear elastic body containing a thin semirigid inclusion. The semirigid inclusion is an anisotropic thin structure that stretches along one direction and moves like a rigid body possessing both rotational and translatory motion along the perpendicular direction. A pre-existing interfacial crack is subject to nonlinear conditions that do not allow the opposite crack faces to penetrate each other. We focus on a variational approach to modelling the physical phenomenon of equilibrium and to demonstrate that the energy release rate associated with perturbation of the crack along the interface is well defined. A higher regularity result for the displacement field is formulated and proved. Then, taking into account this result, we deduce representations for the energy release rates associated with local translation and self-similar expansion of the crack by means of path-independent energy integrals along smooth contour surrounding one or both crack tips. Finally, some relations between the integrals obtained are discussed briefly.