Singular invariant integrals for elastic bodies with thin elastic inclusions and cracks

Abstract

Equilibrium problems for an elastic body with partially delaminated thin elastic inclusions are considered. The inclusions are modeled within the framework of the Euler–Bernoulli and Timoshenko models of elastic beams. The presence of delamination means the existence of a crack between the inclusion and the elastic matrix. Displacements of the opposite crack faces are constrained with nonpenetration conditions. Formulas of the Griffith type giving the first derivatives of energy functionals with respect to the crack length are established. It is shown that the formulas for the derivatives admit representation in the form of invariant integrals independent of the smooth closed curve surrounding the crack tip. The obtained invariant integrals consist of the sum of regular and singular parts and are analogues of the classical Eshelby–Cherepanov–Rice J-integral.