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

Abstract

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.

Optimal location of a thin rigid inclusion for a problem describing equilibrium of a composite Timoshenko plate with a crack

“…Among the studies in this field let us note the recently developed nonlinear models describing the equilibrium state of composite solids having cracks on the interfacial boundary of inclusions. For the mentioned models, the carrier matrix is assumed to be elastic, whereas the inclusion is either absolutely rigid (for instance, see [8-10, 13-18, 20, 23]) or elastic and described by other constitutive relations (in this case, we have the so-called junction problems [26][27][28][29][30][31]). The optimal control problems for the shape of the bodies with with cracks and rigid inclusions are studied, for example, in [10-12, 15, 17, 18, 21, 22].…”

Fictitious Domain Method for Equilibrium Problems of the Kirchhoff-Love Plates with Nonpenetration Conditions for Known Configurations of Plate Edges

“…The method of the fictitious domain has proven useful in establishing the solvability of problems that describe equilibrium of bodies with cracks crossing the external boundary at zero angles [2,4,8,9]. In the last years, within the framework of crack models subject to non-penetration boundary conditions, numerous works have been published, see, for example, [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”

On the connection between two equilibrium problems for cracked bodies in the cases of thin and volume rigid inclusions

“…The use of such boundary conditions, in contrast to the classical formulations of the problems of the crack theory [23,24], does not impose an a priori known zone of contact for the crack faces. The wide range of applicability of variational methods enables successful formulation and investigation of various problems for solids with rigid or elastic inclusions, see, for example [25][26][27][28][29][30][31][32][33][34][35][36][37]. In particular, a foundational reference for two-dimensional elasticity problems with Signorini-type conditions on a part of the boundary of a thin delaminated rigid inclusion is [4].…”