Integral identities for a semi-infinite interfacial crack in 2D and 3D elasticity

TL;DR

This paper develops integral identities for a semi-infinite interfacial crack in 2D and 3D elastic media, providing a fundamental formulation using weight functions and reciprocal identities that aids in fracture mechanics and multiphysics applications.

Contribution

It introduces a formulation of the elasticity problem for interfacial cracks using singular integral equations based on weight functions and Betti's reciprocal theorem.

Findings

Formulation of crack problems via singular integral equations.

Application to heterogeneous media and multiphysics scenarios.

Facilitates analysis of hydraulic fracturing and related processes.

Abstract

The paper is concerned with the problem of a semi-infinite crack at the interface between two dissimilar elastic half-spaces, loaded by a general asymmetrical system of forces distributed along the crack faces. On the basis of the weight function approach and the fundamental reciprocal identity (Betti formula), we formulate the elasticity problem in terms of singular integral equations relating the applied loading and the resulting crack opening. Such formulation is fundamental in the theory of elasticity and extensively used to solve several problems in linear elastic fracture mechanics (for instance various classic crack problems in homogeneous and heterogeneous media). This formulation is also crucial in important recent multiphysics applications, where the elastic problem is coupled with other concurrent physical phenomena. A paradigmatic example is hydraulic fracturing, where the elasticity equations are coupled with fluid dynamics.