# Interfacial crack integral identities incorporating mean displacement

## TL;DR

This paper develops integral identities for interfacial cracks in bi-materials that relate crack face loadings to the full displacement profile, enabling comprehensive analysis of crack behavior.

## Contribution

It introduces new integral identities that incorporate mean displacement, extending previous formulations to obtain full displacement profiles from face loadings.

## Key findings

- Derived boundary integral formulation using Betti's identity
- Established identities linking loadings to full displacement profiles
- Enabling full displacement recovery from face loadings

## Abstract

A semi-infinite crack loaded by a general asymmetric system of forces in an infinite bi-material plane is considered. A boundary integral formulation is derived using the fundamental reciprocal identity (Betti formula). The resulting singular integral equations link the applied loading and the full resulting crack displacement profile (not just the displacement jump across the crack). When used in conjunction with previously derived identities, the new identities allow for the full displacement profile to be derived from crack face loadings and vice versa.