Boundary perturbations due to the presence of small linear cracks in an elastic body

TL;DR

This paper derives an asymptotic formula for how small linear cracks in an elastic body perturb boundary displacement and traction, aiding in crack detection and characterization.

Contribution

It provides a rigorous asymptotic expansion for boundary perturbations caused by small cracks, with explicit formulas for leading order effects.

Findings

Leading order boundary perturbation is proportional to the square of crack length

The third-order term in the expansion vanishes

The results facilitate crack detection and size estimation

Abstract

In this paper, Neumann cracks in elastic bodies are considered. We establish a rigorous asymptotic expansion for the boundary perturbations of the displacement (and traction) vectors that are due to the presence of a small elastic linear crack. The formula reveals that the leading order term is \epsilon^2 where \epsilon is the length of the crack, and the \epsilon^3-term vanishes. We obtain an asymptotic expansion of the elastic potential energy as an immediate consequence of the boundary perturbation formula. The derivation is based on layer potential techniques. It is expected that the formula would lead to very effective direct approaches for locating a collection of small elastic cracks and estimating their sizes and orientations.