Small Perturbations of an Interface for Elastostatic Problems

TL;DR

This paper develops a rigorous asymptotic expansion for how small shape changes in elastic inclusions affect displacement fields and elastic moments tensors, using layer potential techniques and the field expansion method.

Contribution

It introduces a systematic approach to analyze perturbations in elastostatic problems, extending existing methods to relate shape changes with measurable elastic responses.

Findings

Derived asymptotic expansion for displacement perturbations.

Established relationship between measurements and shape of inclusions.

Quantified changes in elastic moments tensors due to interface perturbations.

Abstract

We consider solutions to the Lam\'e system in two dimensions. By using systematic way, based on layer potential techniques and the field expansion (FE) method (formal derivation), we establish a rigorous asymptotic expansion for the perturbations of the displacement field caused by small perturbations of the shape of an elastic inclusion with C2-boundary. We extend these techniques to determine a relationship between traction-displacement measurements and the shape of the object and derive an asymptotic expansion for the perturbation in the elastic moments tensors (EMTs) due to the presence of small changes in the interface of the inclusion.