# Elasticity with Arbitrarily Shaped Inhomogeneity

## TL;DR

This paper introduces a semi-analytic method using conformal maps to determine stress tensors in elastic materials with arbitrarily shaped inhomogeneities, improving understanding of complex inhomogeneity effects.

## Contribution

The paper presents a novel semi-analytic approach employing dual conformal maps to solve elasticity problems with arbitrary inhomogeneity shapes, expanding analytical capabilities.

## Key findings

- Method converges for mild shape distortions
- Provides accurate stress tensor calculations
- Compared successfully with previous results

## Abstract

A classical problem in elasticity theory involves an inhomogeneity embedded in a material of given stress and shear moduli. The inhomogeneity is a region of arbitrary shape whose stress and shear moduli differ from those of the surrounding medium. In this paper we present a new, semi-analytic method for finding the stress tensor for an infinite plate with such an inhomogeneity. The solution involves two conformal maps, one from the inside and the second from the outside of the unit circle to the inside, and respectively outside, of the inhomogeneity. The method provides a solution by matching the conformal maps on the boundary between the inhomogeneity and the surrounding material. This matching converges well only for relatively mild distortions of the unit circle due to reasons which will be discussed in the article. We provide a comparison of the present result to known previous results.