# Inverse antiplane problem on $n$ uniformly stressed inclusions

## TL;DR

This paper addresses the inverse antiplane elasticity problem to determine the shapes of multiple uniformly stressed inclusions using complex analysis and Riemann-Hilbert problems, providing solutions for symmetric and non-symmetric cases.

## Contribution

It introduces a novel method to solve the inverse problem for multiple inclusions via conformal mapping and Riemann-Hilbert problems, extending previous approaches to complex geometries.

## Key findings

- Solutions for two and three inclusions are demonstrated.
- The method handles symmetric and non-symmetric inclusion configurations.
- The approach reduces the problem to quadratures on a Riemann surface.

## Abstract

The inverse problem of antiplane elasticity on determination of the profiles of $n$ uniformly stressed inclusions is studied. The inclusions are in ideal contact with the surrounding matrix, the stress field inside the inclusions is uniform, and at infinity the body is subjected to antiplane uniform shear. The exterior of the inclusions, an $n$-connected domain, is treated as the image by a conformal map of an $n$-connected slit domain with the slits lying in the same line. The inverse problem is solved by quadratures by reducing it to two Riemann-Hilbert problems on a Riemann surface of genus $n-1$. Samples of two and three symmetric and non-symmetric uniformly stressed inclusions are reported.