Slit maps in the study of equal-strength cavities in $n$-connected elastic planar domains

TL;DR

This paper develops a mathematical framework using conformal mappings and Riemann-Hilbert problems to analyze equal-strength cavities in elastic domains, revealing conditions leading to inadmissible poles.

Contribution

It introduces a novel approach linking conformal mappings with hyperelliptic integrals for elastic cavity problems, expanding understanding of boundary value problems in elasticity.

Findings

Existence of conformal mappings for any number of cavities

Zeros of mappings can generate inadmissible poles

Method connects complex analysis with elasticity boundary problems

Abstract

The inverse problem of plane elasticity on $n$ equal-strength cavities in a plane subjected to constant loading at infinity and in the cavities boundary is analyzed. By reducing the governing boundary value problem to the Riemann-Hilbert problem on a symmetric Riemann surface of genus $n-1$ a family of conformal mappings from a parametric slit domain onto the $n$-connected elastic domain is constructed. The conformal mappings are presented in terms of hyperelliptic integrals and the zeros of the first derivative of the mappings are analyzed. It is shown that for any $n\ge 1$ there always exists a set of the loading parameters for which these zeros generate inadmissible poles of the solution.