Riemann-Hilbert problem on an elliptic surface and a uniformly stressed inclusion embedded into a half-plane subjected to antiplane strain

TL;DR

This paper develops a mathematical method using Riemann-Hilbert problems and conformal mappings to determine the shapes of elastic inclusions in a half-plane under shear, with numerical results illustrating parameter effects.

Contribution

It introduces a novel approach linking Riemann-Hilbert problems on hyperelliptic surfaces to shape determination of inclusions in elasticity.

Findings

The method successfully recovers inclusion shapes from stress conditions.

Numerical tests demonstrate how model parameters influence inclusion geometry.

The approach simplifies to scalar problems in the case of a single inclusion.

Abstract

An inverse problem of elasticity of $n$ elastic inclusions embedded into an elastic half-plane is analyzed. The boundary of the half-plane is free of traction. The half-plane and the inclusions are subjected to antiplane shear, and the conditions of ideal contact hold in the interfaces between the inclusions and the half-plane. The shapes of the inclusions are not prescribed and have to be determined by enforcing uniform stresses inside the inclusions. The method of conformal mappings from a slit domain onto the $(n+1)$-connected physical domain is worked out. It is shown that to recover the map and therefore the inclusions shapes, one needs to solve a vector Riemann-Hilbert problem on a genus-$n$ hyperelliptic surface. In a particular case of loading of a single inclusion in a half-plane, the problem is equivalent to two scalar Riemann-Hilbert problems on two slits on an elliptic surface. In addition to three parameters of the model the conformal map possesses a free geometric parameter. Results of numerical tests which show the impact of these parameters on the inclusion shape are presented.