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

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.