Riemann–Hilbert problem on a hyperelliptic surface and uniformly stressed inclusions embedded into a half-plane subjected to antiplane strain

Abstract

An inverse problem of the elasticity of n elastic inclusions embedded into an elastic half-plane is analysed. 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 the shapes of the inclusions, one needs to solve a vector Riemann–Hilbert problem on a genus- n hyperelliptic surface. In a particular case of loading, the vector problem reduces to two scalar Riemann–Hilbert problems on n + 1 slits on a hyperelliptic surface. In the elliptic case, in addition to three parameters of the model, the conformal map possesses a free geometric parameter. The results of numerical tests in the elliptic case show the impact of these parameters on the inclusion shape.