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

DOI10.1098/RSPA.2021.0350arXiv2104.14038WikidataQ114089184 ScholiaQ114089184MaRDI QIDQ6366392

Y. A. Antipov

Publication date: 28 April 2021

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.

Mathematics Subject Classification ID

Boundary value problems in the complex plane (30E25) Riemann surfaces (30F99)

This page was built for publication: Riemann-Hilbert problem on an elliptic surface and a uniformly stressed inclusion embedded into a half-plane subjected to antiplane strain

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6366392)