A free boundary problem of elasticity in an angle and a vector Riemann–Hilbert problem on a torus

Abstract

An inverse problem of elasticity of reconstructing the shape of an inclusion placed in a wedge, when the whole inclusion is subjected to a prescribed antiplane uniform stress, is considered. The doubly connected physical domain is conformally mapped onto a parametric plane cut along two finite segments on the real axis. Determination of such a map requires solving a vector Riemann–Hilbert problem of the theory of holomorphic functions on a symmetric genus-1 Riemann surface. The main steps of the method include factorization of a discontinuous function on a torus, solution of the associated Jacobi inversion problem, representation of the unknown vector-functions in terms of Weierstrass integrals, and reduction of the vector Riemann–Hilbert problem to a single integral kernel of the second kind with a discontinuous kernel on the elliptic surface. An integral representation of the conformal map in terms of the solution to the integral equation is given. In addition to four dimensionless problem parameters, the conformal map possesses five free parameters. Numerical results, which reconstruct the inclusion shape for sample sets of the parameters, are reported and discussed.