Method of automorphic functions for an inverse problem of antiplane elasticity

Abstract

A nonlinear inverse problem of antiplane elasticity for a multiply connected domain is examined. It is required to determine the profile of $n$ uniformly stressed inclusions when the surrounding infinite body is subjected to antiplane uniform shear at infinity. A method of conformal mappings of circular multiply connected domains is employed. The conformal map is recovered by solving consequently two Riemann-Hilbert problems for piecewise analytic symmetric automorphic functions. For domains associated with the first class Schottky groups a series-form representation of a ($3n-4$) parametric family of conformal maps solving the problem is discovered. Numerical results for two and three uniformly stressed inclusions are reported and discussed.