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.

Uniform stresses inside both a non-parabolic inhomogeneity and a non-elliptical inhomogeneity located in the vicinity of a circular Eshelby inclusion

“…It is seen that there contain five non-trivial parameters L, r, p, q, and j 0 in equation (8). Among these parameters, only r is real valued while the remaining four constants are complex valued.…”

“…where l is an additional given complex constant as compared with equation ( 8). The mapping description from the physical plane to the image plane via the use of equation ( 41) is identical to that by using equation (8). The appearance of the second-order poles at j = j À1 0 and j = r À1 j À1 0 in equation ( 41) is to account for the linear eigenstrains imposed on the circular Eshelby inclusion.…”

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

“…Here, A is a complex constant. Then [19], the real boundary conditions (2.2) are equivalent to the following complex conditions on the contours Lj for the analytic function ffalse(zfalse): ffalse(zfalse)+Az+ibj′=κj+12μjfalse(τ¯jz+djμjfalse)−κj−12μjfalse(τjzfalse¯+d¯jμjfalse),1emz∈Lj, j=1,…,n, where κj=μj/μ0, τj=τ1j+iτ2j, dj=aj+iaj′ and bj<...>…”

“…Two methods of conformal mappings from a canonical domain onto the physical multiply connected domain were proposed [18,19] for the inverse problem of antiplane strain of a plane with n uniformly stressed inclusions. In this model, the stresses τ13 and τ23 inside all of the inclusions are equal to constants τ1 and τ2, respectively, and are independent of the stresses prescribed at infinity.…”

“…In this model, the stresses τ13 and τ23 inside all of the inclusions are equal to constants τ1 and τ2, respectively, and are independent of the stresses prescribed at infinity. In [18] the canonical domain was chosen to be a parametric plane with n slits lying in the real axis, while in [19] the canonical domain was a plane with n circular holes. In the former case, the method of the Riemnn–Hilbert problem on a hyperelliptic surface was applied and the conformal map was reconstructed in terms of singular integrals.…”