A circular inhomogeneity interacting with an open inhomogeneity designed to admit internal uniform hydrostatic stresses

Abstract

We use analytic continuation and conformal mapping techniques to prove the existence of an internal uniform hydrostatic stress field inside a two-dimensional non-parabolic open inhomogeneity interacting with a nearby circular elastic inhomogeneity. The circular inhomogeneity and the surrounding matrix have distinct Poisson's ratios but identical shear moduli. The single complex constant appearing in the conformal mapping function is determined analytically. A simple condition is found that ensures that the internal stresses inside the non-parabolic inhomogeneity are uniform and hydrostatic. We also establish the existence of an internal uniform hydrostatic stress field inside a non-parabolic open inhomogeneity near an arbitrary number of circular elastic inhomogeneities. The circular inhomogeneities and the matrix have distinct Poisson's ratios but a common shear modulus. All of the complex constants in the conformal mapping function are uniquely determined by solving a set of non-linear equations via iterations.