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

Abstract

We establish the uniformity of stresses inside both a non-parabolic open inhomogeneity and a non-elliptical closed inhomogeneity interacting with a nearby circular Eshelby inclusion undergoing uniform anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. Our procedure involves the introduction of a conformal mapping function for the doubly connected domain occupied by the matrix and the circular Eshelby inclusion. Two conditions are established in order to achieve the uniformity property inside each of the two inhomogeneities. Our results indicate that: (a) the internal uniform stresses are independent of the specific shapes of the two inhomogeneities and the existence of the nearby circular Eshelby inclusion; (b) the open and closed shapes of the respective inhomogeneities are significantly affected by the presence of the circular Eshelby inclusion. We also consider the two more complex cases involving: (a) an arbitrary number of circular Eshelby inclusions undergoing uniform eigenstrains; (b) a circular Eshelby inclusion undergoing linear eigenstrains. Detailed numerical results demonstrate the feasibility and effectiveness of the proposed theory.