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.

A non-circular Eshelby inclusion near a non-parabolic inhomogeneity admitting internal uniform stresses

“…Wang and Schiavone [18,19], Antipov [20,21], Marshall [22] and Lim and Milton [23]. Recent investigations by the current authors [16,17] have revealed that uniform stresses continue to be attainable inside an uncoated or coated non-parabolic open elastic inhomogeneity despite the presence of a nearby circular Eshelby inclusion undergoing uniform or non-uniform anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses.…”

“…Recent investigations by the current authors [16,17] have revealed that uniform stresses continue to be attainable inside an uncoated or coated non-parabolic open elastic inhomogeneity despite the presence of a nearby circular Eshelby inclusion undergoing uniform or non-uniform anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. Most recently [19], we have solved the inverse problem in anti-plane elasticity associated with the uniformity of stresses inside a non-elliptical inhomogeneity with a closed curvilinear boundary interacting with a non-circular Eshelby inclusion. The non-circular inclusion was represented by a Booth's lemniscate inclusion.…”