A constrained proof of the strong version of the Eshelby conjecture for the three-dimensional isotropic medium

TL;DR

This paper advances the understanding of the Eshelby conjecture in 3D isotropic media by proving the strong version for cases where two eigenvalues of the eigenstress are identical, under specific material constraints.

Contribution

It provides a constrained proof of the strong Eshelby conjecture for the case with two identical eigenvalues of eigenstress in 3D isotropic media, extending previous results.

Findings

Established a necessary condition for convex inclusions to produce uniform elastic stress

Proved that only ellipsoids can satisfy certain material and stress constraints for the uniformity property

Confirmed the strong Eshelby conjecture under specific material parameter constraints for a set of eigenstresses

Abstract

Eshelby's seminal work on the ellipsoidal inclusion problem leads to the conjecture that the ellipsoid is the only inclusion possessing the uniformity property that a uniform eigenstrain is transformed into a uniform elastic strain. For the three-dimensional isotropic medium, the weak version of the Eshelby conjecture has been substantiated. The previous work of Ammari et al. substantiates the strong version of the Eshelby conjecture for the cases when the three eigenvalues of the eigenstress are distinct or all the same, whereas the case where two of the eigenvalues of the eigenstress are identical and the other one is distinct remains a difficult problem. In this work, we study the latter case. To this end, firstly, we present and prove a necessary condition for a convex inclusion being capable of transforming a single uniform eigenstress into a uniform elastic stress field. Since the necessary condition is not enough to determine the shape of the inclusion, secondly, we introduce a constraint that is concerned with the material parameters, and prove that there exist combinations of the elastic tensors and uniform eigenstresses such that only an ellipsoid can have the Eshelby uniformity property for these combinations simultaneously. Finally, we provide a more specifically constrained proof of the conjecture by proving that for the uniform strain fields constrained to that induced by an ellipsoid from a set of specified uniform eigenstresses, the strong version of the Eshelby conjecture is true for a set of isotropic elastic tensors which are associated with the specified uniform eigenstresses. This work makes some progress towards the complete solution of the intriguing and longstanding Eshelby conjecture for three-dimensional isotropic media.