Progress on the Strong Eshelby's Conjecture and Extremal Structures for the Elastic Moment Tensor

TL;DR

This paper advances the proof of the strong Eshelby's conjecture in three dimensions, showing that certain uniform strain conditions imply the inclusion must be ellipsoidal, and characterizes extremal shapes for the elastic moment tensor.

Contribution

It proves that specific uniform strain conditions imply ellipsoidal inclusion shapes and characterizes extremal shapes for the elastic moment tensor in 2D and 3D.

Findings

Ellipsoidal shape is necessary under certain uniform strain conditions.

Extremal elastic moment tensor shapes are ellipses in 2D and ellipsoids in 3D.

Progress towards the strong Eshelby's conjecture in three dimensions.

Abstract

We make progress towards proving the strong Eshelby's conjecture in three dimensions. We prove that if for a single nonzero uniform loading the strain inside inclusion is constant and further the eigenvalues of this strain are either all the same or all distinct, then the inclusion must be of ellipsoidal shape. As a consequence, we show that for two linearly independent loadings the strains inside the inclusions are uniform, then the inclusion must be of ellipsoidal shape. We then use this result to address a problem of determining the shape of an inclusion when the elastic moment tensor (elastic polarizability tensor) is extremal. We show that the shape of inclusions, for which the lower Hashin-Shtrikman bound either on the bulk part or on the shear part of the elastic moment tensor is attained, is an ellipse in two dimensions and an ellipsoid in three dimensions.