Summary: Progress on the Strong Eshelby's Conjecture and
Extremal Structures for the Elastic Moment Tensor
Graeme W. Milton¶
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 in-
side 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 con-
sequence, 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 inclu-
sion 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.