  In this work, we prove that in anisotropic media possessing cubic,
transversely isotropic, orthotropic, and monoclinic symmetries, there exist
non-ellipsoidal inclusions that can transform particular quadratic eigenstrains
into quadratic elastic strain fields in them. Further, we prove that in these
anisotropic media, there exist non-ellipsoidal inclusions that can transform
particular polynomial eigenstrains of even degrees into polynomial elastic
strain fields of the same even degrees in them. A sufficient condition for the
existence of those counter-examples is provided. These results constitute
counter-examples, in the strong sense, to the generalized high-order Eshelby
conjecture (inverse problem of Eshelby's polynomial conservation theorem) for
polynomial eigenstrains in both anisotropic media and the isotropic medium
(quadratic eigenstrain only). In addition, we also show that there are
counter-examples to the strong version of the generalized Eshelby conjecture
for uniform eigenstrains in these anisotropic media. These findings reveal
striking richness of the uniformity between the eigenstrains and the
correspondingly induced elastic strains in inclusions in anisotropic media
beyond the canonical ellipsoidal inclusion.
