Solutions to the generalized Eshelby conjecture for anisotropic media: Proofs of the weak version and counter-examples to the high-order and the strong versions

The Eshelby formalism for an inclusion in a solid is of significant theoretical and practical implications mechanics other fields heterogeneous media. Eshelby's finding that uniform eigenstrain prescribed solitary ellipsoidal infinite isotropic medium results elastic strain field the leads to conjecture ellipsoid only possesses so-called uniformity property. Previously, weak version has been proved medium, whereas general validity anisotropic media three dimensions yet be explored. In this work, firstly, we present proofs generalized possess cubic, transversely isotropic, orthotropic, monoclinic symmetries. Secondly, prove these media, there exist non-ellipsoidal inclusions can transform particular polynomial eigenstrains even degrees into same them. These constitute counter-examples, strong sense, high-order (inverse problem conservation theorem) both (quadratic only). findings reveal striking richness between correspondingly induced strains beyond canonical inclusions. Since embedded inherently quantum dot crystals are effective tuning knobs quality emitted photons by dots, may have technology information, addition materials science.