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 in mechanics and other fields of heterogeneous media. Eshelby's finding that a uniform eigenstrain prescribed in a solitary ellipsoidal inclusion in an infinite isotropic medium results in a uniform elastic strain field in the inclusion leads to the conjecture that the ellipsoid is the only inclusion that possesses the so-called Eshelby...