Properties of the Eshelby tensor and existence of the equivalent ellipsoidal inclusion solution

We show that the Eshelby tensor, SE, when written in the 6 × 6 matrix (Voigt) form, is weakly positive definite, i.e., it can be written as a product of two positive definite matrices. All eigenvalues of SE are real and lie between 0 and 1, for an arbitrary anisotropic elastic medium with a positive definite elastic stiffness tensor C. The weakly positive definiteness property leads to a direct proof of the existence of Eshelby's equivalent inclusion solution for a “transformed” ellipsoidal inhomogeneity in an infinite elastic medium.