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

We show that the Eshelby tensor, S-E, when written in the 6 x 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 S-E 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. (C) 2018 Elsevier Ltd. All rights reserved.