On the existence of Eshelby’s equivalent ellipsoidal inclusion solution

Abstract

The existence of Eshelby’s equivalent inclusion solution is proved for a non-degenerate ‘transformed’ ellipsoidal inhomogeneity in an infinite anisotropic linear elastic matrix. We prove the invertibility of the fourth rank tensor expression, [Formula: see text], where C is the stiffness tensor of the matrix, C′ is the stiffness tensor of the inhomogeneity, I is the Eshelby tensor, and [Formula: see text] is the symmetric identity tensor. Taking advantage of the positive definiteness of certain tensor expressions, a proof-by-contradiction using energy arguments is posited that eliminates the possibility that the above expression is singular. Because the tensor expression is non-singular, it can always be inverted and Eshelby’s equivalent ellipsoidal inclusion method can be used to find the stress and strain fields in both the matrix and inhomogeneity.

A modified volume integral equation for anisotropic elastic or conducting inhomogeneities: Unconditional solvability by Neumann series

“…This work also extends results in recent investigations on Eshelby's equivalent inclusion method [3], in particular [4,9] (addressing the existence of solutions for Eshelby's method in the case of ellipsoidal inhomogeneities) and [5] (where the solvability of the singular VIE for isotropic inhomogeneity and background materials is established using Mikhlin's theory of singular integral operators [11]). …”

“…The solvability of the elastostatic SVIE is established in [5] for the less-general case of isotropic inhomogeneity and background materials, by explicitly computing the symbolic determinant of the singular integral operator and invoking Mikhlin's theory of singular integral operators [11]. Moreover, solvability results for the special case of ellipsoidal inhomogeneities are given in [4,9].…”

The topological gradient in anisotropic elasticity with an eye towards lightweight design

“…Next, we relate the solutions and This is a special case of Eshelby's equivalent inclusion method. Article is dedicated to a different proof. Lemma Let be an ellipsoid.…”

“…This is a special case of Eshelby's equivalent inclusion method. Article [52] is dedicated to a different proof.…”

Dislocation Interactions with (101̅2) Twin Boundary in Mg

“…to the actual total (elastic and transformation) strain of the inclusion when imbedded in a body(Eshelby, 1957;Kuykendall et al, 2012). In the above equation, both tensors are transformed to the same set of spatial coordinate axes ¶ % , ¶ * and ¶ ' .…”