On the Existence of Eshelby’s Equivalent Ellipsoidal Inclusion Solution

The existence of Eshelby’s equivalent inclusion solution is proved for a degenerate “transformed” ellipsoidal inhomogeneity in an infinite anisotropic linear elastic matrix. We prove the invertibility of the fourth rank tensor expression, C S + C(I − SE), where C is the stiffness tensor of the matrix, C ′ is the stiffness tensor of the inhomogeneity, SE is the Eshelby tensor, and I 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 nonsingular, 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.