Equivalent Inclusion Method (EIM) for isotropic and anisotropic spatially oriented spheroidal inhomogeneities: A unified calculation module validated via comparisons to Finite Element (FE) simulations

Abstract

This paper is centered on the Equivalent Inclusion Method due to Eshelby (1957, 1959, 1961) to solve inhomogeneous ellipsoidal inclusion problems. The objective is to develop a unified EIM program for spatially oriented isotropic and anisotropic spheroids and validate each stage of the developments in a controlled and rigorous way. At first, the analytical calculations of spatial derivatives of the elliptic integrals, involved in the expression of the strain field in the isotropic infinite medium, are pushed as far as possible for an Oblate spheroid and coded as it was done by Vincent et al., 2014 for the Prolate shape. Then, spatial orientation of the spheroid with respect to the global axis system attached to the infinite medium is introduced. Anisotropic metal inhomogeneities are finally dealt with, with the possibility to assign different crystallographic orientations. In this final configuration involving both the spatial orientation and anisotropy, three axis systems have to be managed simultaneously. Such a complexity legitimates a progressive evaluation towards this case. Thus, each new functionality introduced in the code is carefully validated by comparisons of the results to Finite Element reference solutions both inside and outside the inhomogeneity along different paths from the interface. This is done for an isotropic inhomogeneity without and with spatial orientation at first, and for an anisotropic inhomogeneity in the same way. These evaluations are presented for different shapes, aspect ratios, property contrasts. Such a complete evaluation involving at each stage various cases and examining the fields inside and outside the inhomogeneity constitutes an original contribution of the present work and allows to be confident in the proposed code (available on request). Another contribution of the paper is to analyze the influence of various factors (shape, aspect ratio, spatial orientation) on the fields distribution when both the anisotropy and the spatial orientation of the spheroidal inhomogeneity are combined. This last part illustrates also the efficiency of the EIM to deal with such an analysis with both accuracy and a low calculation cost.