Stress fields of a spheroidal inhomogeneity with an interphase in an infinite medium under remote loadings

Abstract

This paper presents the elastostatic solution of the problem of an arbitrarily oriented spheroidal inhomogeneity with an interphase embedded in an infinite medium. The latter is under a remote axisymmetric loading. The complete solution of this problem requires three fundamental solutions, which are obtained by the Papkovich–Neuber displacement potentials and the expansion formulae for spheroidal harmonics. New displacement potentials are given when the remote loading is a longitudinal shear. The influence of the orientation and aspect ratio of the inhomogeneity, and of the remote stress ratio on the stress concentrations at the interfaces and the von Mises equivalent stress in the inhomogeneity, are studied. It is found that the interphase between the inhomogeneity and the surrounding medium significantly alters the stress distribution in, and around, the inhomogeneity. In addition to the general solution for an inhomogeneity with an interphase, the stress field exterior to a spheroidal inhomogeneity without an interphase (the Eshelby problem) is presented in a simple form. It is pointed out that the solution of a spheroidal inhomogeneity with an interphase in an infinite medium subjected to an arbitrary uniform eigenstrain, or a combination of a uniform eigenstrain and an arbitrary remote mechanical loading, can be obtained using the procedure developed in this paper.