Spherically layered inclusions in a homogeneous elastic medium

Abstract

A three-dimensional homogeneous and isotropic elastic medium is considered that contains an isolated inhomogeneity (inclusion) in the shape of a sphere. It is assumed that the elastic moduli of the medium within the sphere depend only on the distance r to the centre of the inclusion. It is shown that in the case of a constant external field the problem of the equilibrium of a medium with an inhomogeneity reduces to a system of ordinary differential equations in three scalar functions of the variable r. An inhomogeneity with a piecewise-constant dependence of the elastic moduli on r (a spherically layered inclusion) is examined in detail. In this case, an effective calculational algorithm is proposed to construct the solution of the problem. The solution of the problem of one inclusion is then utilized to determine the effective elastic moduli of a medium with a random set of spherically layered inclusions and the estimates of the stress concentration at individual inhomogeneities. The method of an effective (selfconsistent) field is used to take account of interaction between the inclusions. The problem of a spherically layered inclusion in a homogeneous elastic medium was solved /1–3/ for particular forms of the constant external field. The method proposed below enables us, within the framework of a single scheme, to examine both spherically layered inclusions with practically any number of layers and inclusions with elastic moduli varying continuously along the radius for an arbitrary homogeneous external stress (strain) field.