Although various forms for the Hashin–Shtrikman bounds on the effective elastic properties of inhomogeneous materials have been written down over the last few decades, it is often unclear how to construct and compute such bounds when the material is not of simple type (e.g. isotropic spheres inside an isotropic host phase). Here, we show how to construct, in a straightforward manner, the Hashin–Shtrikman bounds for generally transversely isotropic two-phase particulate composites where the inclusion phase is spheroidal, and its distribution is governed by spheroidal statistics. Note that this case covers a multitude of composites used in applications by taking various limits of the spheroid, including both layered media and long unidirectional composites. Of specific interest in this case is the fact that the corresponding Eshelby and Hill tensors can be derived analytically. That the shape of the inclusions and their distribution can be specified independently is of great utility in composite design. We exhibit the implementation of the computations with several examples.

We consider the homogenization of parabolic systems with Dirichlet boundary conditions when the operators and the domains in which the problems are posed vary simultaneously. We assume the operators do not depend on t. Then we show that the corrector obtained in a previous paper for the elliptic problem still gives a corrector for the parabolic one. From this result, we easily obtain the limit problem in the parabolic case.

The Eshelby inhomogeneity problem plays a crucial role in the micromechanical analysis of the effective mechanical behaviour of inhomogeneous media since it provides a mechanism to predict interior fields associated with ellipsoidal inhomogeneities. In the context of linear elasticity, Eshelby showed that given an isolated elliptical (two dimensions) or ellipsoidal (three dimensions) inhomogeneity embedded in a homogeneous material of infinite extent, then for any uniform strain or traction imposed in the far field, the induced strain inside the inhomogeneity is also uniform. In the case of non-uniform far-field conditions, Eshelby showed that if the loading is a polynomial of order n, the associated interior field is characterized by a polynomial of the same order. This is often called ‘Eshelby's polynomial conservation theorem’. Since then, the problem has been studied by many, but in most cases for the uniform loading scenario, i.e. when strains or tractions in the far field are uniform. However, in many applications, e.g. permittivity, conductivity, elasticity, etc., the case of non-uniform conditions is also of interest and furthermore, methods to deal with non-elliptical and non-ellipsoidal inhomogeneities are required. In this work, for prescribed non-uniform polynomial far-field conditions, we introduce a method to approximate interior fields for isolated inhomogeneities of elliptical shape. This subproblem is relevant for approximating effective properties of numerous composites since constituent inhomogeneities are often of this form, or limiting forms, e.g. layered and fibre reinforced composites. We verify that the obtained results agree with the polynomial conservation property and with results determined using conformal mappings or the classical circle inclusion theorem. We close with a discussion of how the method can be straightforwardly extended to the case of non-elliptical inhomogeneities.