The problem of thermal conduction for two ellipsoidal inhomogeneities in an anisotropic medium and its relevance to composite materials

We consider the problem of thermal conduction for an unbounded medium containing two ellipsoidal inhomogeneities subjected to a remote homogeneous boundary condition of temperature. The constituents are anisotropic and the ellipsoids could be at arbitrary orientations. In the formulation we first introduce some appropriate transformations into the heterogeneous medium and transform the problem into an isotropic matrix consisting of two analogous ellipsoidal inhomogeneities. Next, we replace the effect of inhomogeneities by some polynomial types of equivalent eigen-intensities by the concept of equivalent inclusion. These procedures allow us to write the local fields in terms of harmonic potentials and their derivatives. Numerical results show that linear approximations of eigen-fields yield accurate results in comparison with existing solutions by Honein et al. [2] for moderately separated inhomogeneities. Solutions of this type are used to estimate the overall thermal conductivity of composites with periodic microstructure. Finally, we present results for composites consisting of spherical inclusions with body-centered cubic, face-centered cubic, body-centered orthorhombic, and face-centered orthorhombic arrays.