Analytical treatment on the effective material properties of a composite material with spheroidal and ellipsoidal inhomogeneities in an isotropic matrix

Effective material properties of a composite with spheroidal and ellipsoidal inhomogeneities in an isotropic matrix are investigated analytically using the dilute approximation and the Mori–Tanaka approximation together with the Eshelby's equivalent inclusion method. Both uniaxially aligned and uniformly randomly oriented spheroidal and ellipsoidal inhomogeneities are treated. For a spheroid, both oblate and prolate spheroidal inhomogeneities are considered. It is analytically shown that a composite with uniaxially aligned anisotropic ellipsoidal inhomogeneities in an isotropic matrix is anisotropic in general in thermal conductivity. It is also analytically shown that a composite with uniformly randomly oriented anisotropic ellipsoidal inhomogeneities in an isotropic matrix is exactly isotropic in thermal conductivity. Various special cases are also treated for the effective thermal conductivity of a composite with ellipsoidal and spheroidal inhomogeneities. Similar results are also obtained for the effective elastic moduli. Newly obtained expressions for the effective elastic moduli of a composite with isotropic spheroidal inhomogeneities are rather involved. Conversely, an effective thermal conductivity of a composite with anisotropic ellipsoidal inhomogeneities is relatively simple. An effective thermal conductivity of a composite with isotropic spheroidal inhomogeneities reduces to a known result (Kerner, E. H. [1956] “The electrical conductivity of composite media,” Proceedings of the Physical Society London Section B 69, 802–807; Hashin, Z. and Shtrikman, S. [1962] “A variational approach to the theory of the effective magnetic permeability of multiphase materials,” Journal of Applied Physics 33, 3125–3131.) as the spheroid aspect ratio approaches 1 (i.e., a sphere). The effective thermal conductivity of a composite with uniformly randomly oriented isotropic spheroidal inhomogeneities in an isotropic matrix obtained in this paper as a special case is similar to the one obtained by Hatta and Taya (Hatta, H. and Taya, M. [1985] “Effective thermal conductivity of a misoriented short fiber composite,” Journal Applied Physics 58, 2478–2486.) in some respects, but is different. Numerical results are shown for a composite with oblate spheroidal voids in an isotropic matrix.