An effective medium theory for multi-phase matrix-based dielectric composites with randomly oriented ellipsoidal inclusions

An effective medium approximation is formulated for multi-phase matrix-based dielectric composites with randomly oriented ellipsoidal inclusions. The main idea is based on considering a homogenized effective medium which is subjected to a uniform electric field, embedding in it a finite group of representative sub-elements of the composite, and then demanding that the dominant part of the far-field correction to the uniform field which prevailed in it vanishes. This condition results in an algebraic equation for the sought effective property. The calculation of the dominant part of the far-field correction is achieved in an approximate manner. That disturbance is assumed to be the sum of the disturbances caused individually by the each of the embedded elements that consist of a particle of the inclusion phases, surrounded first by some matrix material and then embedded separately in the effective medium. The volumetric ratio of the matrix shell surrounding a particle of the inclusion phase to the total volume of the embedded entity is determined according to the following strategy: the particle of an inclusion phase is assigned an amount of matrix, in proportion to the volume that this specific phase occupies relative to the total volume of all the inclusion phases in the actual composite. Numerical results are produced for a three-phase composite with randomly oriented ellipsoidal inclusions, and compared with the predictions of the average field approximation, the Mori–Tanaka mean field method, and the differential scheme. It is shown that the predictions of the average field approximation and the Mori–Tanaka model violate the multi-phase Hashin–Shtrikman bounds in several circumstances, whereas those of the effective medium approximation and the differential scheme obey those bounds.