An Examination of the Mori-Tanaka Effective Medium Approximation for Multiphase Composites

The Mori-Tanaka method is considered in the context of both scalar thermal conductivity and anisotropic elasticity of multiphase composites, and some general properties are deduced. Particular attention is given to its relation to known general bounds, and to the differential scheme. It is shown that the moduli predicted by the method always satisfy the Hashin-Shtrikman and Hill-Hashin bounds for two-phase composites. This property does not generalize to multiphase composites. A specific example illustrates that the method can predict moduli in violation of the Hashin-Shtrikman bounds for a three-phase medium. However, if the particle shapes are all spheres, then the prediction for the multiphase composite is coincident with the Hashin-Shtrikman bounds if the matrix material is either the stiffest or the most compliant phase. It is also shown that the generalized differential effective medium method yields the same moduli as the Mori-Tanaka approximation if certain conditions are satisfied in the differential scheme. Thus, it is required that at each stage in the differential process, and for each phase j (j = 1, 2, …, n) of new material, the average field in the incrementally added phase j material must be the same as the average field in the bulk phase j. For two phase media, n = 1, this condition reduces to the less stringent requirement that the ratio of the field in the incrementally added material to the average field in the matrix material is the same as the dilute concentration ratio. The cumulative findings of this paper, particularly those concerning bounds, suggest that the Mori-Tanaka approximation be used with caution in multiphase applications, but is on firmer ground for two-phase composites.