Some reflections on the Mori-Tanaka and Ponte Castaneda-Willis methods with randomly oriented ellipsoidal inclusions

For a two-phase isotropic composite consisting of an isotropic matrix and randomly oriented isotropic ellipsoidal inclusions, Mori-Tanaka's (MT) method and the more recent Ponte Castaneda-Willis (PCW) method are perhaps the only two methods that deliver explicit results for its effective moduli. An attractive feature of the MT method is that it always stays within the Hashin-Shtrikman bounds, while the novel part of the PCW approach is that it has a well defined microstructure. In this paper, we made a comparative study on these two models, for both elasticity and their applications to plasticity. Over the entire range of inclusion shapes, the PCW estimates are found to be consistently stiffer than the MT estimates. An investigation of the possibility of a PCW microstructure for the MT model indicates that the MT moduli could be found from the PCW formulation, but this would require a spatial distribution that is identical to the oriented inclusion shape. Such a requirement implies that the underlying two-point joint probability density function is not symmetric, and thus it is not permissible. One is led to conclude that, unlike the aligned case, the MT model cannot be realized from the PCW microstructure with randomly oriented inclusions.