Multi-step homogenization in the Mori-Tanaka-Benveniste theory

Micromechanical schemes homogenizing elastic properties as well as thermal and electrical conductivities of heterogeneous materials are most often developed as one-step procedures when all inhomogeneities are drawn into calculations at once, leaving no possibility for a multi-step approach. However, in practice, the multi-step homogenization is involved more often than expected, sometimes not on purpose. The reason is unknown microstructure of composite constituents while micromechanical schemes typically require their phases to be homogeneous. Thereby, constituents' properties, submitted to schemes, are effective, not intrinsic, contrary to schemes' requirements. This implicitly introduces preliminary homogenization steps to provide these effective properties, and the actual scheme considered corresponds to the last step of a multi-step homogenization process. However, our knowledge on applicability of multi-step homogenization techniques is more limited in comparison with the well-studied one-step schemes. Our study explores differences among one-, two-, and multi-step procedures in the Mori-TanakaBenveniste theory. For two-phase unidirectional composites, two- and one-step procedures differ by change in effective field related to change in concentration tensors. For porosity, two- and one-step homogenizations become identical if Eshelby tensor is multiplied by a scalar coefficient depending on volume fractions. Finally, applicability of the two-step procedure is verified for anisotropic three-phase composites.