Direct Mori-Tanaka calculations of strains in ellipsoidal inclusions with multiple orientations – Comments on the papers: Naili, G. et al. Comp Sci Tech, 187: 107942, 2020 (https://doi.org/10.1016/j.compscitech.2019.107942) and Jain, A. et al., Comp Sci Tech, 87: 86–93, 2013 (https://doi.org/10.1016/j.compscitech.2013.08.009)

Abstract

Mean field homogenization (MFH) methods are commonly deployed for homogenization of composite materials. Owing to physical admissibility problems and other concerns, one usually resorts to multi-step formulations of the Mori-Tanaka (MT) method involving the use of MT homogenization in more than one step. Using Representative Volume Elements (RVE) consisting of varying proportion of spheres and infinite cylinders, six variants of the Multi-step MT are implemented and benchmarked against FE and single step MT formulation. The physically admissibility conditions are checked for the different schemes, the average stresses in the inclusion and matrix phases and predicted effective modulus are compared with each other. The choice of the RVE has led to clear identification of errors associated with each of the seven mean field implementations. An important consideration for some of the multi-step MT formulations is the choice of the Poisson's ratio of the intermediate effective medium after the first step of homogenization, consequences of this choice are studied in this paper.Single step MT yields physically inadmissible values of the strain concentration tensor, while all the other multi-step MT formulations are able to circumvent this problem. Significant deviations and errors were observed for both the phase average stresses and the effective modulus with no one scheme giving acceptable results for all the cases considered in this paper. The order of homogenization in the multi-step MT formulation has significant effect on the predictions of the effective modulus as well as the phase average stresses. When the volume fraction of one of the phases tends to zero, the predictions of some multistep MT schemes do not always converge to the single step MT.