Evaluation of the Relevance of Global and By-Step Homogenization for Composites and Heterogeneous Materials at Several Scales

Two hypotheses divide experts on determining the effective properties of composite materials using multi–scale homogenization methods. The first hypothesis states that multi-scale homogenization methods can ensure the direct determination of effective properties, at the macro level, of composite materials from a single representation of the medium at the lowest possible scale that allows for a good representation of all heterogeneities. The second hypothesis states that the determination cannot be ensured directly from a single scale but rather through multistep homogenization where each step represents the medium at a different scale from the lowest to the macroscale. To answer this question, a rigorous study is carried out; it includes calculating the two effective elastic properties, bulk, and shear moduli of three phases of a multi–layered sphere composite model by studying three phases. A multistep homogenization method is used to determine the effective properties of the composite and the obtained results are compared with those of the direct homogenization. Two different studies are considered: the first is based on an analytical model and the second on the numerical homogenization based on finite element calculation. To consider the effect of some influential parameters, several situations are treated by the combination of the variation of the volume fractions of the three phases and their property contrasts. The analytical calculations are performed using the Python 3.10 commercial software. It could be concluded that the effective elastic properties obtained either by the multistep or by the direct homogenization show no significant difference.