Homogenization-Based Mechanical Behavior Modeling of Composites Using Mean Green Operators for Infinite Inclusion Patterns or Networks Possibly Co-continuous with a Matrix

The Fourier–Green homogenization method for estimating the behavior of composites was first developed for aggregates and inhomogeneity-reinforced (-weakened) matrices, based on Eshelby (Proc R Soc Lond, A 421:379–396, 1957, Proc R Soc Lond, A 252:561–569, 1959) solution of the isolated inclusion problem. The need to address increasingly complex structures opened fruitful development routes, firstly solving the inhomogeneity pair interaction problem and the one of heterogeneous (double or multilayered) inhomogeneities, in order to account for inclusion dense concentrations and patterns. This work reports recent developments from the authors and co-workers which examined in that framework possibly infinite inclusion patterns, possibly arranged into an infinite network possibly co-continuous with the embedding matrix and possibly evolving under strain. The proposed modeling amounts to determining the representative mean Green operator (mGO) for the infinite pattern or network in its current (strain evolving) state. Once the method foundations being summarized, previously solved “elementary” cases are recalled, concerning infinite coaxial alignments of spheres, spheroids or finite cylinders and planar alignments of infinite parallel cylinders or rectangular beams. It is next shown how other complex patterns or networks could be represented in combining such elementary ones. The mGO solution for a new family of inhomogeneous axial inclusion alignments is reported to support the discussion. Potential other application fields are evoked.