Effective property estimates for n-phase composites with from all to none co-continuous phases

Several papers from the author and co-workers have been dedicated in the last years to estimating effective properties of n-phase composite materials (∀n ≥ 2) with all phases being co-continuous. The proposed estimate started from an old literature on that point and several of the presented formal results provided new highlights for such phase architectures that are not accounted for by the classical homogenization schemes. By duality, the case when none of the n phases in such composites can be taken as an infinite matrix (all discontinuous phases, say) was also treated, with a general explicit form being given for a relevant effective property estimate. Disregarding multi-level phase arrangements, we here treat the arrangement that still has no estimate so far, in terms of effective properties from the general homogenization framework for micro-heterogeneous and macro-homogeneous media: the case with several (p) but not all (n) phases being co-continuous. This assemblage type exists from three-phase arrangements and becomes the most important in number from four assembled phases. The proposed estimate form is based on the determination of the appropriate properties to be given to a reference n+1 matrix phase with infinitesimal volume fraction for satisfying both the co-continuity of the p phases and the embedded nature of the n-p not continuous (included) ones. For more than two co-continuous phases and any number of included ones, these reference matrix properties are shown to result from solving an inverse problem. We provide, up to five co-continuous phases and for several material symmetry types of interest, an analytical solution for these reference matrix properties. We exemplify the results, holding for various (elastic- or dielectric-like, possibly coupled) properties and structure symmetries, for three continuous phases. With these results all possibilities of assembling n phases from all to none continuous ones have relevant theoretical estimates, as exemplified for n=3, including some comparisons with Finite Element calculations.