Analytical mean Green operators/Eshelby tensors for patterns of coaxial finite long or flat cylinders in isotropic matrices

Recently, an analytical solution has been provided for mean and axial Green operators (GO) of circular cylindrical inclusions with finite length in 3D isotropic (elastic-like and dielectric-like) materials, from long rods to truly penny-shaped ones. This result provides a situation for examining the commonly used “shape spheroidal approximation” of same aspect ratio as well as possibly more appropriate alternatives. For cases when such a spheroidal approximation would remain needed, we propose and compare an “operator spheroidal approximation” in terms of best matching GO, which is exactly operator-identical for dielectric-like properties and nearly identical for elastic-like ones. Next, a simple analytical solution is explicated for the mean isotropic GO of all patterns of coaxial circular long or flat cylinders with same radii, obtained from solving the coaxial cylinder pair interaction problem. These patterns can be seen as either fragmented cylinders or as cylinder alignments. The obtained GO solution form is further shown to formally hold for similar patterns of domains having more general shapes. The established general property of such patterns is that the interaction part of their mean GO can be expressed as a linear combination of the mean GO of all the pattern sub-domains. The mean GO solution for one such pattern is further proved to be valid as well for the domains - of the double inclusion type - obtained in connecting the pattern elements by their outer envelop of zero thickness, thanks to the GO blindness to connectivity differences. This GO equivalency is finally used to define a global operator spheroidal approximation in terms of best matching GO for any such coaxial cylinder pattern. Effective properties for an isotropic matrix reinforced by congruent cylinders or patterns are estimated within a mean field approximation context.