The boundary-due terms in the Green operator of inclusion patterns from distant to contact and to connected situations using radon transforms: Illustration for spheroid alignments in isotropic media

We examine the boundary-due components in the mean modified Green operator integral (Green operator for short) of an inclusion pattern in distant-contact and contact-connection transitions. The Direct (RT) and inverse (IRT) Radon Transforms, which allow specification of the different contributions to the mean Green operator of the pattern in simple geometrical terms, are used. The already well-documented case of axially symmetric alignments of equidistant identical oblate spheroids, in an infinite matrix of isotropic (elastic-like or dielectric-like) properties is treated up to infinite alignments and for any aspect ratio from unity (spheres) to infinitesimal (platelets). Simple closed forms for this mean Green operator and for its different parts are newly obtained. These closed forms allow an easy parametric study of the operator variations in terms of the alignment characteristics from distant to contact situations. From contact to connection of the inclusions, the changes in the Green operator’s contributions are pointed, what provides relevant operator forms for the connected patterns. These results are of interest in problems where phase percolation, connectivity inversions or co-continuity are implied.