On the modified Green operator integral for polygonal, polyhedral and other non-ellipsoidal inclusions

A “strange” particularity of polyhedral inclusions and of fibres of regular polygonal cross sections has been recently stressed in the literature. For respectively fully or transversally isotropic elasticity of the embedding material, they have a mean and a central Green operator integral both equal to the uniform one of respectively the sphere or the cylindrical fibre. In using the Radon transform (RT) method, this particularity is here shown to be shared by much larger shape types in the same limits of material elasticity symmetry. As a subcase, even more shape types fulfill the similar particularity for material linear properties of second-rank characteristic tensor, such as thermal conductivity, magnetic or dielectric properties. When calculated using the RT method, the modified Green operator integral at any interior point of a bounded domain (inclusion) takes the form of a weighted average over an angular distribution of a single elementary operator. The weight function is geometrically defined from the characteristic function of the domain, and the four-rank or second-rank elementary operator depends on the material linear property of concern. The RT method simply shows that the noticed particularity is due to matching symmetry between the inclusion shape (through the weight function) and the material property (through the elementary operator). The general geometrical characteristics of the inclusion shapes belonging to these sphere-class and cylindrical-fibre-class are specified, and some remarkable shapes of these classes are commented.