Using the radon transform to solve inclusion problems in elasticity

Abstract

From the Fourier transform method, the modified Green operator integral over a bounded domain in an infinite elastic medium takes, on each domain point, the form of a weighted average over an angular distribution of a single elementary operator. The Radon transform provides a geometric definition of the weight function characteristic of the domain shape, in terms of the domain intersections with all planes passing through the point. It allows a geometrically more meaningful analytical resolution of the general inclusion problem in an infinite medium of general elasticity symmetry, the “inclusion” being any bounded domain possibly made of groups (or distributions) of inclusions. The method is also likely to provide insights in the related problem of effective moduli estimates for heterogeneous microstructures. The determination of the weight functions characteristics of the involved inclusional domain shapes is therefore a key step of the resolution, the mean values of these weight functions being of first-order interest. Here, it is exemplified, on the case of cuboidal domain shapes, that for material morphologies involving shapes of hardly accessible exact mean weight functions, one can make use of approximate (conveniently analytical) expressions, to remain more accurate than using ellipsoidal approximations of the shapes.