On the cluster bounds for the effective properties of microcracked solids

The paper is concerned with the problem of predicting macroscopic properties of solids, containing randomly distributed penny-shaped microcracks. It is proposed to employ the formalism of marked sets of random points, treating the mark as orientation of the crack located at a given point. As a first and simplest application of this approach the so-called first-order cluster bound of Torquato (1986 J. Chem. Phys. 84, 6345‐6359) on the eective scalar conductivity is derived. It turns out that the bound does not depend on the two- and three-point statistics of the distribution of cracks, unlike the case of dispersions of spheres or spheroids. The “optimality” of the bound within a much wider class of “cluster type” trial fields is demonstrated. In the elastic case the cluster bounds are explicitly derived as well, and are again found to be independent of cracks’ statistics. In both scalar and elastic cases the bounds coincide with the so-called “approximation of non-interacting cracks” thus rigorously proving that for the assumed isotropic cracks’ statistics the interactions always decrease the conductivity and the elastic moduli of a microcracked solid.