Morphological properties of random arrays of infinitely long overlapping cylinders for modeling statistically homogeneous and isotropic fibrous media.

Abstract

In this article, we examine random disordered arrays of infinitely long cylinders of revolution, all with the same diameter and allowed to overlap. These arrangements can be considered as simple representations of fibrous media, which are widely used in the field of high-performance thermal insulation. In a cylinder of revolution of infinite length, the expression of the cumulative distribution function of internal i-random chord lengths F_{i}(s) is given in terms of elliptic integrals, and some notable properties are highlighted. It is shown that there is a link between F_{i}(s) and the sphere/solid cylinder intersection area. A statistically homogeneous and isotropic arrangement of infinite cylinders is then studied. An algorithm for generating such a medium is reviewed, and the resulting elementary properties are given. Finally, a study in the field of stochastic geometry makes it possible to determine an analytical expression for the autocorrelation function (two-point probability function) inside the solid phase and, more generally, for the number of overlaps probability function. This makes it possible to deduce, in the Laplace domain, an expression for the cumulative distribution function of i-random chord lengths F_{i}(Π,s) within the solid phase for any porosity Π. The autocorrelation function also allows one to define a representative elementary volume (REV) related to local fluctuations in porosity.