Analytical results for random line networks applications to fracture networks and disordered fiber composites

Abstract

We study the geometrical properties of a continuum model consisting of an ensemble of intersecting lines placed within a finite region in two dimensions. We consider the model in the high-density regime where it emulates fiber composites, filtering elements and fracture networks. We report exact results for the probability distribution of segment lengths produced by line intersections. Segment loop perimeter and area distributions are estimated numerically. For the probability distribution of perimeters, we found the interesting phenomenon of small pore perimeter power law repulsion for pores of four or more sides. Two subsidiary models, with infinite line length, are solved exactly and reveal the salient features of the finite line model. Important applications to fiber filter selectivity and optimal intersection trajectories in fracture networks are discussed.