Iterated Boolean random varieties and application to fracture statistics models

Abstract

Models of random sets and of point processes are introduced to simulate some specific clustering of points, namely on random lines in R2 and R3 and on random planes in R3. The corresponding point processes are special cases of Cox processes. The generating distribution function of the probability distribution of the number of points in a convex set K and the Choquet capacity T (K) are given. A possible application is to model point defects in materials with some degree of alignment. Theoretical results on the probability of fracture of convex specimens in the framework of the weakest link assumption are derived, and used to compare geometrical effects on the sensitivity of materials to fracture.