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Boolean model; Boolean varieties; Cox process; weakest link model; fracture statistics; mathematical morphology
Models of random sets and of point processes are introduced to simulate some specific clustering of points, namely on random lines in $ \mathbb {R}^{2}$ and $\mathbb {R} ^{3}$ and on random planes in $ \mathbb {R}^{3}$. 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.
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