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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.
[1] Delisée, Ch., Jeulin, D., Michaud, F.: Caractérisation morphologique et porosité en 3D de matériaux fibreux cellulosiques. C.R. Académie des Sciences de Paris, t. 329, Série II b French (2001), 179-185.
[2] Dirrenberger, J., Forest, S., Jeulin, D.: Towards gigantic RVE sizes for 3D stochastic fibrous networks. Int. J. Solids Struct. 51 (2014), 359-376. DOI 10.1016/j.ijsolstr.2013.10.011
[3] Faessel, M., Jeulin, D.: 3D multiscale vectorial simulations of random models. Proceedings of ICS13 (2011), 19-22.
[4] Jeulin, D.: Modèles Morphologiques de Structures Aléatoires et de Changement d'Echelle. Thèse de Doctorat d'Etat è s Sciences Physiques, Université de Caen (1991).
[5] Jeulin, D.: Modèles de Fonctions Aléatoires multivariables. Sci. Terre French 30 (1991), 225-256.
[6] Jeulin, D.: Random structure models for composite media and fracture statistics. Advances in Mathematical Modelling of Composite Materials (1994), 239-289.
[7] Jeulin, D.: Random structure models for homogenization and fracture statistics. Mechanics of Random and Multiscale Microstructures D. Jeulin, M. Ostoja-Starzewski CISM Courses Lect. 430, Springer, Wien (2001), 33-91. DOI 10.1007/978-3-7091-2780-3_2 | Zbl 1010.74004
[8] Jeulin, D.: Morphology and effective properties of multi-scale random sets. A review, C. R. Mecanique 340 (2012), 219-229. DOI 10.1016/j.crme.2012.02.004
[9] Jeulin, D.: Boolean random functions. Stochastic Geometry, Spatial Statistics and Random Fields. Models and Algorithms V. Schmidt Lecture Notes in Mathematics 2120, Springer, Cham (2015), 143-169. MR 3330575 | Zbl 1366.60013
[10] Jeulin, D.: Power laws variance scaling of Boolean random varieties. Methodol. Comput. Appl. Probab. (2015), 1-15, DOI: 10.1007/s11009-015-9464-5. DOI 10.1007/s11009-015-9464-5 | MR 3564853
[11] Maier, R., Schmidt, V.: Stationary iterated tessellations. Adv. Appl. Probab. 35 (2003), 337-353. DOI 10.1017/S000186780001226X | MR 1970476 | Zbl 1041.60012
[12] Matheron, G.: Random Sets and Integral Geometry. Wiley Series in Probability and Mathematical Statistics John Wiley & Sons, New York (1975). MR 0385969 | Zbl 0321.60009
[13] Nagel, W., Weiss, V.: Limits of sequences of stationary planar tessellations. Adv. Appl. Probab. 35 (2003), 123-138. DOI 10.1017/S0001867800012118 | MR 1975507 | Zbl 1023.60015
[14] Schladitz, K., Peters, S., Reinel-Bitzer, D., Wiegmann, A., Ohser, J.: Design of acoustic trim based on geometric modeling and flow simulation for non-woven. Computational Materials Science 38 (2006), 56-66. DOI 10.1016/j.commatsci.2006.01.018
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