Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-12T17:17:04.296Z Has data issue: false hasContentIssue false
  • English
  • Français

Random patterns of nonoverlapping convex grains

Part of:Stochastic processesGeometric probability and stochastic geometryInference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Marianne Månsson and
Mats Rudemo
Marianne Månsson*
Affiliation:
Chalmers University of Technology
Mats Rudemo*
Affiliation:
Chalmers University of Technology
*
∗ Postal address: Department of Mathematical Statistics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden.
∗ Postal address: Department of Mathematical Statistics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden.
Get accessRights & Permissions [Opens in a new window]

Abstract

Generalizing Matérn's (1960) two hard-core processes, marked point processes are considered as models for systems of varying-sized, nonoverlapping convex grains. A Poisson point process is generated and grains are placed at the points. The grains are supposed to have varying sizes but the same shape as a fixed convex grain, with spheres as an important special case. The pattern is thinned so that no grains overlap.We consider the thinning probability of a ‘typical point’ under various thinning procedures, the volume fraction of the resulting system of grains, the relation between the intensity of the point processes before and after thinning, and the corresponding size distributions. The study is inspired by problems in material fatigue, where cracks are supposed to be initiated by large defects.

Keywords

Germ-grain processnonoverlapping grainsradius distributionPoisson processvolume fractionmaterial fatigueMatérn's hard-core processes

MSC classification

Primary:60D05: Geometric probability and stochastic geometry
Secondary:60G55: Point processes62M30: Spatial processes

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 4 , December 2002, pp. 718 - 738
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Berwald, L. and Varga, O. (1937). Integralgeometrie XXIV. Über die Schiebungen im Raum. Math. Z. 42, 710–736.CrossRefGoogle Scholar
Betke, U. (1992). Mixed volumes of polytopes. Arch. Math. 58, 388–391.Google Scholar
Blaschke, W. (1937). Integralgeometrie XXI. Über Schiebungen. Math. Z. 42, 399–410.CrossRefGoogle Scholar
Bonnesen, T. and Fenchel, W. (1948). Theorie der Konvexen Körper. Chelsea, New York.Google Scholar
Daley, D. J., Stoyan, H. and Stoyan, D. (1999). The volume fraction of a Poisson germ model with maximally non-overlapping spherical grains. Adv. Appl. Prob. 31, 610–624.CrossRefGoogle Scholar
Eggleston, H. G. (1963). Convexity. Cambridge University Press.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.CrossRefGoogle Scholar
Häggström, O. and Meester, R. (1996). Nearest neighbour and hard sphere models in continuum percolation. Random Structures Algorithms 9, 295–315.Google Scholar
Jeulin, D. (1989). Morphological modeling of images by sequential random functions. Signal Process. 16, 403–431.Google Scholar
Jeulin, D. (1993). Random models for morphological analysis of powder. J. Microscopy 172, 13–21.Google Scholar
Jeulin, D. (1998). Probabilistic models of structures. In PROBAMAT-21st Century: Probabilities and Materials, ed. Frantziskonis, G. N., Kluwer, Dordrecht, pp. 233–257.Google Scholar
Mase, S. (1985). On the possible form of size distributions for Gibbsian processes of mutually non-intersecting discs. J. Appl. Prob. 23, 649–659.Google Scholar
Matérn, B., (1960). Spatial Variation (Meddelanden Statens Skogsforskningsinst. 49). Statens Skogsforskningsinstitut, Stockholm. Second edition: Springer, Berlin, 1986.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Meakin, P. and Jullien, R. (1992a). Random-sequential adsorption of disks of different sizes. Phys. Rev. A 46, 2029–2038.Google Scholar
Meakin, P. and Jullien, R. (1992b). Random sequential adsorption of spheres of different sizes. Physica A 187, 475–488.Google Scholar
Miles, R. E. (1965). On random rotations in R3 . Biometrika 52, 636–639.Google Scholar
Murakami, Y. and Beretta, S. (1999). Small defects and inhomogeneities in fatigue strength: experiments, models and statistical implications. Extremes 2, 123–147.Google Scholar
Reiss, H., Frisch, H. L. and Lebowitz, J. (1959). Statistical mechanics of rigid spheres. J. Chem. Phys. 31, 369–380.Google Scholar
Santaló, L. A. (1976). Integral Geometry and Geometric Probability. Addison-Wesley, Reading, MA.Google Scholar
Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press.CrossRefGoogle Scholar
Stienen, H. (1982). Die Vergroeberung von Karbiden in reinen Eisen-Kohlenstoff Staehlen. Dissertation, RWTH Aachen.Google Scholar
Stoyan, D. (1990). Stereological formulae for a random system of non-overlapping spheres. Statistics 22, 449–462.Google Scholar
Stoyan, D. (1998). Random sets: models and statistics. Internat. Statist. Rev. 66, 1–27.Google Scholar
Stoyan, D. and Schlater, M. (2000). Random sequential adsorption: relationship to dead leaves and characterization of variability. J. Statist. Phys. 100, 969–979.Google Scholar
Stoyan, D. and Stoyan, H. (1985). On one of Matérn's hard-core point process models. Math. Nachr. 122, 205–214.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Talbot, J., Tarjus, G., Van Tassel, P. R. and Viot, P. (2000). From car parking to protein adsorption: an overview of sequential adsorption processes. Colloids Surfaces A 165, 287–324.Google Scholar
Torquato, S. (1995). Nearest neighbor statistics for packings of hard spheres and disks. Phys. Rev. E 51, 3170–3182.Google Scholar
Weil, W. (1990). Iterations of translative formulae and non-isotropic Poisson processes of particles. Math. Z. 205, 531–549.Google Scholar
Weil, W. and Wieacker, J. (1993). Stochastic geometry. In Handbook of Convex Geometry, Vol. B, eds Gruber, P. M. and Wills, J. M., North-Holland, Amsterdam, pp. 1391–1438.Google Scholar
Zoughi, R. et al. (1997). Real-time and on-line microwave inspection of surface defects in rolled steel. In Proc. Asia-Pacific Microwave Conf. (Hong Kong, December 1997), IEEE, New York, pp. 1081–1084.Google Scholar