Random patterns of nonoverlapping convex grains

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.