Non-overlapping Germ-grain Models: Characteristics and Material Modelling

Abstract: We consider three different non--overlapping germ--grain models, two of which are constructed in a similar fashion. These two models are generalisations of Mat'ern's hard--core models. In both cases we start with a homogeneous Poisson process and use the points as centres of convex sets, grains, of the same shape. The process is thinned so that no grains overlap. Two different thinning schemes result in the two models. The pair--correlation functions and the mark--correlation functions for both models are derived. The models are fitted to images of inclusions in cast iron. For one of the models above, if the thinning is performed independently of the grain sizes we show that the volume fraction is at most $1/2^d$ for dimension $d=1$ or $2$. If the thinning is performed dependently of the grain sizes, it is possible to achieve volume fraction arbitrarily close to one for any dimension. The third non--overlapping germ--grain model is a Voronoi tessellation. It is used as a model for the grain structure of the surface of a metal. As an example of this approach, we study the influence of grain structure on fatigue life. A crack growth model is applied to simulated grain structures. The conclusion is that the fatigue life increases, compared to a model with grains of equal size.