J. Lemaitre

Arrangement of cells in Voronoi tesselations of monosize packing of discs

Abstract We describe a two-dimensional mosaic obtained by the Voronoi tesselation of a monosize assembly of discs at different packing fractions. The experimental device (hard discs moving on an air table) produces, for every concentration of the discs, a succession of mosaics in statistical equilibrium, which constitutes a statistical ensemble. This ensemble is large enough for fluctuations from the most probable distributions to be negligible. Both topological and metric properties show deviations from those of a totally random mosaic. These deviations can be ascribed to steric exclusions. In particular, distributions of the numbers of sides, of the perimeters and of the areas of the polygons differ from those observed in biological celi assemblies. The Aboav law holds, but with a slope which can be as Iow as 4–66. The Lewis law is obeyed only for smali packing fractions. The variance of the distribution of the number n of polygon sides is a universal function of p 6, the probability of finding a six-si...

Experimental Study of Densification of Disc Assemblies

An air cushion table that we have built and tested has been used to study the structure of 2D assemblies of identical discs of increasing density, going from very dilute systems up to the most compact crystalline array. We analyse the radial and angular distribution functions, calculated from an image analysis treatment, and the progressive structuration of these assemblies. The disorder-order transition is observed for a packing fraction close to the theoretical value of 0.82. It is confirmed by a study of the geometry of the Voronoi polygons.

Arrangement of discs in 2d binary assemblies

We study arrangements of the two species of discs in binary assemblies at an intermediate scale. Small discs rearrange along large ones in clusters whose mass and compactness are analyzed with the tools of percolation. The assemblies are generated analogically on an air table or numerically from RSA or Powell algorithms. At a given packing fraction, an infinite cluster of small discs exists above a critical composition; a phenomenological expression for this threshold is proposed. Like in usual percolation problems, the number of inner links in a cluster is a linear function of its mass, with a slope depending both on the packing fraction, the composition of the mixture and the building procedure. An approximate expression is derived for it.

Application of the Voronoi Tessellations to the Study of Flow of Granular Materials

This chapter provides a very simple method to simulate the displacement of a sphere in interaction with a bi-dimensional or tri-dimensional structure. These structures are modeled as assemblies of spheres glued on a plane (2D) or packed inside a tank (3D). The displacement mainly occurs in the space in between the spheres, so a subnetwork made by the Voronoi tessellation of the previous structures can describe nicely this open space. Each vertex of this tessellation is linked to its neighbors (3 at 2D and 4 at 3D) and creates a continuous network through which the moving sphere can travel. By using a series of assumptions for the calculation of the probabilities to use each link and a Monte Carlo method for the choice of them after, we have simulated the diffusion process of the moving sphere. These simulations correspond to two experimental setups used in our laboratory and agree very well with the experimental results.

Dense Packings of Hard Grains: Effects of Grain Size Distribution

A packing of grains, spherical or not, is often described by its mean properties: geometrical properties: porosity: φ = Void volume/Total volume coordinance z: mean number of real contacts/grain, etc... physical or physicochemical properties: electrical conductivity (in “pore space” as well as in “grain space”) permeability, etc...