Statistical mechanical considerations on the random packing of granular materials

Abstract

The historical development on the random packing of granular materials is briefly discussed. The geometrical probability space for granular packings is then introduced and the significance and relevance of characteristic microelements, i.e. ‘Voronoi polyhedra’, is elaborated upon. The classical statistical mechanical theory based on Boltzmann's postulate is remodeled and applied to a collection of ‘Voronoi cells’ in the form of routine ensemble phase averages. The exact probability density functions for the distribution of void ratios in a random aggregate of granular materials are then found to be exponential and the associated partition functions are obtained. The statistical entropy is shown to be a global maximum for the ‘loose random packing’ state which is also equivalent to the critical state reached in simple shearing of such aggregates. At such states the distribution density is shown to be uniform. Furthermore, it is shown that for random close packings the distribution density is skewed towards the denser ‘Voronoi cells’. Exact expressions are given for the expected values of the critical void ratios as well as critical porosities for random loose packing states of both three- and two-dimensional granular aggregates. Similar to three-dimensional packings, it is shown that there exist two critical porosities for the random two-dimensional packing; one corresponding to random loose packing for which n cr ≈ 0.160 and another for random close packing for which n cr ≈ 0.111. Finally, in an Appendix, the connections between the statistical mechanical considerations, the information theoretic considerations, and Jayne's postulate on minimally biased distribution are presented.