The manufacture of high resistance concrete or hard ceramics needs extremely dense granular packings. They can only be realised when the size distribution of grains is strongly polydisperse. Typically powerlaw distributions give the best results. We present a simple packing model for polydisperse distributions, namely a generalized reversible parking lot model. We also discuss the perfectly dense limit, namely Apollonian packings in three dimensions and show in particular the existence of space filling bearings rotating without slip and without torsion. I IntroductionThe search for the perfect packing has a long history [1] and although much is known about monodisperse or bidisperse systems, the real challenge lies in polydispersity. Materials of very high resistance made of an originally granular mixture as it is the case for high performance concrete (HPC) [2] and for hard ceramics are manufactured by trying to reach the highest possible densities. From the fracture mechanics point of view, higher densities imply less and smaller microcracks and therefore higher resistance and reliability. This goal can be reached as shown clearly for the case of HPC by mixing grains of very different sizes (gravel, sand, ordinary cement, limestone filler, silica fume), where the size distribution of the mixture follows as closely as possible a powerlaw distribution. In fact it is known that configurations of density one are obtained for spherical particles in so-called Apollonian packings (albeit not yet physically realisable) and constitute the idealized final goal of a completely space filling packing having absolutely no defects.In the studies presented here, on the one hand we generalize a toy model for granular compaction, namely the reversible parking lot model to size distributions following a powerlaw. On the other hand we will discuss possible different realisations and self similar packings in three dimensions. II A reversible parking lot model for polydisperse size distributionsParking lot models have served as simple representations of compaction phenomena. Ben-Naim and Krapivski [3] introduced a reversible parking lot model to describe the compaction dynamics of monodisperse packings. They found an asymptotically logarithmic approach to a final density which was confirmed experimentally by Knight et al. [4]. The model is defined within a one-dimensional interval on which particles of fixed size are randomly absorbed with a rate k + and desorbed with a rate k − . The density reached after an infinite time depends on the ratio k − /k + and is unity when this ratio vanishes. For strongly polydisperse size distributions, this model must be considerably modified in order to still make sense [5].• A finite reservoir of particles must be considered in order to keep the distributions the same and this reservoir must be essentially not larger than the actual particles one would need to fill the interval.• The system must be initialized very carefully by putting first the large particles, otherwise small particles will cre...
Dense packings of granular systems are of fundamental importance in the manufacture of hard ceramics and ultra strong concrete. We generalize the reversible parking lot model to describe polydisperse dynamic packings. The key ingredient lies in the size distribution of grains. In the extreme case of perfect filling of spherical beads (density one), one has Apollonian tilings with a powerlaw distribution of sizes. We will present the recent discovery of 3D packings which also have the freedom to rotate (bearings) in three dimensions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.