We present the first space-filling bearing in three dimensions. It is shown that a packing which contains only loops with even number of spheres can be constructed in a self-similar way and that it can act as a three dimensional bearing in which spheres can rotate without slip and with negligible torsion friction. PACS numbers: 46.55.+d, 61.43.Bn, 91.45.Dh Space-filling bearings have been introduced in several contexts, such as in explaining the so-called seismic gaps [1,2] of geological faults in which two tectonic plates slide against each other with a friction much less than the expected one, without production of earthquakes or of any significant heat. Space-filling bearings have also been used as toy models for turbulence and can also be used in mechanical devices [3]. Two dimensional spacefilling bearings have been shown to exist and a discrete infinity of realizations has been constructed [4,5]. The remaining question still open is: Do they also exist in three dimensions? This question is of fundamental importance to the physical applications.In this letter, we will report the discovery of a selfsimilar space-filling bearing in which an arbitrary chosen sphere can rotate around any axis and all the other spheres rotate accordingly without any sliding and with negligible torsion friction.In two dimensions, different classes of space-filling bearings of disks have been constructed in Refs. [4,5] by requiring the loops to have an even number of disks, since in two dimensions this is obviously a necessary and sufficient condition for disks to be able to rotate without any slip. Successive disks must rotate, in alternation, clockwise and counter-clockwise in order to avoid frustration.The situation in three dimensions is different from two dimensions in two ways; The axes of rotation need not be parallel, and the centers of spheres in a loop may not lie all in the same plane. As a result, even in an isolated odd loop spheres could rotate without friction. But, as we will see, in the packings with an infinite number of interconnecting loops, we could construct unfrustrated configurations of rotating spheres when all loops have an even number of spheres. Such a packing is bichromatic, i.e., one can color the spheres using only two different colors in such a way that no spheres of same color touch each other, as shown in Fig.1.No three dimensional space-filling bearing has been known up to now. The classical Apollonian packing is space-filling and self-similar but not a bearing since at least five colors are needed to assure different colors at each contact. This packing can be constructed in different ways [6,7]. By generalizing the inversion technique used in Ref.[6] to other Platonic Solids than the tetrahedron (the base of 3D Apollonian packing) we were able to construct five new packings including a bichromatic one. Details on the construction algorithm and on the complete set of new configurations will be published elsewhere [8]. We give here only a qualitative description of this technique for the bichr...
We develop an algorithm to construct new self-similar space-filling packings of spheres. Each topologically different configuration is characterized by its own fractal dimension. We also find the first bicromatic packing known up to now.
Self-similar space-filling bearings have been proposed some time ago as models for the motion of tectonic plates and appearance of seismic gaps. These models have two features which, however, seem unrealistic, namely, high symmetry in the arrangement of the particles, and lack of a lower cutoff in the size of the particles. In this work, an algorithm for generating random bearings in both two and three dimensions is presented. Introducing a lower cutoff for the sizes of the particles, the instabilities of the bearing under an external force such as gravity, are studied.
We present a systematic overview of granular deposits composed of ellipsoidal particles with different particle shapes and size polydispersities. We study the density and anisotropy of such deposits as functions of size polydispersity and two shape parameters that fully describe the shape of a general ellipsoid. Our results show that, while shape influences significantly the macroscopic properties of the deposits, polydispersity plays apparently a secondary role. The density attains a maximum for a particular family of non-symmetrical ellipsoids, larger than the density observed for prolate or oblate ellipsoids. As for anisotropy measures, the contact forces show are increasingly preferred along the vertical direction as the shape of the particles deviates for a sphere. The deposits are constructed by means of an efficient molecular dynamics method, where the contact forces are efficiently and accurately computed. The main results are discussed in the light of applications for porous media models and sedimentation processes.
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