Abstract
In this paper, an assembly of disordered packings is considered as a suitable set of packing cells of ordered spheres. In consequence, any of its parameters can be obtained by averaging the values of the set. Namely, the density of a packing of ordered spheres is described by two variables: the angle of the base, and the angle of the inclined edge of the associated parallelepiped. Then, the density of a packing of disordered spheres is obtained by averaging the angle of the base, and the subsequent averaging of the other angle, according to the kind of strain induced by the experiment. The average packing yields the density limits of loose sphere assemblies achieved by a process of fluidization and sedimentation in air, in water, and in viscous liquid at zero gravitational force. It also models the close sphere assemblies shaped by gentle tapping, vertical shaking, horizontal and multidirectional vibrations. The theory allows to elucidate the mechanism of each of the limits, as, for example, the metastable columns of spheres in the loosest packing, as well as the random close packing, and crystallization. The limits obtained coincide very well with the published experimental, numerical and theoretical data.
Highlights
Granular materials have complex mechanical behaviour as a consequence of the complicated interaction between grains
The geometry of ordered packings obeys the laws of lattices or crystalline arrangements, and the volume of their phases can be calculated, since it is always possible to select a parallelepiped, whose vertices are the center of a sphere (Fig. 1a). In this primitive unit cell, the spheres of the base are not necessarily in contact, and the spheres of two successive layers are in contact at the middle point of the inclined edge
The loosest state of the packing is obtained by maximizing the equation (2) with respect to the angle of the inclined edge, and with respect to the angle of the base of the parallelepiped
Summary
Granular materials have complex mechanical behaviour as a consequence of the complicated interaction between grains. The model should allow to solve complicated field problems of stress distribution, strain, and failure. To meet this objectives, the model should be as simple as possible. That is the disorderly and deformable packing of rigid spheres, the rearrangement of which results in the change of shape of the packing For this model, one of the main tasks is the determination of the extreme packings, associated to the loosest and densest states. One of the main tasks is the determination of the extreme packings, associated to the loosest and densest states Another problem is the modelling of the strain response, whose most important component is the displacement of the spheres. Paraphrasing Tolman [7], the proposed method is to be regarded as really statistical in character, and the results which they provide are to be regarded as true on the average for the systems in an appropriately chosen set, rather than as necessarily precisely true in any individual case
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