Internal states of model isotropic granular packings. II. Compression and pressure cycles

This is the second paper of a series of three investigating, by numerical means, the geometric and mechanical properties of spherical bead packings under isotropic stresses. We study the effects of varying the applied pressure P (from 1 or 10 kPa up to 100 MPa in the case of glass beads) on several types of configurations assembled by different procedures, as reported in the preceding paper. As functions of P, we monitor changes in solid fraction \Phi, coordination number z, proportion of rattlers (grains carrying no force) x0, the distribution of normal forces, the level of friction mobilization, and the distribution of near neighbor distances. Assuming the contact law does not involve material plasticity or damage, \Phi is found to vary very nearly reversibly with P in an isotropic compression cycle, but all other quantities, due to the frictional hysteresis of contact forces, change irreversibly. In particular, initial low P states with high coordination numbers lose many contacts in a compression cycle, and end up with values of z and x0 close to those of the most poorly coordinated initial configurations. Proportional load variations which do not entail notable configuration changes can therefore nevertheless significantly affect contact networks of granular packings in quasistatic conditions.Comment: Published in Physical Review E 12 page

Internal states of model isotropic granular packings. III. Elastic properties

In this third and final paper of a series, elastic properties of numerically simulated isotropic packings of spherical beads assembled by different procedures and subjected to a varying confining pressure P are investigated. In addition P, which determines the stiffness of contacts by Hertz's law, elastic moduli are chiefly sensitive to the coordination number, the possible values of which are not necessarily correlated with the density. Comparisons of numerical and experimental results for glass beads in the 10kPa-10MPa range reveal similar differences between dry samples compacted by vibrations and lubricated packings. The greater stiffness of the latter, in spite of their lower density, can hence be attributed to a larger coordination number. Voigt and Reuss bounds bracket bulk modulus B accurately, but simple estimation schemes fail for shear modulus G, especially in poorly coordinated configurations under low P. Tenuous, fragile networks respond differently to changes in load direction, as compared to load intensity. The shear modulus, in poorly coordinated packings, tends to vary proportionally to the degree of force indeterminacy per unit volume. The elastic range extends to small strain intervals, in agreement with experimental observations. The origins of nonelastic response are discussed. We conclude that elastic moduli provide access to mechanically important information about coordination numbers, which escape direct measurement techniques, and indicate further perspectives.Comment: Published in Physical Review E 25 page

Internal states of model isotropic granular packings. I. Assembling process, geometry and contact networks

This is the first paper of a series of three, reporting on numerical simulation studies of geometric and mechanical properties of static assemblies of spherical beads under an isotropic pressure. Frictionless systems assemble in the unique random close packing (RCP) state in the low pressure limit if the compression process is fast enough, slower processes inducing traces of crystallization, and exhibit specific properties directly related to isostaticity of the force-carrying structure. The different structures of frictional packings assembled by various methods cannot be classified by the sole density. While lubricated systems approach RCP densities and coordination number z^*~=6 on the backbone in the rigid limit, an idealized "vibration" procedure results in equally dense configurations with z^*~=4.5. Near neighbor correlations on various scales are computed and compared to available laboratory data, although z^* values remain experimentally inaccessible. Low coordination packings have many rattlers (more than 10% of the grains carry no force), which should be accounted for on studying position correlations, and a small proportion of harmless "floppy modes" associated with divalent grains. Frictional packings, however slowly assembled under low pressure, retain a finite level of force indeterminacy, except in the limit of infinite friction.Comment: 29 pages. Published in Physical Review