Critical and noncritical jamming of frictional grains

Abstract

We probe the nature of the jamming transition of frictional granular media by studying their vibrational properties as a function of the applied pressure p and friction coefficient . The density of vibrational states exhibits a crossover from a plateau at frequencies տ * ͑p , ͒ to a linear growth for Շ * ͑p , ͒. We show that * is proportional to ⌬z, the excess number of contacts per grain relative to the minimally allowed, isostatic value. For zero and infinitely large friction, typical packings at the jamming threshold have ⌬z → 0, and then exhibit critical scaling. We study the nature of the soft modes in these two limits, and find that the ratio of elastic moduli is governed by the distance from isostaticity. DOI: 10.1103/PhysRevE.75.020301 PACS number͑s͒: 45.70.Ϫn, 46.65.ϩg Granular media, such as sand, are conglomerates of dissipative, athermal particles that interact through repulsive and frictional contact forces. When no external energy is supplied, these materials jam into a disordered configuration under the action of even a small confining pressure ͓1͔. In recent years, much new insight has been amassed about the jamming transition of models of deformable, spherical, athermal, frictionless particles in the absence of gravity and shear ͓2͔. The beauty of such systems is that they allow for a precise study of the jamming transition that occurs when the pressure p approaches zero ͑or, geometrically, when the particle deformations vanish͒. At this jamming point J and for large systems, the contact number ͓3͔ equals the so-called isostatic value z iso 0 ͑see below͒, while the packing density J 0 equals random close packing ͓2,4͔. Moreover, for compressed systems away from the jamming point, the pressure p, the excess contact number ⌬z = z͑p͒ − z iso 0 , and the excess density ⌬ = − J 0 are related by power-law scaling relations-any one of the parameters p , ⌬z, and ⌬ is sufficient to characterize the distance to jamming.Isostatic solids are marginal solids-as soon as contacts are broken, extended "floppy modes" come into play ͓5͔. Approaching this marginal limit in frictionless packings as p → 0, the density of vibrational states ͑DOS͒ at low frequencies is strongly enhanced-the DOS has been shown to become essentially constant up to some low-frequency crossover scale * , below which the continuum scaling ϳ d−1 is recovered ͓2,6-11͔. For small pressures, * vanishes ϳ⌬z. This signals the occurrence of a critical length scale, when translated into a length via the speed of sound, below which the material deviates from a bulk solid ͓9͔. The jamming transition for frictionless packings thus resembles a critical transition.In this paper we address the question whether an analogous critical scenario occurs near the jamming transition at p =0 of frictional packings. The Coulomb friction law states that, when two grains are pressed together with a normal force F n , the contact can support any tangential friction force F t with F t Յ F n , where is the friction coefficient. In typical packings, essentially non...

Pushing on a Nonlinear Material

“…One should not be surprised to observe a finite z 0 below jamming: when the vibration is turned off, the structure is quenched abruptly from a vibrational state where the averaged number of contact need not be zero. The sublinear increase of z 0 with packing fraction is compatible with the one obtained in simulations of frictional particles [17,38]. The initial pressure P 0 also increases above jamming from a small residual value below jamming, again inherited from the vibrational state [see Fig.…”

The jamming scenario—an introduction and outlook

“…In simulations of frictional systems, it is indeed indeed found that packings at the jamming threshold with different values of Z and density φ can be made, by varying in particular the quench rate; moreover, there is a clear trend for the average contact number Z to approach the lower lower limit d + 1 -the isostatic value in the presence of friction -in the limit µ → ∞ [7,82]. In line with this, one finds that if one studies the linear vibration-rotation modes of these packings, the density of states only develops a plateau in this large friction limit [81]. However, it was later discovered that the situation is more subtle.…”

“…In the presence of static friction, therefore, the properties of a packing are very history dependent, as the tangential forces are not only dependent on the configuration, but also on how the configuration was made -see e.g. [78][79][80][81].…”

Reflection of Nonlinear Acoustic Waves from the Mechanically Free Surface of an Unconsolidated Granular Medium

“…consistently appears in computer experiments with shear loading near jamming transition [4][5][6][7][8][9] and can be supported by some analytical arguments [7]. The theories relate faster decrease of the elastic shear modulus with pressure in comparison with the bulk modulus to an additional relaxation of the applied shear stress through a non-affine motion of the beads, which does not influence importantly bulk modulus [4][5][6][7].…”

“…In particular, behavior of granular media near the jamming transition at zero pressure, where with increasing pressure the emergence of the solid-like elasticity appears, still contains many puzzles [4][5][6][7][8][9]. In addition, the low-pressure region near the phase transition to the solid-like state in a granular material is extremely difficult to access experimentally.…”