We study the origin of the scaling behavior in frictionless granular media above the jamming transition by analyzing their linear response. The response to local forcing is non-self-averaging and fluctuates over a length scale that diverges at the jamming transition. The response to global forcing becomes increasingly nonaffine near the jamming transition. This is due to the proximity of floppy modes, the influence of which we characterize by the local linear response. We show that the local response also governs the anomalous scaling of elastic constants and contact number.

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...

We present numerical simulations of acoustic wave propagation in confined granular systems consisting of particles interacting with the three-dimensional Hertz-Mindlin force law. The response to a short mechanical excitation on one side of the system is found to be a propagating coherent wavefront followed by random oscillations made of multiply scattered waves. We find that the coherent wavefront is insensitive to details of the packing: force chains do not play an important role in determining this wavefront. The coherent wave propagates linearly in time, and its amplitude and width depend as a power law on distance, while its velocity is roughly compatible with the predictions of macroscopic elasticity. As there is at present no theory for the broadening and decay of the coherent wave, we numerically and analytically study pulse-propagation in a one-dimensional chain of identical elastic balls. The results for the broadening and decay exponents of this system differ significantly from those of the random packings. In all our simulations, the speed of the coherent wavefront scales with pressure as p 1/6 ; we compare this result with experimental data on various granular systems where deviations from the p 1/6 behavior are seen. We briefly discuss the eigenmodes of the system and effects of damping are investigated as well.

Shear induced alignment of elongated particles is studied experimentally and numerically. We show that shear alignment of ensembles of macroscopic particles is comparable even on a quantitative level to simple molecular systems, despite the completely different types of particle interactions. We demonstrate that for dry elongated grains the preferred orientation forms a small angle with the streamlines, independent of shear rate across three decades. For a given particle shape, this angle decreases with increasing aspect ratio of the particles. The shear-induced alignment results in a considerable reduction of the effective friction of the granular material.

The creation of fractal clusters by diffusion limited aggregation (DLA) is studied by using iterated stochastic conformal maps following the method proposed recently by Hastings and Levitov. The object of interest is the function Φ (n) which conformally maps the exterior of the unit circle to the exterior of an n-particle DLA. The map Φ (n) is obtained from n stochastic iterations of a function φ that maps the unit circle to the unit circle with a bump. The scaling properties usually studied in the literature on DLA appear in a new light using this language. The dimension of the cluster is determined by the linear coefficient in the Laurent expansion of Φ (n) , which asymptotically becomes a deterministic function of n. We find new relationships between the generalized dimensions of the harmonic measure and the scaling behavior of the Laurent coefficients.