Fluctuations in granular media

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Abstract
Dense slowly evolving or static granular materials exhibit strong force fluctuations even though the spatial disorder of the grains is relatively weak. Typically, forces are carried preferentially along a network of “force chains.” These consist of linearly aligned grains with larger-than-average force. A growing body of work has explored the nature of these fluctuations. We first briefly review recent work concerning stress fluctuations. We then focus on a series of experiments in both two- and three-dimension [(2D) and (3D)] to characterize force fluctuations in slowly sheared systems. Both sets of experiments show strong temporal fluctuations in the local stress/force; the length scales of these fluctuations extend up to 102 grains. In 2D, we use photoelastic disks that permit visualization of the internal force structure. From this we can make comparisons to recent models and calculations that predict the distributions of forces. Typically, these models indicate that the distributions should fall off exponentially at large force. We find in the experiments that the force distributions change systematically as we change the mean packing fraction, γ. For γ’s typical of dense packings of nondeformable grains, we see distributions that are consistent with an exponential decrease at large forces. For both lower and higher γ, the observed force distributions appear to differ from this prediction, with a more Gaussian distribution at larger γ and perhaps a power law at lower γ. For high γ, the distributions differ from this prediction because the grains begin to deform, allowing more grains to carry the applied force, and causing the distributions to have a local maximum at nonzero force. It is less clear why the distributions differ from the models at lower γ. An exploration in γ has led to the discovery of an interesting continuous or “critical” transition (the strengthening/softening transition) in which the mean stress is the order parameter, and the mean packing fraction, γ, must be adjusted to a value γc to reach the “critical point.” We also follow the motion of individual disks and obtain detailed statistical information on the kinematics, including velocities and particle rotations or spin. Distributions for the azimuthal velocity, Vθ, and spin, S, of the particles are nearly rate invariant, which is consistent with conventional wisdom. Near γc, the grain motion becomes intermittent causing the mean velocity of grains to slow down. Also, the length of stress chains grows as γ→γc. The 3D experiments show statistical rate invariance for the stress in the sense that when the power spectra and spectral frequencies of the stress time series are appropriately scaled by the shear rate, Ω, all spectra collapse onto a single curve for given particle and sample sizes. The frequency dependence of the spectra can be characterized by two different power laws, P∝ω−α, in the high and low frequency regimes: α∼2 at high ω; α<2 at low ω. The force distributions computed from the 3D stress time series are at least qualitatively consistent with exponential fall-off at large stresses.