Fluctuations in granular media

Dense slowly evolving or static granular materials exhibit strong force flu ctuations even though the spatial disorder of the grains is relatively weak . Typically, forces are carried preferentially along a network of "force ch ains." These consist of linearly aligned grains with larger-than-average fo rce. 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) an d (3D)] to characterize force fluctuations in slowly sheared systems. Both sets of experiments show strong temporal fluctuations in the local stress/f orce; the length scales of these fluctuations extend up to 10(2) grains. In 2D, we use photoelastic disks that permit visualization of the internal fo rce structure. From this we can make comparisons to recent models and calcu lations that predict the distributions of forces. Typically, these models i ndicate that the distributions should fall off exponentially at large force . We find in the experiments that the force distributions change systematic ally as we change the mean packing fraction, gamma. For gamma's typical of dense packings of nondeformable grains, we see distributions that are consi stent with an exponential decrease at large forces. For both lower and high er gamma, the observed force distributions appear to differ from this predi ction, with a more Gaussian distribution at larger gamma and perhaps a powe r law at lower gamma. For high gamma, the distributions differ from this pr ediction 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 mode ls at lower gamma. An exploration in gamma has led to the discovery of an i nteresting continuous or "critical" transition (the strengthening/softening transition) in which the mean stress is the order parameter, and the mean packing fraction, gamma, must be adjusted to a value gamma(c) to reach the "critical point." We also follow the motion of individual disks and obtain detailed statistical information on the kinematics, including velocities an d particle rotations or spin. Distributions for the azimuthal velocity, V-t heta, and spin, S, of the particles are nearly rate invariant, which is con sistent with conventional wisdom. Near gamma(c), the grain motion becomes i ntermittent causing the mean velocity of grains to slow down. Also, the len gth of stress chains grows as gamma -->gamma(c). The 3D experiments show st atistical rate invariance for the stress in the sense that when the power s pectra and spectral frequencies of the stress time series are appropriately scaled by the shear rate, Omega, all spectra collapse onto a single curve for given particle and sample sizes. The frequency dependence of the spectr a can be characterized by two different power laws, P proportional to omega (-alpha), in the high and low frequency regimes: alpha similar to 2 at high omega; alpha < 2 at low omega. The force distributions computed from the 3 D stress time series are at least qualitatively consistent with exponential fall-off at large stresses. (C) 1999 American Institute of Physics. [S1054 -1500(99)00803-4].