Logarithmic rate dependence in deforming granular materials
  Rate-independence for stresses within a granular material is a basic tenet of many models for slow dense granular flows. By contrast, logarithmic rate dependence of stresses is found in solid-on-solid friction, in geological settings, and elsewhere. In this work, we show that logarithmic rate-dependence occurs in granular materials for plastic (irreversible) deformations that occur during shearing but not for elastic (reversible) deformations, such as those that occur under moderate repetitive compression. Increasing the shearing rate, \Omega, leads to an increase in the stress and the stress fluctuations that at least qualitatively resemble what occurs due to an increase in the density. Increases in \Omega also lead to qualitative changes in the distributions of stress build-up and relaxation events. If shearing is stopped at t=0, stress relaxations occur with \sigma(t)/ \sigma(t=0) \simeq A \log(t/t_0). This collective relaxation of the stress network over logarithmically long times provides a mechanism for rate-dependent strengthening. 
R. R. Hartley and R. P. Behringer