Rheology of granular materials: dynamics in a stress landscape

We present a framework for analysing the rheology of dense driven granular materials, based on a recent proposal of a stress-based ensemble. In this ensemble, fluctuations in a granular system near jamming are controlled by a temperature-like parameter, the angoricity, which is conjugate to the stress of the system. In this paper, we develop a model for slowly driven granular materials based on the stress ensemble and the idea of a landscape in stress space. The idea of an activated process driven by the angoricity has been shown by Behringer et al. (Behringer et al. 2008 Phys. Rev. Lett. 101, 268301) to describe the logarithmic strengthening of granular materials. Just as in the soft glassy rheology (SGR) picture, our model represents the evolution of a small patch of granular material (a mesoscopic region) in a stress-based trap landscape. The angoricity plays the role of the fluctuation temperature in the SGR. We determine (i) the constitutive equation, (ii) the yield stress, and (iii) the distribution of stress dissipated during granular shearing experiments, and compare these predictions with the experiments of Hartley & Behringer (Hartley & Behringer 2003 Nature 421, 928-931.).