A self - similar model for shear flows in dense granular materials

We propose a model to describe the quasistatic shearing of dry granular materials, which captures the differences in velocity profiles recently observed in 2 and 3-D Couette flow experiments. In our scheme, the steady-state flow is due to the intermittent motion of particle clusters moving together with the wall. The motion of a cluster is associated with the formation of a fracture in the pack under stress. The model is based on the assumption that the persistence length for a fracture is exactly its distance to the moving wall, which imposes a self-similar structure of the clusters. Through a probabilistic approach, we can evaluate the rate of appearance of a cluster of a given size, and obtain a prediction for the the average velocity profiles. This model also predicts the existence of large stress fluctuations at the moving wall, which characteristics are in reasonnable agreement with experimental data.