Analysis of the statistics of sandpile avalanches using soil-mechanics results and concepts

Abstract

Glass-sphere avalanches have been studied experimentally using a drum partly filled with beads and rotating slowly (\ensuremath{\Omega}) around its horizontal axis. The statistics of the avalanche characteristics (i.e., duration D and size \ensuremath{\delta}\ensuremath{\Theta}) have been determined as a function of the rotation speed \ensuremath{\Omega}, the sphere diameter d, and the drum length l. The widths of these statistics are broad, but avalanches do not exhibit either periodicity or 1/f noise. We conclude also that avalanches are governed by inertia and gravity. We recall then classical results of soil mechanics; we will see that triaxial test results, together with the so-called ``critical state'' of granular material and the Granta gravel model of sandpiles will make evident and quantify the well-known effects of friction, caging, and dilatancy in granular samples. Using these established results, especially those on the ``critical'' state of soil, we will demonstrate that the maximum angle of repose of a pile may exceed the angle of friction for initially dense enough materials, but that this leads to a catastrophic event (avalanche). This critical-state approach also allows us to relate the mean angle per avalanche to the mean avalanche duration, using an inertial process, and to predict the avalanche duration. According to our model, the avalanche size is controlled by the difference between the real pile specific volume v and that of the ``critical'' state ${\mathit{v}}_{\mathit{c}}$; macroscopic avalanches are obtained for v${\mathit{v}}_{\mathit{c}}$ (i.e., a first-order process), but we expect critical fluctuations (and probably 1/f noise) when v=${\mathit{v}}_{\mathit{c}}$ (i.e., a second-order transition). This theory makes a link between the theory of self-organized criticality of sandpile avalanches and experimental data; it links also the Coulomb approach of the stability of a free surface and the dilatancy effect discovered by Reynolds.