The motion of a finite mass of granular material down a rough incline

Abstract

Rock, snow and ice masses are often dislodged on steep slopes of mountainous regions. The masses, which typically are in the form of innumerable discrete blocks or granules, initially accelerate down the slope until the angle of inclination of the bed approaches the horizontal and bed friction eventually brings them to rest. The present paper describes an initial investigation which considers the idealized problem of a finite mass of material released from rest on a rough inclined plane. The granular mass is treated as a frictional Coulomb-like continuum with a Coulomb-like basal friction law. Depth-averaged equations of motion are derived; they bear a superficial resemblance to the nonlinear shallow-water wave equations. Two similarity solutions are found for the motion. They both are of surprisingly simple analytical form and show a rather unanticipated behaviour. One has the form of a pile of granular material in the shape of a parabolic cap and the other has the form of anM-wave with vertical faces at the leading and trailing edges. The linear stability of the similarity solutions is studied. A restricted stability analysis, in which the spread is left unperturbed shows them to be stable, suggesting that mathematically both are possible asymptotic wave forms. Two numerical finite-difference schemes, one of Lagrangian, the other of Eulerian type, are presented. While the Eulerian technique is able to reproduce theM-wave similarity solution, it appears to give spurious results for more general initial conditions and the Lagrangian technique is best suited for the present problem. The numerical predictions are compared with laboratory experiments of Huber (1980) involving the motion of gravel released from rest on a rough inclined plane. Although in these experiments the continuum approximation breaks down at large times when the gravel layer is only a few particle diameters thick, the general features of the development of the gravel mass are well predicted by the numerical solutions.

Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane

“…The agreement is good for the runout L of the deposit, whereas the lateral spreading W is overestimated by approximately 20%. It is interesting to compare the above results with the prediction of the simple model using a constant friction coefficient (Savage & Hutter 1989). In this case, no deposit is predicted as soon as the inclination of the plane is higher than the friction angle.…”

“…Assuming in addition that the flow is incompressible of constant density ρ, which is justified for the dense flow regim studied here Savage & Hutter 1989, Ertas et al 2001, we obtain the depth averaged mass and momentum conservations by integrating the full 3D equations (Savage & Hutter 1989). The following equations are written in terms of the local thickness h(x, y, t) and the depth averaged velocity u(x, y, t) = ue x + ve y :…”

“…3.2 represents the pressure force linked to the thickness gradient. The coefficient K represents the ratio of the vertical normal stress to the horizontal one (Savage & Hutter 1989). For quasi-static deformation this coefficient can be calculated from standard Mohr-Coulomb plasticity model as done by Savage and Hutter (1989).…”

“…Moreover, we observe that K = 1 gives the best agreement for the predicted deposits in section 4.1. An example of the prediction with another value of K is shown in Fig 9. We have used the value K = 1+sin 2 δ1 1−sin 2 δ1 corresponding to the Mohr Coulomb prediction when the basal friction is equal to the internal friction coefficient taken equal to tan δ 1 (Savage & Hutter 1989). For our glass beads we obtained K = 1.29.…”

“…In their model, the interaction between the granular material and the rough surface is described by a simple friction law: the shear stress at the bottom is proportional to the normal stress, the coefficient of friction being a constant. The model works well when the surface of the plane is smooth enough: Savage, Hutter and coworkers were able to predicts the motion and spreading of a granular mass on steep slopes in two and three dimensions (Savage & Hutter 1989, Wielandet al 1999 . Experiments have been also carried out on curved beds (Greve & Hutter 1993, and the measurements agree relatively well with the prediction of the depth averaged model.…”