A shallow model including static-flowing transition for viscoplastic Drucker-Prager materials

Observed avalanche flows of dense granular material have the property to present two possible behaviors: static (solid) or flowing (fluid). In such situation, an important challenge is to describe mathematically the evolution of the physical interface between the two phases. In this work we derive a set of equations that is able to manage the dynamics of such interface, in the so-called shallow regime where the flow is supposed to be thin compared to its downslope extension. It is obtained via a shallow asymptotics starting from the Drucker-Prager model for viscoplastic materials, in which we have to make several assumptions. Additionally to the classical ones that are that the curvature of the topography, the width of the layer of material, and the viscosity are small, we assume that the internal friction angle is close to the slope angle, the velocity is small, and the pressure is convex with respect to the normal variable. The assumption that the internal friction angle has to be close to the slope angle is necessary for the velocity to remain small, and the convexity of the pressure is an assumption of stability of the double layer static-flowing configuration. We present here the resulting model, that takes the form of an overdetermined initial-boundary problem in the variable normal to the topography. It handles arbitrary velocity profiles, and is therefore more general than previous models such as the BCRE model. It Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), CNRS, UPEM, UPEC, F-77454, Marne-la-Vallée, France ([email protected]) CNRS & Université Paris 13, Laboratoire des Sciences des Procédés et des Matériaux, 99, Av. J.-B. Clement, F-93430 Villetaneuse, France ([email protected]) Université Paris Diderot, Sorbone Paris Cité, Institut de Physique du Globe de Paris, Equipe de Sismologie, 1 rue Jussieu, 75005 Paris, France ([email protected]) INRIA, CEREMA, Lab. J.-L. Lions, Equipe ANGE, Paris, France