Onset and dynamic shallow flow of a viscoplastic fluid on a plane slope

The shallow flow of a viscoplastic fluid on a plane slope is investigated. The material constitutive law may include two plasticity (flow/no-flow) criteria: Von-Mises (Bingham fluid) and Drucker–Prager (Mohr–Coulomb). Coulomb frictional conditions on the bottom are included, which implies that the shear stresses are small and the extensional and in-plane shear stress becomes important. A stress analysis is used to deduce a Saint-Venant type asymptotic model for small thickness aspect ratio. The 2D (asymptotic) constitutive law, which relates the average plane stresses to the horizontal rate of deformation, is obtained from the initial (3D) viscoplastic model.The “safety factor” (limit load) is introduced to model the link between the yield limit (material resistance) and the external forces distribution which could generate or not the shallow flow of the viscoplastic fluid. The DVDS method, developed in I.R. Ionescu, E. Oudet, Discontinuous velocity domain splitting method in limit load analysis, Int. J. Solids Struct., doi:10.1016/j.ijsolstr.2010.02.012], is used to evaluate the safety factor and to find the onset of an avalanche flow.A mixed finite element and finite volume strategy is developed. Specifically, the variational inequality for the velocity field is discretized using the finite element method while a finite volume method is adopted for the hyperbolic equation related to the thickness variable. To solve the velocity problem, a decomposition–coordination formulation coupled with the augmented lagrangian method, is adapted here for the asymptotic model. The finite volume method makes use of an upwind strategy in the choice of the flux.Several boundary value problems, modeling shallow dense avalanches, for different visoplastic laws are selected to illustrate the predictive capabilities of the model.