Retrogressive failure of a static granular layer on an inclined plane

Abstract

When a layer of static grains on a sufficiently steep slope is disturbed, an upslope-propagating erosion wave, or retrogressive failure, may form that separates the initially static material from a downslope region of flowing grains. This paper shows that a relatively simple depth-averaged avalanche model with frictional hysteresis is sufficient to capture a planar retrogressive failure that is independent of the cross-slope coordinate. The hysteresis is modelled with a non-monotonic effective basal friction law that has static, intermediate (velocity decreasing) and dynamic (velocity increasing) regimes. Both experiments and time-dependent numerical simulations show that steadily travelling retrogressive waves rapidly form in this system and a travelling wave ansatz is therefore used to derive a one-dimensional depth-averaged exact solution. The speed of the wave is determined by a critical point in the ordinary differential equation for the thickness. The critical point lies in the intermediate frictional regime, at the point where the friction exactly balances the downslope component of gravity. The retrogressive wave is therefore a sensitive test of the functional form of the friction law in this regime, where steady uniform flows are unstable and so cannot be used to determine the friction law directly. Upper and lower bounds for the existence of retrogressive waves in terms of the initial layer depth and the slope inclination are found and shown to be in good agreement with the experimentally determined phase diagram. For the friction law proposed by Edwardset al.(J. Fluid. Mech., vol. 823, 2017, pp. 278–315,J. Fluid. Mech., 2019, (submitted)) the magnitude of the wave speed is slightly under-predicted, but, for a given initial layer thickness, the exact solution accurately predicts an increase in the wave speed with higher inclinations. The model also captures the finite wave speed at the onset of retrogressive failure observed in experiments.

Self-channelisation and levee formation in monodisperse granular flows

“…is used in this paper to best match experimental results. This is very close to the value h * /h stop = 1.33 measured in flows of glass beads by Russell et al (2019).…”

“…The surface velocity data for sand (Takagi et al 2011) shown in figure 11 suggest that u s = 2.35ū, which implies λ = 2.05. This is within 15 % of the value u s = 2.7ū (corresponding to λ = 2.45) measured experimentally in a thin unconfined flow of glass beads (Russell et al 2019).…”

Constitutive relations for compressible granular flow in the inertial regime

“…7.3. Extensions to the theory Inertial CIDR is also able to incorporate non-monotonicity of the µ(I) function (DeGiuli & Wyart 2017), which is crucial for modelling hysteretic effects, such as coexisting static and moving regions, in depth-averaged avalanche models (Daerr & Douady 1999;Pouliquen & Forterre 2002;Edwards & Gray 2015;Edwards et al 2017;Russell et al 2019). Non-monotonic µ = µ(I) functions are problematic in the incompressible µ(I)-rheology, because they imply that the theory is always ill-posed in regions of decreasing friction (Barker et al 2015).…”

Mathematical Optimization Problems for Particle Finite Element Analysis Applied to 2D Landslide Modeling

“…This means that the material parameters required for the depth-averaged analysis consist of the density ρ 2000 kg/m 3 and the frictional coefficient of the basal surface μ 0.27. In the simulation by the depth-average model, the common used earth pressure coefficient (Savage and Hutter 1989) is assumed to be unity, like in other recent works (e.g., Russell et al 2019).…”