Modeling sand slides by a mechanics-based degenerate parabolic equation

Abstract

Avalanching plays a crucial role in granular materials dynamics, in particular in the evolution of the shape of the leeward side of sand dunes. This paper presents a physically-based mathematical model capable of reproducing the kinematic evolution of the surface of sand piles and to obtain eventually the stationary configurations, in the presence of external sources as well. Simulation results with different boundary conditions and geometries are reported in order to show the high flexibility of the model. The model is also validated by means of comparison with the experimental results of different authors.

Preface to the Special Issue in Memory of Prof. Gaetano Giaquinta (1945–2016)

“…Mićunović and Kudrjavceva [14] use the effective field approach to retrieve the fourth-order elasticity tensor and to study the damage evolution of a composite with a void phase and a short fibre phase, with an application to the damage-elasto-viscoplastic strain of an industrial steel. Lo Giudice et al [15] present a mathematical model aimed at reproducing the kinematic evolution of the surface of sand piles and dunes and at predicting stationary configurations; the study is corroborated by a comparison with experimental results. Wittum et al [16] study a model of the barrier function of the skin tissue, by treating the extracellular space with full spatial resolution, considering the cells being of both hexagonal prismatic shape and tetrakaidekadehedral shape, and considering vertical cell stacking.…”

A new mathematical model for traveling sand dunes: analysis and approximation

“…On the contrary, rolling grains stop and deposit when they reach a region where the standing layer is less steep than α. It is worth to say that different models for the description of sandpile dynamics have been proposed for example in [18], and more recently in [16]. In order to extend the two layer approach to the dynamics of a desert dune one has to add to the model the effects of the blowing wind.…”

A numerical study of an Heaviside function driven degenerate diffusion equation

“…Other examples of this approach can be found in the literature even without the Heaviside function, for example in cases where the model changes its behavior in a discontinuous way according to the values of the solution or of its partial derivatives: in these cases the Heaviside function is replaced by the positive part function, σ → (σ) + := max {σ, 0}, for all σ ∈ R to describe non-reversible phenomena that can be connected to the avalanche behavior in the sand piles [10] and to damage mechanics models [1].…”