Dry granular column collapse: Numerical simulations using the partially regularized μ(I)-model via stabilized finite elements and phase field formulation

Abstract

We revisit the gravitational collapse of a 2D column of dry granular material surrounded by air, using a continuum mechanics approximation. By employing the Cahn-Hilliard phase-field equation as an interface capturing technique and by coupling it with the Cauchy equation, we numerically simulate this multiphase system, eliminating the need for any ad-hoc numerical adjustment to prevent the finger formation of light fluid between the material and the solid boundary due to the no-slip boundary condition. We implement the $\mu \left(I\right)$-rheology in our stabilized Finite Element method, highlighting the presence of instabilities when using this constitutive law. Our study is characterized by three main goals. First, we address the instability issue by implementing the partially regularized formulation of the $\mu \left(I\right)$-rheology proposed by Barker and Gray (2017). An important outcome is that using shock-capturing terms in the momentum equation can significantly smooth these oscillations by adding dissipation in the direction of the gradients. Second, we systematically study the fluid dynamics under realistic conditions. Our results accurately replicate the material dynamics during collapse, confirming three distinct stages: free-fall, spreading, and cessation. We identify two regions in the material during the spreading phase: a quasi-static zone with negligible velocities and deformations, and a flowing layer exhibiting high shear rates. These observations closely align with experimental data. Additionally, we examine the evolution of the yielded/unyielded regions based on the Drucker-Prager criterion, and we also explore an empirical criterion, based on a critical value of the velocity norm, that satisfactorily separates these regions. Finally, we perform an extensive parametric study covering a wide range of rheological parameters.