Journal Club Theme of November 2009: Steady granular flow
How does sand flow? A surprisingly difficult question. This entry tells the story behind a model detailed in  for dry, dense, steady-flowing granular materials.
As an initial thought experiment, consider a long, 2D simple shear cell filled with viscoelastic disks. The disks are slightly bidisperse (to avoid crystal packings), and have average diameter d and mass m. The rough plates of the cell are compressed together with pressure P and sheared relative to each other at a rate dv/dy. Supposing the disks are always composed of the same material, we ask the question: How does the steady-state packing fraction φ and shear stress τ depend on the given parameters? Dimensional arguments are helpful here. Aside from φ, there are two other dimensionless groups that can be constructed: I = (dv/dy) √(m/Pd) is the normalized shear-rate, and μ = τ/P is the effective friction. The problem should have a unique steady behavior, so it follows that μ = g(I) and φ = h(I), since I is the only dimensionless group determined solely from the problem setup. These dependences generalize "Bagnold scaling" ; Bagnold was the first to observe that at fixed packing fraction, the pressure exerted on the wall of a granular shear cell depends quadratically on the shear-rate.
Disk shear simulations of , revealed that the functions g and h always have a relatively simple form in the "inertial regime" (1e-3 < I < 1e-1), characteristic of shearing in day-to-day flows like an hourglass. Namely, φ = h(I) stays roughly constant at the random-close-packing value, and μ = g(I) ≈ μ_s + β I indicative of a rate-dependent flow stress with static yield criterion μ_s. Here, rate-sensitivity arises from the dominating role of impact dissipation in the inertial regime; increasing the normalized shear-rate increases both the frequency of collisions and the energy loss per collision.
Next, the work of Jop et al.  extended these 2D results into a 3D rheology for monodisperse spheres by trying a straightforward re-interpretation: Replace τ and dv/dy with the equivalent shear stress and equivalent shear-rate, let P be the hydrostatic pressure, presume incompressibility, and enforce codirectionality to relate the direction of the deviatoric stress tensor to the deformation-rate tensor (non-associative). In essence, a Bingham fluid treatment of granular matter was proposed with Drucker-Prager yield criterion μ_s. The model was heralded as a major step forward, as it turned out to be capable of predicting highly inhomogeneous 3D flow profiles with accuracy. The applicability of the continuum rheology was bolstered in , where it was shown that the size-scale for RVE behavior in 3D inertial flow is generally a mere 5 particle diameters.
However, a key ingredient was missing. Bingham models give no stress computation in static regions (below μ_s). Solid-like zones are common in steady granular flow, and must be described in order for the law to be mechanically well-posed under admissible kinematic/traction boundary conditions. Thus, the next step was to splice a granular elasticity law into this framework, converting the Bingham model to an elasto-plastic model. The elasticity of a static granular material element is itself a complicated phenomenon, since even in the small-strain limit, the elastic response cannot be approximated as linear because grain assemblies do not support tension. The work of Jiang and Liu  proposed a nonlinear granular elasticity model, which expanded on successful mean-field theories of Hertzian contacts. With demonstrated experimental validation, both in terms of acoustic and static behavior, the Jiang-Liu elasticity law seemed to fit the bill.
My goal in  was to merge the Jiang-Liu elasticity model with the Jop flow law, to produce a unified granular constitutive law, which can be implemented in FEM, and used to predict steady stress and flow profiles throughout any 3D geometry. Following similar theories for thermodynamically compatible elasto-plasticity in , my approach was to presume a multiplicative Kröner-Lee decomposition of the deformation gradient. Running the model to steady-state as a VUMAT in ABAQUS, several flows were computed in different 3D geometries (e.g. silo, inclined chute, annular shear). The predicted flow and stress profiles compared favorably against the data of a number of experimental and discrete simulation studies.
While there is no shortage of remaining open questions in granular flow, it seems the most important to the study of dense, well-developed flow would be a general description of the "quasi-static" flow regime (I < 1e-3), which characterizes many geological applications. Its rheology appears significantly more complex than the inertial regime, with rate-independent flow stresses and a non-local size dependence.