Extending recent modeling efforts for emulsions, we propose a nonlocal fluidity relation for flowing granular materials, capturing several known finite-size effects observed in steady flow. We express the local Bagnold-type granular flow law in terms of a fluidity ratio and then extend it with a particular Laplacian term that is scaled by the grain size. The resulting model is calibrated against a sequence of existing discrete element method data sets for two-dimensional annular shear, where it is shown that the model correctly describes the divergence from a local rheology due to the grain size as well as the rate-independence phenomenon commonly observed in slowly flowing zones. The same law is then applied in two additional inhomogeneous flow geometries, and the predicted velocity profiles are compared against corresponding discrete element method simulations utilizing the same grain composition as before, yielding favorable agreement in each case.

No abstract

We propose and numerically implement a constitutive framework for granular media that allows the material to traverse through its many common phases during the flow process. When dense, the material is treated as a pressure sensitive elasto-viscoplastic solid obeying a yield criterion and a plastic flow rule given by the µ(I) inertial rheology of granular materials. When the free volume exceeds a critical level, the material is deemed to separate and is treated as disconnected, stress-free media. A Material Point Method (MPM) procedure is written for the simulation of this model and many demonstrations are provided in different geometries. By using the MPM framework, extremely large strains and nonlinear deformations, which are common in granular flows, are representable. The method is verified numerically and its physical predictions are validated against known results. BackgroundGranular materials present several modeling challenges when considering a continuum approach. During dense flow, the material can be characterized as an elasto-viscoplastic material with a frictional yield criterion. Extremely high levels of strain often occur, which challenge certain computational techniques, but the material can also behave as a solid, able to support shear loads in a static configuration. Moreover, because dry grains do not support tension, their constitutive behavior changes from that of a dense plastic media to a gas-like disconnected state during extension, a dramatic switch that is difficult to represent in a unified modeling and numerical framework.Several approaches have been used to simulate granular flow. One of the most accurate methods is the discrete element method (DEM), first described in Cundall & Strack (1979). While accurate, DEM solves the classical equations of motion on each grain individually, resulting in untenable computational expense over the large physical domains in many industrial and geological applications. A recent set of continuum rheological models for granular flow, such as the µ(I) relation in da Cruz et al. (2005) (later extended to 3D in Jop et al. (2006)) and the nonlocal extension in Kamrin & Koval (2012), offer a number of improvements over the commonly used rate-independent Drucker-Prager and Mohr-Coulomb models for problems with zones of dense, rapid flow (as is common in industrial settings) where rate-sensitivity is more pronounced and particle size-effects can play a role. The incompressible Navier-Stokes solver Gerris has been used in Staron et al. (2012) and Staron et al. (2014) with the µ(I) relation, while the commercial finite-element software Abaqus was used in Kamrin (2010) and appended with the nonlocal model in Henann & Kamrin (2013). While both methods can yield good results in certain regimes, the fluid solvers have difficulties with extensional disconnection and truly static zones cannot be represented, while the finite-element method (FEM) has issues when mesh distortion becomes large. Fixes such as Arbitrary-Lagrangian Eulerian (ALE) re-meshing may cause los...

Flows of granular media down a rough inclined plane demonstrate a number of nonlocal phenomena. We apply the recently proposed nonlocal granular fluidity model to this geometry and find that the model captures many of these effects. Utilizing the model's dynamical form, we obtain a formula for the critical stopping height of a layer of grains on an inclined surface. Using an existing parameter calibration for glass beads, the theoretical result compares quantitatively to existing experimental data for glass beads. This provides a stringent test of the model, whose previous validations focused on driven steady-flow problems. For layers thicker than the stopping height, the theoretical flow profiles display a thickness-dependent shape whose features are in agreement with previous discrete particle simulations. We also address the issue of the Froude number of the flows, which has been shown experimentally to collapse as a function of the ratio of layer thickness to stopping height. While the collapse is not obvious, two explanations emerge leading to a revisiting of the history of inertial rheology, which the nonlocal model references for its homogeneous flow response.

A recent granular rheology based on an implicit 'granular fluidity' field has been shown to quantitatively predict many nonlocal phenomena. However, the physical nature of the field has not been identified. Here, the granular fluidity is found to be a kinematic variable given by the velocity fluctuation and packing fraction. This is verified with many discrete element simulations, which show the operational fluidity definition, solutions of the fluidity model, and the proposed microscopic formula all agree. Kinetic theoretical and Eyring-like explanations shed insight into the obtained form.

This work proposes a model for granular deformation that predicts the stress and velocity profiles in well-developed dense granular flows. Recent models for granular elasticity (Jiang and Liu 2003) and rate-sensitive fluid-like flow (Jop et al. 2006) are reformulated and combined into one universal elasto-plastic law, capable of predicting flowing regions and stagnant zones simultaneously in any arbitrary 3D flow geometry. The unification is performed by justifying and implementing a Kröner-Lee decomposition, with care taken to ensure certain continuum physical principles are necessarily upheld. The model is then numerically implemented in multiple geometries and results are compared to experiments and discrete simulations.

Based on discrete element method simulations, we propose a new form of the constitutive equation for granular flows independent of packing fraction. Rescaling the stress ratio μ by a power of dimensionless temperature Θ makes the data from a wide set of flow geometries collapse to a master curve depending only on the inertial number I. The basic power-law structure appears robust to varying particle properties (e.g., surface friction) in both 2D and 3D systems. We show how this rheology fits and extends frameworks such as kinetic theory and the nonlocal granular fluidity model.

A recently proposed nonlocal rheology for dense granular flow, based on the concept of nonlocal granular fluidity, has demonstrated predictive capabilities in multiple geometries. This work is concerned with determining how the parameters of this continuum model arise from the properties of the grains themselves. We perform a controlled study investigating how the surface friction of the grains influences the continuum parameters, with a focus on how the nonlocal amplitude, the model's one new parameter, is affected. This is achieved comparing two-dimensional discrete-element simulations of flowing disks to numerical solutions of the model in planar shear and several annular shear geometries. A multistep calibration scheme for the continuum parameter extraction is developed and implemented. Results indicate the nonlocal amplitude varies less than an order of magnitude over a wide range of surface frictions, with a slight tendency to increase as surface friction decreases, particularly in a regime of small surface friction. Our data also show that the stress and flow-rate variables deviate little from a local relationship as surface friction vanishes, which corroborates certain existing experimental findings.