Classical and Cosserat plasticity and viscoplasticity models for slow granular flow

We consider models for the rheology of dense, slowly deforming granular materials based of classical and Cosserat plasticity, and their viscoplastic extensions that account for small but finite particle inertia. We determine the scale for the viscosity by expanding the stress in a dimensionless parameter that is a measure of the particle inertia. We write the constitutive relations for classical and Cosserat plasticity in stress-explicit form. The viscoplastic extensions are made by adding a rate-dependent viscous stress to the plasticity stress. We apply the models to plane Couette flow, and show that the classical plasticity and viscoplasticity models have features that depart from experimental observations; the prediction of the Cosserat viscoplasticity model is qualitatively similar to that of Cosserat plasticity, but the viscosities modulate the thickness of the shear layer.

The hard-particle model for a dense granular flow down an inclined plane

Perfectly hard particles are those which experience an infinite repulsive force when they overlap, and no force when they do not overlap. In the hard-particle model, the only static state is the isostatic state where the forces between particles are statically determinate. In the flowing state, the interactions between particles are instantaneous because the time of contact approaches zero in the limit of infinite particle stiffness. Here, we discuss the development of a hard particle model for a realistic granular flow down an inclined plane, and examine its utility for predicting the salient features both qualitatively and quantitatively. We first discuss Discrete Element simulations, that even very dense flows of sand or glass beads with volume fraction between 0.5 and 0.58 are in the rapid flow regime, due to the very high particle stiffness. An important length scale in the shear flow of inelastic particles is the `conduction length' delta = (d/(1 - e(2))(1/2)), where d is the particle diameter and e is the coefficient of restitution. When the macroscopic scale h (height of the flowing layer) is larger than the conduction length, the rates of shear production and inelastic dissipation are nearly equal in the bulk of the flow, while the rate of conduction of energy is O((delta/h)(2)) smaller than the rate of dissipation of energy. Energy conduction is important in boundary layers of thickness delta at the top and bottom. The flow in the boundary layer at the top and bottom is examined using asymptotic analysis. We derive an exact relationship showing that the a boundary layer solution exists only if the volume fraction in the bulk decreases as the angle of inclination is increased. In the opposite case, where the volume fraction increases as the angle of inclination is increased, there is no boundary layer solution. The boundary layer theory also provides us with a way of understanding the cessation of flow when at a given angle of inclination when the height of the layer is decreased below a value h(stop), which is a function of the angle of inclination. There is dissipation of energy due to particle collisions in the flow as well as due to particle collisions with the base, and the fraction of energy dissipation in the base increases as the thickness decreases. When the shear production in the flow cannot compensate for the additional energy drawn out of the flow due to the wall collisions, the temperature decreases to zero and the flow stops. Scaling relations can be derived for h(stop) as a function of angle of inclination.

Kinetic theory for the density plateau in the granular flow down an inclined plane

The striking lack of observable variation of the volume fraction with height in the center of a granular flow down an inclined plane is analysed using constitutive relations obtained from kinetic theory. It is shown that the rate of conduction in the granular energy balance equation is O(delta(2)) smaller than the rate of production of energy due to mean shear and the rate of dissipation due to inelastic collisions, where the small parameter delta = (d/(1 - e(n))H-1/2), d is the particle diameter, en is the normal coefficient of restitution and H is the thickness of the flowing layer. This implies that the volume fraction is a constant in the leading approximation in an asymptotic analysis in small delta. Numerical estimates of both the parameter delta and its pre-factor are obtained to show that the lack of observable variation of the volume fraction with height can be explained by constitutive relations obtained from kinetic theory.

Dense granular flow down an inclined plane A comparison between the hard particle model and soft particle simulations

The granular flow down an inclined plane is simulated using the discrete element (DE) technique to examine the extent to which the dynamics of an unconfined dense granular flow can be well described by a hard particle model First, we examine the average coordination number for the particles in the flow down an inclined plane using the DE technique using the linear contact model with and without friction, and the Hertzian contact model with friction The simulations show that the average coordination number decreases below 1 for values of the spring stiffness corresponding to real materials, such as sand and glass, even when the angle of inclination is only 10 larger than the angle of repose Additional measures of correlations in the system, such as the fraction of particles with multibody contact, the force ratio (average ratio of the magnitudes of the largest and the second largest force on a particle), and the angle between the two largest forces on the particle, show no evidence of force chains or other correlated motions in the system An analysis of the bond-orientational order parameter indicates that the flow is in the random state, as in event-driven (ED) simulations V Kumaran, J Fluid Mech 632, 107 (2009), J Fluid Mech 632, 145 (2009)] The results of the two simulation techniques for the Bagnold coefficients (ratio of stress and square of the strain rate) and the granular temperature (mean square of the fluctuating velocity) are compared with the theory V Kumaran, J Fluid Mech 632, 107 (2009), J Fluid Mech 632, 145 (2009)] and are found to be in quantitative agreement In addition, we also conduct a comparison of the collision frequency and the distribution of the precollisional relative velocities of particles in contact The strong correlation effects exhibited by these two quantities in event-driven simulations V Kumaran, J Fluid Mech 632, 145 (2009)] are also found in the DE simulations (C) 2010 American Institute of Physics doi 10 1063/1 3504660]

Effect of base dissipation on the granular flow down an inclined plane

The effect of base dissipation on the granular flow down an inclined plane is examined by altering the coefficient of restitution between the moving and base particles in discrete element (DE) simulations. The interaction laws between two moving particles are kept fixed, and the coefficient of restitution (damping constant in the DE simulations) between the base and moving particles are altered to reduce dissipation, and inject energy from the base. The energy injection does result in an increase in the strain rate by up to an order of magnitude, and the temperature by up to two orders of magnitude at the base. However, the volume fraction, strain rate and temperature profiles in the bulk (above about 15 particle diameters from the base) are altered very little by the energy injection at the base. We also examine the variation of h(stop), the minimum height at the cessation of flow, with energy injection from the base. It is found that at a fixed angle of inclination, h(stop) decreases as the energy dissipation at the base decreases.

Transition due to base roughness in a dense granular flow down an inclined plane

Particle simulations based on the discrete element method are used to examine the effect of base roughness on the granular flow down an inclined plane. The base is composed of a random configuration of fixed particles, and the base roughness is decreased by decreasing the ratio of diameters of the base and moving particles. A discontinuous transition from a disordered to an ordered flow state is observed when the ratio of diameters of base and moving particles is decreased below a critical value. The ordered flowing state consists of hexagonally close packed layers of particles sliding over each other. The ordered state is denser (higher volume fraction) and has a lower coordination number than the disordered state, and there are discontinuous changes in both the volume fraction and the coordination number at transition. The Bagnold law, which states that the stress is proportional to the square of the strain rate, is valid in both states. However, the Bagnold coefficients in the ordered flowing state are lower, by more than two orders of magnitude, in comparison to those of the disordered state. The critical ratio of base and moving particle diameters is independent of the angle of inclination, and varies very little when the height of the flowing layer is doubled from about 35 to about 70 particle diameters. While flow in the disordered state ceases when the angle of inclination decreases below 20 degrees, there is flow in the ordered state at lower angles of inclination upto 14 degrees. (C) 2012 American Institute of Physics. http://dx.doi.org/10.1063/1.4710543]

The effect of base roughness on the development of a dense granular flow down an inclined plane

The development of the flow of a granular material down an inclined plane starting from rest is studied as a function of the base roughness. In the simulations, the particles are rough frictional spheres interacting via the Hertz contact law. The rough base is made of a random configuration of fixed spheres with diameter different from the flowing particles, and the base roughness is decreased by decreasing the diameter of the base particles. The transition from an ordered to a disordered flowing state at a critical value of the base particle diameter, first reported by Kumaran and Maheshwari Phys. Fluids 24, 053302 (2012)] for particles with the linear contact model, is observed for the Hertzian contact model as well. The flow development for the ordered and disordered flows is very different. During the development of the disordered flow for the rougher base, there is shearing throughout the height. During the development of the ordered flow for the smoother base, there is a shear layer at the bottom and a plug region with no internal shearing above. In the shear layer, the particles are layered and hexagonally ordered in the plane parallel to the base, and the velocity profile is well approximated by Bagnold law. The flow develops in two phases. In the first phase, the thickness of the shear layer and the maximum velocity increase linearly in time till the shear front reaches the top. In the second phase, after the shear layer encompasses the entire flow, there is a much slower increase in the maximum velocity until the steady state is reached. (C) 2013 AIP Publishing LLC.

Granular flow in the marginal ice zone

The region of sea ice near the edge of the sea ice pack is known as the marginal ice zone (MIZ), and its dynamics are complicated by ocean wave interaction with the ice cover, strong gradients in the atmosphere and ocean and variations in sea ice rheology. This paper focuses on the role of sea ice rheology in determining the dynamics of the MIZ. Here, sea ice is treated as a granular material with a composite rheology describing collisional ice floe interaction and plastic interaction. The collisional component of sea ice rheology depends upon the granular temperature, a measure of the kinetic energy of flow fluctuations. A simplified model of the MIZ is introduced consisting of the along and across momentum balance of the sea ice and the balance equation of fluctuation kinetic energy. The steady solution of these equations is found to leading order using elementary methods. This reveals a concentrated region of rapid ice flow parallel to the ice edge, which is in accordance with field observations, and previously called the ice jet. Previous explanations of the ice jet relied upon the existence of ocean currents beneath the ice cover. We show that an ice jet results as a natural consequence of the granular nature of sea ice.

Granular flow in polymineralic rocks bearing sheet silicates: New evidence from natural examples

Natural deformation in carbonate mylonites bearing sheet silicates occurs via a complex interaction of granular flow and solution transfer processes and involves continuous cycles of dissolution, grain boundary diffusion, nucleation and growth. In this way, new sheet silicates (a) nucleate within voids formed by grain boundary sliding of calcite grains. (b) grow, and (c) rotate towards the shear plane. As a consequence, small mica grains show a wide range of orientations with respect to the shear plane, but moderate to large grains are subparallel both to each other and to the shear plane. Increases of average grain sizes with increasing temperature of sheet silicates in mica-rich layers is more pronounced than in mica-poor layers. In the calcitic matrix however, sheet silicates can only grow via solution-precipitation and mass transfer processes. Therefore, the observed grain size variability indicates drastic differences in mass transfer behavior between the individual layers, which might be related to differences in the fluid flux. Based on these observations, a conceptual model for the microfabric evolution in sheet silicate bearing mylonites is presented. © 2001 Elsevier Science B.V. All rights reserved.

Kinematic flow patterns in slow deformation of a dense granular material

The kinematic flow pattern in slow deformation of a model dense granular medium is studied at high resolution using in situ imaging, coupled with particle tracking. The deformation configuration is indentation by a flat punch under macroscopic plane-strain conditions. Using a general analysis method, velocity gradients and deformation fields are obtained from the disordered grain arrangement, enabling flow characteristics to be quantified. The key observations are the formation of a stagnation zone, as in dilute granular flow past obstacles; occurrence of vortices in the flow immediately underneath the punch; and formation of distinct shear bands adjoining the stagnation zone. The transient and steady state stagnation zone geometry, as well as the strength of the vortices and strain rates in the shear bands, are obtained from the experimental data. All of these results are well-reproduced in exact-scale non-smooth contact dynamics simulations. Full 3D numerical particle positions from the simulations allow extraction of flow features that are extremely difficult to obtain from experiments. Three examples of these, namely material free surface evolution, deformation of a grain column below the punch and resolution of velocities inside the primary shear band, are highlighted. The variety of flow features observed in this model problem also illustrates the difficulty involved in formulating a complete micromechanical analytical description of the deformation.

Force chains, friction, and flow: Behavior of granular media across length scales

<p>We study the behavior of granular materials at three length scales. At the smallest length scale, the grain-scale, we study inter-particle forces and "force chains". Inter-particle forces are the natural building blocks of constitutive laws for granular materials. Force chains are a key signature of the heterogeneity of granular systems. Despite their fundamental importance for calibrating grain-scale numerical models and elucidating constitutive laws, inter-particle forces have not been fully quantified in natural granular materials. We present a numerical force inference technique for determining inter-particle forces from experimental data and apply the technique to two-dimensional and three-dimensional systems under quasi-static and dynamic load. These experiments validate the technique and provide insight into the quasi-static and dynamic behavior of granular materials.</p> <p>At a larger length scale, the mesoscale, we study the emergent frictional behavior of a collection of grains. Properties of granular materials at this intermediate scale are crucial inputs for macro-scale continuum models. We derive friction laws for granular materials at the mesoscale by applying averaging techniques to grain-scale quantities. These laws portray the nature of steady-state frictional strength as a competition between steady-state dilation and grain-scale dissipation rates. The laws also directly link the rate of dilation to the non-steady-state frictional strength. </p> <p>At the macro-scale, we investigate continuum modeling techniques capable of simulating the distinct solid-like, liquid-like, and gas-like behaviors exhibited by granular materials in a single computational domain. We propose a Smoothed Particle Hydrodynamics (SPH) approach for granular materials with a viscoplastic constitutive law. The constitutive law uses a rate-dependent and dilation-dependent friction law. We provide a theoretical basis for a dilation-dependent friction law using similar analysis to that performed at the mesoscale. We provide several qualitative and quantitative validations of the technique and discuss ongoing work aiming to couple the granular flow with gas and fluid flows.</p>

Steady flow dynamics during granular impact.

We study experimentally and computationally the dynamics of granular flow during impacts where intruders strike a collection of disks from above. In the regime where granular force dynamics are much more rapid than the intruder motion, we find that the particle flow near the intruder is proportional to the instantaneous intruder speed; it is essentially constant when normalized by that speed. The granular flow is nearly divergence free and remains in balance with the intruder, despite the latter's rapid deceleration. Simulations indicate that this observation is insensitive to grain properties, which can be explained by the separation of time scales between intergrain force dynamics and intruder dynamics. Assuming there is a comparable separation of time scales, we expect that our results are applicable to a broad class of dynamic or transient granular flows. Our results suggest that descriptions of static-in-time granular flows might be extended or modified to describe these dynamic flows. Additionally, we find that accurate grain-grain interactions are not necessary to correctly capture the granular flow in this regime.