Session B8: Focus Session: Granular Materials Near Jamming

Elastic Granular Flows

There is no fundamental understanding of the mechanics of granular solids. Partially this is because granular flows have historically been divided into two very distinct flow regimes, (1) the slow, quasistatic regime, in which the bulk friction coefficient is taken to be a material constant, and (2) the fast, rapid-flow regime, where the particles interact collisionally. But slow hopper flow simulations indicate that the bulk friction coefficient is not a constant. Rapidly moving large scale landslide simulations never entered the collisional regime and operate in a separate intermediate flow regime. In other words, most realistic granular flows are not described by either the quasistatic or rapid flow models and it is high time that the field look beyond those early models. This talk will discuss computer simulation studies that draw out the entire flowmap of shearing granular materials, spanning the quasistatic, rapid and the intermediate regimes. The key was to include the elastic properties of the solid material in the set of rheological parameters; in effect, this puts solid properties back into the rheology of granular solids. The solid properties were previously unnecessary in the plasticity and kinetic theory formalisms that respectively form the foundations of the quasistatic and rapid-flow theories. Granular flows can now be divided into two broad categories, the Elastic Regimes, in which the particles are locked in force chains and interact elastically over long duration contact with their neighbors and the Inertial regimes, where the particles have broken free of the force chains. The Elastic regimes can be further subdivided into the Elastic-Quasistatic regime (the old quasistatic regime) and the Elastic-Inertial regime. The Elastic-Inertial regime is the ``new'' regime observed in the landslide simulations, in which the inertially induced stresses are significant compared to the elastically induced stresses. The Inertial regime can also be sub-divided into an Inertial-Non-Collisional where the stresses scale inertially, but the particles interact in clusters through long duration contacts, and the Inertial-Collisional (or the old rapid-flow) regime. Finally, the simulations show that Stress-Controlled flows are rheologically different from Controlled-Volume flows. Physically, there is a range of dense concentrations (0.5$<\nu <$0.6) in which it is possible, but not necessary to form force chains and demonstrate elastic behavior. (In other words it is possible for the material to exhibit two different states at the same concentration.) By forcing the material to support an applied loads across force chains, Stress-Controlled flows may behave elastically through this range of concentrations while, at the same shear rates rate Controlled-Volume flows, fixed at the average concentration of the Stress-Controlled flow, behave inertially.   

Force distributions and stress fluctuations in a triangular lattice of rigid bars

We study the uniformly weighted ensemble of force balanced configurations on a triangular network of nontensile contact forces as a model of force distribution on a hyperstatic granular material. For periodic boundary conditions corresponding to isotropic compressive stress, the probability distribution for single-contact forces, $P(f)$, decays faster than exponentially, and a field closely related to the lattice version of the Airy stress function is found to have fluctuations characterized by a structure factor $S(q) \sim 1/q^4$. The super-exponential decay of $P(f)$ persists in lattices diluted to the rigidity percolation threshold. On the other hand, for anisotropic imposed stresses, a broader tail emerges, becoming a pure exponential in the limit of infinite lattice size and infinitely strong anisotropy.   

Measurement of Forces inside Three-Dimensional, Frictionless, Disordered Matter

We directly measured individual forces inside the concentrated piles of frictionless droplets. We report on the distribution of contact forces normalized by the droplet mean, in addition to the distribution of forces normalized by sample mean, as well as the distribution of contact angles. We compare these results to existing models and to a new, numerical calculation that treats the droplets as independent particles and derive the contact-force probability distribution that arises self-consistently from balancing forces. The force chain architecture was visualized, and quantified with a new definition based on long-range correlations. The obtained chain persistence length helps to establish a connection between microscopic force network and the modulus of meso- or macroscopic piles. This work is supported by NSF (DMR-0305395).   

Effects of Particle Size Dispersity on the Response to Compressive Strains

Particle packings found in nature and industry are rarely comprised of single components, in terms of particle size. These packings are generated under a variety of circumstances which influence its response to an external load or strain. We explore both the effect of packing history prior to application of compressive strain, and the variation in the response with the size distribution of the component particles. We generate the packings by allowing the particles to settle under gravity for a fixed interval of time, or until a cut-off packing fraction is attained, followed by application of a compressive strain for a fixed interval of time. We repeat these studies using numerical experiments for samples of discrete size (200 microns, 195-225 microns , 170-260 microns, 150-295 microns) and random (100-300 microns, 100-400 microns, 100-500 microns) size distributions. We find the number of particles with fewer than 4 contacts to increase with size dispersity of the sample after the particles settle under gravity. In addition, the fraction of plastic contacts decreases with increasing variation in particle size during the compression. We also present correlations between the populations of low and high force bearing contacts, particle size and the yield state of the contacts.   

Measurements of the Yield Stress in Repulsive Athermal Systems

We performed molecular dynamics of dry frictionless granular media to gain a deeper understanding of the yield shear stress in these materials. The measurements were obtained by shearing the systems in both the constant shear force and constant shear velocity ensembles. At fixed shear force, we identified the yield shear stress as the shear stress $\Sigma_{yf}$ required to maintain steady flow in an initially unsheared static state. At fixed shear velocity, we identified the yield shear stress as the average shear stress $\Sigma_{yv}$ in the limit of zero shear velocity. At finite system size, $\Sigma_{yf} > \Sigma_{yv}$, which implies that there is a shear rate discontinuity when the system begins flowing in the constant shear force ensemble. However, the difference between the two measures of the yield shear stress decreases with increasing system size; $\Sigma_{yf}$ and $\Sigma_{yv}$ become identical in the infinite system size limit. Thus, the jump discontinuity in the shear rate at the unjamming threshold is a finite-size effect in frictionless granular systems.   

Experiment test of a Janssen formula in a dense granular column

The stresses inside a tall column of either static or flowing granular material saturate with depth, because the weight of the material is borne by friction with the walls. In the static case, the height dependence of the stress is traditionally described by the Janssen formulation, in which the shear stress at the wall is assumed to be proportional to the normal stress. We report measurements of all three components of force at the wall of a dense, gravity-driven flow of glass beads. We find that the depth dependence of the stress in this slow flow is well-described by a Janssen-like formula. We are also able for the first time to directly test the Janssen assumption, and find that the fluctuations in the shear and normal forces at the wall are highly correlated. The measured friction angle is independent of flow rates for the slow flows we have examined, and is surprisingly close to the ensemble average of the friction angle measured when the flow is stopped.   

Impact of Particle Elasticity on Granular Force Networks

We investigate the distribution of force within vertically-confined granular assemblies using photoelastic techniques that allow determination of both geometric configuration and force upon each particle. By binning multiple realizations with depth, we are able to compare our results with the simple, continuum model of Janssen. Recent experimental studies of the force at the boundaries of such assemblies have largely confirmed Janssen's prediction that mean force saturates exponentially with depth due to frictional contacts at the boundaries. [Ovarlez, Fond and Clement, PRE 67, 060302 (2003)] We have observed deviations from these predictions in our system, which we quantify in terms of the structure of the network that distributes forces. We examine the role of internal elasticity of the particles in causing these deviations. This research is supported by NSF grants DMR-0137119 and DMS-204677 and NASA grant NNC04GB08G.   

Freezing and Melting in Granular Materials

From bowls of nuts to eroding soil, granular materials are all around us. In spite of the fact that granular materials are dissipative and athermal, statistical mechanics allows considerable insight into their behavior. I will present experiments on particles which are vibrated from below and sheared from above within an annular channel. The vibrations have the remarkable effect of crystallizing the material, rather than melting it as temperature would an ordinary material. This freezing/melting transition is hysteretic, with the critical line corresponding to equal kinetic energies for vibration and shear. We characterize the transition between these two states, and observe features reminiscent of both a jamming transition and critical phenomena. Another remarkable property is the increase of pressure with volume over a continuum of partially and/or intermittently melted states, in contrast to standard thermodynamic behavior.   

Plastic Failure Events in 2D Sheared Granular Systems

We present experimental measurements of plastic failure events in a two dimensional granular system consisting of polymer photoelastic disks, placed horizontally, and confined within a rectangular biaxial cell. The bi-refringence of these disks allows us to determine the normal and tangential components of contact forces. We image the system at various deformation states and measure the stress changes and displacements of the disks during one complete shear cycle. The stress changes are found by computing the stress tensor of each disk and the displacements are measured by particle tracking. We obtain bulk stress-strain curves by spatial averaging and find that the system exhibits regions of reversible deformation interrupted by irreversible plastic failure events. We also obtain the behavior of shear modulus of the system. The spatial distribution of reversible and plastic deformations found by studying the displacements of the disks show that in two corners, the disks move uniformly but in a central band aligned along a principal strain direction, we observe multiple vortices. Reversing the direction of shear causes maximum plastic deformation which results in disruption of the vortex structure. We compare our results to the shear transformation zone (STZ) theory.   

Statistical Properties of Granular Solid to Liquid Transition in Small Systems under Shear

The fluidization transition of a dense granular assembly under shear is studied numerically using soft particle molecular dynamics simulations in two dimensions using a previously verified predictor-corrector algorithm. We focus on small systems in a thin Couette cell, examining the bistable region while increasing shear, with varied amounts of random noise, and determine the statistics of shear required for fluidization. We find an approximately linearbetween noise and fluidization shear threshold over the transition regime, and that the variance in the threshold decreases as the system size increases.   

Granular shear flow with imposed vibrations

We present results on a 2D photoelastic shearing experiment in which we impose force fluctuations by vibrating the shearing surface. The experiment consists of a dense assembly of 2D photoelastic grains between two belts moving in opposite directions, such that the central region approximates planar shear. The granular medium lies horizontally between the belts such that gravity does not compact the grains. One of the shearing surfaces is vibrated at a known frequency and amplitude during shear. We measure properties of the particle flow and characterize the force network by placing the photoelastic grains between crossed polarizers. We find that as vibration amplitude is increased, the number and magnitude of these force chains decreases drastically. The vibration also leads to increased slip at the shearing surface and decreased particle flow at both shearing surfaces.