Granular Packings: Nonlinear elasticity, sound propagation and collective relaxation dynamics

Experiments on isotropic compression of a granular assembly of spheres show that the shear and bulk moduli vary with the confining pressure faster than the 1/3 power law predicted by Hertz-Mindlin effective medium theories (EMT) of contact elasticity. Moreover, the ratio between the moduli is found to be larger than the prediction of the elastic theory by a constant value. The understanding of these discrepancies has been a longstanding question in the field of granular matter. Here we perform a test of the applicability of elasticity theory to granular materials. We perform sound propagation experiments, numerical simulations and theoretical studies to understand the elastic response of a deforming granular assembly of soft spheres under isotropic loading. Our results for the behavior of the elastic moduli of the system agree very well with experiments. We show that the elasticity partially describes the experimental and numerical results for a system under compressional loads. However, it drastically fails for systems under shear perturbations, particularly for packings without tangential forces and friction. Our work indicates that a correct treatment should include not only the purely elastic response but also collective relaxation mechanisms related to structural disorder and nonaffine motion of grains.Comment: 21 pages, 13 figure

Rheology and Contact Lifetime Distribution in Dense Granular Flows

We study the rheology and distribution of interparticle contact lifetimes for gravity-driven, dense granular flows of non-cohesive particles down an inclined plane using large-scale, three dimensional, granular dynamics simulations. Rather than observing a large number of long-lived contacts as might be expected for dense flows, brief binary collisions predominate. In the hard particle limit, the rheology conforms to Bagnold scaling, where the shear stress is quadratic in the strain rate. As the particles are made softer, however, we find significant deviations from Bagnold rheology; the material flows more like a viscous fluid. We attribute this change in the collective rheology of the material to subtle changes in the contact lifetime distribution involving the increasing lifetime and number of the long-lived contacts in the softer particle systems.Comment: 4 page

Indeterminacy, Memory, and Motion in a Simple Granular Packing

We apply two theoretical and two numerical methods to the problem of a disk placed in a groove and subjected to gravity and a torque. Methods assuming rigid particles are indeterminate -- certain combinations of forces cannot be calculated, but only constrained by inequalities. In methods assuming deformable particles, these combinations of forces are determined by the history of the packing. Thus indeterminacy in rigid particles becomes memory in deformable ones. Furthermore, the torque needed to rotate the particle was calculated. Two different paths to motion were identified. In the first, contact forces change slowly, and the indeterminacy decreases continuously to zero, and vanishes precisely at the onset of motion, and the torque needed to rotate the disk is independent of method and packing history. In the second way, this torque depends on method and on the history of the packing, and the forces jump discontinuously at the onset of motion.Comment: 11 pages, 7 figures, submitted to Phys Rev

Piling and avalanches of magnetized particles

We performed computer simulations based on a two-dimensional Distinct Element Method to study granular systems of magnetized spherical particles. We measured the angle of repose and the surface roughness of particle piles, and we studied the effect of magnetization on avalanching. We report linear dependence of both angle of repose and surface roughness on the ratio $f$ of the magnetic dipole interaction and the gravitational force (\emph{interparticle force ratio}). There is a difference in avalanche formation at small and at large interparticle force ratios. The transition is at $f_c \approx 7$. For $f < f_c$ the particles forming the avalanches leave the system in a quasi-continuous granular flow (\emph{granular regime}), while for $f > f_c$ the avalanches are formed by long particle clusters (\emph{correlated regime}). The transition is not sharp. We give plausible estimates for $f_c$ based on stability criteria.Comment: 9 pages, 7 figure

Mechanism for Surface Waves in Vibrated Granular Material

We use molecular dynamics simulations to study the formation of surface waves in vertically vibrated granular material. We find that horizontal movements of particles, which are essential for the formation of the waves, consist of two distinct processes. First, the movements sharply increase while the particles are colliding with a bottom plate, where the duration of the collisions is very short compared to the period of the vibration. Next, the movements gradually decrease between the collisions, during which the particles move through the material. We also find that the horizontal velocity field after the collisions is strongly correlated to the surface profile before the collisions.Comment: 6 pages, 3 figures (included

Measurements of the Yield Stress in Frictionless Granular Systems

We perform extensive molecular dynamics simulations of 2D frictionless granular materials to determine whether these systems can be characterized by a single static yield shear stress. We consider boundary-driven planar shear at constant volume and either constant shear force or constant shear velocity. Under steady flow conditions, these two ensembles give similar results for the average shear stress versus shear velocity. However, near jamming it is possible that the shear stress required to initiate shear flow can differ substantially from the shear stress required to maintain flow. We perform several measurements of the shear stress near the initiation and cessation of flow. At fixed shear velocity, we measure the average shear stress $\Sigma_{yv}$ in the limit of zero shear velocity. At fixed shear force, we measure the minimum shear stress $\Sigma_{yf}$ required to maintain steady flow at long times. We find that in finite-size systems $\Sigma_{yf} > \Sigma_{yv}$, which implies that there is a jump discontinuity in the shear velocity from zero to a finite value when these systems begin flowing at constant shear force. However, our simulations show that the difference $\Sigma_{yf} - \Sigma_{yv}$, and thus the discontinuity in the shear velocity, tend to zero in the infinite system size limit. Thus, our results indicate that in the large system limit, frictionless granular systems are characterized by a single static yield shear stress. We also monitor the short-time response of these systems to applied shear and show that the packing fraction of the system and shape of the velocity profile can strongly influence whether or not the shear stress at short times overshoots the long-time average value.Comment: 7 pages and 6 figure

Microscopic origin of granular ratcheting

Numerical simulations of assemblies of grains under cyclic loading exhibit ``granular ratcheting'': a small net deformation occurs with each cycle, leading to a linear accumulation of deformation with cycle number. We show that this is due to a curious property of the most frequently used models of the particle-particle interaction: namely, that the potential energy stored in contacts is path-dependent. There exist closed paths that change the stored energy, even if the particles remain in contact and do not slide. An alternative method for calculating the tangential force removes granular ratcheting.Comment: 13 pages, 18 figure

Amorphous Systems in Athermal, Quasistatic Shear

We present results on a series of 2D atomistic computer simulations of amorphous systems subjected to simple shear in the athermal, quasistatic limit. The athermal quasistatic trajectories are shown to separate into smooth, reversible elastic branches which are intermittently broken by discrete catastrophic plastic events. The onset of a typical plastic event is studied with precision, and it is shown that the mode of the system which is responsible for the loss of stability has structure in real space which is consistent with a quadrupolar source acting on an elastic matrix. The plastic events themselves are shown to be composed of localized shear transformations which organize into lines of slip which span the length of the simulation cell, and a mechanism for the organization is discussed. Although within a single event there are strong spatial correlations in the deformation, we find little correlation from one event to the next, and these transient lines of slip are not to be confounded with the persistent regions of localized shear -- so-called "shear bands" -- found in related studies. The slip lines gives rise to particular scalings with system length of various measures of event size. Strikingly, data obtained using three differing interaction potentials can be brought into quantitative agreement after a simple rescaling, emphasizing the insensitivity of the emergent plastic behavior in these disordered systems to the precise details of the underlying interactions. The results should be relevant to understanding plastic deformation in systems such as metallic glasses well below their glass temperature, soft glassy systems (such as dense emulsions), or compressed granular materials.Comment: 21 pages, 18 figure

Phase transition in inelastic disks

This letter investigates the molecular dynamics of inelastic disks without external forcing. By introducing a new observation frame with a rescaled time, we observe the virtual steady states converted from asymptotic energy dissipation processes. System behavior in the thermodynamic limit is carefully investigated. It is found that a phase transition with symmetry breaking occurs when the magnitude of dissipation is greater than a critical value.Comment: 9 pages, 6 figure

Shear strength properties of wet granular materials

We investigate shear strength properties of wet granular materials in the pendular state (i.e. the state where the liquid phase is discontinuous) as a function of water content. Sand and glass beads were wetted and tested in a direct shear cell and under various confining pressures. In parallel, we carried out three-dimensional molecular dynamics simulations by using an explicit equation expressing capillary force as a function of interparticle distance, water bridge volume and surface tension. We show that, due to the peculiar features of capillary interactions, the major influence of water content over the shear strength stems from the distribution of liquid bonds. This property results in shear strength saturation as a function of water content. We arrive at the same conclusion by a microscopic analysis of the shear strength. We propose a model that accounts for the capillary force, the granular texture and particle size polydispersity. We find fairly good agreement of the theoretical estimate of the shear strength with both experimental data and simulations. From numerical data, we analyze the connectivity and anisotropy of different classes of liquid bonds according to the sign and level of the normal force as well as the bond direction. We find that weak compressive bonds are almost isotropically distributed whereas strong compressive and tensile bonds have a pronounced anisotropy. The probability distribution function of normal forces is exponentially decreasing for strong compressive bonds, a decreasing power-law function over nearly one decade for weak compressive bonds and an increasing linear function in the range of tensile bonds. These features suggest that different bond classes do not play the same role with respect to the shear strength.Comment: 12 page