Challenges in Granular Physics

The following sections are included:

• Preface

• CONTENTS

We have examined extended structures, bridges and arches, in computer generated, non-sequentially stabilized, hard sphere deposits. The bridges and arches have well-defined distributions of sizes and shapes. The distribution functions reflect the contraints associated with hard particle packing and the details of the restructuring process. A subpopulation of string-like bridges has been identified. Bridges are fundamental micro-structural elements in real granular systems and their sizes and shapes dominate considerations of structural properties and flow instabilities such as jamming.

We consider a vertically shaken granular system interacting elastically with the vibrating boundary, so that the energy injected vertically is transferred to the horizontal degrees of freedom through inter-particle collisions only. This leads to collisions which, once projected onto the horizontal plane, become essentially stochastic and may have an effective restitution coefficient larger than unity. We therefore introduce the model of inelastic hard spheres with random restitution coefficient α (larger or smaller than unity) to describe granular systems heated by vibrations. In the non-equilibrium steady state, we focus in particular on the single particle velocity distribution f(υ) in the horizontal plane, and on its deviation from a Maxwellian. We use Molecular Dynamics simulations and Direct Simulation Monte Carlo (DSMC) to show that, depending on the distribution of a, different shapes of f(υ) can be obtained, with very different high energy tails. Moreover, the fourth cumulant of the velocity distribution (which quantifies the deviations from Gaussian statistics) is obtained analytically from the Boltzmann equation and successfully tested against the simulations.

We discuss two athermal types of dynamics suitable for spin-models designed to model repeated tapping of a granular assembly. These dynamics are applied to a range of models characterized by a 3-spin Hamiltonian aiming to capture the geometric frustration in packings of granular matter.

We consider the free evolution of systems of granular particles whose dynamics is characterized by a collision rule which preserves the total momentum, but dissipates the kinetic energy. Starting from an inelastic version of a minimal model proposed by Ulam for a gas of Maxwell molecules, we introduce a new lattice model aimed at investigating the role of dynamical correlations and the onset of spatial order induced by the inelasticity of the interactions. We study, in one- and two-dimensional cases, the velocity distribution, the decay of the energy, the formation of spatial structures and topological defects. Finally, we relate our findings to other models known in other fields.

We consider a tapping dynamics, analogous to that in experiments on granular media, on the simple one-dimensional ferromagnetic Ising model. When unperturbed, the system undergoes a single spin flip falling dynamics where only energy lowering moves occur. With this dynamics the system has an exponentially large number of metastable states and gets stuck in blocked or jammed configurations as do granular media. When stuck, the system is tapped, in order to make it evolve, by flipping in parallel each spin with probability p (corresponding to the strength of the tapping). Under this dynamics the system reaches a steady state regime characterized by an asymptotic energy per spin E(p), which is determined analytically. Within the steady state regime we compare certain time averaged quantities with the ensemble average of Edwards based on a canonical measure over metastable states of fixed average energy. The ensemble average yields results in excellent agreement with the dynamical measurements.

This paper discusses a simple two-species ripple model with avalanching. The effect of the avalanching term is investigated numerically, and is found to be crucial in producing realistic ripple profiles.

The coarsening of an array of vortex ripples prepared in an unstable state is discussed within the framework of a simple mass transfer model first introduced by K. H. Andersen et al. (see Ref. 1). Two scenarios for the selection of the final pattern are identified. When the initial state is homogeneous with uniform random perturbations, a unique final state is reached which depends only on the shape of the interaction function f(λ). A potential formulation of the dynamics suggests that the final wavelength is determined by a Maxwell construction applied to f(λ), but comparison with numerical simulations shows that this yields only an upper bound. In contrast, the evolution from a perfectly homogeneous state with a localized perturbation proceeds through the propagation of wavelength doubling fronts. The front speed can be predicted by standard marginal stability theory. In this case the final wavelength depends on the initial wavelength in a complicated manner which involves multiplication by powers of two and rational ratios such as 4/3.

We briefly describe how mean-field glass models can be extended to the case where the bath and friction are non-thermal. Solving their dynamics, one discovers a temperature with a thermodynamic meaning associated with the slow rearrangements, even though there is no thermodynamic temperature at the level of fast dynamics. This temperature can be shown to match the one defined on the basis of a flat measure over blocked (jammed) configurations. Numerical checks on realistic systems suggest that these features may be valid in general.

The initiation and steady-state dynamics of granular shear flow are investigated experimentally in a Couette geometry with independently moveable outer and inner cylinders. The motion of particles on the top surface is analyzed using fast imaging. During steady state rotation of both cylinders at different rates, a shear band develops close to the inner cylinder for all combinations of speeds of each cylinder we investigated. Experiments on flow initiation were carried out with one of the cylinders fixed. When the inner cylinder is stopped and restarted after a lag time of seconds to minutes in the same direction, a shear band develops immediately. When the inner cylinder is restarted in the opposite direction, shear initially spans the whole material, i.e. particles far from the shear surface are moving significantly more than in steady state.

Simulation results of dense granulates with particles of different sizes are compared with theoretical predictions concerning the mixture pressure. An effective correlation function is computed which depends only on the total volume fraction and on the dimensionless width of the size-distribution function. From simulation data of elastic and weakly dissipative systems, one can predict how much disorder (size-dispersity) is necessary to avoid ordering effects due to crystallization. Finally, a global equation of state is proposed, which unifies both the dilute, disordered gas/fluid and the dense, solid regime.

We report measurements of the density of a vibrated granular material as a function of time or taps. The material studied consists of monodisperse spherical glass beads confined to a long, thin cylindrical tube. Changes in vibration intensity are used to induce transitions between two steady state densities that depend on the intensity of the vibrations. We find a complex time evolution similar to previous work on the irreversible relaxation from a loose state toward a steady state. In addition, frequency dependent third order moments of the density fluctuations are measured. The data indicate a coupling between large variations in density on one time scale and noise power over a broad range of higher-frequency scales.

Many features of real granular fluids under rapid flow are exhibited as well by a system of smooth hard spheres with inelastic collisions. For such a system, it is tempting to apply standard methods of kinetic theory and hydrodynamics to calculate properties of interest. The domain of validity for such methods is a priori uncertain due to the inelasticity, but recent systematic studies continue to support the utility of kinetic theory and hydrodynamics as both qualitative and quantitative descriptions for many physical states. The basis for kinetic theory and hydrodynamic descriptions is discussed briefly for the special case of a low density gas.

Granular surface flows are common in industrial practice and natural systems, however, theoretical description of such flows is at present incomplete. Two prototype systems involving surface flow are compared: heap formation by pouring at a point and rotating cylinders. Continuum models for analysis of these flows are reviewed, and experimental results for quasi-2D systems are presented. Experimental results in both systems are well described by continuum models.

We have performed numerical studies of dense granular flows on an incline with a rough bottom in two and three dimensions. This flow geometry produces a constant density profile that satisfies scaling relations of the Bagnold, rather than the viscous, kind. No surface-only flows were observed. The bulk and the surface layer differ in their rheology, as evidenced by the change in principal stress directions near the surface; a Mohr–Coulomb type failure criterion is seen only near the surface. In the bulk, normal stress anomalies are observed both in two and in three dimensions. We do not observe iso-staticity in static frictional piles obtained by arresting the flow.

Our model of shaken sand, presented in earlier work, has been extended to include a more realistic ‘glassy’ state, i.e. when the sandbox is shaken at very low intensities of vibration. We revisit some of our earlier results, and compare them with our new results on the revised model. Our analysis of the glassy dynamics in our model shows that a variety of ground states is obtained; these fall into two categories, which we argue are representative of regular and irregular packings.

A simple model is presented for the description of steady uniform shear flow of granular material. The model is based on a stress fluctuation activated process. The basic idea is that shear between two particle layers induces fluctuations in the media that may trigger a shear at some other position. Based on this idea, a minimum model is derived and applied to different configurations of granular shear flow.

Although there is a vast engineering literature for granular materials, as a subject for physicists it has seen great growth in recent years. This is because, when stripped down to the fundamental problem, it is quite novel, and demands a rethink of the kind of laws familiar elsewhere in physics. We consider the transmission of stress in granular materials and investigate the simplest statically determinate problem.

We formulate a phenomenological model for the dynamical evolution of granular mixtures in a rotating drum. In particular, we present a dynamical basis for an “effective surface tension,” which penalizes sharp interfaces between regions enriched in different components of the mixture. We also present representative results which demonstrate that our model captures many experimental features for this problem.

We discuss the application of synchrotron X-ray microtomography (XMT) to granular matter, foams, crumpled membranes, and paper. XMT provides rapid, high-resolution, fully three-dimensional characterization of each of these classes of material. In some cases, subsequent three-dimensional image processing allows the virtual reconstruction of the disordered material as a specified assemblage of idealized basic structural units. This allows measurement of otherwise inaccessible correlation functions and can also be used as the starting point for data-initiated simulations.

The elastic properties of granular materials can be enormously nonlinear as compared with the properties of non-porous materials. Experiments on isotropic compression of a granular assembly of spheres show that the shear μ, and bulk k, moduli vary with the confining pressure faster than the 1/3 power law predicted by Hertz–Mindlin elasticity theory. Moreover, the ratio between the experimental bulk and shear moduli is found to be constant but with a value larger than the theoretical prediction. Numerical simulations resolve the question as to whether the problem lies with the treatment of the grain-grain contact or with the elastic framework. We find that the problem lies principally with the latter; the affine assumption (which underlies the elastic formulation) is found to be valid for k but to breakdown seriously for μ. This explains why the experimental and numerical values of μ(p) are much smaller than the elastic predictions. In this paper we review recent progress on the understanding of this problem based on microscopic simulations, elasticity theory and more innovative ideas such as jamming, fragility and thermodynamics of granular materials.

Granular flow is all around us. Examples include hourglasses, avalanches, flows of grains and stress distributions in silos and sandpiles, mixing of granular matter such as pharmaceutical components, tumbler drying, segregation of heterogeneous components such as in dry food products, etc. Indeed, the majority of industries have to deal with granular flow problems and the monetary loss due to these problems is unimaginable…

Granular surface flows have still to be fully modelled. Here, we present the four types of front that can be observed in avalanches. These strongly inhomogeneous and unsteady flows are very sensitive test cases for the different types of model. We show that, at least qualitatively for the moment, the model we propose, based on the analysis of the motion of a single grain and layers of grains, can reproduce the different characteristics of these four fronts.

The contact network of a frictionless polydisperse granular packing is isostatic in the limit of low applied pressure. It is argued here that, on disordered isostatic networks, displacement–displacement and stress–stress static Green functions are described by random multiplicative processes and have a truncated power-law distribution, with a cut-off that grows exponentially with distance. If the external pressure is increased sufficiently, excess contacts are created, the packing becomes hyperstatic, and the abovementioned anomalous properties disappear because Green functions now have a bounded distribution. Thus, the low-pressure, isostatic, limit is a critical point.

The following sections are included:

• INDEX