Volume 9, Issue 3, September 1999
 FOCUS ISSUE: GRANULAR MATERIALS


Introduction to the focus issue on granular materials
View Description Hide DescriptionIn a review paper [H. M. Jaeger, S. R. Nagel, and R. P. Behringer, “Granular solids,liquids and gases,” Rev. Mod. Phys. 68, 1259–1273 (1996)] a few years ago, we wrote about granular material as a distinctive form of matter that exhibits behavior rather different from that of ordinary solids,liquids, or gases. We traced this distinction to three characteristic properties. First, the individual particles making up a granular material are typically large so that thermal energy is irrelevant compared to gravitational energy. Consequently, concepts from equilibrium statistical mechanics are often not applicable. Second, the interactions between particles are frictional and can be mobilized to different degrees depending on the preparation history, giving rise to memory effects, i.e., a static pile will remember how it was formed. Third, when particles collide they do so inelastically so that a “gas” of particles will slow down and come to rest in clumps. In the intervening years, the research on granular matter has progressed rapidly and this may be a good time to ask what we have learned since that article was written. In this spirit, the present special issue of the journal Chaos assembles a spectrum of papers discussing recent developments in the field.

Jamming and static stress transmission in granular materials
View Description Hide DescriptionWe have recently developed some simple continuum models of static granular media which display “fragile” behavior: They predict that the medium is unable to support certain types of infinitesimal load (which we call “incompatible” loads) without plastic rearrangement. We argue that a fragile description may be appropriate when the mechanical integrity of the medium arises adaptively, in response to a load, through an internal jamming process. We hypothesize that a network of force chains (or “granular skeleton”) evolves until it can just support the applied load, at which point it comes to rest; it then remains intact so long as no incompatible load is applied. Our fragile models exhibits unusual mechanical responses involving hyperbolic equations for stress propagation along fixed characteristics through the material. These characteristics represent force chains; their arrangement expressly depends on the construction history. Thus, for example, we predict a large difference in the stress pattern beneath two conical piles of sand, one poured from a point source and one created by sieving.

Modeling of stress distribution in granular piles: Comparison with centrifuge experiments
View Description Hide DescriptionThe classical method to compute stress and strain distributions in granular materials is recalled using continuum mechanics approach, and different rheological laws described. It is recalled that granular materials exhibit highly nonlinear response such as nonlinear elasticity, dilatancy and plastic flow.Finite element technique is used to predict the stress field distribution below a conic and a triangular pile. The dependence of the stress distribution on the rheological law, the bottom boundary condition and the building process (horizontal or inclined strata) is demonstrated. These results are compared to experimental data obtained in centrifuge.

Contact forces in a granular packing
View Description Hide DescriptionWe present the results of a systematic numerical investigation of force distributions in granular packings. We find that all the main features of force transmission previously established for twodimensional systems of hard particles hold in threedimensional systems and for soft particles, too. In particular, the probability distribution of normal forces falls off exponentially for forces above the mean force. For forces below the mean, this distribution is either a decreasing power law when the system is far from static equilibrium, or nearly uniform at static equilibrium, in agreement with recent experiments. Moreover, we show that the forces below the mean do not contribute to the shear stress. The subnetwork of the contacts carrying a force below the mean thus plays a role similar to a fluid surrounding the solid backbone composed of the contacts carrying a force above the mean. We address the issue of the computation of contact forces in a packing at static equilibrium. We introduce a model with no local simplifying force rules, that allows for an exact computation of contact forces for given granular texture and boundary conditions.

Compactivity and transmission of stress in granular materials
View Description Hide DescriptionWe outline a statisticalmechanical theory of granular materials. Stress propagation and force fluctuations in static granular media are still poorly understood. We develop the statisticalmechanical theory that delivers the fundamental equations of stress equilibrium. The formalism is based on the assumptions that grains are rigid, cohesionless, and that friction is perfect. Since grains are assumed perfectly rigid, no strain or displacement field can enter the equations for static equilibrium of the stress field. The complete system of equations for the stress tensor is derived from the equations of intergranular force and torque balance, given the geometric specification of the material. These new constitutive equations are indeed fundamental and are based on relations between various components of the stress tensor within the material, and depend on the topology of the granular packing. The problem of incorporating into the formalism the “no tensile forces” constraint is considered. The compactivity concept is reviewed. We discuss the relation between the concept of compactivity and the problem of stress transmission.

Fluctuations in granular media
View Description Hide DescriptionDense slowly evolving or static granular materials exhibit strong force fluctuations even though the spatial disorder of the grains is relatively weak. Typically, forces are carried preferentially along a network of “force chains.” These consist of linearly aligned grains with largerthanaverage force. A growing body of work has explored the nature of these fluctuations. We first briefly review recent work concerning stress fluctuations. We then focus on a series of experiments in both two and threedimension [(2D) and (3D)] to characterize force fluctuations in slowly sheared systems. Both sets of experiments show strong temporal fluctuations in the local stress/force; the length scales of these fluctuations extend up to grains. In 2D, we use photoelastic disks that permit visualization of the internal force structure. From this we can make comparisons to recent models and calculations that predict the distributions of forces. Typically, these models indicate that the distributions should fall off exponentially at large force. We find in the experiments that the force distributions change systematically as we change the mean packing fraction, γ. For γ’s typical of dense packings of nondeformable grains, we see distributions that are consistent with an exponential decrease at large forces. For both lower and higher γ, the observed force distributions appear to differ from this prediction, with a more Gaussian distribution at larger γ and perhaps a power law at lower γ. For high γ, the distributions differ from this prediction because the grains begin to deform, allowing more grains to carry the applied force, and causing the distributions to have a local maximum at nonzero force. It is less clear why the distributions differ from the models at lower γ. An exploration in γ has led to the discovery of an interesting continuous or “critical” transition (the strengthening/softening transition) in which the mean stress is the order parameter, and the mean packing fraction, γ, must be adjusted to a value to reach the “critical point.” We also follow the motion of individual disks and obtain detailed statistical information on the kinematics, including velocities and particle rotations or spin. Distributions for the azimuthal velocity, and spin, of the particles are nearly rate invariant, which is consistent with conventional wisdom. Near the grain motion becomes intermittent causing the mean velocity of grains to slow down. Also, the length of stress chains grows as The 3D experiments show statistical rate invariance for the stress in the sense that when the power spectra and spectral frequencies of the stress time series are appropriately scaled by the shear rate, Ω, all spectra collapse onto a single curve for given particle and sample sizes. The frequency dependence of the spectra can be characterized by two different power laws, in the high and low frequency regimes: at high ω; at low ω. The force distributions computed from the 3D stress time series are at least qualitatively consistent with exponential falloff at large stresses.

Axial segregation of granular materials
View Description Hide DescriptionWhen mixtures of granular materials are rotated, it is often found that they segregate into bands, along the axis of rotation, which are rich in the various components. This effect is discussed experimentally and theoretically, with emphasis on a mechanism based on surfaceflow. The complimentary phenomenon of radial segregation is reviewed, and a mechanism is proposed. Finally, we consider the longtime behavior of rotating mixtures, particularly their anomolous coarsening.

Measurement of particle motions within tumbling granular flows
View Description Hide DescriptionFlowing granular materials are complex, industrially important, and scientifically provocative. In this paper we report measurements of granular transport in 3dimensional tumbling containers. We use magnetic resonance imaging techniques for direct tracking of particles and measure the interior flows of granular materials. One goal is to measure industrial mixer performance over a wide range of conditions. As the mixer geometries are relatively simple, such measurements could serve as incisive tests during development of better granular equations of motion.

Mixing and segregation of granular materials in chute flows
View Description Hide DescriptionMixing of granular solids is invariably accompanied by segregation, however, the fundamentals of the process are not well understood. We analyze density and size segregation in a chute flow of cohesionless spherical particles by means of computations and theory based on the transport equations for a mixture of nearly elastic particles. Computations for elastic particles (Monte Carlo simulations), nearly elastic particles, and inelastic, frictional particles (particle dynamics simulations) are carried out. General expressions for the segregation fluxes due to pressure gradients and temperature gradients are derived. Simplified equations are obtained for the limiting cases of low volume fractions (ideal gas limit) and equal sized particles. Theoretical predictions of equilibrium number density profiles are in good agreement with computations for mixtures of equal sized particles with different density for all solids volume fractions, and for mixtures of different sized particles at low volume fractions when the particles are elastic or nearly elastic. In the case of inelastic, frictional particles the theory gives reasonable predictions if an appropriate effective granular temperature is assumed. The relative importance of pressure diffusion and temperature diffusion for the cases considered is discussed.

Chaotic granular mixing
View Description Hide DescriptionSeveral models for convective mixing of coarse, freely flowing in granular tumblers have been proposed over the past decade. Powders of practical interest, by contrast, are frequently fine and cohesive, and cannot be analyzed with these models. Moreover, even in the freely flowing regime, mixing transverse to the dominant, convective, direction is typically slow and inefficient. In this paper, we examine two chaotic mixing mechanisms, the first of which can be intentionally applied to increase transverse mixing rates severalfold, with new prospects for further improvements in threedimensional mixing through judicious process design. The second mechanism occurs spontaneously in fine grains, resulting in mixing rates overwhelmingly exceeding what would be possible in freely flowing grains. Finally, we show that the same chaotic mixing mechanisms seen in simple drum mixers are also found to be at work in more complex blender configurations widely used in batch industrial operations.

Segregation induced instabilities of granular fronts
View Description Hide DescriptionExperimental investigation of granular flows containing particles of several sizes and moving down slopes shows that segregation of coarsegrained, irregularly shaped particles induces a fingering instability at the propagating front. The sizesegregation mechanism involves percolation of small particles downward and a corresponding migration of large ones toward the flowsurface. Large particles at the flowsurface experience velocities that are greater than average so that they migrate forward and begin to collect at the flow front. In the case of dry cohesionless flows, the instability depends upon these large particles at the flow perimeter being more angular and thus more resistant to flow than the smaller rounder ones in the interior. A simple analytical model predicts the fingering instability when friction of the flow front is greater than that of the following flow. The presence of viscous liquid inhibits both sizesegregation and the development of the instability. Fluidization of dry flows permits segregation of large particles to flow perimeters, thus increasing permeability and permitting a similar instability that owes its development to the dry frictional perimeter that surrounds a partly fluidized interior.

The rotating bucket of sand: Experiment and theory
View Description Hide DescriptionThe surface shape of a bucket of sand rotating about its cylindrical axis is studied experimentally and theoretically. Focusing on fast time scales on which surface shape is determined by avalanches, we identify three regimes of behavior. At intermediate and high frequencies, the surface shape is always at its critical shape determined by the Coulomb yield condition. The low frequency behavior displays an unexpected subcritical region at the center of the bucket. To understand this central region, we adapt a continuum model of surface flow developed by Bouchaud et al. and Mehta et al. The model indicates that the subcritical region is due to a nonlinear instability mechanism.

Motion of grains down a bumpy surface
View Description Hide DescriptionWe summarize in this article an extensive experimental and theoretical effort carried out to understand the behavior of a single ball when rolling down a bumpy surface. This may appear to be a simple problem but in fact is one that displays a rich variety of different behaviors which allow us to understand better dissipative systems such as granular media. Studies performed previously have shown that the motion of the single ball on the rough surface can be characterized by three different dynamic regimes according to the different values of the two control parameters, the inclination angle θ and the ratio where R is the radius of the rolling ball and r the radius of the glass beads which make up the rough surface. The three regimes are a decelerated regime A, a stationary regime B, characterized by a constant average velocity and a jumping regime C. This result was found to be independent of the composition of the rolling ball and the rough surface. It has been demonstrated that regime B is characterized by a viscouslike friction force that appears for specific parameter values. This friction force can be explained by a model whose central ingredient is the geometry of the surface. The trajectory of the ball in regime B can be pictured as a driven random walkmotion where the fluctuations of the local velocities are due to collisions of the moving sphere and the surface grains. A detailed analysis of diffusive properties of the motion is discussed.

Hysteretic transition between avalanches and continuous flow in rotated granular systems
View Description Hide DescriptionExperiments in drums or cylinders partly filled with a granular system and rotated constantly about their horizontally aligned axis of symmetry show a hysteretic transition from discrete avalanches to continuous flow if the rotation rate is adiabatically changed. Herein, we show that this hysteresis can be explained by the impact of global Langevintype fluctuations in a recently proposed minimal model for surfaceflow along granular piles. For too large magnitudes of the fluctuations corresponding to almost elastic grains, the hysteresis vanishes. This might explain why molecular dynamical simulations were not yet able to detect the hysteretic transition.

Hydraulic theory for a debris flow supported on a collisional shear layer
View Description Hide DescriptionWe consider a heap of grains driven by gravity down an incline. We assume that the heap is supported at its base on a relatively thin carpet of intensely sheared, highly agitated grains that interact through collisions. We adopt the balance laws, constitutive relations, and boundary conditions of a kinetic theory for dense granular flows and determine the relationship between the shear stress, normal stress, and relative velocity of the boundaries in the shear layer in an analysis of a steady shearing flow between identical bumpy boundaries. This relationship permits us to close the hydraulic equations governing the evolution of the shape of the heap and the velocity distribution at its base. We integrate the resulting equations numerically for typical values of the parameters for glass spheres.

Scales and kinetics of granular flows
View Description Hide DescriptionWhen a granular material experiences strong forcing, as may be the case, e.g., for coal or gravel flowing down a chute or snow (or rocks) avalanching down a mountain slope, the individual grains interact by nearly instantaneous collisions, much like in the classical model of a gas. The dissipative nature of the particle collisions renders this analogy incomplete and is the source of a number of phenomena which are peculiar to “granular gases,” such as clustering and collapse. In addition, the inelasticity of the collisions is the reason that granular gases, unlike atomic ones, lack temporal and spatial scale separation, a fact manifested by macroscopic mean free paths, scale dependent stresses, “macroscopic measurability” of “microscopic fluctuations” and observability of the effects of the Burnett and superBurnett “corrections.” The latter features may also exist in atomic fluids but they are observable there only under extreme conditions. Clustering, collapse and a kinetic theory for rapid flows of dilute granular systems, including a derivation of boundary conditions, are described alongside the mesoscopic properties of these systems with emphasis on the effects, theoretical conclusions and restrictions imposed by the lack of scale separation.

Clustergrowth in freely cooling granular media
View Description Hide DescriptionWhen dissipative particles are left alone, their fluctuation energy decays due to collisional interactions, clusters build up and grow with time until the system size is reached. When the effective dissipation is strong enough, this may lead to the “inelastic collapse,” i.e., the divergence of the collision frequency of some particles. The cluster growth is an interesting physical phenomenon, whereas the inelastic collapse is an intrinsic effect of the inelastic hard sphere (IHS) model used to study the cluster growth—involving only a negligible number of particles in the system. Here, we extend the IHS model by introducing an elastic contact energy and the related contact duration This avoids the inelastic collapse and allows to examine the longtime behavior of the system. For a quantitative description of the cluster growth, we propose a burninglike algorithm in continuous space, that readily identifies all particles that belong to the same cluster. The criterion for this is here chosen to be only the particle distance. With this method we identify three regimes of behavior. First, for short times a homogeneous cooling state (HCS) exists, where a meanfieldtheory works nicely, and the clusters are tiny and grow very slowly. Second, at a certain time which depends on the system’s properties, cluster growth starts and the clusters increase in size and mass until, in the third regime, the system size is reached and most of the particles are collected in one huge cluster.

Velocity statistics in excited granular media
View Description Hide DescriptionWe present an experimental study of velocity statistics for a partial layer of inelastic colliding beads driven by a vertically oscillating boundary. Over a wide range of parameters (accelerations 3–8 times the gravitational acceleration), the probability distribution deviates measurably from a Gaussian for the two horizontal velocity components. It can be described by in agreement with a recent theory. The characteristic velocity is proportional to the peak velocity of the boundary. The granular temperature, defined as the mean square particle velocity, varies with particle density and exhibits a maximum at intermediate densities. On the other hand, for free cooling in the absence of excitation, we find an exponential velocity distribution. Finally, we examine the sharing of energy between particles of different mass. The more massive particles are found to have greater kinetic energy.

Convection in horizontally vibrated granular material
View Description Hide DescriptionWe report observations of convective motion in a container filled with granular material when it is vibrated in the horizontal direction. We find that the roughness of the boundaries and the container dimensions play an important role in determining the shape and number of the convection cells. When the container bottom and lateral walls are rough, the system typically exhibits four counterrotating rolls stacked in two pairs on top of each other; for very low filling height, it is possible to observe a single row of rolls arranged laterally along the bottom of the container. With smooth walls, on the other hand, we find that the system typically forms only a single pair of counterrotating convection rolls that originate in the two upper corners of the vibrated material; when the filling height is increased to a level that depends on the container width, we observe a transition to the fourroll state.
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 REGULAR ARTICLES


Islands of accelerator modes and homoclinic tangles
View Description Hide DescriptionIslands are divided according to their phase space structure—resonant islands and tangle islands are considered. It is proved that in the nearintegrable limit these correspond to two distinct sets, hence that in general their definitions are not trivially equivalent. It is demonstrated and proved that accelerator modes of the standard map and of the web map are necessarily of the tangle island category. These islands have an important role in determining transport—indeed it has been demonstrated in various works that stickiness to these accelerator modes may cause anomalous transport even for initial conditions starting in the ergodic component.
