Granular Flow Material Overview Campbell 2003

(2006) 208–229www.elsevier.com/locate/powtec

Powder Technology 162

Granular material flows – An overview

Charles S. Campbell ⁎

Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1453, USA

Received 6 June 2005; received in revised form 19 August 2005

Abstract

The paper attempts to give a critical overview of the field of granular flow with attention both to the history and the underlying physics thatgovern the field. It starts with a discussion of the basic transport mechanisms in a granular flow. It continues with a discussion of contactmechanics – the way that individual particles see each other mechanically. It then discusses the historical limiting regimes of granular flow, theQuasistatic and the Rapid-Flow regimes. Finally, it concludes with a review of the Elastic picture of granular flow, which both unifies theQuasistatic and Rapid regimes and fills in the intervening space. It shows that the rheological behavior of granular systems changes with systemscale constraints, and, in particular, that the materials behave differently under controlled-stress and controlled-concentration conditions. TheElastic model defines an entire flowmap of granular flow and thus allows one to place boundaries on where the Quasistatic and Rapid-Flowmodels (sometimes called kinetic theory models) are something of a red herring and cannot be applied to common granular flows.© 2005 Elsevier B.V. All rights reserved.

Keywords: Transport mechanism; Quasistatic regime; Rapid-Flow regime; Granular flow

1. Introduction

Under the correct conditions, a granular solid can flow like afluid. This was probably first recorded be Lucretius (ca. 98–55 B.C.), who wrote “One can scoop up poppy seeds with aladle as easily as if they were water and, when dipping the ladle,the seeds flow in a continuous stream,” (quotation taken fromJacques [1]). As long as there has been mining and agriculture,man has attempted to exploit the flowability of granular solidsto ease handling and storage problems. In particular, the abilityof gravity to drive a granular flow, as noted by Lucretius, greatlysimplifies and provides a cost-free mechanism of transport. As aresult, the most common granular handling devices, chutes andhoppers, are gravity-driven flows. Yet the design of granularsystems is still something of a black art, in part because even themost basic flow mechanisms of granular materials are not wellunderstood. In fact, science has not identified the set of materialproperties that control the flow behavior.

For the purposes of this article, a granular solid is taken to bea collection of discrete solid particles. In general the spacesbetween the particles are filled with an interstitial fluid, usually

⁎ Corresponding author.E-mail address: [email protected].

0032-5910/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.powtec.2005.12.008

air. However, it will be assumed herein that the particles arelarge and heavy in the sense that they are immune to effects ofthe interstitial fluid. For the most part we will also ignorecohesion between particles; cohesion arises from surface forcesor related phenomena such as liquid bridges, both of which acton the surface area and thus can generally be neglected for largeparticles with small surface area to volume ratios. Note thatthese requirements collectively define what is meant by “large”although those criteria cannot yet be quantitatively defined by aset of dimensionless parameters.

This paper grew out of a long lecture given to the Ohio Statesummer course on Powder Technology. It is an attempt to putthe state of knowledge of granular flows into perspective. It isnot intended to be a review article, in the sense that I am nottrying to mention every paper written on the subject, but insteadattempt to hit the highpoints and give a critical and balancedview to the whole subject.

2. Internal force transmission

The unique features of granular material arise from themanner in which force is internally transmitted. In continuummechanics this is represented by a stress tensor τ, eachcomponent of which τij represents the force in the i-direction

Fig. 1. The two mechanisms of internal momentum transport. (a) Contacttransport: Here momentum crosses the imaginary surface (the dashed line) as theresult of the contact force Fc which can be thought of as transporting momentumbetween the centers of the particles along the vector l. (b) Streaming transport:Here the momentum of the particles is carried across the imaginary surface dueto the random motion of the particles in a manner analogous to the transport ofmomentum in the kinetic theory of dilute gases.

Fig. 2. A schematic of the contact between two spheres of radius R generated bythe application of a force F. Here A is the area of the contact and δ is thedeformation. As shown the deformation is greatly exaggerated.

209C.S. Campbell / Powder Technology 162 (2006) 208–229

on a surface with outward pointing normal unit vector in the jdirection. There are two internal modes of stress transmission.The first or Contact Stress, τc is due to force transmission acrossinterparticle contacts. Thus a force Fc applied at a contact can bethought of as being transmitted in the direction of the vectorl that connects the centers of mass of the two particles involved.(The length of l is the distance between the particle centers.)When averaged ⟨ ⟩ over time and volume, this yields theContact stress tensor:

tc ¼ hFcli ð1Þ(If the forces are transmitted collisionally, this is sometimescalled the Collisional Stress tensor). This is shown schemati-cally in Fig. 1a. Note that Fc need not point in the direction of l;when coupled with the fact that the averaging volume must belarger than a particle (and not allowed to shrink to infinitesimalsize as in standard continuum mechanics), this means that thecontact stress tensor need not be symmetric. Any symmetry isbalanced by gradients in a couple-stress tensor that governs thetransmission of torques internal to the material (see [2]).

If the particles are moving, there will be some degree ofinternal momentum transport due to the motion of an individualparticle as it moves relative to the bulk material, carrying itsmomentum with it as illustrated in Fig. 1b. If u′ represents thevelocity of that relative motion, then one defines a StreamingStress Tensor τs analogous to the Reynolds stress tensor inturbulent flow.

ts ¼ qpvhuVuVi ð2ÞHere ρp is the density of the solid material and ν is the “solid-fraction” or solid concentration, the fraction of a unit volumeoccupied by the particles (so that ρpν is the bulk density of thesolid phase). The streaming stresses will only be significant atsmall concentrations when contacts are infrequent and in caseswhere the random particle velocity u′ is large.

Common granular flows, such as hoppers, chutes andlandslides are densely packed with solid concentrations wellabove 50% by volume. It is possible to obtain flows at a smallconcentration, but they are limited to high-speed laboratoryshear cells, computer simulations and perhaps the rings ofSaturn. As such, the contact stresses will dominate and thestreaming stresses can usually be neglected.

3. Contact forces

In a dense granular flow, forces are largely generated byinterparticle contacts. The contact forces are how the particles“see” one another mechanically. Imagine the two sphericalparticles shown in Fig. 2. As long as linear elasticity applies,the normal force on the contact will proportional to EAε, E isthe Young's modulus, A is the contact area and ε is the localstrain. The strain, ε= δ/L where δ is the depth of the contactdeformation (the distance the contact has been compressed)and L is an appropriate length scale. For the contact betweentwo spheres shown in Fig. 2, note that the contact area A isthe zero on the unloaded contact to the left and increaseswhen the particles are pressed together as on the right. Thefact that both the area A and the strain ε, changesimultaneously as the particle is deformed, leads to non-linearity of the contact response.

In 1882, Hertz [3] derived an elastic solution for the contactbetween bodies. The solution is not exact as often thought, butcontains an implicit assumption that the square root of thecontact area is small compared to both the local radius ofcurvature and the overall dimensions of the body. The normalforce exerted on a contact between two bodies of local curvatureR is:

fn ¼ 43R1=2 E

1� m2d3=2 ð3Þ

(from Johnson [4]), where E is the Young's modulus, and υ isPoisson's ratio. This corresponds to a normal stiffness:

k ¼ dfndd

¼ 2R1=2 E1� m2

d1=2 ð4Þ

Substituting for δ from (2), one can write the stiffness in termsof fn.

k ¼ 61=3R1=3 E1� m2

� �2=3

f 1=3n ð5Þ

210 C.S. Campbell / Powder Technology 162 (2006) 208–229

Now Bathurst and Rothenburg [5] derived the bulk elasticmodulus of a random granular material from the contactstiffness and showed that:

Ebulk~f nð Þ kR

ð6Þ

where n is the coordination number. (The coordination numberis the number of contacts between a particle and its neighbors; itappears in the bulk modulus since the larger the number ofcontacts on a particle, the larger the number of contactsavailable to resist an applied force and, consequently, the stifferthe material.) Note first that the bulk modulus depends on thestiffness, not directly on the modulus E of the material thatmakes up the particles because it is through the stiffness than theparticle see one another elastically. While the stiffness is linearlydependent on E, it also depends on R, the local radius ofcurvature and thus depends on the geometry of the contact.

The bulk modulus in Eq. (6) makes it possible to use thesoundspeed in a static granular material as a way to probe thecontact stiffness. The soundspeed varies as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEbulk=qbulk

pand is

thus proportional toffiffiffik

p. Fig. 3 shows the soundspeed as a

function of the confining pressure p, applied to uniform-sizedstainless steel spheres arranged in a regular face-centered cubicpacking (which, being nearly spherical, smooth and placed in anordered arrangement, is about as nice a granular material as isavailable). Data are taken from Goddard [6], who in turn tookthe data from Duffy and Mindlin [7].

The pressure p, which is applied to the bulk assembly, mustbe balanced by the forces on individual particle contacts. FromEq. (5) one expects the stiffness k to vary as the cube root of thenormal force on the contact fn

1/3, and thus to the cube root of thepressure, p. Thus the soundspeed should go as p1/6. As Fig. 3 isa log–log plot, this means that the sound speed as a function ofpressure should have a slope of 1/6, represented by the solidlines. Fig. 3 shows that this is true only at large confiningpressures. Surprisingly at low pressures the soundspeed variesas p1/4, and only assumes the p1/6 predicted for Hertziancontacts for large pressures. Working backwards through theabove calculation, this would imply that the stiffness varies as

Fig. 3. The sound speed as function of hydrostatic confining pressure in an FCC packtolerance (▵, ±10×10−6 in.). The solid lines have a slope of 1/6, indicative of a Hecontact. From Goddard [6] based on the data of Duffy and Mindlin [7].

fn1/2 or is linearly proportional to the deformation δ. That isbehavior reminiscent of the interaction between the point of aconical contact and a surface. (Conical contacts are non-Hertzian because they have zero radius of curvature at thepoint.) Goddard [6] pointed out that the observed behavior canbe explained if the particles initially interact across near-conicalasperities on the surface, thus accounting for the conicalbehavior at low pressures. As the pressure increases, theasperities are compressed until the spherical surfaces of theparticles come into direct contact eliciting a Hertzian response.Goddard also presents a model that encompasses both limits andthe transitional region between. There have been attempts toexplain this behavior in terms of an increase in the coordinationnumber n with pressure (e.g. Makse et al. [8]) and such anincrease has been observed (see for example, Potapov andCampbell [9]), but the Duffy and Mindlin data used materialscarefully assembled in an FCC packing so that the coordinationnumber was fixed. It is a bit surprising to find a significant effectof asperities on high tolerance stainless steel balls, but ifexamined on a small enough scale, any surface will exhibitsome asperities. One can only expect more severe behaviorfrom common granular materials and indeed soundspeedmeasurements in sands by Richart and coworkers [10,11] (Seealso [6]) show a pressure dependence more characteristic ofconical contacts.

All of the above results indicate a purely elastic contact.Different behavior can be expected if the material yieldsplastically under the application of the contact force. Analysesof Hertzian contacts with plastic yielding performed by Walton[12] and Thornton [13] show that, as the load on a contact isincreased, beyond an initial period of elastic behavior, thenormal force fn, varies nearly linearly with δ, indicating a nearlyconstant normal stiffness k. However, the unloading follows adifferent curve, again nearly linear, but with a steeper slope,indicating a larger but still constant k. (This last means that theforce drops to zero before the particle centers are separated bythe sum of their radii. Physically, this occurs because the plasticdeformation leaves a flat indentation in the surface of theparticles so that they lose contact early.) This bi-linear behavior

ing of 1/3 in. diameter steel balls with low tolerance (○, ±50×10−6 in.) and highrtzian contact while the dashed lines have a slope of 1/4, indicative of a conical


lead to the “latched-spring” contact model used in manycomputer simulations starting with Walton and Braun [14].

In addition, the result of many successive contacts on aparticle surface may work harden or in other ways evolve thecharacter of the surface. For example, in one of the earliest(1918) studies of impact behavior, Raman [15] observed thatrepeatable results could only be achieved if the surfaces of theparticles were polished between experiments – presumablyremoving any plastic damage and work hardening of thesurface. Thus, each contact can change the local surfaceproperties so that the properties on the surface may change withboth position and time.

All of the above discussion involves only the normal forceon the contact. As a result of interparticle friction, each contactwill also experience a tangential force which will evolve evenmore dramatically with its history. Mullier et al. [16] showedthat, when a contact is loaded tangentially until it reaches “grosssliding” (when the surfaces of the two particles slip relative toone another), the effect of gross sliding is to shear off asperitiesfrom the particle's surface, thus changing its frictionalproperties. Mullier et al. found that the behavior before grosssliding could be well described by the complex theory ofMindlin and Deresiewicz [17], but not the behavior followinggross sliding. Presumably, this occurs because the removal ofthe asperities during gross sliding changes the surface frictioncoefficient (Mindlin and Deresiewicz assume the frictioncoefficient remains unchanged throughout the process).

4. Quasistatic or slow-flow theories

Granular flow modeling began with the 1773 paper byCoulomb [18] who first described the yielding of granularmaterials as a frictional process. He was not interested in flow,per se, but in the prediction of soil failures for Civil Engineeringapplications. As such, the onset of failure in the soil, usuallymeant that the structure collapsed; compared to such acatastrophe, the subsequent motion of the soil, that part thatinterests those of us working on granular flows, was of littleinterest.

Fig. 4. A schematic illustration of the critical stress concept. (a) Demonstrates howshear, γt. An over-consolidated material starts above the critical concentration whilecollapse to νc at large strains. (b) The variation of the critical concentration with applieand increases only for very large loadings when the applied stress is large stress is l

The Mohr–Coulomb failure criterion is usually expressed inthe form:

sVcþ r tan/ ð7ÞHere c represents the cohesion of the material (for this paper, theparticles are assumed to be large and dry so that c is assumed tobe zero.), σ is the normal stress, τ is the shear stress, and ϕ is the“friction angle”. When τ=cmσ tanϕ, the material yields andbegins to flow. The two constants c and ϕ are assumed to bematerial properties that are measured in standard shear cell tests.

Coulomb yield, could be used to construct a plastic yieldcriterion, and only the adoption of a flow rule was required toemploy the methods of metal plasticity to granular flow. As aresult, it is not necessary to consider the behavior on the level ofindividual particles (as in Eqs. (1) and (2)), but the material canbe treated as a continuous plastic solid. The general principlesand governing equations are laid out in Sokolovski [19].

The problem is further simplified by the idea of a “CriticalState”, the observation that a shearing granular material willapproach a “critical” concentration, νc, i.e. the fraction of a unitvolume filled with solid material, whose value depends only onthe applied load and is again assumed to be a material property.(The critical state concept is probably due to Casagrande [20]and is explored in detail in Schofield and Wroth [21].) This isshown schematically in Fig. 4. Fig. 4a shows a representativeplot of the approach of a soil towards the critical concentration.A material that is “under-consolidated” (i.e. starting with aconcentration below the critical concentration νc) will increaseits concentration as it sheared until it reaches the critical value.Conversely, an “over-consolidated” material will decrease itsconcentration as it sheared until it reaches the criticalconcentration. As slow granular flows usually involve largeshear strains, it is reasonable to assume that the material isshearing at the critical concentration. Fig. 4b shows thedependence of the critical concentration on the applied stressσ. Note that the critical concentration is nearly constant over awide range of σ, and only increases at large σ. The increase in νcat large σ can be attributed to the compressibility of theparticles. At large applied stress, the solid particles are

the overall concentration approaches the critical concentration νc, at large totalan under-consolidated material starts below the critical concentration, yet bothd stress σ. Note that the critical concentration is nearly constant at small loadingsarge enough to compress the particles.

Fig. 5. A contour diagram of the apparent friction coefficient, tanϕ, from a two-dimensional simulation of a hopper with a 60° angle and a polydisperse granularmaterial, from Potapov and Campbell [27]. The annotations max and minindicate the areas where tanϕ takes its maximum and minimum values,quantitative values of which are written at the bottom of the plot. Note that tanϕis far a constant, but changes by a factor of more than three.

Fig. 6. A photoelastic image of the force chains generated in the two-dimensional shear cell of Howell et al. [29,30]. Here, the inner cylinder isrotating counter-clockwise to force the particles together into chains.


compressed together due to solid deformation at the contactpoints and are squeezed into the interparticle pore space. But fora wide range of smaller loadings, the critical concentration isindependent of the applied stress. In many soil mechanicsapplications, the applied stress can be large, (for examplebeneath a large building). But in most granular flows, theapplied stresses are relatively small and as the total strains arelarge, it is reasonable to assume that the flow is incompressibleand fixed at the critical concentration, νc.

These plasticity-derived techniques have been used widely insoil mechanics to predict the failures of soils below foundationsand structures such as retaining walls and earthen dams. Therewere also problems that became apparent. For example, the firstversions of these theories predict the material would continu-ously expand with shearing and never approach a critical state(e.g. [22]).

When extended to study granular flows, this technique hashad partial success in predicting the flow from hoppers (e.g.Jenike and Shield [23], Davidson and Nedderman [24], andBrennen and Pearce [25]). As that material flows within thehopper, it is assumed that the material is always yielding so that:

s ¼ r tan/ ð8Þeverywhere within the hopper. Furthermore, as the materialexperiences large shear strains, it is always assumed to be at thecritical concentration, υc, and it is treated as incompressible.There were many successes of these theories. In particular theyshowed that the flowrate from a hopper was independent of thedepth of material, a characteristic that makes sand hourglassesan easily built method of timekeeping. (This is a directreflection of the 1895 analysis of Janssen [26] – perhaps the

second great work in granular flow –which showed that beyonda certain height the weight of a bed within a bin is supported byfriction on the sidewalls. Thus, the pressure on the bottom of thebin is independent of bed depth. As that pressure controls theflowrate through the orifice, the flowrate is depth-independent.)But the techniques suffered from mathematical problems ofapplying boundary conditions and the flowrate predictionscould have been better. Jackson [22] examines this in somedetail.

A likely source of the problems, is the assumption that ϕ is aconstant material property. Fig. 5 shows measurements of tanϕin two dimensional hopper flow simulation by Potapov andCampbell [27]. In it, tanϕ can be seen to vary by more than afactor of 3, violating the fundamental assumptions of quasistaticflow theory. This variation of tanϕ can explain the discrepanciesbetween the theory and experiment. However, it is notunderstood why tanϕ changes, as simple shear simulations onsimilarly constituted materials indicate that tanϕ is a constant atsmall shear rates (e.g. [28]).

4.1. The “frictional” nature of granular materials

Eq. (8) indicates that tanϕ is the ratio of shear to normalforces in the material and thus can be understood as an apparantfriction coefficient. Recently, it has become popular to refer toquasistatic flows as “frictional”. However, this is misleading asthe internal behavior of the material is not what one wouldclassically call “frictional”.


To see this requires understanding how the particles interactinside a granular material. Fig. 6 shows a photoelastic picture ofthe interparticle forces inside a two-dimensional shear cell[29,30]. For the photoelastic technique, the brightness of thelight surrounding a contact point is proportional to the force onthe contact. This allows the force distribution within thematerial to be visualized. Notice that the forces are not evenlydistributed throughout the material, but are concentrated in“Force Chains” (e.g. [31–33]). These are quasi-liner structuresthat support the bulk of the internal stress within the material.(Note that many of the interparticle contacts are unloaded, ornearly unloaded.) In a shearing material, these force chains aredynamic structures. When the material shears, particles arepushed together to form the chains. After it is formed, the chainwill be rotated slightly by the shear motion, but will quicklybecome unstable and collapse.

While observing failure within a granular material, it wasnoticed that the failure occurs along narrow planes within thematerial. This lead to a picture of two blocks of particles slidingover one another as shown in Fig. 7a. That picture evokes thestandard model of sliding friction and thus accounts for theMohr–Coulomb behavior (7). However, the slip planes are nottrue planes of infinitesimal thickness, instead are zones on theorder of ten particles across called “shear bands”. Inside theshear band, the particles exist within force chains such as thoseseen in Fig. 6. However, these still behave globally in a“frictional” manner, in the sense that the shear and normalstresses are related. Consider the idealized force chain in asimple shear flow, shown in Fig. 7b. Note that the x- and y-direction forces, Fx and Fy, are related by the force F actingalong the chain. Eq. (1) gives,

sxysyy

¼ Fxly� �Fyly� � ð9Þ

where lx and ly are the x- and y-direction components of thevector l connecting the centers of the contacting particles. Thus

Fig. 7. The source of the “frictional” nature of granular materials: (a) thestandard view that the material deforms as large blocks that interact frictionallyand slip occurs along slip planes within the material, thus interacting in a truefrictional manner, (b) the forces are generated by the compression of ForceChains. As by Eq. (1), the contact stresses are generated as averages of theseforces, one can see that the ratio of shear to normal stress will be related to theratio of Fx to Fy, which are related through F and the angle of the chain. As thechain angles do not vary dramatically, this results in an apparent frictionalbehavior.

the stress ratio τxy/τyy, which in a non-cohesive material is theapparent friction coefficient or tanϕ, is related to the ratio Fx/Fy,which depends only on the geometry of the force chain. Aschains form in the direction best suited to resist the appliedforces, and as they collapse before they have rotated to anysignificant degree, their geometry is roughly fixed andcontrolled by the applied force. Thus it is not surprising thatquasistatic flows demonstrate a friction-like response. However,this response is not the result of frictional sliding as shown inFig. 7a, but a result of the internal structure of force chainsshown in Fig. 7b.

Interestingly, Rapid Granular Flow theories, which will bedescribed in the next section, also predict that the bulk frictioncoefficient τxy/τyy is a constant, (e.g. Lun et al. [34]), althoughexperiments (e.g. [35]) and computer simulations (e.g. [36])show it to be a weak function of the solid fraction ν. (The νdependence is attributed to internal microstructure development[37,38]). As the concentration increases, the particles arrangethemselves in a regular order that allows the material to shear ata large concentration. That structure restricts the orientations ofthe available contacts between particles, the vector l in thecontact stress tensor in Eq. (1) and thus affects the relativemagnitude of the stress tensor components.) This occursalthough there are no long duration solid–solid contacts inRapid Flows and thus, like Quasistatic flows, no true frictionalbehavior. Thus Quasistatic and Rapid Flows are equally,frictional flows.

5. Collisional or rapid granular flows

Bagnold [39] was the first to try and model a granularmaterial from the point of view of individual particles. Heimagined particles of radius d and density ρp at a solidconcentration ν, in a shear flow with shear rate γ. (From here on,it is assumed that γ is the magnitude of the gradient of x-direction velocity u that points in the y-direction within a simpleshear flow. In common notation, γ=du/dy.) The stress tensormust then vary as:

sij ¼ f ðm;qp;d;gÞ ð10ÞAs ν is already dimensionless, the Buckingham Pi theoremallows only one other dimensionless parameter:

sijqpd2g2

¼ fij mð Þ ð11Þ

or

sij ¼ fijðvÞqpd2g2: ð12Þ

This is Bagnold's famous result indicating that the stressesshould vary as the square of the shear rate γ. Bagnold justifiedthe result with a simple model in which the first γ controlled thedegree of momentum exchange between particles and thesecond power of γ reflected the collision rate or the number ofmomentum exchanges per unit time. The stresses then reflectthe internal momentum transport due to interparticle collisions.This is a valid interpretation of Eq. (12), but the equation itself is

Fig. 8. The two modes of granular temperature production: (a) collisional mode;(b) streaming mode.


a result entirely of dimensional analysis and is completelyindependent of any underlying model. In other words, anymodel that uses the same dimensional quantities in Eq. (10), willyield a result in the form of Eq. (12). Bagnold justified hisresults with experiments performed on a suspension of particlesin a shear cell, but a recent re-examination of that data, Hunt etal. [40], throws doubt on the interpretation of the experimentalresults.

5.1. The granular temperature

Bagnold's picture of the individual granules moving in ashear flow invariably brings comparison with the motion ofmolecules in the kinetic theory of gases. Furthermore,interparticle collisions will induce random velocities that arereminiscent of the thermal motion of molecules. The magnitudeof these fluctuating velocities is called the “granular tempera-ture”.

T ¼ 13j uV2� �j ¼ 1

3uV2� �þ vV2

� �þ wV2� �� ð13Þ

Notice that this is related to the trace of the streaming stresstensor described in Eq. (2).

T ¼ 13qv

TraceðtsÞ ð14Þ

As the streaming stresses arise from unsteady velocities, itshould not be surprisingly that there is this close connectionbetween τs and the granular temperature. The concept of agranular temperature was first introduced by Ogawa [41].

If one takes the granular temperature as the exact analogue ofthe molecular temperature in the kinetic body theory, then onecan use the formalisms of kinetic theory, for example theChapman–Enskog method (e.g. [42]), to derive a set ofgoverning equations for granular flow. From these ideas camethe field of “Rapid Granular Flows”. As in the kinetic theory, itassumes that the granules interact by instantaneous collisionsand that all transport rates are governed by the granulartemperature. As will be shown, both of these are problematicassumptions. But central to the whole issue is the granulartemperature, which will be examined here in some detail.

Just like the stresses in Eqs. (1) and (2), granular temperatureis produced by a collisional mechanism (schematically shownin Fig. 8a) and a streamingmechanism (Fig. 8b). The collisionalmechanism is the most intuitively obvious. Because of thegeometry of the impact, any collision between particles will actto randomize the impact velocity, thus converting the meanmotion of the flow into granular temperature. The streamingmechanism (Fig. 8b) results from the motion of a particlerelative to a velocity gradient.

Consider the streaming mechanism illustrated schematicallyin Fig. 8b. Imagine a particle that starts in a high velocity regionof the flow and imagine also that its random motion has acomponent moving parallel to the velocity gradient, towards thelower velocity regions of flow. In the time before its nextcollision, the difference the between the mean flow velocity atits original and current positions appears as granular temper-

ature. While a product of the mean flow field, this appears as arandom velocity, because on the average, for every particle thatmoves downwards in the velocity gradient – producing apositive “random” velocity – another moves upwards –producing a negative “random” velocity. Although the gener-ated velocities are random in sign, they are not random indirection. This mechanism only produces “random” velocitycomponents in the direction perpendicular to the velocitygradient, which is usually the direction of the mean flowvelocity (here, the x-direction). Thus the granular temperaturesare large in the mean flow direction. This results in normal-stress differences, with the largest normal-stresses in the flowdirection; at low concentrations, the flow direction temperature,can be several times larger than in the other directions.Naturally, the streaming mechanism dominates at a smallconcentrations while the collisional mechanism dominates atlarge concentrations.

As described above, granular temperature is only producedby velocity gradients. There are other mechanisms of granulartemperature production. For example, shock waves, such asthose produced in vibrated granular flows can generate granulartemperature [9,43]. In multiphase flows, interactions betweenparticles and the interstitial fluid can also generate granulartemperature. However, this paper will only consider shear-induced temperatures.

Granular temperature is one of the most confusing conceptsin granular flow. Typically, when granular temperature ismeasured, it is assumed to be a measure of the unsteadycomponents of velocity. But granular temperature reflects thevelocity fluctuations on top of the mean velocity, and if themean velocity is itself unsteady, then one must be careful toseparate the unsteady mean velocity from the granulartemperature. To clarify these issues, one must understand therole of granular temperature in kinetic theory modeling. There,the temperature drives the transport rate in the two principalmodes of internal transport. Granular temperature provides therelative velocity that drives particles together to collide,

Fig. 9. The energy flow pattern in a rapid granular flow (after Campbell [53]).

Fig. 10. The parameter S=dγ/T1/2 relating the magnitude of the granulartemperature and shear rate as a function of the solid concentration, v. The linesare the predictions of Lun et al. [34]. While the symbols are from the computersimulations: (from Campbell [36], used with permission).


resulting in the collisional transport mode (the contact stressesin Eq. (1)). Also, granular temperature causes the diffusiveparticle mixing that results in the streaming mode of transport(Eq. (2)). Both arise because the granular temperaturegenerates relative motion between particles. But if the meanflow is unsteady (for example, particles resting on a surfacethat is gently vibrated with accelerations well below 1−g), allthe particles will move together with the same unsteadymotions. But that unsteadiness does not drive relative motionbetween particles and is thus not a granular temperature. Aslightly more confusing case is fluid turbulence, in whichparticles may become entrained in turbulent eddies. As all theparticles within a given eddy move with the same mean, if anunsteady mean, velocity, the turbulence does not generateinterparticle collisions and, in that sense, is not a granulartemperature. On the other hand, the eddies do inducestreaming transport of the particle momentum and, in thatsense, the turbulence acts like a granular temperature. In otherwords, the unsteady motion in an eddy is only partially agranular temperature as it generates streaming stresses but nocollisional stresses. Also, when two eddies collide thedifference in velocity between the eddies will induce collisionsbetween particles in different eddies, and will generatecollisional stresses. Thus, it is almost impossible to quantifi-ably the portion of turbulence that fills the role of a granulartemperature. In general, it is difficult, if not impossible, todetermine the granular temperature in unsteady systems.

Unlike molecular systems, the interactions between macro-scopic granules are inelastic. Hence the energy reflected in thegranular temperature is continually being dissipated away toheat and must be re-supplied from the mean flow energy. Asboth temperature generation mechanisms described in Fig.8 derive from the shear motion, the granular temperature iscreated by shear work. Thus rapid granular flows follow anenergy flow pattern shown in Fig. 9. Driving forces, e.g. gravityor the motion of walls and other boundaries are converted intothe kinetic energy of the mean flow, which is converted togranular temperature through shear work, and from there tosensible heat through inelastic collisions or another dissipationmechanism such as air drag. However, most rapid flow theoriesonly assume collisional dissipation. And, as collisions areassumed to be instantaneous, the dissipation is represented as acoefficient of restitution, ε, the ratio of the approach to recoilvelocities in the center of mass. Since granular temperature isproduced by shear work one might expect a relationshipbetween the shear rate γ and the magnitude of the granulartemperature. As given in Eq. (13), the granular temperature Thas units of (velocity)2. Thus a convenient dimensionlessscaling of the granular temperature and shear rate is theparameter

S ¼ dg

T1=2ð15Þ

first proposed by Savage and Jeffrey [44].Measurements of S viathe computer simulations of Campbell [36] and comparison withthe predictions of Lun et al. [34] are shown in Fig. 10. Thegeneral trend of the simulation data is predicted by the Lun et al.

Fig. 11. The ratio of the x- and y-direction temperatures, Tx/Ty. Note theanisotropy at small ν (after [36]).


theory, although the simulations are frictional while the theoryignores friction. (Particle surface friction also dissipates energy;this allows a simulation to produce steady state temperatures atan otherwise elastic ε=1.) There are a few interesting featureshere. The first is that over most of the range of solid fractions ν, Sis of order 1. This means that the relative particle velocitiesinduced by the mean shear dγ, is of the same order as thatproduced by the granular temperature T1/2. Also note that thevalue of S generally decreases as the coefficient of restitutionincreases, reflecting the fact that that the larger ε, the smaller thedissipation rate and thus the larger the granular temperatureproduced by a given shear rate. Perhaps the most striking featureis that S→0 as ν→ 0, indicating that at small solidsconcentration, the granular temperature becomes infinite. Thiscan be understood by first noting that, at small concentrations,the stresses are largely generated by the streaming mechanism(Eq. (2)) so that in a simple shear flow with flow in the x-direction and the velocity gradient in the y-direction, the granulartemperature is generated by the shearwork, τxyγ=ρρνbu′ν′Nγ∼νand is directly proportional to the solid fraction ν. The dissipationof the granular temperature occurs through collision. Theprobability of a collision is proportional to finding two particlesin contact, and at small concentrations, where the position of oneparticle has little effect on the position of any other particle, theprobability of finding a particle at any location is proportional to ν;thus the probability of finding two particles in contact isproportional to ν2, so that the collision rate (and the dissipationrate) are proportional to ν2. Consequently, the ratio of granulartemperature production to dissipation ∼ν/ν2∼1/ν and becomesinfinite as ν→0. Thus, the granular temperature becomes infiniteas ν→0. Note that this only occurs because collisions are assumedto be the only dissipation mechanism. If there are other ways ofdissipating energy (air-drag for example), this limit would notappear. In fact, there are no documented observations of this limitother than in computer simulations.

It should also be noted that, unlike the thermodynamictemperature, the granular temperature is not isotropic. Thisshould be obvious because the streaming mechanism oftemperature generation (Fig. 8b) only generates temperaturein the direction perpendicular to the velocity gradient. Fig. 11shows the ratio of Tx and Ty the x- and y-direction temperatures,from Campbell [36]. Note that at small ν, where the streamingmode of temperature generation is largest, Tx/Ty is significantlylarger that unity reflecting the streaming generation of x-direction temperature. But at large ν, where the collisional modedominates, Tx/Ty is near unity indicating the more isotropictemperatures produced collisionally. Note also that thecoefficient of restitution ε has a strong effect, with the largestanisotropies appearing at the smallest ε. The larger dissipationat the smaller ε reduces the collisional generation of temperatureand thus enhances the relative importance of the streamingmechanism, leading to larger temperature anisotropies. Notethat while higher order kinetic theories can produce normalstress differences [46,47], they are strictly only valid in the limitν→0; furthermore, they predict that the differences arefunctions only of the inelasticity and are independent of theconcentration.

5.2. Rapid granular stresses

Rapid granular flows provide a forum in which it is possibleto observe the interplay between the collisional stresses (Eq. (1))and streaming stresses (Eq. (2)). (This is largely because theflow will be in the rapid regime whenever the concentration issmall enough that streaming stresses become significant.) Fig.12 shows the dimensionless shear stresses, scaled as in Eq. (11),generated in a simple shear flow. Note that the simulation dataasymptotes to ∞ both as ν→0 and as ν approaches a “randomclose pack” (the concentration of a randomly assembled volumeof spheres) at ν≈0.64. The latter limit occurs because in arandom close pack, the particles are in intimate contact and thecollision rate is approaching ∞; thus this asymptote occurs as asingularity in the contact stress tensor (Eq. (1)). It is also abyproduct of the rigid particle assumptions that lie at the heart ofrapid flow ideas; because the particles cannot deform, it wouldrequire an infinite stress to shear a material near the randomclose pack. But the singularity would disappear for real particleswith finite elastic moduli, as by deforming their shape, theparticles can be forced to shear at any concentration. In addition,the stresses demonstrate a singular behavior as ν→0. This is areflection of the fact that the granular temperature becomesinfinite in that limit (as seen in Fig. 10). As the granulartemperature is closely related to the streaming stresses, obviousfrom Eq. (14), the stresses demonstrate the same asymptoticbehavior as the granular temperature.

5.3. Rapid granular flow models

The formalisms of gas kinetic theory can be used to derive aset of equations for granular flow if the particles are assumed tobe rigid. In turn, the rigid particle assumption implies that all

Fig. 12. The dimensionless shear stress in a rapid flow from computersimulation. The open symbols are from the rough particle simulations [36]. Thesolid symbols are from smooth particle simulations [45]. The lines arepredictions of rapid flow theory [35] (after [36]).


contacts occur instantaneously. Thus, there is vanishingprobability of multiple simultaneous contacts and only binarycontacts need be considered. From there, kinetic theoryformalisms yield a set of Navier–Stokes like equations. Thisshould not be surprising as the same formalism applied to gasesalso yield the Navier–Stokes equations. While there are minorvariations between theories, the basic equations are:

Conservation of mass:

DqmDt

þ qmjdu ¼ 0 ð16Þ

Conservation of momentum:

qmDuDt

¼ jp q;m;T ;eð Þ þjdðgðq;m;T ;eÞjuÞ ð17Þ

Conservation of granular energy (granular temperature):

qmDTDt

¼ jdðaðq;m;T ;eÞjTÞ þ s:ju� Cðq;m;T ;eÞ: ð18Þ

In this last equation, α is the conductivity, τ :∇u is thetemperature production by shear work, and Γ is the dissipationby inelastic collisions. Haff [48] gave a wonderful heuristicanalysis that through dimensional analysis gives a set of scalinglaws for the various constitutive coefficients. As the pressurehas units of (mass)/(length-time2):

p ¼ qfpðm;eÞT ð19Þ

As the viscosity η has units of (mass)/(length-time), one canwrite the viscosity as:

g ¼ qdfgðm;eÞT1=2 ð20ÞFollowing the same ideas, the conductivity of the granulartemperature is:

a ¼ qdfaðm;eÞT1=2 ð21Þand the dissipation is:

C ¼ qfCðm;eÞd

T3=2 ð22Þ

Where fp, fη, fα and fΓ are unknown functions of thedimensionless, concentration ν, and coefficient of restitution,ε, that must be determined from experiment or analysis.

There are several problems that should be immediatelyapparent with this formulation. The most obvious is that therange of applicability of rapid flow theory is limited. Note thatfrom Eq. (17), the solid phase stresses are viscous in nature.Thus systems such as hoppers cannot be modeled by rapid flowtheory. A key feature of hoppers is the frictional support ofmaterial by the vertical walls of the bin (the effect explained byJanssen [26]). A viscous material produces no forces unless thematerial is in motion and can thus provide no such support. AsJanssen's frictional support appears in static materials, it cannotbe modeled the type of viscous material predicted by rapid-flowtheory. There have been attempts to add a frictional response torapid flow models, most notably Johnson et al. [49,50]. Butthese are ad hoc models based on self-contradictory assump-tions as they are formed by simply adding a rapid-flow model,based on instantaneous collisions, to a frictional model, basedon long duration contacts.

Also, the gas kinetic theory on which rapid-flow models isbased, assumes that the molecular collisions are elastic in thesense that they do not dissipate energy. In converting the theoryto granular materials, the inelasticity of granular impacts islargely accounted for by the granular energy Eq. (18). But onemust compute integrals over a velocity distribution functions inorder to compute the constitutive relationships, (19–22) and themethods for computing the distribution function restrict thesystem to “nearly elastic particles”, roughly ε=0.9 and above.This severely limits the materials that may be modeled withthese methods.There is also a more subtle problem. Notice thatall of the constitutive laws in (19–22) obtain their rates throughthe granular temperature T. This implicitly assumes that themagnitude of the thermal velocities (T1/2) is much larger thanthe relative velocities induced by the shear (dγ). In terms of Fig.10 this means that S≪1, which is only observed at extremelysmall solid concentrations. However, Fig. 10 shows that overmost of the range of solid concentrations, S≈1, indicating thatdγ∼T1/2. Thus, the mean shear and the temperature are equallyimportant in driving the relative motion between particles, thecollision rate, and thus the transport rates in a rapid granularflow. As the kinetic theories depend only on the temperature togovern transport, they most likely to either underpredict thetransport rate or overpredict the granular temperature.


Furthermore, as the relative velocities induced by the shear rateare only in the direction perpendicular to the velocity gradient,this introduces anisotropies in the angular distribution ofcollisions about a particle (as measured by Campbell andBrennen [37]). Interestingly, the collisional anisotropy wasincluded in the earliest, albeit incomplete, rapid flow models,Savage and Jeffrey [44] and Jenkins and Savage [51], whichonly considered contact stresses. To include streaming stresses,required modifying the velocity distribution function, whichproved intractable if the collisional anisotropy was included. Asthe theories predict that S≈1 (Fig. 10), they are not self-consistent in that their predictions conflict with their implicitassumptions. Goldhirsch [52] cites Sela and Goldhirsch's [47]comparison with normal stress difference data as evidence thatthis effect is unimportant; but this argument is not applicable asthe comparison is done at ν=0, the only point where the Selaand Goldhirsch calculation is valid. There, Fig. 10 shows S≈0(T=∞) so, of course γd≪T1/2, and, while there may becollisional anisotropy due to the anisotropic granular temper-ature, there will be no shear-induced collisional anisotropyunder the conditions of the Sela and Goldhirsch analysis.

Finally, at the heart of all kinetic theories is theassumption of Boltzmann's “Stosszahlansatz” or molecularchaos, that there are no correlations in the velocities orpositions of colliding particles. This is troubling becausecommon granular flows occur at such large concentrationsthat any given particle will interact many times with itsneighbors and it is likely their velocities will be stronglycorrelated. In addition, the aforementioned microstructures[37,38] correlate the relative positions of particles. Thus truemolecular chaos is unlikely in real granular systems althoughit is difficult to estimate the degree of error introduced by thisassumption.

In 1990, I wrote a review article on the field of rapid granularflows [53]. The article ended with a list of “Pressing Concerns”designed to push the field towards more realistic systems and itis worth a paragraph to comment on the progress of the last 15years. The concerns were: Material properties, Microstructure,Non-spherical particles, Non-uniform particle size and segre-gation, Interstitial fluid effects and Solid/Fluid behavior ofgranular systems. However, it should have been obvious, evenin 1990, that the first 3 topics would be almost intractable, eitherbecause they complicated the collision integrals from which theconstitutive properties are derived or because they violate theassumptions of molecular chaos. For example, even simpleproperties such as a stick–slip surface friction make adiscontinuity in the collision integrals; as a result, friction isonly approximately incorporated in Rapid-Flow theoriesthrough a tangential coefficient of restitution. Also, frictiondissipates energy and as discussed above, if the energydissipation is large enough, it may be possible to accuratelyassess the velocity distribution function. Like the collisionalanisotropy, the development of internal microstructure affectsthe contact angle between particles and it is difficult to includein the kinetic theories, partially because of the complications tothe collision integrals and partially because it violates theStosszahlansatz. Non-round particle shapes bring the particle

orientation into the problem, which similarly complicates thenotion of molecular chaos.

Rapid-Flow theory has been used to generate models forbinary mixtures, pioneered by Jenkins and Mancini [54].Kinetic theory based models of segregation have also beendeveloped, but the results are quite mixed (see the review ofOttino and Khakhar [55]). Being probabilistic models, theyagree well with probabilistic Monte Carlo simulations, but theagreement breaks down if compared against more realisticdeterministic simulations. Some degree of agreement could behad only if the granular temperature is used as a fittingparameter. Khakhar et al. [56] argue that this is due to abreakdown in the underlying kinetic theory, in that the frictionaldissipation is so large. But if kinetic theory assumptionsbreakdown in determining the granular temperature, it isdifficult to argue that the same assumptions work well inpredicting segregation within the same flow. Besides, if thetheory cannot handle frictional particles, then it can handle norealistic materials.

Much of the recent effort in the areas of rapid granular flowor kinetic theory have been directed towards issues that are oflargely academic interest, either because of unrealistic assump-tions or because they are only of interest at small particleconcentrations that are never found outside the laboratory.These include items such as the “cooling” of homogeneouslythermalized granular “gas” (first introduced by Haff [48]) whichcan of course never be found in reality because there is no wayto create a homogeneously thermalized granular gas. Some ofthe higher order kinetic theories (e.g. Sela and Goldhirsch [47])are valid only in the ν→0 limit and thus inapplicable to anyrealistic granular flow. The development of “inelastic micro-structure”, a clustering instability, first observed by Hopkins andLounge [57] has received much attention. At low concentra-tions, particles are observed to not be homogeneouslydistributed but to form higher concentration clusters surroundedby regions that are near voids. But this has little effect for thelarge concentrations of common granular flows, simply becausethe particles are already so tightly packed, there is no room forthe clusters and voids to grow. (It is somewhat inappropriate toeven refer to these disordered clusters as “microstructures”, asthat name implies an ordered “structure” of particles; as a result,these are often confused, e.g. Goldhirsch, [52], with the orderedhigh concentration microstructures that strongly affect therelative magnitude of the stress tensor components [37,38].)

In fact, much of the progress has been negative in the sensethat we are learning that more and more granular systemscannot be described by rapid-flow theory. For example, manymodels have been developed that use rapid granular flow ideasto model the solid phase stresses in multiphase systems (one ofthe “Pressing Concerns” from Campbell [53]) mostly in gas-fluidized systems (e.g., [58–60]). But direct measurements influidized beds [61] show that they cannot be modeled by kinetictheory [62,63]. The transition from solid-like to fluid-likebehavior, such as is seen at the boundaries of funnel flows inhoppers (the last Pressing Concern) has been shown not to bethe phase change suggested by Campbell [53], but insteadoccurs outside the realm of rapid granular flows as a quasistatic-


like failure [64,65]. And even vibrated beds, at least those ofcommercially significant depth, cannot be described by kinetictheory [66].

6. Elastic granular flows

Recently, Campbell [28,67,68] has been able to unify thevarious granular flow theories and, in particular, fill in the gapbetween the quasistatic and rapid flow regimes, and drawcomplete flowmaps for shearing granular materials. Themissing link was to include the elastic properties of the particlesinto the models – in effect to put the solid back into granularsolids. The principle quantity is the interparticle stiffness k, as itgoverns how particles “see” one another mechanically and, asmentioned previously, determines the bulk elastic properties ofthe granular material. It requires the exercise of littleimagination to see the importance of the stiffness to therheology of dense granular flows. At the large concentrations ofcommon granular flows, particles are locked into force chains,such as those seen in the shear cell in Fig. 6. Assuming that thewalls the cell are rigid, then the degree of compression of theforce chains and thus the deformation of each contact, isdetermined only by the need for the material to shear at a givenconcentration. Thus, if each particle in Fig. 6 was removed andreplaced with one with, say, twice the stiffness, the forces oneach contact and thus the contact stresses would double. As thestreaming stresses are insignificant at such large concentration,this means that the stresses are proportional to the contactstiffness.

Now imagine a high-concentration granular material, withthe forces distributed in force chains, undergoing shear atconstant volume. The shear will force particles together to formthe chain, cause the force chain to rotate until it becomesunstable and collapses. As the chain rotates, it will want to dilatethe bulk material, but is prevented from doing so by the constantvolume constraint; instead, the rotation compresses the chainand generates an elastic response. If Δ is the overalldeformation of the chain, then the deformation of each contactis δ=Δ/N=Δd/L where N=L/d is the number of contacts in achain of length L composed of particles with characteristic sized. Thus the force F=kδ=kΔd/L. The generated stress τ∼F/d2∼k/d. Hence an appropriate dimensionless scaling for thestress in the regime is τd/k. Note that following the aboveargument, τd/k∼F/kd=δ/d; that is, τd/k can be interpreted as theparticle deformation δ represented as a fraction of the particlediameter, d.

Campbell [28] divided the entire granular flow field into twobroad regimes the Elastic and the Inertial. The Elastic regimesencompass all flows in which force is transmitted principallythrough the deformation of force chains for which the naturalstress scaling is τd/k. Continuing from the above arguments,follow a force chain through its life cycle. The chain will formwhen particles are driven together by the shear rate γ and thusthe rate of chain production is proportional to γ. The chain thenrotates and is compressed. The degree of compression and thusthe force generated in the chain is determined by the necessityof meeting the constant volume constraint; hence the magnitude

of the force is independent of the shear rate. But the chainrotates at a rate ∼γ and eventually become unstable and self-destructs. Thus, the lifetime of the chain is proportional to 1/γ.Consequently, the product of (formation− rate)× (lifetime) for achain is γ-independent and because the force generated is alsoγ-independent, The stresses generated are quasistatic. This isthe Elastic–Quasistatic regime. It is the same as the oldQuasistatic regime; the word Elastic is added to indicate that itis a subregime of the global Elastic regime.

But at high shear rates, the elastic forces in the chain mustabsorb the inertia of the particles that are gathered in the chain,requiring an extra force required to accelerate the particles in thechain so that it rotates at a rate proportional to the shear rate.Thus, even though the particles are locked in force chains, theforces generated must reflect the particle inertia. The force Fgenerated in the chain must have the form F=a+bγ where a isthe baseline elastic force and bγ is the inertial augmentation.Still the (formation− rate)× (lifetime) of the chain is γ-independent so that the resultant stresses τ∼F∼a+bγ increaselinearly with the shear rate. (This is shown in Campbell [67].)Naturally, there will be some inertial effect throughout theElastic regime, but for a wide range of flows, bγ≪a and theflows appear quasistatic. However, when bγ becomes of thesame the same order as a, i.e. the inertial forces become of thesame order as the elastic forces, the flow transitions into theElastic–Inertial regime in which the forces are linearlyproportional to the shear rate γ.

The ratio of elastic to inertial effects is govern by adimensionless parameter:

k⁎ ¼ kqd3g2

ð23Þ

Note that k / (ρd3γ2)= (τ /ρd2γ2) / (τd /k) and is thus the ratio ofBagnold's inertial to the elastic stress scalings. A similarparameter was first proposed by Babic et al. [69]. Campbell [28]gives several interpretations for this parameter, but the mostuseful is that k⁎ represents (d /δi)

2 where δi is degree ofdeformation expected from the impact by a particle moving atthe shear velocity, dγ, making k⁎ a measure of inertially induceddeformation, much as (τd/k) is a measure of elastic deformation.As such the value of k⁎ reflects the relative effects of elastic toinertial forces, i.e. in principle at large k⁎ elastic forces dominateand at small k⁎, inertial forces dominate.

Fig. 13 shows the dimensionless normal stress, scaledelastically as τyyd/k plotted against the stiffness parameter. Theplot is marked to show the division into the Elastic–Quasistaticand Elastic–Inertial regimes, which are differentiated solely bythe fact that the stresses are independent (Elastic–Quasistatic)or dependent (Elastic–Inertial) on the shear rate γ, and thus onthe parameter k / (ρd3γ2). As expected, the flow transitions fromElastic–Quasistatic to Elastic–Inertial as the shear rateincreases, (k/(ρd3γ2) decreases). Each point was taken forthree particle diameters and as many as three stiffnesses andthus each point represents up to 9 overlapping points illustratingthe robustness of the scaling. These data are for a single solidfraction ν=0.6, which lies below a random close-pack

Fig. 13. The dimensionless normal stress τyyd/k, as a function of the stiffness parameter k/(ρpd3γ2), showing the separation into the Elastic–Quasistatic and Elastic–

Inertial regimes. This data was taken in shear flow simulations with constant stiffness at a concentration v=0.6 and particle surface friction μ=0.5 (after [28]).


concentration; hence, these stresses are a byproduct of the shearflow. In the absence of shear, the particles need not to be incontact and no stress would be generated. Note that thedissipation rate reflected through the coefficient of restitution ε,is only of importance for the smallest k / (ρd3γ2) (or the largestshear rates, γ), making these flows very different from RapidFlows in which the granular temperature is critically dependenton the dissipation rate. (This should be apparent from the strongeffect of the coefficient of restitution in Fig. 12.)

Fig. 14 shows the effect of the particle surface frictioncoefficient μ on the dimensionless normal stress, τyyd /k. Whileμ has only a weak effect on the shear to normal stress ratio τxy /τ yy (the bulk friction coefficient), it has a major effect on thenormal stresses [28]. The μ=0.1 data in Fig. 14, appears to gozero at large k / (ρd3γ2), but actually the values are about twoorders of magnitude smaller than the corresponding points for

Fig. 14. The effect of the particle surface friction coefficient μ on the dimensionless nconcentration, ν=0.6. Surprisingly, the surface friction has a strong effect on the normtransitions from Elastic to Inertial behavior at large k / (ρd3γ2) (after [28]).

μ=0.5. At μ=0.1, force chains are weak and the flowtransitions from Elastic to Inertial behavior. The Inertialregime encompasses flows where force chains cannot formand the momentum is transported largely by particle inertia as inRapid-Flow theory (although as shall be shown, inertialbehavior does not necessarily mean a Rapid Flow). In Inertialflows, the stresses are independent of the stiffness k and thenatural scaling is the Bagnold scaling τ /ρd2γ2.

But notice that the μ=0.1 data in Fig. 14, is only Inertial atlarge k/(ρd3γ2) and transitions back to Elastic–Inertial behavioras γ increases (as k/(ρd3γ2) is reduced). This leads to the ironicconclusion that Inertial flows occur at only small shear rates. Italso implies that increasing the shear rate can compel theformation of force chains at lower concentrations than one mightnormally expect. This can be understood from an alternateinterpretation of k / (ρd3γ2), in particular that k / (ρd3γ2)∼1

ormal stress τyyd/k, as a function of the stiffness parameter k / (ρd3γ2) for a fixedal stress because it affects the strength of force chains. In fact, for μ=0.1, the flow


(γTbc)2 where Tbc is the binary contact time, the duration of a

contact between two freely colliding particles. (Physically, thislast can be thought of as the ratio of 1 /γ, the time scale relevantto how quickly particles are drawn together by the shear flow, toTbc, a scale characteristic of how quickly the elastic contactforces push the particles apart). In other words, at small k /(ρd3γ2), the shear rate is driving particles together at ratescomparable to those at which the elastic forces are driving themapart, so that at large shear rate, force chains may form at smallerconcentrations than in relatively quiescent systems.

Fig. 15. Normal stress data for a variety of solid concentrations, ν. (a) Elastic scaling, τa slope of −1. (b) Inertial Scaling, τyy/ρd2γ2: Here Inertial flows plot as horizontal lineμ=0.5 (after [28]).

The stresses for Elastic Flows, dominated by force chains,naturally scale as τd /k. Thus, if an Inertial Flow, for which thescaled stress τ /ρd2γ2 is independent of k/(ρd3γ2), is plotted onan elastically scaled log–log plot, τyyd/k vs. k/(ρd3γ2), theinertial flows should plot with a slope of −1. At the same time ifan Quasistatic–Elastic flow, τd/k=const., were plotted on aninertially scaled plot τyy/ρd

2γ2 vs. k/(ρd3γ2), it would have aslope of 1. Fig. 15 shows plots of the normal stress τyy, forvarious solids concentrations with both scalings. It is easy todistinguish the Elastic and Inertial regimes in each figure.

yyd /k: here Elastic–Quasistatic flows plot as flat lines and Inertial flows plot withs and Elastic–Quasistatic flows have a slope of 1. In all cases, the surface friction


But notice in Fig. 15 that the change from elastic to inertialbehavior occurs quite abruptly with as little as 1% change inconcentration. At large k/(ρd3γ2), the flow is elastic at ν=0.59,but inertial at ν=0.58. And that change is accompanied by hugechanges in stress. At k/(ρd3γ2)=107, the stress changes by morethan two orders of magnitude from Elastic flow at ν=0.59, tothe Inertial flows at ν=0.58. This reflects the fact that the elasticforce chains can support much larger forces than can particleinertia. Notice also that for each concentration shown, the flowbehaves elastically for small k/(ρd3γ2) even those that behaveinertially at large k/(ρd3γ2), demonstrating once again that highshear rates can force a transition from Inertial to Elastic–Inertial behavior.

For the constant stiffness models used here, the binarycontact time Tbc is a fixed constant. In other words, for anycollision between two particles, the contact time must be exactlyTbc. The only way that a contact can endure for a longer time isif, during the contact period, one or more additional particlesmake contact with one of the colliding particles and preventsthem from moving apart. On the other hand, if particles arelocked in force chains, each particle is interacting simulta-neously with, at least, its near neighbors in the chain and thecontact will endure for the lifetime of the chain: as force chainsare created, rotated and destroyed by the shear, the inverse shearrate 1 /γ, is a characteristic time of the chain lifetime. Thus,another property that can be used to probe the flow is theaverage contact time, tc. If the ratio, tc/Tbc=1, the particles areinteracting collisionally, i.e., it is a rapid flow. On the otherhand, if tc/Tbc∼1/γ(∼[k / (ρd3γ2)]1/2), the flow is dominated byforce chains, i.e. it is an Elastic flow. Fig. 16 shows tc /Tbc forthe same data plotted in Fig. 15. It is easy to see the Elasticbehavior as the points line up with slope 1/2. The pointscorresponding to inertial behavior deviate from the sloppinglines. But note that for all the data shown, tc/TbcN1. Thus eventhe Inertial flows are not Collisional – they are not RapidFlows. Instead, they are dominated by conditions in which

Fig. 16. The ratio of the average contact time to the binary contact time, tc /Tbc as a fuin Fig. 15 (after [28]).

many particles are interacting simultaneously. This means thatthe Inertial regime is divided into two sub-regimes, Inertial–non-Collisional and Inertial–Collisional (the Rapid-flowregime).

Thus granular flows can be divided into two global regimes,Elastic and Inertial. The Elastic regime is dominated by forcechains and is divided into the Elastic–Quasistatic regime andthe Elastic–Inertial regime depending on whether there is anoticeable dependence of the stresses on the shear rate. BothElastic subregimes have the same physical underpinnings and itis difficult to draw a precise division between the two. TheInertial regime, which is free of force chains and had hasstresses that scale with the square of the shear rate, can bedivided into the Inertial–non-Collisional regime and theInertial–Collisional (or Rapid-Flow) regime depending onwhether the dominant particle interaction is binary collision.Campbell [70] examines the confinement of a contact byinfinitely chains of surrounding particles (a non-collisionalcase) and found that as long as the chain is unloaded, theconfinement only the extended the contact time by a fewpercent. Thus, one can conclude that tc/Tbc must be almostexactly one for the flow to be truly collisional. With thisinformation one can draw the entire flowmap encompassing allthe regimes of granular flow, such as that shown in Fig. 17.

There are several interesting features of this figure. Note thatit is generally assumed that, at small shear rates, a flow willbehave quasistatically, and that by increasing the shear rate, onewill eventually end up in the Rapid-Flow regime. But his flowmap shows that picture to be false. If one starts in the Elastic–Quasistatic regime and increases the shear rate (decreasing k /(ρd3γ2)) at a fixed ν, one eventually reaches Elastic–Inertialbehavior. The Rapid-Flow, or that matter any of the Inertialflow regimes, will never be reached except by reducing theconcentration. This makes sense because at such largeconcentrations, there will always be force chains, and increasingthe shear rate will not make them go away. Thus at constant

nction of the stiffness parameter, k / (ρd3γ2). The data is from the same set shown

Fig. 17. A flowmap showing the division into the four sub-regimes as a function of the concentration ν, and the stiffness parameter, k/(ρd3γ2). The data are for μ=0.5(after [28]).


volume, one cannot leave the Elastic regimes by changing theshear rate, one can only cause transitions between Elastic–Quasistatic and Elastic–Inertial behaviors.

Also note that Rapid-Flows (the Inertial–Collisional regime)ironically appear at low-shear rates, so that, in effect, Rapid-Flows are the least rapid of the flow regimes. Starting with aRapid Flow, and increasing the shear rate at constant ν, oneprogresses through Inertial–non-Collisional flow and finally toElastic–Inertial flow. This transition reflects the phenomenadiscussed earlier that very large shear rates can force particlestogether at a rate comparable to the time it takes the elasticforces to break them apart, making it possible to form forcechains at remarkably small concentrations if the shear rate islarge enough.

But most granular flows do not occur at fixed concentration.Common granular flows such as chutes and hoppers, as well aslandslides, all have a free surface. Thus the stress level iscontrolled by an overburden of material. But the concentrationis not fixed because the materials are free to expand toward thefree surface as needed to balance the applied stress. Theconcentration may vary slightly, even immeasurably so, but itstill may have an important effect on the rheological behavior.Thus it is possible that the behavior of stress-controlled systemsis different from those for which the concentration is controlled.There are indicators of this from the long history of soilmechanics. There it is common to perform “drained” and“undrained” test on water-saturated soil samples – whichdemonstrate remarkably different behaviors even when per-formed on the same soil samples. In a drained test the water isallowed to leak out as the material is sheared, while in undrainedtest, the water is confined within the sample. The significance isthat the water is a nearly incompressible fluid and if the pore-space is filled with water (i.e. the soil is saturated), then thewater keeps the volume of the sample from changing. In suchcases, an applied load can be supported by a combination of the

pressure on the granular contacts and the pressure in the water.In a drained test, the water is allowed to leak out, so that all theforces are supported across the particle contacts. And thereinlies a possible difference between controlled-stress andcontrolled-volume systems. In controlled-stress systems, thematerial must, through the action of elastic and/or inertialforces, support the applied load under all conditions; at smallshear rates, even a small applied stress forces particles intocontact, and form force-chains. Under controlled volume, it ispossible for the stress to fall by orders of magnitude with a smallconcentration change, as seen in the transition from Elastic toInertial behavior at large k/(ρd3γ2), because there is nothing toforce the particles into contact.

Fig. 18 shows the dependence of the solid concentration ν,on the dimensionless applied stress τ0d/k, and the stiffnessparameter k/(ρd3γ2). Remember that τ0d/k represents thedeformation of a particle as a fraction of its diameter. Thus,the large concentrations seen at large τ0d/k are simply areflection of the compressibility of the particles; i.e. at τ0d/k=0.1, the particles are pressed until their centers are 10%closer together. Note that for a range of the stiffnessparameter, the concentration is independent of k/(ρd3γ2).This is Casagrande's critical state behavior illustrated in Fig.2. Then one by one, starting with the smallest k/(ρd3γ2) (thelargest shear rates), the lines diverge from the critical stateline. The line marked “Low-Stress Critical state Concentra-tion” is particularly interesting. It shows that there is a rangeof τ0d/k and k/(ρd3γ2) for which the concentration isindependent of either parameter. This is partially becausethe stresses are too small to cause a noticeable change in theconcentration; for example one would expect an insignificantchange in concentration between τ0d/k=10

−3 (0.1% defor-mation) and τ0d/k=10

−6 (0.0001% deformation). And at thesame time, the shear rate is too small (k/(ρd3γ2) is too large)for the inertia of the particles to support the applied load.

Fig. 18. The relationship between the solid concentration ν, applied stress, τ0d/k, and k⁎=k/(ρd3γ2) in a stress-controlled simulation with μ=0.5 (from Campbell [68],

reprinted with permission).


One interesting correspondence between controlled stressand controlled volume flows is that the Low-Stress CriticalState concentration in controlled stress flows corresponds to thebounding concentration between the Elastic–Quasistatic–Elastic and Inertial regimes in controlled-volume flows. Thismakes sense, if one thinks of the transition as the minimumconcentration at which force chains can form without help fromthe particle inertia. Thus in controlled volume flows, itrepresents the transition between flows with and without forcechains, that is between Elastic and Inertial Flows. In controlled-stress flows, it represents the minimum concentration that cansupport the applied load without inertial help. But this will turnout to be one of the few correspondences between controlled-volume and stress flows. And indeed the difference makes itselfapparent even here. As it is possible to create elastic structureswith a wide range of concentrations (Fig. 17 shows elasticbehavior as low as ν=0.45), the first appearance of inertialeffects in controlled-stress flows – the first deviation from thelow-stress critical state line – does not correspond to thedisappearance of force chains, only the first sign of inertialsupport. Thus, it is an Elastic–Quasistatic/Elastic–Inertialtransition instead of the Elastic–Quasistatic/Inertia–non-Col-lisional transition seen in controlled volume tests. Hence,although the transitions occur at the same concentrations incontrolled-stress and volume, they are physically distincttransitions.

In controlled-stress flows, the magnitude of all the stressesare more or less fixed by the applied stress τ0. It is not possibleto examine, say the effect of strain-rate γ on the stresses,because the magnitude of the stress is fixed by τ0 and changingγ will only change the concentration. Thus indirect means areused to determine the flow-regime by making analogies to theobservations made in controlled-volume flows. For example,the deviation from critical state behavior in Fig. 18 is taken asone indicator of Elastic–Quasistatic to Elastic–Inertial behav-

ior. (Another indicator can be found from changes in the ratio ofshear to normal stresses.) And as in Fig. 16, contact time data isused to determine the transition to Inertial behavior. (See [68]for details.)

This allows flowmaps, like that in Fig. 19, to be drawn forcontrolled-stress flows. (Note that vertical ordinate in Fig. 19 isthe dimensionless applied stress, τ0d/k, instead of the solidfraction, ν used in Fig. 17.) The behavior is very different fromthe controlled volume case in Fig. 17. Most noticeable is thatcollisional flows are found at small k/(ρd3γ2) instead of at thelarge k/(ρd3γ2) in Fig. 17. This means that increasing the shear-rate causes the expected transition from Elastic–Quasistatic→Elastic–Inertial→ Inertial flows, nearly the opposite of what isseen at constant volume. Here one sees this behavior becausethe increased shear rate causes changes, often small changes, inthe concentration. One might also notice that there are noInertial–non-Collisional flows delineated in Fig. 19. The data[68] indicate that the transition from Elastic–Inertial toInertial–Collisional behavior occurs very rapidly in controlledstress flows. The Inertial–non-Collisional regime is verynarrow and can only be observed in a few isolated cases.

In addition, Elastic–Inertial behavior appears over a muchwider range of k/(ρd3γ2) at constant stress, than at constantvolume. This is also easy to understand. Remember thatElastic–Inertial flows appear when the inertial forces becomecomparable to the elastic forces. At constant volume, theelastic forces are generally so large that it takes very largeshear rates to generate comparable inertial forces. But undercontrolled stress, the magnitude of the elastic force at zeroshear rate, is the applied load and can be as large or small asdesired. Hence at small applied loads, it is easy for the inertialforces to become of the same order as the elastic forces andthus possible to have Elastic–Inertial behavior at relativelysmall shear rates. For the fixed concentration data in Campbell[28] at ν=0.6, Elastic–Inertial behavior was observed for k/

Fig. 19. A flowmap of controlled-stress granular flows. The particle coefficient of restitution, ε=0.7 and particle surface friction, μ=0.5 (from Campbell [68], reprintedwith permission).


(ρd3γ2)b104 which, for 1 mm sand (which Campbell [68]estimates to correspond to about k=1.5×104 N m−1), requiredabout 775 s−1 of shear – an unrealistically large shear rateindicating that Elastic–Inertial behavior will never be observedfor sand under controlled-volume conditions. (However, 2-mmplastic beads should assume elastic inertial behavior at a moreaccessible, γ=10 s−1.) But that is not so for the stress-controlled studies shown here. For example, Fig. 19 indicatesthat for μ=0.5 and τ0d/k=10

−5 (ten particles deep), Elastic–Inertial behavior can first be observed k/(ρd3γ2)∼5×107 orabout γ=14 s−1 in 1 mm sand-like materials. The reason forthis is obvious from the underlying physics. Elastic–Inertialbehavior occurs when the inertial forces are of the same orderas the elastic forces. In fixed-concentration flows [28], themagnitude of the elastic forces were dictated by therequirement that the particles undergo shear at the largeconcentrations where force chains form. Reducing theconcentration to the point that the force chains disappearedeliminates the elastic forces and the stresses drop by orders ofmagnitude; this is possible because the concentration wasindependent of the stresses generated. In constant-stress flows,as long as the shear rate is small enough, force chains canalways form even for very small applied stresses, simplybecause if the inertia effects are small, force chains are the onlymethod available to the balance the applied load. And for smallapplied stresses, the corresponding elastic forces are small andit only takes a small shear rate to generate inertial forces thatare comparable to the elastic forces, pushing the flow into theElastic–Inertial regime. As a result, Elastic–Inertial behavioris much more accessible in controlled stress flows. Further-more, under controlled stress, when progressing from Elastic–Quasistatic to Inertial behavior, the flow must always gothrough an intermediate Elastic–Inertial regime.

One advantage of having complete flowmaps, is that onecan set bounds on that various flows regimes. For example,

the maps indicate why rapid granular flows are seldomobserved. Fig. 19 indicates that, for τ0d/k=10

−5 (whichcorresponds to an overburden of a 10-particle-deep layer), theflow becomes collisional at k/(ρd3γ2)=106. Continuing withthe 1 mm sand numbers, this corresponds to a very large shearrate of about 100 s−1. This is similar to the minimum shearrates observed for larger glassbeads in 10 particle deep shearcell experiments [71]. Now, Campbell [68] points out that thetransition shear rate should scale with particle diameter asγd=0.1 m s−1. While there are no direct experiments tocompare these numbers to, Wang and Campbell [71] foundfrom measurements on three sizes of glassbeads, that theirminimum achievable shear rates were somewhat higher atγd=0.34 m s−1 (requiring a minimum of about 340 s−1 ofshear for 1 mm particles). Deeper depths would require evenlarger shear rates and while such shear rates can be obtained inlaboratory shear cells and computer simulations, they are notfound in practice. One should note that these limiting shearrates are relatively insensitive to the particle stiffness. Going toa softer material would decrease k and with it k/(ρd3γ2),pushing the system towards an inertial flow, but at the sametime would proportionally increase τ0d/k, pushing the systemaway from inertial behavior, so that, from Fig. 19, a change ink is unlikely to cause a change in flow regime. While highspeed laboratory shear cells can achieve the hundreds ofinverse seconds of shear rate required to achieve Rapid Flow,shear rates in the hundreds of inverse-seconds, will not befound in any common granular flows, such as chutes andhoppers, all of which are gravity driven. One might argue thatunder reduced gravity conditions, say on the Moon or Mars,smaller shear rates are required. While this is certainly true,reducing gravity also reduces the driving force for the bulkmotion and reduces the available shear rates proportionally.Hence, reducing gravity does not make rapid flows any moreaccessible.


7. Conclusions

This paper is an attempt to summarize what is currentlyknown about granular material flows. It began with adiscussion of basic transport mechanisms. It then goes on todiscuss contact laws and points out that real contact behavioris much more complex than Hertz and Hertz–Mindlin models.This was followed by a discussion of the two limiting granularflow models, the Slow-Flow or Quasistatic model, and theRapid-Flow or Collisional model. It finally describes theElastic model, which, by including particle elasticity, allowsthe entire flowmap for granular flow to be drawn, includingthe Quasistatic and Rapid-Flow regions and the area inbetween.

The Quasistatic models were derived from metal plasticitytheory using a Mohr–Coulomb frictional yield condition. Asapplied to granular flows, these generally assume that thefriction angle ϕ (tanϕ is the apparent friction coefficient ofthe material) is a constant material property, and that thematerial is in the critical-state and thus behaves incompres-sibly with the density fixed at the critical-state concentration.While there are mathematical problems in using these models,particularly in the application of boundary conditions, theyhave been used to model hoppers and are able to predict thequalitative features of hopper flow and the predicted flowrates are in the same ballpark as experiments. The deviationsmay be attributed to the aforementioned problems inboundary conditions, etc., but simulations of hopper flowsuggest that the error lies in the assumption of a constantfriction angle ϕ. It is not yet understood why ϕ changeswithin the hopper.

The Rapid-Flow models that have been so popular in thelast few decades, are also derived using models from anotherbranch of science, in this case the kinetic theory of gases.They assume that the granules behave like molecules and, toapply kinetic theory ideas, assume that the particles interact byinstantaneous collision. Superimposed on the velocity field isanother field quantity, the granular temperature representingthe random, thermal-like, kinetic energy of the particles.Because the particle collisions dissipate energy, the granulartemperature must be continually re-supplied from the energyof the mean flow field through the mechanism of shear work.The granular temperature is assumed to control all thetransport rates within the granular material; in particular, thepressure and all the internal normal stresses are proportional tothe granular temperature. However, there is a problem withthis assumption; as the granular temperature is producedthrough shear work, it has roughly the same magnitude as thesquare of γd, the shear rate times the particle diameter. Thus,the transport rates are equally controlled by the shear rate asby the granular temperature. As such, Rapid-Flow theory isnot self-consistent with its basic assumptions. Also, just askinetic theory derives the equations of fluid motion, rapid flowtheory predicts a viscous-like response from a granularmaterial, meaning that rapid-flows cannot demonstrate anyof the solid-like properties (such as Janssen's frictionalsupport) commonly associated with bulk granular material.

But the largest problem with the rapid flow theory is that hugegranular temperatures are required to support any reasonableoverburden of material. And as the granular temperaturederives from the shear rate, huge shear rates are required toproduce enough pressure to support even small overburdens ofmaterial. Estimates of the required granular temperaturederived from the Elastic models, and by the limitedexperimental evidence that is available, indicate that rapidflows are unobtainable outside of high-speed laboratory shearcells and gravity-free computer simulations.

Surprisingly, despite its 27-year history, as of this writing,there have been no other attempts to validate the fundamentalassumptions of rapid granular flow theory – a long silence that,in itself, speaks volumes. Monte Carlo simulations provide andthus assume rapid flow behavior; they cannot be used to probeits limits. Neither can rigid particle (or event-driven) computersimulations, such as those used to derive Figs. 10–12, as theylikewise, presuppose a collisional flow. As it stands, Rapid Flowis a theory in search of an application as steadily, more and moresystems are shown not to be rapid flows. In addition to the factthat common granular flows cannot generate large enough shearrates to enter the rapid flow regime, there have also been directmeasurements of the particle forces in bubbling fluidized beds,indicate that they cannot be modeled by kinetic theory ideas[61–63]. Even vibrated beds, at least those of commercialdepth, several hundred to thousands of particle deep, cannot bethermalized and are not rapid flows [66]. After 27 years, it is notoutrageous to ask for some verification of the Rapid-Flowtheories. Thus, it probably best to abandon rapid-flow studies, atleast until a practical application for the results of the theory canbe found.

By including the particle stiffness – which governs howparticles see each other mechanically and thus governs theelastic properties of the bulk material – it is possible to drawout the entire flowmap that connects the Quasistatic andRapid-Flow regimes, at least for simple granular materials.Flows are divided into two global regimes, the Elastic and theInertial. Elastic flows are dominated by force chains and thestresses scale directly with the interparticle stiffness as τd/k.which is interpreted as the elastic contact deformation as afraction of the particle diameter. This regime is furthersubdivided into the Elastic–Quasistatic (the old Quasistaticregime) and the Elastic–Inertial regimes, depending onwhether the shear rate significantly affects the magnitude ofthe stresses. The relative magnitude of inertial and elasticeffects are governed by a parameter, k⁎=k/(ρd3γ2), interpretedas the inverse-square elastic deformation on a contact duesolely to the particle inertia, again as a fraction of the particlediameter. Inertial Flows are free of force chains and internallytransmit force by interparticle inertia and the stresses scalewith the Bagnold scaling, τ/ρd2γ2. They can also be dividedinto two flow regimes, the Inertial–non-Collisional for whichmany particles interact simultaneously, and the Inertial–Collisional regime (the old Rapid-Flow regime), whereparticles interact by binary collisions.

Somewhat surprisingly the rheological behavior appears tochange with system-scale constraints, in particular whether the


concentration is determined by controlling the volume of thesystem or by controlling the applied stress. This has a directanalog in drained and undrained tests on water-saturated soils. Itis generally assumed that at small shear rates, the behavior isquasistatic and, by increasing the shear rate, will eventuallybecome rapid. However, that behavior was only observed undercontrolled-stress conditions. For controlled-volume flows, thereis no path between quasistatic and rapid flows, except bydecreasing the density. By simply varying the shear rate, onecan only move between Elastic–Quasistatic and Elastic–Inertial flows or between Elastic–Inertial and Inertial flowsdepending on the specified concentration. Surprisingly, RapidFlows (Inertial–Collisional flows) appear only at small shearrates under controlled volume conditions; increasing the shearrate forces particles into force chains and causes a transition toElastic–Inertial behavior. In controlled-stress flows, the volumecan change to balance the applied stress. Thus starting with aquasistatic flow and increasing the shear rate, one can end upwith a rapid flow, because the transition brings about a slightincrease in volume.

At this point, one can only speculate whether the elasticeffects can explain the variation in tanϕ seen in the hopper flowplots shown Fig. 5. Campbell [53] shows that the apparentfriction coefficient τxy/τyy (the equivalent of tanϕ in simpleshear), decreases with k/(ρd3γ2), throughout the Elastic–Inertial regime and becomes constant in the Elastic–Quasistaticregime. An examination of the data of Babic et al. [69],indicates that effect is even more severe for two-dimensionaldisc flows such as those used in the hopper simulations in Fig. 5.Thus, if the hopper flow were in the Elastic–Inertial regime, thisindicates that tanϕ should increase with shear rate. This isconsistent with Fig. 5 as the largest tanϕ are found in the hopperthroat where the shear rates are the largest, and the smallest tanϕare in the bin section where the shear rate is small. But thesensitivity to system-scale constraints may provide anotherpossible explanation. A hopper flow is far from a simple shearflow and it may not be possible to directly apply simple shearresults to hoppers.

It is interesting that the elastic properties of the particles donot appear in either the quasistatic or rapid flow theories. Butremember that both quasistatic and rapid-flow modeling werebased on the formalisms derived from other branches of science,of metal plasticity and kinetic theory respectively, and neitherformalism allows the introduction of elastic properties. Forquasistatic theory, all that is required is a yield surface and aflow rule. In rapid flow theory, particle interactions are assumedto be instantaneous collisions and thus assume a infinitely rigidsolid with no allowance for finite elasticity. Thus, it would bemost difficult, if not impossible to adapt these theories toinclude elastic effects.

8. Some lessons

As noted in the Introduction, this paper was derived from alecture given to the summer Powder Technology class at OhioState. Hence, it is appropriate to highlight some importantlessons learned in the course of the paper.

8.1. Rigid sphere models are inapplicable to dense systems

Dense systems interact by force chains and transmit forcealong the chain by elastically deforming the interparticlecontacts. The elasticity of the particles is intrinsically importantto the internal processes of the system. Such systems cannot bemodelled as rigid spheres and any such model would be missingessential physics. Also, contrary to intuition, granular materialsare, in the bulk, quite soft. This is evidenced in their soundspeed which is of the order of 100 m s−1, roughly 50 timesslower than the soundspeed in their constituent solid material,and indicates that the bulk granular material has an apparentelastic modulus, more than three orders of magnitude smallerthan its constituent solid.

8.2. Particle surface friction is essential to modelling densesystems

Fig. 14 shows that the simple act of removing surfacefriction can cause a transition between an Elastic and Inertialflow regime. Friction is vital to the strength of force chainsand force chains are essential for the Elastic flow regimes.Thus, while it is convenient to remove surface friction inmany analyses or computer simulations, it is removingessential physics from the problem and will lead to erroneousbehavior. At the same time, the importance of surface frictionmay be dwarfed by the importance of particle shape. Angularparticles have a greater tendency to lock together and willthus both more easily form force chains and form strongerforce chains. The effect of particle shape has yet to beexplored in detail. However, results on frictional spheres havesome value. One can at least buy frictional spherical particles;frictionless particles, spherical or otherwise do not appear innature.

8.3. Dense flows are not “frictional”

This was dealt with in detail in the text, but is worthrepeating here. While it is true that slow granular flows (and forthat matter rapid flows) appear frictional because in the bulk, theratio of shear to normal stress is a constant, it is not a result ofsolid–solid friction. Instead the frictional behavior of densegranular materials is a reflection of the evolution of the forcechain structure.

8.4. All important granular flows are dense

This is not so much an observation of this paper, as much asan observation of real life. Hoppers, reasonably deep chuteflows and even commercial scale vibrated boxes, all operate in anear packed state.

8.5. Rapid Flows are seldom, if ever encountered outside thelaboratory

This has also been dealt with in great detail above, butdeserved repeating here if only because the scarcity of rapid


granular flows is the physical reason behind the last lesson,that all important flows are dense. Simply put, the shear ratesobtained in common flows cannot generate enough granulartemperature to support the material at small concentrations,leaving all flows in or near the Elastic regimes.

8.6. System-scale constraints can affect the rheologicalbehavior

This is reflected in the different rheological behaviorsobserved for controlled-volume and controlled stress flows aswell in the difference between drained and undrained tests insoil mechanics. There may be many other system constraints ofimportance that are yet to be found.

Acknowledgements

The author would like to thank Professor L. S. Fan forinviting me to give the lecture that inspired this article and forhis hospitality while I was at Ohio State.

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Granular Flow Material Overview Campbell 2003