Granular Flows

J. Goddard

University of California San Diego

1 Introduction

Granular materials represent a majorobject of human activities: as measuredin tons, the first material manipulatedon earth is water; the second is granu-lar matter. . . This may show up in verydifferent forms: rice, corn, powders forconstruction. . . In our supposedly mod-ern age, we are extraordinarily clumsywith granular systems. . .

P. G. de Gennes, “From Rice to Snow”,Nishina Memorial Lectures, 2008

The epigrammatic quote by a French Nobel lau-reate is familiar to many in the field of granu-lar mechanics and reflects a long-standing scien-tific fascination and practical interest. This is ac-knowledged by many others and summarized inthe articles listed under Further Reading.

As a general definition, we understand by gran-ular medium a particle assembly dominated bypairwise nearest-neighbor interactions and usu-ally limited to particles larger than 1 microme-ter in diameter, for which the direct mechanicaleffects of Van der Waals and ordinary thermal(“Brownian”) forces are negligible. This includesa large class of materials, such as cereal grains,pharmaceutical tablets and capsules, geomateri-als such as sand, and the masses of rock and icein planetary rings.

Most of the present article is concerned withdry granular materials in which there are negli-gible effects of air or other gases in the intersti-tial space, or else with granular materials com-pletely saturated by an interstitial liquid. In thiscase, liquid surface-tension at “capillary necks”between grains, as in the wet sand of sand castles,or related forms of cohesion are largely negligible.

There are several important and interrelatedaspects of granular mechanics: (i) experiment andindustrial or geotechnical applications, (ii) ana-lytical and computational micromechanics (grain-level dynamics), (iii) homogenization (“upscal-ing” or “coarse graining”) to obtain smoothed

continuum models from the Newtonian mechan-ics of discrete grains, (iv) mathematical classifi-cation and solution of the continuum field equa-tions for granular flow, and (v) development ofcontinuum models (“constitutive equations”) forstress-deformation behavior. The present articlefocuses on (v), since the study of most of the pre-ceding items either depends on it, or is stronglymotivated by it. The following discussion of (iv)sets the stage for the subsequent coverage of (v).

2 Field Equations and ConstitutiveEquations

By field equations, we mean the partial differen-tial equations (PDEs), with time t and spatial po-sition x = [xi] as independent variables, that rep-resent the continuum-level mass and linear mo-mentum balances governing the density ρ(x, t),velocity v(x, t) = [vi], and symmetric stress ten-sor T(x, t) = [τij ], for any material,

ρ = −ρ∇ · v and ρv = ∇ ·T. (1)

The indices refer to components in Cartesian co-ordinates, and the notation “= [ ]” indicates com-ponents of vectors and tensors in those coordi-nates. Written out, (1) becomes ∂tρ+∂j(ρvj) = 0and ∂t(ρvi) = ∂j(τij−ρvivj) for i = 1, 2, 3, wherew = ∂tw + vj∂jw for any quantity w, sums fromj = 1 to 3 are taken over terms with repeatedindices, ∂iw = ∂w/∂xi, ∂tw = ∂w/∂t, and theproduct rule has been used. For more informa-tion on (1), see continuum mechanics [4.x].

There are ten dependent variables, ρ, vi, andτij = τji, but only four equations in (1). In or-der to “close” them (to make them soluble), weneed six more equations. For example, the clo-sure representing the constitutive equations foran incompressible Newtonian fluid (see Navier–Stokes equations [3.x]) is

T′ = 2ηD′ and ρ = constant, (2)

where η is a coefficient of shear viscosity and Ddenotes the rate of deformation, also a symmetricsecond-rank tensor, D = sym∇v = 1

2 [∂ivj +∂jvi],where “sym” denotes the symmetric part of asecond-rank tensor and the prime denotes thedeviator (“traceless” or “shearing” part), X′ =

1

2

X− 13 (tr X)I, X ′ij = Xij− 1

3Xkkδij , where δij arethe components of the identity I.

For rheologically more complicated (“com-plex”) fluids, such as viscoelastic liquids, thestress at a material point may depend on the en-tire past history of the velocity gradient ∇v atthat point. Viscoelasticity is exemplified by thesimplest form of the Maxwell fluid giving the rateof change of T′ as a linear function of D′ and T′,

◦T′ = 2µD′ − λT′ = λ(2ηD′ −T′), (3)

where◦X = X + sym(XW), XW = [XikWkj ],

W is the skew part of the velocity gradient,W = 1

2 [∂ivj − ∂jvi], λ is the relaxation rate orinverse of the relaxation time, and µ is the elasticmodulus. In this simple model, µ can be identi-fied with the elastic shear modulus G to be dis-cussed below.

According to (3), which was proposed in aslightly different form by the famous physicistJames Clerk Maxwell in 1867, the coefficient ofviscosity and the elastic modulus are connectedby η = µ/λ, and a material responds elasticallyon a time scale λ−1 with viscosity and viscousdissipation arising from relaxing elastic stress.

One obtains from (3) elastic-solid or viscous-fluid behavior in the respective limits λ → 0 orλ → ∞. This is illustrated by the quintessentialviscoelastic material “silly putty” which bounceselastically but undergoes viscous flow in slow de-formations. Rheologists employ a Deborah num-ber of the form γ/λ to distinguish between rapid,solid-like and slow fluid-like deformations.

The superposed “◦” in (3) denotes a Jaumannderivative, which gives the time rate of change ina frame translating with a material point and ro-tating with the local material spin. The Jaumannderivative is the simplest objective time derivativethat embodies the principle of material frame in-difference. This principle, which is not a funda-mental law of mechanics, is tantamount to theassumption that arbitrary rigid-body rotationssuperimposed on a given motion of a materialmerely rotate stresses in the same way. Statedmore generally, accelerations of a body relativeto an inertial (“Newtonian”) frame do not affectthe stress-deformation behavior. The principle isalready reflected in the simpler constitutive equa-

tion (2), where only the symmetric part of ∇v isallowed.

While frame-indifference is a safe assumptionfor most molecular materials, which are domi-nated at the molecular level by random thermalmotion, it could conceivably break down for gran-ular flows, where random granular motions arecomparable to those associated with macroscopicshearing. Nevertheless, the assumption will beadopted in the constitutive models consideredhere.

Note that, for a fixed material particle x◦, (3)represents a set of five simultaneous ordinary dif-ferential equations, which we call Lagrangian or-dinary differential equations (LODEs). In theEulerian or spatial description, they must becombined with (1) to provide nine equations inthe nine dependent variables, p = − 1

3 tr T, v andT′, governing the velocity field of the Maxwellfluid. The solution, subject to various initial andboundary conditions, is no easy task and usuallyrequires advanced numerical methods.

We identify below a broad class of plausibleconstitutive equations, which basically amountto generalizations of (3), and include the so-called hypoplastic models for granular plasticity.It turns out that all these can be obtained fromelastic and inelastic potentials, which are famil-iar in classical theories of elastoplasticity. Theabove models can be enlarged to include modelsof visco-elastoplasticity that are broadly applica-ble to all the prominent regimes of granular flow.Following the brief review in the following sectionof phenomenology and flow regimes, a systematicoutline of the genesis of such models will be pre-sented.

The present article does not cover a large bodyof literature dealing with numerical methods, nei-ther the direct simulation of micromechanics bythe distinct element method (DEM), nor solutionof the continuum field equations based on finiteelement methods (FEM) or related techniques.

3 Phenomenological Aspects

We consider here the important physical param-eters and dimensionless groups that character-ize the various regimes of granular mechanicsand flows. The review article of Forterre and

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Pouliquen provides more quantitative compar-isons for fairly simple shearing flows.

Key parameters and dimensionless groups

Apart from various dimensionless parameters de-scribing grain shape, the most prominent physicalparameters for non-cohesive granular media are:grain elastic (shear) modulus Gs, intrinsic graindensity ρs, representative grain diameter d, in-tergranular (Coulomb) contact-friction coefficientµs or macroscopic counterpart µC , and confiningpressure ps. These parameters define the key di-mensionless groups that serve to delineate vari-ous regimes of granular flow, namely an elasticitynumber , an inertia number , and a viscosity num-ber , given respectively by

E = Gs/ps, I = γd√ρs/ps, H = ηsγ/ps,

and involving γ, a representative value of theshear rate |D′|. In addition, we will have occasionto refer briefly below to a Knudsen number basedon the ratio of microscopic to macroscopic lengthscales.

Note that I is the analog of the Deborah num-ber mentioned above, but now involving a re-laxation rate that represents the competition be-tween grain inertia and Coulomb friction µCps.The quantity I2 represents the ratio of represen-tative granular kinetic energy ρsd

2γ2 to frictionalconfinement.

The transition from a fluid-saturated granularmedium to a dense fluid-particle suspension oc-curs when the viscosity number H ≈ 1, whereviscous and frictional contact forces are compa-rable.

In the case of fluid-particle suspensions, it iscustomary to identify I2/H as the “Stokes num-ber” or the “Bagnold number” (after a pioneerin granular flow) representing the magnitude ofgrain-inertial to viscous forces.

In the following sections, we shall focus atten-tion on dry granular media, including later onlythe briefest mention of fluid-particle systems.

Granular flow regimes

Although granular materials are devoid of intrin-sic thermal motion at the grain level, they never-theless exhibit states that resemble the solid, liq-

X

a

b

I II III

τ/ps

μC

Figure 1: Schematic diagram of granular-flow regimes

uid and gaseous states of molecular systems. Allthese granular states may co-exist in the sameflow field, as the analogs of “multiphase flow.”There are several open questions as to the propermatching of solid-like immobile states with therapidly sheared states that cannot be addressedin this brief article.

With τ denoting a representative shear stressand γ a representative shear strain relative to arest state (in which τ/ps = 0 and I = 0), thevarious flow regimes are shown in the qualitative,highly simplified sketch in Fig. 1, and in Table 1.In Fig. 1, the dimensionless ratio τ/ps is repre-sented as a function of a single dimensionless vari-able representing an interpolating form,

X ∼ µCEγ/(Eγ + µC) + I2. (4)

(A compound representation, closer to the con-stitutive models considered below and illustratedin Fig. 5, is τ/ps = µC + I

2 with γE = (τ/ps)E,where the first of these represents a Coulomb–Bagnold interpolation found in certain constitu-tive models and γE is elastic deformation at anystress state.)

Figure 1 fails to capture the strong nonlinearityand history-dependence of granular plasticity inregime Ib, a matter addressed by the constitutivemodels to be discussed below.

Like the liquid states of molecular systems,dense-rapid flow, represented as a general dimen-sionless function f(I) in Table 1, may be the mostpoorly understood regime of granular mechanics.It involves important phenomena such as the fas-cinating granular size-segregation. There is some

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Table 1: Granular-flow regimes in Fig. 1. The last

column gives the scaling of the stress τ

I. Quasi-static: (Hertz–Coulomb)elastoplastic “solid”

Ia. (Hertz) elastic GsγIb. (Coulomb) elastoplastic µC

II. Dense-rapid: psf(I)viscoplastic “liquid”

III. Rarified-rapid: (Bagnold) ρsd2γ2

viscous “granular gas”

evidence that this regime may involve an addi-tional dependence on E for soft granular materi-als.

The elastic regime. The geometry of contactbetween quasi-spherical linear (Hookean) elasticparticles should lead to nonlinear elasticity of agranular mass at low confining pressures ps and toan interesting scaling of elastic moduli and elasticwave speeds with pressure.

Based on Hertzian contact mechanics, a rough“mean-field” estimate of the continuum shearmodulus G in terms of grain shear modulus Gs

is given by G/Gs ∼ E−1/3, which indicates a 13 -

power dependence on pressure. For example, withps ∼ 100 kPa andGs ∼ 100 GPa (a rather stiff ge-omaterial), one finds E ∼ 106 and G ∼ 10−2Gs,amounting to a huge reduction of global stiffnessdue to relatively soft Hertzian contact.

In principle, we should replace Gs by G in Ta-ble 1 and (4), and E = Gs/ps by G/ps = E

−2/3.In so doing, we obtain an estimate of the limitingelastic strain for the onset of Coulomb slip (withµC ∼ 1) to be γE ∼ E

−2/3 ∼ 10−4, given theabove numerical value. Although crude, this pro-vides a reasonable estimate of the small elasticrange of stiff geomaterials such as sand.

The elastoplastic regime. Given its vener-able history, dating back to the classical worksof Coulomb, Rankine, and others in the 18thand 19th centuries, and its enduring relevanceto geomechanics, the field of elastoplasticity isthe most thoroughly studied area of granular me-

(a) Shear bands in slow hopper flow

(b) Shear bands in axial compression

Figure 2: DEM simulations (Courtesy of W. Ehlers)

chanics. Here, we touch on a few salient phenom-ena and the theoretical issues surrounding them,as background for the discussion of constitutivemodeling to follow.

Figure 2 shows the results from the 2D DEMsimulations of Prof. W. Ehlers (University ofStuttgart) of a quasi-static hopper discharge anda biaxial compression test, both of which illus-trate a well-known localization of deformationinto shear bands. A hallmark of granular plas-ticity, this localized slip (or “failure”) may be im-plicated in dynamic “arching,” with large tran-sient stresses on bounding surfaces such as hopperwalls or structural retaining walls. Similar phe-nomena are implicated in large-scale landslides.

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(a) Before compression

(b) After compression

Figure 3: Experimental axial compression of dry Hos-

tun sand specimen (Courtesy of W. Ehlers)

Figure 3 shows the development of shear bandsin a standard experimental quasi-static compres-sion test on a sand column surrounded by a thinelastic membrane. The literature abounds withmany interesting experimental observations andnumerical simulations; see, e.g., Tejchman’s book.

The occurrence of shear bands can be viewedmathematically as material bifurcation and insta-bility arising from loss of convexity in the under-lying constitutive equations, accompanied by achange of type in the PDEs involved in the fieldequations. We recall that similar changes of type,e.g., from elliptic to hyperbolic PDEs, are asso-ciated with phenomena such as the gas dynamictransition from subsonic to supersonic flows withformation of thin “shocks.”

σ

εV

ε0

Figure 4: Schematic diagram of triaxial stress/dila-

tation-strain curves for initially dense (solid curves)

and loose (dashed curves) sands

To help understand the elastoplastic instability,Fig. 4 presents a qualitative sketch of the typi-cal stress-strain/dilatancy behavior in the axialcompression of dense and loose sands, with σ de-noting compressive stress, ε compressive strain,and εV = log(V/V0) volumetric strain (where avolume V0 has been deformed into V ). While nonumerical scale is shown on the axes, the peakstress and the change of εV from negative (con-traction) to positive (dilation) typically occur atstrains of the order of few percent (ε ∼ 0.01–0.05)for dense sands.

Although tempting to regard the initial growthof σ with ε as elastic in nature, the elastic regimeis represented by much smaller strains (corre-sponding to the estimates of order 10−4 givenabove), corresponding to a nearly vertical un-loading from any point on the σ-ε curve. It istherefore much more plausible that the initialstress growth represents an almost completely dis-sipative plastic “hardening” associated with com-paction (εV < 0) accompanied by growth of con-tact number density nc and contact anisotropy,whereas the maximum in stress can be attributedto the subsequent decrease in nc accompanyingdilation (εV > 0).

As will be discussed below, one can rational-ize the formation of shear bands as the result of“strain softening” or unstable “wrong-way” be-

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psμC

TμE

μPηB

μE~

Figure 5: Slide-block/spring/dashpot analog of visco-

elastoplasticity

havior (decrease of incremental force with incre-mental displacement) following the peak stress.According to certain analyses, this can arise asa purely dissipative process associated with thedecrease of plastic stress, whereas according toothers it may result from a peculiar quasi-elasticresponse.

Whatever the precise nature of the materialinstability, it generally calls for multipolar or“higher-gradient” constitutive models involvingan intrinsic material length scale. Hence, to ourlist of important dimensionless parameters wemust now add a Knudsen number K = `/L asdiscussed further below. Suffice it to say here thata material length scale ` is necessary to regularizefield equations in order to avoid sharp discontinu-ities in strain rate by assigning finite thickness tozones of strain localization. Such a scale, clearlyevident in the DEM simulations and experimentsof Figs. 2 and 3, cannot be predicted with thenon-polar models that constitute the main sub-ject of this article.

Springs and slideblocks. Setting asidelength-scale effects, the simple mechanical modelin Fig. 5 provides a useful intuitive view of thecontinuum model considered below. There, theapplied force T′ is the analog of continuum-mechanical shear stress, and the rate of extensionof the device represents deviatoric deformationrate D′.

The serrated slide-block at the top of Fig. 5is a modification of the standard flat plasticslide-block, where sliding stress τ represents apressure independent yield stress, or a frictionalslide-block with τ/ps = µC representing the(Amontons–Coulomb) coefficient of sliding fric-tion. The serrated version represents the granu-lar “interlocking” model of Taylor (1948) or the“sawtooth” model of Rowe (1962) based on theconcept of granular dilatancy introduced in 1885by Osborne Reynolds (who is also renowned forhis work in various fields of fluid mechanics). Ac-cording to Reynolds the shearing resistance of agranular medium is partly due to volumetric ex-pansion against the confining pressure. Thus, ifthe sawtooth angle vanishes, then one obtains astandard model of plasticity, with resistance duesolely to yield stress or pressure-dependent fric-tion.

At the point of sliding instability in the model,where the maximum volume expansion occurs,part of the stored volumetric energy must be dis-sipated by subsequent collapse and collisional im-pact. This may be assumed to occur on extremelyshort time scales giving rise to an apparent slidingfriction coefficient µC even if there is no slidingfriction between grains (µs = 0). This rapid en-ergy dissipation is emblematic of tribological andplastic-flow processes, where, owing to topologi-cal roughness and instability in the small , storedenergy is thermalized on negligibly small timescales giving rise to rate-independent forces in thelarge.

As a second special feature of the model, thespring with constant µE converts plastic defor-mation into “frozen elastic energy” and also pro-vides an elementary model of plastic work hard-ening. It is intended to illustrate the fact thatsome of the stored elastic energy can never be en-tirely recovered solely by the mechanical action ofT′, which leads to history effects and associatedcomplications in the thermodynamic theories ofplasticity.

Viscoplasticity. For rapid granular flow, theviscous dashpot with (Bagnold) viscosity ηB (seeFig. 5) adds a rate-dependent force associatedwith granular kinetic energy and collisional dis-sipation. This leads to a form of Bingham plas-

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ticity discussed below and described by certainrheological models, providing a rough interpola-tion between regimes I and III in Fig. 1. Theviscous dashpot can also represent the effects ofinterstitial fluids. Note that removal of the plasticslide block gives the Maxwell model of viscoelas-ticity (3) whenever the spring and dashpot arelinear.

Apart from presenting constitutive models thatencompass those currently employed, we do notdeal directly with the numerous issues and chal-lenges in modeling the dense-rapid flow regime.

4 Constitutive Models

Inevitably, constitutive models for granular floware complicated because of the wide range of phe-nomenology observed. We use a class of modelsthat relate Cauchy stress T to deformation rateD, often referred to as the Eulerian description.In particular, we focus on a class of generalizedhypoplastic models, based on a stress-space de-scription that allows for nonlinear functions ofstress. Although the approximation of linear elas-ticity is suitable for many granular materials, par-ticularly stiff geomaterials, the nonlinear theorymay find application to soft granular materialssuch as pharmaceutical capsules and clayey soils.

Hypoplasticity

Let us start with an isotropic nonlinearly elasticmaterial. This is one for which the stress T andthe finite (Eulerian) strain tensor E = 1

2 (FFT−I)are connected by isotropic relations of the form

T = t(E) or E = t−1(T). (5)

Here, F = ∂x/∂x◦, where x◦ and x(x◦) denotereference position and current placement, resp-ectively, of material points. Hyperelasticity (or“Green elasticity”) is based on the thermody-namic consistency condition that the functions in(5) be derivable from elastic potentials. Trues-dell’s hypoelasticity , a generalization of elasticitythat allows for a more general rate-independentbut path-dependent relation between stress andstrain, is represented by the LODE

◦T = µµµE(T) : D, (6)

where µµµE is the fourth-rank hypoelastic modu-lus. With suitable integrability conditions on thismodulus, it is possible to show that (6) implies anelastic relation of the form (5).

To obtain visco-elastoplasticity from (6), weadapt the first fundamental postulate of incremen-tal plasticity, that the deformation rate can be de-composed into elastic and inelastic (or “plastico-viscous”) parts, D = DE + DP . Then replaceD in the relations above by DE = D −DP andprovide a constitutive equation for DP .

As the second fundamental postulate, we as-sume that the elastic stress T conjugate to DE

is identical to the inelastic stress conjugate toDP , which follows from an assumption of internalequilibrium of the type suggested by the simplemodel in Fig. 5.

Finally, as the third fundamental postulate, weassume that the system is strongly dissipativesuch that T and DP are related by dual inelas-tic potentials. Viscoelasticity follows and yields anonlinear form of the Maxwell fluid (3).

Plasticity, as defined by rate-independentstress, represents a singular exception to theabove: the magnitude |DP | becomes indetermi-nate. The assumption of overall independence ofrate and history implies a relation of the form

|DP | = |D|ϑ(T, D),

where D = D/|D|, and, hence, to a generalizedform of isotropic hypoplasticity ,

◦T = µµµH(T, D) : D, (7)

with a formula for µµµH in terms of the “inelasticclock” function ϑ. In the standard theory of hy-poplasticity, ϑ(T, D) is independent of D, there isno distinction between elastic and plastic defor-mation rates, and the constitutive equation re-duces to (7) with hypoplastic modulus taking theform

µµµH = µµµE(T)−K(T)⊗ D, (8)

where A ⊗ B = [AijBkl]. Such models are pop-ular with many in the granular mechanics com-munity, particularly those interested in geome-chanics, mainly because these models do not relydirectly on the concept of a yield surface or in-elastic potentials (whence the qualifier “hypo”),

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although they do involve limit states where Jau-mann stress rate vanishes asymptotically underthe action of a constant deformation rate D. Thisasymptotic state may be identified with the so-called “critical state” of soil mechanics, wheregranular dilatancy vanishes and the granular ma-terial flows essentially as an incompressible liquid.

The problem with the general form (8) is thatit is exceedingly difficult to establish mathemat-ical restrictions that will guarantee its thermo-dynamically admissibility, e.g., such that steadyperiodic cycles of deformation do not give posi-tive work output. By contrast, models built upfrom elastic and dissipative potentials are muchmore likely to be satisfactory in that regard.

The treatise by Kolymbas provides an excellentoverview and history of hypoplastic modeling.

Anisotropy, internal variables andparametric hypoplasticity

Initially isotropic granular masses exhibit flow-induced anisotropy, particularly in quasi-staticelastoplastic flow, since the grain kinetic en-ergy or “temperature” is insufficient to randomizegranular microstructure. This anisotropy is oftenmodeled by an assumed dependence of elastic andinelastic potentials, and associated moduli, on asymmetric second-rank fabric tensor , A = [Aij ],which is subject to a rate-independent evolutionequation of the form

◦A = ααα(T, D,A) : D.

This represent a special case of the“isotropicextension” of anisotropic constitutive relations,where the anisotropic dependence on stress T orstrain E is achieved by the introduction of a setof “structural tensors”, consisting in the simplestcase of a set of second rank tensors and vectors.The structural tensors represent in turn a spe-cial case of a set XXX of evolutionary internal vari-ables consisting of scalars, vectors and second-order tensors.

At present, the origins of the evolution equa-tions are unclear, although there may be a pos-sibility of obtaining them from a generalized bal-ance of internal forces derived on elastic and in-elastic potentials which depend on the internalvariables or their rate of change.

Whatever the origin of their evolution equa-tions, it is easy to see that one can enlarge theset of dependent variables from T to {T,XXX} withthe evolution equations leading to parametric hy-poplasticity defined by a generalization of (7).This gives a set of LODEs which describe the ef-fects of initial anisotropy, density or volume frac-tion, etc., represented as initial conditions, andtheir subsequent evolution under flow. This for-mulation includes most of the current non-polarmodels of the evolutionary plasticity of granularmedia.

Multipolar effects

In strongly inhomogeneous systems, such as theshear field associated with shear bands, we ex-pect to encounter departures from the responseof a classical simple (non-polar) material havingno intrinsic length scale. The situation is gener-ally characterized by non-negligible magnitudesof a Knudsen number K = `/L, where ` is acharacteristic microscale and L is a characteristicmacroscale.

In the case of elastoplasticity, various empiricalmodels suggest taking ` to be about five to tentimes the median grain diameter, a scale is mostplausibly associated with the length of the ubiq-uitous force chains in static granular assemblies.These are chain-like structures in which the con-tact forces are larger than the mean force, andit is now generally accepted that these representthe microscopic force network that supports gran-ular shear stress and rapidly reorganize under achange of directional loading. The micromechan-ics determining the length of force chains is stillpoorly understood.

Whatever the physical origins, a microscopiclength scale serves as a parameter in various en-hanced continuum models, including various mi-cropolar and higher-gradient models, all of whichrepresent a form of weak non-locality referred toby the blanket term multipolar. Perhaps the sim-plest is the Cosserat model, often referred to asmicropolar. This model owes its origins to ahighly influential treatise on structured continuaby the Cosserat brothers, Eugene and Francois,celebrated internationally in 2009 on the occasionof the centenary of its original publication. Tejch-man’s book provides a comprehensive summary

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of a fairly general form of Cosserat hypoplastic-ity.

Particle migration and size-segregation.Another fascinating aspect of granular mechan-ics is the shear-driven separation or “unmixing”of large particles from an initially uniform mix-ture of large and small particles. Various mod-els of particle migration in fluid-particle suspen-sions or size-segregation in granular media in-volve diffusion-like terms that suggest multipolareffects. While some models involve gravitation-ally driven sedimentation opposed by diffusionalremixing, other models involve a direct effect ofgradients in shearing akin to those found in fluid-particle suspensions.

It may be significant that many granular size-segregation effects are associated with dense flowin thin layers, which again suggests the likelihoodof Knudsen-number or multipolar effects. What-ever the origins of particle migration, it can prob-ably be treated as a strictly dissipative process,implying that it can be represented as a general-ized velocity in a dissipation potential, thus sug-gesting a convenient way of formulating properlyinvariant constitutive relations.

Relevance to material instability

There is a bewildering variety of instabilitiesin granular flow. These range from the shear-banding instabilities in quasi-static flow discussedabove to gravitational layering in moderately-dense rapid flow and clustering instabilities ingranular gases.

The author advocates a distinction betweenmaterial or constitutive instability, representingthe instability of homogeneous states in the ab-sence of boundary influences, and the dynamicalor geometric instability that occurs in materiallystable media, such as elastic buckling and inertialinstability of viscous flows. With this distinction,it is easier to assess the importance of multipolarand other effects.

Past studies reveal multipolar effects on elasto-plastic instability, not only on post-bifurcationfeatures such as the width of shear bands, butalso on the material instability itself. This repre-sents an interesting and challenging area for fur-ther research based on the parametric viscoelas-

tic or hypoplastic models of the type discussedabove. The general question is whether and howthe length scales that lend dimensions to subse-quent patterned states enter into the initial insta-bility leading to those patterns.

Further Reading

1. Forterre Y., and O. Pouliquen. 2008, Flows ofdense granular media. Annual Review of FluidMechanics 40:1–24.

2. Goddard, J. 2014. Continuum modeling of gran-ular media. Applied Mechanics Reviews, submit-ted.

3. Kolymbas, D. 2000. Introduction to Hypoplas-ticity. Rotterdam: A. A. Balkema.

4. Lubarda, V. A. 2002. Elastoplasticity Theory.Boca Raton: CRC Press.

5. Ottino, J. M. and D. V. Khakhar. 2000. Mixingand segregation of granular materials. AnnualReview of Fluid Mechanics 32:55–91.

6. Rao, K. and P. Nott. 2008. An Introduction toGranular Flow. Cambridge: Cambridge Univer-sity Press.

7. Tejchman, J. 2008. Shear Localization inGranular Bodies with Micro-polar Hypoplastic-ity. Berlin: Springer.

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Biography of contributor

Joe Goddard received his Ph.D. in chemical engi-neering from the University of California, Berke-ley in 1962. He joined the chemical engineeringfaculty of the University of Michigan in 1963,and in 1976 he accepted a position in the De-partment of Chemical Engineering of the Univer-sity of Southern California. He has been Profes-sor of Applied Mechanics and Engineering Sci-ence in the University of California, San Diego,since 1991. His professional distinctions includeNATO, NSF and Fulbright Postdoctoral and Se-nior Postdoctoral Fellowships, Fluor Professorof Chemical Engineering, University of South-ern California, D.L. Katz Lecturer, University ofMichigan, President, U.S. Society of Rheology,G.I. Taylor Medalist of the Society of Engineer-ing Science, and visiting scholar and researcherat several universities in Europe and the U.S., in-cluding a visit as Springer Visiting Professor inthe University of California, Berkeley.

September 5, 2013 pcamJG3.tex