Constitutive law of dense granular matter

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Constitutive law of dense granular matterTo cite this article Takahiro Hatano 2010 J Phys Conf Ser 258 012006

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Constitutive law of dense granular matter

Takahiro Hatano

Earthquake Research Institute University of Tokyo 1-1-1 Yayoi Bunkyo Tokyo 113-0032Japan

E-mail hatanoeriu-tokyoacjp

Abstract The frictional properties of dense granular matter under steady shear flow areinvestigated using numerical simulation Shear flow tends to localize near the driving boundaryunless the coefficient of restitution is close to zero and the driving velocity is small The bulkfriction coefficient is independent of shear rate in dense and slow flow whereas it is an increasingfunction of shear rate in rapid flow The coefficient of restitution affects the friction coefficientonly in such rapid flow Contrastingly in dense and slow regime the friction coefficient isindependent of the coefficient of restitution and mainly determined by the elementary frictioncoefficient and the rotation of grains It is found that the mismatch between the vorticity offlow and the angular frequency of grains plays a key role to the frictional properties of shearedgranular matter

1 IntroductionGranular flow is ubiquitous in solid earth sciences and engineering eg avalanches landslidesdebris flow silo flow etc These phenomena are essentially dominated by the frictional propertiesof granular matter Thus to find a law that describes the behavior of friction coefficient (ratioof the shear stress to the normal stress) of granular matter is an essential problem [1]

The mechanical properties of granular matter generally depend on many ingredients densityshear rate pressure temperature humidity interstitial fluid etc These ingredients mustbe carefully controlled in investigating the frictional properties as they affect the result inunexpected ways Because such potentially important ingredients are not known a priori despiteextensive experimental efforts the frictional properties of dense granular matter are still notclear

In order to rule out chemical processes that potentially affect the frictional propertiesnumerical simulation plays an important role Through extensive simulations [23] it turnsout that the behavior of friction coefficient of dry granular matter is well described by anondimensional number I = γ

radicmPd where γ denotes shear rate m is the mass of a grain

P is the normal pressure and d is the grain diameter This nondimensional number is referredto as the inertial number which seems to be successful in stating a constitutive law for fastgranular flow [4]

To clarify a constitutive law for slow granular flow da Cruz et al performed an extensivesimulation on a two dimensional system in a wide range of the inertial number 6times 10minus4 le I le03 They find a friction law that reads micro = micro0 + aI where micro is the kinetic friction coefficientand micro0 and a are positive constants However in three dimensional systems somewhat differentfriction law is found where the inertial-number dependence of friction coefficient is described

International Conference on Science of Friction 2010 (ICSF2010) IOP PublishingJournal of Physics Conference Series 258 (2010) 012006 doi1010881742-65962581012006

ccopy 2010 IOP Publishing Ltd 1

by a power law which is of the same form as the Herschel-Bulkley model ie micro = micro0 + sIφwhere s is a positive constant and φ 03 [56] This power law behavior is confirmed in anumerical simulation of a certain class of granular matter particularly smooth (frictionless)beads However the validity of this power-law friction in a wide class of granular materialsas well as the theoretical understanding and derivation are still open In addition beforetheoretical consideration some questions regarding phenomenology are still open How do thecoefficient of restitution rolling resistance and other material properties affect the frictionalproperties of granular matter In this paper these questions shall be answered

2 Model21 force modelHere each grain is assumed to be sphere The interaction between grains is described by thediscrete element method (DEM) [7] which is the standard model used in powder engineering andsoil mechanics Consider a grain i of radius Ri located at ri with the translational velocity vi andthe angular velocity Ωi This grain interacts with another grain j whenever they are in contactie |rij | lt Ri + Rj where rij = ri minus rj The force acting on two particles is decomposed intotwo direction normal and transverse to rij Introducing the normal unit vector nij = rij|rij |the normal force Fij is given by [khij minus ζnij middot rij ]nij where hij = Ri + Rj minus |rij | and k is aconstant The coefficient of restitution e can be calculated using the relation

e = exp

[minus πradic

4mkζ minus 1

] (1)

where m = mimj(mi + mj) In order to define the transverse force we define the relativetangential velocity as

Ξij = rij minus nij middot rij +RiΩi +RjΩj

Ri +Rjtimes rij

and introduce the relative tangential displacement vector Θ asintroll dsΞ(s) The subscript

of the integral indicates that the integral is performed only when the contact is rolling iekt|Θij | lt microe|Fij | or Ξij middotΘij lt 0 Otherwise the contact is said to be sliding The magnitude ofthe tangential force depends on the state of the contact microe|Fij | for sliding contact and kt|Θij |for rolling contact

22 configurationThe particles are bidisperse the diameters are 08d and 10d respectively The number of eachparticle is the same For simplicity the mass of the grains is set to be the same which is denotedby m

The dimensions of the system are LtimesLtimesH where we use periodic boundary conditions alongthe x and the y axes In the z direction there exist two rigid walls that consist of the largerparticles One of the walls is displaced along the x axis at constant velocity V to realize plainshear flow where the velocity gradient is formed in the z direction The grains that constitutewalls interact with the bulk grains via the force described above This wall is also allowed tomove along the z axis so that the pressure is kept constant at P while it is immobile along they axis Namely the z dimension of the system denoted by H depends on the velocity of thewall Here we define that H0 is the layer thickness at V = 0 Here the dimensions are set tobe H0 = 12d and L = 10d in which approximately 2 000 particles are intervened between thewalls See Fig 1 for schematic of the computational system The equation of motion of thewall along the z axis is given as MH = Fz minus PL2 where M denotes the mass of the wall andFz is the repulsive force given by the grains The mass of the wall M is set to be M = 100mIt should be remarked that the gravity is not taken into account because the pressure P shallbe chosen to be large enough


2

Figure 1 Schematic of the computational model

23 parametersThe transverse spring coefficient is set to be kt = k5 throughout this study This relation isempirically known to reproduce experimental results The other parameters microe and ζ generallydepends on the material of grains It is thus important to investigate how these parameters affectthe frictional properties of granular matter The pressure P is kept fixed as Pdk = 86times 10minus4For de-dimensionalization we set d = 1 k = 1 and m = 1 Here we estimate the materialconstants based on the sound velocity and the Youngrsquos modulus which roughly corresponds todradickm and kd respectively (Here the numerical factors are neglected) Note also that any

length scale in the simulation is scaled with the grain diameter d Thus the unit velocity andthe unit pressure in DEM are of the order of kilometer per second and several tens of Gigapascalrespectively The pressure applied to the system is thus on the order of 10 MPa In this paperwe limit ourselves to the frictional property of steady states The steady state is realized aftercertain strain which can be confirmed by observing the friction coefficient and the height of thewall H Here the computational data is taken only after the system is subject to 100 strainThe traction acting on the moving wall is monitored which is denoted by Fx so that the bulkfriction coefficient of the system is defined as FxPL

2

3 Velocity profileIn a certain class of systems including granular matter the uniform shear deformation can beunstable and deformation tends to localize within a narrow band This spontaneous structureformation is referred to as shear-banding Although it is speculated that some healing processes(or thixotropy velocity-weakening friction etc) may play a certain role in shear-banding [8] thegeneral mechanism for shear banding is still not clear

Thus the internal flow structure (velocity profile) should be measured before discussing thefrictional properties In order to exclude the fluctuation the instantaneous flow velocity isaveraged over a certain strain (100 ) Fig 2 shows the averaged flow velocity as a functionof the depth Although any healing mechanism is not modeled in the present computationalsystem the behavior that is similar to shear-banding is observed Indeed shear flow tends tolocalize near one of the wall It is found that the flow localization is more apparent for systemswith larger coefficient of restitution (ie with smaller damping coefficient ζ)

The flow localization is also affected by the driving velocity of the wall In the right panelof Fig 2 where the effect of damping is significant (ζ = 1) the shear flow is uniform exceptfor the highest velocity case V = 10minus1 Thus flow localization is more likely to occur at larger


3

0 2 4 6 8 10 12z

0

02

04

06

08

1V

1e-11e-21e-31e-41e-5

0 2 4 6 8 10 12z

0

02

04

06

08

1

V

1e-11e-21e-31e-41e-5

Figure 2 Flow velocity profiles along the depth at various wall velocities The legends denotethe wall velocity The flow velocity is normalized by the maximum velocity (the wall velocity)(Left) ζ = 001 (Right) ζ = 10

driving velocity of the wallIt should be remarked that the localization near the wall is also observed in a two dimensional

system [9] where the localization is more frequently observed in a system at lower pressure Thistendency is also observed in the present three-dimensional system

4 Behavior of the bulk friction coefficient41 general descriptionShear stress σ in steady shear flow is determined by the balance between power input and thedissipation rate D

γσ = D (2)

where γ denotes shear rate Thus the nature of shear stress (and thus friction coefficient)involves the mechanism of dissipation Note that the present computational model has twodifferent sources of dissipation at the particle level damping in the normal force and frictionin the tangential force The former is proportional to the damping coefficient ζ and the latteris proportional to the elementary friction coefficient microe Therefore the dependence of frictioncoefficient on each parameter may be a key to the understanding the dissipation mechanism insheared granular matter

42 shear rate dependenceFirst the shear rate dependence of the friction coefficient is discussed focusing the effect ofthe damping coefficient ζ Fig 3 shows the bulk friction coefficient as a function of theinertial number I (ie nondimensional shear rate) with several values of ζ Here three cases areinvestigated ζ = 10minus2 10minus1 1 whereas the elementary friction coefficient is fixed at 06 Interms of the coefficient of restitution e these values of correspond to e = 080 049 and 0043respectively (Note Eq (1)) The bulk friction coefficient is insensitive to shear rate and tothe damping coefficient in the lower I regime I le 10minus2 This indicates that the collision in thenormal direction is irrelevant to the bulk friction coefficient in dense and slow granular flowNote also that it is much less than the elementary friction coefficient

The bulk friction coefficient then sharply increases around I sim 10minus1 In such high I regionthe friction coefficient apparently depends on the coefficient of restitution it increases as thecoefficient of restitution decreases (For example at I = 028 micro = 044 for e = 080 micro = 050


4

10-5

10-4

10-3

10-2

10-1

100

I

03

04

05

microe=080e=048e=0043

Figure 3 The friction coefficient as a function of the inertial number defined by Eq (1) Thelegends denote the coefficient of restitution The solid and dotted lines are empirical constitutivelaws Eq (3) where φ = 05 for the solid line and φ = 10 for the dotted line

for e = 049 and micro = 057 for e = 0043) This implies that the dissipation due to normalcollision is dominant in the rapid flow region where I ge 10minus1 The shear rate dependence canbe empirically described by the Hershel-Bulkley model

micro = micro0 + sIφ (3)

Here the exponent φ ranges from 05 to 10 (See Fig 3) It should be remarked that thesevalues are larger than previously estimated [56] where 03 le φ le 04 The difference may bedue to the elementary friction coefficient In the previous studies [56] the elementary frictioncoefficient is set to be zero or 02 while it is 06 here Thus the exponent φ depends on theelementary friction coefficient Although microe = 06 case is also investigated in Ref [5] the dataare insufficient for the estimate of the exponent

43 effect of elementary friction coefficientIn dense and slow granular flow the elementary friction coefficient between grains microe is moreimportant parameter than the coefficient of restitution Fig 4 shows how the bulk frictioncoefficient of a slowly-sheared system (I = 28times10minus5) depends on elementary friction coefficientThe bulk friction coefficient increases as the elementary friction coefficient increases at smallermicroe but then becomes independent of microe (There is a maximum value around 035) It should benoted that this insensitivity of the bulk friction coefficient to the elementary friction coefficientis also found in two dimensional granular systems [3]

5 Rotation of grainsAs shown in Figs 3 and 4 in dense and slow flow the extent of dissipation in normal interaction(ie the coefficient of restitution) is irrelevant to the bulk friction coefficient but sliding frictionbetween grains plays an important role Now let us estimate the bulk friction coefficient in thelow I regime taking the sliding friction between grains into account

For simplicity we consider the two dimensional case Neglecting the fluctuation in thetranslational velocity of each particle the sliding velocity at the contact of two particles isapproximately

d (γ cos θij minus (Ωi + Ωj)2) (4)


5

0 02 04 06 08 1microe

01

02

03

04

micro

Figure 4 The bulk friction coefficient taken at I = 28times 10minus5 (the slowest data in Fig 3) asa function of the elementary friction coefficient

10-7

10-6

10-5

10-4

10-3

10-2

shear rate

05

055

06

065

c

Figure 5 (Left) Schematic of grain rotation and shear flow (Right) The frustration parameteras a function of shear rate Here ζ = 10 and microe = 06

where cos θij = nij middot nz (See the left panel of Fig 5) Note that θij le 0 without the loss ofgenerality For simplicity the diameter is assumed to be the same for the all particles Using theaverage normal force Pnbd where nb is the number of contacts in unit volume the frictionalforce acting between particles i and j is microePnbd Then the energy dissipation rate per contactreads

microePnminus1b |γ cos θij minus (Ωi + Ωj)2| (5)

Using the power balance Eq (2) the bulk friction coefficient is estimated as

micro = microe

lang∣∣∣∣cos θij minusΩi + Ωj

2γ

∣∣∣∣rang (6)

where the bracket denotes the average over the contacts Equation (6) indicates that the frictioncoefficient vanishes if the grains have their neighbors only in the velocity gradient direction (thuscos θij = 1) and Ωi = γ Similarly the friction coefficient can also vanish if the grains have theirneighbors only in the flow direction (thus cos θij = 0) and Ωi = Ωj However the orientations


6

0 02 04 06 08 1microe

0

02

04

06

08

1

micro

Figure 6 The bulk friction coefficient taken at as a function of the elementary friction coefficientbetween particles Here the grains are not allowed to rotate

of contacts in granular matter are of course random This is a reminiscence of spin-glassesthe interaction in the flow direction is antiferromagnetic and the one in the gradient directionis ferromagnetic In addition there are intermediate interactions and these interactions arecoupled to translational velocity

As the problem is very complicated let us make further simplification that the angularfrequency of particles are the same ie Ωi = ω and cos θij is replaced by unity Then Eq (6)becomes

micro sim microelang∣∣∣∣1minus ω

γ

∣∣∣∣rang (7)

The quantity 1 minus ωγ which we shall refer to as the frustration parameter is thus importantto the nature of friction in dense and slow granular flow The right panel of Fig 5 shows thetypical behavior of the frustration parameter c as a function of shear rate In slow and denseflow regime the frustration parameter takes the value around 05 This means micro sim 05microe whichis not a very bad prediction for microe = 06 case where micro 035 (See Fig 4) Note that the valueof frustration parameter may depend on microe At this point we do not know how this quantity isdetermined from the contact network of grains

In order to test the validity of Eq (7) one can consider a model in which the grains do notrotate at all This leads to ω = 0 so that it is expected from Eq (7) thatmicro microe Fig 6 showsthe bulk friction coefficient of this model which increases as the elementary friction coefficientincreases up to 10 This behavior makes a quite contrast to that of the previous model withthe particle rotation

6 ConclusionsThe friction coefficient of dense granular matter is an increasing function of shear rate Thecoefficient of restitution affects the friction coefficient only in rapid flow Contrastingly indense and slow regime the friction coefficient is mainly determined by the elementary frictioncoefficient and the rotation of grains Mismatch between the vorticity of flow and the angularfrequency of grains plays a key role to the friction coefficient

References[1] C Marone Annu Rev Earth Planet Sci 26 643 (1998)


7

[2] GDR MiDi Eur Phys J E 14 341 (2004)[3] F da Cruz S Emam M Prochnow J-N Roux F Chevoir Phys Rev E 72 021309 (2005)[4] P Jop Y Forterre O Pouliquen Nature 441 727 (2006)[5] T Hatano Phys Rev E 75 060301 (2007)[6] P-E Peyneau Roux J-N Phys Rev E 78 011307 (2008)[7] P A Cundall O D L Strack Geotechnique 29 47 (1979)[8] L Isa R Besseling and W C K Poon Phys Rev Lett 98 198305 (2007)[9] E Aharonov and D Sparks Phys Rev E 65 051302 (2002)


8

Constitutive law of dense granular matter

Takahiro Hatano

Earthquake Research Institute University of Tokyo 1-1-1 Yayoi Bunkyo Tokyo 113-0032Japan

E-mail hatanoeriu-tokyoacjp

Abstract The frictional properties of dense granular matter under steady shear flow areinvestigated using numerical simulation Shear flow tends to localize near the driving boundaryunless the coefficient of restitution is close to zero and the driving velocity is small The bulkfriction coefficient is independent of shear rate in dense and slow flow whereas it is an increasingfunction of shear rate in rapid flow The coefficient of restitution affects the friction coefficientonly in such rapid flow Contrastingly in dense and slow regime the friction coefficient isindependent of the coefficient of restitution and mainly determined by the elementary frictioncoefficient and the rotation of grains It is found that the mismatch between the vorticity offlow and the angular frequency of grains plays a key role to the frictional properties of shearedgranular matter

1 IntroductionGranular flow is ubiquitous in solid earth sciences and engineering eg avalanches landslidesdebris flow silo flow etc These phenomena are essentially dominated by the frictional propertiesof granular matter Thus to find a law that describes the behavior of friction coefficient (ratioof the shear stress to the normal stress) of granular matter is an essential problem [1]

The mechanical properties of granular matter generally depend on many ingredients densityshear rate pressure temperature humidity interstitial fluid etc These ingredients mustbe carefully controlled in investigating the frictional properties as they affect the result inunexpected ways Because such potentially important ingredients are not known a priori despiteextensive experimental efforts the frictional properties of dense granular matter are still notclear

In order to rule out chemical processes that potentially affect the frictional propertiesnumerical simulation plays an important role Through extensive simulations [23] it turnsout that the behavior of friction coefficient of dry granular matter is well described by anondimensional number I = γ

radicmPd where γ denotes shear rate m is the mass of a grain

P is the normal pressure and d is the grain diameter This nondimensional number is referredto as the inertial number which seems to be successful in stating a constitutive law for fastgranular flow [4]

To clarify a constitutive law for slow granular flow da Cruz et al performed an extensivesimulation on a two dimensional system in a wide range of the inertial number 6times 10minus4 le I le03 They find a friction law that reads micro = micro0 + aI where micro is the kinetic friction coefficientand micro0 and a are positive constants However in three dimensional systems somewhat differentfriction law is found where the inertial-number dependence of friction coefficient is described


ccopy 2010 IOP Publishing Ltd 1



e = exp

[minus πradic

4mkζ minus 1

] (1)



Ri +Rjtimes rij






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0 2 4 6 8 10 12z

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02

04

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1e-11e-21e-31e-41e-5





γσ = D (2)





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0 02 04 06 08 1microe

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0 2 4 6 8 10 12z

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γσ = D (2)





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γ

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10-7

10-6

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micro = microe


2γ

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γ

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8

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Constitutive law of dense granular matter