arX

iv:1

808.

1015

7v1

[co

nd-m

at.s

tat-

mec

h] 3

0 A

ug 2

018

Universal Scaling Laws for Shear Induced Dilation in Frictional Granular Media

Prasenjit Das1, Oleg Gendelman2, H. George E. Hentschel1 and Itamar Procaccia11Department of Chemical Physics, the Weizmann Institute of Science, Rehovot 76100,

Israel, 2 Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel.

Compressed frictional granular matter cannot flow without dilation. Upon forced shearing togenerate flow, the amount of dilation may depend on the initial preparation and a host of materialvariables. Here we show that as a result of training by repeated compression-decompression cyclesthe amount of dilation induced by shearing the system depends only on the shear rate and on thepacking fraction. Relating the rheological response to structural properties allows us to derive ascaling law for the amount of dilation after n cycles of compression-decompression. The resultingscaling law has a universal exponent that for trained systems is independent of the inter-granulesforce laws, friction parameters and strain rate. The amplitude of the scaling law is analyticallycomputable, and it depends only on the shear rate and the asymptotic packing fraction.

I. INTRODUCTION

Compressed granular media, with or without friction,are jammed, and cannot flow without dilation [1, 2]. Sub-jected to shear rate by external forces, such media dilate,reducing the packing fraction in regions that participatein flows. The dilation may be very inhomogeneous, andmay depend on a host of parameters that characterize thegranular assembly. Understanding the resulting rheologyis complicated due to the inherent properties of granu-lar matter, like frictional losses, arching, segregation andthixotropy [3]. These complications result in a paucity ofuniversal results, and the literature of frictional granularrheology at finite strain rate offers a bewildering arrayof particular examples that are not easy to comprehend,resisting attempts to organize and systematize [4–7].

In recent studies it became apparent that some uni-versal results can be gleaned by training the system un-der repeated cycles of compression-decompression [8–10],building a memory that “cleans” the system from randomeffects present in “as compressed” frictional granular sys-tems. For example it was shown that the packing frac-tion converges under repeated cycles to an asymptoticvalue following a universal law [11]. Another example isthe universal giant friction slip event that occurs whenthe pressure goes to zero upon unjamming [12]. Herewe follow on this line of reasoning and study the dila-tion induced by shear rate after training the system byn compression-decompression cycles. Indeed we find anenormous simplification resulting in a universal powerlaw that characterizes the amount of dilation observedafter training with n cycles. The power law indicatesthat training and memory result in the amount of di-lation becoming a function of the strain rate and thepacking fraction only. The exponent of the scaling law isindependent of the working pressure, the strain rate, thefriction parameters and the force laws between granules.To introduce these results we review briefly some recentresults on training and memory formation.

II. TRAINING BY

COMPRESSION-DECOMPRESSION CYCLES

When frictional granular media are trained by cyclicloading and unloading [10, 11, 13, 14] memory is intro-duced in the system. Here we refer to training by uniaxialcompression until the pressure reaches a maximal valuePmax after which the the system is decompressed backto zero pressure, and then compressed again. The com-pression and decompression are achieved by one movingwall (upper wall in our simulations) and “pressure” al-ways refer to the external pressure on this moving wall.In each cycle the packing fraction is increased until itreaches an asymptotic limit. During compression anddecompression dissipation leads to hysteresis, but withrepeated cycles the dissipation diminishes to a finite limitand the system retains memory of an asymptotic loadedstate that is not forgotten even under complete unload-ing. An example of such training protocol as observed innumerical simulations [11] is shown in Fig. 1.

Associated with the reduced dissipation and the in-crease in memory one finds a universal power law in thepacking fraction Φn after the nth cycle. This scalings isexpected to hold irrespective of the details of the micro-scopic interactions. In every compression leg of the cyclethe system compactifies, until a limit Φ value is reachedfor the chosen maximal pressure. To quantify this pro-cess we can measure the volume fraction Φn(Pmax) atthe highest value of the pressure in the nth cycle. Definethen a new variable

Xn ≡ Φn+1(Pmax)− Φn(Pmax) . (1)

This new variable is history dependent in the sense thatXn+1 = g(Xn) where the function g(x) is unknown atthis point. This function must have a fixed point g(x =0) = 0 since the series

∑

n Xn must converge; for anygiven chosen maximal pressure there is a limit volumefraction that cannot be exceeded. Near the fixed point,assuming analyticity, we expect the form

Xn+1 = g(Xn) = Xn − CX2n + · · · . (2)

2

0.846 0.848 0.85 0.852 0.854 0.856 0.858 2 4 6 8 10 12 14 160

20

40

60

80

100

P

Φn

P

FIG. 1. Typical hysteresis loops obtained numerically uponuniaxial n cycles ofcompression and decompression of anamorphous configuration of frictional disks. Here an assemblyof disks with binary sizes are compressed from above in thepresence of gravity. The pressure P is on the upper piston,measured in dimensionless units of mg/d, see text for details.The packing fraction Φ is dimensionless. Compression legsare in red and decompression in green. The n axis is providedfor clarity, since the hysteresis curves are more compressedthan in cyclic training without gravity, see [11].

The solution of this equation for n large is

Xn =C−1

n. (3)

A direct measurement of Xn as a function of n in thepresent simulations which are recorded below is shownin the log-log plot presented in Fig. 2. In Ref. [11] onecan find arguments and evidence for the generality of thispower law.

III. DILATION UNDER SHEAR

A. preparation

The granular system that we simulate consists of disksof mass m = 1 and diameters d = 1 and 1.4d in equalnumbers. To prepare the system for shear and dilationwe begin with a “box” of fixed horizonal length (in thex direction) of 40d and a height (in the y direction) of100d. To start, 1000 small and 1000 large disks are placedrandomly without overlaps. The upper wall has a massM = 100 that is free to move; gravity is chosen such thatg = 1. The moving upper wall and the fixed lower wallare made of particles of identical properties and diame-ters in the continuous range of [d, 2d]. Applying periodicboundary condition in the horizontal direction we nowapply a small pressure P on the upper wall. We simulatethe system using molecular dynamics with Hertz normalforces and Mindlin tangential forces as described below.We solve Newton’s equations of motion with linear damp-ing in the velocities of the disks. For a given pressurethe simulation continues until mechanical equilibrium isreached. The pressure is then increased in small stepsfollowed by equilibration until the desired final pressure

1 10n

10-5

10-4

Xn

n-1

FIG. 2. Log-log plot of Xn vs. n. The black dots are the data,the dashed line is the theoretical inverse power law prediction.The data corroborates Eq. (3).

FIG. 3. An example of a typical initial configuration in thenumerical simulation. The blue disks are the large ones andthe green the small ones. Red disks are glued to the walls.

is obtained. An example of an initial configuration isshown in Fig. 3.The contact fores (both the normal and tangential

forces which arises due to friction) are modeled accord-ing to the DEM (discrete element method) developed byCundall and Strack [15]. Implementation of static fric-tion is done via tracking the elastic part of the shear dis-placement from the time contact was first formed. Whenthe disks are compressed they interact via both normaland tangential forces. Particles i and j, at positions ri, rjwith velocities vi,vj and angular velocities ωi,ωj will ex-perience a relative normal compression on contact givenby ∆ij = |rij −Dij |, where rij is the vector joining thecenters of mass and Dij = Ri + Rj ; this gives rise to

a normal force F(n)ij . The normal force is modeled as a

Hertzian contact, whereas the tangential force is givenby a Mindlin force [15]. Defining R−1

ij ≡ R−1i +R−1

j , theforce magnitudes are,

F(n)ij =kn∆ijnij −

γn2vnij

, F(t)ij =−kttij −

γt2vtij (4)

kn = k′

n

√

∆ijRij , kt = k′

t

√

∆ijRij (5)

γn = γ′

n

√

∆ijRij , γt = γ′

t

√

∆ijRij . (6)

Here δij and tij are normal and tangential displace-

ment; nij is the normal unit vector. k′

n = 2 × 105 and

3

k′

t = 2k′n/7 are spring stiffness for normal and tangential

mode of deformation: γ′

n = 50 and γ′

t = 50 are viscoelas-tic damping constant for normal and tangential deforma-tion. vnij

and vtij are respectively normal and tangentialcomponent of the relative velocity between two particles.The relative normal and tangential velocity are given by

vnij= (vij .nij)nij (7)

vtij = vij − vnij−

1

2(ωi + ωj)× rij . (8)

where vij = vi − vj . Elastic tangential displacement tijis set to zero when the contact is first made and is calcu-lated using

dtijdt

= vtij and also the rigid body rotationaround the contact point is accounted for to ensure thattij always remains in the local tangent plane of the con-tact [16].The translational and rotational acceleration of parti-

cles are calculated from Newton’s second law; total forcesand torques on particle i are given by

F(tot)i =

∑

j

F(n)ij + F

(t)ij (9)

τ(tot)i = −

1

2

∑

j

rij × F

(t)ij . (10)

The tangential force varies linearly with the relative tan-gential displacement at the contact point as long as thetangential force does not exceed the Coulomb limit

F(t)ij ≤ µF

(n)ij , (11)

where µ is a material dependent coefficient. When thislimit is exceeded the contact slips in a dissipative fashion.In our simulations we keep the magnitude of tij so that

F(t)ij = µF

(n)ij . The direction of tij is allowed to change if

further slip takes place.

B. Shearing and Dilating

Having compacted the granular medium through a cer-tain number of cycles, we next examine what happens ifthis same medium is subjected to a shear strain at a rateγ on its upper surface. Flow is possible only by dilatingthe material especially close to the upper moving wall[4–7]. Denoting the rest height of the box by Ly(0) wemeasure the actual height of the upper wall which is afunction of time and the shear rate, denoted as Ly(t, γ).The dilation is now denoted by δ(t, γ) where

δ(t, γ) ≡ Ly(t)− Ly(0) . (12)

The time dependence of δ(t) is quite complex. A typicaltrajectory of this quantity is shown in Fig. 4. Obviously,the trajectory indicates some noisy periodicity aroundsome average. To extract the dominant frequency of the

0 200 400 600t

0

0.1

0.2

0.3

δ(γ,

t).

FIG. 4. A typical trajectory of δ(t) after 9 training cycles.Here P = 5, γLy(0) = 0.1 and the friction coefficient µ = 0.1.

0.2 0.3 0.4 0.5 0.6f

0

0.001

0.002

0.003|S

(f)|

FIG. 5. A Fourier transform of the trajectory shown in Fig. 4.

response of the upper wall we can compute the Fouriertransform of this trajectory,

S(f) ≡1

500

∫ 700

t=200

dt [δ(t, γ)− 〈δ〉(γ)] ei2πft , (13)

〈δ〉(γ) ≡1

500

∫ 700

t=200

dt δ(t, γ) . (14)

The limits of integration were chosen to eliminate theinitial rise to a ‘steady state’ and to ensure convergenceof the result. Averaging such spectra over 50 indepen-dent initial configurations results in a typical spectrumas seen in Fig. 5. The spectrum is dominated by onetypical frequency. The nature of this frequency and itsdependence on shear rate are interesting by themselves,but they fall outside the scope of the present paper. Weonly note in passing that the principal frequency (themain peak in Fig. 5) is fully understandable as a result

4

5 10 15 20 25 30 35 40 45 50γL

y

0

5

10

15

20

25

30

35

<δ>

(γ)

µ=0.0µ=0.1µ=0.2µ=0.3µ=0.5µ=0.7µ=1.0

.

.

FIG. 6. The dependence of the average dilation on shear rateand friction coefficient.

of the excitation of the primary bulk elastic mode of thesystem.Having computed the average dilation 〈δ〉(γ) we can

next examine its dependence on the parameters of themodel. The average dilation obviously depends on themany variables, including the packing fraction, the shearrate, the friction coefficient etc. For a given packing frac-

tion we can study the dependence on the shear rate andthe friction coefficient. Typical results are presented inFig. 6. One sees that the average dilation increases withshear rate. In fact the dependence of the dilation on theshear rate when γ → 0 appears to be non-analytic. Thediscussion of this non-analyticity is interesting but willbe defer to a future publication. We also note that theamount of dilation reduces with the friction coefficient,leading to convergence in the plot for µ > 0.3.The question that remains is how can we obtain data

collapse and universal statements about the dilation un-der shear. To this aim we need to develop a bit of theory.

IV. DATA COLLAPSE AND UNIVERSAL LAWS

A. Cyclic Training

Motivated by the universal scaling laws for the pack-ing fraction as described in Sect. II and Fig. 2 we studynext the physics of dilation in cyclically trained systems.The cyclic training is achieved by uniaxial straining suchthat the pressure is increased by pushing down the up-per wall in quasistatic fashion until we reach a maximalchosen pressure; in the present simulations this pressureis Pmax = 100. After each compression step, the systemis allowed to relax to reach a new mechanical equilib-rium. After a full compression leg, a cycle is completedby decompressing back to zero pressure, where the next

compression cycle begins. The packing fraction Φ is mon-itored throughout this process. Each such cycle traces ahysteresis loop in the P − Φ plane, see Fig. 1 as an ex-ample.The measurements of average dilation will be made

now after n − 1 cycles. The system is decompressed tozero at the n−1’th cycle, and then compressed again to achosen value of the pressure Pw. At that pressure we thenstrain the system at a given strain rate γ to measure theaverage dilation. To get better statistics we repeat thewhole procedure to obtain 〈δn〉(γ) averaged over manyrealizations.To achieve universal results it is always prudent to

work with dimensionless quantities. Thus instead ofworking with 〈δn〉(γ) we opt to define a new, relatedquantity which is dimensionless. To define this dimen-sionless quantity denote by Φn(Pw) the packing fractionassociated with the unstrained systems in the nth cycle.After settling into the steady state with a give shear rateγ the asymptotic average packing fraction is denoted asΦ∗

n (Φn(Pw), γ). The dimensionless dilation is then

D(Φn(Pw), γ) ≡[ Φn(Pw)

Φ∗

n(Φn(Pw), γ)− 1

]

. (15)

Needless to say, besides being dimensionless the depen-dence of this measure on the shear rate and on the fric-tion coefficient remains identical to the data shown inFig. 6. To simplify the notation we use below Dn(γ) ≡D(Φn(Pw), γ).

B. Universal scaling law

Having at our disposal the universal scaling law for theseries Xn it is natural to consider the series of differencesin dimensionless dilations Dn+1−Dn. The main result ofthis subsection will be the series of these differences canbe re-scaled to become (for large n) independent of γ, theinitial pressure, the friction coefficient etc. To see how toachieve this simplification we note that after many cycles,when Φn → Φ∞, we can write

Dn+1(γ)−Dn(γ) ≈ D′(Φ∞, γ)Xn ≈D′(Φ∞, γ)/C

n.

(16)where D′(Φ∞, γ) = dD(Φ, γ)/dΦ|Φ=Φ∞

. Besides theimmediate consequence that dilation difference tends tozero as 1/n, we also predict that the amplitude of thisscaling appears to be a universal coefficientD′(Φ∞, γ)/C.Since C is known from the data on the packing fractionitself, we need here to examine the coefficient D′(Φ∞, γ).To compute the coefficient we start from Eq. (15) and

write

ΦnD′(Φ, γ) = [1 +D(Φn, γ)]− [1 +D(Φn, γ)]

2dΦ∗

n/dΦn.(17)

Now if we assume that as n → ∞ the granular mediumlooses its memory of its initial condition then we would

5

1 10n

10-2

10-1

100

A[D

n+1(γ

) - D

n(γ)]

γLy=0.10

γLy=1.00

γLy=5.00

γLy=10.0.

.

.

..

.

FIG. 7. A[Dn+1 −Dn] vs n. The line is not a fit, but simplec/n with c = 1.2. The data shown are for µ = 0.1 but for othervalues of µ the re-scaled data collapse on the same curve.

expect that dΦ∗

n/dΦn → 0 and asymptotically we willfind

D′(Φ∞, γ) ≈[1 +D(Φ∞, γ)]

Φ∞

. (18)

and the asymptotic scaling of the dilation can be writtenas

Dn+1(γ)−Dn(γ) ≈[1 +D(Φ∞, γ)]/(Φ∞C)

n. (19)

Finally, denoting

A−1 ≡ [1 +D(Φ∞, γ)]/(Φ∞C) , (20)

we expect that A[Dn+1 − Dn] should become indepen-dent of any parameter in the problem yielding a universal

power law 1/n. This prediction is tested against the nu-merical simulations and the results are shown in Fig. 7.We find that instead of unity the pre-factor is close to 1.2.Of course constants of the order of unity are permissiblein this theory.

V. CONCLUDING REMARKS

The main point of this paper is that training a fric-tional granular system by compression-decompression cy-cles can “clean” an “as compressed” system from ran-dom effects that complicate the interpretation of rheolog-ical properties. In the present example we examined theamount of dilation caused by shearing the system witha give shear rate γ. After training with n compression-decompression cycles the scaled dilation Dn(γ) could bepredicted since the series converges with a universal scal-ing exponent n−1. The pre-factors could be also esti-mated from the knowledge of the equally generic n−1

dependence of the associated series Φn of the packingfraction after n cycles. One should note that in our sim-ulations finite size effects are significant, introducing er-rors of the order of unity in the coefficient C of Eq. (3), inthe value of Φ∞ and in the values of D(Φ∞, γ). All theseenter the coefficient A of Eq. (20). Together they con-tribute to the scatter seen in Fig. 7. Taking all this intoaccount we consider the agreement between theory andmeasurement quite satisfactory. It appears quite worth-while to continue in the future to examine the effectsof training and memory on disordered systems and theirrheology.

[1] R. A. Bagnold, Proc. Roy. Soc. A 295, 219 (1966).[2] H. M. Jaeger, S. R. Nagel, and R. P. Behringer,

Rev. Mod. Phys. 68, 1259 (1996).[3] P. A. Thompson and G. S. Grest,

Phys. Rev. Lett. 67, 1751 (1991).[4] .[5] H.-J. Tillemans and H. J. Herrmann,

Physica A: Statistical Mechanics and its Applications 217, 261 (1995).[6] E. Aharonov and D. Sparks,

Phys. Rev. E 65, 051302 (2002).[7] K. A. Lrincz and P. Schall, Soft Matter 6, 3044 (2010).[8] J. Zhang, T. S. Majmudar, A. Tordesillas, and R. P.

Behringer, Granular Matter 12, 159 (2010).[9] J. Ren, J. A. Dijksman, and R. P. Behringer,

Phys. Rev. Lett. 110, 018302 (2013).

[10] M. M. Bandi, M. K. Rivera, F. Krzakala, and R. E. Ecke,Phys. Rev. E 87, 042205 (2013).

[11] M. M. Bandi, H. G. E. Hentschel, I. Procaccia, S. Roy,and J. Zylberg, Europhys. Lett. 122, 38003 (2018).

[12] H. G. E. Hentschel, I. Procaccia, and S. Roy, arXiv:1805.11439v1.

[13] I. Regev, T. Lookman, and C. Reichhardt,Phys. Rev. E 88, 062401 (2013).

[14] D. Fiocco, G. Foffi, and S. Sastry,Phys. Rev. Lett. 112, 025702 (2014).

[15] P. A. Cundall and O. D. L. Strack,Gotechnique 29, 47 (1979).

[16] L. E. Silbert, D. Ertas, G. S. Grest, T. C.Halsey, D. Levine, and S. J. Plimpton,Phys. Rev. E 64, 051302 (2001).