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Training, Memory and Universal Scaling in Amorphous Frictional Granular Matter

M. M. Bandi1, H. George E. Hentschel2, Itamar Procaccia2,3, Saikat Roy2, and Jacques Zylberg21 Collective Interactions Unit, OIST Graduate University, Onna, Okinawa, 904-0495 Japan.

2Dept of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel.3The Niels Bohr International Academy, University of Copenhagen,

Blegdamsvej 17, DK-2100 Copenhagen, Denmark

We report a joint experimental and theoretical investigation of cyclic training of amorphousfrictional granular assemblies, with special attention to memory formation and retention. Measuresof dissipation and compactification are introduced, culminating with a proposed scaling law for thereducing dissipation and increasing memory. This scaling law is expected to be universal, insensitiveto the details of the elastic and frictional interactions between the granules.

“Memory” in materials physics is usually associatedwith the existence of macroscopic hysteretic responses[1]. Two distinct states, separated by a potential barrierlarger than the thermal energy scale, can be used as amemory encoding mechanism. Here we focus on memorythat is induced by training a frictional granular matterby cyclic loading and unloading [2–9]. In each such cycledissipation leads to hysteresis, but with repeated cyclesthe dissipation diminishes until the system retains mem-ory of an asymptotic loaded state that is not forgotteneven under complete unloading. We report and explaina universal power law associated with the reduced dissi-pation and increase in memory which is expected to holdirrespective of the details of the microscopic interactions.

FIG. 1: Typical hysteresis loops obtained experimentallyupon uniaxial compression and decompression of an amor-phous configuration of frictional disks as seen in Fig. 2. Thepressure P is measured in N/m, and Φ is dimensionless. Com-pression legs are in black and decompression in red.

The phenomenon under study is best introduced bythe experimental plots of pressure vs. packing fractionobtained by compressing and decompressing uniaxiallyan array of frictional disks [6, 9], cf. Fig. 1. The exper-imental set up is detailed in the SI. A typical exampleof the experimental cell is shown in the upper panel ofFig. 2. Two opposing boundaries separated by chamberlength L were movable while the other two (transverse)

boundaries were held fixed. The two opposing movableboundaries provided uni-axial pack compression, throughwhich the packing fraction Φ was controlled. Accord-ingly, we define the packing fraction Φ as the ratio oftotal area occupied by the disks to the chamber areabounded within the four boundaries, two of which aremovable. Fig. 1 displays a typical series of consecutiveloops of compression and decompression. The qualita-tive experimental observation is that the area in consec-utive hysteresis loops diminishes monotonically while thepacking fraction is increasing with every loop. This in-dicates that the system is compacted further with everyloop and this process is accompanied by a reduction inthe dissipation. The experimental results left howevertwo open questions: (i) whether asymptotically the dis-sipation vanished, such that every compression becamepurely elastic and the decompression to zero pressure leftthe system with perfect memory of the stressed config-uration; and (ii) whether there is anything universal inthe way that the areas of the loops approaches its asymp-tote, be them finite or zero. To answer these question weperformed numerical simulations that lead to the conclu-sion that (i) asymptotically the hysteresis loops are stilldissipative due to frictional losses, but the structural re-arrangements disappear and the neighbor list becomesinvariant; and (ii) that the area An under the nth hys-teresis loop (which is a direct measure of the dissipation)decays as a power law to an asymptotic value accordingto

An = A∞ +Bn−θ , θ ≈ 1 . (1)

Here A∞ represents the dissipation due to frictional slipsthat exist even in the asymptotic loop, and it dependson the material properties. The second term in Eq. (1)is due to the successive compactification of the sample,and the constant B is also expected to depend on mate-rial properties. The form of this law however is universal,expected to hold independently of the details of the mi-croscopic interaction between the granules.The details of the numerical set up are provided in

the SI. An example of an initial configuration is shownin the lower panel of Fig. 2. We assign Hertzian normal

2

FIG. 2: Upper panel: an example of a typical initial config-uration in the experiment. Lower pannel: an example of atypical initial configuration in the numerical simulation.

force F(n)ij and a Mindlin tangential force F

(t)ij [6] to each

binary contact ij. The tangential force is always limitedby the Coulomb law

F(t)ij ≤ µF

(n)ij . (2)

In uniaxial straining the pressure is increased by push-ing two opposite walls of the system towards each other.In each cycle we first reach a chosen maximal pressureby quasi-static steps. After each compression step, thesystem is allowed to relax to reach a new mechanicalequilibrium in which the global stress tensor is measuredby averaging the dyadic products between all the binarycontact forces and the vectors connecting the centers ofmass in a given volume. The trace of this stress tensor isthe new pressure P . After a full compression leg, a cy-cle is completed by decompressing back to zero pressure,where the next compression cycle begins. The packingfraction Φ is monitored throughout this process. Eachsuch cycle traces a hysteresis loop in the P − Φ plane,see Fig. 3 as an example. The area within each hystere-sis loop is a measure of the dissipation, which in generalstems from two sources. One is plastic events in whichthe neighbor lists change in an irreversible fashion, andthe other is due to frictional losses when the frictionaltangential force exceeds the allowed Coulomb limit. Thetraining of the system is exemplified by the fact that thedissipation as measured by the area An of the nth cyclereduces with n and reaches an asymptotic value when

φ

0.825 0.83 0.835 0.84

P

50

100

150

FIG. 3: A succession of hysteresis loops as measured in thenumerical simulation. Blue symbols are compression legs andred symbols decompression legs. Here µ = 0.1.

n

100

101

102

An

-A∞

10-3

10-2

10-1

FIG. 4: The power law for the decaying areas under the hys-teresis loops as measured in the numerical simulation. Hereµ = 0.1, Black dots are data and the red line is the best fittingpower law y = 0.095X−1.03 . The observed lot is independentof µ, cf. the SI.

n → ∞. Measuring the area in the nth loop we find thatit follows a power law decay in the form of Eq. (1) Thedata supporting this power law are exhibited in Fig. 4.From this data we can conclude that θ ≈ 1 and that thescaling law appears universal with respect to changes inthe value of µ. A further evidence of universality is ob-tained by changing the size distribution of disks, choosinga multi-dispersed system with readii ratios 1, 1.1, 1.2 and1.4. Identical power laws were found.To understand the scaling law we need to identify the

two important processes that take place during the cyclictraining. One is compactification. In every compressionleg of the cycle the system compactifies, until a limitφ value is reached for the chosen maximal pressure. Toquantify this process we can measure the volume fractionΦn(Pmax) at the highest value of the pressure in the nthcycle. Define then a new variable

Xn ≡ Φn+1(Pmax)− Φn(Pmax) . (3)

This new variable is history dependent in the sense thatXn+1 = g(Xn) where the function g(x) is unknown at

3

n

101

102

Xn

10-5

10-4

10-3

FIG. 5: Log-log plot of Xn vs. n. The black dots are the data,the blue line is the best fitting scaling law y = 0.02x−1.02. Thedata corroborates Eq. (5).

this 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 must have the form

Xn+1 = g(Xn) = Xn − CX2n + · · · . (4)

The solution of this equation for n large is

Xn =C−1

n. (5)

This is the source of the second term in Eq. (1),which stems from the compactification which reducesthe amount of dissipation due to irreversible plastic rear-rangements. A direct measurement ofXn as a function ofn is shown in the log-log plot presented in Fig. 5, support-ing the generality of this power law. Without any reasonfor non-analyticity in the function g(Xn) this conclusionis firm. It should be noted at this point that the scal-ing laws Eqs. (1) and (5) must contain some logarithmiccorrections, since the harmonic series does not converge,but the series

∑

n Xn must converge to get an asymp-totic value of Φmax. Indeed, the scaling laws measuredabove consistently show exponents slightly smaller than-1, and therefore the series of Φn converges. It is verylikely that this small difference is due to logarithmic cor-rections to the scaling law which cannot be computedfrom the simple theory presented here.The first term in Eq. (1), on the other hand, is due to

the frictional dissipation. In uniaxial compression thereis a shear component, and the shear stress loads the tan-gential contacts. Whenever the tangential force exceedsthe Coulomb limit Eq. (16), the system dissipates someenergy to a frictional slip. Even if the neighbor list be-comes invariant at large values of n, the loading of thesystem is always accompanied by frictional slips. In oursimulations we can measure the energy dissipated by fric-

tion slips, denoted as ∆E(f)n in the nth hysteresis loop.

n

10 20 30 40

Sn

0.6

0.7

0.8

0.9

1

1.1

FIG. 6: The normalized frictional energy loss in each hys-teresis loop. This energy loss drops to a stable value that isresponsible for the asymptotic dissipation that is encoded bythe area A∞ in Eq.(1).

To have a non-dimensional measure we normalized thisquantity by ∆E

(f)1 , denoting the result as Sn. The de-

pendence of this normalized dissipated energy on n ispresented in Fig. 6. It is clear that the normalized dissi-pated energy due to frictional slips reduces rapidly to astable value; this is the first term in Eq. (1).Encouraged by this theory we returned to the experi-

mental data to measure both Xn and An. To obtain thelogarithmic n dependence of the area we needed to knowthe value of A∞. A direct measurement of this area isunfeasible. But one recognizes that the total dissipationunder the asymptotic loop should be the same with orwithout memory. Accordingly, we applied an acousticperturbation to the configuration after each quasi-staticstep to destroy memory and training, and force the sys-tem to go to asymptotic state. The result is shown in theupper panel of Fig. 7, showing A∞ alone in blue squares.Indeed A∞ is finite and flat as a function of n. Usingthe measured value of A∞ we get the power low scalingfor An −A∞ as expected and as shown by the data withblack rhombi. To underline the fact that this scaling lawis governed by the memory during the training protocolwe demonstrate the change that is caused by destroyingthe memory midway through the cycles; this is a fur-ther validation that the power law is indeed coming fromtraining and memory formation. For the data shown ingreen triangles, there are no acoustic perturbations forn = 1 - 10; there we see the power-law behavior. Fromn = 11 - 50, we apply acoustic perturbations after eachquasi-static step. The loop areas now fall drastically andbecome flat. Note that the magnitude is below A∞ sincewe are plotting the difference An −A∞.It should be commented that the simple scenario dis-

cussed in the Letter requires a subtle change in the shapeof the hysteresis loops. The low order loops are increasingthe volume fraction, such that the compression leg startswith at a lower value of Φ than the end of the decompres-sion loop, see the upper panel in Fig. 8. This continuesto be the case as long as the systems can be compactified

4

n

100

101

102

An

-A∞

10-5

10-4

10-3

10-2

10-1

100

n

100

101

102

Xn

10-5

10-4

10-3

FIG. 7: Upper panel: The areas An − A∞ measured experi-mentally as a function of n, agreeing to Eq. (1) with θ ≈ 0.97;black rhombi. Blue squares: the estimate of A∞ obtained byforcing the system to the asymptote by acoustic perturba-tions. Green triangles: the areas resulting from the destruc-tion of memory after 10 regular loops. Lower panel: experi-mental results for Xn shown in log-log plot vs. n; the slopeis approximately 0.96.

further. The high order loops must begin and end at thesame value of Φn, see the lower panel in Fig. 8. Thus forlarge value of n the hysteresis loops become repetitive,with an invariant trace in the P − Φ plane, even thoughthey have frictional dissipation in the limit n → ∞. Thissubtle change in the shape of the hysteresis loops allowsthe function g(x) to have the fixed point at x = 0 aroundwhich the analytic expansion dictates the universality ofthe power law.In conclusion, we presented and explained a univer-

sal scaling law in the context of the cyclic training of anamorphous assembly of frictional disks. The protocol ex-hibits a reduction in the dissipation per loop until thesystem reaches an asymptotic configuration with max-imal volume fraction (for the maximal pressure chosenin the cyclic protocol). Once achieved, the system hasa perfect memory of the stressed state even when it iscompletely decompressed to zero pressure. Interestingly

enough, repeated compressions are not without dissipa-tion, since shear always induces frictional slips. But the

φ

0.824 0.826 0.828 0.83 0.832 0.834

P

20

40

60

80

100

120

140

160

φ

0.832 0.834 0.836 0.838 0.84

P20

40

60

80

100

120

140

160

FIG. 8: Upper panel: examples of low order hysteresis loopsin the P − Φ plane. The compression legs are blue and thedecompression legs red. Lower panel: examples of higher or-der hysteresis loops in the P − Φ plane with the same colorconvention. The high order loops are no longer able to com-pactify the system further, and the compression leg begins atthe same volume fraction where the decompression leg ends.

beginning and final volume fractions become invariantand the system repeats exactly the same hysteresis loopin the P − Φ space. The amount of dissipation in thecyclic loops is governed by the scaling law Eq. (1) whichhas a universal form with material dependent coefficients.The two terms in this equation were identified and relatedto the dissipation due to changes in the neighbor list andthe frictional slips respectively.

Acknowledgments

This work had been supported in part by the US-IsraelBSF and the ISF joint grant with Singapore. IP is in-debted to the Niels Bohr International Academy for theirhospitality under the Simons Fellowship where this paperwas written. MMB and the experiments were supportedby the Collective Interactions Unit, OIST Graduate Uni-versity.

5

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6

I. DETAILS OF THE EXPERIMENTAL SYSTEM

A. Experimental Design

The basic experimental design (see Fig. 2 upper panelof main text for schematic) follows the setup reportedin Ref. [1] with several design improvements to be elab-orated below. It consisted of a quasi two-dimensionalchamber constructed from a steel frame of inner dimen-sions 0.6 m × 1.1 m × 0.02 m, with a transparent acrylicbottom plate. A transparent Teflon sheet was glued tothe top side of the acrylic bottom plate facing into thechamber to reduce friction between disks and the bottomplate. Whereas our primary measurements are conductedwith disks comprised of stress birefringent (photoelas-tic) polymer, the current experiments were conductedwith disks machined out of Lexan Polycarbonate. Wedo not detail photoelastic measurements here. The topand bottom of each disk was glued with a transparentteflon sheet, but the sides were intentionally machinedwith high roughness to achieve inter-disk contact frictionresponsible for friction-induced hysteresis studied here.The machined roughness yielded a static friction coef-ficient of µ = 0.21 between disk contacts as measuredusing the method detailed in Ref. [1]. The friction be-tween disks and bottom plate was measured to have astatic friction coefficient of µ = 0.04.Our experiments were designed to achieve high pre-

cision in translation using feedback control via capaci-tive displacement sensors. The setup therefore demandedvery high machining tolerance in dimensions of the cham-ber frame as well as the internal and external movableand immovable boundaries. Since it is impossible toachieve machining tolerance of 100 nm over part lengthsof meter dimensions, the frame and boundary parts weremachined in individual pieces of 10 cm length on Com-puter Numerically Controlled mills. These individualparts were then assembled in an interlocking French cleatmechanism with a 37.5◦ ratchet geometry and held byscrews to assure structural rigidity to obtain a contigu-ous structure. The assembled chamber was clamped rigidto an optical table standing on graded concrete founda-tion and floated by compressed air to isolate extraneousground vibrations. A bidisperse set of large (diameterDL = 1.5 cm) and small (diameter DS = 1 cm) disksof thickness 0.975 cm in equal number ratio, were placedin the quasi two-dimensional experimental chamber withtwo opposing movable boundaries (Compression Axis)and two transverse, immovable boundaries (TransverseAxis). Four acoustic transducers placed asymmetricallyas shown in Fig. 2 upper panel of main text (also seeFig. 9a), provided spatially homogeneous acoustic exci-tation to the pack as explained later.An internal steel boundary extended 5 cm into the

chamber from each chamber wall, thus providing actualinterrogation chamber dimensions of 0.5 m (L) × 1 m

(W ) × 0.02 m (D). The internal boundaries are what weshow in the schematic presented in Fig. 2 upper panelof main text marked “Movable” and “Fixed”, and infig. 9a and b as the “Internal movable boundary” forthe compression axis perimeter. As schematically shownin fig. 9a, the internal boundaries maintain a 1 mm ver-tical gap with the bottom acrylic plate, but do contactdisks whose height extends to 0.975 cm. Four steel rodsconnect each internal boundary to an external boundaryby passing through high precision linear bearings placedwithin the steel chamber frame. A precision force sensorwas located at the terminating point of each steel rodwithin the external boundary to measure boundary pres-sure of the granular pack. A total of 16 sensors (4 perboundary) provided the boundary pressure read out. Inthe absence of frictional contact between internal bound-aries and acrylic bottom, and high lubrication transla-tion provided by linear bearings, the forces measured bythe sensors were predominantly those experienced by thegranular pack alone, with no systematic errors in mea-sured signals as will be explained in the following.The external boundaries along the transverse axis were

clamped rigid to the floating optical table thereby ren-dering the transverse axis boundaries immovable. On theother hand, the external boundaries along the compres-sion axis connected to a high precision motorized trans-lation stage to achieve quasi-static compression. A cir-cular DC light source illuminated with cold LEDs wasplaced under the experimental chamber to provide back-lit illumination through the transparent acrylic bottom.Since measurements reported here only concern bound-ary pressure measurements, and do not involve imaging,the details will be presented elsewhere.

B. Movable Boundary Translation

The compression axis boundary positions were con-trolled by very high precision motorized translationstages as schematically shown in fig. 9a and b. A New-port CONEX MFA-CC servo translation stage with 250mm translational distance and 100 nm minimum stepsize was employed for automated translation control. Wefound the CONEX-CC controller usually supplied withthe stage did not offer the precision we desired. Wetherefore constructed a precision displacement sensor inhouse to measure quasi-static compressive displacementsalong the movable boundary. Our choice of inhouse de-sign for the displacement sensor was dictated by experi-mental considerations. We found optical interferometricdisplacement sensors failed due to unavoidable mechan-ical vibrations induced during the experimental proto-col. Instead we employed a set of two translation stagesseparated by 25 nm, and a homebuilt parallel plate ca-pacitor was installed as shown in fig. 9b. The parallelplace capacitor was used as a capacitive displacement

7

FIG. 9: (Color online) Compression axis schematic (a) Side view: A linear bearing passes through the interrogation chamberframe and connects an internal boundary to an external one. The internal boundary avoids frictional contact with the acrylicbottom (1 mm gap), but makes contact with, and moves disks. A force sensor (red) located within the external boundarymakes contact with the linear bearing rod and is held rigid in place by a screw. A circular polarized DC LED light sourceilluminates the pack from below. (b) Top view: The primary translation stage moves the outer and inner boundaries. Asecondary translation stage is capacitively coupled to the movable boundary through a parallel plate capacitor (light blue).Both primary and secondary stages are motorized by LabView. After adjusting initial capacitor gap to 50 nm, the secondarystage is held stationary while the primary stage displaces boundaries and increases the capacitor gap. A 5 V DC signal acrossthe capacitor is sampled at 10 KHz, constantly monitors displacement and achieves a minimum precise quasi-static step of∆L/2 = 250 nm per boundary through closed loop LabView control.

FIG. 10: (Color online) (a) The force sensor (red) within the external boundary is enclosed in a heat insulating ceramic jacket(yellow). Its nipple makes contact with the linear bearing rod with a ceramic heat insulating disk glued to its end. Nitrogen(N2 gas) at 68K flows through to cool the sensor. (b) Electronic circuit schematic for correlated noise removal has two inputs,one from the DC regulated voltage source which drives the sensor and the sensor output. Noise in sensor output due to voltagedriver is spectrally filtered out and fed to LabView data acquisition system. (c) Probability density function (PDF) of forcewith correlated noise (solid black circle) follows the Generalized Extreme Value (GEV) distribution (solid black line) whereasthe noise post cross-correlator stage (solid red squares) is fit very well by a Gaussian (dashed red line). Force fluctuationsmeasured post cross-correlation follows Johnson-Nyquist form and (d) its standard deviation σF (solid black circle) falls assquare root of Temperature (solid black line). The linear fit (dashed red line) is plotted for comparison.

8

sensor and was interfaced with a LabView data acquisi-tion and control system to monitor displacements at 10KHz sampling frequency and control compression axisquasi-static displacements down to steps of ∆L/2 = 250nm. Our tests showed the real achievable displacementresolution was 17.6 nm with an uncontrollable backlashintroduced through the stage manufacturing process of2.35 nm, both being far below our desired displacementresolution of 250 nm.We employed the following quasi-static compression

protocol:1) At the start of each quasi-static step, the secondarystage was moved into position to achieve a parallel platecapacitor gap of 50 nm between the primary and sec-ondary stages.2) Keeping the secondary stage position fixed, the pri-mary stage was translated at a speed of 1 mm/s, withcapacitance being monitored at 10 KHz sampling fre-quency by the LabView system.3) The closed loop control circuit implemented in Lab-View automatically stopped the primary translationstage once a distance of ∆L/2 was achieved, where theminimum achievable ∆L/2 was 250 nm.4) This quasi-static compressive displacement transformsto a quasi-static step in packing fraction ∆Φ = V/(∆L×W ), since both opposing movable boundaries were trans-lated simultaneously. This latter point albeit subtle, be-comes important in light of the strong protocol depen-dence observed in frictional measurements.

C. Boundary Force Measurement

The boundary pressure was measured with subminia-ture force sensors (LCMKD-50N, Omegadyne). Eachforce sensor was placed within the external boundary andheld in place by a screw from one end. At the other end,the nipple of the button sensor made contact during com-pression with the linear bearing rod which terminatedwithin the external boundary. The sensor was encasedwithin a 3D printed (Visijet PXL-Core 3D printer) heatinsulating ceramic jacket and a ceramic tip was glued tothe teminating end of the linear bearing rod as shownin Fig. 10a. A small gap was maintained between thesensor and linear bearing rod through which Nitrogengas at 68K flowed through a hole drilled in the externalboundary to cool down the force sensor.The raw force sensor output exhibits correlated noise

from several sources, including ground loops, capacitivecoupling, and correlated noise from the regulated DVvoltage circuit that drives the force sensor. After ac-counting for ground loops and capacitive coupling in thesystem, a noise cross-correlator circuit designed inhousewas employed (Fig. 10b). Output from the regulated DCvoltage circuit was split with one line driving the forcesensor and the other forming one of two inputs to the

cross-correlator circuit. The force sensor output formedthe second input to the cross-correlator, which spectrallyfiltered the noise in force output arising from fluctuationsin the DC regulated voltage supply. The output from thecross-correlator was then sent to a LabView data acqui-sition system for Analog to Digital Conversion (ADC).

Figure 10c shows the probability density function(PDF) of force sensor output with (solid black circles)correlated noise, i.e. prior to entering cross-correlatorcircuit, and is fit very well by the Generalized ExtremeValue (GEV) or the Fischer-Tippett distribution (solidblack line in Fig. 10c) of the form:

Π(F ) =1

σFt(F )ξ+1e−t(F ) (6)

where t(F ) =(

1 +(

F−〈F 〉σF

)

ξ)−1/ξ

if ξ 6= 0 and t(F ) =

e−(F−〈F 〉)/σF if ξ = 0. Here F is measured force inNewtons, 〈F 〉 is mean force time-averaged over the du-ration of signal acquisition, σF is the standard devia-tion, with ξ forming the only fit parameter for the data,which was found to be ξ = 0.5 ≃ 0, and accordinglyt(F ) = e−(F−〈F 〉)/σF .

On the other hand, the force sensor output postcross-correlator stage (solid red squares in Fig. 10c) ex-hibits nearly Gaussian fluctuations (dashed red curve inFig. 10c). This classical Johnson-Nyquist form for postcross-correlator noise permits one to employ noise re-duction by exploiting the fluctuation-dissipation theoremby virtue of the fact that the force sensor output is avoltage which is linearly proportional to the measuredforce. Just as Johnson-Nyquist noise follows the formVrms =

√4kBTR∆f where kB is Boltzmann constant, T

is temperature, R is resistance and ∆f is the frequencybandwidth, we have for the force measurement:

σF ∝√

4kBTR∆f (7)

Indeed, cryogenically cooling the electronics and forcesensors exhibits noise reduction with a square-root de-pendence on temperature T as shown in Fig. 10d. There,the standard deviation in force σF (solid black circles)fell as we cooled the circuits from room temperature(T = 291 K) down to T = 68 K. The square-root fitwith temperature (solid black line) is decidedly betterthan a linear fit (dashed red line). We found flicker noise(Shot Noise) if the circuits were cooled below T = 68 K.Rather than employ further electronic measures, we in-stead adopted noise averaging by sampling the force at 1KHz for 10 second duration and taking its average. Sincethe uncorrelated noise is expected to average as 1/

√N

where N is the number of force measurement samples,one achieves a further noise reduction. The final preci-sion we achieved in force measurement was 5.3 mN.

9

D. Acoustic Perturbation

Mechanical perturbation of granular media is usuallyachieved by applying vibrations at the boundaries, butsuch schemes are not spatially homogeneous because thedissipative collisions among particles cause a gradientin the perturbation magnitude as one moves from theboundary into the system interior. Spatially homogenousperturbation being necessary for the current experiments,we stuck four acoustic transducers (SD1G from SolidDrive) to the bottom of the acrylic plate (see Fig. 9a)with an asymmetric placement as shown in Fig. 2 ofmain text. With acoustic energy being transferred tothe acrylic bottom plate, it acted as the speaker andthus provided homogeneous perturbation. Each trans-ducer was powered by an amplifier (SD250 from SolidDrive) connected to an independent function generatorwhich output white noise with a frequency cutoff of 15KHz, thereby providing a reasonable approximation forδ-correlation in time. However, acoustic waves have longcorrelation lengths in the transmitted medium as onetrivially observes in Chaldni patterns; achieving a rea-sonable approximation for δ-correlation in space is there-fore not so straightforward. In a beautiful study, Cadotet al demonstrated [2] nonlinear response when an elas-tic medium is subjected to random acoustic forcing. Inaddition to approximately Gaussian in time acoustic ex-citation, we additionally scrambled the amplitudes of allfour transducers with a fifth function generator that pro-vided band-limited (to 15 KHz) white noise. The fourtransducer amplitudes were scrambled in a manner suchthat their sum was always constant at any given instant.

Following the protocol of Cadot et al, we used laservibrometry (see Fig. 11a for schematic) to measure spa-tial cross-correlations in surface deformations. The cross-correlation function is defined as:

X(r) =〈h(~r1, t)h(~r2, t)〉σh(~r1)σh( ~r2)

(8)

where h(~r1, t) and h(~r2, t) are instantaneous (at time t)height variations at positions ~r1 and ~r2 respectively, andσh(~r1) and σh(~r2) are standard deviations of heights at po-sitions ~r1 and ~r2 respectively, where the height variationh was recorded by high-speed cameras (Phantom v640)that captured laser beams reflected off the acrylic plate.This cross-correlation function X(r) (solid red circles) isplotted versus distance r = |~r1 − ~r2| in mm in log-linearscale in Fig. 11. X(r) exhibits an exponential decay withdecay length of 3.7 mm obtained from fit to data (solidblack line in Fig. 11). This decay length of 3.7 mm beingless than small disk diameter of 1 cm, we treat this asapproximately δ-correlated perturbation in space.

FIG. 11: (Color online) (a) Laser vibrometry schematic: Twolaser beams incident onto the bottom acrylic plate underacoustic excitation at two locations separated by distance r.Height variations caused by acoustic waves cause perturba-tions in the reflected laser beams. These perturbations arecaptured by two high-speed cameras (Phantom v640) at 1500frames per second, and the images are analysed to obtain theinstantaneous spatial cross-correlation X(r). (b) X(r) (solidred circles) versus r in log-linear scale shows exponential de-cay with a decay length of 3.7 mm obtained from data fit(solid black line).

II. DETAILS OF THE NUMERICAL

SIMULATIONS

Conjugate gradient method is generally used to studythe frictionless granular materials [3]; but when the par-ticles have friction, MD simulation is preferred as it cor-rectly keeps track of both normal and history dependenttangential forces. So we employ MD simulation for thecurrent study. Simulation of uniaxial compression of twodimensional granular packings are performed using opensource codes, LAMMPS [4] and LIGGGHTS [5]. To sim-ulate the experimental system the particles are taken asbi-disperses disks of unit mass with diameters 1 and 1.4respectively. The particles are placed randomly in a threedimensional box of dimension, 57 (along x), 102 (alongy) and 1.4 (along z). Quasistatic compression is imple-mented by displacing the boundary particles. A side wallmade of particles is placed in the direction perpendicularto the compression direction.

The contact fores (both the normal and tangentialforces which arises due to friction) are modeled accord-ing to the DEM (discrete element method) developed byCundall and Strack [6]. Implementation of static frictionis done via tracking the elastic part of the shear displace-ment from the time contact was first formed. When thedisks are compressed they interact via both normal andtangential 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 the

10

centers 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 [6]. Defining R−1

ij ≡ R−1i + R−1

j , theforce magnitudes are,

F(n)ij = kn∆ijnij −

γn2vnij

, F(t)ij =−kttij −

γt2vtij(9)

kn = k′

n

√

∆ijRij , kt = k′

t

√

∆ijRij (10)

γn = γ′

n

√

∆ijRij , γt = γ′

t

√

∆ijRij . (11)

Here δij and tij are normal and tangential displacement;

nij is the normal unit vector. k′

n and k′

t are spring stiff-

ness for normal and tangential mode of deformation: γ′

n

and γ′

t are viscoelastic damping constant for normal andtangential deformation. vnij and vtij are respectivelynormal and tangential component of the relative velocitybetween two particles. The relative normal and tangen-tial velocity are given by

vnij= (vij .nij)nij (12)

vtij = vij − vnij− 1

2(ωi + ωj)× rij . (13)

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 rotation

around the contact point is accounted for to ensure thattij always remains in the local tangent plane of the con-tact [7].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 (14)

τ(tot)i = −1

2

∑

j

rij × F

(t)ij . (15)

The tangential force varies linearly with the relativetangential displacement at the contact point as long asthe tangential force does not exceed the limit set by theCoulomb limit

F(t)ij ≤ µF

(n)ij , (16)

where µ is a material dependent coefficient. When thislimit is exceeded the contact slips in a dissipative fash-ion. In our simulations we reset the value of tij so that

F(t)ij = 0.8µF

(n)ij . This choice is somewhat arbitrary, but

recommended on the basis of frictional slip events mea-sured in experiments in the laboratory of J. Fineberg[8]. A global damping is implemented to reach the staticequilibrium in reasonable amount of time. After eachcompression step, a relaxation step is added so that the

n

100

101

102

An

-A∞

10-3

10-2

10-1

FIG. 12: The power law for the decaying areas under the hys-teresis loops as measured in the numerical simulation. Hereµ = 0.3, Black dots are data and the red line is the best fittingpower law with θ = −1.005.

n10 1 10 2

Xn

10 -5

10 -4

10 -3

FIG. 13: Log-log plot of Xn vs. n. Here µ = 0.3, the blackdots are the data, the blue line is the best fitting scaling lawwith the exponents being -1.005.

system reaches the static equilibrium and then the globalstress tensor is measured by taking averages of the dyadicproducts between the contact forces and the branch vec-tor over all the contacts in a given volume,

σαβ =1

V

∑

j 6=i

rαijFαij

2(17)

The pressure is determined from the trace of the stressand it is measured as a function of the packing fraction.After a full compression cycle, the packing is again de-compressed to zero pressure and then again the next com-pression cycle begins. The area between the compression

11

and decompression curve is also calculated as a functionof number of cycles. We also calculate the total energyloss due to sliding events in each cycle. The system sizeis N = 4000 and the data are averaged over ten differ-ent initial configurations. We used two different frictioncoefficient 0.1 and 0.3. In Fig. 12 we plot loop area aftersubtracting it from asymptote area as a function of cy-

cles for friction coefficient of 0.3 and the exponent of thepower law remains the same as that of low friction case(See Fig. 4 main paper). We also plot Xn as a functionof number of cycles for the same friction coefficient inFig. 13 and the measured exponent (See Fig. 5 mainpaper) indicates the universality in the scaling law.

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

[2] O. Cadot, C. Touze, and A. Boudaoud. Phys. Rev. E. 82046211 (2010).

[3] T. G. Mason, M. D. Lacasse, G. S. Grest, D. Levine, J.Bibette, and D. A Weitz, Phys. Rev. E. 56, 3150 (1997).

[4] S. Plimpton, J. Comput. Phys. 117.1, 119 (1995).

[5] C. Kloss, C. Goniva , A. Hager , S. Amberger , and S.Pirker, Prog. Comput. Fluid Dyn. 12, 140152 (2012).

[6] P. Cundall and O. Strack, Geotechnique 29, 4765 (1979).[7] L. E. Silbert, D. Ertas, G. S. Grest, T. C. Halsey, D. Levine

and S. J. Plimpton, Phys. Rev. E. 64, 051302 (2001).[8] J. Fineberg, Private Communication.