PHYSICAL REVIEW E 87, 042205 (2013)

Fragility and hysteretic creep in frictional granular jamming

M. M. Bandi,1,2,* M. K. Rivera,3 F. Krzakala,2,4 and R. E. Ecke2

1MPA-10, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA2T-CNLS, Los Alamos, National Laboratory, Los Alamos, New Mexico 87545, USA

3D-4, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA4CNRS and ESPCI ParisTech, 10 rue Vauquelin, UMR 7083 Gulliver, Paris 75000, France

(Received 15 July 2012; revised manuscript received 11 February 2013; published 18 April 2013)

The granular jamming transition is experimentally investigated in a two-dimensional system of frictional,bidispersed disks subject to quasistatic, uniaxial compression without vibrational disturbances (zero granulartemperature). Three primary results are presented in this experimental study. First, using disks with differentstatic friction coefficients (μ), we experimentally verify numerical results that predict jamming onset atprogressively lower packing fractions with increasing friction. Second, we show that the first compressioncycle measurably differs from subsequent cycles. The first cycle is fragile—a metastable configuration withsimultaneous jammed and unjammed clusters—over a small packing fraction interval (φ1 < φ < φ2) and exhibitssimultaneous exponential rise in pressure and exponential decrease in disk displacements over the same packingfraction interval. This fragile behavior is explained through a percolation mechanism of stressed contacts wherecluster growth exhibits spatial correlation with disk displacements and contributes to recent results emphasizingfragility in frictional jamming. Control experiments show that the fragile state results from the experimentalincompatibility between the requirements for zero friction and zero granular temperature. Measurements withseveral disk materials of varying elastic moduli E and friction coefficients μ show that friction directly controlsthe start of the fragile state but indirectly controls the exponential pressure rise. Finally, under repetitive loading(compression) and unloading (decompression), we find the system exhibits pressure hysteresis, and the criticalpacking fraction φc increases slowly with repetition number. This friction-induced hysteretic creep is interpretedas the granular pack’s evolution from a metastable to an eventual structurally stable configuration. It is shownto depend on the quasistatic step size �φ, which provides the only perturbative mechanism in the experimentalprotocol, and the friction coefficient μ, which acts to stabilize the pack.

DOI: 10.1103/PhysRevE.87.042205 PACS number(s): 45.70.−n

I. INTRODUCTION

The “jamming” framework [1] has been proposed as anoverarching, unifying description governing the behavior ofa wide variety of disordered materials, including glasses,colloids, foams, and granular media via the jamming phasediagram with three axes representing temperature T , shearstress �, and inverse packing fraction 1/φ. Of particularinterest is a point J at zero temperature and zero shear stressalong the inverse packing fraction axis (1/φ) predicted tohave special properties. This point J is the critical packingfraction φc at which a frictionless granular pack undergoes asharp transition from an athermal, particulate gas to a stiff,disordered solid. Numerics [2], mean-field theories [3,4], andexperiments [5] all show that many interesting properties ariseat this transition.

Being amenable to theoretical treatment, several early stud-ies concentrated on idealized systems composed of frictionlessparticles and have contributed to the understanding of granularjamming. A growing body of numerical [6–10] and experimen-tal [11–13] studies, however, conclusively demonstrate thatinterparticle friction substantially alters jamming behavior.Friction plays a significant role in granular materials: notonly is it technologically relevant (e.g., in compression and

*Present address: Collective Interactions Unit, OIST GraduateUniversity, 1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa,Japan 904-0495; bandi@oist.jp

sintering of ceramics [14]), but friction also radically altersthe mechanical behavior of materials (e.g., in sedimentaryrocks [15]). Understanding the role of friction in granularjamming therefore becomes relevant and important.

This article, which contributes to this growing body of re-sults, presents an experimental study of frictional granular jam-ming of a loose, granular pack comprising a two-dimensional,bidisperse system of disks subjected to quasistatic, uniaxialcompression (loading) and decompression (unloading). Thefollowing experimental results and their interpretations arepresented.

(i) By employing disks with different static friction coeffi-cients, we verify numerical predictions [7,16] for the effect offriction on the onset of jamming.

(ii) We show that the pressure scaling differs remarkablybetween the first and subsequent loading cycles. In the firstcycle, as the system’s boundaries are moved in to achievean increasingly tighter packing, a fragile state is observedwhere the pressure exhibits an exponential rise P ∝ eφ/χP

over a range of packing fractions (φ1 < φ < φ2), followedby a deviation from exponential scaling for φ > φ2. Thisexponential rise in pressure is simultaneously reflected inan exponential decrease in particle displacements over thesame range of packing fractions, implying the simultaneousexistence of jammed and unjammed clusters in the evolv-ing granular pack. This fragile state is characterized bydeveloping contact stresses being spatially correlated withdisk displacements. It is shown to arise as a consequenceof nonzero friction which is known to introduce protocol

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M. M. BANDI, M. K. RIVERA, F. KRZAKALA, AND R. E. ECKE PHYSICAL REVIEW E 87, 042205 (2013)

dependence on experimental measurements and is in accordwith prior experimental observations [5,11]. These resultsare interpreted as a percolation of stressed contacts whichexponentially decrease the fractional area enclosed withinstress chains (defined by a threshold stress) over the rangeof packing fractions (φ1 < φ < φ2). The fragile state and itsassociated stress percolation mechanism fall within a broaderset of recent results that show existence of a fragile regime inthe approach to jamming, including contact percolation [17],contact dynamics in granular glasses [13], and shear inducedjamming that causes force network percolation [12].

We note the term fragile here denotes a mechanicallymetastable configuration—it is easily destroyed under theslightest external perturbation such as nonzero granular tem-perature, as we show later in the article. This state should notbe confused with fragility in glass formers as defined by Angell[18] to distinguish between strong and fragile glasses whichexhibit Arrhenius and Vogel-Fulcher behavior, respectively[19]. Although related to glasses within the broader contextof disordered systems, this work does not concern itself withglass phenomenology in particular.

(iii) Under repetitive loading and unloading, we observe thecritical packing fraction φc at which the granular pack jamsprogressively increases to higher values, thereby exhibitingcreep. At the same time, the pressure curves for the loading-unloading cycles exhibit hysteretic responses. Relevant inthe geophysical context [20,21], this hysteretic creep isexperimentally shown to arise from interparticulate contactfriction.

In Sec. II, we present a brief review of frictionless granularjamming and how friction changes the predictions expectedto hold under idealized conditions. In Sec. III, we explain ourexperimental setup and the experimental protocol we followin conducting our measurements. The main results of thisstudy are presented in Sec. IV, followed by discussion andinterpretation of these results in Sec. V, with a brief summaryof results in Sec. VI.

II. BACKGROUND: GRANULAR JAMMING

A. Ideal jamming

The primary motivation for the jamming proposal [1] wasto provide a common framework to describe the nonequi-librium behavior of a wide variety of disordered materials.O’Hern et al. [2] conducted extensive numerical studies withfrictionless particles interacting via soft, finite-range, repulsivepotentials at zero temperature and zero applied shear stress;henceforth, we refer to this set of specifications as idealgranular jamming. They reported many interesting propertiesof the ideal jamming transition around φc that have since beenverified by mean-field theories [3,4] and experiments [5]. Fora finite number of particles N , φc is a configuration-dependentrandom variable, the full width at half maximum of whosedistribution was empirically determined [2] to follow theformula w = w0N

−� (where w0 = 0.16 ± 0.04 and � =0.55 ± 0.03). O’Hern et al. found φc is sharply defined inthe limit of infinite system size (N → ∞), where φc coincideswith random close packed density (φRCP = 0.64 in 3D and 0.84in 2D), a concept first introduced by J. D. Bernal [22–24]

to understand liquids which are structurally disordered byconstruction. We note, however, that difference of opinionexists within the community both with respect to the definitionof random close packing [25] itself, as well as its coincidencewith φc in granular jamming [26,27]. These divergent opinionsnotwithstanding, the existence of a φc where the jammingtransition occurs is not in question.

The behavior of two quantities is of particular interest forthe ideal jamming transition. The pressure P is zero below φc

and rises continuously above φc as a power law [P ∝ (φ −φc)ψ for φ > φc], whereas the coordination number Z = 0below φc and undergoes a discontinuous jump to a criticalvalue Z = Zc at φ = φc, followed by a power-law increaseabove φc, (Z − Zc) ∝ (φ − φc)β . The critical coordinationnumber Zc = 2D (D being the system’s dimensionality) forfrictionless particles, since φc is the system’s isostatic point atwhich the total number of degrees of freedom equal the totalnumber of constraints providing force balance.

The definition of a contact plays a central role here. Agranular contact is said to exist when two particles come inphysical contact and propagate a stress; albeit necessary, astress-free physical contact is deemed insufficient. This re-quirement gives rise to a crucial and, perhaps, little-appreciatedinterpretation of the jamming transition. The discontinuousjump in Z from 0 to Zc at φc implies that there are no contactsin the system up to φc—it may lose floppy modes and becomerigid, but it is not stressed and, therefore, has zero granularcontacts. At φc, a pressure rise above zero simultaneouslyactivates contacts system wide due to stress propagation,and Z discontinuously jumps from 0 to Zc. Because of theadditional stress requirement in contact definition, granularjamming ideas make no allowance for a system to lie inan intermediate regime where part of the system is jammedand the rest is not—the only two states permitted are totalsystemwide jamming or lack thereof.

B. Frictional jamming

An increasing number of studies [6–9,11–13,16,28] haveexplored the role of friction and how it changes the idealjamming predictions. Although a general framework forfrictional packs has yet to emerge, studies indicate deviationsfrom ideal jamming predictions. First, in addition to the normalcomponent of force (FN ), friction gives rise to a tangentialcomponent (FT ). Second, the preparation method and historyof the pack become important for frictional packs [28].Treating the tangential force component as a new independentdegree of freedom (at least for low friction coefficients) sets thecritical coordination number at φc to lie in the range D + 1 �Zc � 2D, i.e., hypostatic configurations are permissible. Thecondition Zc < 2D implies that frictional packs can jam atφc < φRCP, with φc progressively decreasing with increasingμ before asymptoting to a constant value that has beentermed random loose packing density φRLP [16]. φRLP is anempirically determined value from numerical simulations; itstheoretical underpinnings are not well understood [10]. Thesame arguments made against the definition of φRCP [25] applyto φRLP.

Recent experiments in frictional systems also demonstratea percolation mechanism in stress [12] en route to jamming

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as well as in the steady-state contact dynamics [13] within thejammed regime. An experimental study by Cheng et al. [11]has also reported jamming of particles into a metastable config-uration due to interparticle friction, which is readily destroyedunder external perturbation or, alternatively, application ofnonzero granular temperature.

III. EXPERIMENT

A. Description of the experimental setup

Figure 1(a) shows a schematic of the experimental setup.The system consists of a bidisperse mixture of 950 large(diameter dL = 0.9525 ± 0.0025 cm) and 950 small (diameterdS = 0.635 ± 0.0025 cm) disks of thickness 0.508 cm. Thedisks are placed in a chamber, with dimensions L = 25.9 cmand W = 48.2 cm (L is the compression direction), consistingof two glass plates held 0.635 cm apart by means of an acrylicframe that runs along the system’s perimeter. Two movableboundaries are placed on the acrylic frame with aluminumplates that can slide back and forth within the chamber fromopposite ends. The transverse boundaries are held fixed. Thepositions of the movable boundaries are controlled by twomicrometers with a precision of 0.001 cm. Taking variationsin radii into account for the given system of 1900 disks, thistranslates to a precision �φ = 1 × 10−5 in the packing fractionand, hence, serves as the lower bound on the quasistatic stepsize (�φ). All measurements reported in this article, however,were made at a quasistatic step size of �φ = 1 × 10−4,3.5 × 10−4, or 7 × 10−4. The packing fraction (φ) is defined asthe ratio of the area occupied by the disks to the total chamberarea. The packing fraction is, therefore, controlled by changingthe chamber area in this experiment. A set of six sensors[labeled A through F in Fig. 1(a)] placed along the boundariesmeasure the global two-dimensional pressure (N/m). Visualmeasurements using a Nikon D-90 camera (12.3 megapixelresolution) yield positions and displacements of individualdisks.

Measurements were performed with disks machined fromdifferent materials spanning a range of experimentally de-termined static friction coefficients. The majority of themeasurements were conducted with polymer disks that ex-hibit stress birefringence or photoelastic response (PE, staticfriction coefficient μ = 0.19, elastic modulus E = 2.5 GPa).Measurements were also performed with photoelastic diskslubricated with graphite dust (PEG, μ = 0.14, E = 2.5 GPa),photoelastic disks soaked in ethanol for 24 h which changed themodulus (PEA, μ = 0.19, E = 0.004 GPa), lexan polycarbon-ate disks intentionally machined rough to obtain a high frictioncoefficient (LEX, μ = 0.22, E = 2 GPa), and Teflon diskswith an intrinsically low friction coefficient (TEF, μ = 0.06,E = 0.5 GPa).

B. Measurement of static friction coefficient μ

A schematic of the apparatus used to measure the staticfriction coefficient is shown in Fig. 1(c). Four disks (of thesame material) of diameter dL = 0.9525 cm are arranged asshown in Fig. 1(c). The upper disk and the two bottom disks areheld fixed and not allowed to rotate. A mass M is suspended

FIG. 1. (a) Experimental schematic: The system consists of 950large and 950 small disks (ratio of radii 1:1.5). Two movableboundaries at opposite ends control the system’s packing fractionφ and are used to provide uniaxial, quasistatic compression. Forcesensors labeled A through F in the schematic measure the boundarypressure. The image is contrast enhanced data for φ = 0.8113.(b) Side view schematic of force sensor placement within thecompression boundary. When the micrometer (left) pushes on theboundary plate to its right, the force sensor registers the cumulativeforce of the boundary plate and granular pack. (c) Schematic ofthe static friction coefficient measurement apparatus. Four disks ofdiameter dL = 0.9525 cm are placed in contact with each other asshown in the schematic. The three outer disks are held fixed andhave no translational or rotational degrees of freedom. The middledisk can be rotated by means of an external lever. A normal forceFN = MNg is applied by a suspended weight of mass MN on the topdisk. This force is transferred to the disk sandwiched in the middle,which in turn transfers it to the two bottom disks at an angle of 30◦.A tangential force FT = MT g applied on the middle disk allows itto slip at a critical force, which allows determination of the staticfriction coefficient μ for the disks.

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TABLE I. Experimentally determined values of μ for indicatedmaterials (abbreviations defined in text) and their elastic modulii E

(supplied by manufacturer).

Material Friction Coefficient μ Modulus E (GPa)

PE 0.19 2.5PEG 0.14 2.5PEA 0.19 0.004LEX 0.22 2.0TEF 0.06 0.5

from the upper disk such that a force Fa = Mg is appliedvertically down by the upper disk onto the central disk. Atangential force Fm = mg is applied on the central disk (freeto rotate) via a pulley system where a mass m is attached tothe pulley with diameter dp = 1.887 cm. Mass is added tom until slip events occur. In practice, the weights are placedin a receptacle of mass 29 g, hence Fm = (m + 29)g. Dueto the torque ratio, the actual tangential force applied on thecentral disk is FT = Fmdp/d. The normal force on the centraldisk arises from the three (one top and two bottom) disksin contact with it. The vertical disks apply a normal forceFNu = Fa , whereas the bottom disks must balance this verticalforce. Hence, Fa is equally divided between the two bottomdisks giving Fa = 2FNb cos θ . The total normal force is FNu +2FNb = Fa + Fa/ cos θ = Fa(1 + 1/ cos θ ).

The ratio of the total normal force applied to the totaltangential force required for a slip to occur in the central diskprovides a measure of the static friction coefficient μ. Pluggingin values for dP /dL = 1.98 and (1 + 1/ cos θ ) = 2.15 withθ = 30 ◦, we get

μ = FT

FN

= FmdP /d

Fa(1 + 1/ cos θ )= 1.98Fm

2.15Fa

= 0.92Fm

Fa

. (1)

Values of the static friction coefficients for each of the materialsused in this study are listed in Table I.

C. Experimental protocol

We initially place all disks in the interrogation chamberin random positions and move the boundary plates to aninitial packing fraction chosen a priori to fall comfortablyin an unjammed state. The disks are subject to friction (withthe glass bottom, the boundary and with each other) andthe system jams below the random close packed densityφRCP ≈ 0.84. After each quasistatic step, a 10-s time-trace ofall six boundary sensors is collected at a sampling frequencyof 1 KHz followed by a digital image of the whole system.The time-averaged value of the trace constitutes the pressuremeasured by the boundary sensor at a particular value ofφ. The boundaries are then moved through a quasistaticstep (�φ), and the procedure is repeated. The only controlparameter in this experiment is the quasistatic step �φ. Noother external excitation or perturbation is applied. As aconsequence, this experiment studies the pack evolution at“zero granular temperature.” Unlike ideal jamming, real-worldexperiments have to contend with friction. Usually, frictionaleffects are circumvented by applying a granular temperature

via acoustic excitation or vibration of boundaries that relaxfrictional stresses. In the absence of any such mechanismin this experiment, frictional effects become fully manifest.The ideal jamming requirements of zero temperature and zerofriction are, experimentally, mutually incompatible. Meetingthe zero friction requirement violates the zero temperaturerequirement and vice versa. Second, we do not tap the systemafter each quasistatic step to mimic annealing in simulations[5]. Tapping the system in experiments (or annealing insimulations via gradient minimization or alternative methods),no matter for how short a duration or how weak in amplitude, istantamount to application of an effective granular temperaturewhich violates the zero temperature requirement. In order tounderstand the role of granular temperature, we performedone experimental run, discussed below, where the system issubjected to gentle (but systematically uncontrolled) tappingafter each quasistatic step.

D. Experimental systematics

The noise floor (instrumental noise) of the boundarypressure sensors is about 0.1 N/m. Accordingly, sensorsC-F [see Fig. 1(a)] placed along the immovable transversedirection register 0.0 ± 0.1 N/m pressure when the systemis unjammed. On the other hand, sensors A and B [seeFig. 1(a)] placed along the movable, compression directionregister an offset of about 11 N/m. As shown in Fig. 1(b),the compression axis sensors were placed within the movableboundary plate. As per the experimental protocol, the initialboundaries were set to a predetermined packing fraction thatlies well below the jamming threshold. At this point themicrometers were detached from the boundary, and a baselinepressure trace was recorded to measure the instrumental noise.This was considered prudent, despite the sensors being drivenby homebuilt voltage regulator circuits designed for stabilityagainst thermal drift. Furthermore, the 10-s pressure time-tracerecorded at 1 KHz averages the instrumental noise. Like thetransverse axis sensors (C–F), the compression axis sensors (Aand B) also register 0.0 ± 0.1 N/m pressure reading when themicrometers are detached from the boundary plates. When,however, a micrometer comes in contact with the boundaryplate and pushes against it, a pressure offset of 11 N/m isregistered. This pressure jump for compression axis pressurecan be seen in the inset of Fig. 4(a). When the micrometersare advanced through a quasistatic step �φ, they directly pushagainst a stainless steel plug that protects the sensor housed ina groove within the boundary plate. The sensor in turn pushesagainst the boundary plate. As a consequence, the boundaryplate’s friction with the bottom glass plate gives rise to the11-N/m offset.

In the data presented in the following [in particular, pleasesee the inset of Fig. 4(a)], a 11-N/m offset was added tothe pressure signals for immovable transverse axis sensors torender them co-incident with the compression axis pressurescaling. Offset addition to transverse axis pressure waspreferred over offset subtraction of compression axis signalsfor a practical reason. Offset subtraction of compression signalcauses some fluctuations to go negative, causing chopping ofnegative fluctuations when the data is plotted on a log-linearscale [Fig. 4(b)].

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In order to ascertain the magnitude of this systematicerror in measured compression pressure, we modified thearrangement of pressure sensors. In particular, rather thanmeasure the pressure at a position between the compressionmicrometer and the movable boundary, two sensors (asopposed to one) at the two ends of the movable plate wereplaced near a very small and light frame located near the disks.This did reduce the offset to 0.25 ± 0.01 N/m, but the need fora small and light frame also means it bends at the center dueto the slight, unavoidable bulge of the sensors required so theyalone make contact with the frame. This creates a differentsystematic error in the experiment. The results obtained werevery similar in this case, but we place more confidence in themore extensive measurements made with the first design.

As the compression boundary moves in and mobilizes anincreasing fraction of disks, the friction between disks andthe bottom glass plate is registered through a linear ramp ofsmall, but measurable, slope in pressure. A portion of this rampis discernible in the prejamming threshold data presented inFig. 4(b). The presence of a top glass confining plate wasalso considered. The number and variety of runs with thismodified apparatus were small compared to those taken withthe primary apparatus whose results we discuss here. Thegeneral characteristics of the results are the same for bothexperiments.

Although photoelastic disks were employed to discernthe spatial stress distribution, the photoelastic threshold wasrather high due to high stiffness of the material used. As aresult, photoelastic stress signals were not visible well into therise in the system’s global pressure measured with boundarysensors. Due to this design property, we were unable to providemeasurements of the coordination number Z in this study.

IV. RESULTS

A. First loading cycle: Jamming onset in presence of friction

Here we discuss the role of friction on the onset of jammingwhen the pack is compressed the first time from a randominitial configuration. Silbert et al. have shown [7,16] that thecritical coordination number Zc as well as the critical packingfraction φc at which jamming occurs shift to progressivelylower values with increasing friction coefficient, due to theadditional structural stability provided by the tangential forces.Analysis for jamming [7] and for the unjamming transition [16]reached similar conclusions. In Fig. 2 we plot the pressure P

versus packing fraction φ for disks with different friction coef-ficients (Z is not measured for reasons discussed in Sec. III D).Treating commencement of pressure rise as the jamming onsetcondition, we confirm the numerical prediction of Silbert et al.that jamming onset does occur at lower packing fractions withincreasing friction. Whereas jamming onset also representsthe jamming transition in ideal jamming, the same is not truein this experiment. Here the term jamming onset marks thenucleation of the first jammed cluster within the system. Unlikeideal jamming, which is marked by an abrupt transition, in thisexperiment the jamming transition proceeds smoothly througha stress percolation mechanism, as we discuss in Sec. IV B.

The range of friction coefficients 0.06 � μ � 0.22 fordisks employed in this study yields friction-dependent values

0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85Packing Fraction φ

0

50

100

150

200

250

300

Pre

ssur

e (N

/m)

PEPEAPEGLEXTEF

FIG. 2. (Color online) P vs φ (quasistatic step size �φ =1 × 10−4). The packing fraction φ at which system pressure startsincreasing monotonically falls with increasing static friction coeffi-cient μ. The change in pressure slopes arises from different elasticmoduli of the materials used.

of the critical packing fraction φc that are greater thanthe asymptotic steady value corresponding to random loosepacking (φRLP); one would need higher μ to reach RLPconditions. Nevertheless, the experimental values of φc wemeasure for the experimental values of μ are close to thoseobtained in numerical simulations of Silbert et al. (see Fig. 1and Table 1 of Ref. [16]). Since we use different materials withvarying elastic moduli, the slopes of the pressure curves varybetween the materials. As a counterpoint, a quick comparisonof PE and PEG data shows that the two plots have the sameslope since they have the same modulus. Since PEA disks aresofter, however, they have a shallower slope in comparisonto PE data, but the pressure rise does approximately coincidefor both PE and PEA data since they share the same frictioncoefficient. The slight difference in the φ value of pressure risestart between PE and PEA can be attributed to configuration-dependent fluctuations that naturally arise among differentexperimental runs.

In Fig. 3, we show the variation of the critical packingfraction φc as a function of the disk friction coefficient μ. Inparticular, the quantity 1 − φc/φRCP, the fractional deviationfrom the expected zero friction random close packed valueφRCP, increases consistently with a linear dependence on μ,again agreeing with numerical simulations [16].

B. First loading cycle: Fragile behavior

We next consider the shape of the pressure curve itself bycomparing the pressure signal for the first and second jammingcycles. Figure 4(a) plots P from one compression boundarysensor versus φ for the first and second jamming cycles forphotoelastic disks (PE). The continuous increase in P for thefirst cycle differs qualitatively from the more abrupt change inslope for the second cycle. The lateral shift in P is a signatureof a friction-induced hysteretic response which progressivelyshifts φc to higher values as the system is repeatedly jammedand unjammed (discussed below in Sec. IV C). The vertical

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0 0.05 0.1 0.15 0.2 0.25μ

0.00

0.02

0.04

0.06

0.08

0.101-

φ c/φR

CP

FIG. 3. (Color online) Deviation of fractional φc with respectto φRCP vs μ for materials described in Fig. 2. The error bars areestimates as opposed to statistical averages over many runs.

shift in P within the flat (unjammed) regime for the first cycle[black circles in Fig. 4(a)] is traced to the friction between themovable boundary plates plus the mobilized fraction of disksand the glass bottom as explained in Sec. III B. At the end ofthe first loading cycle, when the system is decompressed, theboundary plates only move back until stresses in the pack arerelieved. Further quasistatic reverse stepping of micrometersdoes not cause continued backward motion of boundary platesto their initial positions. Instead, the micrometers decouple(loss of physical contact) from the boundary plates. For thisreason, the unjammed regime observed during the secondcompression cycle [red squares in Fig. 4(a)] does not showthe vertical shift in P . During the first cycle, the boundariesmove in and push the disks towards a jammed configuration.When unjammed and jammed again, the contacts that weredeveloped at the end of the first jamming cycle are reactivated,and the stresses build up as the system is subjected again touniaxial compression.

To better understand the smooth increase in P for the firstjamming cycle, Fig. 4(b) plots P versus φ on a log-linearscale. One sees three distinct regimes in the pressure curve.In the unjammed (consolidation regime) P is essentially flat,modulo a shallow ramp due to friction between disks andbottom glass plate. The second regime is characterized byan exponential increase in P beginning at φ1 ≈ 0.8093. Thesolid line in Fig. 4(b) is an exponential fit to the data P ∝ eφ/χP

with χP = 0.00195. Note that although the range of φ for theexponential regime is small, the increase in P over that intervalis almost one decade. The pressure eventually deviates fromthis exponential regime and settles to a regime that seemsto fit well with linear scaling [dashed line in Fig. 4(b)] P ∝(φ − φ2). Due to a limited range of pressure data for φ > φ2, itis not clear whether this regime scales linearly or algebraically(P ∝ φψ ) with the exponent ψ being marginally greater than 1.The inset in Fig. 4(a) plots P from all four boundaries against φon a linear scale. The remarkably good collapse of the pressurecurves atop one another strongly suggests that anisotropiceffects arising from uniaxial compression are not detected

FIG. 4. (Color online) (a) P vs φ (linear scale) for the first(black circles) and second (red squares) compression cycles. Thepressure scaling is gradual for the first cycle as compared to themore abrupt transition during the second cycle where φc is indicated.The inset plots the pressure from all four boundaries (same verticaland horizontal scale). No anisotropy is observed in the pressuresignal. (b) P vs φ for the first jamming cycle (log-linear scale)shows the existence of an intermediate regime where pressure scalesexponentially. The solid line is an exponential fit to the data P ∝ eφ/χP

with χP = 0.00195, and the dashed line is a linear fit. The values ofφ1 and φ2 are indicated in the plot.

by the pressure sensors. This isotropy may result from ourgeometry for which the aspect ratio is L/W = 0.54 and wouldprobably not persist for large aspect ratios, i.e., L/W � 1.

Although this exponential pressure scaling for the firstloading cycle seems to contradict the predicted [2,3,5] powerlaw for P across the jamming transition, as we will discussbelow, it can be explained through a stress percolationmechanism. The predicted power-law scaling (approximatelylinear1) is, however, recovered for subsequent jamming cycles

1The best fit of the data to the form P ∝ (φ − φc)β yields β = 1.1,consistent with Ref. [5].

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0.1

0.2

0.3

0.4

0.5N

m/N

T

0.807 0.808 0.809 0.810 0.811 0.812 0.813

φ0

0.01

0.02

0.03

0.04

0.05

σ/d~

0.0

(c)

φ1

φ1

φ2

(b)

(a)

FIG. 5. (Color online) (a) Fraction of disks moving Nm/NT vsφ showing a constant region for φ < φ1 and exponential decay forthe fragile regime (φ1 < φ < φ2) with χN ≈ 0.001. (b) Normalizeddisplacement variance σ/d relative to a state near φ2 showing a lineardecrease for φ < φ1 and an exponential decrease with χσ = 0.0007for the fragile regime (φ1 < φ < φ2). (c) Image of the superpositionof the difference in the stress network (black lines) and the magnitudeof disk displacements between φ = 0.8095 and φ = 0.8105. Arrowsindicate φ1 and φ2.

as shown for the second jamming cycle in Fig. 4(a), whereφc = 0.8118 is determined by the point at which P starts torise.

We now look at the behavior of disk displacements.Figure 5(a) shows the fraction of disks moving a distancegreater than about 1% of a mean disk diameter2 as a functionof φ where Nm is the number moving and NT = 1900 is thetotal number of disks. The fraction is constant at about 0.3up to φ1 = 0.8094, after which it decreases rapidly up tothe jamming value φ2 = 0.8124. The decrease is consistentwith an exponential e−φ/χN with χN ≈ 0.001. The varianceof individual disk displacements σ [normalized by d =(dL + dS)/2 = 0.794 cm] relative to a state near the jamming

2There are smaller displacements below our threshold that maybehave differently as a function of φ.

threshold3 is shown in Fig. 5(b) as a function of φ. One againobserves two distinct regimes: a linear one that correspondsto the unjammed (consolidation) regime where the pressurecurve in Fig. 4(b) is flat and a second one in which the variancedecreases exponentially with σ ∝ e−φ/χσ with χσ = 0.0007.This exponential drop in displacement variance occurs over thesame interval in φ as the exponential regime of the pressurecurve in Fig. 4(b) as indicated by the arrows depicting φ1 andφ2. The presence of disk displacements in the exponentialregime, where a percolating force network already exists,suggests that the particles that are part of the force networkstill undergo small displacements and deformations, which inturn allows visible displacement inside the network region,eventually leading to the refining of the network.

The above picture is reinforced by the data in the insetFig. 5(c), where the difference in the force chain networkbetween φ = 0.8095 and φ = 0.8105 is shown as black linesand the spatial distribution of the magnitude of particledisplacements is shown. The correlation of new stress chaincreation (the differences) and the particle displacements, albeitstriking, is not quantitative. Qualitatively, the formation of anew stress chain is not spatially correlated with disk displace-ment magnitudes but rather with the presence or absence ofdisplacements. We followed the simple procedure describedbelow to quantify this spatial correlation. For each new stresscontact formed between two disks during a quasistatic step, weconsidered the center of the line joining the two disk centers.An annulus of inner radius r and outer radius (r + �r) wasconsidered [�r = d = (dL + dS)/2, the average diameter ofone small and one large disk]. Please see the schematic insetin Fig. 6. The ratio of the number of displaced disk centers Nd

to the total number of disk centers counted within the annulusNc was measured for increasing distances of r in multiplesof d . The procedure was repeated for each new stress chainformed during a quasistatic step at several radial distances r

and averaged over.Figure 6 plots the average ratio 〈Nd/Nc〉 versus the

dimensionless radial distance (r + �r)/d. The bar graphrepresents the value within the annular region of width �r

for increasing inner radial distance r (in multiples of d). Theintegrated value of this quantity is plotted in (red) solid circlesrepresenting the cumulative fraction 〈Nd/Nc〉 for the circleof radius (r + �r). The value at r/d = 1 is biased because itcan only include small disk centers (d < d) within that radialdistance. A little over 40% of the disks are mobilized within(r + �r) = 3d from the center of the newly formed stresschain. 〈Nd/Nc〉 abruptly falls for (r + �r) > 3d , implying amajority of disk displacements occur within the vicinity ofa newly formed stress chain. This boundary is denoted by avertical dotted line in Fig. 6; this dotted vertical line is merelya visual indicator. The ratio 〈Nd/Nc〉 = 0 for (r + �r) > 3d

for two reasons. First, there are displacements in unjammedregions of the granular pack where no new stress chainsform. Second, these displacements do contribute indirectly to

3Picking different reference states near or beyond jamming causesa finite offset in variance of different magnitudes, but the exponentialdecay is preserved with the same value of χ = 0.0007 to withinexperimental precision.

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FIG. 6. (Color online) Bar graph shows the number of displaceddisk centers Nd normalized by total number of disks Nc countedwithin an annulus of width �r at a radial distance r from the centerof a newly formed stress chain (see schematic in the inset) vs thedimensionless radial distance (r + �r)/d from the center of a newlyformed stress chain [d = (dL + dS)/2 is the average diameter of thelarge and small disk]. The solid circles (red) present the cumulativefraction 〈Nd/Nc〉 within the radius r + �r . The vertical dotted lineis a marker indicating the radial distance beyond which an abrupt fallin 〈Nd/Nc〉 is observed.

new stress chain formation at distances (r + �r) > 3d in thesense that they communicate displacements from the movingboundary to the location where the stress chain forms.

To further study the nature of the different regimes observedin global pressure measurements, we consider local propertiesof disk configurations, namely the structure of stress chains.The identified stress chains enclose domains with no measuredstress as illustrated in the insets of Fig. 7. We characterize thestress chain networks by the mean am and variance av of thefractional domain area (relative to the total cell area on the leftaxis) and also by the mean am and variance av of the domainarea but now normalized (right axis) by the average of the areaenclosed by a triangular arrangement of small and large disks(all large, all small, two large, two small). As indicated in theinset images, the mean and variance decrease rapidly between0.810 (the lowest value of φ for which we could determinethe stress chain network) and φ2 = 0.8124. For higher φ >

0.8124, the mean and variance decrease linearly with a smallslope. Fits to an exponentially decreasing function e−φ/χs asshown in Fig. 5 yields the same value of χs = 0.00035. Theexponential decrease in am and av is another signature ofthe fragile jammed state. The transition from exponential tolinear behavior occurs over a short range of packing fractionshighlighted in grayscale in Fig. 7. The end of this transitionregime coincides with φ2 = 0.8124. Solid line fits of the linearregime are included to highlight the deviation from transitionregime to linear behavior at φ2 (please see Fig. 7).

As a control, we also conducted one experimental runwhere the system was subjected to gentle (but systematicallyuncontrolled) tapping after each quasistatic step during thefirst compression cycle for multiple reasons. First, we wantedto verify that our granular system reproduced the results of

FIG. 7. (Color online) Log-linear plot of the mean (•, blacksolid circles in lower curve) and variance (◦, red circles in uppercurve) of stress chain domain area normalized by total area, am,av (left axis) or by the mean area formed by connections betweendifferent combinations of three disks in a close packed triangulararray, am, av (right axis), respectively. Insets show stress networksand corresponding domains for φ = 0.8104 (left) and φ = 0.8148(right). The dashed (lower curve) and dotted (upper curve) lines arefits to a combined linear dependence with a decaying exponential (seemain text) for am and av , respectively. The grayscale region highlightsthe transition from exponential to linear behavior under compression.The solid red and black lines are markers included to indicate the endof transition from exponential to linear behavior near φ2 = 0.8124.

Majmudar et al. [5] since the essential point of departurebetween the two measurements lies in the protocol—namelytapping the system to mimic annealing. Accordingly, werecovered their result for the jamming transition (see Fig. 8)although the corresponding φc is still below φRCP = 0.84. Weattribute the discrepancy to two possibilities. First, we do not

FIG. 8. P vs φ for first compression cycle where the system isgently tapped (albeit with no systematic control) after each quasistaticstep.

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systematically control the tapping process. We merely tappedthe system along the four boundaries gently with a malletafter each quasistatic cycle. Hence, the plot in Fig. 8 onlyserves as qualitative verification. Second, the discrepancy canbe attributed to our system size, N = 1900 disks. Based on thenumerical studies of O’Hern et al. [2], where φc was shown tobe a configuration dependent random variable for finite systemsize, it is possible that the control experiment we performedhad a configuration that jammed marginally below φRCP.Finally, as noted in Sec. II A, the location of φc relative to φRCP

not withstanding, the scaling properties about φc are robust.The control run for the first loading cycle with tapping is

an excellent example of how the presence of friction makespack evolution strongly protocol dependent: In the absence ofexternal perturbations the pack exhibits exponential scaling,but with tapping it exhibits power-law scaling. In addition, thiscontrol run also points to another subtle feature of the jammingparadigm. The ideal jamming predictions are predicated onthe requirements of zero friction, zero temperature, and zeroapplied stress. The results observed with and without tappingshow us zero friction and zero temperature are incompatiblerequirements in the real world where one cannot escapefrictional effects, save in strong exceptions like jamming infoam. One then begs to ask, if tapping [5] is consideredequivalent to annealing processes invoked in simulations, isthe zero temperature requirement being adhered to? Also, ifone interprets tapping as a thermal kick (rather than a constantthermal agitation), it is tantamount to destroying the system’sevolution history after each quasistatic step. Then are the idealjamming predictions a result of a protocol that renders the packmemoryless? Whereas the hysteresis results (a memory effect)to follow in Sec. IV C, suggest tapping destroys pack memoryin our system, we note that tapping could presumably alsodestroy a developing heterogeneous configuration where thegranular solid and liquid phases coexist. It is not clear from ourexperiments whether tapping destroys memory, homogenizesan evolving heterogeneous configuration, or both. We believethese are important issues that merit further work, sincethermalization can act both to render a system memoryless asin the present instance or help to retain system memory [29].

Finally, we present a result that bridges the fragile behaviorjust discussed with the third primary result of this study,namely friction-induced hysteresis and creep. In Fig. 9, weplot the pressure from the compression boundary sensoragainst packing fraction for photoelastic (PE) disks (μ = 0.19and E = 2.5 GPa) for three different quasistatic step sizes,�φ = 1 × 10−4, 3.5 × 10−4, and 7 × 10−4. The data belongto the first compression cycle as evidenced by the smoothpressure increase expected for the fragile regime. Figures 4, 9,and 10 present data from different experimental runs of thefirst compression cycle to underscore the robustness of fragilebehavior we have argued for in this subsection.

The same experimental protocol, as detailed in Sec. III Cwas followed to obtain the result plotted in Fig. 9. In particular,we emphasize that no tapping was employed (as done inthe control run described immediately above) after eachquasistatic step. All three curves follow the same slope forthe exponential χP ∼ 0.00195. These experimental runs wererepeated with varying wait times between quasistatic steps toexplore the possibility of interstep annealing; no measurable

FIG. 9. (Color online) Pressure P from the compression sensorfor the first compression cycle versus φ for PE disks (μ = 0.19)jams at higher packing fractions with increasing step magnitude ofquasistatic compression �φ = 1 × 10−4, 3.5 × 10−4, and 7 × 10−4.

change in signal was observed. Due to the strongly athermalnature of the system in question, the system state can only bechanged through one of two mechanisms: perturbations thatintroduce a granular temperature such as the brief thermal kickprovided by tapping or the quasistatic step �φ. The result inFig. 9 falls in the latter category.

The pressure curves in Fig. 9 exhibit a monotonic increasein the jamming onset as the corresponding control parameter�φ is increased, thereby suggesting the absence of a uniquepacking fraction at which a frictional pack jams into anirregular solid. As noted in Secs, II A and II B, the notion ofa uniquely defined critical packing fraction for jamming onset(namely random close packing φRCP and random loose packingφRLP, respectively) is not universally accepted. Evidence to thiseffect comes from several studies for both frictionless [26] andfrictional [28] systems that demonstrate a continuous range of

0.806 0.808 0.81 0.812 0.814Packing Fraction φ

0

50

100

150

200

250

300

Pre

ssur

e (N

/m)

Loading CycleUnloading Cycle

FIG. 10. (Color online) P vs φ (equivalent to stress-strainmeasurement) for PE disks at �φ = 1 × 10−4 exhibits hysteresis.

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packing fractions over which jamming occurs. Therefore, theresult we present in Fig. 9 is neither new, nor surprising initself; yet a crucial distinction exists, as we explain in thefollowing.

Of several sources that induce fluctuations in a granularsystem, including experimental and thermal noise protocols,configurations, annealing protocols, particle geometry andpolydispersity, and friction coefficient, the sources relevantto this study are the experimental protocol, configurationdependent fluctuations, and friction coefficient. Unlike sim-ulations, it is difficult for an experiment to cycle through alarge set of configurations to seek out the average thresholdpacking fraction for jamming onset. For the five independentexperimental runs presented in Fig. 9, all runs exhibited themonotonic dependence on �φ. The configuration-dependentfluctuations in jamming onset for a given run at a specified �φ

were never large enough to encroach on the jamming onset fora different �φ. Whereas we are unable to entirely discountthe possibility that this effect arises from configurationalfluctuations, our tests suggest this may not be the case. Thiseffect is clearly protocol dependent. In addition to the tappingmechanism which changes the nature of the fragile regime, thedependence of jamming threshold on �φ suggests a new typeof protocol dependence.

In the thermal counterpart of jamming in liquids, namely theglass transition, the dependence on cooling rate of the approachto the transition is well known [30,31]. Further, in simulationsof frictional jamming, Inagaki et al. [28] established thedependence of jamming onset φc on the annealing rate. Allgranular and glass studies, however, show that the approach torespective φc or Tg is delayed as the cooling rate is slowed. Thisis in contrast with our observation that increasing �φ leads tojamming onset at higher packing fractions. In analogy with thecooling rate in liquids, if an analogous quasistatic rate �φ

m(m

is the number of quasistatic steps) were considered, it remainsconstant across all three runs in Fig. 9; doubling �φ halves thenumber of experimental points, keeping their ratio constant.Further, we do not apply the experimental equivalent of anannealing protocol (interstep tapping) in these experiments.As we explain in the following subsection on friction-inducedhysteretic creep, this result finds its origin in friction. Thisresult does merit the question whether φ1 → constant (modulofluctuations from other sources) as �φ → 0. This question isbeyond our experimental purview and is best explored throughnumerical simulations which offer access to infinitesimal stepsizes.

C. Repetitive loading: Friction-induced hysteresis and creep

Following the first loading cycle, we now turn our attentionto the role of friction under repetitive loading and unloading.Here we report two frictional effects that strongly deviatefrom current jamming predictions. When the granular packis subjected to loading and unloading, hysteresis is observedin the pressure versus packing fraction plot (see Fig. 10). Ifsubjected to repetitive loading-unloading cycles, the systemexhibits creep, whereby the packing fraction at which thesystem jams progressively shifts and the hysteresis curvesevolve towards higher packing fractions. Friction-inducedhysteresis [32], as well as the rate-dependent behavior of φc

0 1 2 3 4 5 6 7 8 9 10

Hysteresis Cycle n0.000

0.001

0.002

0.003

0.004

φ cn - φ

c0

3.75x10-3

(1-e(-n/3.8)

)

2.8x10-3

(1-e(-n/4.1)

)

2.2x10-3

(1-e(-n/4.3)

)

Δφ = 7x10-4

Δφ = 3.5x10-4

Δφ = 1x10-4

0 1 2 3 4 5 6 7 8 9 10

Hysteresis cycle n0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

φ cn - φ

c0

10.75x10-3

(1-e(-n/1.85)

)

7.5x10-3

(1-e(-n/2.2)

)

6x10-3

(1-e(-n/2.2)

)

3.75x10-3

(1-e(-n/4.2)

)TEFPEGPELEX

FIG. 11. (Color online) Top: Difference between φnc and its initial

onset value φ0c as a function of the number of cycles of loading and

unloading n for different materials. Bottom: Same quantities for thePE material and for different step sizes �φ. Materials and step sizesare labeled in the plot legends as are the coefficients in the solid-lineexponential fits.

[28], have been reported recently. Because frictional jammingis heavily dependent on preparation protocol, we are unableto offer a comparison between our study and prior works.Nevertheless, since our experimental setup design shares closecorrespondence with standard mechanical load cell designs, itallows us to compare our results with relevant amorphoussolids in the geophysical context (e.g., certain forms ofsandstones and sedimentary rocks).

In Fig. 11 we plot the difference in φc between the nth (φnc )

and 0th (φ0c ) loading-unloading cycles against the cycle number

n. In the top panel we vary the friction coefficient μ whilekeeping the quasistatic step-size constant at �φ = 1 × 10−4,whereas in the bottom panel we keep the friction coefficientconstant at μ = 0.19 for PE disks, while varying the quasistaticstep size �φ. The rate at which the pack evolves fromφc to φRCP exhibits monotonic dependence on the frictioncoefficient with the pack evolution being quickest for TEFwith lowest friction coefficient, followed by PEG, PE, and,finally, LEX with the highest friction coefficient. Also, asshown in the bottom panel in Fig. 11, the quasistatic step-size

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�φ, which controls the magnitude of perturbation provided tothe pack also controls the pack evolution rate monotonicallyat step sizes �φ = 1 × 10−4, 3.5 × 10−4, and 7 × 10−4,thereby exhibiting rate dependence in the pack evolution ina quasistatic sense. This rate dependence in hysteretic creepdoes not imply a reduction in width of the hysteresis loop withincreasing cycle n. The hysteresis curves for all cycles, andfor all quasistatic step sizes �φ, exhibit the same area withintheir loops for a given friction coefficient—this area within thehysteresis loop is the energy dissipated due to friction. Instead,this rate dependence implies, the lateral shift in the hysteresisloop decreases with increasing hysteresis cycle number n.Furthermore, once the granular pack eventually stabilizes andno further creep is measurable, the hysteresis loop tracesover itself, while maintaining the same area (and thereforedissipation) within the loop. The existence of hysteresis evenafter creep has stopped under quasistatic perturbation implies:(a) the friction coefficient alone controls the area within thehysteresis curve; (b) to our best knowledge, the hysteresis doesnot vanish at a finite, yet infinitesimal �φ; and (c) as we showbelow, both μ and �φ control the “quasistatic creep rate.”

Given the uncertainty in the effect of parasitic friction, itis difficult to be confident about the functional form of therelaxation. Significant care would be required to tease outthese relationships accurately but the qualitative dependenceon μ and �φ seems solid.

V. DISCUSSION

Having presented the results, we now discuss them vis-a-vis ideal granular jamming, explain how they are relatedto each other, and establish how they are related to priorworks. We start with the first compression cycle where fragilebehavior is observed. The critical packing fraction φc in idealjamming is that packing fraction at which two conditions aresimultaneously met: (i) the packing fraction at which pressurestarts rising above zero and (ii) the isostatic point where totaldegrees of freedom equal the total number of constraints, i.e.,the number of floppy modes equals D(D + 1)/2 (rigid bodyrotation and translation), and all individual disk displacementsare strongly impeded. The first condition is met at φ1, thepacking fraction at which pressure starts rising above zero(albeit exponentially and not power law). The second conditionis met, however, only at φ2 where all displacements becomevery small; ergo the two conditions defining φc are wellseparated by a fragile, exponential regime characterized bysimultaneous existence of nonzero pressure (jammed clusters)and nonzero displacements (unjammed clusters). As noted inSec. II A, however, the definition of a contact in ideal jammingpermits the system exist in any one of two discrete states,completely unjammed or completely jammed; an intermediateregime as demonstrated by the fragile state is not allowed.This anomalous behavior therefore raises several questionsvis-a-vis the ideal jamming paradigm, which we explain below.

(i) Why have prior studies that have successfully verifiedthe jamming predictions not reported this anomalous scalingbehavior? All prior experimental [5,33] and numerical [2]studies to our knowledge study the unjamming transition,i.e., they approach φc from the jammed state towards theunjammed state. An analysis of the unjamming over jamming

transition is favored for technical reasons. Precise detectionof pressure rise commencement around φc is very difficult todetect over numerical or instrumental noise and fluctuationsfrom discrete configurational adjustments during compression.We, therefore, believe the fragile state exists in their systemsbut forms part of the experimental preparation phase, whichmay not have been systematically analyzed.

Two exceptions lend support to this possibility. In recentnumerical work on contact percolation transition (CPT), Shenet al. [17] analyzed the approach to jamming transition andshow deviations (discussed below) occur prior to jammingonset. But perhaps of greater relevance to the present studyis the earlier experimental work of X. Cheng [11] where thejamming transition was studied by swelling tapioca pearls inwater. Of particular interest is Fig. 14 of Ref. [11] where thestructural factor (pair correlation), measured boundary force,and mean-square particle displacements are plotted againstpacking fraction. The force exhibits two distinct regimes atpacking fractions labeled φ1 and φ2, where φ2 is shown tocoincide with random close packing. The pair correlationfunction exhibits two distinct peaks at φ1 and φ2. Finally, themean-square displacement goes through a maximum betweenφ1 and φ2 and falls to zero at φ2. This behavior is relatedto existence of local jammed clusters starting at φ1 whichgrow until global jamming is achieved at φ2. That study tracesthe source of this anomalous behavior to friction, the proofin support being vibrational disturbances (nonzero granulartemperature) relieve these frictional contacts and recover idealjamming predictions (see Fig. 16 in Ref. [11] and relateddiscussion). Given frictional jamming exhibits sensitive de-pendence on experimental protocol and preparation history, thecorrespondence between Cheng’s experiment and the presentstudy is noteworthy, particularly since they follow differentexperimental protocols.

(ii) Does an alternative physical mechanism explain thefragile state? In the fragile regime, part of the system isjammed as evidenced by nonzero pressure [Fig. 4(b)], whilethe remainder is unjammed as evidenced by nonzero dis-placements (Fig. 5). The co-incidence of exponential pressurerise and fall in displacements with increasing φ points to thepercolation of jammed clusters across the system. Evidence ofstress percolation comes in two parts: first, the strong spatialcorrelation between local disk displacements and nucleationof new stressed contacts [Figs. 5(c) and 6] and, second,exponential decrease in the fractional area enclosed by stresschains (Fig. 7). The present work is not isolated in its claim of apercolation route to jamming. Several recent studies [12,13,17]have shown some form of percolation mechanism precedingthe jamming transition.

(iii) How does friction control the fragile state? Figure 2presents clear evidence that the start of fragile state (or φ1) isdirectly dependent on the friction coefficient. With increasingvalue of the friction coefficient, the fragile state commencesat a lower packing fraction. In Fig. 12 we replot the datapresented in Fig. 2 on a log-linear scale for PE, PEG, andPEA disks. The data for PEG and PEA are horizontallyshifted so their φ1 values coincide with that of PE disks. Itis apparent from Fig. 12 that PE and PEG disks have the sameexponential slope (P ∝ eφ/χP , χP = 0.00195). They share thesame modulus (E = 2.5 GPa) but different friction coefficients

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0.8 0.81 0.82 0.83Packing Fraction φ

10

100

Pre

ssur

e P

(N

/m)

PE μ=0.19; B=2.5 GPaPEG μ=0.14; B=2.5 GPaPEA μ=0.19; B=0.004 GPa

P = 10-178.92

*e(φ/0.00195)

P = 10-54.575

*e(φ/0.0063)

FIG. 12. (Color online) P vs φ in log-linear scale for the firstcompression cycle. All plots have been horizontally shifted forcoincidence of φ1. The exponential pressure scaling (P ∝ eφ/χP ) forPE and PEG exhibits same slope (χP = 0.00195) as demonstratedby the exponential fit (solid black line), implying variation in frictioncoefficient has no effect on χP . On the other hand, PEA disks withsame friction coefficient as PE, but lower modulus exhibit a shallowerexponential scaling (with χP = 0.0063, long dashed blue line) overapproximately same range of pressure as PE and PEG.

(μ = 0.19 for PE μ = 0.14 for PEG), suggesting friction hasno measurable effect on the slope of exponential pressurescaling. In contrast, the fragile regime for PEA disks whichhave same friction coefficient as PE disks (μ = 0.19) buthave much lower modulus (E = 0.004 GPa) show shallowerscaling with χP = 0.0063, implying χP depends on themodulus. The values of χP for all materials are provided inTable II.

The above results do not mean, however, that friction has noeffect on χP . Frictionless disks (ideal jamming conditions) un-der compression only jam under normal stresses. Any tangen-tial stress would cause them to slip and disturb local jammedclusters. Therefore, nonzero friction is an essential componentfor the metastability of the fragile state, even though it doesnot reveal itself in the preceding discussion, because there aretwo modulii entering the pressure measurement, the modulusof the disk material and the effective pack modulus. We haveonly varied the disk material modulus; the pack modulus, onthe other hand, is a function of coordination number, which inturn depends on the friction coefficient. Hence, the effectivepack modulus is nonlinear due to coordination number; a fact

TABLE II. Experimentally determined values of the exponentialslope χP for indicated materials (abbreviations defined in text).

Material φ1 μ E (GPa) Exponential slope χP

PE 0.8093 0.19 2.5 0.00195PEG 0.8131 0.14 2.5 0.00195PEA 0.8079 0.19 0.004 0.0063LEX 0.7778 0.22 2.0 0.00135TEF 0.8298 0.06 0.5 0.0021

also evident from power-law scaling of P versus φ curves,equivalent to stress-strain relations, in ideal jamming. Fromthat relation [P ∝ (φ − φc)ψ ] one can discern the nonlineareffective modulus must be ψ − 1. We note two subtletiesthat arise here. First, all experimental, and most numerical,P versus φ curves are measured for finite systems, andthe asymptotic approach of ψ in the thermodynamic limit(large system size) is not understood. This would be theideal limit at which to study the nonlinear elastic constants.Albeit subtle, nonlinear elastic constants play a central role inelastoplastic responses of amorphous solids [34,35]. Further,since the Coulomb yield criterion (FT � μFN ) providesonly a lower bound on the value of the tangential stresscomponent, it is not possible to experimentally or theoreticallylearn how friction controls the coordination number and,therefore, the effective pack modulus. Empirical deductionfrom numerical simulations may be able to shed some light onthis relationship but, at this point, one can only say that frictioncontrols the exponential slope χP indirectly via effective packmodulus.

(iv) Why is fragile behavior not observed during subsequentcompression cycles? When we decompress the system after thefirst compression cycle, the stresses in the pack are relievedand the boundaries move just enough to relax the system. Thedisks, however, are left in the final configuration in which theyfound themselves at the end of the first compression cycle.If the granular pack is subjected to a second compression,the contacts that existed at end of the first compressioncycle are immediately activated everywhere across the systemsimultaneously at a critical packing fraction φc. This situationexactly corresponds to the sudden system wide emergence ofstressed contacts at φc.

In Fig. 13, we plot the pressure P against the packingfraction φ for the second compression cycle. We recall thatjamming theory predicts zero pressure below φc. At φc

when the system satisfies the isostatic condition, and allconstraints are activated simultaneously across the system,a rise in pressure is recorded with a power-law scaling[P ∝ (φ − φc)ψ ]. Prior experiments by Majmudar et al. [5]have demonstrated the power-law increase in pressure with anexponent of 1.1. Our experimental data are fit very well withan exponent of 1.15 (see solid line fit for the experimental datain Fig. 13) and are in very good agreement with the results inRef. [5]. Given that Majmudar et al. tapped their system aftereach quasistatic step, which we do not, the agreement in theexponent is indeed remarkable.

The incompatibility between zero friction and zero temper-ature raises another important question about what it meansfor a frictional granular pack to be structurally stable. Asargued in Ref. [7], with increasing friction a granular packcan jam at φc < φRCP, and the isostatic point can occur atD + 1 � Zc � 2D. At zero temperature but nonzero friction,repetitive loading data (Fig. 11) exhibit evolution in φc,implying there must be an increase in Zc at each cycle whichwe are unable to measure due to experimental shortcomings.Nevertheless, a straightforward physical interpretation forhysteretic creep may be presented from the granular jammingperspective. Due to friction, the system jams at φc into ametastable configuration and will remain so indefinitely unlessperturbed externally (recall, the disks are macroscopic and

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FIG. 13. (Color online) P vs φ for the second compression cyclein linear scale shows the predicted power law scaling for granularjamming transition for all four boundaries. The pressure rise is abruptand is in contrast with the gradual (exponential) rise observed for thefirst compression cycle. The solid line is the best power-law fit tothe experimental data with a power law exponent of 1.15. The long-dashed line representing the best linear fit to the data demonstratesthat the experimental data does not follow linear scaling.

not susceptible to thermal fluctuations). Any external driving(e.g., tapping or jiggling) relieves frictional stresses in thesystem (at least partially if not all of them) and destroys thismetastable configuration. In the absence of such external drive,the only perturbative mechanism available to the system is themagnitude of the quasistatic step (�φ) which is equivalentto strain. Hence, it follows that with each loading-unloadingcycle, the system evolves ever so slightly through a series ofmetastable configurations towards a final, stable configuration.The magnitude of �φ therefore directly controls the packevolution as shown in the bottom panel of Fig. 11. Furthermore,whether or not a given value of �φ can evolve the systemfrom one metastable configuration to the next depends on thestatic friction coefficient μ which controls the degree of aconfiguration’s stability—the higher the friction coefficient,the more stable a configuration is. The rate at which φc evolvestherefore must be a function of μ and �φ. Indeed, as shownin Fig. 11, the difference (φn

c − φ0c ) between φc for the nth and

0th cycles depends on �φ and μ.Interestingly, strain-dependent creep and hysteresis are

also observed in viscoelastic materials, which represent anentirely different class of amorphous media. Viscoelasticmedia (unlike purely elastic materials) are characterized by thepresence of both an elastic and viscous component. For thisreason, whereas purely elastic materials dissipate no energyon application and subsequent removal of a load (reversibleprocess), a viscoelastic medium does dissipate energy viaviscosity (irreversible process) [21]. This is seen through thepresence of hysteresis in the stress-strain curve, with the areaunder the hysteresis loop being equal to the energy lost duringthe loading cycle. Viscosity being resistance to thermallyactivated plastic deformation, viscous and viscoelastic media

lose energy through a loading cycle, i.e., plastic deformationresults in energy dissipation, a property uncharacteristic of apurely elastic material’s reaction to a loading cycle.

More specifically, viscoelasticity is a molecular rearrange-ment. Application of stress to a viscoelastic material, e.g.,a polymer, cause parts of the long polymer chain to changetheir positions. This rearrangement is termed viscoelastic creep[36]. Polymers retain their solid properties, even as parts oftheir chains undergo rearrangement in order to accompanythe stress, and as this occurs, it creates a back stress inthe material. When the magnitude of back stress equalsthat of the the applied stress, the material ceases to exhibitcreep. Unlike viscoelastic media where creep and healing ofmetastable polymer configurations are thermally activated, inthe athermal granular system considered here, the quasistaticstrain is the only perturbative mechanism available by whichthe system creeps towards its ultimate stable configurationat φRCP. Whereas the dissipative mechanism available toviscoelastic amorphous media is supplied by viscosity, inthe granular system it comes about through friction. Suchviscoelastic behavior has been observed in naturally occurringgranular packs in the geophysical context, namely sandstoneand sedimentary rocks [15,20]. Particularly noteworthy is thatour loading protocol is very similar to loading proceduresfollowed in measuring mechanical properties of geophysicalrock samples in standard load cells.

VI. SUMMARY

In summary, we have presented experimental results fora system of bidispersed, frictional disks subjected to uniaxialcompression. We verify the numerical predictions for frictionaljamming [7,16], whereby jamming is shown to occur atprogressively lower packing fractions with increasing frictioncoefficient. We also show the first compression cycle exhibitsexponential increase in pressure and a corresponding expo-nential fall in displacements over a range of packing fractionsφ1 < φ < φ2. We show this exponential scaling separates thetwo conditions that define the critical packing fraction φc.We compare our data against published experimental andnumerical results and delve into how friction controls thisregime in a nontrivial manner. To put our work in perspective, itfalls within a class of recent results that demonstrate some formof percolation mechanism arising prior to jamming transition,with stress percolation presenting the route to jamming in thepresent case. Finally, we find hysteretic creep under repetitiveloading-unloading cycles and experimentally trace its source tofriction. Despite our inability to reliably measure coordinationnumbers, our experiments help explain the various regimesarising in frictional granular jamming.

In conclusion, this article shows jamming in the presence offriction demonstrates rich behavior beyond the ideal jammingscenario. Several subtle, yet important questions arise, asdiscussed throughout this article. Some of them lie beyondexperimental purview where numerical simulations can lendcrucial support. Resolution of these questions should aid indevelopment of a tractable theory of frictional jamming, inturn, aiding advances in both fundamental (statistical andcondensed matter physics) and applied (materials physics andgeophysics) realms alike.

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M. M. BANDI, M. K. RIVERA, F. KRZAKALA, AND R. E. ECKE PHYSICAL REVIEW E 87, 042205 (2013)

ACKNOWLEDGMENTS

This work was carried out under the auspices of the NationalNuclear Security Administration of the US Department of

Energy at Los Alamos National Laboratory under Contract No.DE-AC52-06NA25396. The authors gratefully acknowledgehelpful discussions with O. Dauchot.

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