VOLUME 81, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 24 AUGUST 1998

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Isostatic Phase Transition and Instability in Stiff Granular Materials

Cristian F. Moukarzel*Instituto de Fı´sica, Universidade Federal Fluminense, 24210-340 Niterói RJ, Brazil

(Received 10 March 1998)

Structural rigidity concepts are used to understand the origin of instabilities in granular aggregIt is first demonstrated that the contact network of a noncohesive granular aggregate becomesexactlyisostatic when I ­ keyfl ¿ 1, wherek is the stiffness,e is the typical interparticle gap, andfL isthe typical stress induced by loads. Thus random packings of stiff particles are typically isostFurthermore isostaticity is responsible for the anomalously large susceptibility to perturbation obsein granular aggregates. The load-stress response function of granular piles iscritical (power-lawdistributed) in the isostatic limit, which means that slight overloads will produce internal rearrangeme[S0031-9007(98)06949-X]

PACS numbers: 61.43.Gt, 05.70.Jk, 46.10.+z

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Photoelastic visualization experiments [1–3] showclearly defined stress-concentration paths in noncohesgranular materials under applied load. These often suffsudden rearrangement on a global scale when the loconditions are slightly changed, evidencing a degreof susceptibility to perturbation not usually present ielastic materials. It is rather possible that this intrinsiinstability is responsible for much of the interestingphenomenology of granular materials [3,4]. Recentlynumber of phenomenological models [2,5–8] have beeput forward, which succeed to reproduce several aspeof stress propagation in granular systems, and the issof instability has been addressed by noting that the loastress response function may take negative values [6].is the purpose of this Letter to show that structural rigiditconcepts help us understand theorigin of instability ingranular materials, linking it to the topological propertieof the system’s contact network.

Structural rigidity [9] studies the conditions that anetwork of points connected by rotatable bars (reprsenting central forces) has to fulfill in order to sustaiapplied loads. A network with too few bars isflexible,while if it has the minimum number required to berigid it is isostatic. Networks with bars in excess ofminimal rigidity areoverconstrained, and are in generalself-stressed. Concepts from structural rigidity were firstintroduced in the study of granular media by Guyonet al.[10], who stressed that granular systems are not etirely equivalent to linear elastic networks since in thformer only compressive interparticle forces are posible. We next show that this constraint has far-reachinconsequences for the static behavior of stiff granulaggregates.

Consider a d-dimensional frictionless granular pilein equilibrium under the action of external forces$Fi

(gravitational, etc.) on its particles. Imagine building aequivalent linear-elastic central-force network (thecontactnetwork), in which two sites are connected by a bond iand only if there is a nonzero compression force betwe

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the two corresponding particles. Because of linearitstressesfij on the bonds of this equivalent system cabe decomposed asfij ­ fself

ij 1 floadij where fself

ij areself-stresses, andfload

ij are load-dependent stresses. Thelast are linear in the applied load and do not changeall stiffnesses are rescaled. Self-stresses in turn dodepend on the applied load, but are linear combinatioof terms of the formkijeij wherekij are the stiffnesses ofthe bonds, andeij their length mismatches. In a granulapile, length mismatches are due to interparticle gaps, atherefore will depend on the distribution of radii and on thcharacteristics of the packing. Furthermore, self-strescan arise only withinoverconstrainedsubgraphs [9,10],i.e., those with more contacts than strictly necessarybe rigid. It is easy to see that a bounded overconstrainsubgraph with nonzero self-stresses must have at leone negative (traction) self-stress. It suffices to consida joint belonging to the envelope of the overconstraincluster: since bonds can reach it from only one side of tfrontier, stresses of both signs are necessary in orderthe joint to be equilibrated. Now rescale all stiffnesses acording tok ! lk. In doing so, self-stresses are rescaleby l, but load-dependent stresses remain constant. Tif self-stresses were nonzero, in the limitl ! ` at leastone bond of the network would have negativetotal stress,which is a contradiction. Therefore stiff granular pilemust either (a) have zero length mismatches or (b) hano overconstrained graphs at all. Condition (a) cannbe satisfied if the particles have random polydispersino matter how small,or if the packing is disordered.Therefore there can be no overconstrained subgraphspolydisperse or disordered packings in the large-stiffnelimit. In other words: The contact network of a granulapacking becomes isostatic when the stiffness is so lathat the typical self-stress, which is of orderke, would bemuch larger than the typical load-induced stressfL [11].

The isostaticity condition above is perhaps simplto understand when cast in the following terms: granlar packings will only fail to be isostatic if the applied

© 1998 The American Physical Society

VOLUME 81, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 24 AUGUST 1998

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compressive forces are strong enough to close interpartgaps establishing redundant contacts.

Thus, real disordered packings will typically be isostatic (interparticle gaps are large) unless its particlesstrongly deformed by the load. This explains why thaverage coordination number of random sphere packinis usually close to 6 (see [10] and references thereiIsostaticity was also reported in numerical simulationsrigid disks [12].

Finally we note that an adimensional “isostaticitparameter” can be defined asI ­ keyfL, and thatI ¿ 1corresponds to the isostatic limit.

We now discuss the consequences of isostaticitythe static behavior of a pile. It is possible to obtaiuseful insight from recent studies of the related probleof central-force rigidity percolation [13]. Rigid backboneare found to be composed of large overconstrained clustisostaticallyconnected to each other bycritical bonds(alsocalled red bonds). Cutting one critical bond is enoughproduce the collapse of the entire system, because eacthese is by definition essential for rigidity. In percolatiobackbones though, the number of such critical bondsnot extensive, but scales atpc as L1yn where n is thecorrelation-length exponent [14]. Thus, if we pertur(cut or stretch) a randomly chosen bond in a percolatibackbone, most of the times the effect will be only locsince no critical bond will be hit. The new element istiff granular contact networks is the fact thatall contactsare isostatic or critic; i.e., there isextended isostaticity.Thus we may expect stiff granular systems to have a larsusceptibility to perturbation since cutting (stretching)bond will often produce a large part of the systemcollapse (move).

Let us now quantify these ideas. We perturb thsystem by introducing aninfinitesimal change dl inthe length of a randomly chosen bond, and record tinduced displacementd $xi suffered by particle centers inequilibrium. The system’s susceptibility to perturbatiois then defined asD ­

PNi­1 jd $xiydlj2. We propose

to measureDsOyd as a function of the densityOy ofoverconstraints(excess contacts) randomly located on thnetwork. Isostatic piles haveOy ­ 0.

A simple one-dimensional model for the propagationperturbations can be analytically solved [15] for arbitrarvalues of the densityOy of overconstraints. For anynonzeroOy, D as defined above takes a finite value, bdiverges asO21y2

y for Oy ! 0. Therefore there is a phasetransition at the isostatic pointOy ­ 0.

We now analyze a two-dimensional system numecally. In the spirit of previously studied models [16]we consider a triangular packing ofH layers height, withone of its principal axes parallel to gravity, and madedisks with small random polydispersitydR and weightW . Since dR is small, disk centers are approximatellocated on the sites of a regular triangular lattice. If thstiffnessk is large enough (I ­ kdRyWH ¿ 1), the con-

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tact network will be isostatic. We enforce isostaticity iour model by letting each site be supported from beloby only two out of its three neighbors. This gives threpossible local configurations which are depicted in Fig.By appropriately choosing among these [15], random isstatic networks with only compressive stresses are genated. In order to study the effect of a finite densityOy

of overconstraints (which would appear if the stiffnesslowered), we furthermore letall three bondsbe presentwith probabilityOy at each site.

After building a disordered network in this way, a randomly chosen bond in the lowest layer is stretched, athe induced displacement field is measured. After avaging over disorder, the stress distribution [15] is founddecay exponentially for large stresses, in accordance wprevious work [2,17].

The results for the susceptibilityDysH, Oyd ­PNi­1 dy2

i are shown in Fig. 2a. Heredyi is the verticaldisplacement of sitei due to a unitary bond stretchingas measured on packings ofN ­ H 3 H particles. ForOy . 0, Dy goes to afinite limit for large sizesH, butdiverges with system size ifOy ­ 0. Measurements onisostatic packings of up toH ­ 2000 layers [15] showthat logDy ~ H; i.e., the divergence ofDy is exponentialwith size when Oy ­ 0. Thus there is a surprisingphase transitionat Oy ­ 0, where anomalously largesusceptibility sets in.

In order to understand how displacements propagupwards, we measure the probability distributionPhsdydto have a vertical displacementdy, h layers above theperturbation. Numerical results for isostatic systems wiH ­ 2000 are shown in Fig. 2b. For largeh, Phsdyddecays as a power law with anh-dependent cutoff:Phsdyd , h2rjdyj2u, dy , dMshd. As seen in Fig. 2b,dMshd grows exponentially with heighth. This producesthe observed exponential divergence ofDy. Similarmeasurements were done on systems with a finite denof overconstraintsOy, in which case the distribution ofdisplacements presents a height-independent bound [1

The puzzling appearance of exponentially large diplacements on isostatic piles can be explained as duethe existence of “lever configurations” or “pantographswhich amplify displacements. Figure 3 shows aexample of a pantograph with amplification factor 2Given that this and similar mechanisms appear wi

L S R

FIG. 1. Appropriately choosing among these three isostaconfigurations for each site, only compressive stressesproduced on a triangular packing. FirstS is chosen withprobability 1y2. If S is not chosen then eitherR or L are,depending on the sign of the horizontal force acting on the si

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VOLUME 81, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 24 AUGUST 1998

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FIG. 2. (a) Total susceptibilityDy versus system heightH in layers, as numerically measured on two-dimensiontriangular packings. The fraction of overconstraints (fractioof sites supported by three lower neighbors) is as follow0.00 (open circles), 0.01 (squares),0.02 (diamonds), 0.05(triangles), and1.00 (full circles). (b) The probabilityPhsdydto have an induced vertical displacementdy, h layers abovethe perturbation, as obtained in a numerically exact fashifor the isostatic (Oy ­ 0) triangular piles described in thetext. Results are shown forh ­ 200, 400, 600, . . . , 2000. Onlypositive values ofdy have been plotted here.

a finite density per layer, it is clear that the seconmoment ofPhsdyd will grow exponentially with systemheight. Furthermore, it is easy to understand why thamplification effect only exists in theisostatic limit:Pantographs as the one in Fig. 3 are no longer effectiveblocked by overconstraints, for example, if an additionbond is added between siteA and the site below it. Inthis case, a stretching of bondB would induce stresses inthe whole pantograph, and only a small displacementsiteA.

In order to formalize the relationship between thesfindings and the observed unstable behavior [1,2]

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FIG. 3. The observed exponential growth of induced displacments is due to the existence of random “pantographs” asone shown in this figure. Upon stretching bondB by an amountd, siteA moves vertically by an amount2d. Conversely a uni-tary weight atA produces a stress of magnitude 2 on bondB.This is a consequence of the general equivalence betweenduced displacements and the load-stress response function.

granular materials, we now demonstrate the equivalenbetween induced displacements and the load-stresssponse function [6] of the stretched bond. The nework’s total energy can be written asE ­

PNi­1 Wiyi 1

1y2P

b kbslb 2 l0bd2 where the first term is the potentia

energy and the second one is a sum over all bonds andcounts for the elastic energy.lb are bond lengths in equi-librium andl0

b their repose lengths. Upon infinitesimallystretching bondb0, equilibrium requires that

Pi Wi

≠yi

≠lb01P

oy koysloy 2 l0oyd ≠loy

≠lb0­ 0, where the second sum goe

over bondsoy that belong to the sameoverconstrainedgraph asb0 does. This is so since bonds not overcostrained with respect tob0 do not change their lengthsasa result of stretchingb0. Since stressfb on bondb isfb ­ kbsl0

b 2 lbd this may be rewritten asXoy

foy

≠loy

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≠lb0

. (1)

If b0 does not belong to an overconstrained graph, tleft hand sum contains only bondb0 itself, thereforefb ­

Pi Wi

≠yi

≠lbshowing that, in the isostatic case (n

overconstrained graphs at all), the induced displacemdy

sbdi ­

≠yi

≠lbis equal to theresponse functionof stressfb

with respect to an overload on sitei.Taking averages with respect to disorder we obta

kfbl ­P

i Wikdysbdi l and since average stresses grow

H we must havekdysbdi l , H21. We thus see that there

must be delicate cancellations inPsdyd, since its secondmoment diverges as exphHj while its first moment goesto zero with H. This shows thatdy (and therefore theresponse function) takes exponentially large valuesboth signs[Psdyd is approximately symmetric]. Thusa positive overload at sitei would often produce a

VOLUME 81, NUMBER 8 P H Y S I C A L R E V I E W L E T T E R S 24 AUGUST 1998

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(very large) negative stress on bondb, implying theneed for rearrangement since negative stresses areallowed.

The existence of negative values for the responfunction was first discussed in relation with instabilityin the context of a phenomenological vectorial modefor stress propagation [6]. The results of the presework demonstrate that the response function takes eponentially large negative values, and the system is ustable, because of the isostatic character of the contnetwork.

To conclude, we have shown that granular packinare exactly isostaticwhen I ­ keyfL is much largerthan one, which holds for typical disordered packingand also for stiff enough regular packings with randompolydispersity.

For isostatic packings, the distribution of displacemeninduced by a perturbation is power law with an exponentially large cutoff. A susceptibility to perturbationcan be defined, which diverges upon increasingI. Thus,an isostatic phase transition takes place in the limitlargeI.

Induced displacements were furthermore shown toequivalent to the load-stress response function of tperturbed bond. Our results for induced displacementhus mean that response functions take exponentialarge values, as well as positive and negative, in thisostatic limit. This explains why stiff granular piles areunstable. Any nonzero density of overconstraints destrocriticality, and therefore instabilities will not be presenwhen the isostaticity parameterI is small. I can bereduced by reducing the stiffness, the interparticle gapor by increasing the load.

I thank H. J. Herrmann, J. L. Goddard, P. M. Duxburyand J. Socolar for useful discussions and comments, aH. Rieger for an important suggestion. HLRZ Jülichprovided supercomputer access and warm hospitaliThe author is supported by CNPq, Brazil.

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*Email address: cristian@if.uff.br[1] P. Dantu, in Proceedings of the 4th International Con-

ference on Soil Mechanics and Foundation Engineering (Butterworths, London, 1957); P. Dantu, Ann. PontsChaussees4, 144 (1967); M. Ammi, D. Bideau, and J. P.Troadec, J. Phys. D20, 424 (1987); B. Miller, C. O’Hern,and R. Behringer, Phys. Rev. Lett.77, 3110 (1996).

[2] C. Liu et al., Science269, 513 (1995).[3] H. M. Jaeger and S. R. Nagel, Rev. Mod. Phys.68, 1259

(1996).[4] Friction, Arching, Contact Dynamics,edited by D. E.

Wolf and P. Grassberger (World Scientific, Singapore1997).

[5] S. N. Coppersmithet al., Phys. Rev. E53, 4673 (1996).[6] P. Claudin, J.-P. Bouchaud, M. E. Cates, and J. P. Wittme

Phys. Rev. E57, 4441 (1998).[7] P. Claudin and J. P. Bouchaud, Phys. Rev. Lett.78, 231

(1997).[8] M. Nicodemi, Phys. Rev. Lett.80, 1340 (1998).[9] Henry Crapo, Struct. Topology1, 26–45 (1979); T. S. Tay

and W. Whiteley, Struct. Topology11, 21–69 (1985).[10] E. Guyon, S. Roux, A. Hansen, D. Bideau, J.-P. Troade

and H. Crapo, Rep. Prog. Phys.53, 373 (1990).[11] Exceptions are packings with periodic boundary cond

tions because they are not bounded, and idealized reguclose-packings, because they have no length mismatchand therefore no self-stress even on overconstraingraphs.

[12] S. Ouaguenouni and J.-N. Roux, Europhys. Lett.35, 449(1995); Europhys. Lett.39, 117 (1997); see also Ref. [4].

[13] C. Moukarzel and P. M. Duxbury, Phys. Rev. Lett.75,4055 (1995); see also C. Moukarzel, P. M. Duxbury, anP. L. Leath, Phys. Rev. Lett.78, 1480 (1997).

[14] A. Coniglio, J. Phys. A15, 3829 (1982).[15] C. Moukarzel (to be published); see also cond-ma

9807004.[16] S. Roux, D. Stauffer, and H. J. Herrmann, J. Phys. (Pari

48, 341 (1987); J. Hemmingsson, H. J. Herrmann, anS. Roux, J. Phys. I (France)7, 291 (1997).

[17] F. Radjai, M. Jean, J. J. Moreau, and S. Roux, Phys. ReLett. 77, 3110 (1996).

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