Self-organization, Localization of Shear Bands and Aging in Loose Granular Materials

J´anosT¨or¨ok1,2,SupriyaKrishnamurthy2,J´anosKert´esz1andSt´ephaneRoux3

DepartmentofTheoreticalPhysics,InstituteofPhysics,

TechnicalUniversityofBudapest,Budafokiu´t8,Budapest,H-1111,Hungary

2

LaboratoiredePhysiqueetM´ecaniquedesMilieuxH´et´erog`enes,

ESPCI,10rueVauquelin,Paris75231,France.3

SurfaceduVerreetInterfaces,UMRCNRS/Saint-Gobain,39QuaiLucienLefranc,93303AubervilliersCedex,France

(February6,2008)

Weintroduceamesoscopicmodelfortheformationandevolutionofshearbandsinloosegran-ularmedia.Numericalsimulationsrevealthatthesystemundergoesanon-trivialself-organizationprocesswhichisgovernedbythemotionoftheshearbandandtheconsequentrestructuringofthematerialalongit.Highdensityregionsarebuiltup,progressivelyconfiningtheshearbandsinlocalizedregions.Thisresultsinaninhomogeneousagingofthematerialwithaveryslowincreaseinthemeandensity,displayinganunusualglassylikesystem-sizedependence.PACSnumbers:45.70.-n,45.70.Mg,05.65.+b

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arXiv:cond-mat/0003070v1 [cond-mat.stat-mech] 6 Mar 2000Alargeclassofmaterialsarehandledintheformofdispersedsolidgrainsatsomestageoftheirprocessing.Thusthedescriptionoftherheologicalpropertiesofsus-pensions,pastesanddrygranularmediaisakeyquestionwhichcontrolstheabilityofmixing,storing,transport-ingetc,thesedispersemedia[1–3].Granularsystemsconstituteanintermediatestateofmatterbetweenflu-idsandsolids[4,5]:theyflowlikefluidsbuttheyalsobuildpilesindicatingthatanon-vanishingstaticshearstressispresentwhichischaracteristicofsolids.Fromthispointofviewitisalsoofmajorinteresttounder-standtheshearingprocessinthesesystems.Anumberofexperimentshavebeencarriedoutontheshearprocessingranularmaterials[6,7].Mostofthesearetriaxialtests[7,8]todeterminemacroscopicpropertiessuchastheshearstressorthevolumetricstrain,asafunctionoftheshearstrain.

Theintimateinterplaybetweenthegeometricalar-rangementsandthefrictionalpropertiesofthegrainsde-terminesthepreciseformoftherheologicalbehaviortobeusedatacontinuumlevel.Theunderlyingquestionistheidentificationofrelevantinternalvariables.Themostobviousoneisthedensityofthesample,whichcanbemadetovaryoverawiderangebythemethodofpreparation.Comparedtootherparametersdescribingthetexture(e.g.fabrictensorsaccountingforthedistri-butionofcontactorientations)thedensityhasthemostdrasticimpactonthestressneededtoshearthemate-rialaswellasonthemodeofshearing;fromanappar-enthomogeneousstrainforloosepackingstoalocalizedsteadyshearbandfordenseassemblies[9].Thecouplingofthedensitytotheshearpropertiescanbeunderstoodthroughtheconceptofdilatancy[6].Arelatedquestioniswhetherstatisticalfluctuationshaveanimpactonmacroscopicproperties.Lately,therehasbeenanupsurgeofinterestintryingtocharacterizethelargestressfluctuations[10–13]insilos,Couettefloworsliderblockgeometries,ortounderstandthestatistics

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ofinterparticlecontactforces[14].Recently,spectacu-larexperimentsintwo-dimensionalCouetteshearcellswerecarriedout[13]wherethemovementandstressofindividualparticlesweremonitoredinordertodescribetheinnerstructureandtheforcenetworkintheshearedgranularmaterial.Itwasdemonstratedthatstationarymotionisaccompaniedbylargestressfluctuationsduetotheformationandbreakdownofarches.Largefluctu-ationswerealsofoundinthreedimensionalsteadystateshearcells[15].Thisissuehasalsobeenraisedbytheresultsofre-centnumericalsimulationsofrigidgrainassemblies[16],whereevenatlowdensities,theshearingwhichappearsashomogeneousoverlongtimes,infactconsistsofasuccessionofsuddenchangesofquasi-instantaneousandlocalizedstrainfields.Thisobservationsuggeststhatthetransitionfromtheparticlebaseddescriptiontothecon-tinuumonerequiresthedetailedunderstandingofthestatisticalfeaturesassociatedwiththesesuddenchanges.InthisLetterwepresentasimplemodelfortheshear-ingofagranularmediuminloosesamples.Wedescribethestrainfieldateveryinstantasashearband,chosenthroughaglobaloptimizationprocedure,whichisequiv-alent,asweshallseelater,tosearchingforthegroundstateofadirectedpolymerinarandompotential[17,18].However,thispotentialisnotapriorifrozeninbuthasaself-organizeddevelopmentduetoourprocedureofchoosingandchangingtheshearband.Thoughverysim-pleandwithonlytheminimumofingredients,themodelshowsthatthedensityofthemediumincreasesanoma-louslyslowly.Furtherwearealsoabletopredictonthebasisofthismodel,thatlargescaleinhomogeneitiesbuildupinasystemsubjecttoasteadyshear.Thiscouldbeaninterestingfeaturetocomparewithexperiments.Letusconsiderashearprocess,assumedtobeinvari-antalongthesheardirection(zinFig.1).Thisgeom-etryisappropriateforinstance,inanannularshearcelloflargeradius[15].Weconsidermoreoveracontinuum

yxzShearbandShetionceirar dFIG.1.Schematicpictureoftheshearprocess.Theshearband

isparalleltothesheardirectionzduetoperiodicboundarycondi-tionsinthisdirection.

description,validonscalesmuchlargerthanthatofindividualgrains.Wenowintroduceafundamentalas-sumptionofourmodel:Weassumethattheinstanta-neousstrainfieldisalwayslocalizedonasingleshearband[4,19].Experimentally,itisknownthatshearbandshaveatypicalwidthofabouttengraindiameters.Thusatacontinuumlevel,thevelocityfieldisindeeddiscon-tinuousacrosstheshearband.Fromthegeometryofourset-up,theshearbandmustbeacontinuoussurfaceduetotopologicalconstraints(Fig.1).Further,weas-sumethatbecauseofthetranslationalinvariancealongthezaxis,thesystemcanbereducedtoatwodimen-sionaloneinthex-y(cross-section)plane,throughanaveragingoverthezdirection.

Thebasichypothesisofthelocalizationoftheshearontheshearbandatalltimes,isnotasrestrictiveasitmayappear.Weonlyreferheretoinstantaneousshearrates,andprovidedtheshearbandchangesrapidlyenough,coarse-grainingthestrainfieldintimemayproduceauniformshearrate.Experimentally,thoughitisverydifficulttohavedirectaccesstotheinstantaneousshearrate,largefluctuationsfoundintheshearstressmayin-dicatethattheshearisneverquiteuniform,evenatearlytimes.Asmentionedearlier,thisseemsindicatedalsobynumerics[16].

Initiallyweconsideraloose-packedsample.Atasuit-ablycoarse-grainedscalethemediumcanbedescribedasacontinuum,wherethedensityisarandomfunctiondisplayingfluctuationsaroundameanvalue.Underaconstantnormalload,athresholdshearforce(ortorqueforanannularshearcell)hastobeappliedtoimposeanon-zerostrain.Locally,afterintegrationalongthezaxis,thedensitycontrolsthethresholdshearforce.Al-thoughthisisinessential,forsimplicityweassumethattheratioofsheartonormalstress,i.e.thefrictionco-efficient,increaseslinearlywithdensity.Asmentionedearlier,thetextureofthemediumalsocontributestothefrictioncoefficient.However,sinceweconsideronlyshearinafixedorientation,asinglescalarparametercombin-ingdensityandtextureshouldsuffice.Thisparameteriscalled“density”forshortandisdenotedby̺(x,y).Thusatanytimethestateofthemediumischaracterizedbythisfield.

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Wedeterminetheshearband(pathinthe(x,y)plane)bythefollowingthreeconditions:a)itiscontinuous,b)itspansthesampleinthexdirectionwithoutoverhangsandc)thesumofthedensityalongitisminimalamongallpossiblepathssatisfyinga)andb).Onecanrecognizethatthisisthewellknownproblemoffindingthegroundstateofadirectedpolymerinarandompotential[17].Relativemotionoftheparticlestakesplacewithintheshearbandwhiletherestofthesampleremainsstill.Smallmovementscantotallyrearrangethelocalstruc-ture[15,20]andthusmayinducelargechangesinthelocaldensity.Wesimplifythiscomplexbehaviorbyre-newingthedensity̺onlyalongtheshearband,byinde-pendentrandomvaluestakenfromafixeddistribution.Afterthis,anewshearbandisagainlocatedasdescribedabove.Thustheshearprocessconsistsofasuccessionoflocalizedslipsoccurringatverysmalltimescales.Wenotethatincharacterizingthisprocess,inthespiritofacontinuummodeling,weignorepotentialstressinhomo-geneitiesinthemedium.Itisasimplifyingassumptionofthemodeltorelatetheshearbandlocalizationonlytothedensity,andnottothefullsolutionofthelocalstressdistribution.

Inordertobeabletosimulatetheabovemodelwedis-cretizeditonasquarelatticeeitherwithprincipleaxisparalleltoxandyandconsideringfirstandsecondnear-estneighbours,ortiltedby45oconsideringonlynearestneighbours.Periodicboundaryconditionsareimposedintheydirection.Simulationswithsiteandbondversionswerealsocarriedoutleadingessentiallytothesamere-sults.WeconsiderheresquaresampleswithsystemsizeN×NwithNvaryingfrom32to512.Initiallyaden-sity̺i(arandomnumberuniformlydistributedbetween0and1)isassignedtoeverybondi.Wedefinethein-stantaneous󰀁shearbandasthespanningdirectedpathalongwhich̺iisminimal(applyingtheusualtrans-fermatrixmethod[17]).Oncetheshearbandisfoundthebondsbelongingtoitareassignednewvaluestakenfromthesameuniformdistributionasusedinitially.Werepeatthisprocessandmonitordifferentpropertiesofthesystem[21].

Wedefinetheaveragedensity󰀇̺󰀈asthemeanvalueofthedensityofthesitesnotbelongingtotheshearband.Thisdefinition,aswellasourprocedureofchoosingtheleastandchangingit,guaranteesthattheaveragedensityisamonotonicallyincreasingfunctionoftime.

Themonotonicbehaviorandtheboundednatureoftheaveragedensity(̺≤1)ensurethatithasanasymp-toticvalue.Infinitesamplesthisisequalto1.InFig.2wehaveplottedthedeviationoftheaveragedensityfromthisasymptoticvalue.Atearlytimes(t/N<∼2)therescaledcurvesgotogetherindependentlyofthesystemsize;laternon-trivialsystemsizeeffectscanbeobserved.Therelaxationtotheasymptoticvaluegetsslowerasthesystemsizeincreases.

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110101010

−1−2

110

−1−2−3−4−5−6

1010101010

−3−4−5

a)

0.10.04−410

10−310

b)

−310

−110

110

310

5

1010

−110

110

310

5

FIG.3.a)Log-logplotofthetimedependenceofthedistance

10

−210

010

210

4FIG.2.Thedifferenceoftheaveragedensity󰀂̺󰀃fromits

asymptoticvalueisplottedasafunctionoftimetrescaledbythe

systemsizeN.Thesystemsizesare32,,128,256and512fromtobottomtotoprespectively.

dforsystemsizes32to512scaledtogether.BoththedistancedandthetimetscalewiththesystemsizeN.b)Log-logplotofthedeviationofthedensityoftheshearbandbeforetheupdatefromitsasymptoticvalueasafunctionoftimeforsystemsizes32to512scaledtogether.

Sincethesystemevolvesentirelythroughtheprocessofchoosingandchangingtheshearband,wehavemon-itoredthefollowingtwoimportantquantitiesrelatedtotheshearband:TheHammingdistanced(whichisthenumberofdifferentsitesbetweensuccessiveshearbands)(Fig.3a)andtheaveragedensityofthesitesalongtheshearband󰀇̺SB󰀈beforechange(Fig.3b).Itisappar-entfromthefigurethatthereisacharacteristictimeoftc1≃N,belowwhichthedistanceisessentiallyconstantandequaltothesystemsizeandthedensityoftheshearbandisroughlyconstant.Thiscanbeunderstoodquali-tativelyfromthefollowingconsiderations.Sincetheveryfirstshearbandisequivalenttothegroundstateconfor-mationofadirectedpolymerinarandompotential,weknowfromthisanalogythatthemeandensityalongthisshearbandismuchlessthan0.5[17].Oncethepathisrefreshed,itsmeandensityincreasesto0.5.Thenextshearbandtendstoberepelledbythepreviousonesincetherestillexistmanyspanningpathswithalowerden-sity.Thusatearlytimes,twosuccessiveshearbandsdiffercompletely(Fig.3a)andthedensityoftheshearbandremainsmoreorlessthesame(Fig.3b).Thisini-tialphaseshouldlastuntilonaverageallsiteshavebeenrefreshedafewtimes,anumberoftimestepsoftheorderofN.

Theabsenceofoverlapbetweensuccessiveshearbandsinthisearlytimeregimereflectsthefactthatnowellde-finedshearbandcanbeobservedinloosegranularme-dia.Experimentallythisisconnectedtothedifficultyinquantifyingfluctuations,whenthemeanshearstrainisofsmallmagnitude.Sowhatisobservedisseeminglyahomogeneousshear.

Thereisatransitionregimeuptotc2≃20Nwherewestillhaveagoodqualitydatacollapse.Inthisregimebothcurvesdandr≡0.5−̺SBstarttofalloff.Thedecreasingdistanceindicatesanincreasingpersistenceof

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theshearband.Astheaveragedensityofthesystemincreases(Fig.2)thedensityoftheminimalpathalsogrowsandthustherepulsiveinteractionbetweentwocon-secutiveshearbandsprogressivelyfadesaway.Finally,bytheendofthetransitionregime,theinteractionbe-comesattractiveandamuchslowerrelaxationprocesstakesplace.

Theabovemeasurementspointtoalocalizationoftheshearband,inducedbytheimposeddynamics.Inordertounderstandbetterhowthiscomesaboutwepresentdensitysnapshotsofthesystematfourdifferentin-stances(Fig.4)varyingfromt/N∼4to4000.Weobservethatinitially(Fig.4a)thedensityappearshomo-geneouslydistributed.Thenprogressivelyhighdensityregionsbecomeapparent.Themechanismfortheforma-tionoftheseregionsisthefollowing:Astheaverageden-sityincreases,theinteractionbetweensuccessiveshearbandsbecomesattractiveandthepathgetsrestrictedinspace.Smallfluctuationsoftheshearbandthenleadtoadensityincreaseinthisregion.Thepresenceofthesesur-roundingareasofhighdensitiesincreasestheattractionofsuccessiveshearbands,thusleadingtoapositivefeed-backprocessresultinginregionsoffinitewidthandveryhighdensitywheretheshearbandistrappedinthemid-dle,ina“canyon-like”structure(blacklinessurroundedbywhiteinFigs.4candd).

Theescapefromtheabovedescribedtrapisonlypos-sibleviaajumptoanotherlocalminimum.Theproba-bilityofsuchajumpdecreasesfasterthanexponentiallywithincreasingdensity.Thusastimegrows,theaver-agejumpsizedecreaseseventhoughlargeregionswithrelativelysmalldensitiesremain.Theprogressiveself-quenchingoftheshearbandinthesystemisresponsiblefortheanomalousslowincreaseintheaveragedensity.Thisinhomogeneousagingandextremelyslowdynamicsisreminiscentofaglassybehavior.

Inordertogetsomemoreinsightintotheslowdy-namicsofthesystemwehavestudiedthesamemodelonahierarchicallattice.Thesimplegeometryallowsforadetailedanalytictreatmentofthemodel.Thisstudywill

a)b)

c)d)

00.50.750.91

FIG.4.Snapshotsofdensitiesatdifferenttimeonasystem

ofsize256by256:a)t=103≃4N,b)t=104≃40N,c)t=105≃400N,d)t=106≃4000N.Thegreyscaleisindicatedatthebottomofthefigure.

bereportedelsewhere[22].Hereweonlysummarizethemainfeaturesofthisanalysis.Theslowdensityin-creaseandstrongsystemsizedependenceseenonthesquarelatticearealsoseeninthehierarchicalone.Herewecanshowthat1−󰀇̺󰀈decreasesasasumofpower-lawswithavanishingexponentdependingonthelatticesize,i.e.thenumberofgenerationsofthehierarchicallattice.Further,theearlytimeregimeisasinglefunctionoft/Nasforthesquarelattice,whilethelatetimeregimescalesinsteadast/Nα,whereα=1/log(2).

Inspiteofitssimplicity,themodelwehaveintroduceddisplayssomeinterestingconsequencesofcollectiveorga-nizationofdensityfluctuationsinagranularassembly.Althoughonlytime-independentrulesareintroduced,thesimulationsrevealaslowdensificationwhichoccurstogetherwithanon-trivialpatterningofthedensityinthesample.Simultaneously,theshearstrainislocalizedonshearbandswhichacquireprogressivelyalongerandlongerpersistence.Theoccurrenceofhighdensityre-gionsconfiningtheshearbandisafeaturewhichshouldbeobservableusingX-raytomographyasrecentlyper-formedintriaxialtestsbyDesruesetal[9].

Acknowledgment:Thisworkwaspartiallysup-portedbyOTKAT024004andT029985.