Effect of Crack Length and Orientation on the MixedMode Fracture Behavior of Graphene.docx

Effect of Crack Length and Orientation on the MixedMode Fracture Behavior of Graphene.docx

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EffectofCrackLengthandOrientationontheMixedModeFractureBehaviorofGraphene

SUPPLIMENTARYINFORMATION

EffectofCrackLengthandOrientationontheMixed-ModeFractureBehaviorofGraphene

DibakarDatta1,SivaP.V.Nadimpalli2*,YinfengLi3,VivekB.Shenoy4,5,*

Allauthorscontributedequallytothiswork

1SchoolofEngineering,BrownUniversity,Providence,RI02912,UnitesStates

2DepartmentofMechanicalEngineering,NewJerseyInstituteofTechnology,Newark,NJ,UnitedStates

3DepartmentofEngineeringMechanics,SchoolofNavalArchitecture,OceanandCivilEngineering（StateKeyLaboratoryofOceanEngineering,CollaborativeInnovationCenterforAdvancedShipandDeep-SeaExploration）,ShanghaiJiaoTongUniversity,Shanghai,China

4DepartmentofMaterialsScienceandEngineering,UniversityofPennsylvania,Philadelphia,PA19104,UnitedStates

5DepartmentofMechanicalEngineeringandAppliedMechanics,UniversityofPennsylvania,Philadelphia,PA19104,UnitedStates

*CorrespondingAuthors:

Dr.VivekB.Shenoy;Email:

vshenoy@seas.upenn.edu

Dr.SivaP.V.Nadimpalli;Email:

siva.p.nadimpalli@njit.edu

1.IsAIREBOpotentialgoodforthisproblem?

TheAIREBOpotentialwehaveusedisthesameoneusedinthepioneeringworkbyGrantabetal.Science330,946（2010）1,wheretheyhaveinvestigatedstrengthandcrackpropagationpathwayofgraphenewithdifferentsymmetricgrainboundaries.Theycarefullyvalidatedbydeformingpristinegrapheneandcomparingtheresultstoexperiments.Simulationresultsshowanelasticmodulusof1TPa,anultimatestrengthof120GPa,andastrain-at-failureof20%.Thepredictionswerewithin5-10%oftheexperimentalvaluesreportedbyLeeetal.Science321,385（2008）2.Grantabetal.discoveredmechanismsbehindtheanomalousstrengthcharacteristicsofgraphenewithgrainboundariesbystudyingthecrackpropagationpathways.

AIREBOpotentialhasbeenextensivelyusedforinvestigatingfractureandcrackpathpropagationofgrapheneunderdifferentloadingconditionsandsurfacetopology.Weietal.NatureMaterials,11,759-763（2012）3studiedfracturebehavioroftiltGBsforvariouspentagon-heptagondefects.Bystudyingthecrackpathpropagation,theydiscoveredthatmechanicalfailurealwaysstartsfromthebondsharedbyhexagon-heptagonrings.Inarecentpaper,Zhangetal.NatureCommunications5,（2014）4computedthefracturetoughnessofgraphemeandverifiedtheirresultswithexperiments.

Inaddition,therehavebeenmanyotherstudieswherepeoplehaveverifiedsimulationsresultsfromAIREBOwithexperiments.Zhangetal.NanoLetters,12（2012）5performedseriesofmoleculardynamicssimulationtoshowthattensilefracturebehaviorofananocrystallinegraphene（nc-graphene）nanostripcanbecomeinsensitivetoapre-existingflaw.TheyverifiedthatresultsobtainedfromAIREBOpotentialsperfectlymatchwithexperiments.Inaveryrecentstudy,Yinetal.NanoLetters,15（2015）6systematicallystudiedthecrackpathpropagationfordifferentcracklength.Theirestimatedvaluesoffracturetoughnessmatchwellwithexperiments4.

Huangetal.PHYSICALREVIEWB85,195453（2012）7discussedaboutthetearingofmonolayergraphene.Theydeterminedthefracturepathofmonolayergrapheneundertearingi.e.outofplandisplacementofgraphenesheet.Intheirstudy,theyconsideredtwobasicfracturemodes:

opening（modeI）andout-of-planeshear（modeIII）.Forconsistency,modeIIwasnotincluded（KII=0）.Themodemixity,

isdefinedby

rangingfrom0（puremodeIII）to

（puremodeI）.Ontheotherhand,ourworkisconfinedinin-planedisplacement.WeconsideredmodeIandmodeII.Mixed-modeintensityfactorwasdefinedas

.Wehaven'tconsideredanyout-of-planeshear.Intheirstudy,Huangetal.didn'tmentionanythingaboutapplicabilityofREBOpotential.Zhangetal.AppliedPhysicsLetters101,121915（2012）8alsostudiedstrengthandcrackpropagationofgrapheneundermixed-modeloadingwithAIREBOpotential.

Moreover,basedonREBOpotential,Zhangetal.NatureCommunications5,20144verifiedthatthefracturestressisthelowestforthezigzagcrackedge,andthelargestalongthearmchairone;itvariesmonotonicallyforintermediateorientationsofcrack.Theseresultscanbecorrelatedtotheorientationdependenceoftheedgeenergy,

whichisthelowestof11.8Jm-2forthezigzagedgeandthehighestof12.5Jm-2forthearmchairedge.Thisresultedisexpectedandwellaccepted.However,theywrote:

thistrendisoppositetothatpredictedbyreactiveforcefield（ReaxFF）calculations9.Hence,thepotentialwehaveusedcanaccuratelysimulatethecrackpropagationpathwayofgrapheneunderdifferentloadingandsurfacetopologyandresultsareingoodagreementwithexperimentalresults.

2.Effectoflatticetrapping

AsmentionedbyZhangetal.10thelattice-trappingstrengthmonotonicallydecreaseswithincreasinginteractionrangeofthepotentialandbecomenegligiblysmallforlong-rangepotentials.TheAdaptiveIntermolecularReactiveEmpiricalBondOrderPotential（AIREBO）11,includedintheLAMMPSpackage,wasusedtomodeltheinter-atomicforcesinthisstudy.TheAIREBOpotentialisanimprovedversionofBrenner'swell-knownsecondgenerationReactiveEmpiricalBondOrderPotential（REBO）thatincludesaLenhard-Jones（LJ）potentialformtodescribetheVan-der-Waalslong-rangeinteractionsandatorsionaltermforthe

-bondtorsion.ThegeneralformoftheAIREBOpotentialis

（S-1）

Thelong-rangedinteractionsbetweensp2sheetsaredescribedthroughLJinteractions:

the

termprovidesthemissingrepulsion,whilethe

termcapturesthevanderWaalsinteraction.TheLJterminAIREBOisgraftedontotheREBOformalism,employingswitchingfunctionsto（de）activatetheLJinteractionsonthebasisofdistanceandbond-order.Becauseofthislong-rangeinteraction,wedidn'tconsiderlattice-trappingeffect.

3.Effectofedgestressandsurfaceenergy

TherehavebeenseveralstudiesonthevariousaspectsofmechanicalpropertiesofGNRespeciallyabouttheedgelengthbeyondwhicheffectoflengthdiminishes.Reddyetal.12wrote:

`asthewidthofsheetincreases,theedgeeffectgraduallydiminishes,andtheeffectiveelasticmodulusEeffapproachesaconstantvalue,whichisthebulkelasticmodulusE（1900eV/nm2or0.9TPa,assumingthegraphenethicknessof0.335nm）ofgraphene.Thisvalue,whichisindependentofthewidthandedgestructureofgraphenesheet,isingoodagreementwiththeexperimentallymeasuredvalue2’.Luetal.13wrote`theedgeeffectontheinitialYoung'smodulusofGNRsdiminishesastheribbonwidthincreases.'

Reddyetal.12showedthatforGNRfreeedgelengthofover8nm,effectiveelasticmodulusconvergesto≈1900eV/nm2forallkindsoffreeedges–zigzag,armchair,hydrogenatedzigzag,hydrogenatedarmchair.Luetal.13alsoshowedthatbeyond8nmoffreeedgelength,2DYoung’smodulusconvergesto≈243N/m.Zhaoetal.showedthatforgraphenenanoribbon’sdiagonallengthover≈12nm,Young’smodulusandPoissonratioconvergestoperiodicvaluesi.e.1.01±0.03TPaand0.21±0.01respectively.

That’swhyweconsideredribbonwidth≈10nmasatthislength,edgeeffectdiminishes.Wedidn’tconsiderperiodicityaswefollowedthepioneeringworkofGrantabetal.1

4.AIREBOpotentialforsimulationathightemperature

AIREBOpotentialwehaveusedhasbeenverifiedandmatchedwithexperimentalresults.Thispotentialhasbeenusedtostudyfractureofgrapheneathighertemperature.Zhaoetal.14investigatedthevariationinfracturestrengthofgraphenewithtemperature,strainrate,andcracklengthusingmoleculardynamics（MD）simulations.Theystudiedfractureofpristinegrapheneundervarioustemperatures–300K,600K,900K,1200K,1500K,1800K,2100K,and2400K.Inaddition,graphenewithslitswerestudiedathightemperaturee.g.1200K.Thisstudyiswellacceptedandhasbeencitedover100times.

Recently,Dewapriyaetal.15studiedgraphenesheetwithdifferentcracklengthsatvarioustemperatures.Inaddition,AIREBOpotentialhasbeenwidelyusedtostudyfractureofgraphenewithdifferenttopologicalandloadingconditionsathightemperaturese.g.Zhangetal.16,Baimovaetal.17.

5.ArmchairandZigzagDirection

FigureS1:

GNR（withcrackoflength2a）issubjectedtoloading（markedinred）inboth（a）zigzagand（b）armchairdirection.

Inthisstudy,wehaveconsideredpullinginarmchairandzigzagdirections.WehaveclearedourdefinitionofarmchairandzigzagdirectionsinFigureS1.WementionedinourmanuscriptthatGNRwasloadedinbothzigzagandarmchairdirectionsasshowninFigureS1aandS1brespectively.Figure1and2showtheModel-1andModel-2respectively.TherewehaveshownonlyonedirectionofloadinginordertoclarifythemeaningofloadingangleΦ,slitangleθandotherdefinitions.GiventhatdefinitionofarmchairandzigzagdirectionsareclearedinFigureS1,whenwespeakaboutloadinginthesetwodirections,weassumethatreaderswillunderstandwhatwemeanforit.

6.HowtocomputethefarfieldstressinGrapheneNanoribbon（GNR）?

FigureS2:

PristineGNRissubjectedtotensileloading.Stresscomputedbyaveragingoverthewholesystemgivescorrectstrengthandyoungmodulus.Stresscomputedbyaveragingovertheatomsattheboundaryyieldserroneousresults.

FigureS3:

ComputationoffarfieldstressforSIFcalculation.Farfieldstresscomputedbyaveragingovertheentiresystemgivescorrectmeasurement.

7.StressIntensityFactors（SIF）withrespecttoLoadingAnglefordifferentcracklength

FigureS4:

StressIntensityFactors（SIF）forcrackswitharmchair（a-c）