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