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1、HiddenareaandmechanicalnonlinearitiesinfreestandinggrapheneRyanJ.T.Nicholl1,NickolayV.Lavrik2,IvanVlassiouk3,BernadetaR.Srijanto2,andKirillI.Bolotin1,41DepartmentofPhysicsandAstronomy,VanderbiltUniversity,Nashville,TN37235,USA2CenterforNanophaseMaterialsSciences,OakRidgeNationalLaboratory,Oak
2、Ridge,TN37831,USA3Energy&TransportationScienceDivision,OakRidgeNationalLaboratory,OakRidge,TN37831,USA4DepartmentofPhysics,FreieUniversitätBerlin,Arnimallee14,Berlin14195,GermanyWeinvestigatedtheeffectofout-of-planecrumplingonthemechanicalresponseofgraphenemembranes.Inourexperiments,stresswas
3、appliedtographenemembranesusingpressurizedgaswhilethestrainstatewasmonitoredthroughtwocomplementarytechniques:interferometricprofilometryandRamanspectroscopy.Bycomparingthedataobtainedthroughthesetwotechniques,wedeterminedthegeometrichiddenareawhichquantifiesthecrumplingstrength.Whilethedevic
4、eswithhiddenarea~?%obeyedlinearmechanicswithbiaxialstiffness???±??N/m,specimenswithhiddenareaintherange?.?−?.?%werefoundtoobeyananomalousnonlinearHooke’slawwithanexponent~?.?.Athinmembraneisalwayscrumpledduetoitslowbendingrigidityandresultinginabilitytosustaincompressiveforces.Suchcrumplingha
5、sbeenactivelyinvestigatedduringthelastthreedecadestodescribethebehaviorsofwrinkledskin[1,2],biologicallipidmembranes[3,4],andsolarsails[5].Theadventofgrapheneandother2Dmaterialsallowedtestingthemodelsofcrumplingincrystallinemembranesattheultimateatomicthicknesslimit[6].Ingraphenespecifically,
6、crumplingoriginatesfromstaticwrinkling[7–9]andout-of-plane(flexural)phonons[10–12]andpersistsinbothsupportedandfree-standingsamples[13].Recenttheoreticalworkshowedthateverymechanicalpropertyofgrapheneisrenormalizedduetocrumpling[14–20].Inparticular,crumplingcausesthereductionofthestiffness[21
7、,22],increasedbendingrigidity[11,23],variable(andnegative)Poisson’sratio[24,25],andnegativethermalexpansion[26,27].Atthesametime,thecontributionduetocrumplingisalmostuniversallyignoredintheexperimentsprobingmechanicsofthesematerials.Thismayle
