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ORIGINALRESEARCHTheeffectofmulti-directionalnanocompositematerialsonthevibrationalresponseofthickshellpanelswithnitelengthandrestedontwo-parameterelasticfoundationsVahidTahouneh1MohammadHasanNaei2Received:1April2015/Accepted:31January2016/Publishedonline:24February2016TheAuthor(s)2016.
ThisarticleispublishedwithopenaccessatSpringerlink.
comAbstractThemainpurposeofthispaperistoinvestigatetheeffectofbidirectionalcontinuouslygradednanocom-positematerialsonfreevibrationofthickshellpanelsrestedonelasticfoundations.
TheelasticfoundationisconsideredasaPasternakmodelafteraddingashearlayertotheWinklermodel.
Thepanelsreinforcedbyrandomlyorientedstraightsingle-walledcarbonnanotubesarecon-sidered.
ThevolumefractionsofSWCNTsareassumedtobegradednotonlyintheradialdirection,butalsoinaxialdirectionofthecurvedpanel.
Thisstudypresentsa2-Dsix-parameterpower-lawdistributionforCNTsvolumefrac-tionof2-Dcontinuouslygradednanocompositethatgivesdesignersapowerfultoolforexibledesigningofstruc-turesundermulti-functionalrequirements.
Thebenetofusinggeneralizedpower-lawdistributionistoillustrateandpresentusefulresultsarisingfromsymmetric,asymmetricandclassicproles.
ThematerialpropertiesaredeterminedintermsoflocalvolumefractionsandmaterialpropertiesbyMori–Tanakascheme.
The2-Ddifferentialquadraturemethodasanefcientnumericaltoolisusedtodiscretizegoverningequationsandtoimplementboundarycondi-tions.
Thefastrateofconvergenceofthemethodisshownandresultsarecomparedagainstexistingresultsinlitera-ture.
Somenewresultsfornaturalfrequenciesoftheshellareprepared,whichincludetheeffectsofelasticcoefcientsoffoundation,boundaryconditions,materialandgeometricalparameters.
Theinterestingresultsindi-catethatagradednanocompositevolumefractionintwodirectionshasahighercapabilitytoreducethenaturalfrequencythanconventional1-Dfunctionallygradednanocompositematerials.
KeywordsThickshellpanelsRandomlyorientedstraightsingle-walledCNTsTwo-parameterelasticfoundationsVibrationanalysisofstructuresIntroductionLayeredcompositematerials,duetotheirthermalandmechanicalmeritscomparedtosingle-composedmaterials,havebeenwidelyusedforavarietyofengineeringappli-cations.
However,owingtothesharpdiscontinuityinthematerialpropertiesatinterfacesbetweentwodifferentlayers,theremayexiststressconcentrationscausingseverematerialfailure(Weissenbeketal.
1997).
Functionallygradedmaterialsareheterogeneouscompositematerials,inwhichthematerialpropertiesvarycontinuouslyfromoneinterfacetotheother.
Theadvantageofusingthesemate-rialsisthattheycansurviveinhighthermalgradientenvironment,whilemaintainingtheirstructuralintegrity.
Typically,anFGMismadeofaceramicandametalforthepurposeofthermalprotectionagainstlargetemperaturegradients.
Theceramicmaterialprovidesahigh-tempera-tureresistanceduetoitslowthermalconductivity,whiletheductilemetalconstituentpreventsfractureduetoitsgreatertoughness.
FGMsarenowdevelopedforgeneraluseasstructuralelementsinextremelyhightemperatureenvironments.
AlistingofdifferentapplicationscanbefoundinForum(1991).
MostofthestudiesonFGMshave&VahidTahounehvahid.
tahouneh@ut.
ac.
ir;vahid.
th1982@gmail.
comMohammadHasanNaeimhnaei@ut.
ac.
ir1YoungResearchersandEliteClub,IslamshahrBranch,IslamicAzadUniversity,Islamshahr,Iran2SchoolofMechanicalEngineering,TehranUniversity,Tehran,Iran123IntJAdvStructEng(2016)8:11–28DOI10.
1007/s40091-016-0110-4beenrestrictedtothermalstressanalysis,thermalbuckling,fracturemechanicsandoptimization(ChoandTinsleyoden2000;Chunyuetal.
2001;Lanhe2004).
Someresearches(Loyetal.
1999;Pradhanetal.
2000;Hanetal.
2001;Ngetal.
2001)arebasedontheclassicalshellthe-ory,i.
e.
,neglectingtheeffectoftransversesheardefor-mation.
Theapplicationofthistheorytomoderatelythickorthickshellstructurescanleadtoseriouserrors.
Usingtherstandhigherordersheardeformationtheories,somemodicationsaredonetoincludetheeffectsoftransversesheardeformation.
Inthisstudy,theproblemformulationswerebasedonthehigherordersheardeformationshelltheories.
YangandShen(2003)proposedasemi-analyticalapproachbasedonReddy'shigherordersheardeformationshelltheory,forfreevibrationanddynamicinstabilityofFGMcylindricalpanelsundercombinedstaticandperiodicaxialforcesandthermalloads.
Freevibrationandstabilityoffunctionallygradedshallowshellsaccordingtoa2-DhigherorderdeformationtheorywereinvestigatedbyMatsunaga(2008).
FreevibrationanalysisoffunctionallygradedcurvedpanelswascarriedoutusingahigherorderformulationbyPradyumnaandBandyopadhyay(2008).
TheyusedaC0niteelementformulationtocarryouttheanalysis.
Usinga2-Dhigherordersheardeformationthe-ory,vibrationandbucklinganalysesofsimplysupportedcircularcylindricalshellsmadeoffunctionallygradedmaterials(FGMs)werestudiedbyMatsunaga(2009).
Heusedthemethodofpowerseriesexpansionofcontinuousdisplacementcomponentstosolvetheproblem.
Inalloftheabovestudiesthevariationoftheradiusthroughthethicknesswasnotconsideredandtheproblemformulationswerebasedontheconstantmeanradiusofcurvature.
Two-dimensionaltheoriesreducethedimensionsofproblemsfromthreetotwobyintroducingsomeassump-tionsinmathematicalmodelingleadingtosimplerexpressionsandderivationofsolutions.
However,thesesimplicationsinherentlybringerrorsandthereforemayleadtounreliableresultsforrelativelythickpanels.
Asaresult,three-dimensionalanalysisofpanelsnotonlypro-videsrealisticresults,butalsoallowsfurtherphysicalinsights,whichcannototherwisebepredictedbythetwo-dimensionalanalysis.
Therearesomestudiesonfreevibrationanalysisofisotropicandcompositepanelsandshellsbasedonthethree-dimensionalelasticityformulation(ChernandChao2000).
Structuresrestingonelasticfoundationswithdifferentshapes,sizes,andthicknessvariationsandboundarycon-ditionshavebeenthesubjectofinvestigations,andthoseplayanimportantroleinaerospace,marine,civil,mechanical,electronicandnuclearengineeringproblems.
Forexample,platesandshellsareusedinvariouskindsofindustrialapplicationssuchastheanalysisofreinforcedconcretepavementofroads,airportrunwaysandfoundationsofbuildings.
ThePasternakmodel(alsoreferredtoasthetwo-parametermodel)waswidelyusedtodescribethemechanicalbehaviorofthefoundation,inwhichthewell-knownWinklermodelisaspecialcase.
ThemostseriousdeciencyoftheWinklerfoundationmodelistohavenointeractionbetweenthesprings.
Inotherwords,thespringsinthismodelareassumedtobeindependentandunconnected.
TheWinklerfoundationmodelisfairlyimprovedbyadoptingthePasternakfoun-dationmodel,atwo-parametermodel,inwhichtheshearstiffnessofthefoundationisconsidered.
Theevidentimportanceinpracticalapplications,investigationsonthedynamiccharacteristicsofFGMplatesandpanelsonelasticfoundationsarestilllimitedinnumber.
YasandTahouneh(2012)investigatedthefreevibrationanalysisofthickFGannularplatesonelasticfoundationsviadiffer-entialquadraturemethodbasedonthethree-dimensionalelasticitytheoryandTahounehandYas(2012)investigatedthefreevibrationanalysisofthickFGannularsectorplatesonPasternakelasticfoundationsusingDQM.
Tahounehetal.
(2013)studiedfreevibrationcharacteristicsofannularcontinuousgradingberreinforced(CGFR)platesrestingonelasticfoundationsusingDQM.
Morerecently,(TahounehandNaei2014)achievedthenaturalfrequen-ciesofthickmulti-directionalfunctionallygradedrectan-gularplatesrestingonatwo-parameterelasticfoundationvia2-Ddifferentialquadraturemethod,Theproposedrectangularplateshadtwooppositeedgessimplysup-ported,whileallpossiblecombinationsoffree,simplysupportedandclampedboundaryconditionswereappliedtotheothertwoedges.
Faridetal.
(2010)presentedfreevibrationanalysisofinitiallystressedthicksimplysup-portedfunctionallygradedcurvedpanelrestingontwo-parameterelasticfoundation(Pasternakmodel),subjectedinthermalenvironmentwasstudiedusingthethree-di-mensionalelasticityformulation.
Tahouneh(2014)inves-tigatedfreevibrationanalysisofcontinuousgradingberreinforced(CGFR)FGannularsectorplatesontwo-pa-rameterelasticfoundationsundervariousboundarycon-ditions,basedonthethree-dimensionaltheoryofelasticity.
Theplateswithsimplysupportedradialedgesandarbitraryboundaryconditionsontheircircularedgeswereconsidered.
Recently,nanocompositeshavesignicantimportanceforengineeringapplicationsthatrequirehighlevelsofstructuralperformanceandmulti-functionality.
Carbonnanotubes(CNTs)havedemonstratedexceptionalmechanical,thermalandelectricalproperties.
Thesematerialsareconsideredasoneofthemostpromisingreinforcementmaterialsforhighperformancestructuralandmulti-functionalcompositeswithvastapplicationpotentials(EsawiandFarag2007;Thostensonetal.
2001).
Moststudiesoncarbonnanotube-reinforcedcomposites12IntJAdvStructEng(2016)8:11–28123(CNTRCs)havefocusedontheirmaterialproperties(EsawiandFarag2007;Thostensonetal.
2001;Dai2002;Kangetal.
2006;Lauetal.
2006).
Gojnyetal.
(2005)focusedontheevaluationofthedifferenttypesoftheCNTsapplied,theirinuenceonthemechanicalpropertiesofepoxy-basednanocompositesandtherelevanceofsur-facectionalization.
Fidelusetal.
(2005)investigatedthermo-mechanicalpropertiesofepoxy-basednanocom-positesbasedonlowweightfractionofrandomlyorientedsingle-andmulti-walledCNTs.
HanandElliott(2007)determinedtheelasticmodulusofcompositestructuresunderCNTsreinforcementbymoleculardynamicsimula-tionandinvestigatedtheeffectofvolumefractionofSWNTsonmechanicalpropertiesofnanocomposites.
Manchadoetal.
(2005)blendedsmallamountsofarc-SWNTintoisotacticpolypropyleneandobservedthemodulusincreasefrom0.
85to1.
19GPaat0.
75wt%.
Inaddition,thestrengthincreasedfrom31to36MPaby0.
5wt%.
Bothpropertieswereobservedtofalloffathigherloadinglevels.
Theseinvestigationsand(Mokashietal.
2007;Zhuetal.
2007)haveshownthattheadditionofsmallamountofcarbonnanotubeinthematrixcancon-siderablyimprovethemechanical,electricalandthermalpropertiesofpolymericcomposites.
Thisbehavior,com-binedwiththeirlowdensitymakesthemsuitablefortransportindustries,especiallyforaeronauticandaero-spaceapplicationswherethereductionofweightiscrucialinordertoreducethefuelconsumption.
ThepropertiesoftheCNT-reinforcedcomposites(CNTRCs)dependonavarietyofparametersincludingCNTgeometryandtheinter-phasebetweenthematrixandCNT.
Interfacialbondingintheinter-phaseregionbetweenembeddedCNTanditssurroundingpolymerisacrucialissuefortheloadtransferringandreinforcementphenom-enaShokriehandRaee(2010).
Thetraditionalapproachtofabricatingnanocompositesimpliesthatthenanotubeisdistributedeitheruniformlyorrandomlysuchthattheresultingmechanical,thermal,orphysicalpropertiesdonotvaryspatiallyatthemacroscopiclevel.
ExperimentalandnumericalstudiesconcerningCNTRCshaveshownthatdistributingCNTsuniformlyasthereinforcementsinthematrixcanachievemoderateimprovementofthemechanicalpropertiesonly(SeidelandLagoudas2006).
ThisismainlyduetotheweakinterfacebetweentheCNTsandthematrixwhereasignicantmaterialpropertymis-matchexists.
TheconceptofFGMcanbeutilizedforthemanagementofamaterial'smicrostructure,sothatthevibrationalbehaviorofaplate/shellstructurereinforcedbyCNTscanbeimproved.
Accordingtoacomprehensivesurveyofliterature,theauthorsfoundthattherearefewresearchstudiesonthemechanicalbehavioroffunctionallygradedCNTRCstructures.
Forthersttime,Shen(2009)suggestedthatthenonlinearbendingbehaviorcanbeconsiderablyimprovedthroughtheuseofafunctionallygradeddistributionofCNTsinthematrix.
HeintroducedtheCNTefciencyparametertoaccountloadtransferbetweenthenanotubeandpolymericphases.
Duetointrinsiccomplexityoftheformulationsbasedonthethree-dimensionalelasticity,powerfulnumericalmethodsareneededtosolvethegoverningequations.
Thedifferentialquadraturemethod(DQM)isarelativelynewnumericaltechniqueinstructuralanalysis.
Areviewoftheearlydevelopmentsinthedifferentialquadraturemethodcanbefoundinpapersby(BertandMalik1997).
Thispaperismotivatedbythelackofstudiesinthetechnicalliteratureconcerningtothethree-dimensionalvibrationanalysisofthickbidirectionalnanocompositecurvedpanelsrestingonatwo-parameterelasticfounda-tionreinforcedbyrandomlyorientedstraightsingle-wal-ledcarbonnanotubesCNTs.
Totheauthors'bestknowledge,researchonthevibrationofthickcurvedpanelsreinforcedbyrandomlyorientedstraightsingle-walledcarbonnanotubeswhicharegradedinbothdirectionincludingaxialandradialdirectionshasnotbeenseenuntilnow.
Thevolumefractionsofrandomlystraightsingle-walledcarbonnanotubesSWCNTsareassumedtobegradedinthethicknessandalsoaxialdirectionsofthecurvedpanels.
ThedirectapplicationofCNTpropertiesinmicromechanicsmodelsforpredictingmaterialpropertiesofthenanotube/polymercompositeisinappropriatewithouttakingintoaccounttheeffectsassociatedwiththesignicantsizedifferencebetweenananotubeandatypicalcarbonber(Odegardetal.
2003).
Inotherwords,continuummicromechanicsequationscannotcapturethescaledifferencebetweenthenanoandmicro-levels.
Inordertoovercomethislimitation,avir-tualequivalentberconsistingofnanotubeanditsinter-phasewhichisperfectlybondedtosurroundingresinisapplied.
Thisstudypresentsanovel2-Dsix-parameterpower-lawdistributionforCNTsvolumefractionof2-Dfunc-tionallygradednanocompositematerialsthatgivesdesignersapowerfultoolforexibledesigningofstruc-turesundermulti-functionalrequirements.
Variousmate-rialprolesalongtheradialandaxialdirectionsareillustratedbyusingthe2-Dpower-lawdistribution.
TheeffectivematerialpropertiesatapointaredeterminedintermsofthelocalvolumefractionsandthematerialpropertiesbytheMori–Tanakascheme.
Asensitivityanalysisisperformed,andthenaturalfrequenciesarecal-culatedfordifferentsetsofboundaryconditionsanddif-ferentcombinationsofthegeometric,material,andfoundationparameters.
Therefore,verycomplexcombi-nationsofthematerialproperties,boundaryconditions,andfoundationstiffnessareconsideredinthepresentsemi-analyticalsolutionapproach.
IntJAdvStructEng(2016)8:11–2813123ProblemdescriptionInthissection,avirtualequivalentberconsistingofananotubeanditsinter-phasewhichisperfectlybondedtosurroundingresinisintroducedtoobtainthemechanicalpropertiesofthecarbonnanotube/polymercompositebyusingtheresultsofmulti-scaleFEMShokriehandRaee(2010).
TheequivalentberforSWCNTwithchiralindexof(10,10)isasolidcylinderwithdiameterof1.
424nm.
Theinverseoftheruleofmixtureisusedtocalculatematerialpropertiesofequivalentber(Tsaietal.
2003):ELEFELCVEFEMVMVEF;1ETEF1ETCVEFVMEMVEF;1GEF1GCVEFVMGMVEF;tEFtCVEFtMVMVEF;1whereELEF,ETEF,GEF,tEF,ELC,ETC,GC,tC,EM,GM,tM,VEFandVMarelongitudinalmodulusofequivalentber,transversemodulusofequivalentber,shearmodulusofequivalentber,Poisson'sratioofequivalentber,longi-tudinalmodulusofcomposites,transversemodulusofcomposites,shearmodulusofcomposites,Poisson'sratioofcomposites,modulusofmatrix,shearmodulusofmatrix,Poisson'sratioofmatrix,volumefractionoftheequivalentberandvolumefractionofthematrix,respectively.
ELC,GC,andETCareobtainedfrommulti-scaleFEMormoleculardynamics(MD)simulations.
Itshouldbementionedthatthevolumefractionoftheequivalentberisassumedtobe7.
5%(ShokriehandRaee2010)andPoly{(mphenylenevinylene)-co-[(2,5-dioctoxy-p-phenyle)vinylene]},referredtoas(PmPV),isselectedasamatrixmaterial:Em2:1Gpa;qm1150kg/m3;tm0:34:Thematerialpropertiesadoptedforequivalentberare(ShokriehandRaee2010):Ecn1649:12Gpa;Ecn111:27Gpa;t0:284;Gcn5:13Gpa;qcn1400kg/m3Compositesreinforcedwithaligned,straightCNTsFollowingthestandardMTderivation,onecandeveloptheexpressionforeffectivecompositestiffnessC.
ThisisobtainedbyusinganequivalentberhavingtheeffectiveCNTpropertiesintheMTapproachwhichisgivenas(Shietal.
2004):CCmfrCrCmArhifmIfrArhi1;2wherefrandfmaretheberandmatrixvolumefractions,respectively.
Cmisthestiffnesstensorofthematrixmaterial;Cristhestiffnesstensoroftheequivalentber;IistheforthorderidentitytensorandAristhedilutestrain-concentrationtensoroftherthphasefortheberwhichisgivenas:ArISCm1CrCmhi1;3whereSisEshelby'stensor,asgivenby(Eshelby1957)and(Mura1982).
ThetermsenclosedbyanglebracketsinEq.
(2)representtheaveragevalueofthetermoverallorientationsdenedbytransformationfromthelocalbercoordinates(O-x01x02x03)totheglobalcoordinates(O-x1x2x3)(Fig.
1).
Assumeaxisx2asthedirectionalongthealignednanotube.
Theelasticpropertiesofthenanocompositearedeterminedfromtheaveragestrainobtainedintherepresentativevolumeelement.
Thematrixisassumedtobeelasticandisotropic,withYoung'smodulusEmandPoisson'sratiotm.
EachstraightCNTismodeledasalongberwithtransverselyisotropicelasticpropertiesandhasastiffnessmatrixgivenbyEq.
(1).
Therefore,thecompositeisalsotransverselyiso-tropic,withveindependentelasticconstants.
Thesubstitu-tionofnonvanishingcomponentsoftheEshelbytensorSforastraight,longberalongthex2-direction(Shietal.
2004)inEq.
(3)givesthedilutemechanicalstrainconcentrationten-sor.
Then,thesubstitutionofEq.
(3)intoEq.
(2)givesthetensorofeffectiveelasticmoduliofthecompositereinforcedbyaligned,straightCNTs.
TheaxialandtransverseYoung'smodulusofthecompositecanbecalculatedfromtheHill'selasticmodulusas(Shietal.
2004):E1nl2k;E24mknl2knl2mn;4Fig.
1Representativevolumeelement(RVE)withrandomlyori-ented,straightCNT14IntJAdvStructEng(2016)8:11–28123wherek,l,mandnareitsplane-strainbulkmodulusnor-maltotheberdirection,cross-modulus,transverseshearmodulus,axialmodulusandaxialshearmodulus,respec-tively,andcanbefoundintheAppendix.
Asmentionedbefore,theCNTsaretransverselyisotropicandhaveastiffnessmatrixgivenbelow:Cr1ELtTLETtZLEZ000tLTEL1ETtZTEZ000tLZELtTZET1EZ0000001GTZ0000001GZL0000001GLT266666666666666666437777777777777777755whereEL;ET;EZ;GTZ;GZL;GLT;tLT;tLZ;tTZarematerialpropertiesoftheequivalentberwhichcanbedeterminedfromtheinverseoftheruleofmixture.
Compositesreinforcedwithrandomlyoriented,straightCNTsTheeffectivepropertiesofcompositeswithrandomlyori-entednon-clusteredCNTs,suchasinFig.
1,arestudiedinthissection.
Theresultingeffectivepropertiesfortheran-domlyorientedCNTcompositeareisotropic,despitetheCNTshavingtransverselyisotropiceffectiveproperties.
TheorientationofastraightCNTischaracterizedbytwoEuleranglesaandb,asshowninFig.
1.
WhenCNTsarecompletelyrandomlyorientedinthematrix,thecompositeisthenisotropic,anditsbulkmoduluskandshearmodulusGarederivedas:kkmfrdr3Kmar3fmfrar;GGmfrgr2Gmbr2fmfrbr;6wherear;br;drandgrcanbefoundintheAppendix.
TheeffectiveYoung'smodulusEandPoisson'sratiotofthecompositeisgivenby:E9KG3KG;t3K2G6K2G7Functionallygradedcarbonnanotube-reinforcedConsiderabidirectionalnanocompositecurvedpanelres-tedontwo-parameterelasticfoundationsasshowninFig.
2.
Acylindricalcoordinatesystem(r,h,z)isusedtolabelthematerialpointofthepanel.
TheinnersurfaceiscontinuouslyincontactwithanelasticmediumthatactsasanelasticfoundationrepresentedbytheWinkler/PasternakmodelwithKwandKgthatareWinklerandshearcoef-cientsofPasternakfoundation,respectively.
Oneofthewell-knownpower-lawdistributionswhichiswidelyconsideredbytheresearchersisthree-orfour-pa-rameterpower-lawdistribution.
Thebenetofusingsuchpower-lawdistributionsistoillustrateandpresentusefulresultsarisingfromsymmetricandasymmetricproles.
ConsiderVc(volumefractionoftheCNTs)informoff(z)9g(r),f(z)andg(r)areboththethree-parameterpower-lawdistribution.
Theycanbeusedtoillustratesymmetric,asymmetricandclassicalprolesalongtheaxialandradialdirectionsofthecurvedpanels,respec-tively.
SobyconsideringVcasf(z)9g(r),onecanpresenta2-Dsix-parameterpower-lawdistributionwhichisusefultoillustratedifferenttypesofvolumefractionproles,includingclassical–classical,symmetric–symmetricandclassical–symmetricinbothdirections.
Inordertoinvestigate3-Ddynamicresponseofthickbidirectionalnanocompositecurvedpanelsrestingonatwo-parameterelasticfoundation,itisassumedthatthevolumefractionoftheCNTsfollowsa2-Dsix-parameterpower-lawdistribution:VcVbVa12rRhar12rRhbr!
crVa!
1zLzazzLzbz!
cz;8wheretheradialvolumefractionindexcr,andtheparametersar,brandtheaxialvolumefractionindexcz,andtheparametersaz,bzgovernthematerialvariationprolethroughtheradialandaxialdirections,respectively.
ThevolumefractionsVaandVb,whichhavevaluesthatrangefrom0to1,denotethemaximumandminimumvolumefractionofCNTs.
WithassumptionVb=1andVa=0.
3,somematerialprolesintheradialFig.
2Thesketchofanelasticallysupportedthickbidirectionalnanocompositecylindricalpanelrestingonatwo-parameterelasticfoundationandsetupofthecoordinatesystemIntJAdvStructEng(2016)8:11–2815123[gr=(r-R)/h]andaxial(gz=z/Lz)directionsareillus-tratedinFigs.
3,4and5.
AscanbeseenfromFig.
3,theclassicalvolumefractionprolesintheradialandaxialdirectionsarepresentedasspecialcaseofthe2-Dpower-lawdistributionbysettingcrcz4;andaraz0.
InFig.
3,TheCNTsvolumefractiondecreasesintheaxialdirectionfrom1atgz=-0.
5to0atgz=0.
5.
Withanotherchoiceoftheparametersaz,bz,arandbr,itispossibletoobtainvolumefractionprolesalongtheradialandaxialdirectionsofthepanelasshowninFig.
4.
Thisgureshowsaclassicalproleversusgrandasymmetricproleversusgz.
Asobserved,volumefractionontheloweredge(gz=-0.
5)isthesameasthatontheupperedge(gz=0.
5).
Figure5illustratessymmetricprolesthroughtheradialandaxialdirectionsobtainedbysettingbrbz2;andaraz1.
Inthefollowing,wehavecomparedseveraldifferentvolumefractionprolesofconventional1-Dand2-Dcontinuouslygradednanocom-positewithappropriatechoiceoftheradialandaxialparametersofthe2-Dsix-parameterpower-lawdistribu-tion,asshowninTable1.
Itshouldbenotedthatthenotationclassical–symmetricindicatesthatthe2-Dnanocompositecurvedpanelhasclassicalandsymmetricvolumefractionprolesintheradialandaxialdirections,respectively.
ThebasicformulationsThemechanicalconstitutiverelationthatrelatesthestres-sestothestrainsisasfollows:rrrhrzszhsrzsrh2666666437777775C11C12C13000C12C22C23000C13C23C33000000C44000000C55000000C662666666437777775erehezczhcrzcrh2666666437777775:9Fig.
3Variationsoftheclassicalvolumefractionproleintheradialandaxialdirectionscrcz4;araz0Fig.
4Variationsofthevolumefractionprolealongtheradialandaxialdirectionsofthecurvedpanelscrcz3;ar0;az1;bz2Fig.
5Variationsofthesymmetricvolumefractionprolealongtheradialandaxialdirectionsofthecurvedpanelscrcz3;araz1;bzbr2Table1Variousvolumefractionproles,differentparameters,andvolumefractionindicesof2-Dpower-lawdistributionsVolumefractionproleRadialvolumefractionindexandparametersAxialvolumefractionindexandparametersClassical–classicalar0az0Symmetric–symmetricar1;br2az1;br2Classical–symmetricar0az1;bz2Classicalradiallyar0cz0Symmetricradiallyar1;br2cz016IntJAdvStructEng(2016)8:11–28123Intheabsenceofbodyforces,thegoverningequationsareasfollows:orror1rosrhohosrzozrrrhrqo2urot2;osrhor1rorhohoshzoz2srhrqo2uhot2;osrzor1roshzohorzozsrzrqo2uzot210Strain–displacementrelationsareexpressedas:erouror;ehurr1rouhoh;ezouzoz;chzouhoz1rouzoh;crzourozouzor;crh1rourohouhoruhr11whereur,uhanduzareradial,circumferentialandaxialdisplacementcomponents,respectively.
UponsubstitutionEq.
(11)into(9)andtheninto(10),theequationsofmotionintermsofdisplacementcomponentswithinnitesimaldeformationscanbewrittenas:c11o2uror2c121r2ouhoh1ro2uhoroh1rouror1r2urc13o2uzorozoc11orouroroc12orurr1rouhohoc13orouzozc66ro2uhohor1ro2uroh21rouhohc55o2uroz2o2uzozor1rc11ourorc12urr1rouhohc13ouzozc12ourorc22urr1rouhohc23ouzozqo2urot212c661r2ouroh1ro2uroroho2uhor2uhr21rouhoroc66or1rourohouhoruhr1rc12o2urohorc221rouroh1ro2uhoh2c23o2uzohozc44o2uhoz21ro2uzozoh2c66r1rourohouhoruhrqo2uhot213c55o2urorozo2uzor2oc55orourozouzorc44ro2uhohoz1ro2uzoh2c13o2urozorc231rouroz1ro2uhohozc33o2uzoz2c55rourozouzorqo2uzot214Theboundaryconditionsattheconcaveandconvexsurfaces,r=riandro,respectively,canbedescribedasfollows:–Atr=ro,risrzsrh0;rrkwurkgo2uroz21r2o2uroh2&'atrri0atrro8>>>>>>>>>>>>>>>>:23TheboundaryconditionsstatedinEqs.
(16,17,18)canalsobesimplied;however,forthesaleofbrevity,theyarenotshownhere.
2-DDQMsolutionofgoverningequationsItisdifculttosolveanalyticallytheequationsofmotion,ifitisnotimpossible.
Hence,oneshoulduseanapprox-imatemethodtondasolution.
Here,thedifferentialquadraturemethod(DQM)isemployed.
OnecancompareDQMsolutionprocedurewiththeothertwowidelyusedtraditionalmethodsforplateanalysis,i.
e.
,Rayleigh–RitzmethodandFEM.
ThemaindifferencebetweentheDQMandtheothermethodsishowthegoverningequationsarediscretized.
InDQM,thegoverningequationsandboundaryconditionsaredirectlydiscretized,andthuselementsofstiffnessandmassmatricesareevaluateddirectly.
ButinRayleigh–RitzandFEMs,theweakformofthegoverningequationsshouldbedevelopedandtheboundaryconditionsaresatisedintheweakform.
Generallybydoingsolargernumberofintegralswithincreasingamountofdifferentiationshouldbedonetoarriveattheelementmatrices.
Inaddition,thenumberofdegreesoffreedomwillbeincreasedforanacceptableaccuracy.
ThebasicideaoftheDQMisthederivativeofafunc-tion,withrespecttoaspacevariableatagivensamplingpoint,isapproximatedasaweightedlinearsumofthesamplingpointsinthedomainofthatvariable.
InordertoillustratetheDQapproximation,considerafunctionfn;gdenedonarectangulardomain0naand0gb.
Letinthegivendomain,thefunctionvaluesbeknownordesiredonagridofsamplingpoints.
AccordingtoDQMmethod,therthderivativeofthefunctionfn;gcanbeapproximatedas:orfn;gonrjn;gni;gjXNnm1Anrimfmjfori1;2;Nnandr1;2;Nn124whereNnrepresentsthetotalnumberofnodesalongthen-direction.
FromthisequationonecandeducethattheimportantcomponentsofDQMapproximationsaretheweightingcoefcientsAnrijandthechoiceofsamplingpoints.
Inordertodeterminetheweightingcoefcients,asetoftestfunctionsshouldbeusedinEq.
(24).
Theweightingcoefcientsfortherst-orderderivativesinn-directionarethusdeterminedas(BellmanandCasti1971):18IntJAdvStructEng(2016)8:11–28123Anij1aMnininjMnjfori6jPNnj1i6jAnijforij8>>>>>>>:;i;j1;2;Nn;25whereMniYNnj1;i6jninj26Theweightingcoefcientsofthesecond-orderderivativecanbeobtainedinthematrixform(BellmanandCasti1971):BnijhiAnijhiAnijhiAnijhi227Inasimilarmanner,theweightingcoefcientsfortheg-directioncanbeobtained.
Thenaturalandsimplestchoiceofthegridpointsisequallyspacedpointsinthedirectionofthecoordinateaxesofcomputationaldomain.
Itwasdemonstratedthatnon-uniformgridpointsgivesabetterresultwiththesamenumberofequallyspacedgridpoints(BellmanandCasti1971).
Itisshown(ShuandWang1999)thatoneofthebestoptionsforobtaininggridpointsisChebyshev–Gauss–Lobattoquadraturepoints:nia121cosi1pNn1!
&';gjb121cosj1pNg1!
&'fori1;2;Nn;28:1;2j1;2;Ng;whereNnandNgarethetotalnumberofnodesalongthen-andg-directions,respectively.
Atthisstage,theDQmethodcanbeappliedtodiscretizetheequationsofmotion(20,21,22).
Asaresult,ateachdomaingridpoint(ri,zj)withi=2,…,Nr-1andj=2,…,Nz-1,thediscretizedequationstakethefol-lowingforms.
Equation(20):Equation(21):c66ij1r2impUUrijmpUriXNrn1ArinUrnjXNrn1BrinUhnjUhijr2i1riXNrn1ArinUhnj!
oc66orijmpUriUrijXNrn1ArinUhnjUhijri!
1ric12ijmpUXNrn1ArinUrnjc22ijmpUriUrij1rimpU2Uhijc23ijmpUXNzn1AzjnUzin!
c44ijXNzn1BzjnUhinmpUriXNzn1AzjnUzin!
2c66ijrimpUriUrijXNrn1ArinUhnjUhijri!
qijx2Uhij30c11ijXNrn1BrinUrnjc12ijmpUr2iUhijmpUriXNrn1ArinUhnj1riXNrn1ArinUrnjUrijr2i!
c13ijXNrn1XNzv1AzjvArinUznvoc11orijXNrn1ArinUrnjoc12orij1riUrijmpUriUhijoc13orijXNzn1AzjnUzinc66ijrimpUXNrn1ArinUhnj1rimpU2UrijmpUriUhij!
c55ijXNzn1BzjnUrinXNrn1XNzv1AzjvArinUznv!
1ric11ijXNrn1ArinUrnjc12ijUrijrimpUriUhijc13ijXNzn1AzjnUzinc12ijXNrn1ArinUrnjc22ijUrijrimpUriUhijc23ijXNzn1AzjnUzin!
qijx2Urij29IntJAdvStructEng(2016)8:11–2819123Equation(22):c55ijXNrn1XNzv1AzjvArinUrnvXNrn1BrinUznj!
oc55orijXNzn1AzjnUrinXNrn1ArinUznj!
c44ijrimpUXNzn1AzjnUhin1rimpU2Uzij!
c13ijXNrn1XNzv1AzjvArinUrnvc23ij1riXNzn1AzjnUrinmpUriXNzn1AzjnUhin!
c33ijXNzn1BzjnUzinc55ijriXNzn1AzjnUrinXNrn1ArinUznj!
qijx2Uzij31whereArij,AzijandBrij,Bzijaretherst-andsecond-orderDQweightingcoefcientsinther-andz-directions,respec-tively.
TheDQmethodcanalsobeappliedtodiscretizetheboundaryconditionsatr=riandroasfollows.
Equation(23):XNzn1AzjnUrinXNrn1ArinUznj0;mpUriUrijXNrn1ArinUhnjUhijri0;c11ijXNrn1ArinUrnjc12ijUrijrimpUriUhijc13ijXNzn1AzjnUzinkwUrijkgXNzn1BzjnUrinmpUri2Urij!
()d1i032wherei=1atr=riandi=Nratr=ro,andj=1,2,…,Nz;alsodijistheKroneckerdelta.
Theboundaryconditionsatz=0andLzstatedinEqs.
(16,17,18),becomeEq.
(16):Simplysupported(S):UrijUhij0;c13ijXNrn1ArinUrnjc23ijUrijrimpUriUhijc33ijXNzn1AzjnUzin033Equation(17):Clamped(C):UrijUhijUzij034Equation(18):Free(F):c13ijXNrn1ArinUrnjc23ijUrijrimpUriUhijc33ijXNzn1AzjnUzin0;XNzn1AzjnUhinmpUriUzij0;XNzn1AzjnUrinXNrn1ArinUznj035Intheaboveequationsi=2,…,Nr-1;alsoj=1atz=0andj=Nzatz=Lz.
Inordertocarryouttheeigenvalueanalysis,thedomainandboundarynodaldisplacementsshouldbeseparated.
Invectorforms,theyaredenotedas{d}and{b},respec-tively.
Basedonthisdenition,thediscretizedformoftheequationsofmotionandtherelatedboundaryconditionscanberepresentedinthematrixformas:Equationsofmotion,Eqs.
(29,30,31):KdbKddbfgdfg()x2Mdfg0fg36Boundaryconditions,Eq.
(32)andEqs.
(33,34,35):KbddfgKbbbfg0fg37EliminatingtheboundarydegreesoffreedominEq.
(36)usingEq.
(37),thisequationbecomesKx2Mdfg0fg;38whereKKddKdbKbb1Kbd.
Theaboveeigen-valuesystemofequationscanbesolvedtondthenaturalfrequenciesandmodeshapesofthecurvedpanel.
NumericalresultsanddiscussionConvergenceandcomparisonstudiesDuetolackofappropriateresultsforfreevibrationofCGCNTRcylindricalpanelsreinforcedbyorientedCNTsfordirectcomparison,validationofthepresentedformu-lationisconductedintwoways.
Firstly,theresultsare20IntJAdvStructEng(2016)8:11–28123comparedwiththoseofFGMcompositecylindricalpanelsandthen,theresultsofthepresentedformulationsaregivenintheformofconvergencestudieswithrespecttoNrandNz,thenumberofdiscretepointsdistributedalongtheradialandaxialdirections,respectively.
Tovalidatetheproposedapproachitsconvergenceandaccuracyaredemonstratedviadifferentexamples.
Theobtainednaturalfrequenciesbasedonthethree-dimensionalelasticityfor-mulationarecomparedwiththoseofthepowerseriesexpansionmethodforbothFGMcurvedpanelswithandwithoutelasticfoundations(Matsunaga2008;PradyumnaandBandyopadhyay2008;Faridetal.
2010).
Inthesestudiesthematerialpropertiesoffunctionallygradedmaterialsareassumedasfollows:Metal(Aluminum,Al):Em70109Pa;qm2702Kg=m3;tm0:3Ceramic(Alumina,Al2O3):Ec380109Pa;qc3800Kg=m3;tc0:3SubscriptsMandCrefertothemetalandceramicconstituentswhichdenotethematerialpropertiesoftheouterandinnersurfacesofthepanel,respectively.
Tovalidatetheanalysis,resultsforFGMcylindricalshellsarecomparedwithsimilaronesintheliterature,asshowninTable2.
Thecomparisonshowsthatthepresentresultsagreedwellwiththoseintheliteratures.
BesidesthefastTable2ComparisonofthenormalizednaturalfrequencyofanFGMcompositecurvedpanelwithfouredgessimplysupportedX11x11RUqmh=Dp;DEmh3121t2mP(volumefractionindex)R/Lz0.
51510500Nr=Nz=569.
977452.
105242.
720242.
371742.
2595Nr=Nz=769.
972252.
105242.
715842.
371842.
2550Nr=Nz=969.
969852.
100342.
715942.
370042.
2553Nr=Nz=1169.
970052.
100342.
716042.
367742.
2552Nr=Nz=1369.
970052.
100342.
716042.
367742.
2553PradyumnaandBandyopadhyay(2008)68.
864551.
521642.
254341.
90841.
79630.
2Nr=Nz=565.
147047.
939339.
128238.
801038.
7020Nr=Nz=765.
444948.
045639.
100838.
736638.
6834Nr=Nz=965.
452648.
134039.
083638.
756838.
6581Nr=Nz=1165.
430448.
134039.
083538.
756838.
6580Nr=Nz=1365.
430448.
134039.
083538.
756838.
6581PradyumnaandBandyopadhyay(2008)64.
400147.
596840.
162139.
847239.
74650.
5Nr=Nz=560.
119643.
553936.
126435.
820234.
7341Nr=Nz=760.
276943.
712836.
140135.
796435.
0677Nr=Nz=960.
357443.
768936.
094435.
789035.
7032Nr=Nz=1160.
357443.
768836.
094335.
789135.
7032Nr=Nz=1360.
357443.
768936.
094435.
789135.
7032PradyumnaandBandyopadhyay(2008)59.
439643.
301937.
28736.
999536.
90881Nr=Nz=554.
103438.
518031.
986030.
706530.
6336Nr=Nz=754.
603939.
147732.
114031.
698231.
5397Nr=Nz=954.
714139.
162032.
040131.
760831.
6877Nr=Nz=1154.
714139.
162132.
040131.
760831.
6878Nr=Nz=1354.
714139.
162132.
040131.
760831.
6877PradyumnaandBandyopadhyay(2008)53.
929638.
771533.
226832.
958532.
8752Nr=Nz=546.
901634.
770227.
665727.
429527.
3725Nr=Nz=747.
986534.
698027.
573327.
338927.
2669Nr=Nz=948.
525034.
685227.
561427.
323827.
2663Nr=Nz=1148.
525034.
685127.
561427.
323927.
2663Nr=Nz=1348.
525034.
685127.
561427.
323927.
2662PradyumnaandBandyopadhyay(2008)47.
825934.
333827.
444927.
178927.
0961IntJAdvStructEng(2016)8:11–2821123rateofconvergenceofthemethodbeingquiteevident,itisfoundthatonly13gridpoints(Nr=Nz=13)alongtheradialandaxialdirectionscanyieldaccurateresults.
Fur-thervalidationofthepresentresultsforisotropicFGMcylindricalpanelisshowninTable3.
Inthistable,com-parisonismadefordifferentLz/RandLz/hratios,anditisobservedthereisgoodagreementbetweentheresults.
Asanotherexample,theconvergenceandaccuracyofthemethodareinvestigatedbyevaluatingtherstthreenaturalfrequencyparametersoftheFGcurvedpanelrestingonPasternakfoundations.
Thenon-dimensionalformsoftheelasticfoundationcoefcientsaredenedasKw=kwR/GcandKg=kg/(GcR)inwhichGcistheshearmodulusofelasticityoftheceramiclayer.
Theresultsarepreparedfordifferentthickness-to-meanradiusratiosanddifferentval-uesoftheDQgridpointsalongtheradialandaxialdirections,respectively,areshowninTable4.
Also,onecanseethatexcellentagreementexistsbetweentheresults.
ParametricstudiesAfterdemonstratingtheconvergenceandaccuracyofthepresentmethod,parametricstudiesfor3-Dvibrationanalysisofthickcurvedpanelsrestingonatwo-parameterelasticfoundationreinforcedbyrandomlyorientedstraightsingle-walledcarbonnanotubesforvariousCNTsvolumefractiondistribution,length-to-meanradiusratio,elasticcoefcientsoffoundationanddifferentcombinationsoffree,simplysupportedandclampedboundaryconditionsalongtheaxialdirectionofthecurvedpanel,arecomputed.
Theboundaryconditionsofthepanelarespeciedbythelettersymbols,forexample,S–C–S–FdenotesacurvedTable3ComparisonofthenormalizednaturalfrequencyofanFGMcompositecurvedpanelforvariousLZ/RandLZ/hratiosP(volumefractionindex)00.
51410LZ/h=2LZ/R=0.
5Matsunaga(2008)0.
93340.
82130.
74830.
60110.
5461Faridetal.
(2010)0.
91870.
80130.
72630.
52670.
5245Nr=Nz=50.
93420.
80010.
71490.
58780.
5133Nr=Nz=70.
92490.
80110.
72500.
57830.
5298Nr=Nz=90.
92500.
80180.
72530.
57900.
5301Nr=Nz=110.
92490.
80170.
72530.
57890.
5300Nr=Nz=130.
92500.
80180.
72520.
57900.
5301Matsunaga(2008)LZ/R=10.
91630.
81050.
74110.
59670.
5392Faridetal.
(2010)0.
86750.
75780.
68750.
54750.
4941Nr=Nz=50.
89420.
75310.
67460.
57410.
4913Nr=Nz=70.
88510.
76710.
69120.
55990.
5074Nr=Nz=90.
88570.
76660.
69350.
55310.
5065Nr=Nz=110.
88570.
76670.
69340.
55310.
5063Nr=Nz=130.
88560.
76670.
69350.
55320.
5064LZ/h=5LZ/R=0.
5Matsunaga(2008)0.
21530.
18550.
16780.
14130.
1328Faridetal.
(2010)0.
21130.
18140.
16390.
13670.
1271Nr=Nz=50.
22300.
19970.
15420.
13740.
1373Nr=Nz=70.
21760.
18230.
16240.
13620.
1233Nr=Nz=90.
21300.
18170.
16390.
13740.
1296Nr=Nz=110.
21280.
18160.
16400.
13770.
1296Nr=Nz=130.
21290.
18170.
16400.
13740.
1295Matsunaga(2008)LZ/R=10.
22390.
19450.
17690.
14830.
1385Faridetal.
(2010)0.
21640.
18790.
16760.
13940.
1286Nr=Nz=50.
20660.
17650.
15670.
14760.
1409Nr=Nz=70.
21330.
18430.
16880.
13770.
1288Nr=Nz=90.
21540.
18480.
16710.
13920.
1301Nr=Nz=110.
21550.
18470.
16750.
13920.
1299Nr=Nz=130.
21550.
18470.
16710.
13920.
130222IntJAdvStructEng(2016)8:11–28123panelwithedgesh=0andUsimplysupported(S),edgez=0clamped(C)andedgez=Lzfree(F).
Thenon-dimensionalnaturalfrequency,Winklerandshearinglayerelasticcoefcientsareasfollows:Xmnxmn10hqm=Emp;KwkwRGm;KgkgGmR;39whereqm,EmandGmrepresentthemassdensity,Young'smodulusandshearmodulusofthematrix,respectively.
TheeffectoftheWinklerelasticcoefcientonthefundamentalfrequencyparametersfordifferentboundaryconditionsisshowninFigs.
6,7and8.
ItisobservedthatthefundamentalfrequencyparametersconvergewithincreasingWinklerelasticcoefcientofthefoundation.
Accordingtothesegures,thelowestfrequencyparameterisobtainedbyusingclassical–classicalvolumefractionprole.
Onthecontrary,the1-DFGpanelwithsymmetricvolumefractionprolehasthemaximumvalueofthefrequencyparameter.
Therefore,agradedCNTsvolumefractionintwodirectionshashighercapabilitiestoreducethefrequencyparameterthanconventional1-Dnanocom-posite.
ItisalsoobservedfromFigs.
6,7and8,forthelargevaluesofWinklerelasticcoefcient,theshearinglayerelasticcoefcienthaslesseffectandtheresultsbecomeindependentofit,inotherwordsthenon-dimen-sionalnaturalfrequenciesconvergewithincreasingWin-klerfoundationstiffness.
Theinuenceofshearinglayerelasticcoefcientonthenon-dimensionalnaturalfrequencyforS–C–S–C,S–C–S–SandS–F–S–Fbidirectionalnanocompositecurvedpanelrestingonatwo-parameterelasticfoundation,isshowninFigs.
9,10and11.
ItisobservedthatthevariationofWinklerelasticcoefcienthaslittleeffectonthenon-di-mensionalnaturalfrequencyatdifferentvaluesofshearinglayerelasticcoefcient.
Itisclearthatwithincreasingtheshearinglayerelasticcoefcientofthefoundation,thefrequencyparametersincreasetosomelimitvaluesandforTable4Comparisonoftherstthreenon-dimensionalnaturalfre-quencyparametersofpanelonanelasticfoundation-mnxmnhqC=ECp;P1;U60;NrNz13LZ/Rh/RKw,Kg-11-22-3310.
11,0.
1Present0.
22010.
44110.
6462Faridetal.
(2010)0.
22000.
44030.
6427100,10Present0.
22410.
44750.
6679Faridetal.
(2010)0.
22430.
44750.
66810.
51,0.
1Present0.
80431.
86012.
9796Faridetal.
(2010)0.
80411.
85992.
9796100,10Present0.
95001.
89642.
9956Faridetal.
(2010)0.
95031.
89632.
995620.
11,0.
1Present0.
17150.
34300.
5121Faridetal.
(2010)0.
17120.
34340.
5122100,10Present0.
1740.
34770.
5202Faridetal.
(2010)0.
1740.
34750.
52000.
51,0.
1Present0.
57691.
34082.
1825Faridetal.
(2010)0.
57721.
34092.
1827100,10Present0.
76641.
40342.
2027Faridetal.
(2010)0.
76641.
40372.
2023Fig.
6Variationsoffundamentalfrequencyparametersofabidirec-tionalS–C–S–Cnanocompositecurvedpanelsrestingonatwo-parameterelasticfoundationwithWinklerelasticcoefcientfordifferentvolumefractionproles(Kg=100,R/h=Lz/R=3.
5,cr=2,U=135°)Fig.
7Variationsoffundamentalfrequencyparametersofabidirec-tionalS–C–S–Snanocompositecurvedpanelsrestingonatwo-parameterelasticfoundationwithWinklerelasticcoefcientfordifferentvolumefractionproles(Kg=100,R/h=Lz/R=3.
5,cr=2,U=135°)IntJAdvStructEng(2016)8:11–2823123thelargevaluesofshearinglayerelasticcoefcient,thefrequencyparametersbecomeindependentofit.
Thevariationsoffundamentalfrequencyparametersofbidirectionalnanocompositecurvedpanelsrestingonanelasticfoundationwithlength-to-meanradiusratio(Lz/R)fordifferenttypesofvolumefractionprolesaredepictedinFigs.
12,13and14.
Itcanalsobeinferredfromtheseguresthatthefrequencyisgreatlyinuencedinthatfundamentalfrequencyparameterdecreasessteadilyaslength-to-meanradiusratio(Lz/R)becomeslargerandremainsalmostunalteredforthelargevaluesoflength-to-meanradiusratio.
Ascanbeseenfromthisgure,forthealllength-to-meanradiusratio(Lz/R),classical–classicalvolumefractionprolehasthelowestfrequenciesfollowedbyclassical–symmetric,classical,symmetric–symmetricandsymmetricproles.
Thevariationsoffundamentalfrequencyparametersofbidirectionalnanocompositecurvedpanelswithlength-to-Fig.
8Variationsoffundamentalfrequencyparametersofabidirec-tionalS–F–S–Fnanocompositecurvedpanelsrestingonatwo-parameterelasticfoundationwithWinklerelasticcoefcientfordifferentvolumefractionproles(Kg=100,R/h=Lz/R=3.
5,cr=2,U=135°)Fig.
9Variationsoffundamentalfrequencyparametersofabidirec-tionalS–C–S–Cnanocompositecurvedpanelsrestingonatwo-parameterelasticfoundationwiththeshearinglayerelasticcoefcient(R/h=Lz/R=3.
5,cr=cz=2,ar=az=0,U=135°)Fig.
10Variationsoffundamentalfrequencyparametersofabidi-rectionalS–C–S–Snanocompositecurvedpanelsrestingonatwo-parameterelasticfoundationwiththeshearinglayerelasticcoefcient(R/h=Lz/R=3.
5,cr=cz=2,ar=az=0,U=135°)Fig.
11Variationsoffundamentalfrequencyparametersofabidi-rectionalS–F–S–Fnanocompositecurvedpanelsrestingonatwo-parameterelasticfoundationwiththeshearinglayerelasticcoefcient(R/h=Lz/R=3.
5,cr=cz=2,ar=az=0,U=135°)24IntJAdvStructEng(2016)8:11–28123meanradiusratio(Lz/R),andthevolumefractionindexthroughtheradialdirectionofthepanelsforS–F–S–FboundaryconditionsareshowninFig.
15,byconsideringaraz0;cz2;KwKg100forclassical–classi-cal2-Dnanocompositecurvedpanels.
Conrmingtheeffectoflength-to-meanradiusratioonthenaturalfre-quencyalreadyshownintheFigs.
12,13and14,itisfoundthatthefrequencyparameterdecreasesbyincreasingtheradialvolumefractionindexcr.
Thisbehaviorisalsoobservedforotherboundaryconditions,notshownhereforbrevity.
ConclusionremarksInthisresearchwork,freevibrationofthickbidirectionalnanocompositecurvedpanelsrestingonatwo-parameterelasticisinvestigatedbasedonthree-dimensionaltheoryofFig.
12Variationsoffundamentalfrequencyparametersoftwo-dimensionalcontinuouslygradedS–C–S–CnanocompositecurvedpanelsrestingonanelasticfoundationwithLz/Rratiofordifferentvolumefractionproles(Kw=Kg=100,R/h=3.
5,cz=2,U=135°)Fig.
13Variationsoffundamentalfrequencyparametersoftwo-dimensionalcontinuouslygradedS–C–S–SnanocompositecurvedpanelsrestingonanelasticfoundationwithLz/Rratiofordifferentvolumefractionproles(Kw=Kg=100,R/h=3.
5,cz=2,U=135°)Fig.
14Variationsoffundamentalfrequencyparametersoftwo-dimensionalcontinuouslygradedS–F–S–FnanocompositecurvedpanelsrestingonanelasticfoundationwithLz/Rratiofordifferentvolumefractionproles(Kw=Kg=100,R/h=3.
5,cz=2,U=135°)Fig.
15Variationsoffundamentalfrequencyparametersofbidirec-tionalnanocompositecurvedpanelswithlength-to-meanradiusratio(Lz/R),andthevolumefractionindexthroughtheradialdirectionofthepanelsforS–F–S–Fboundarycondition(Kw=Kg=100,R/h=3.
5,cz=2,ar=az=0,U=135°)IntJAdvStructEng(2016)8:11–2825123elasticity.
TheelasticfoundationisconsideredasaPasternakmodelwithaddingashearlayertotheWinklermodel.
Threecomplicatedequationsofmotionforthecurvedpanelunderconsiderationaresemi-analyticallysolvedbyusing2-Ddifferentialquadraturemethod.
Usingthe2-Ddifferentialquadraturemethodalongtheradialandaxialdirections,allowsonetodealwithcurvedpanelwitharbitrarythicknessdistributionofmaterialpropertiesandalsotoimplementtheeffectsoftheelasticfoundationsasaboundaryconditiononthelowersurfaceofthecurvedpanelefcientlyandinanexactmanner.
Thevolumefractionsofrandomlyorientedstraightsingle-walledcar-bonnanotubes(SWCNTs)areassumedtobegradednotonlyintheradialdirection,butalsoinaxialdirectionofthecurvedpanel.
ThedirectapplicationofCNTspropertiesinmicromechanicsmodelsforpredictingmaterialpropertiesofthenanotube/polymercompositeisinappropriatewith-outtakingintoaccounttheeffectsassociatedwiththesignicantsizedifferencebetweenananotubeandatypicalcarbonber.
Inotherwords,continuummicromechanicsequationscannotcapturethescaledifferencebetweenthenanoandmicro-levels.
Inordertoovercomethislimitation,avirtualequivalentberconsistingofnanotubeanditsinter-phasewhichisperfectlybondedtosurroundingresinisapplied.
Inthisresearchwork,anequivalentcontinuummodelbasedontheEshelby–Mori–Tanakaapproachisemployedtoestimatetheeffectiveconstitutivelawoftheelasticisotropicmedium(matrix)withorientedstraightCNTs.
Theeffectsofelasticfoundationstiffnessparame-ters,variousgeometricalparametersonthevibrationcharacteristicsofCGCNTRcurvedpanel,areinvestigated,andalso,differenttypesofvolumefractionprolesalongtheradialandaxialdirectionsofthepanelsandelasticcoefcientsoffoundationofbidirectionalcurvedpanelsrestingonatwo-parameterelasticfoundationarestudied.
Moreover,vibrationbehaviorof2-Dcontinuouslygradednanocompositepanelsarecomparedwithconventionalone-dimensionalnanocompositepanels.
Fromthisstudy,someconclusionscanbemade:Itisobserved,forthelargevaluesofWinklerelasticcoefcient,theshearinglayerelasticcoefcienthaslesseffectandtheresultsbecomeindependentofit,inotherwordsthenon-dimensionalnaturalfrequenciesconvergewithincreasingWinklerfoundationstiffness.
TheresultsshowthatthevariationofWinklerelasticcoefcienthaslittleeffectonthenon-dimensionalnaturalfrequencyatdifferentvaluesofshearinglayerelasticcoefcient.
Itisclearthatwithincreasingtheshearinglayerelasticcoefcientofthefoundation,thefrequencyparametersincreasetosomelimitvaluesandforthelargevaluesofshearinglayerelasticcoefcient;thefrequencyparametersbecomeindependentofit.
Thefrequencyparameterdecreasesrapidlywiththeincreaseofthelength-to-meanradiusratioandthenremainsalmostunalteredforthelongcylindricalpanel(Lz/R[5).
Theinterestingresultsshowthatthelowestmagnitudefrequencyparameterisobtainedbyusingaclassical–classicalvolumefractionprole.
ItcanbeconcludedthatagradedCNTsvolumefractionintwodirectionshashighercapabilitiestoreducethenaturalfrequencythanaconventional1-Dnanocomposite.
Itisfoundthatthefrequencyparameterdecreasesbyincreasingtheradialvolumefractionindexcr.
Forthealllength-to-meanradiusratio(Lz/R),classical–classicalvolumefractionprolehasthelowestfre-quenciesfollowedbyclassical–symmetric,classical,symmetric–symmetricandsymmetricproles.
Basedontheachievedresults,using2-Dsix-parameterpower-lawdistributionleadstoamoreexibledesignsothatmaximumorminimumvalueofnaturalfrequencycanbeobtainedinarequiredmanner.
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0InternationalLicense(http://creativecommons.
org/licenses/by/4.
0/),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedyougiveappropriatecredittotheoriginalauthor(s)andthesource,providealinktotheCreativeCommonslicense,andindicateifchangesweremade.
AppendixTheHill'selasticmoduliarefoundas(Shietal.
2004):erkEmfEmfm2kr1tm1fr12tmg21tmEm1fr2tm2fmkr1tm2t2m;lEmftmfmEm2kr1tm2frkr1t2mg1tmEm1fr2tm2fmkr1tm2t2m;nE2mfm1frfmtm2fmfrkrnrl2r1tm212tm1tmEm1fr2tm2fmkr1tm2t2mEm2f2mkr1tmfrnr1fr2tm4fmlrtmEm1fr2tm2fmkr1tm2t2m;pEmEmfm2pr1tm1fr21tmEm1fr2fmpr1tm;kEmEmfm2mr1tm3fr4tm21tmfEmfm4fr1tm2fmmr3tm4t2mg;26IntJAdvStructEng(2016)8:11–28123ar3KmGmkrlr3krGmbr154Gm2krlr3Gmkr4GmGmpr2Gm3KmGmGm3Km7GmhiGm3KmGmmr3Km7Gm2435dr13nr2lr2krl3Km2GmlrGmkr!
gr1523nrlr8GmprGmpr2krlr2Gmlr3Gmkr!
158mrGm3Km4Gm3KmmrGmGm7mrGm!
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