Determination of thermal expansion coefficients for unidirectional fiber-reinforced composites

ChineseSocietyofAeronauticsandAstronautics&BeihangUniversityChineseJournalofAeronauticscja@buaa.edu.cnwww.sciencedirect.comDeterminationofthermalexpansioncoefficientsforunidirectionalfiber-reinforcedcomposites

RanZhiguo,YanYing*,LiJianfeng,QiZhongxing,YangLei

SchoolofAeronauticsScienceandEngineering,BeihangUniversity,Beijing100191,ChinaReceived3September2013;revised13November2013;accepted10February2014Availableonline19March2014

KEYWORDS

Analyticalsolution;Coefficientofthermalexpansion;

Thermo-elastic;

Transverselyisotropic;Unidirectionalcomposites

AbstractInthepresentwork,thecoefficientsofthermalexpansion(CTEs)ofunidirectional(UD)fiber-reinforcedcompositesarestudied.First,anattemptismadetoproposeamodeltopredictbothlongitudinalandtransverseCTEsofUDcompositesbymeansofthermo-elasticmechanicsanalysis.Theproposedmodelissupposedtobeaconcentriccylinderwithatransverselyisotropicfiberembeddedinanisotropicmatrix,anditissubjectedtoauniformtemperaturechange.ThenaconciseandexplicitformulaisofferedforeachCTE.Finally,somefiniteelement(FE)modelsarecreatedbyafiniteelementprogramMSC.Patranaccordingtodifferentmaterialsystemsandfibervolumefractions.Inaddition,theavailableexperimentaldataandresultsofotheranalyticalsolutionsofCTEsarepresented.Comparisonsaremadeamongtheresultsofthecylindermodel,thefiniteelementmethod(FEM),experiments,andothersolutions,whichshowthatthepredictedCTEsbythenewmodelareingoodagreementwiththeexperimentaldata.Inparticular,transverseCTEsgenerallyofferbetteragreementsthanthosepredictedbymostofothersolutions.

ª2014ProductionandhostingbyElsevierLtd.onbehalfofCSAA&BUAA.

Open access under CC BY-NC-ND license.1.Introduction

Asweknow,compositematerialshavebeenundergoingextraordinarytechnologicaladvancesandenjoyingwidespreadapplicationsindifferentfields.However,asaresultoftheircomplexproperties,suchaswettability,chemicalcompatibil-ity,anisotropicmechanics,heatabsorptionandconductivity

*Correspondingauthor.Tel.:+861082315947.

E-mailaddresses:mfdickl1@163.com(Z.Ran),yingyan@buaa.edu.cn(Y.Yan).

PeerreviewunderresponsibilityofEditorialCommitteeofCJA.Production and hosting by Elsevierabilities,theircompletecharacterizationhasnotbeenachievedsofar.

Coefficientofthermalexpansion(CTE)isdefinedasthefractionalchangeinlengthofabodyunderheatingorcoolingthroughagiventemperaturerange,1anditisusuallygivenasacoefficientperunittemperatureintervalatagiventempera-ture.Itisakeymaterialpropertyespeciallywhenacompositestructureworksinatemperature-changingenvironment.Here,thefocuswasplacedupononstudyingthelongitudinalandtransverseCTEsofcontinuousfiber-reinforcedunidirectional(UD)composites.

Theproblemofrelatingeffectivepropertiesofafiber-rein-forcedmaterialtoitsconstituentpropertieshasdrawngreatattention.Asaresult,manyanalyticalsolutionshavebeenmadetopredicttheupperandlowerboundsofCTEsofUDcomposites,whicharecomposedofisotropicoranisotropic

http://dx.doi.org/10.1016/j.cja.2014.03.010

1000-9361ª2014ProductionandhostingbyElsevierLtd.onbehalfofCSAA&BUAA.Open access under CC BY-NC-ND license.Determinationofthermalexpansioncoefficientsforunidirectionalfiber-reinforcedcomposites1181

fibersandmatrices.2–12InaseriesofstudiesbyvanFoFy,3–5analyticalsolutionswerepresentedtopredictbothaxialandtransverseCTEsofaUDcompositethroughitsconstituentproperties.However,theresultswereverysensitivetotheelas-ticmodulusandPoisson’sratiooftheUDmaterial.Levin6expandedHill’smethodandgavetheupperboundsofacer-tainglassfiber-reinforcedcomposite’sCTEs,andtheresultswereinmuchbetteragreementwiththedatainVanFoFy’sstudy3thanotherpredictionsinthatpaper.Schapery2hasderivedexpressionsforlongitudinalandtransverseCTEsofcompositeswithisotropicfibersembeddedinisotropicmatri-cesbyadoptingextremeenergyprinciples.Sideridis12andChamis9applieddifferentmethodsandobtainedthesamelon-gitudinalCTEexpression,whilethetransverseexpressionswerequitedifferent.Ingeneral,thepredictionsoflongitudinalCTEswerealwaysingoodagreementwithexperimentaldata,whilethoseoftransverseCTEsfailedtoagree.AnexceptionwasRosenandHashin’sprediction10asanextensionoftheworkofLevin.6However,itisinconvenienttoobtainresultsbyRosenandHashin’ssolution,becausetosolvetheCTEsofacomposite,themechanicalpropertiesofboththecompos-iteanditsconstituentsmustbedeterminedfirst.

Atthesametime,ascomputationcapabilityhasgrowndra-maticallyoverthelastthreedecades,numericalsolutionssuchasthefiniteelementmethod(FEM)arebeingextensivelyappliedtodeterminetheCTEsofcompositematerials.Islam13andRupnowskietal.14investigatedthelinearCTEsofUDcompositessystematicallybytheFEM.Karadenizetal.15exploredtheCTEsofdifferentmaterialsystemsbymicrome-chanicalmodelingusingtheFEM,andcomparisonswerecar-riedoutamongtheirresults,analyticalsolutions,andexperimentaldata.However,discrepanciesstillexistbetweenFEMresultsandexperimentaldata.Generally,thetransverseCTEpredictionofaUDcompos-itewasnotasgoodasthatofthelongitudinalCTE.However,anexacttransverseCTEofaUDcompositeisratherimportantindesigninghigh-dimensionalstablestructures.Therefore,wetriedtoachieveapracticalsolutionofCTEsbydoingthermo-elasticanalysisinthispaper,especiallythetransverseCTEwaspaidmuchattentionto.Inaddition,resultsofanalyticalsolutions,theFEM,andexperimentsavailableinliteratureswerecomparedforjustifications.2.Theoreticalanalysis2.1.ProposedmodelThecross-sectionofaUDfiber-reinforcedcompositeisshowninFig.1,andatypicalrepresentativevolumeelement(RVE)Fig.1Cross-sectionofUDfiber-reinforcedcomposite.couldbeacylinderfiberembeddedinacube,inwhichthecylinderstandsforthefiber,whilethecubesymbolizesthematrix.

Tomakethisthermo-elasticanalysiseasier,thecubicRVEistransformedintoaconcentriccylindermodel(seeFig.2)accordingtothefollowingassumptions:(1)boththecubicmodelandthecylindermodelhavethesamefiberradius;(2)twomodelswiththesamefibervolumefraction;(3)twomod-elswiththesamelengthinthelongitudinaldirection.

Considerthattheradiusofthefiberr1,thefibervolumefractionVf,andthelengthoftheRVEhareallknown.

Thecross-sectionofthecubicmodelisdefinedbyacirclewitharadiusofr1surroundedbya2l·2lsquare(seeFig.2(c)),whilethenewmodeliscomposedoftwoconcentriccylinderswithradiiofr1andr2respectively(seeFig.2(d)).ThefibercontentsinthetwomodelsareVpr2fðaÞ¼

1hð2lÞ2h

¼pr21ð2lÞ2ð1Þ

V¼pr21hr2fðbÞ

r2¼1p2hr2ð2Þ

2

Accordingtotheassumption,bothmodelshavethesame

fibervolumefractionVf,thatisVpfðaÞ¼VfðbÞ)2l¼ffiffiffipr2ð3ÞWhenthereisachangeofr2byDr2,thestraininthetransversedirectionofthecubicmodel(seeFig.2(a))iseDð2lÞpffiffiffipðrp2þDffiffiffitðaÞ¼2l¼pffiffiffir2ÞÀpr2pr¼Dr2¼etðbÞð42rÞ2Fig.2Transformationofcubicmodelintocylindermodel.1182

Obviously,bothofthetwomodelshavethesametransversestrain.Foraxialstrains,theyareprovedtobethesameaswellinthefollowingsection.2.2.EffectiveCTEanalysis

ThetheoreticalanalysisofeffectiveCTEsforaUDcompositeisbaseduponthefollowingassumptions:

(1)Thecylindermodeliscomposedoftwophases(fiberand

matrix),ignoringtheeffectoftheirinterface.

(2)Thefiberisregardedastransverselyisotropicwhilethe

matrixisisotropic.

(3)Thecylindermodelundergoesauniformtemperature

changefromT0(stress-freetemperature)toT0+DT.(4)ComparedtotheheightoftheRVEh,thelengthofthe

UDcompositeisinfinitelylong.Therefore,anycross-sectionoftheRVEremainsplanarwhentemperaturechanges,whichmeansthatthecylindermodelisinanequalstrainstatusinthelongitudinaldirection.Thetransverselyisotropicfiberischaracterizedbythefol-lowingsevenindependentthermo-elasticconstants:Ef1(theaxialelasticmodulus(ZdirectioninFig.2(b))),Ef2(thetrans-verseelasticmodulus(RTplane)),Gf12(theaxialshearmodu-lus),t12(theaxialPoisson’sratio),t23(thetransversePoisson’sratio),af1(theaxialCTE),andaf2(thetransverseCTE).

Fortheisotropicmatrix,itischaracterizedbythefollowingconstants:Em(theelasticmodulus),tm(thePoisson’sratio),andam(theCTE).

Thesubscripts‘‘f’’and‘‘m’’denotefiberandmatrix,while1,2,3refertoaxesZ,R,T,respectively.Meanwhile,axialandradialCTEsoftheconcentricmodelaredesignatedasa1anda2.

Accordingtotheassumptionthatthetransversesectionoftheconcentriccylindermodelkeepsplanarwhenthetemper-aturechangesbyDT,togetherwiththataf1andamarealwaysdifferentinvalue,thusthereexiststressesbetweenthefiberandthematrixintheaxialdirection,whicharedenotedbyrf1andrm1,andthefollowingequationsareobtained:

Axial(Z)directionbalanceequationis

rf1pr222

1þrm1pðr2Àr1Þ¼0

ð5Þ

Physicsequationsare

8

>>>e1¼a1DT<

erf1f1¼þaf1D>>ET

f1

ð6Þ

>:erm1m1¼EþamDT

m

Tosimplifythiswork,physicsequationsignorethePoisson’seffectthatiscausedbythetransversestressesofboththefiberandthematrix.

Geometricsequationsise1¼ef1¼em1

ð7Þ

8

>>¼r2>>>ð8Þ

>:V1ÀVr2Àr2m¼f¼21

r22

Z.Ranetal.

Eq.(7)istheformulizationofAssumption4thatboththe

fiberandthematrixhavethesamestrainintheaxialdirection,whileEq.(8)representsaconcisemethodofcalculatingthevolumefractionsofthefiberandthematrix.

SolvingEqs.(5)–(8),theaxialCTEoftheUDcompositeisaf1Vfaf1þEmVmam

1¼

EEf1VfþEmVð9Þ

m

ExplicitanalyticalformulaEq.(9)presentedhereenablestheaxialCTEoftheconcentriccylindermodeltobepredictedintermsofthepropertiesofitsconstituentsandthefibervol-umefraction.Wecanalsofindthatthisformulabehavessim-ilarlyasSchapery’sequation.7ForthetransverseCTEofthecylindermodel,itwillbeanalyzedbythree-dimensionalthermo-elasticmechanics.

AccordingtoAssumption(3),thespatialaxialsymmetrymodel(seeFig.2(b))undergoesauniformtemperaturechange,andthematerialsaretransverselyisotropic.Asaresult,thereisnodisplace-mentintheTdirectionforboththefiberandthematrix,supposeanglehisanarbitrarilyangleinTdirection,theradialdisplacementfunctionsufandumareindependentofanglehandonlyrelatedtoradiusr,andsrh=0forboththefiberandthematrix.

FromAssumption(4),wecanseethatthefiberandthematrixsharethesameaxialdisplacementfunctionx.BecauseofspatialaxialsymmetryandAssumption(4),wehaveszr=0andshz=0forboththefiberandthematrix.

2The3physics2

equationforthistransverselyisotropicfiber6rfr

c11c12323is

4rfh75¼6

c134c12c11c13756efrÀaf2DT4efhÀaf2DT75

ð10Þrfzc13c13c11efzÀaf1DTwherecij(i,j=1,2,3)isthefiberdegradationstiffness.

2

The3physics2equationfortheisotropicmatrix6

rmr

q11q12q1232emrÀamDT3is4rmh75¼64q12q11q127564emhÀamDT75

ð11Þrmzq12q12q11emzÀamDTwhereqij(i,j=1,2,3)isthematrixdegradationstiffness.

Meanwhile,bysubstitutingthepropertiesofthefiberandthematrix,theresultsare

2c11

32

1Àt12t213666c1276ttþt2374c77¼k61221

71356

4t12ð1þt23Þ75ð12Þc33

t12ð1Àt223Þ=t21

wherek¼

Ef2

ð1þt23Þð1Àt23À2t12t21Þ

q11!Em

1Àtm!q¼

1þtð13Þ

12mtm

Thegeometricequationforthisspatialaxialsymmetrymodelis

2@u

i

32

eir

3

6@r

7666eih766u

i

74e776

775¼66r6@x

7iz

77ði¼f;mÞð14Þ

s6

i

izr64

@z7@x7i@ui

5@rþ@zDeterminationofthermalexpansioncoefficientsforunidirectionalfiber-reinforcedcomposites1183

Forthisspecialmodel,becauseoflowdensitiesofbothconstituents,thevolumeforcecanbeignored,sotheequiva-lentequationsare

@rir@sizrrirÀrih

@rþ@zþ

r¼0ði¼f;mÞð15Þ@riz@rþ@sizrsirz

@zþ

r

¼0ði¼f;mÞð16Þ

FromEqs.(10),(14)and(15),weobtain@rfr@sfzrrfrÀrfh

@rþ@zþr¼0)

@rfr@rþrfrÀrfh

r¼0ð17Þ)c@2ufc1111@r2þ

r:@uf@rÀc11

r

2uf¼0Sincec11=k(1Àt12t21)„0andufisindependentofh

andz,thefollowingresultcanbeobtained:d2ufdr2þ1r:dufdrÀuf

r2

¼0ð18Þ

)urþ

A2

f¼A1r

HereandthefollowingAj(j=1,2,...,8)areconstantstobespecified.

FromEqs.(10),(14)and(16),weget@rfz@sfzrsfrz

@rþ@zþ

r

¼0ð19Þ

)xf¼A3zþA4

Likewise,fromEqs.(11),(14),(15)and(16),wehave@rmr@smzrrmrÀr@rþ@zþmh

r

¼0ð20Þ

)uAA6

m¼5rþ

r@rmz@rþ@smzr@zþsmrz

r¼0ð21Þ

)xm¼A7zþA8

Asdiscussedabove,thefiberandthematrixsharethesameaxialdisplacementfunctionx,andthusxm¼xf¼A3zþA4

ð22Þ

Boundaryconditions:

Atr=0,forthisspatialaxialsymmetrymodel,ufmustbezero.ThusA2=0anduf=A1r.

Atr=r1,hereistheinterfacebetweenthefiberandthematrix.Firstly,theirradiusdisplacementsmustbecontinuous,thatis,ufðr¼r1Þ¼umðr¼r1Þ.AA61r1¼A5r1þ

r1

ð23Þ

)A6¼ðA1ÀA5Þr21

Secondly,intheinterface,thepressurebetweenthefiber

andthematrixmustbeequivalent,thatis,rfrðr¼r1Þ¼rmrðr¼r1Þ.BysubstitutingEqs.(10)and(11),theresultis

232

T2323TAA6

3

5À2ÀamDT6c11A1Àaf2DT4c127564A1Àaf2

DT75¼6q114q1276566r1

76A777c13A3Àaf1DTq1264

A65þr2ÀamDT17

5A3ÀamDT

)ðc11þc12þq11Àq12ÞA1þðc13Àq12ÞA3À2q11A5

¼ðc11þc12Þaf2DTþc13af1DTÀðq11þ2q12ÞamDT

ð24Þ

Atr=r2,theoutercylindersurfaceofthematrixisafreeface,anditsuffers3nothing,sothat

22

q3Trmrðr¼r6

AA65Àr2ÀamDT

1162772Þ¼4q7q12561264AA65þr2ÀamDT75.AÀa2

3mDT

BysubstitutingEqs.(8)and(23),theresultisðq12Àq11ÞVfA1þq12A3þðq11þq11Vfþq12Àq12VfÞA5

¼ðq11þ2q12ÞamDT

ð25Þ

Intheaxialdirection,theforcesthatareperpendiculartothecross-sectionoftheconcentricmodelmustbeequivalent,thatXisFfzþX

Fmz¼0ð26ÞX

r1

p

FZ

2fz¼

Z

rfzrdrdh

00

22p

c133T2A1Àaf2DT3ð27Þ

¼

Z

r1

r

Z

60

0

4c13756

4A1Àaf2DT75drdhc33

A3Àaf1DT

X

Z

r2

Z

2pFmz¼

rmzrdrdh

r1

0

ð28Þ

¼Z2

3T2

3rAÀA6ÀamDT2rZ2p6q1265r74q127562

6q64AA75þ6r2ÀamDT77drdh

r1011

5A3ÀamDT

XThusFfzþX

Fmz¼0

)2c13VfA1þ½c33Vfþð1ÀVfÞq11󰀃A3þ2q12ð1ÀVfÞA5¼2c13Vfaf2DTþc33Vfaf1DTþð1ÀVfÞð2q12þq11ÞamDT

ð29Þ

Eqs.(24),(25)and(29)canbecombinedintothefollowingform:BA¼Dð30Þ

Therein,B¼½B1

B2

B3󰀃

A¼½2

A1A3A5󰀃T

c11þc12þq11Àq12

3B1¼6

4ðq12Àq11ÞVf75

2c13Vf

11842c13Àq123B2¼64q1275c33Vfþq11ð1ÀVfÞ2À2q113B3¼64q11ð1þVfÞþq12ð1ÀVfÞ752q12ð1ÀVfÞ2ðc11þc12Þaf2þc13af1Àðq11þ2q12Þam3D¼DT64ðq11þ2q12Þam752c13Vfaf2þc33Vfaf1þð2q12þq11Þð1ÀVfÞam

So;A¼BÀ1D:

Atr=r2,theradiusdisplacementofthisconcentricmodel

is

uA6ðA1ÀA5Þr2mðr¼r2Þ¼A5r2þr¼A5r2þ

1

2r2

ð31Þ¼r2½A5þðA1ÀA5ÞVf󰀃¼r2½A1VfþA5Vm󰀃

TheCTEisdefinedasa¼Dl

foradimensionoftransverseCTEofthisRVEislDT

l,sothe

aDr2

umðr¼r2¼

2ÞA1VfþA5Vmr¼T¼2DTr2DDTð32Þ

¼ad1þd2

mþ

n1þn2þn38Therein,

>>>>d1¼Ef1Ef2Vft12Âðt2

mþtmVmÞðamÀaf1Þ>>>>>þ2V>fðaf2ÀamÞþt2mVfðamþaf1À2af2ÞÃ>>>>d2¼Ef1EmVf½2t21Vmð1Àtmt12ÞðamÀaÞðamÀaf1ÞÃ>>>>>n1¼EmEf1Vm½t21þðt21þt12ÞVf>>>>>þtmt21VmÀð1þ4t21Þtmt12Vf󰀃>>>>:

n2¼E22mVmt12ð1ÀtmÀ2t12t21Þn3¼Ef1Ef2Vft12ð1þVfÀ2t2mVfþtmVmÞ

AccordingtoEq.(4)andthedefinitionofCTE,boththecubicmodelandtheconcentricmodelhavethesamestrainandCTEinthetransversedirection.Eq.(9)showsthattheaxialCTEisindependentoftheshapeofthecross-sectionofthisRVE.Therefore,theaxialandtransverseCTEsforFig.2(a)area1anda2whichareshowninEqs.(9)and(32)thatarederivedbythemodelinFig.2(b).3.MaterialandFEmodel

HeretheFEMisadoptedtodeterminebothaxialandtransverseCTEsoftheUDfiber-reinforcedcompositethroughthecubicmodel(seeFig.2(a))andtheconcentricmodel(seeFig.2(b)).Fortheadvantagesofsymmetry,onlyaquarterofeachRVEismodeled.Asshown,Fig.3(a)and(b)correspondtothecubicandconcentricmodels,respectively.CTEpredictionsbytheanalyticalmethodandtheFEMarecomparedwithexperimentaldatatojustifythesimplificationofcubictocylindermodelandanalyticalresults.

TheFEmodelsarecreatedbyMSC.Patran2010withan8-nodehexahedralelement,andthensubmittedtoMD.Nastranforcomputation.Eachmodeliscreatedaccording

Z.Ranetal.Fig.3FEmodels.todifferentfiber–matrixcombinationsandfibervolumefractions.ThedimensionofeachFEmodelinZdirectionhasaunitlength,whileXandYdimensionsmaychangewithdifferentmaterialsystems.However,foragivenfiber–matrixcombination,Fig.3(a)and(b)sharethesamefiberradiusandfibervolumefraction.

BoundaryconditionsfortheFEmodelsareasfollows:(1)nodesthatintheplaneXOYandZ=0arerestrictedtomoveinXandYdirections;(2)nodeatX=Y=Z=0isfixed,andallowsnodisplacement;(3)theplanesthatareparalleltoX,Y,andZ=0keepplanarandremainparalleltotheiroriginalpositionswhenindeformation;(4)theinitialtemperatureisassumedtoberoomtemperatureandDTis1°C.

MaterialpropertiesfortheFEmodelsareshowninTables1and2.ThedataareextractedfromthestudybyBowlesandTompkins.16Intheirinvestigation,allmatricesareisotropicwhilefibersaretransverselyisotropic.Meanwhile,alldataareusedinanalyticalsolutionsEqs.(9)and(32).4.Resultsanddiscussion

FromthepropertiesofconstituentslistedinTables1and2,itcanbefoundthattheUDfiber-reinforcedcompositematerialsystemshaveanaxialfibertomatrixstiffnessratio(Ef1/Em)rangingfrom6to140,andanaxialfibertomatrixCTEratio(af1/am)rangingfromÀ0.01toÀ0.30.Asaresult,thepresentstudycoversawiderangeoffiber/matrixcombinations.

ComparisonsamongpredictedCTEsoftheconcentriccylin-dermodel,experiments,andsomeotheranalyticalsolutions,includingmodifiedSchapery’sequation,2Chamberlain’sequa-tion,1ChamisandScheider’sequation,1arecarriedout.Fig.4showsacomparisonofpredictedlongitudinalCTEsofeightfiber–matrixcombinations,whicharepredictedbytheconcen-triccylindermodel.Experimentaldata,theFEM,andsomeana-lyticalmethodspredictedresultsareshowninTable3.Theconcentriccylindermodel(CM),Schapery(SH),Chamberlain(CB),andChamis&Scheider(CH)solutionsallsharethesameEq.(9)topredictlongitudinalCTEs,andtheirpredicteddataarelistedinthesamecolumninTable3.Inthelasttwocolumns,FEM-cubicandFEM-cylinderstandfortwodifferentFEmod-elswhichareshowninFig.3.ThedifferencebetweenFEM-cubicandFEM-cylinderresultsofeachmaterialcombinationisalmostnegligibleexceptT300/5208,giventheaccuracyofnumericalcomputation,whichmeansthattheFEmodelsinFig.3(a)and(b)areequivalentinpredictingtheaxialCTE.Themeansquareerrorofcylindermodelpredictionsrelativetoexperimentalresultsis8.55·10À8/°C,whichismuchsmallercomparedtothepredictedCTEs.Thatmeansallthepredictions

Determinationofthermalexpansioncoefficientsforunidirectionalfiber-reinforcedcomposites

Table1FiberT300C6000HMSP75P1001185

Propertiesoffibersatroomtemperature.16E1(GPa)233.04233.04379.21550.20796.34E2(GPa)23.1023.106.219.517.24G12(GPa)8.968.967.586.896.89G23(GPa)8.278.272.213.382.62m120.200.200.200.200.20m230.400.400.400.400.40a(10À6/°C)a1À0.54À0.54À0.99À1.35À1.404a210.0810.086.846.846.84Table2MatrixPropertiesofmatricesatroomtemperature.16E(GPa)4.344.344.344.343.4573.0862.74G(GPa)1.591.591.591.591.3127.5826.20m0.370.370.370.370.350.330.20a(10À6/°C)43.9243.9243.9263.3636.0023.223.24934epoxy5208epoxy930epoxyCE339epoxyPMR15polyimide2024aluminumBorosilicateglassareingoodagreementwithexperimentalresults.Althougherrorsexist,thegeneralresponsebehavessimilarly(decreasinglongitudinalCTEswithincreasingfibervolumefraction).Itindicatesthattherelativemagnitudesoffiber/matrixstiffnessratioandCTEratiodonotsignificantlyaffectthegeneraltrendinlongitudinalCTEs.Furthermore,wecanfindthatforthesamematerialsystem,suchasT300/5208,T300/934,andC6000-Pi,thelargerCTEofthematrix,themoredramaticallytheCTEofcompositedecreaseuntilVfexceeds0.6.Forthesamematrix,thelargermodulusandabsolutevalueoftheaxialCTEofthefiber,themoredramaticdecreaseinthelongitudinalCTEofsuchamaterialcombination(e.g.,T300/934,P75/934).ItcanbeseenfromEq.(9)thatthelongitudinalCTEisdeterminedbyboththefiberandthematrix.Usually,theCTEofthematrixismuchlargerthanthatofthefiber,sotheupperandlowerboundsofa1aretheCTEofthematrixandthelongitudinalCTEofthefiber,respectively.Whenthefibercontentislow,e.g.,Vf60.15,a1isdominatedbytheCTEofthematrix,becausethevolumefractionandCTEofthematrixaremuchlargerthanthoseofthefiber;whilethefibercontentissomewhathigher,e.g.,VfP0.5,a1ismostlydeterminedbythethermalpropertyofthefiber,becausethestiffnessofthefiberinthelongitudinaldirectionisquitelargerthanthatofthematrix.SuchaphenomenoncouldbefoundinFig.4directlyaswell.

Table3

Fig.4PredictedlongitudinalCTEvsfibervolumefraction.Fig.5showstheresponseoftransverseCTEsasafunctionoffibervolumefractionfordifferentfiber–matrixcombina-tions.ExperimentaldataandFEMresultsarealsoshowninthefiguresmentionedaboveaswellasinTable4.InTable4,itcanbefoundthattheFEM-cubicandFEM-cylinderresultsfordifferentmaterialcombinationsarealmostthesame,themaximumerrorbetweentheFEM-cubicandFEM-cylinderpredictionsis0.489·10À6/°CforC6000-Pimaterial,andthemaximumrelativeerrorislessthan2.5%.Suchgoodagree-mentsmeanthattheequivalentprocessofFig.2(a)to(b)inthetransversedirectionisreasonable.

ErrorsbetweentheFEMresultsandtheconcentriccylindermodelpredictionsstillexist.Apossiblereasonliesintheassumptionsandsimplificationsoftheconcentriccylindermodel.However,thepredictionsbytheconcentricmodelareinmuchbetteragreementwithexperimentaldatathanthosebyothersolutions.Themaximumerrorbetweentheconcentriccylindermodelandexperimentalresultsis5.672·10À6/°C,whilethemaximumerrorbetweentheFEM-cubicandexperi-mentalresultsis6.748·10À6/°C.Theminimalerrorsofthecon-centriccylindermodelandFEMpredictionsare0.079·10À6/

ComparisonofexperimentalandanalyticaldataforthelongitudinalCTE.

LongitudinalCTE(10À6/°C)Experiment

CH,CM,SH,CBÀ0.1530.076À0.965À1.157À0.916À0.225À0.3241.575

FEM-cubicÀ0.0680.171À0.920À1.125À0.857À0.175À0.3241.634

FEM-cylinderÀ0.0920.166À0.922À1.129À0.859À0.1870.3241.634

À0.113À0.002À1.051À1.076À1.021À0.212À0.4141.440

MaterialsystemsT300/5208T300/934P75/934P75/930P75/CE339C6000/PiHMS/GlassP100/Al

1186Z.Ranetal.Fig.5Table4

TransverseCTEofdifferentcompositematerials.ComparisonofexperimentalandanalyticaldataoftransverseCTE.

TransverseCTE(10À6/°C)Experiment

CM25.31530.74935.35026.04444.51823.8545.01128.492

SH27.51832.74735.50726.69644.61725.6615.98321.375

CH18.84122.29823.18017.16128.19018.0045.88115.336

CB-Hex16.43219.90820.91814.76924.84716.0515.46314.549

CB-Sq13.36017.09818.13111.48920.30413.8085.72813.448

FEM-cubic23.96729.45734.04224.96842.70922.0394.48626.865

FEM-cylinder24.43329.58334.04725.02542.70322.5294.47727.002

25.23629.03434.52431.71647.41222.4283.78026.118

MaterialsystemsT300/5208T300/934P75/934P75/930P75/CE339C6000/PiHMS/GlassP100/Al

Determinationofthermalexpansioncoefficientsforunidirectionalfiber-reinforcedcomposites1187

°Cand0.101·10À6/°C.Thereareeightmaterialcombinationsintotalinthisresearch,andthecomparedresultsinTable4showthatsixoutofeightpredictionsbytheconcentriccylindermodelareinbetteragreementwithexperimentaldatathanthosebyothersolutions;fortheothertwomaterialcombinations(P75/934andP75/930),theresultspredictedbyconcentriccylindermodelandSchaperymodelarealmostthesame,andthepredictionsarequiteclosetotheexperimentaldataaswell.

AsseeninFig.5,unliketheaxialCTE,theresponseofthetransverseCTEasafunctionoffibervolumefractionisaffectedbyboththefibertomatrixstiffnessratio(Ef2/Em)andthetransverseCTEratio(af2/am).P100-AlhasasimilartransverseCTEratio(af2/am<1)withT300-5208/934,P75-934/930,P75-CE339,andC6000-Pi,sotheresponseofa2decreaseswithincreasingfibervolumefraction.Meanwhile,P100-Alhasafibertomatrixstiffnessratio(Ef2/Em<1)whichiscontrarytotheotherfiber–matrixcombinations,andthusthedifferencebetweenthecylindermodelandothersolutionsinFig.5(e)ismoreapparentthanthoseinFig.5(a)–(d).ThesamephenomenoncanbefoundinFig.5(f).FormaterialHMS-glass,thetransverseCTEofthefiberislargerthanthatofthematrix(af2/am>1),whichiscontrarytotheothermaterialcombinations.Asaresult,theresponseofa2increaseswithincreasingfibervolumefraction.

Generally,thetransverseCTEofaUDfiber-reinforcedcompositeisdeterminedbyboththematrixandthefiber,withtheCTEofthematrixasabase,ascanbeseeninEq.(32).ThetransverseCTEisalmostlinearasthefibercontentincreases,forwhichmaybethereasonliesinthatEf2af2%Emam,andtheproductoftheCTEandstiffnessofthefiberinthetransversedirectionisofthesamemagnitudeorderasthatofthematrix,whichmeansthecontributionstothetransverseCTEoftheUDcompositebythefiberandthematrixareatthesamelevel.Therefore,thetransverseCTEexhibitsquitelinearity.5.Conclusions

(1)Theequivalentprocessoftransformingthemodelin

Fig.2(a)totheoneinFig.2(b)isreasonable.

(2)ExplicitformulaEqs.(9)and(32)forCTEpredictions

areeasytouseandresultsareingoodagreementwithexperimentaldata.

(3)ThelongitudinalCTEofaUDfiber-reinforcedcompos-itecanbemoreaccuratelypredictedthanthetransverseCTE.

(4)Aswouldbeexpected,bothlongitudinalandtransverse

CTEsaremostlyaffectedbythefibertomatrixstiff-nessratio(Ef2/Em)andthefibertomatrixCTEratio(af2/am).

Acknowledgement

ThankMissYIXiyuanforherkindhelpinrevisingthispaper.

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YanYingisaprofessorandPh.D.advisorinSchoolofAeronauticScienceandEngineeringatBeihangUniversity.ShereceivedherPh.D.degreefromUniversityofSouthamptonin1994.Hercurrentresearchinterestsareflyingvehiclecompositestructuredesign&optimization,aircraftvibrationmodeling,helicoptercrashless,andcompositestructurerepairtechnology.