introduction to composite materials & structures

8/9/2019 Introduction to Composite Materials & Structures

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MaterialsandStructures

IndianInstituteofTechnologyKanpur


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Lecture17

BehaviorofUnidirectionalComposites


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re c vemo e s or ransverses ness

ShearmodulusandPoissonsratio

Estimatesfortransversestrength

Predictivemodelsforcoefficientofthermal

expansion

Thermalconductivity


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PredictingTransverseModulusofUnidirectionalLamina

Figure17.1showsasimplemodelforpredictingtransversemodulusof

unidirectionallamina.Here,themodelconstitutesoftwoslabsof

, , f m, .

overallthicknessofcompositeslabistc,whichissumoftfandtm.Itmay

benotedherethatthesethicknessesoffiberandmatrixaredirectlypropor ona o e rrespec vevo ume rac ons.

Fig.17.1:ASlabLike

ModelforPredicting

TransversePropertiesof

nsuc asys em,ex erna y mpose s resson ecompos e c s

assumedtobesameasthatseenbyfiber(f)andalsobymatrix(m).

Thisisincontrasttothemodeldevelopedforpredictinglongitudinal

modulus,wherewehadassumedthatstrains,andnotstresses,incomposite,fiberandmatrixareequal.


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Further,insuchamodel,whichisakintospringsinseries,theoverall

dis lacementincom osite intransversedirectionduetoexternalload

isasumofdisplacementinfiber(f)anddisplacementinmatrix(m).

c=f+m

Further,recognizingtherelationbetweenstrainsineachconstituent,and

theirthicknesses,aboveequationcanberewrittenas:

t = t + t

Dividingaboveequationbythicknessofcomposite(tc),andrealizingthat

tf tc,an tm tcequa Vfan Vm,respective y,weget:

c=mVm+fVf

Inlinearelasticrange,strainisaratioofstressandthemodulus.Hence,

aboveequationcanbefurtherrewrittenas:

(c

/Ec

)=(m

/Em

)Vm

+(f

/Ef

)Vf


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However,wehadearlierassumedthatexternallyappliedstressonthe

composite(c)issameasthatseenbyfiber(f)andalsobymatrix(m).

,

1/Ec=Vm/Em+Vf/Ef (Eq.17.1a)

Oralternatively,

Ec =(EfEm)/([(1Vf)Ef+VfEm] (Eq.17.1b)

Equation17.1givesusanestimatefortransversemodulusof

.

fibervolumefractionisrequiredtoraiseoveralltransversemodulusin

moderateamounts.Thisisinstarkcontrastwithlongitudinalmodulus,.

Equation17.1,eventhoughbasedonasimplemodel,isnotborneout

wellbeexperimentaldata.TOaddressthisinconsistency,severalalternativemodelshavebeendeveloped.


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However,inthislecturewewillusesimpleandgeneralizedexpressionsfor

transversemodulusasdevelo edb Hal in andTsai.Thesearerelativel

simplerelations,andhenceeasytouseindesignpractice.Theresults

fromHalpin andTsaiarealsoquiteaccurateespeciallyiffibervolume.

AsperHalpin andTsai,transversemodulus(ET)canbewrittenas:

ET/Em=(1+Vf)/(1 Vf) (Eq.17.2)where,

= Ef Em 1 Ef Em +

, ,

loadingcondition.Itsvaluesaregivenbelowfordifferentfibergeometries.

=2forfiberswithsquareandroundcrosssections.

= a o e sw ec angu a c osssec on. e ea s ec osssec ona mens on

offiberindirectionofloading,whilebistheotherdimensionoffiberscrosssection.


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ShearModulusandPoissonsRatio

Aperfectlyisotropicmaterialhastwofundamentalelasticconstants,Eand

.Itsshearmodulusandbulkmoduluscanbeex ressedintermsofthese

twoelasticconstants.

Likewise,atransverselyisotropiccompositeplyhasfourelasticconstants.

Theseare:

EL,i.e.elasticmodulusinlongitudinaldirection.

ETi.e.elasticmodulusintransversedirection. GLTi.e.longitudinalshearmodulus.

LTi.e.Poissonsratio

Adetaileddiscussiononthemathematicallogicunderlyingexistenceof

.

Tillsofar,wehavedevelopedrelationsforEL,andET.Nowwewilllearn

aboutsimilarrelationshipsforGLT

andLT

.


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ShearModulusandPoissonsRatio

Halpin andTsaihavedevelopedrelationssimilartoEq.17.2whichcanbe

usedto redictlon itudinalshearmodulus G .Thisisshownbelow.

GLT/Gm=(1+Vf)/(1 Vf) (Eq.17.3)

where,

=[(Gf/Gm)1]/[(Gf/Gm)+1]

ForpredictingPoissonsratioLT,weexploitthefactthatalongitudinal

tensilestraininfiberdirection,willgeneratePoissoncontractionin

transversedirectioninboth,matrixandfibermaterials.

Inthiscontext,wealsousethefactthatrelativestrainvaluesforsucha

fraction.Thus,overallPoissonsratioLT forthecompositecanbewritten

as:

LT =fVf+fVm (Eq.17.4)


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TransverseStrength

Suchaconstraintonmatrixdeformation,tendstoincreaseplystransversemodulus,thoughonlymarginally(unlessfibervolumefractionishigh).

However,thestoryisevenmorestarklydifferentincaseoftransversestrength.

Thedeformationconstraintsimposedonmatrixbyfiberstendtogeneratestrain

andstressconcentrationsinmatrixmaterial.

Thesestressandstrainconcentrationscausethematrixtofailatmuchlesser

values of stress and strain than a sam le of matrix material which has no fibers at

all.Thus,unlikelongitudinalstrength,transversestrengthtendstogetreducedforcompositesduetopresenceoffibers.

Thisreductionintransversestrengthofaunidirectionalplyischaracterizedbya

factor,S,thestrengthreductionfactor. Theexactvalueofthisfactorcanbe

numericalsolutiontechniques.

es reng o un rec ona p y n ransverse rec on,uT,can ewr enas:

uT=uf/S (Eq.17.5)


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SomeOtherPropertiesofUnidirectionalPlies

Usingapproachesasdescribedearlier,thermalconductivityinL(kL)

directioncanbewrittenas:

kL=Vfkf +Vmkm (Eq.17.6)

Similarly,transverseconductivity,kT,canbewrittenas:

kT/km=(1+Vf)/(1 Vf) (Eq.17.7)

w ere,

=[(kf/km)1]/[(kf/km)+], where, log=1.732log(a/b)

Finally,longitudinalandtransversethermalexpansioncoefficientshave

beenshowninengineeringliteraturetobe:

L= EfVf f +EmVmm EL Eq.17.8

= m m m . .


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re c vemo e s or ransverses ness

ShearmodulusandPoissonsratio

Estimatesfortransversestrength

Predictivemodelsforcoefficientofthermal

expansion

Thermalconductivity


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e erences

1. Analysis and Performance of Fiber Composites, Agarwal,. . an rou man, . ., o n ey ons.

. ec an cs o ompos te ater a s, ones, . ., c raw

Hill.

3. Engineering Mechanics of Composite Materials, Daniel, I.

. , ., .

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