introduction to composite materials & structures
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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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PredictingTransverseModulusofUnidirectionalLamina
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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PredictingTransverseModulusofUnidirectionalLamina
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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PredictingTransverseModulusofUnidirectionalLamina
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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