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IntroComposite�Applns.PropertiesVoigt, �Reuss,�Hill

Anistrpy.CTECellular�Matls.Wood

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Microstructure-Properties:Composites

Microstructure Properties

Processing

Performance27-301A.D.Rolle/,M.DeGraef

Last modified: 2nd Nov. ‘15�



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Lecture Objectives: Composites•  Themainobjec?veofthislectureistointroduceyouto

microstructure-propertyrela?onshipsincompositematerials.

•  Compositematerialscons?tuteahugeclassofmaterials.Theobjec?veofthislecturewillthereforebetoprovidesomedefini?onsanddescribesomeofthebasicrela?onships.

•  Cellularmaterialswillbeemphasizedbecauseoftheirconnec?ontonaturalmaterials(biomaterials)andespeciallywood,whichsomeofyouwillstudyinthesecondLab.



Questions & Answers for Part 11.  Whatarethegeneraladvantagesofcomposite

materialsovermonolithicmaterials?Givebothbiomaterialandman-madeexamples.Compositesgenerallyhavehigherspecificproper?es.Woodandcarbon-fiberreinforcedplas?csareexamples.

2.  Whatistheruleofmixturesasappliedtocomposites?Integratethepropertyofinterestoverthevolumeofthecomposite.

3.  Whatdothetermsisostressandisostrainmean?Asimplied,iso-stressmeanssamestressinallmaterials;iso-strainmeanssamestraininallmaterials.Foriso-stressyoucanthinkofthephasesasbeingconnectedinseriesbetweentheplanesacrosswhichtheloadistransmi/ed(andviceversaforiso-strain).

4.  Derivetheisostrainmodel.Seethenotes;deriva?onreliesonaveragingthestressesinthedifferentphases.

5.  Derivetheisostressmodel.Seethenotes;deriva?onreliesonaveragingthestrainsinthedifferentphases.

6.  Sketchthevaria?onsinmodulusexpectedforcompositesinwhichthecomponentshavestronglydifferentmoduli.Seethenotes;iso-strainmodelgiveslinearvaria?on(sameasRuleofMixturesinthiscase)whereasiso-stressmodelgivesnon-linearvaria?on.

7.  ExplainwhatismeantbytheVoigt,ReussandHillaveragemoduli.Voigt=iso-strain,Reuss=iso-stress,Hillaveragesthesetwo.

8.  Whichmodelfors?ffnessappliestoacompositematerialwithacompliantmatrixandawelldispersedpar?culatesecondphasethatiss?ffer(thanthematrix)?Inthiscase,theReuss(iso-stress)modelappliesbecausetheindividualpar?clesarenotconnectedandthusthereisli/leloadtransferbetweenthem.

9.  Whichmodelfors?ffnessappliestoacompositematerialwithacompliantmatrixandawelldispersed,parallel,s?fffibersthatisloadedalongthefiberdirec?on?Inthiscase,theVoigt(iso-strain)modelappliesbecausetheindividualfibersarestrainedequallywiththematrix.

10.  Whyarecellularorfoammaterialsusefulforachievinglowmodulus?Bymakingasubstan?alfrac?onofthe“material”emptyspace(airortrappedgas),onecanreducethemodulustothevolumeaverageofthesolidmaterialandgas.Thisaccessesmodulusvaluesthatareinaccessibletofullydensematerials.

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Key points•  Compositesareregardedasar?ficial(man-made)mixturesof

phases.•  Classifica?onofcompositesbyreinforcementtype(dimensionality)-

par1cles,fibersandlaminated.•  Applica?onoftheRuleofMixtures.•  Dependenceofcompositeproper?esonthespa?alarrangementof

thephases.•  Upperandlowerboundsonproper?es-exampleofelas?cmodulus,

VoigtandReussapproxima?ons.•  Highproperty:densityra1osachievablewithcomposites.•  Engineeringwithresidualstressincomposites.•  Anisotropyofcompositeproper?es,e.g.elas?cmodulus.•  Proper?esofwoodasacellularmaterial.•  Cellular/foammaterialsasshockabsorbers.

Examinable�



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Examples



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What are Composites?•  Compositematerialscontainmorethanonephase.•  Almostallmaterialscontainmorethanonephase,so

what’sthedifference?•  Thetermcompositeistypicallyappliedtoamaterialwhen

themul?-phasestructureisconstructedbydirectinterven?on(externaltothematerial).

•  CompositeMaterialExamples:glassfiberreinforcedplas?c(GRP),wood,clamshell,Marsbar.

•  Mul?-phaseMaterialExamples:precipita?onstrengthenedaluminumalloys,Ti-6Al-4V,dual-phasesteel,transforma?ontoughenedalumina(Al2O3-CeO2).

•  Cau?on!Thereissomeoverlapbetweenthecategories!

Examinable�



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Properties•  Itisusefultoreviewthebasicproper?esofthedifferent

typesofmaterialsthatareusedincomposites.•  Polymers-long[carbon]chainmoleculeswithanything

fromvanderWaalsbondingbetweenthechains(thermoplas?cs)tocovalentlinks(thermosets).Lowdensity,lowmoduluscomparedtoothermaterials.Onenhighlyformable(duc?le).

•  Ceramics-ionicorcovalentbonding,lowersymmetrycrystalstructures,highmel?ngpointandmodulus,resistanttodegrada?on,bri/le,highmodulus.

•  Metals-metallicbonding,symmetriccrystalstructures,mediummel?ngpoint,mediummodulus,duc?le,formable,variableresistancetodegrada?on.

Examinable�



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Why Use Composites? [Biomaterials]•  Innature,thebasicmaterialstendtobeweakand/orbri/le.

Evolu?onhasresultedinstructuresthatcombinematerialstogetherforproper?esthatfarexceedthosethatcouldbeobtainedinthebasicmaterials.

•  Thebasicinorganiccons?tuentofbone,forexample,iscalciumphosphateintheformofcrystallineCa10(PO4)6(OH)andamorphousCaPO3.Thisceramicisbri/leandnotpar?cularlys?ff.Thematrixoffibrouscollagenistoughbutevenlesss?ff.Whenembeddedarrangedintheformofacellularmaterial,however,remarkablevaluesofs?ffness:densityandtoughness:densityareachieved(andland-basedmul?-tonnecreaturesarepossiblesuchaselephants).

•  Asimilarsitua?onexistsinwoodwherethebasicmaterialsarequitecompliantbutarrangedinthemul?-levelcompositeformsthatweknow,highvaluesofstrength:densityandtoughness:densityresult.



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Why Use Composites? [Man-made]•  Thebasicreasonfortheuseofcompositesisalwaysthe

same:somecombina?onofproper?escanbeachievedthatisimpossibleinamonolithicmaterial[foragivencost].

•  InSiC-reinforcedaluminumforbrakerotors,forexample,thecombina?onoflightweight,toughness(fromtheAlmatrixat~2.7Mgm/m3),ands?ffness(fromtheSiCaddi?ons)isnotpossibleineithercons?tuentbyitself.

•  InCu-Nbforhighstrengthelectricalconductors,thecombina?onof>1GPayieldstrengthandhighelectricalconduc?vity(intheCu)couldneverbeachievedineithercons?tuentbyitself.Inthiscasethehighstrengthisasynergis?cpropertyofthecomposite.



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Key aspects of composites

•  Compositesareexpensivetomake,ascomparedtomonolithicmaterials,especiallyiftheshapeandarrangementofthephasesmustbecontrolled.

•  Thereforetheremustbeastrongmo?va?onformakingacompositestructuretooffsetthecost.

•  Thesimplestcompositesarepar?culatecomposites.Laminatesarenext,followedbyfibercomposites.Wovenstructuresarethemostcomplex.

Examinable�



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Typical Microstructures

•  Weshownextsometypicalmicrostructures.•  Inbiomaterials,manyarecellularcompositesatsomelengthscale(typicallyaround1µm).

•  Man-madecompositesaremoreonenfullydense.Thethreemajor[structural]materialtypesareallusedsotheabbrevia?onsMMC[metalmatrixcomposite],CMC[ceramicmatrixcomposite],andPMC[polymermatrixcomposite]arecommonlyused.

Examinable�



12 Cellular Biomaterials

Gibson & Ashby: Cellular Solids

Notethevaria?onindensity;alsothepresenceofdis?nctlayersofcellsinsomewoods,andinbone.Notealsothattheshapeofthecellsandtheirwallsmakesadifferencetotheirproper?es.



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Man-made Examples

SiCfibersinTi3Almatrix

SiCfibersinaCASceramicmatrixDowling: Mech. Behavior Materials

Notethetypicallengthscaleof~100µm,andtheuseoffibersforreinforcement.Thisbasictypeoffiber-reinforcedcompositeisstronglyanisotropic.ThetoughnessofsuchcompositesandtheneedforlimitedadhesionbetweenfiberandmatrixisdiscussedinthelectureonFracture.



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Food!

Fromlentoright,toptobo/om:a)Breadb)Meringuec)Chocolatebard)Chipe)Malteser(Candy)f)Jaffacake(cookie,seebelow)

Gibson & Ashby: Cellular SolidsMaltesersimage:commons.wikimedia.org/MaltesersOpen.jpg�

Jaffa:thetas?ngbuds.com�



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Food for Thought!

•  Howdoesicecreamrepresentamaterialinwhichthethermal-mechanicalhistoryiscri?caltoitsmicrostructurewhich,inturn,controlsitsproper?es?

•  Hint:thisinvolvesboththeproper?esofcompositematerials(ice,cream,voids)andpar?clecoarsening(theice).



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Examples of composites•  Theclassicalexampleofacompositeisconcrete.•  Itismorecomplexthanitappears.Therearetypicallycoarse

andfinepar?cles(rocks!)embeddedinamatrixofsilicatesandsulfates.Thereisahighfrac?onofporesofallsizes.Thisisanexampleofapar1culatecomposite.

•  Ordinaryconcrete(properlymade)hasexcellentcompressivestrengthbutpoortensilestrength.Thusreinforcedconcretewasinventedtocombinethetensilestrengthofsteelwiththecompressivestrengthofconcrete.Thisisanexampleofamul?scalepar1culateandfiberreinforcedcomposite.Itispar1culatebecausetheaggregate(coarsegravel)reinforcesthecement,andfiberbecausethesteelrodsreinforcetheconcrete.

•  Asubtlebutveryimportantvariantofreinforcedconcreteispre-stressedconcreteinwhichthereinforcingrodsareplacedintensionbeforetheconcreteisallowedtoset.Seefollowingslidesonresidualstresses.

Examinable�



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Glass-ceramics

•  Glassceramicsareusefulmaterialsthatcombinechemicalinertnesswiththermalstability.Theytypicallyarestrongerthanamorphousglasses.

•  Thismaterialclasswasinvented(bytheSandiaNa?onalLaboratories)forthespecificpurposeofmakingamaterial(insulator)thatwouldhaveagoodmatchforthethermalexpansioncharacteris?csofmetals(stainlesssteel,nickelalloys),i.e.arela?velyhighCTEwithvaluesintermediatebetweenceramics(typicallylow)andmetals(typicallyhigh).

•  Typicalphasemixtureincludeslithiumsilicate(s),cristobaliteandresidualglassphase.



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Property Ranges

Amuchwiderrangeofproper?esispossibleincompositesthaninmonolithicmaterials.Foams

permitmuchsmallermodulianddensi?esthanfullydensematerials.Thefollowingchartillustratesafewbasicproper?es.

Gibson & Ashby: Cellular Solids

Examinable�



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NotationA, B, C phasesAandB,CompositeVA volumefrac?onofphaseAPA propertyofphaseAEA (Young’s)modulusofphaseAεC (average)strainincompositeσA stressinphaseAK bulkmodulusG shearmodulusα coefficientofthermalexpansion(CTE)ρs,ρcell cellwalldensityρ* rela?vedensity(cellularmaterial)l length(ofabeam)t thicknessb depthδ displacementKIC fracturetoughness[planestrain]P loadx posi?on(orloca?on,inamaterial)

Examinable�



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Simple Models: Rule of Mixtures•  Whatisthesimplestmodelthatcanbeusedtopredicta

materialpropertyinacomposite?Answer:RuleofMixtures

•  Definethevolumefrac?ons,V,ofthevariousmaterialscomprisingthecomposite.Theaveragepropertyofthecompositeisthengivenby,discrete:PC = VAPA + VBPB + … =ΣViPi

con1nuum:PC = ∫ P(x)dV•  TheRuleofMixturesisanacceptablefirstapproxima?on

fores?ma?ngcompositematerialproper?es.Itis,however,onenconsiderablyinerrorandbe/ermethodsarerequired.

Examinable�



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Limits, bounds•  Therearesomecircumstancesunderwhichonewould

liketobeabletomakequan?ta?vepredic?onsoftheproper?esofacompositebutanexactsolu?onisnotavailable.

•  Underthesecircumstances,itiss?llpossibletosetlimitsontheproperty.Inaformalsensetheselimitsareknownasboundsbecausetheyaretheresultofanalysisusingtheprinciplesofsolidmechanics.Suchanalysiscandemonstratethataneitheranupperoralower(orboth)boundexistsforagivenstructureandloading.

•  Anupperboundmeansthatthevalueofthepropertycannotgoanyhigherthanacertainvalueandviceversaforalowerbound.

Examinable�



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Exact versus bounds

•  Exactsolu?onsareusuallyavailableforsimplegeometries.Reinforcedconcretewithparallelrodsissuchanexample.

•  Complexgeometriesarealmostalwayslimitedtoapproximatesolu?onsandboundsprovidethebestes?mate.Mostpar?culatecomposites,especiallythosewithcracksfallinthiscategory.

•  Themostinteres?ngproper?esforthislecturearethoseassociatedwithmechanicalbehaviorsuchass?ffness,strength,thermalexpansion.

•  Inthemostgeneralsense,weareseekingmethodsforaveragingapropertyovertheheterogeneouselementsofthemicrostructure.

Examinable�



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Isostress, isostrain

•  “Iso-”isaprefixmeaning“same”.Isostressisanassump?onthatthephasesexperiencethesamestress.Bycontrast,isostrainmakestheassump?onthatthephasesaresubjecttothesamestrain.

•  Eachassump?onleadstoverydifferentresults,especiallywhentheproper?esofeachphasearedivergent,asweseefromtheexampleofthebrickandthefoam.

Examinable�



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Isostrain•  Imagineparallelslabsofmaterial

betweenplatensthatapplyaload.

•  PhaseAhasvolumefrac?onVAandmodulusEA;PhaseBhasvolumefrac?onVBandmodulusEB.

•  Compositemodulus,EC?•  Weassumeisostrainbecause

eachphaseseesthesamechangeinlength.

•  Thestrain,ε=εC,isthereforethefield;thestressistheresponse(andthes?ffnessistheproperty).

Phase BPhase A

Examinable�



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Isostrain: 2•  Eachphasegivesadifferentstress:

σA=EAεC; σB=EBεC.•  Weaveragethestressesoverthe

compositeinpropor?ontothevolumefrac?onofthephase:σC=VAσA+ VBσB = VAEAεC + VBEB εC.

•  Themodulusisthera?oofthestresstothestraininthecomposite:EC =σC/ εC= VAEA + VBEB

•  Thismodulusisthusthearithme1cmean

ofthemoduliofeachphase,weightedbythevolumefrac?ons.Ineffect,theruleofmixtureshasbeenappliedtothes?ffnesses.

•  Exercise:provetoyourselfthatthiscanbeextendedtoanynumberofphases.

Phase BPhase A

Examinable�



26

Isostress•  Imagineparallelslabsofmaterial

betweenplatensthatapplyaload.

•  PhaseAhasvolumefrac?onVAandmodulusEA;PhaseBhasvolumefrac?onVBandmodulusEB.

•  Compositemodulus,EC?•  Weassumeisostressbecause

eachphaseseesthesamestress(assumingsamecross-sec?onalarea).

•  Thestress,σ=σC,istherefore

thefield;thestrainistheresponse(andthecomplianceistheproperty).

Phase B

Phase A

Examinable�



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Isostress: 2•  Eachphasegivesadifferentstrain:εA= σC/EA;εB= σC/EB.

•  Weaveragethestrainsoverthecompositeinpropor?ontothevolumefrac?onofthephase:εC=VAεA+ VBεB = VAσC/EA + VBσC/EB.

•  Themodulusofthecompositeisthera?oofthestresstothestraininthecompositeasbefore,exceptthatitiseasiertoworkwithinversemoduli,i.e.compliances:1/EC =εC/σC= VA/EA + VB/EB

•  Thecompositemodulusisthustheharmonicmeanofthemoduliofeachphase,weightedbythevolumefrac?ons.Ineffect,theruleofmixtureshasbeenappliedtothecompliances.

Phase B

Phase A

Examinable�



28

Example of Cu-W Composites

•  Thegraphsshow(a)examplesofthedifferenceinthecalculatedmodulusbasedonthe2differentassump?ons[parallelisequivalenttoisostrain,andseriestoisostress];(b)anexampleofthemeasureddifferenceinmodulusofCu-Wcomposites,contras?ngwire[=fiber]withpar?clereinforcement.Thefibercompositecorrespondsverycloselytotheisostraines?mate;thepar?culatecompositeisclosetotheisostress,althoughnotquitesoprecisely.

•  Notehowtheisostressandisostraines?matesaresimilarwhenthemodulidifferbyonlyafactoroftwo.Whenthemodulidifferbyanorderofmagnitude,however,thetwoes?matesdifferwidely.

•  Here,isostrainhappenstobethesameastheRuleofMixtures.

“Structural Materials”, Weidemann, Lewis and Reid

Examinable�



29

Voigt, Reuss, Hill•  Thesesimplees?matesofmodulushavenamesassociatedwiththem.•  TheisostressapproachisknownastheReussmodulus.•  TheisostrainapproachisknownastheVoigtmodulus.•  Hillproposedthatareasonableaverageofthetwowouldbe

appropriateinmaterialswheretheloadingisintermediatebetweenthetwoextremecases.HencetheaverageoftheIsotressandIsostrainvaluesisknownastheHillAverageModulus.

•  Wecanalsotreatthecompositeproperty(forelas?cmodulus)intermsofanarithme1cmean(isostrain)versusaharmonicmean

(isostress),whichisthereciprocaloftheaverageofthereciprocalvalues.

•  Aretherebe/eres?mates?Yes,lookintheSupplementalslidesforadescrip?onoftheHashin-Shtrikmanes?matesofmodulus.TherearealsotextbooksbyMura,Nemat-Nasser,Miltonandseveralothers.

Examinable�



Homework Questions•  Noworkedexampleisprovidedhereontheiso-strainandiso-stress

models.•  Exampleswerequotedoftheore?calcombina?onsofmaterialsand

forCu-W.•  Homework/examques?onsarelikelytoaskyoutocalculatemodulus

valuesatdifferentvolumefrac?ons(oftwophases),toplottheresults(linearorlogscale)andtocompareagainstexperimentaldata.

•  Youmaybeaskedtora?onalizedevia?onsofmeasuredmodulusvaluesfromcalculatedonesbyconsideringmicrostructure.Forexample,ifapar?culatecomposite(withs?ffpar?clesinacompliantmatrix,e.g.SiCinAl)hashighermodulusthanyoucomputefromtheiso-stressmodel,thenthismaybebecausethepar?clesarenotperfectlydispersedandtheyformnetworksthroughinter-par?clecontacts.

30 Examinable�



Summary: Part 1•  Compositesareman-mademixturesofphases,onenwith

differentmaterialtypes,e.g.glass(ceramic)asas?ffeningreinforcementinepoxy(polymer).

•  Thesimplestwaytoes?mateproper?esistousetheRuleofMixtures.Suchsimplevolumeaveragingisalsovalidforfieldquan??essuchasstressorstrain,dependingonboundarycondi?ons.

•  Thenextsimplestapproachtocompu?ngtheproper?esofacompositeistolookforupperandlowerbounds.Fortheexampleofelas?cmodulus,theiso-strainandiso-stressmodelsweredeveloped.Theiso-strainmodelhappenstogivethesameresultastheRuleofMixturesbuthasaphysicalbasis.

31 Examinable�



Part 2

•  InthisPart,weconsidertheproper?esofwood.•  Woodisamul?scalecompositematerial,inthesensethatitisself-evidentlyacellularmaterialbutthecellwallsarethemselvescompositestructures.

•  Woodisanaturalexampleofacellularmaterial.•  Wealsoexaminetheanisotropyofcompositematerials,partlyasawayoftyingtogetherwhatwelearnedaboutanisotropywithwhatwelearnaboutcompositestructures.

32 Examinable�



Questions & Answers for Part 21.  Whatmakeswoodamul?scalecompositematerial?Woodis

amul?scalecompositematerialbecausethereisiden?fiablestructureatthescaleoffilaments,microfibrils,cellwalllayersandcellorganiza?on(“grain”).

2.  Whatarethemainchemicalcomponentsofwood?Woodcontainsmostlycellulose(invariousforms)andlignin.

3.  Whatdis?nguisheswoodfromotherplants?Themaindifferencebetweenwoodandotherplantsisthatitscellwallscontainlignin,whichmakesitstrongerandmoreresistanttopests.

4.  Whatisthemacrostructureofwood?Woodcontainshighlyelongatedcells,thatdefinethe“grain”ofwood.Cellsaredepositedonanearlycon?nuousbasisbuttheirdiametervariesduringtheyearwithlargerdiametercellsduring?mesofrapidgrowth.Therearealsoradialstructuresknownas“rays”.

5.  Whatisthestructureofthecellwalls?Thereareseverallayers.ThereisaPrimaryouterlayer(P),outsideofwhichthereisa“middlelamella”thatcontainsmostofthelignin.InsidethePlayer,therearetheS1,S2&S3layers,withdifferentlayupsofthemicrofibrils.

6.  Whatare“rings”inwood?Asnotedabove,thecellsizevariesonanannualbasiswhichmeansthatacross-sec?onthroughatrunkrevealswhatlooklikeringsinthestructure;eachringcorrespondstooneyear,whichpermitstheageofatreetobees?matedwithgoodreliability.Thevaria?onsincellsizealsorevealchangesinlocalclimate.

7. Whatmicrostructuralcharacteris?ccorrelatesmoststronglywiththemechanicalproper?esofwood?Proper?essuchasmodulus,strengthandfracturetoughnesscorrelatemoststronglywithdensity.

8.  Howdoeselas?cmodulusdependondensityinwood?Theelas?cmodulusvarieseitherinpropor?ontodensityfortheaxial/longitudinaldirec?onorwiththedensitysquaredacrossthegrain(radialorcircumferen?al).

9.  Explainthestructureandcomponentsofmicro-fibrils.Eachmicrofibrilisabundleofcellulosefibersinamatrixofhemicelluloseandlignin.

10. Explainthestructureofcellwallsinwood.Seethenotes;Severallayersarepresentinthecellwall,eachwithitsowncharacteris?clay-upangleofthemicro-fibrils.Inpar?cular,theanglebetweenthemicrofibrilsandtheaxialdirec?onintheS2layerisstronglyan?-correlatedwiths?ffness.

11. Describetheanisotropyofthemechanicalproper?esofwood.Woodismuchs?ffer,strongerandtoughparalleltothegrainthanacrossthegrain.

12. Basedonthechemistryofwood,commentonitssensi?vitytomoisture.Certaincomponentsofthewood(esp.cellulose)arehydrophilicandabsorbwater.Increasedmoisturecontentincreasess?ffnessandstrength.

13. Whydoesthemodulusvaryfasterthanlinearacrossthegrain?Crucially,woodisacellularmaterialanddeformsprimarilyviabendingofthecellwalls.

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•  Notethevaria?onincellsizeduringtheyearfromEarlywood(spring-summer)toLatewood(summer-autumn).Thisvaria?onincellsizeproducesthecharacteris?c“rings”thatindicatetheageofthewoodbecauseoftheyearlycycleincellsize(andthemagnitudesofthecellsizescorrelatewithclima?ccondi?ons).The“Rays”arealignedwiththeradialdirec1on.Thelongdirec?onofthecellsistheaxialorlongitudinaldirec1on.Followaringaroundthetrunkandthisisthecircumferen1alortangen1aldirec1on.

Wood: Macro-structureExaminable�



35

Wood: Microstructure

•  Columbianpine-3orthogonalsec?ons

•  T:transverseR:radialL:longitudinal(ver?calinbothimages)

T-L

T-R

R-L

Examinable�



36

Itisimportanttounderstandwoodasacellular,compositestructure.Itisone,however,thathasseveraldifferentlengthscalesfromthatofthecellulosemoleculetothemacrostructureoflumberasweaccustomedtolookingatitatthevisualscale.Thefigureillustratesthehierarchyoflengthscalesfromtheatomicstructureofcellulose(A)tothestructureofatreetrunk(E).Thebasicbuildingblockofwoodisthepolymerofglucoseknownascellulose,whichoccursasa(mostly)crystallinefiber.Theothercri?calcomponentofwoodislignin,whichisacomplex,amorphousmaterialcontainingphenylgroups.Ligninsetswoodapartfromotherplants;itsoccurrenceintheouterandinnerliningsofthecellwallsiscri?calforbothstructuralproper?esandforwood’s(rela?ve)insensi?vitytoenvironment.

A B

C

D

E

F

Wood: multiscaleExaminable�



37

Wood: cell structure•  Eachcellwallcontainsmicrofibrils,eachofwhichisabundleof

cellulosefibersinamatrixofhemicelluloseandlignin.Thereforeitisafiber-reinforcedcomposite!ThePlayeris5%ofthethicknesswithrandomfiberdirec?ons;theS

1layeris9%withfibersat50-70°w.r.t.

theaxis;theS2layeris85%,fibersat10-30°;theS

3layeris1%,with

fibersat90°totheaxis.NotethedependenceofthetensilestrengthonthemicrofibrilangleintheOuterWall,labeled“S1”.Eachcellisalongtube,someofwhichareusedfortranspor?ngwater(butnotall).

C

Examinable�



38

Wood: Microfibril structure•  Thefilamentsorfibersofcellulose,(C6H10O5)n,wheren~104,are

organizedinbundles(togetherwithligninsurroundingthefibers)calledmicrofibrilswhosesizeisabout10nm.Eachsetofmicrofibrilsformsabundlethatisitselfastructuralmemberofthewallofacell(nextslide).

•  Sonwoodshavelongercellulosefibersthanhardwoods(whichma/erstothemanufactureofpaper).

AB

Examinable�



39

Wood: Constituents: Cellulose•  Cellulose:ahighmolecularweight,stereoregular,andlinearpolymer

ofrepea?ngbeta-D-glucopyranoseunits.Itisthemainstructuralelementandmajorcons?tuentofthecellwalloftreesandplants.Theempiricalformulaforcelluloseis(C6H10O5)nwhere'n'isdegreeofpolymeriza?on(DP).[h/p://www.paperonweb.com/wood.htm]

Substance Degree of Polymerization (DP)

Molecular Weight

Native Cellulose >3500 >570,000 Purified Cotton 1000 - 3000 150,000 - 500,000 Wood Pulp 600 - 1000 90,000 - 150,000 Commercial Regenerated Cellulose (e.g. Rayon)

200 - 600 30,000 - 150,000

Cellulose 15 - 90 3000 - 15,000 Y Cellulose <15 <3000 Dynamite Nitro-Cellulose 3000 - 5000 750,000 - 875,000 Plastic Nitro-Cellulose 500 - 600 125,000 - 150,000 Commercial Cellulose Acetate

175 - 360 45,000 - 100,000



40

•  Takingwoodasanexample,itisfoundempiricallythatthemodulivarywith(rela?ve)densityanisotropically.

•  Notethediscrepancybetweenthe

empiricalequa?onandtheslopeintheplot.Thetheore?calpredic?ongoesas(ρ/ρcell)3.

[Gibson & Ashby: Cellular Materials]

€

Eaxial ∝ Ecellρρcell

E transverse ∝ Ecellρρcell

$

% &

'

( )

2

Cellular Materials: Young’s Modulus Examinable�



41

Wood: Young’s Modulus

(1)

(2)

• Thefirstequa?onquan?fiestheideathatthetensilemodulusofwoodparalleltothegrainisjustthevolumeaverageoftheareafrac?onoccupiedbycellwall.

€

Eaxial ∝ Ecellρρcell

E transverse ∝ Ecellρρcell

$

% &

'

( )

2

• Tounderstandwhatcontrolstheelas?cmodulus(Young’smodulus)ofwood,wehavetoconsiderbendingofthecellwallsinthemicrostructure.



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Wood: Modulus, contd.

•  Thesecondequa?on(modulustransversetothegrain)ismoresubtleandstatesthattheelas?cmodulusvariesmorerapidly-withthesquareofthedensity-thantheaxialmodulus.Thereasonforthiscanbeunderstoodverysimplyintermsofthecellularstructure.Whenwoodisloadedacrossthegrain,thecellwallsbendlikeminiaturebeams.Thisresponsecanbequan?fiedbyuseofbeamtheorytoarriveatthefunc?onaldependenceofequa?on2.



43

Summary: Part 2A•  Woodcanbeunderstoodasacompositematerialor,

moreusefully,asacellularmaterial.•  Woodisamul?-scalecompositematerial.•  Thecellwallsofwoodarethemselvescomposite

structures.•  Eventhefibersinthecellwallsarealsocomposites.•  Theelas?cproper?esofwoodarehighlyanisotropic:

woodiss?fferintheaxialdirec?onandmorecompliantinthetransversedirec?on.

•  Thevaria?oninmoduluswithrela?vedensityislinearintheaxialdirec?onbutvariesasthesquareoftherela?vedensityinthetransversedirec?on.

•  Inpart2B,weintroducebeambendingtheorytoquan?fytheseeffects.



44

2B: Introduction to beam theory

€

Δll0

=(R + y)θ − Rθ

Rθ but ε =

Δll0

∴ε =yR

=8δmaxyl2

=3FlyEwt 3

€

Moment of Inertia, I :

I = y 2dAy= 0

y= t∫ =

wt 3

12Moment on the beam, M :

M = σ ydAy= 0

y= t∫

Stress varies linearly with strain :σy

=Eεy

=E(y /R)

y=ER

This shows that stress varies linearly with yso σ/y is a constant :

M =σ y y 2dAy= 0

y= t∫ =σ y I

Thus this double equality is true :MI

=σy

=ER

For a force, F, at the center of the beamthe maximum deflection, δmax is :

δmax =l2

8R=l2M8EI

=l2Fl /48EI

=l3F

32EI=

3Fl3

8Ewt 3

w

tδmax

•  Considera3-pointbeamwithlength,l:supportedateitherendandloadedinthecenterwithaforce,F.Themostimportantpointisthatthereisaneutralpointinthebeam,n,atwhichthestressiszero;abovethisitiscompressive,andbelowitistensile.Thestressispropor?onaltodistancefromtheneutralplane.

[Dowling]�



45

Beam theory applied to wood•  Themechanicalbehaviorcanbemodeledbyaframeworkofbeams.

Thedeflec?on,δ,ofabeamoflengthlandthicknesst,underaloadF,isgivenbystandardbeamtheory(seepreviousslide)asδ= F l3/ 32EcellI,whereEcellistheYoung’smodulusofthebeammaterial(i.e.thecellwall)andIisthebendingmomentwhichispropor?onaltot4(recallthatI = wt3/12 ,soforw=t,I = t4/12).Theforceisstress,σ,mul?pliedbyarea,= l2,i.e. F = σ l2. Thestrain, ε,isthedisplacement,δ, dividedbythecelllength, ε = δ / l = F l3/ 32 l EcellI = F l2 / 32 EcellI.



46

Wood: modulus, contd.•  ThuswecanobtainEq.2asthera?oofstresstostrain.

€

E transverse =σε

= σFl3 32EcellIl( )

= σσl2 l3 32EcellIl( )

= 32EcellIl4( )

E transverse = 8 3Ecelltl( )

4 forw=t,I = t4/12�



47

Wood: modulus, contd.•  Butwealsorelatethedensitytothecelldimensionsbywri?ng

ρ ∝ (t/l)2andobtainEq.2(wherethepropor?onalityconstant,C”~1,basedonexperimentaldata),Etransverse = C” Ecell ρ2.!

•  Notethatthisderiva?onisageneraloneforopen-celledfoamsandhappenstobeasimple,easy-to-understandapproach.Woodshavemorecomplexstructuresthantheopencellmodelwhichhelpstoexplainthesca/erinthedata.

•  Notethatthetheoryforclosed-celledfoams(seesupplementalslides),whichisclosertotheactualstructureofwood,showsadependenceon(ρ/ρcell)3,not(ρ/ρcell)2asderivedhere.



48

Wood: strength

•  Here,thestoryisverysimilartothatofmodulus.Theaxialmodulusisdeterminedbytheareafrac?onofcellwallmaterial,hencethelineardependenceondensity.Thetransversestrength,however,islimitedbybendingandplas?chingebehaviorofthecellularstructure,hencethequadra?cdependenceondensity.Thedifferencebetweenaxialandtransverseproper?esissogreatforbothmodulusandmostothermechanicalproper?esthatitisalwaysnecessarytobeawareoftheanisotropyofwood,i.e.thattheproper?esvarymarkedlywithdirec?on.Moresuccinctly,woodismuchstrongerands?fferalongthegrainthanacrossthegrain.Thelowerthedensity,themoreobviousthedifference.

€

σ axial ∝σ cellρρcell

σ transverse ∝σ cellρρcell

%

& '

(

) *

2



49 Wood: fracture toughness

KIC:axial ∝KICcell (ρ/ρcell)3/2KICtransverse∝KICcell(ρ/ρcell)3/2KICtransverse»KIC:axial

•  Forfracturetoughness,theresultisgivenwithoutproofthatthecellularstructureleadstoa3/2exponentinthedensitydependence,regardlessofdirec?on.Thecrucialpointisthatpropaga?ngacrackparalleltothegrainismucheasierthantransverse,byafactorof~10!Morethanonemicrostructuralfeaturecontributestothehightransversetoughness,includingfiberpull-out,propaga?onofsecondarycracksperpendiculartotheprimarycrack,andelonga?onofthepolymerchainsinthecellwalls.Again,therearemanydifferentdirec?onsandplanesforcrackpropaga?oninthisanisotropicmaterialwhichfurtherincreasesthevariabilityofthetoughness.

More detailed figure available in Gibson & Ashby, fig. 10.17�



50

Wood: moisture content

•  Waterisfoundinwoodbothinchemicallyboundform,andstoredinvessels(“lumin”).

•  Theboundformofwaterstronglyaffectsproper?esofallkinds.

•  Thefreewaterhasonlyaminoreffect.•  The“fibersatura?onpoint”isthewatercontentthatcorrespondstosatura?onoftheboundwater.TheFSPisabout28%ofthefullydrywood.



Bone�•  Similarstrong

sensi?vityofproper?estomoisturecontentasobservedforwood.

•  Dependenceofmodulusondensityislesscleareventhanforwood.

•  Compressivestrengthvariesasthesquareofthedensity�

51

Note:bonevariesconsiderablyinstructure,dependingonthelocalloadingthatthebodyputsonit.



52

Future Composites

“Carbonnanotubecomposites”,PJ.F.Harris,Intl.Matls.Reviews,49,31(2004)

•  Carbonnanotubecomposites:currentlybasedonpolymer-nanotubematerials,butcombina?onsofnanotubeswithceramicsarebeingfabricated.

•  (a)Nanotubetypes(b)TEMmicrographofnanotubes(notefringesinthewallsindica?ngmul?plewalls);(c)TEMimageofmul?wallednanotube(MWNT)-polystyrenethinfilmcomposite.



53

Impact Protection for Space Vehicles

•  h/p://hi|.jsc.nasa.gov/hi|pub/main/index.html•  h/p://see.msfc.nasa.gov/mod/modtech.htm-shielddesign.

•  h/p://oea.larc.nasa.gov/PAIS/MISSE.html-materialstes?ng.

•  h/p://www.nasa.gov/lb/missions/science/spinoff9_nextel_f.htmluseofNextelasashieldmaterial.



54

Summary: Part 2B•  Woodcanbeunderstoodasacompositematerialor,

moreusefully,asacellularmaterial.•  Woodisamul?-scalecompositematerial.•  Thecellwallsofwoodarethemselvescomposite

structures.•  Eventhefibersinthecellwallsarealsocomposites.•  Wecanes?matetheirproper?esbasedonthe

applica?onofbeambendingtheorytothewayinthecellwallsdeformunderload.

•  Bonehasproper?esthatresemblewoodinsomerespectsi.e.asimilardependenceofmodulusondensity.



Part 3

•  InthisPart,weconsiderthebasiccharacteris?csoffibersforfibercomposites.

•  Weexaminehowtoengineercompositeproper?esbyexploi?ngresidualstress.

•  Wealsoexaminetheanisotropyoftheproper?esofcompositeproper?es,whichbuildsonwhatwelearnedabouttensorproper?es.

55 Examinable�



56

Fiber Composites•  Animportantclassofcompositesisthatoffibercomposites.•  Thematerialsinvolvedmaybemetal,ceramicorpolymer.Glass-fibercompositeis

typicalinlow-coststructuressuchasboathulls.Carbon-fibercompositesareusedinhigherperformancestructuressuchasairplaneswheretheirhighercostisjus?fiedbytherequirements.Ceramiccompositesareusedtypicallyforhightemperatureservice,suchasheatexchangers.

•  Thebasicideaistotakeadvantageofhighstrengthands?ffnessofthefibersandtoobtaindamagetolerance(andspecificshapes)byembeddingtheminasuitablematrix.Morespecifically,thefibermaterial(e.g.graphite,glass)isamaterialthatwouldnotgenerallybeconsideredtobeastructuralmaterial.

•  Solidmechanicsoffibercomposites:thekeytounderstandingthemechanicalproper?esoffibercomposites(forfiberswhoselengthisshortcomparedtothesizeofthecomponent)isloadtransferbetweenthematrixandthefibers.Thismeansthatthestressoneachfibervariesalongitslength.Also,thecompositematerialsarestronglyanisotropic(sotensorsareusefulagain).Seediscussioninthesupplementalslides.

•  Moderndevelopments:carbonnanotubesofferexcep?onals?ffnessandstrength,nottomen?oninteres?ngelectricalproper?esinsomecases.Ifwecanfigureouthowtoseparateoutthevariousdifferentconforma?onsandhowtoalignthenanotubes,thereshouldbeawiderangeofexci?ngmaterialspossible.

Examinable�



57

Fibers for Polymer Matrix Composites

•  Manytypesoffibersareavailable:carbon,glass,aramid,quartz,polyethylene,boron,siliconcarbide,alumina,aluminosilicate.

•  Thepolymermatrixcompositebusinessisdominatedbyvolumebycarbon,glassand

aramidfibersbecausetheyofferthebestperformance:pricera?o.

“MechanicsofFibrousComposites”,C.T.Herakovich

Examinable�



58

Carbon Fibers•  Modulusrangesfrom200-750GPa

(comparewithsteel:210GPa)•  Strengthrangesfrom2-6GPa•  Breakingstrainrangesfrom0.2-2%•  Densityrangesfrom1.76-2.15•  Highestcostcomparedtoglassoraramid,butgreatest

rangeofproper?es.•  Internalstructureconsistsofradially-alignedgraphite

platelets,whichleadstosomeanisotropyinproper?esinthefibers.Boththermalandelectricalconduc?vityaregenerallygood(buttheninsula?onrequiredwheremetalsmightbeincontactforcarbon-fibercomposite).



59

Glass Fibers•  Glassfibersproducedbyspinningliquidglassdirectlyto

finefibers.JustasintheGriffithexperiments,thestrengthisbasedonsmalldiameter.

•  Modulusrangesfrom70-90GPa.•  Strengthrangesfrom1.7-5GPa•  Breakingstrainfrom2to5%•  Density~2.5gm/cc.•  “Eglass”[electrical,borosilicateglass]isthecheapestand

mostcommon.“Rglass”and“Sglass”ismoreexpensivebutmorecorrosionresistant,forexampleandhigherstrength.



60

Aramid Fibers•  Aramidfibersareproducedbydrawingliquidcrystalpolymersbased

on,e.g.polyparabenzamideorpolyparaphenyleneterephthalamide.•  Polymerchainsarrangedinradiallyoriented,kinkedsheets.

Bondingbetweenthemoleculesislargelyhydrogenbondingsothetransverseproper?esareweakcomparedtoon-axis.Thereforedifficulttopropagateacrackalongafiber.

•  Modulusrangesfrom55-120GPa•  Strengthrangesfrom3to3.6GPa•  Breakingstrainrangesfrom2.5to4%•  Density~1.45gm/cc.•  Aramidfibersvulnerabletoenvironmentaldegrada?on(sunlight).



61

Residual Stresses and Composites•  Inasta?onarybodythatisfreeofexternalloads,theaveragestress(andmoment)mustbezero

because(Newton’sLaws)theremustbenonetforceonit.•  Thestressstateinsidethebody,however,canvaryarbitrarily.Suchvariableinternalstressesare

onenknowasresidualstressesbecausetheyarethelen-overfrompreviousprocessing.•  Thesimplicityofelas?cstressesisthattheycanbesuperimposed.Thereforeonecanassumeinbeam

loadingthatthestressesimposedbyexternalloadingcanbeaddedtotheinternalvaria?ons.•  Aswithallphenomena,thereareengineeringapplica?ons.Reinforcedconcrete,forexample,isa

fiber-reinforcedcompositewithabri/lematrix(concrete)andaduc?lefiberreinforcement(steelbarsorcable).Thesteelistypicallyheldintensionduringthese~ng-upoftheconcrete,resul?nginacompositeforwhichthesteelisinastateoftensionandtheconcreteisincompression.

•  Forfiber-reinforcedmaterials,forexample,adifferenceinthermalexpansioncoefficientcanproducearesidualstressstateinacomposite.Forexample,ifthefiberhasasmallerCTEandthecompositeiscooledfromazerostressstateathightemperature,thenthematrixshrinksmorethanthereinforcingfibers,pu~ngthematrixintensionandthefibersincompression.

•  SafetyGlassascommonlyusedforthewindshieldsofcarsrelyonresidualstressdevelopedthroughheattreatment.Acompressiveresidualstressnearthesurface(s)isbalancedbyatensileresidualstressinthecenter.Furthermore,theheattreatmentisdoneinsuchafashionastodevelopafinepa/ernsothat,ifthewindshielddoesbreak,itsha/ersintomanysmallbutcompactpiecesthatarefarlesshazardousthanthetypicalshardsofwindowglass.

Examinable�



62

Reinforced Concrete•  Stepsrequired:

1.  Stretchreinforcingsteelcables(i.e.placethemintension)2.  Pourconcretearoundthecables;allowconcretetoset3.  Removetensioningforcefromsteelcables4.  Thesteelcablescontractelas?callybuttheconcretematrixresiststhe

contrac?on5.  Steelremainsintension(didnotshrinkbacktozerostrain)whereasthe

concreteisincompressiontobalancethetensilestressinthesteelcables•  Ques?on:isthereanop?mumloca?onforthereinforcementwithinthe

beam?Atthetop?Bo/om?•  LoadingofReinforcedConcreteBeams:

–  Asthebeamisloaded(e.g.3-pointbending),theconcreteunderneaththeloadingpointexperiencesthesumofitsresidualcompressivestress,plusthetensilestressfromthebendingload.Formoderateloads,thestressremainscompressive,protec?ngagainstbri/lefailure.

•  Thecompositeishighlyanisotropic,ofcourse.•  Famousexample(localtoPi/sburgh):thecan?leveredterracesofFrank

LloydWright’shouse,Fallingwater(imageabove).•  h/p://structsource.com/analysis/types/concrete.htm

Examinable�



Pre-stressed Reinforced ConcreteRemember:intheabsenceofexternalloads

(trac;ons)thenetstressinthematerialmustbezero.

63

Steel rod: large tensile stress from external load

Add concrete, allow to set, no stress in concrete

Remove external load on steel; compressive stress in concrete increases to balance the decreased tensile stress in the steel

€

0 = σdVVolume∫

Examinable�



Homework Questions•  Aworkedexampleisverysimpleinthiscase.•  Ifthefracturetoughness,KIC,ofconcreteismeasuredtobe2 MPa√m,and

themaximumflawsizeis5 mm (basedontheaggregatesizes),whatisthemaximumtensilestressthatitcanwithstand?Answer:applytheGriffithEq.withthemaximumflawsizeasthecracksize(sincethisrepresentstheweaklinkinthematerial),whichsuggeststhatthebreakingstress=√{KIc/πc} = √{2.106 / π / 5.10-3} = 11.28 kPa,whichisverysmallindeed.

•  Ifthevolumefrac?onofreinforcingsteelinconcreteislimitedto10%,itsyieldstressis1.5 GPa andyoucanstressthesteelto80%ofitsyield(represen?ngthesafetyfactor),whatapproximatetensilestrengthcanyoudevelopintheconcreteviapre-stressing?Answer:assumethatyoucanneglecttheinherenttensilestrength.Assumethatyoucanapply1500 * 0.8 MPa tensilestressinthesteel,whichisbalancedby1500*0.8*0.1/0.9 = 133 MPacompressivestressintheconcrete.Thisresidualcompressivestressintheconcreterepresentsthemaximumtensilestressthatyoucanapplybeforeyouexpecttheconcretetobreak.

64 Examinable�



Anisotropy of Cell Wall65

cij =

0

BBBBBB@

16 11 11 0 0 011 16 11 0 0 011 11 16 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

1

CCCCCCA

De Graef HW 4 2009 (adapted)

Examinable�



Cell Wall: Young’s Modulus: Anisotropy

•  Thefirstdecisioniswhichmodeltouse.•  Inthiscontextitmeans,doweuseiso-strainoriso-stress?

•  Sincewearelookingatloadingthematerialintheplaneofthelayers,thenitisappropriatetousetheiso-strainmodel.

•  Thismeansthatwecanusetheruleofmixturesforthe3phasesthatcontributetotheYoung’smodulus:σC= V1σ1 + V2σ2 + V3σ3 = V1E!εC + V2E2εC + V3E3εC.

•  Thenextstepistocomputethemoduli.

66 Examinable�



67

S in terms of CInordertocomputeYoung’smodulus,weneedtousethe

reciprocalcompliances.Therela?onshipsfors(compliance)intermsofc(s?ffness)

aresymmetricaltothosefors?ffnessesintermsofcompliances(asimpleexerciseinalgebra!).s11 = (c11+c12)/{(c11-c12)(c11+2c12)}

= (16+11)/{(16-11)(16+22)}� = 0.1421 �s12 = -c12/{(c11-c12)(c11+2c12)}

= -11/{(16-11)(16+22)}� = -0.05789 �s44 = 1/c44� = 1/1 = 1.

Examinable�



68

Rotated compliance (matrix)•  Thestandardrela?onshipisasfollows:

•  Nowwejustneedtospecifythedirec?oncosines,ofwhichonlythe1stterm,(α1α2)2,isnon-zero.FortheS3layer,itiseasybecausethevalueiszero,soonlys11isused!ForS2(α1α2)2=cos2(20)cos2(70)=0.1033;forS1the(α1α2)2=cos2(60)cos2(30)=0.1875.Thecombina?onofcompliances=2*(0.1421+0.05789-0.5)=-0.3001.

! s 11 = s11 −

2 s11 − s12 − 12s44( ) α12α22 +α 22α3

2 +α32α1

2{ }

Examinable�



Compliance values; Young’s Modulus

•  s11forS1:0.1421•  s11forS2:0.1421+0.1033*-0.3001=0.1111•  s11forS3:0.1421+0.1875*-0.3001=0.08583•  Makethevolume-basedaverage:•  1/Ecell=0.1*0.1421+0.8*0.1111+0.1*0.08585=0.111675

•  Ecell=1/0.111675=8.954

69 Examinable�



Cell Wall: Young’s Modulus: Anisotropy

•  Whatifthefibershave,saytetragonalsymmetry,asismorelikelythancubic?Thenthes?ffnesstensorwilltakethefollowingform.

•  Herethechallengeistoinverttheproper?esofatetragonalmaterialsothatweoughttousecompliancesratherthans?ffnesses.

70

cij =

0

BBBBBB@

16 5 11 0 0 05 16 11 0 0 011 11 4 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 3

1

CCCCCCA



Tetragonal Fibers�

•  Let’sfurtherassumethatthe4-foldsymmetryaxisisparalleltothelongdirec?onofthefibers.

•  Inver?ngthecompliance-s?ffnessrela?on,however,isnon-trivialfornon-cubics.ThisisfoundinNyeorNewnham.Therela?onshipsarewri/enoutforcintermsofs,buttheyaresymmetricalsoscanbesubs?tutedforc,andviceversa.

•  c11+ c12= s33 / s ; c11- c12=1/(s11- s12); c13 = -s13/s33 c33 = (s11 + s12) /s ; c44 = 1 / s44 ; s = s33 (s11 + s12) - 2s2

13 . •  Nextweneedtofindtheformulaeforthevaria?onins11

withdirec?on.�

71



Tetragonal Fibers, contd.�

•  Again,asfoundinNye:�

72

€

" s 11 = s11 α14 +α2

4( ) + s33α34 + s12 + s44( )α12α2

2

+α22 1−α3

2( ) s13 + s44( ) + 2s16α1α2 α12 −α2

2( ){ }•  Thecomputa?onisthensimilarbutlongerandmoredetailed.

•  Whatemergesistheconclusionthatthecellwallcanbes?ffer,ormorecompliant,thanispossiblebyaligningthefibersinonlyonedirec?on.�



Summary: Part 3

•  Inthispart,welearnedabouttheproper?esoffiber-reinforcedcomposites.

•  Wealsolearnedabouthowimportanttheanisotropyofcompositesonenis,andhowtorepresentthatanisotropyintermsoftensorproper?esofmaterials.Furtherinforma?ononanisotropyofcompositescanbefoundinthesupplementalslides.

73 Examinable�



Part 4

•  InthisPart,weconsiderthebasiccharacteris?csofcellularmaterials.

•  Weexaminetheproblemofshockabsorbingmaterialsasanexampleoftheapplica?onofcompositeproper?esforfoams(cellularmaterials).

74 Examinable�



75

Cellular Materials

•  Thisnextsec?onprovidessomebasicinforma?ononcellularmaterials.

•  Whystudycellularmaterials?Answer:cellularmaterialsprovidearangeofproper?esthatarenotachievableinbulkmaterials.Especiallywhenloadcarryingcapacityatverylowdensi1esisrequired,onlycellularmaterialscansa?sfytherequirements.Shockresistanceisalsoavitalcharacteris?cofcellularmaterials.

•  Cellularstructuresarefeasible(andusedforengineeringapplica?ons)withallmaterialstypes.Metalhoneycombsareusedintransportapplica?ons.Ceramicfoamsareusedininsula?on.Cellularstructuresareubiquitousinbiomaterials(wood,bone,shells…).

Examinable�



76

Honeycombs: properties

[Gibson & Ashby: Cellular Materials]

•  Notethecontrastbetweentensionandcompression(plateaupresent),4.2avs.4.2b.

•  Evenbri/lewallmaterialsexhibitprogressivefailureincompression,4.2e.

•  Thestress-straincurvesarelabeledbytheircharacteris?cstages.

•  Veryimportantconsequencesforenergyabsorbingstructures(seelaterslides)

Examinable�



77

Energy Absorption

•  Whyarefoamsuseful?!Onereasonistheircapacitytoabsorbenergy.

[Gibson]

Examinable�



78

Energy Absorption: 2•  Howdothesetwographsconnect?Eachlineonthe2ndgraphcorrespondtoa

locusofpointsfromthe1stgraph,forapar?cularrela?vedensity.Notetheturn-overinthecurveofenergyversusstress:thisisthemostefficientuseofthematerial.

[Gibson]

Examinable�



79

Energy Absorption: 3

Elas?c

WallBuckling

FullyDensified

Duringwallbuckling,densifica?onproceedsataapproximatelyconstantexternalstress.

[Gibson]

Notethat,oncethefoamstartstodensify(steepupturninthestress-straincurve)thenthestressriseswithli/leincreaseinenergyabsorbed.

Examinable�



80

•  Asseenbefore,thestress-strain(8.4a)canbere-plo/edasenergyabsorbedversusstress(8.4b).Varyingthedensityvariesthemaximumenergythatcanbeabsorbedattheplateaustress.

•  Wecandrawanenvelopethroughthepointsofmaximumenergy÷plateaustress.

•  Varia?onsinotherparameterssuchasstrainratecanalsobeshownonsuchanenergy-stressdiagrambyplo~ngonlytheseenvelopes.

[Gibson]

Energy Absorption: 4Examinable�



81

Shock Cushions

•  Onceoneknowstheenergy-stresscharacteris?cofamaterial,itispossibletocalculatetheop?mumthickness.

•  Giventhekine?cenergytobeabsorbed,U,andtheareaofcontactbetweenobjectandfoam,A,thethickness,t,isgivenby

t = U / W A (Eq. 1)

whereWistheenergyabsorbedperunitvolumeinthefoam.•  Typically,themassoftheobject,m,andthepeakdecelera?on,a,is

alsospecified(asamul?pleofgravita?onalaccelera?on,g)whichdeterminesthemaximumstress,σ,

σ = m a / A (Eq. 2)

Examinable�



82

Shock Cushion: 2•  Inaddi?on,adropheightisspecifiedwhichinturnsetsthevelocity,

v,andtheenergy,U,thatmustbeabsorbed;U = m v2 / 2.Thusthethickness,t,isgivenbyt = m v2 / (2 W A) (Eq. 3) �

•  Thisinturnspecifiesthestrainrate,dε/dt,inthefoamwhichaffectstheenergy-stressrela?onship(seeFig.8.4c):dε/dt=v / t (Eq. 4) �

•  Agoodplacetostartistoiden?fythemaximumallowablestressandreadofftheassociatedenergyatahighstrainrate.Theenergyis,however,afunc?onofbothstressandstrainrate,sosomeitera?onisrequiredtoiden?fyasuitablethickness.

Examinable�



Shock Cushion: 3�WorkedExampleProblemspecifica;onMassofpackagedobject:500 gms.Areaofcontactbetweenobjectandfoam:A = 0.01 m2

Velocityofpackageonimpact,v = 4.5 m/s(dropheight,h=1m)Energytobeabsorbed,U = mv2/2 = 5 JMax.allowableforceonpackage(10gdecelera?on),F = ma = 50 N Max.allowablepeakstress(Eq.2),σp = F/A = 5 kPa Solidmodulusofpolyeurethanefoam,Es = 50 MPa Max.allowablepeakstress,normalized=σp/Es = 0.0001 WeuseGibson-Ashby,fig.8.8(nextslide).�

83

Gibson & Ashby: Table 8.2, p. 231�

Examinable�



Shock Cushion: 4�

Choiceofthickness,t:1 m 0.001 m Strainrate,dε/dt=v/t(Eq4):4.5 s-1 4500 s-1

Energy/modulus(W/Es)atσp/Es= 0.0001:(Fig.8.8)5.25 10-5 7.4 10-5

Energyabsorbed/unitvolume:2.62 kJ/m3 3.70 kJ/m3�

84

Tostartworkingontheproblem,wehavetomakesomeratherarbitrarychoicesofthicknessthatbracketthelikelyresult.�

Tocompletetheproblem,wehavetoiterateonthethicknessun?lweconvergeonaself-consistentresult.�

Gibson & Ashby�

Examinable�



Shock Cushion: 5�

Thickness,t = U/WA:0.19 m 0.14 m Strainrate,dε/dt=v/t(Eq4):24 s-1 32 s-1

Energy/modulus(W/Es)atσp/Es = 0.0001: (Fig.8.8)6.6 10-5 6.7 10-5

Energyabsorbed/unitvolume:3.30 kJ/m3 3.35 kJ/m3�

85

Tocon?nuewiththeproblem,were-calculatethethicknessesfromEq.1.�

Clearlywehavenearlyconverged,sowehavetoiterateonthethicknessonemore?me,usingt = U/WA,whichgivest= 150 mmandanop?mumrela?vedensity=0.01.�

Gibson & Ashby�

Examinable�



Summary: Part 4

•  Foamsorcellularmaterialsareanexampleofcompositematerials.

•  Wedevelopedanexampleofhowcellularmaterialsareusefulasshockcushions.

•  Thisleadtoworkedexampleofhowcalculatetheop?mumthicknessofsuchasshockcushion.

86 Examinable�



87

Summary: Overall•  Compositematerialshavebeendescribedwithrespectto

theirmicrostructure-propertyrela?onships.•  Useofthecompositeapproachenablesmuchlarger

varia?onsinproper?estobeachievedwithinagivenmaterialtype.

•  Carefulop?miza?onofthematerialwithrespecttoallthepropertyrequirements[foragivenapplica?on]isessen?al.

•  CTEofacompositecanbees?mated(supplementaryslides)fromtheCTEsofthecons?tuentphases.



88

References•  CellularSolids,Pergamon,L.J.GibsonandM.F.Ashby(1988),ISBN0-08-036607-4.•  MaterialsPrinciples&Prac?ce,Bu/erworthHeinemann,editedbyC.Newey&G.

Weaver.•  MechanicalMetallurgy,G.E.Dieter,3rdedi?on,McGrawHill.•  MechanicalBehaviorofMaterials,T.H.Courtney(2000),Boston,McGraw-Hill.•  MechanicalBehaviorofMaterials,N.E.Dowling(1999),Pren?ce-Hall.•  StructuralMaterials,Bu/erworthHeinemann,editedbyG.Weidmann,P.LewisandN.

Reid.•  PhysicalCeramics,Y.-T.Chiang,D.P.BirnieIII,W.D.Kingery(1997),Wiley,NewYork,

0-471-59873-9.•  TheNewScienceofStrongMaterials,J.E.Gordon,Princeton.•  AnIntroduc?onofCompositeProducts,Chapman&Hall,K.Po/er(1997),ISBN

0-412-73690-X.•  AnIntroduc?ontotheMechanicalProper?esofSolidPolymers,Wiley,I.M.Wardand

D.W.Hadley(1993),ISBN0-471-93887-4.•  Varia?onalMethodsinMechanics,OxfordUniversityPress,USA,1992,ToshioMura,

ISBN0195068300.•  Plas?city:ATrea?seonFiniteDeforma?onofHeterogeneousInelas?cMaterials,

CambridgeUniversityPress,2009,S.Nemat-Nasser,ISBN0521108063.•  TheTheoryofComposites,CambridgeUniversityPress,2001,G.F.Milton,

ISBN0521781256.



89

Supplemental Slides

•  Thefollowingslidescontainsupplementalmaterialthatwillbeofinteresttothosewhoarecurioustoobtainmoredetail.



90

Improved bounds

•  UpperandlowerboundsformodulushavebeendevelopedbyHashin&Shtrikmanthatnarrowtherangebetweenthetwobounds.

•  Differentformulaeestablishedforbulk,K,andshearmoduli,G.

•  Nota?on:bulkmoduliKAandK

B;shearmoduliG

A

andGB.Klower = KA +

VB1

KB − KA

+3 1 −VB( )3KA + 4GA( )



91

Hashin-Shtrikman

Kupper = KB +1 − VB

1KA − KB

+3VB

3KB + 4GB( )

Gupper = GB +1− VB

1GA −GB

+6 KB + 2GB( )VB5GA 3KB + 4GB( )

Glower =GA +VB

1GB −GA

+6 KA + 2GA( ) 1 − VB( )5GA 3KA + 4GA( )



92

Examples•  Thisexamplefrom

Green’stextshowshowthebulkandshearmodulivarywithvolumefrac?onfortwophaseswhosemodulidifferbyafactorof10.

•  TheresultshowsthattheH-Sboundsaregenerallymoreuseful.



93

Anisotropy in Composites

•  Thesamemethodsdevelopedinlecture4fordescribingtheanisotropyofsinglecrystalscanbeappliedtocomposites.

•  Anisotropyisimportantincomposites,notbecauseoftheintrinsicproper?esofthecomponentsbutbecauseofthearrangementofthecomponents.

•  Asanexample,consider(a)auniaxialcomposite(e.g.tennisrackethandle)and(b)aflatpanelcross-plycomposite(e.g.wingsurface).



94

Fiber Symmetry

x

y

z



95

Fiber Symmetry

•  Wewillusethesamematrixnota1onforstress,strain,s?ffnessandcomplianceasforsinglecrystals.

•  Thecompliancematrix,s,has5independentcoefficients.

€

s11 s12 s13 0 0 0s12 s11 s13 0 0 0s13 s13 s33 0 0 00 0 0 s44 0 00 0 0 0 s44 00 0 0 0 0 2 s11 − s12( )

#

$

% % % % % % %

&

'

( ( ( ( ( ( (



96

Relationships

•  Forauniaxialstressalongthez(3)direc?on,

•  Thisstresscausesstraininthetransverseplane:e11 = e22 = s12σ33.ThereforewecancalculatePoisson’sra?oas:

•  Similarly,stressesappliedperpendiculartozgiverisetodifferentmoduliandPoisson’sra?os.

€

E3 =σ 3ε3

=1s33

=σ zz

εzz

$

% &

'

( )

€

ν13 =e1e3

=s13s33

=exxezz

#

$ %

&

' (

€

E1 =σ1ε1

=1s11, ν 21 =

−s12s11

, ν 31 =−s13s11



97

Relationships, contd.

•  Similarlythetorsionalmodulusisrelatedtoshearsinvolvingthezaxis,i.e.yz orxzshears:

s44 = s55 = 1/G

•  Shearinthex-y planeisrelatedtotheothercompliancecoefficients:

s66 = 2(s11-s12) = 1/Gxy



98

Plates: Orthotropic Symmetry

•  Again,weusethesamematrixnota1onforstress,strain,s?ffnessandcomplianceasforsinglecrystals.


€

s11 s12 s13 0 0 0s12 s22 s23 0 0 0s13 s23 s33 0 0 00 0 0 s44 0 00 0 0 0 s55 00 0 0 0 0 s66

"

#

$ $ $ $ $ $ $

%

&

' ' ' ' ' ' '



99

Plates: 0° and 90° plies•  Ifthecompositeisalaminatecompositewithfiberslaidin

at0°and90°inequalthicknessesthenthesymmetryishigherbecausethexandydirec?onsareequivalent.


€

s11 s12 s13 0 0 0s12 s11 s13 0 0 0s13 s13 s33 0 0 00 0 0 s44 0 00 0 0 0 s44 00 0 0 0 0 s66

"

#

$ $ $ $ $ $ $

%

&

' ' ' ' ' ' '



100

Anisotropy: Practical Applications

•  Theprac?calapplica?onsofanisotropyofcomposites,especiallyfiber-reinforcedcompositesarenumerous.

•  Thes?ffnessoffibercompositesvariestremendouslywithdirec?on.Torsionalrigidityisveryimportantincarbodies,boats,aeroplanesetc.

•  Eveninmonolithicpolymers(e.g.drawnpolyethylene)thereexistslargeanisotropybecauseofthealignmentofthelong-chainmolecules.



101

Closed Cell Wall Bending•  LHS:responseto

compressiveloadinginthexdirec?on;RHS:responsetocompressiveloadingintheydirec?on.

•  Considerloadinginthexdirec?on:eachobliquesegmentexperiencesbendingateachend.Theload,P,isP=σ1(h+lsinθ)b-seefig.4.8b [Gibson: Cellular Materials]



102

Modulus(relative density)

•  Treateachsegmentasabeamoflengthl,thicknesst,depthb,andYoung’sModulusEs.

•  Theforce,C,resolvedonthey(ver?cal)direc?onmustbezeroinordertosa?sfyequilibrium.

•  Themoment,M,onthesegment: M = P lsinθ / 2

•  Thedeflec?on,δ,ofthesegment: δ= P l3 sinθ / 12EcellI

whereIisthesecondmomentofiner?a: I = bt3 / 12



103

Cell Geometry (general hexagonal)

θ

l

t

h

x1

x2 or y€

relativedensity=ρ *ρs

=t l( ) h l + 2( )

2 cosθ h l + sinθ( )

Regular honeycomb:�h = l, θ = 30°�ρ*/ρs = 2t/√3l

b: depth of cell�(out-of-plane)

h+lsinθ



104

Modulus(relative density): E1

•  Weneedthecomponentofthedeflec?onthatisparalleltotheXaxis,δ sinθ. Thusthestrainis:

€

ε1 =δ sinθl cosθ

=σ1 h + l sinθ( )bl 2 sin2θ

12EsI cosθ

E1 =σ1ε1

∴E1Es

=tl'

( ) *

+ , 3 cosθh l + sinθ( ) sin2θ



105

Modulus(relative density): E2

•  Themodulusintheperpendiculardirec?onissimilar.

€

ε2 =δ cosθh + l sinθ

=σ 2bl

4 cos4 θ12EsI h + l sinθ( )

E2 =σ 2ε2

∴E2Es

=tl'

( ) *

+ , 3 h l + sinθ( )

cos3θ



106

Modulus(relative density): regular hex

Forregularhexagons,thereducedmoduliinthetwodirec?onsarethesame:

E1 / Ecell = E2 / Ecell = 2.3 (t/l)3

Wealreadyestablishedthattherela?vedensityforaregularhexagonis2/√3 (t / l) ~ 2.3 (t / l),sowecanwrite:

E1 / Ecell = E2 / Ecell = 2.3 (ρ/ρcell)3



107

Wood Deformation



108

Moisture, CTE



109

Wood: anisotropy



110

Strength of Fiber Composites•  Justasformodulus,thesimplestmodelforcomposite

strengthistheRuleofMixtures,whereσm isthetensilestrengthofthematrix.

σc = σmVm + σfVf

•  Abe/ermodeltakesaccountoftheactualstress-straincharacteris?csofthecomponentphases.

•  InMMCs,forexample,thefiberreinforcementisonenquitebri/lecomparedtothematrix(e.g.graphitefibersinMg,SiCfibersinTi).

•  Thebri/lenessofthefiberslimitsthestrainthatcanbeappliedtoacomposite.



111

Ductile matrix + brittle fibers

•  Ifthecompositeisdeformedbeyondthebreakingstrainofthefibers,thenthebrokenfibersnolongersupportloadandtheirstrengtheningcontribu?onislost.Inthiscase,thestrengthisjustthis: σc = σmVm �



112

Ductile matrix + brittle fibers, contd.•  Athighenoughvolumefrac?ons,however,thehardeningin

thematrixisexhaustedbeforethefailurestrengthofthefibersisreached.Thematrixthenfailsata(constant)stress, � σ*

m = Em ε*f,whichcorrespondstothefailurestrain,ε*f,ofthefibers.Underthesecondi?ons,thestrengthofthecompositeisanaverageofthestrengthofthefibersandthestrengthofthematrixatthefailurestrainofthefibers.Thestrengthofthecompositethenincreaseswithvolumefrac?onofreinforcingfibersandisgivenby:

σc = σ�mVm + σfVf



113

Ductile matrix + brittle fibers, contd.•  Thusthereisacross-overbetweenthetwotypesofbehavior.•  Aminimumvolumefrac?onoffibersisrequiredinorderforthe

strengthofthefibercompositetoexceedthatofthematrix.

0Vf1

σc

σc = σ�mVm + σfVf

σc = σmVm



114

Coefficient of Thermal Expansion

•  Thenextsec?onrelatesthecoefficientofthermalexpansion(CTE)tothemicrostructureofcomposites,usingglass-ceramicsasanexample.



115

CTE versus modulus

•  Thethermalexpansioncoefficientofacomposite,αcomp,canberelatedtotheexpansioncoefficientsandbulkmoduliofthecons?tuentphasesbythefollowing.Obviously,thecompositebulkmodulusmustbedeterminedbyothermeans.

αcomposite = α A +KB α B −α A( ) KA − Kcomposite( )

Kcomposite KA − KB( )



116

Quartz•  Thecompressibilityfor

cristobaliteisgivenas100.10-6K-1(alpha-cristobalite)and4.8.10-6K-1(beta-cristobalite).

•  TheCTEisgivenas25.2.10-6foralpha-cristobaliteand11.2.10-6forbeta-cristobalite.

•  Comparetotherangeof12-20.10-6K-1claimedfortheglass-ceramic.

Cristobalite structure:�[Chiang et al.]

α

β



117

Li-Zn glass ceramics

•  Notethevaria?oninexpansionatthealpha-betatransi?on(displacive)incristobalite.

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