Modeling bamboo as a functionally graded material
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AbstractNaturalfibersarepromisingforengineer-
ingapplicationsduetotheirlowcost.Theyare
abundantlyavailableintropicalandsubtropical
regionsoftheworld,andtheycanbeemployedas
constructionmaterials.Amongnaturalfibers,bamboo
hasbeenwidelyusedforhousingconstructionaround
theworld.Bambooisanoptimizedcompositethat
exploitstheconceptofFunctionallyGradedMaterial
(FGM).Biologicalstructuressuchasbamboohave
complicatedmicrostructuralshapesandmaterial
distribution,andthustheuseofnumericalmethods
suchasthefiniteelementmethod,andmultiscale
methodssuchashomogenization,canhelptofurther
understandingofthemechanicalbehaviorofthese
materials.Theobjectiveofthisworkistoexplore
techniquessuchasthefiniteelementmethodand
homogenizationtoinvestigatethestructuralbehavior
ofbamboo.Thefiniteelementformulationusesgra-
dedfiniteelementstocapturethevaryingmaterial
distributionthroughthebamboowall.Toobserve
bamboobehaviorunderappliedloads,simulationsare
conductedundermultipleconsiderationssuchas
aspatiallyvaryingYoung’smodulus,anaveragedYoung’smodulus,andorthotropicconstitutive
propertiesobtainedfromhomogenizationtheory.The
homogenizationprocedureuseseffective,axisymmet-
ricpropertiesestimatedfromthespatiallyvarying
bamboocomposite.Three-dimensionalmodelsof
bamboocellswerebuiltandsimulatedundertension,
torsion,andbendingloadcases.
Introduction
Biologicalsystemssuchasplantandtreestems,animal
bonesandotherbiologicalhardtissuestendtobe
optimizedfortheloadingconditionstheyaresubjected
to.Theirgeometrychangeswithloadingconditionsto
matchstress-orstrain-dependentrequirements,and
theirmaterialpropertiesarealsooptimallydistributed.
Forexample,theinteriorstructureofbonechanges
dependingontheprincipalstressdirectionsandthe
magnitudeofshearstresstheycarry[1].
Biologicalstructuresareusuallymadeofcomposite
materialswhicharemultifunctionalandhaveliving
organismswhichprovidesadaptability.Thisoccursdue
tothefactthatbiologicalsystemsmustbeableto
performavarietyoffunctionswell,andthus,theyare
optimizedformultifunctionalpurposes.Asaconse-
quence,biologicalstructuresarecomplicatedandnon-
uniform,whichmakestheirrealisticmodelingdifficult
andinvolved.
Amongbiologicalstructures,thenaturalfibersare
veryinterestingforengineeringapplicationsdueto
theirlowcostandconvenientavailability.Theygrow
abundantlyintropicalandsubtropicalregionsofthe
world,andtheycanbeusefullyemployedas
con-E.C.N.SilvaDepartmentofMechatronicsandMechanicalSystemsEngineering,EscolaPolite´cnicadaUniversidadedeSa˜oPaulo,Av.ProfessorMelloMoraes,2231,Sa˜oPaulo,SP05508-900,Brazil
M.C.WaltersÆG.H.Paulino(&)DepartmentofCivilandEnvironmentalEngineering,UniversityofIllinoisatUrbana-Champaign,NewmarkLaboratory,205NorthMathewsAvenue,Urbana,IL61801,USAe-mail:paulino@uiuc.eduJMaterSci(2006)41:6991–7004DOI10.1007/s10853-006-0232-3
123Modelingbambooasafunctionallygradedmaterial:lessons
fortheanalysisofaffordablematerials
Emı´lioCarlosNelliSilvaÆMatthewC.WaltersÆ
GlaucioH.Paulino
Publishedonline:16September2006ÓSpringerScience+BusinessMedia,LLC
2006structionmaterials[2–4].Examplesofnaturalfibers
arebamboo,coconutfibers,sisal,etc.Amongthe
naturalfibers,bamboofindswidespreaduseinhousing
constructionaroundtheworld,andisconsideredasa
promisinghousingmaterialinunderdevelopedand
developedcountries.Recently,somestudiesinvesti-
gatingcompositesmadeofbambooandconcrete[5]
andbambooandpolymer[6,7]havebeenconducted.
Bambooisatree-likeplant(seeFig.1(a))that
belongstothesubfamilyBambusoideaeofthegrass
familyPoaceae.Bamboostalksareoptimizedcom-
positematerialsthatnaturallyexploittheconceptof
FunctionallyGradedMaterials(FGMs)[1,9–13].Such
materialspossesscontinuouslygradedpropertiesand
arecharacterizedbyspatiallyvaryingmicrostructures
createdbynon-uniformdistributionsoftheconstitu-
entphases.Inthesematerials,theroleofreinforce-
mentandmatrix(base)materialinterchangesina
continuousmanner[14].Thesmoothvariationof
propertiesmayofferadvantagessuchasreductionof
stressconcentrationandincreasedbondingstrength
[15,16].
Thebambooculmisanapproximatelycylindrical
shellthatisdividedperiodicallybytransversaldia-phragmsatnodes.Between20%and30%ofthecross-
sectionalareaoftheculmismadeoflongitudinalfibers
thataredistributednon-uniformlythroughthewall
thickness,theconcentrationbeingmostdensenearthe
exterior(seeFig.1(b)).Theorientationofthesefibers
makesbambooanorthotropicmaterialwithhigh
strengthalong,andlowstrengthtransversaltofibers[5,
17,18].
Mostworkintheliteraturethatcharacterizes
bambooisexperimental,dedicatedtoestimating
strengthandstiffnessproperties[17–23].Fewworks
treatingthemodelingofnaturalfibershavebeen
foundintheliterature[24,25],andthesedealpri-
marilywithsimplifiedanalyticalmodels.Inthese
studies,comprehensiveexperimentalcharacterization
ofcertainspeciesofbamboowereperformedby