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