Role of crystal arrangement on the mechanical performance of enamel

BingbingAna,b,RaoraoWangc,DongshengZhangb,d,⇑aShanghaiInstituteofAppliedMathematicsandMechanics,Shanghai200072,People’sRepublicofChinaShanghaiKeyLaboratoryofMechanicsinEnergyEngineering,Shanghai200072,People’sRepublicofChinacTheTenthPeople’sHospitalofTongjiUniversity,Shanghai200072,People’sRepublicofChinadDepartmentofMechanics,ShanghaiUniversity,Shanghai200444,People’sRepublicofChinabarticleinfoabstract

Thesuperiormechanicalpropertiesofenamel,suchasexcellentpenetrationandcrackresistance,arebelievedtoberelatedtotheuniquemicroscopicstructure.Inthisstudy,theeffectsofhydroxyapatite(HAP)crystalliteorientationonthemechanicalbehaviorofenamelhavebeeninvestigatedthroughaser-iesofmultiscalenumericalsimulations.Amicromechanicalmodel,whichconsiderstheHAPcrystalarrangementinenamelprisms,thehierarchicalstructureofHAPcrystalsandtheinelasticmechanicalbehaviorofprotein,hasbeendeveloped.Numericalsimulationsrevealedthat,undercompressiveload-ing,plasticdeformationprogressiontookplaceinenamelprisms,whichisresponsiblefortheexperimen-tallyobservedpost-yieldstrainhardening.Bycomparingthemechanicalresponsesfortheuniformandnon-uniformarrangementofHAPcrystalswithinenamelprisms,itwasfoundthatthestiffnessforthetwocaseswasidentical,whilemuchgreaterenergydissipationwasobservedintheenamelwiththenon-uniformarrangement.Basedontheseresults,weproposeanimportantmechanismwherebythenon-uniformarrangementofcrystalsinenamelrodsenhancesenergydissipationwhilemaintainingsuf-ficientstiffnesstopromotefracturetoughness,mitigationoffractureandresistancetopenetrationdefor-mation.Furthersimulationsindicatedthatthenon-uniformarrangementoftheHAPcrystalsisakeyfactorresponsiblefortheuniquemechanicalbehaviorofenamel,whilethechangeinthenanostructureofnanocompositescoulddictatetheYoung’smodulusandyieldstrengthofthebiocomposite.Ó2012ActaMaterialiaInc.PublishedbyElsevierLtd.Allrightsreserved.Articlehistory:Received18March2012Receivedinrevisedform2June2012Accepted19June2012AvailableonlinexxxxKeywords:EnamelHAPcrystalMechanicalbehaviorHierarchyNumericalmodeling1.IntroductionAstheoutermosthardtissueofteeth,enamelpossessesacom-plexhierarchicalstructure[1].Onthemicronlengthscale,enamelisabiocompositeconsistingofkeyhole-likeprismswithadiame-terofapproximately5lm[2,3]andprotein-richprismsheathswhichhaveathicknessof800–1000nm[4]andconstitutetheboundariesofprisms[5].Theindividualprismsalsoexhibitacom-positestructurecomposedofhydroxyapatite(HAP)crystallitesembeddedinasoftproteinlayerwithathicknessof$2nm[6,7],whichformsthenanostructureofenamel.TheHAPcrystalshaveathicknessof20–120nm[8–10]andexhibitdifferentorientationswithinenamelprisms.Intheprismhead,HAPcrystalsareparalleltotheprismaxis,whileinthetail,anangleof60°betweenthecrystalc-axisandtheprismaxisisobserved[11].Inregionsbetweentheprismheadandtail,thec-axesofcrystalsgraduallyinclinetotheprismaxiswithanangleupto60°.Themicrostructureandmechanicalperformanceofenamelhavebeenstudiedthroughexperimentalandnumericalapproaches.⇑Correspondingauthorat:DepartmentofMechanics,ShanghaiUniversity,Shanghai200444,People’sRepublicofChina.Tel.:+862166135258.E-mailaddress:donzhang@staff.shu.edu.cn(D.Zhang).Owingtothesmallsize,fractureofHAPcrystallitesisgovernedbythetheoreticalstrengthratherthanthecrackpropagation,whichimpliesthatthemineralcrystallitesinnanocompositesareinsensi-tivetoflaws[12].Itwasfoundthroughtheindentationtechniquethat,owingtothehighpercentage(95%)ofmineralsinenamel[13],theelasticmodulusofenamelwas70–120GPa[14–16],whichismuchgreaterthanthatforotherhardtissues(boneanddentin).Whilethecontentofproteinislowcomparedwithminerals,ithasasignificantinfluenceonthemechanicalbehaviorofenamel.HeandSwain[17,18]reportedthat,whenenamelwassubjectedtocon-tactloading,theproteinunderwentlargesheardeformation,andtheinelasticdeformationbehavioroftheproteinresultedinthenonlin-earstress–strainresponseforenamel.StudiesbyXieetal.[19,20]re-vealedthat,whentheproteinbecamethicker,whichwastypicalinhypomineralizedenamel,theabilityofenameltoresistinelasticdeformationwasreduced,providinganexplanationforthede-gradedmechanicalpropertiesofhypomineralizedenamel[21,22].Duetotheexistenceofsacrificialbonds,whichmayrupturewhenproteinexperienceslargedeformation,proteincanalsoleadtotheirreversibledeformationbehaviorofenamel[23,24].HeandSwain[10]reportedthattheindentationstress–straincurveforenamelwassimilartothatformetals,ratherthanitsmajorconstituents(HAPcrystals).Theplasticdeformation,whichwasdemonstrated1742-7061/$-seefrontmatterÓ2012ActaMaterialiaInc.PublishedbyElsevierLtd.Allrightsreserved.http://dx.doi.org/10.1016/j.actbio.2012.06.026Pleasecitethisarticleinpressas:AnBetal.Roleofcrystalarrangementonthemechanicalperformanceofenamel.ActaBiomater(2012),http://dx.doi.org/10.1016/j.actbio.2012.06.0262B.Anetal./ActaBiomaterialiaxxx(2012)xxx–xxxbytheresidualimpressionafterunloading,wasfurtherrevealed.Morerecently,Angetal.[25]utilizedsphericalindenterswithdiffer-entradiitoevaluatetheindentationstress–strainresponseforen-amelatmultiplelengthscales,andfoundthattheconstitutiverelationshipofenamelwasdependentonthelengthscales.Theplasticityanddamagecausedbymicrocrackingweretheprimarymechanismsresponsiblefortheinelasticdeformationbehaviorofenamel.Proteinalsoplaysanimportantroleinthefractureproper-tiesofbiocomposites.Thickproteinlayerscouldreducestresscon-centrationatthecracktip,therebyenhancingthefracturetoughnessofbiocomposites[26].Besidestheconstituentsofenamel,themechanicalperformanceofenamelisalsodependentontheorganizationoftheseconstitu-ents.JiandGao[27]declaredthattheYoung’smoduliandshearmodulusofbiocompositesarehighlyanisotropic.Thestaggeredarrangementofmineralcrystalsadoptedbynaturalbiologicalmaterialsleadstohighstiffnessandstrength,andgoodenergystor-ageabilities[28].Liuetal.[29]derivedanalyticalsolutionsofstressandstraindistributionforfurtherstudyingthecomplexmechanicalpropertiesofbiologicalmaterials.Basedonthefactthatmanybio-logicalmaterialspossesshierarchicalstructures[30–32],Gao[33]andYaoandGao[34]developedaself-similarhierarchicalmodeltostudythemechanicalpropertiesofhierarchicalbiologicalmate-rials.Basedonthetheoryforhierarchicalmaterials,Bechtleetal.[1]analyzedtheexperimentaldataforenamelandconcludedthatthestiffness,strengthandtoughnessofenamelwerefunctionsoflevelsofhierarchy.Anetal.[35]studiedtheplasticdeformationandgradientpropertiesofenamelusingnanoindentationandnumeri-calsimulation.Asthefoundationforenamel,dentinalsoplaysanimportantroleintoothfunction.Thelowmodulusandhardnessofdentincanreducestressinteeth[16].Imbenietal.[36]conductedinden-tationexperimentsonteeth,andfoundthatcracksinitiatedinen-amelpropagatedthroughthedentin–enameljunctionandstoppedindentin,implyingthatdentinexhibitedstrongresistancetocrackgrowthandtherebypromoteddamagetoleranceofteeth.ThestudybyBraueretal.[37]indicatedthatitwastheasymmetriesofstiffnessandhardnessofdentinthatenabledthetoothtoresistimpactloadingeffectively.Previousstudieshaveidentifiedthehierarchicalmicrostructureofenamelandmechanicalbehaviorsofenamel.However,therela-tionshipbetweenthemechanicalpropertiesandtheHAPcrystal-litearrangementisstillunderinvestigation.Forinstance,asthehighpercentageofmineralmayleadtobrittlenessinenamel[38],howdoesenamelachievedamagetoleranceduringmastica-tiontoretainstructuralintegrity?Whyisitthenon-uniformarrangement,ratherthantheuniformarrangement,ofHAPcrystal-litesinprismsthatispresentbyenamel?Thesequestionscallforabetterunderstandingoftherelationshipbetweenthestructure,mechanicalpropertiesandfunctionsofenamel.Theprimarygoalofthisinvestigationistoexplorethemechanicaladvantagesofthenon-uniformdistributionofHAPcrystallites,aswellastoiden-tifythemechanicaldesignprinciplesofenamel.Aseriesofhierar-chicalnumericalmodelingwhichconsideredtheHAPstructureanditsorientationatdifferentregionsinenamelrodswerecon-ducted.TheinfluencesofHAPcrystalarrangementonthestiffness,energydissipationandmitigationoffractureareelucidated.Theunderlyingdeformationmechanismofenamelundercompressiveloadingisdiscovered.2.1.NanocompositemodelingEnamelisananocompositeconsistingofhexagonalHAPcrys-tallitesandprotein,tightlypackedandorientedatdifferentanglesfromtheprismheadtothetail.Themechanicalpropertiesofthenanocompositecanbeevaluatedbyconstructingarepresentativevolumeelement(RVE)andexploringthemechanicalbehavioroftheRVE.InthisstudythreeRVEswereconsidered.RVE(A)takesintoaccounttheimperfectionofHAPcrystallites,suchasthecurvatureandbranch[39].Wangetal.[40]andHanetal.[41]createdanRVEforthenanocompositeganoinescalesofPolypterussenegalus,inwhichthetransversemineralelementwasintroducedtocapturetheeffectsofdiscontinuityofthepro-teinandtheirregulararrangementofmineralcrystallites.Consid-eringthatatthenanoscaleenamelandganoinehaveidenticalbuildingblocks,i.e.HAPcrystallitesandprotein,andthatthecrys-tallitespossessthesamevolumefractionandgeometryinboththosebiologicalmaterials[40],inthisstudytheRVEforganoinewasadoptedandreferredtoasRVE(A)(Fig.1a).RVE(B)wasaccordinglyintroducedbasedontheassumptionoftheregularalignmentofHAPcrystalliteswithoutconsideringtheimperfectionofthenanocomposite.ThisRVEisusedtoinvestigatetheinfluenceofcrystalimperfectiononthemechanicalpropertiesofenamel.Itisreportedthatthethicknessofcrystallitesinenamelis40–50nm[19,42].Here,inordertoinvestigatetheinfluenceofcrystal-litethickness(orinfluenceofthevolumefractionofminerals)onthemechanicalresponseofenamel,RVE(C),whichisalsobasedontheassumptionoftheregularalignmentofHAPcrystallites,wasdeveloped.RVE(B)hasthesamemorphologyasRVE(C),whilethethick-nessesofmineralcrystallitesinRVE(B)andRVE(C)aredistinct.AsshowninFig.1,theproteinsheetsinRVE(A)becomediscontinuousduetotheintroductionoftransversemineralelements,whereastheproteinlayersinRVE(B)andRVE(C)arecontinuous.NotethatthelengthforthethreeRVEsistakentobe220nm,whichisconsistentwiththeobservedvalue,100nm–100lm[9,39,43],andtheRVEsarealsosuitableforthecasewheretheHAPcrystallitelengthreachesupto100lm,wherethenanocompositecanbeconsideredasalong-fiber-reinforcedcomposite.ThethicknessforproteininthethreeRVEswasassignedtobe2nm,andtheHAPcrystallitethick-nessinRVE(A)wastakenas40nm,whilethoseforRVE(B)andRVE(C)were50and40nm,respectively[19,42].2.2.MicroscopicmodelingThemechanicalpropertiesofenamelcanbedeterminedbyinvestigatingthemechanicalperformanceofabiocomposite2.MethodsSinceenamelpossessesahierarchicalmicrostructure,multiscalenumericalsimulations,i.e.atthenano-andmicroscales,wereconducted.Fig.1.RVEsfornanocompositesconsistingofhexagonalHAPcrystallitesandprotein.Pleasecitethisarticleinpressas:AnBetal.Roleofcrystalarrangementonthemechanicalperformanceofenamel.ActaBiomater(2012),http://dx.doi.org/10.1016/j.actbio.2012.06.026B.Anetal./ActaBiomaterialiaxxx(2012)xxx–xxx3composedofenamelprismsandprismsheaths,whilethemechan-icalbehaviorforenamelprismscanbedeterminedaccordingtoitsnanostructure.Itisobservedthat,withintheenamelprism,thereisagradualchangeintheorientationofHAPcrystallitesintheareabetweenthehead(centralregion)andthetailoftheprism[44].Tofacilitatenumericalsimulation,thekeyhole-likeenamelprism(Fig.2(a))canbesimplifiedasthegeometryshowninFig.2(b)[45–47].Inthemodel,thecross-sectionoftheenamelprismwasdividedintosevenregions,withtheorientationsoftheHAPcrys-tallitesbeingidenticalwithineachregion,butdistinctbetweenthedifferentregions.Intheregionoftheprismhead,thec-axesoftheHAPcrystalliteswereparalleltothelongaxisoftheprism,whereasintheregionofprismtail,theanglebetweenthec-axesofthecrystallitesandtheprismaxiswasassignedtobe60°accord-ingtothemicroscopicobservation[11].A12°anglewasincremen-tallyaddedtotheadjacentregionfromtheheadtothetail.Afterconstructionoftheenamelprism,theenamelbiocompos-itemodelcanbedevelopedbyaddingprismsheaths.Enamelprismswereencasedbyprismsheathswithathicknessof800nm[4],whichconstitutestheRVEfortheenamelbiocompos-ite,asshowninFig.2(c).Theprismsheathshaveasignificantlyhigherproteincontentcomparedwiththeprisms[4],andthelow-percentagemineralcrystallitesinthesheathsalsoexhibitdif-ferentorientations[5].InordertoevaluatetheimportanceofthearrangementofHAPcrystallitesinsidetheenamelprism,twoRVEsweregenerated.RVE(D)istherepresentativevolumeelementofFig.2.REsforenamelconsistingofenamelprismsandprotein.(a)Thestructureofasingleprism.(b)Simplifiedcross-sectionalgeometryofaprism.(c)Schematicsofsimplifiedenamelbiocompositewithprisms(black)andproteinsheaths(white).Pleasecitethisarticleinpressas:AnBetal.Roleofcrystalarrangementonthemechanicalperformanceofenamel.ActaBiomater(2012),http://dx.doi.org/10.1016/j.actbio.2012.06.02B.Anetal./ActaBiomaterialiaxxx(2012)xxx–xxxthenon-uniformarrangementofcrystal(NA),inwhichthechangeinorientationofHAPcrystalliteswasconsidered,whileinRVE(E)allthecrystalliteswerepackedparalleltotheprismaxistopresentauniformarrangementofcrystallite(UA)withintheenamelprisms.2.3.FiniteelementanalysisNumericalanalysiswasfirstconductedforthenanocompositeconsistingofhexagonalHAPcrystallitesandprotein.Themechan-icalpropertiesofnanocomposites,suchasYoung’smodulus,yieldstressandPoisson’sratio,couldresultfromthestress–straincon-stitutivebehavior,andwereincorporatedintothenumericalsim-ulationsofmechanicalresponsesofenamelatmicroscale.TheHAPcrystallitewasmodeledasaperfectlyelastic–plasticmaterialwithanelasticmodulusof80GPa,aPoisson’sratioof0.3andayieldstressof2.6GPa[40,41];theproteinwasalsorepresentedbyaper-fectlyelastic–plasticmaterialwithanelasticmodulusof4.3GPa[40,41,48],aPoisson’sratioof0.3andayieldstressof400MPa[40,41,49].Previousstudieshavedetermined,bymeansoftheindentationmethod[25,40,41],thattheyieldstrengthofHAPcrys-tallitesshowswidevariation(2.6–20GPa).Thisisreasonableduetothecomplexstressandstrainstatesinthevicinityoftheinden-ter[50].SincetheimpurityandporosityofHAPcrystallitesweretakenintoaccountbyWangetal.[40]andHanetal.[41],themorerealisticvalueof2.6GPawasadoptedinthisstudy.RVE(A),RVE(B)andRVE(C)weremeshedwithan8-nodelinearbrickelement(C3D8inABAQUS6.9)andconstrainedbyaperiodicboundarycondition[40].Approximately16,000elementsforeachRVEwereobtained.Tocalculatetheelasticparametersandyieldstressesofthenanocompositeinalldirections,eachRVEwassub-jectedtocompressiveloadingsalongdirections1,2and3definedinFig.1andshearingloadingsalongthethreeaxes,respectively.Thematerialparameters,suchasYoung’smodulusandPoisson’sratio,werecalculatedfromstress–straincurvesanddefinedasthemechanicalpropertiesofnanocomposites.TheRVEsofthemicrostructureofenamel(Fig.2(c))werealsomeshedwithan8-nodelinearbrickelement(C3D8inABAQUS6.9)andconstrainedbyaperiodicboundarycondition.Atotalof34,000elementswereobtained.TheelasticconstantsandyieldstressesofnanocompositesdeterminedbyRVE(A),RVE(B)andRVE(C)wereincorporatedintotheenamelRVEs.Eachregioninthecross-sectionoftheenamelprismwasmodeledasananiso-tropicelastic–plasticmaterialobeyingHill’sanisotropicyieldlocus,whichcancapturetheanisotropicyieldbehaviorsofbiologicalhardtissueseffectively[40].Compressiveloadswereappliedalongtheprismaxisanddeformationresponseswerenumericallysimulated.TheoverallstiffnessandenergydissipationofenamelwereassessedbasedontheindividualRVEs.Thecapabilityofenergydis-sipationwascharacterizedbytherelationshipbetweentheappliedstressandtheenergydissipationdensity,definedasrffiffiffiffiffiffiffiffiffiffiffiffiffi2pp󰀂ep¼ee3ijijparametersofthenanocompositeresultingfromthenumericalanalysiswereidenticalbyusingthesamemechanicalconstitutiverelationshipforHAPcrystallitesandprotein.ThemechanicalparametersinTable1werefurtherincorporatedintoRVE(D)andRVE(E)toevaluatetheinfluencesofstructuralchangeofnanocompositesandthearrangementofcrystallitesinenamelprismsonthemechanicalbehaviorsofenamel.Thestress–strainresponsesofenamelwiththenon-uniform(RVE(D))anduniform(RVE(E))orientationsofcrystalsareshowninFig.3,andthecorrespondingdensitiesofplasticenergydissipationasafunctionofappliedstressesforthethreecasesaredepictedinFig.4.Thepost-yieldstrainhardeningbehaviorisobservedintheNAcase(RVE(D)),consistentwiththeexperimentalobservation[25],whilethemechanicalbehaviorintheUAcase(RVE(E))exhib-itsperfectlyelastic–plasticdeformationbehavior,withgreatermagnitudesofyieldstress.Itisfoundthatthestiffness,assessedbytheslopeofthestress–straincurve,forthetwocasesisidentical,whiletheplasticdeformationbehaviordiffersdistinctly.Indetail,theelasticmoduliresultingfromtheNAandUAcaseswiththeuseofRVE(A),RVE(B)andRVE(C)are29.6,66.5and60.54GPa,respectively.Theyieldstressinbothcaseswasdeterminedattheonsetofplasticdeformationidentifiedinthebiocomposites.Be-yondthisstage,theplasticdeformationwasobservedinthebuild-ingblocksofenamel,i.e.prismsandsheaths.IntheNAcase,theyieldstresseswereidentifiedas350,293and327MPaforRVE(A),RVE(B)andRVE(C),respectively,whileintheUAcasetheywerefoundtobe794MPa,1.78GPaand1.74GPaforRVE(A),RVE(B)andRVE(C),respectively.Thedifferentplasticdeformationbehaviorsforthetwotypesofarrangementledtoremarkablecapabilitiesofenergydissipation(Fig.4).Thedensitiesofplasticenergydissipationforenamelwithanon-uniformarrangementofcrystals(NA)increasewithincreas-ingappliedloadregardlessofthetypesofRVEofthenanocompos-ites,whilethedensityofplasticenergyremainszerofortheUAarrangement.Forinstance,thedensitiesofplasticenergydissipa-tioncorrespondingtostressof0.6GPaintheNAcaseusingRVE(A),RVE(B)andRVE(C)are1.92,1.31and1.42JcmÀ3,respectively.TheplasticstraindistributionsforthemicroscopicRVEswereestimated.Theequivalentplasticstraincontoursatthecross-sec-tionintwocasesbasedonRVE(A)areshowninFig.5.Notethattheequivalentplasticstrainisdefinedas:1W¼VZVrijdepijwhererijarethecomponentsofthestresstensor,VisthevolumeoftheRVEandepijarecomponentsoftheplasticstrainwiththesameindicessatisfyingthesummationconvention.3.ResultsThepredictedmaterialparametersofdifferentRVEsofnano-compositeswerelistedinTable1.Notethat,asananisotropicstructure,thesubscripts1,2and3inTable1denotethedirectionsalongthecorrespondingaxesdefinedinFig.1.AsthegeometryofRVE(A)wasadoptedfromapreviousstudy[40],themechanicalwhereepijistheplasticstrain,withthesameindicesfollowingthesummationconvention.Accordingtotheoverallstress–strainre-sponsesinFig.3(a),thetwocurvesagreewitheachotherwhentheappliedstressisbelow350MPaanddeviatewithincreasingload.Thus,theplasticstraindistributionswereevaluatedatspecificstrainsofenamel,suchase=0.014,0.018and0.033.Itisobservedthat,inthenon-uniformorientationcase,anon-uniformplasticdeformationappearsintheprismtailattheonsetofplasticdefor-mation(Fig.5(a)).Withincreasingload,theplasticdeformationspreadsintothemiddleregionoftheprisms(Fig.5(b)),andeventu-allyplasticdeformationoccursinthewholeprismsatlargestrain,asshowninFig.5(c).Ontheotherhand,plasticdeformationpro-gressionisnotobservedintheUAcase.Atsmallstrains(e=0.014ore=0.018),noplasticdeformationoccurs(Fig.5(d)),whileatlargestrain(e=0.033),uniformplasticstraintakesplaceinthewholeprismsafteryielding(Fig.5(e)).Similarplasticstraindistributionswerealsoobservedatthecross-sectionsofenamelprismswhenRVE(B)andRVE(C)wereadopted.Thestressandstrainresponsesofthesoftproteinwhichformstheprismsheathwerealsoanalyzed.TheshearstressdistributionsintheprismsheathfortheNAandUAcasesaredisplayedinFig.6(a)and(b).Atastressof357MPa,theshearstresscomponentsofprismPleasecitethisarticleinpressas:AnBetal.Roleofcrystalarrangementonthemechanicalperformanceofenamel.ActaBiomater(2012),http://dx.doi.org/10.1016/j.actbio.2012.06.026B.Anetal./ActaBiomaterialiaxxx(2012)xxx–xxxTable1

ThepredictedmechanicalparametersfromRVE(A),RVE(B)andRVE(C).RVEABCE1(GPa)41.0041.4138.70E2(GPa)41.0045.4542.80E3(GPa)51.6073.4572.11G12(GPa)15.404.763.95G23(GPa)13.5075.3080.G31(GPa)13.5027.33.45l120.280.330.27l32=l310.250.300.30rY1(GPa)1.050.540.54rY2(GPa)1.050.540.60rY3(GPa)1.212.392.35sY12(GPa)0.610.160.19sY23(GPa)0.40.290.24sY31(GPa)0.40.210.24Thesubscripts1,2and3denotethedirectionsalongthecorrespondingaxesdefinedinFig.1.E:elasticmodulus;G:shearmodulus;l:Poisson’sratio;rYandsY:yieldstressfornormalandshearstress.sheathsintheNAcaseareS12=4.63MPa,S13=4.65MPaandS23=4.MPa,whicharesignificantlygreaterthanthoseintheUAcase,whereS12=0.74MPa,S13=1.48Â10À3MPaandS23=1.4Â10À3MPa.Thisindicatesthat,withanon-uniformarrangementofcrystallites,theprismsheathscanjoinenamelprismstoresistexternalstressesandundergolargeplasticdeformationtoconsumeexternalenergies.Thedifferencesarealsodisplayedwiththeequiv-alentplasticstraindistributionshowninFig.6(c)and(d).Whentheappliedstressreaches700MPa,plasticdeformationoftheprismsheathsoccursintheNAcase(Fig.6(c))butnotintheUAcase(Fig.6(d)).4.DiscussionInthisstudy,themechanicalbehaviorsofenamelwereinvesti-gatedbyintroducingamultiscalenumericalsimulationwhichcon-sideredthedetailstructureoftheHAPnanocompositesandthearrangementofcrystalsinenamelprisms.ThreetypicalRVEsweregeneratedaccordingtonanoscopicobservationsfromenamel.Themechanicalpropertiesofthenanocompositeswerefurtherincor-poratedintoamodeloftheenamelmicrostructure.Theoverallstress–strainresponses,densityofplasticenergydissipationandplasticstraindistributionoftheenamelprismsweredetermined.Angetal.[25]measuredthecompressivestress–straincurveforenamelexperimentally,andreportedthattheelasticmodulusandyieldstresswere30GPaand400MPa,respectively.Thepre-dictedelasticmodulusandyieldstress(Fig.3(a))fortheNAcaseare29.6GPaand350MPa,respectively,whichisingoodagree-mentwiththeexperimentaldata.ComparedwiththemodelspreviouslydevelopedbySpears[48]andMachoetal.[45],thepresentmodelpossessesthefollowingadvantages.First,theprismsheaths,whichformtheinterfacesamongprisms,areexplicitlymodeledinthestudy,whichmakesthemodelmorerealisticandrational.Secondly,inthepresentstudy,theplasticmodelisemployedtocapturetheinelasticdefor-mationbehaviorofprotein.Itisexperimentallyreportedthat,whenenamelissubjectedtoindentationloading,theproteinunderwentinelasticdeformation[25],whichdissipatedmoreenergybeforethecatastrophicfailureofenamelandsignificantlycontributedtoenamel’sfracturetoughness.Thisimportantfeaturewasnotcon-sideredinpreviousstudies[45,48].Finally,accordingtothestudiesbyWangetal.[40]andHanetal.[41],thetransversemineralele-mentwasutilizedtocapturetheeffectsofimperfectionofHAPcrystalsinenamel,suchasthecrystalcurvatureandcrystalbranchobservedusingtransmissionelectronmicroscopy(TEM)[39],whicharenottakenintoaccountinpreviousmodelsforenamel[45,48].Spear’smodel,whichpredictedtheYoung’smodulusofprismstobe60–110GPa[48],agreeswiththeresultsobtainedinthepresentmodel,whichdoesnotconsidertheimperfectionofHAPcrystalsinenamel.Thecrystalarrangementintheenamelprismsapparentlyaffectsthemechanicalbehaviorofenamel.Angetal.[25]conductedexperimentsonenamelandfoundthatitexhibitedpost-yieldbehavior.However,theunderlyingfactscontributingtothisbehav-iorwerenotclearlyexpatiated.ByutilizingmicromechanicalmodelingintheNAandUAcases,themechanismsresponsiblefortheoverallstress–strainresponsesforthetwotypesofarrange-mentwerediscovered.Whenenamelissubjectedtocompressiveloading,non-uniformstraindistributionisidentifiedinenamelprisms.Plasticdeformationfirstoccursinspecificregions,suchasintheprismtail,andtheplasticdeformationzoneexpandstomoreareaswithincreasingload.Incontrast,whentheHAPcrystalliteswereorienteduniformlyalongtheprismdirection(theUAcase),uniformequivalentplasticstraindistributionwasobservedandanoverallidealelastic–plasticmechanicalbehaviorwasfound.Therefore,itisbelievedthattheplasticdeformationprogressionmechanismcausedbythenon-uniformarrangementofnanocom-positeswithinenamelprismsisresponsiblefortheexperimentallyobservedpost-yieldstrainhardeningbehavior.Thispropertyofplasticdeformationprogressionprovidesanadvantageousmecha-nismfordamagetoleranceandmitigationoffracture.Withtheuni-formarrangementofcrystallitesinprisms,plasticdeformationappearsinthewholeprismregionabruptlywithincreasingstress,increasingthepropensityforlocalizeddeformationandtherebycausingdamageorcatastrophicfailuretooccureasily.Conversely,theplasticdeformationprogressionmechanismfortheNAcasecanmitigatefracturebyspreadingdeformationintootherregions,henceretainingstructuralintegrity.Themechanicaladvantageofthenon-uniformarrangementofcrystalsalsoliesintheroleofprotein.Whenabiocompositeissub-jectedtoloading,theproteintransfersstresstothemineralsbyshearingdeformation,andundergoeslargeinelasticdeformationtodissipateenergy[17],whichisthemajorcontributiontothetoughnessofbiocomposites[51].Accordingtothepresentsimula-tions,theshearstresscomponentsofprismsheathsintheNAcasearelargerthanthoseintheUAcase,whichleadstotheoccurrenceofgreaterplasticdeformationinsheathsintheNAcase(Fig.6(c)and(d)).Therefore,thenon-uniformarrangementofHAPcrystal-litesalsofacilitatesstresstransfer,promotesenergydissipationandtherebyenhancesthetoughnessofenamel.Thestructureofnanocompositesofcrystalsandproteinalsoinfluencesthemechanicalpropertiesofenamel.Sinceenamelnano-compositesconsistofhexagonalHAPcrystallitesandprotein,RVE(A)wasestablishedbyconsideringtheimperfectionofthecrys-tallites,whichhasbeenshowntobecommonbyTEMobservations[39].Thenumericalresultsindicatethatthechangeinthenanocom-positestructuredoesnotinfluencethepost-yieldstrainhardeningbehaviorofenamel,thoughitdoesinfluencethemagnitudesofstiff-nessandyieldstrengthintheNAcase.Forinstance,thepredictedelasticmodulusandyieldstresswere29.6GPaand350MPa,respectively,basedonRVE(A)(Fig.3(a)),whiletheywere66.5GPaand293MPaforRVE(B)(Fig.3(b)).However,forallthreetypesofnanostructuresdefinedasRVE(A),RVE(B)andRVE(C),thestiffnessoftheuniformandnon-uniformarrangementsofcrystallitesinprismsisidentical,whereastheenergydissipationforthenon-uni-formarrangementisgreaterthanthatfortheuniformarrangement.Basedontheseresults,weproposeanimportantmechanismwhere-bythenon-uniformarrangementofcrystalsinprismsenhancesen-ergydissipationwhileretainingsufficientstiffness;thismechanismisindependentofthenanostructureoftheprisms.Pleasecitethisarticleinpressas:AnBetal.Roleofcrystalarrangementonthemechanicalperformanceofenamel.ActaBiomater(2012),http://dx.doi.org/10.1016/j.actbio.2012.06.0266B.Anetal./ActaBiomaterialiaxxx(2012)xxx–xxxFig.3.Overallstress–strainresponsesofNAandUAwiththemechanicalpropertiesdeterminedfrom(a)RVE(A),(b)RVE(B)and(c)RVE(C).Fig.4.DensitiesofplasticenergydissipationofNAandUAwiththemechanicalpropertiesdeterminedfrom(a)RVE(A),(b)RVE(B)and(c)RVE(C).Therelationshipbetweenthestructure,mechanicalpropertiesandfunctionmayprovidesuggestionsastowhythenon-uniformarrangementofHAPcrystalsinprismsisselectedbynaturalevolu-tion.Enamel,atypeofbiologicalhardtissue,servesastheouterlayerofteeth.Itsmainfunctionistocutandgrindupfood.Althoughtheunderlyingdentin,whichactsasasoft,elasticbase,canreducethestressesinatooth[16],theenamel–andespeciallytheouterenamel–stillsuffersfromwear,impactandfatigueloadsintheharshoralenvironment.Tofulfillitsfunction,enamelneedstopossesshighstiffness,whichcanbothcutfoodefficientlyandPleasecitethisarticleinpressas:AnBetal.Roleofcrystalarrangementonthemechanicalperformanceofenamel.ActaBiomater(2012),http://dx.doi.org/10.1016/j.actbio.2012.06.026B.Anetal./ActaBiomaterialiaxxx(2012)xxx–xxx7Fig.5.EquivalentplasticstraindistributionofenamelinNAandUAcasesbasedonRVE(A).(a–c)TheNAcase;(d,e)theUAcase.NotethateisthestrainofenamelshowninFig.3.Pleasecitethisarticleinpressas:AnBetal.Roleofcrystalarrangementonthemechanicalperformanceofenamel.ActaBiomater(2012),http://dx.doi.org/10.1016/j.actbio.2012.06.0268B.Anetal./ActaBiomaterialiaxxx(2012)xxx–xxxFig.6.ShearstressdistributionandplasticdeformationinprismsheathsinNAandUAcasesbasedonRVE(A).(a)ShearstressintheNAcase.(b)ShearstressintheUAcase.(c)EquivalentplasticstrainintheNAcase.(d)EquivalentplasticstrainintheUAcase.Sijstandsfortheshearstressintheplaneperpendiculartothei-axis;thepositivedirectionofSijisalongthej-axis.reducewear.Fractureresistanceisalsoanimportantproperty,sincetheservicedurationofatoothshouldbealifetime.Asaconsequence,thenon-uniformdistributionofcrystallitesinprismsisselectedbynature,sincethisstructureretainssufficientstiffnessandenhancestoughnesssimultaneously,accordingtotheresultsofthisstudy.Theproposedmechanismforenamelalsoanswersthequestionwhythebrittleenamelcansurviveinsuchaharshenvironmentwithoutfailure[52,53].Thisstudyhasestablishedagoodunderstandingoftherelationshipbetweenthemicrostruc-tureandmechanicalpropertiesofenamel.Fromthemechanicalpointofview,previousstudiesindicatethatenamelisagradedmaterial:theouterenamelexhibitshighstiffness[38],buthasapoorcapabilityforenergydissipation[35],whereastheinnerenamelshowsgreatfracturetoughness[54]butlowstiffness[55].Fromthemicroscopicstructuralpointofview,ithasbeenwidelyacceptedthatprismsexhibittwistedPleasecitethisarticleinpressas:AnBetal.Roleofcrystalarrangementonthemechanicalperformanceofenamel.ActaBiomater(2012),http://dx.doi.org/10.1016/j.actbio.2012.06.026B.Anetal./ActaBiomaterialiaxxx(2012)xxx–xxx9andwavyarrangementsfortheinnerenamelandgnarledenamel,respectively,whichcanpromotethefracturetoughness[54,56,57].Fromthechemicalcompositionalpointofview,previousstudieshaveshownthattheinnerenamelhasalargervolumetricfractionofproteinthantheouterenamel[58,59].AccordingtoGaoetal.’sfindings[33,60],themoretheprotein,thegreaterthefracturetoughnessabiocompositepossesses.Therefore,itiscommonlyac-ceptedthattheinnerenamelshowsgreaterfracturetoughnessandenergydissipationcomparedwiththeouterenamel.However,thepresentstudyhaselucidatedhowthebalancebetweenstiffnessandtoleranceoffractureoftheouterenamelisachieved.Theben-efitofthedesignisthatitimprovesthefracturetoughnesswithoutreducingthestiffness.Thisspecialmechanicaldesignprincipleofenamelalsoprovidesagoodideaforthedevelopmentoffuturebio-inspiredmaterials.5.ConclusionsThisstudyexplorestheeffectsofthecrystallitearrangementinprismsonthemechanicalperformanceofenamelusingaseriesofmultiscalenumericalsimulations.Thenon-uniformarrangementofHAPcrystallitesinenamelprismscancausegradualprogressionofregionalplasticdeformationandrevealstheoverallpost-yieldstrainhardeningbehaviorwhenenamelissubjectedtocompres-siveloads.Basedonnumericalsimulations,weproposeanimpor-tantmechanismwherebythenon-uniformarrangementofHAPcrystallitesinprismsenhancestheenergydissipationandretainssufficientstiffnessfortheouterenamel.Thismechanismsuccess-fullyexplainsthedamagetoleranceofbrittleenamel,andpoten-tiallyprovidesguidelinesforthedesignofbio-inspiredmaterials.AcknowledgementsTheauthorsaregratefulfortheNSFCGrant#11172161,theShanghaiLeadingAcademicDisciplineProject#S30106,theInno-vationProgramofShanghaiMunicipalEducationCommission#12ZZ092,theStateKeyLaboratoryofOralDiseases(SichuanUni-versity)grantSKLODSCU2009KF03,andtheScienceandTechnol-ogyCommissionofShanghaiMunicipalitygrants#10410701900,#11195820900and#10ZR1423400.AppendixA.FigureswithessentialcolourdiscriminationCertainfiguresinthisarticle,particularlyFigs.5and6,aredifficulttointerpretinblackandwhite.Thefullcolourimagescanbefoundintheon-lineversion,athttp://dx.doi.org/10.1016/j.actbio.2012.06.026.References[1]BechtleS,AngSF,SchneiderGA.Onthemechanicalpropertiesofhierarchicallystructuredbiologicalmaterials.Biomaterials2010;31:6378–85.[2]HabelitzS,MarshallSJ,MarshallJrGW,BaloochM.Mechanicalpropertiesofhumandentalenamelonthenanometrescale.ArchOralBiol2001;46(2):173–83.[3]JengYR,LinTT,HsuHM,ChangHJ,ShiehDB.Humanenamelrodpresentsanisotropicnanotribologicalproperties.JMechBehavBiomedMater2011;4(4):515–22.[4]GeJ,CuiFZ,WangXM,FengHL.Propertyvariationsintheprismandtheorganicsheathwithinenamelbynanoindentation.Biomaterials2005;26(16):3333–9.[5]BechtleS,HabelitzS,KlockeA,FettT,SchneiderGA.Thefracturebehaviourofdentalenamel.Biomaterials2010;31(2):375–84.[6]CuiFZ,GeJ.Newobservationsofthehierarchicalstructureofhumanenamel,fromnanoscaletomicroscale.JTissueEngRegenMed2007;1:185–91.[7]HannigM,HannigC.Nanomaterialsinpreventivedentistry.NatNanotechnol2010;5:565–9.[8]JohansenE.Microstructureofenamelanddentin.JDentRes19;43:1007–20.[9]RobinsonC,KirkhamJ,BrookesSJ,ShoreRC.Chemistryofmatureenamel.In:RobinsonC,KirkhamJ,ShoreRC,editors.Dentalenamel:formationtodestruction.BocaRaton,FL:CRCPress;1995.p.167–91.[10]HeLH,SwainM.Enamel—a‘‘metallic-like’’deformablebiocomposite.JDent2007;35:431–7.[11]WhiteSN,LuoW,PaineML,FongH,SarikayaM,SneadML.Biologicalorganizationofhydroxyapatitecrystallitesintoafibrouscontinuumtoughensandcontrolsanistropyinhumanenamel.JDentRes2001;80:321–7.[12]GaoH,JiB,JagerIL,ArztE,FratzlP.Materialsbecomeinsensitivetoflawsatnanoscale:lessonsfromnature.ProcNatlAcadSci2003;100(10):5597–600.[13]ZhouJ,HsiungL.Biomolecularoriginoftherate-dependentdeformationofprismaticenamel.ApplPhysLett2006;:051904.[14]XuHHK,SmithDT,JahanmirS,RombergE,KellyJR,ThompsonVP,etal.Indentationdamageandmechanicalpropertiesofhumanenamelanddentin.JDentRes1998;77:472–80.[15]MahoneyE,HoltA,SwainM,KilpatrickN.Thehardnessandmodulusofelasticityofprimarymolarteeth:anultra-micro-indentationstudy.JDent2000;2:5–94.[16]FongH,SarikayaM,WhiteSN,SneadML.Nano-mechanicalpropertiesprofilesacrossdentin–enameljunctionofhumanincisorteeth.MaterSciEngC2000;7:119–28.[17]HeLH,SwainMV.Contactinduceddeformationofenamel.ApplPhysLett2007;90:171916.[18]HeLH,SwainMV.Understandingthemechanicalbehaviourofhumanenamelfromitsstructuralandcompositionalcharacteristics.JMechBehavBiomedMater2008;1:18–29.[19]XieZH,SwainM,MunroeP,HoffmanM.Onthecriticalparametersthatregulatethedeformationbehaviouroftoothenamel.Biomaterials2008;29:2697–703.[20]XieZ,SwainM,HoffmanMJ.StructuralIntegrityofenamel:experimentalandmodeling.JDentRes2009;88(6):529–33.[21]MahoneyE,RohanizadehR,IsmailFSM,KilpatrickNM,SwainM.Mechanicalpropertiesandmicrostructureofhypomineralisedenamelofpermanentteeth.Biomaterials2004;25:5091–100.[22]XieZ,MahoneyE,KilpatrickN,SwainM,HoffmanM.Onthestructure–propertyrelationshipofsoundandhypomineralizedenamel.ActaBiomater2007;3:865–72.[23]HeLH,SwainMV.Influenceofenvironmentonthemechanicalbehaviourofmaturehumanenamel.Biomaterials2007;28:4512–20.[24]HeLH,SwainMV.Energyabsorptioncharacterizationofhumanenamelusingnanoindentation.JBiomedMaterResA2007;81(2):484–92.[25]AngSF,BortelEL,SwainMV,KlockeA,SchneiderGA.Size-dependentelastic/inelasticbehaviorofenamelovermillimeterandnanometerlengthscales.Biomaterials2010;31:1955–63.[26]JiB,GaoH.Astudyoffracturemechanismsinbiologicalnano-compositesviathevirtualinternalbondmodel.MaterSciEngA2004;366:96–103.[27]JiB,GaoH.Elasticpropertiesofnanocompositestructureofbone.CompSciTech2006;66:1212–8.[28]ZhangZQ,LiuB,HuangY,HwangKC,GaoH.Mechanicalpropertiesofunidirectionalnanocompositeswithnon-uniformlyorrandomlystaggeredplateletdistribution.JMechPhysSolids2010;58:16–60.[29]LiuG,JiB,HwangK,KhooBC.Analyticalsolutionsofthed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