Modeling-of-the-effect-of-particles-size-particles-distribution-and-particles-number-on

CompositesPartB86(2016)135e142

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Modelingoftheeffectofparticlessize,particlesdistributionandparticlesnumberonmechanicalpropertiesofpolymer-claynano-composites:Numericalhomogenizationversusexperimentalresults

Y.Djebaraa,A.ElMoumenb,T.Kanitb,*,S.Madania,A.Imadb

ab

LaboratoiredeM ecaniquedesStructuresetMat eriaux,LMSM,Universit khdar,Batna,AlgeriaLaboratoiredeM ecaniquedeLille,LML,CNRS/UMR8107,Universit eLille1,Villeneuved'Ascq,France

articleinfo

Articlehistory:

Received22June2015Receivedinrevisedform3September2015

Accepted22September2015

Availableonline11November2015Keywords:

A.Polymer-matrixcompositesA.Particle-reinforcementB.Microstructures

putationalmodellingNano-composites

abstract

Themaingoalofthispaperistopredicttheelasticmodulusofpartiallyintercalatedandexfoliatedpolymer-claynano-compositesusingnumericalhomogenizationtechniquesbasedonthe niteelementmethod.Therepresentativevolumeelementwasemployedheretocapturenano-compositesmicro-structure,wherebothintercalatedexfoliatedandclayplateletscoexistedtogether.Theeffectivemacroscopicpropertiesofthestudiedmicrostructureareobtainedwithtwoboundaryconditions:pe-riodicboundaryconditionsandkinematicuniformboundaryconditions.Theeffectofparticlevolumefractions,aspectratio,numberanddistributionofparticlesandthetypeofboundaryconditionsarenumericallystudiedfordifferentcon gurations.ThispaperinvestigatealsotheperformanceofseveralclassicalanalyticalmodelsasMoriandTanakamodel,HalpinandTsaimodel,generalizedselfconsistentmodelthroughtheirabilitytoestimatethemechanicalpropertiesofnano-composites.Acomparisonbetweensimulationresultsofpolypropyleneclaynano-composites,analyticalmethodsandexperi-mentaldatahascon rmedthevalidityofthesetresults.

©2015ElsevierLtd.Allrightsreserved.

1.Introduction

Polymericcompositesreinforcedwithnanoscalere-inforcementssuchasnanotube-reinforced,silicananoparticle-reinforcedandnanoclay-reinforcedhaverecentlyattractedatremendousattentioninresearchersandindustrials,sincetheyexhibitenhancedmechanicalproperties.AccordingtoKojimaetal.[17];claynanoparticlesareclassi edbestcandidatestostrengthenpolymersmaterials,duetotheirmechanicalandphysicalproper-ties,theirhighaspectratio,theirhighavailabilityinnatureandproductionlowcost.

Generally,therearethreedifferenttechniquestocharacterizethebehaviorofnanocomposites:experimentalapproaches,analyticalmethodsbasedonthetheoriesofboundsandmodelsandnumericalmethodsbasedontherepresentativevolumeelement(RVE)coupledwith niteelementmethods(FEM).Itshouldbementionthatinexperimentalworksitisverydif culttocontrolthein uenceoftheparticlesize,particlesshapeandits

*Correspondingauthor.Tel.:þ33320434243;fax:þ33320337088.E-mailaddress:tou k.kanit@univ-lille1.fr(T.

Kanit)./10.1016/positesb.2015.09.0341359-8368/©2015ElsevierLtd.Allrightsreserved.

distributionsonthemacroscopicbehaviorofpolymerclaynano-composites(PCN).Forthat,someworksconfronttheexperimentaldatawithnumericalandanalyticalmethodstodeterminetheeffectofmorphologicalparameters.Theelasticpropertiesarethendeterminedbyapplyinganalyticalornumericalmethods.Themostimportantonesare:MoriandTanaka[21](MT),HalpinandTsai[11](HT)andgeneralselfconsistent(SC)method.Foranalyticalbounds,theusedmicromechanicalmethodsare:the rstorderboundsofVoigt[24];thesecondorderboundsofHashinandShtrikman[10]andthethirdorderboundsofBeranandMolyneux[1].Fornu-mericalcharacterization,thetechniqueofthehomogenizationbasedonRVEandFEMisintroducedinmanysituationsinordertoestimatetheeffectivepropertiesofnanocomposites.Forexample,FornesandPaul[9],Shengetal.[23],Hbaiebetal.[12],DongandBhattacharyya[3],FigielandBuckley[8]andPahlavanpouretal.[22].

FornesandPaul[9]proposedanexperimentalworktounder-standtheoriginofthesuperiorreinforcingef ciencyobservedinwellexfoliatedpolymerclaynanocompositescomparedtocon-ventionalreinforcementsusingcompositetheory.TheyfoundthatcompositetheoriesofHTandMTwereemployedtobetterunder-standingofthesuperiorreinforcementobservedforwell-exfoliated

136Y.Djebaraetal./CompositesPartB86(2016)135e142

nanocompositesrelativetoconventionalglass berscomposites.Shengetal.[23]employed2DalignedmicrostructurescombinedwithmicromechanicalmodelsofMT,HTandFEMtopredictthestiffnessofpolymerclaynanocomposites.Theparticleswereassumedtobeallalignedandisotropic.Hbaiebetal.[12]haveused2Dand3DFEMmodelsofthepolymer/claynanocompositeswithalignedandrandomlyorientedparticlestodeterminetheelasticpropertiesofthismaterial.TheycalculatedtheeffectiveYoung'smodulusforrandomandalignedparticlesandconfronttheresultstoMTmodel.TheauthorsconcludedthattheMTmodeldidnotpredictaccuratelythestiffnessofthecomposites.DongandBhat-tacharyya[3]predictedtheelasticmoduliofpolymerclaynano-compositesusingnumericaltechniquebasedonmappingthereal2Dmicro-nanostructuresofclayplatelets.Theresultswereveri edbythecomparisonofnumericalresultstoexperimentaldataandtheconventionalcompositestheoriesasHTandHuiandShia[13]models.TheresultsshowthatthenumericalsimulationsprovidethegreatinsighttowellpredicttheelasticmodulusofPoly-propylene(PP)/claynanocompositesincomparisontoconventionalcompositestheories.

Recently,Pahlavanpouretal.[22]evaluatedtheperformanceofcommonlyusedanalyticalmicromechanicalmodelstopredicttheelasticpropertiesofpolymer/claynanocompositeswiththehelpofnumericalsimulationsbasedFEM.Theresultsshowthatthecom-parisonbetweenanalyticalandsimulationsrevealedthattheMTmodelisthemostreliablemethodtobeusedforthepossiblerangesofmoduluscontrast,aspectratioandvolumefraction.Lie-lensetal.[18]giveabestpredictioncomparedtoMTmodelathighvolumefractionswhentherigiditycontrastbetweeneffectiveparticleandpolymerisalsohigh.TheSCschemeoverestimatesthe

axialYoung'smodulusforallstudiedcasesofPCN(polymerclaynanocomposites).

Themajorityofmicromechanicalanalyticalmodelsdonottakeintoaccountthein uenceoftheparticleshapeontheeffectivepropertiesofnanocomposites.Theclassicalmodels,theiraccuracyandtheirrangeofapplicability,basedonmoreorlesssuitablehypotheses,cannotbeestablishedintheabsenceofanexactso-lution,seeElMoumenetal.[7].ThiscanonlybeobtainedbysolvingnumericallytheboundaryvalueproblemforaRVEofnanocomposites.

Inthepresentpaper,themaingoalistopredicttheeffectofparticlesize,particlesnumberandparticlesdistributiononme-chanicalpropertiesofrandomlypartiallyintercalatedandexfoliatedpolymerclaynanocompositesusingnumericalhomogenizationtechniques.Severalmicrostructureswithdifferentvolumefractionsandaspectratiorangingfrom5%to40%aregenerated.FEMsimu-lationsofdetailedmicrostructuresareperformedwithdifferentboundaryconditions.Theeffectofboundaryconditionsinme-chanicalpropertiesofclaynanocompositesisalsoinvestigated.TheRVEsizeofmicrostructuresandtheireffectiveelasticpropertiesarecomparedandrelatedwithnumberofparticles.Theresultsarethencomparedwithbothofexperimentaldataofpolypropylenemont-morillonitenanocomposites(PP/MMT)andconventionalcompos-itestheoriesofanalyticalmodel.

2.Generationofmicrostructuresand niteelementmesh2.1.Morphologyofmicrostructures

NanocompositesmorphologieswerereconstructeddigitallyusingPoissonprocess.RandomlydistributedparticlesweregeneratedinRVEwithanalgorithmimplementedinMATLABsoftware.ThealgorithmisbasedoncompositemicrostructuresandiselaboratedbyElMoumenetal.[7]forthecaseofmicrostructureswithellipsoidalparticles.Thealgorithmwasadaptedtointerca-latedandexfoliatedpolymerclayparticles.Thisprocessiswidelyusedforgeneratingofcompositesreinforcedwithsphericalorcy-lindricalparticles,seeElMoumenetal.[4].Theideaistoembedpointsrandomlyina2DplaneaccordingtoaPoissonlaw.Thesepointsrepresentthecenterofeachinclusionandthenastraightlinewitharandomorientationisgeneratedfromeachofthesepoints.

Inthisstudy,wehaveconsidereda2Dmicrostructureofpartiallyintercalatedandexfoliatedpolymerclaynanocomposites.Themorphologyofclayplatelets(MMT)werecarriedwithdifferentvaluesofaspectratios:z¼5,z¼10,z¼20,z¼40,embeddedandrandomlyorientedinaPPmatrixwithvariousvolumefractionsof4.5%,6%and10%.Fig.1showssomeexamplesofthegeneratedmicrostructuresofclayplateletsinthematrixincludingexfoliatedandintercalateddistribution.Thephysicalandmechanicalprop-ertiesaffectedtoeachphasearegivenbyShengetal.[23]andKimetal.[16]andlistedinTable1.Itshouldmentionthatbothofthematrixandclayparticlesareisotropicandtheparticlesareassumedtobeperfectlyboundtothematrix.2.2.Finiteelementmeshing

Oncethegeometryofthemicrostructureisperformed,ameshcanbegenerated.Theregular niteelementmeshissuperimposedontheimageofthemicrostructureusingtheso-calledmultiphaseelementtechnique.ThistechniquewasdevelopedbyLippmannetal.[19]andextensivelyusedbyElMoumenetal.[5]andElMoumenetal.[7]forhomogenizationofrealandvirtualcompositemicrostructuresrespectively.Indeed,theimageofthemicrostruc-tureisusedtoattributetheproperphasepropertyto

each

Y.Djebaraetal./CompositesPartB86(2016)135e142137

Fig.1.RVEincludingrandomlyorientedparticlesforanaspectratioz¼40andvolumefractionof:(a)4.5%,(b)6%and(c)10%ofparticles.

Table1

PhysicalpropertiesofPP/Claynano-composites.Wheren,E,kandmarethePoissonratio,Young'smodulus,bulkmodulusandshearmodulusrespectively.

computemacroscopicin-planebulkkappandshearmappmoduliaregivenby:

n

Matrix(PP)

Clayparticles(MMT)

0.350.26

E(MPa)176048,300

k(MPa)217339,931

m(MPa)

65219,167

Density(g/cm3)0.91.8

E¼

k

1001

andE¼

m

00:50:50

(1)

Therefore,wecande netheapparentmacroscopicbulkmoduluskappandtheapparentmacroscopicshearmodulusmappas:

integrationpointofaregularmesh,accordingtothecoloroftheunderlyingelement.Fig.2showsanexampleofaregularmeshwithgridof500by500 niteelements.

kapp¼

1

traceð<s>Þandmapp¼<s12>4

(2)

Thesign<s>meansaveragevalueofthelocalstresss.3.2.Convergenceofmacroscopicelasticproperties

Theconvergenceofthemacroscopiceffectivepropertiesisob-tainedstudyingthewill-knowmeshdensity.Itisde nedasthenumberof niteelementsnecessarytomeshelementaryvolumeofnanocomposites.Forthatpurpose,aspeci c2Dmicrostructuremadeofrandomclayparticlesisconsidered.Thenumberofpar-ticlesandthegeometryofthemicrostructureareunchanged,butdifferentmeshresolutionsareused.Fig.3showstheresultsofcomputationsofthemacroscopicelasticpropertiesasafunctionofthenumberofused niteelementsforeachmeshresolution.Thenumberofnodesincreasesfrom341(meshcontains100elements)to2,51,001(meshcontains250,000elements),keepingthesamemicrostructurecontaining100particlesoccupying10%oftotalsurface.Fig.3showsthatthehomogenizedproperties rstrapidlydecreasefor nermeshesandtendstostabilizeforlarge

volumes.

putationalhomogenization3.1.Boundaryconditions

Twotypesofboundaryconditionsareconsideredtobepre-scribedontheboundaryofthedomain.Inthecaseoflinearelas-ticity,theseconditionsare:kinematicuniformboundaryconditions(KUBC)andperiodicityconditions(PBC).ItshouldbenotedthattheminimalsizerequiredtoestimatetheeffectivepropertiesismuchlowerforPBCcomparedtoKUBC.TheresultsproducedbytheperiodicboundaryconditionsconvergemorerapidlythantheonesobtainedbyKUBC,seeKanitetal.[14].

Inheterogeneousmaterials,2Dmacroscopiceffectivebulkandshearmoduliwerecalculatedbysolvingtwofundamentalbound-aryvalueproblemswithimposedaveragestrain,Kanitetal.[14].Themacroscopicimposedstraintensors,EkandEm,usedto

Fig.2.Meshingtechnique:(a)polymerclaynano-compositesimageand(b)associatedmesh.

138Y.Djebaraetal./CompositesPartB86(2016)135e142

3.3.Numericalresults

Inthispart,theeffectofclayparticlesdistributionandparticlesnumber,aspectratioandthetypeofboundaryconditionsontheeffectiveelasticpropertiesofPP/nano-clayparticlesispresented.Foreachmicrostructureofnanocomposites,containingrandomparticles,ndifferentrealizationswerecreatedtoestimatetheelasticmoduliandtostudytheeffectofdistribution.Thedifferencebetweencreatedrealizationsis:orientationanddispositionofparticles,itsdistributionandthesizeofmicrostructures.Thenumberofrealizations,consideredforeachvolumesizecontainingNparticles,isgiveninTable2.Forillustration,Fig.4givesanexampleofthegeneratedpartiallyintercalatedandexfoliatedrealizationwithsamevolumefractionsbutdifferentdistributions.Thesimulationsareperformedwiththe niteelementmethod.Dependingonboundaryconditionsanddifferentrealizations,macroscopiclinearproperties,in-planebulkandshearmoduli,werecalculatedto ndtheelasticbehavior.ThesetobtainedresultsfortheapparentandeffectivepropertiesofeachaspectratioarepresentedinFigs.5e7.ThesecurvesexplainthevariationofthemacroscopicpropertiesofPP/claynano-compositeswiththechangeinnumberofparticlesN.These guresshowalsothe uc-tuationoftheresultsfordifferentrealizations,themeanvaluesanditsintervalofcon dencefortheapparentproperties.Itappearsthat,forbothboundaryconditions,theerrordecreaseswhenthenumberofparticlesincreases,andtendstozeroforlargevolumes.Forthesereasons,onerealizationisenoughtodescribethe

elastic

Fig.3.Variationofmacroscopicbulkmoduliwithchangingthenumberof niteel-ementsforvariousmeshgrids.

Table2

Numberofdifferentrealizationsusedforeach xednumberofparticles.Nn

1050

3050

5040

10030

15020

20020

This gurealsoshowsthatthenumberof niteelementsinwhichitbeginsconvergenceisapproximately81,000elements/100par-ticles.Itappearsthatameshdensityof81,000elementsper100particlesisnecessarytogetaprecisionof1%.

Fig.4.Examplesofconsideredrealizationsntostudytheeffectofparticlesdistributionforanaspectratioof40and10%ofvolumefraction:(a)n¼30,(b)n¼50and(c)n¼100particles.

Y.Djebaraetal./CompositesPartB86(2016)135e142139

Fig.5.Variationofthehomogenizedelasticpropertiesversusnumberofparticlesfordifferentboundaryconditionsand4.5%of

particles.

behaviorofnano-compositesforvolumeslargerthanthedeter-ministicRVE.ThesameresultsareshownbyElMoumenetal.[5]forrandomcompositesreinforcedwithnaturalparticles.Foreachmodulus,thetwovaluesofboundaryconditions,KUBCandPBC,convergetowardstothesamelimitstartingfrom50particles.Itappearsalsothatthedistributionofparticlesdoesnotaffecttheelasticpropertiesinvolumeslargerthan50particlesfordifferentaspectratios.Itshouldbementionthattheparticlevolumefrac-tionsandtheaspectratioaretheprincipalmorphologicalparam-eterswhichin uencetheelasticpropertiesofnano-compositeestimatedwithnumericalsimulations.Forillustration,theeffectivepropertiesfound,inthecaseofPBC,arereportedinTable3andcomparedwithanalyticalmethodssuchasVoigt(V)andReuss(R)bounds,HashinandShtrikman(HS)boundsandgeneralizedselfconsistentestimatesofChristensenandLo[2].Thistablegivesacomparisonoftheanalyticalapproximationmethodsbrie yrecalledintheintroductiontothe niteelementresultsregardingthesetgeneratedmicrostructuresfordifferentratios.Itisclearfromthistableand guresthattheestimationsprovidedbytheapproximationschemesareclosetothecorresponding niteelementsimulationsinthecaseoflowvolumefractions.Themaximumdifferencebetweennumericalandanalyticalresultsisobtainedinthecaseofheightvolumefractionofparticleswithlargeaspectratio.Thisdifferenceisaroundof15%.Forillustration,anexampleofthedeformedmicrostructuresofnano-compositesispresentedinFig.8forcomputationsofthebulkandshearmodulus.

Fig.6.Variationofthehomogenizedelasticpropertiesversusnumberofparticlesfordifferentboundaryconditionsand6%ofparticles.

4.Experimentalresultsversusnumericalandanalyticalapproaches

Theobjectiveofthissectionistopresentaconfrontationbe-tweennumerical,analyticalandexperimentalresultsoftheesti-matedelasticmoduli.Thenumericalresultsareobtainedusingthehomogenizationtechniqueandtheanalyticalonesbymicro-mechanicalmodels.ThemostimportantmicromechanicalmodelsaretheVR,HSboundsandHTmodel.Thesemodelsgenerallyconsiderthevolumefractionandtheaspectratioofparticlesinsidethematrix.Fortheexperimentalresults,Table4presentsthevaluesoftheelasticmodulus,extractedfromtheliterature,seeDongandBhattacharyya[3];asafunctionofparticleweightfractions.

Sincethevolumefractionisaveryimportantparameterin niteelementsimulationsandmicromechanicalmodels,itisveryimportanttoestablishaquantitativerelationbetweentheweightratioandthevolumefraction,toproperlycomparethesimulationresultswiththoseofexperimentaldata.Foratwo-phasecompositematerialconsistingofamatrixandparticles,thevolumefractioncanbecalculatedfromtheweightfractionwf,matrixdensityrm,andclayparticledensityrfasfollows:

P¼

ww.frf

r(3)

ffþ1Àwf

rm

140Y.Djebaraetal./CompositesPartB86(2016)135e142

4.1.NumericalpredictionofnanocompositesYoung'smodulusThesamemethodologyusedinthelastsectionfornumericalhomogenizationofbulkandshearmoduliisusedinthispartforelasticmodulus.Numericalbulkandshearmoduliaredeterminedfordifferentvolumefractionsofclayparticleswithdifferentas-pectsratio.Thus,theeffectiveelasticmodulusEeffandeffectivePoissonrationeffofthepolymerclaynanocompositesarecalculatedaccordingtothe2DisotropyrelationsasgivenbyMeilleandGar-boczi[20]:

E

eff

¼

4

þn

eff

keffÀmeff

¼k(4)

Table5givesthesetobtainednumericalresultsconfrontedtotheresultsofmicromechanicalmodels.Itappearsthattheanalyt-icalmodelsgiveagoodpredictionofYoung'smodulusinthecaseoflowervolumefractionswithsmallaspectratio.However,thedif-ferencebetweenanalyticalandnumericalresultsbecomesimpor-tantbyincreasingthevolumefractionandtheaspectratioofparticles.

4.2.Confrontationofexperimental,numericalandanalyticalresultsFig.9showstheconfrontationofnumericalresultswiththepredictionsoftheHTmodel,HSboundsandtheexperimentaldata.Thevariationisgivenasafunctionofthevolumefractionofclayparticlesatvariousaspectratiorangingfromz¼5toz¼40.Fromthis gureitappearsthat:

Thereisagoodagreementbetweenbothofnumericalsimula-tionandexperimentaldatafordifferentvolumefractions.AthighervolumefractionsthereisasmalldifferencethatmaybeassociatedtothesizeofexperimentalsamplesasshownbyKanitetal.[15]andElMoumenetal.[5].Inexperimentalcharacterization,withincreasingofthevolumefractionsthenumbersofparticlesisalsoincreasedandleadstocreation

of

Fig.7.Variationofthehomogenizedelasticpropertiesversusnumberofparticlesfordifferentboundaryconditionsand10%ofparticles.

Table3

Comparisonbetweennumericalresultsanddifferentanalyticalmodels.TheerroriscalculatedbetweenGSCandtheresultsofnumericalsimulations.z5

P(%)4.5610

10

4.5610

20

4.5610

40

4.5610

Effectivepropertieskeffkeffkeffkeff

R226968123046922400722227068123046922400722227068123046922400722227068123046922400722

HSÀ229669923407162463763229669923407162463763229669923407162463763229669923407162463763

GSC229669923407162463764229669923407162463764229669923407162463764229669923407162463764

εr(%)2.3132.954.334.556.154.185.726.078.808.9714.677.019.168.8012.9912.5117.2810.1913.4512.6515.9218.5118.46

Numericalresults234972024097472575811239273924827792684873245776325468092771896253079326368302919905

HSþ28059613023106836291365280596130231068362913652805961302310683623136528059613023106836291365

V387214854438176389492503387214854438176389492503387214854438176389492503387214854438176389492503

meffmeffmeff

keffkeffkeff

meffmeffmeff

keffkeffkeff

meffmeffmeff

keffkeff

meffmeffmeff

Y.Djebaraetal./CompositesPartB86(2016)135e142141

Fig.8.Exampleofdeformedmicrostructuresforcomputationtest:(a)initialmicrostructure,(b)computationofbulkmodulus,(c)computationofshearmodulus.

Table4

ExperimentalYoung'smodulusofPP/claynanocompositesversusvolumefractions,see.YuDongetal.(2009).wf00.030.050.0850.10

P0

0,0150,0260,0420,053

z010

5<z<10z5<5

ExperimentalE(MPa)17601970204321312097

TheHTmodelresultsclearlyoverestimatetheelasticmodulus,becausethedifferencebetweenthismodelandthenumericalsimulationandexperimentaldataincreaseswithincreasingthevolumefractionandtheaspectratioofparticles,excludingthecaseofthesmallaspectratios(z¼5andz¼10),wheretheHTmodelgivesareasonablepredictionoftheelasticmodulus.TheresultsoftheHTmodelarewelllocatedbetweenupperandlowerHSbounds,butinthecaseofhighratio,z¼40,theresultsareclearlydivergedouttheHSbounds.5.Conclusion

Inthispaper,wehavepresentedaconfrontationbetweennu-merical,experimentalandanalyticalresultsofsamplesrepresent-ingPPmatrix/claynanocomposites.Numericalhomogenizationishandledby niteelementmethodbasedontherepresentativevolumeelement(RVE)andcomparedtotheclassicalanalyticalmodelsandexperimentaldata.Anoriginalmethod,basedonPoissonprocess,hasbeenusedtogenerateautomaticallythemicrostructureofPP/claynanocompositesrepresentingthisma-terial.Thismethodconsistsinrandomlyplacingnon-overlappingexfoliation/intercalationparticlesintheRVE.Severalmicrostruc-turesweregenerateddependingontheaspectratioofparticlesrangingfrom5to40fordifferentvolumefractions.Themeshofthemicrostructuresisobtainedusingtheso-calledmulti-phaseelementmethod.Thevalidityoftheobtainedresultsreliesmainlyonthehypothesisthatthemicrostructuresarestatisticallyrepre-sentativeandthenumberofusedelementisenoughtoachieveagoodconvergenceofmacroscopicproperties.Theseconditionswereperformedbystudyingthemeshdensity.Itappearstheex-istenceofasmalldifference(0.01%)betweentheresultsobtainedbydifferentmicrostructuresofnanocomposites.TwodifferentboundaryconditionsaswellaskinematicuniformandperiodicboundaryconditionswereimposedinordertoassesstheeffectoftheseconditionsonthemicrostructuresofPP/claynanocomposites.Itappearsthattheoverallpropertiesestimatedonthevolumecontainingmorethan50particlesareindependentofboundaryconditions.Thissizecanbeconsideredastheminimumvolumeforwhichthenumericalresultsarerepresentatives.Thein uenceoftheshapewashighlightedthroughthecomparisonofnumericalresultsperformedonthemicrostructureswithdifferentaspectratio.Itappearsadifferenceintheelasticmoduliregardingtheaspectratio,especiallyinthecaseofhighparticlevolumefractions.Theconfrontationofthenumericalresultstothesomeofthemostusedanalyticalmodels(asVRandHSbounds,HTandGSCmodels)andexperimentalapproachesispresented.Globally,inthecaseoflowervolumefractionandsmallaspectratio,alltheanalyticalmodelsprovidegoodapproximationsfordifferentstudiedcon g-urations.However,whentheaspectratioand/orparticles

volume

Table5

ComparisonbetweenanalyticalpredictionsandnumericalresultsofPP/MMT.TheerrorrepresentsthedifferencebetweennumericalresultsandHTanalyticalmodel.z

P(%)

Eeff(MPa)R

5

4.56104.56104.56104.5610

184018681948184018681948184018681948184018681948

HSÀ214321922329214321922329214321922329214321922329

Numericalresults220522802467225723712605232824552738241625262763

Error(%)1.51.138.735.929.5419.614.2519.3830.7420.9427.8841.28

HT216523062703239926213240271530453953305634984705

HSþ286331573967286331573967286331573967286331573967

V385445526414385445526414385445526414385445526414

10

20

40

Fig.9.Confrontationofnumerical,experimentalandanalyticalYoung'smodulusofPP/claynanocomposites.

clusteredparticles.ThisparticleclusteringcausesadecreaseinelasticmodulusasclearlyshowninFig.9,seeforexampleHbaiebetal.[12].

142Y.Djebaraetal./CompositesPartB86(2016)135e142

fractionsbecomeimportant,theVRboundsandHTmodelsare

largelydivergescomparedtoexperimentalresultsand niteelementsimulatedresults.ThisstudythenshowsthattheexplicitanalyticalmodelsmaycorrectlyestimatetheeffectivemechanicalpropertiesofPP/ClaynanocompositesbasedMMT,expectthecaseofhighvolumefractionsandaspectratiolessthan10.References

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