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Anissa EDDHAHAK, Mathieu VANDAMME, V. T. VU - Micromechanical contribution for the
prediction of the viscoelastic properties of high rate recycled asphalt and influence of the level
blending - Archives of Civil and Mechanical Engineering - Vol. 15, n°4, p.1037-1045 - 2015
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Micromechanical
contribution
for
the
prediction
of
the
viscoelastic
properties
of
high
rate
recycled
asphalt
and
influence
of
the
level
blending
A.
Eddhahak
a,*
,
M.
Vandamme
b,
V.T.
Vu
baLaboratoryPIMM,CNRSUMR8006,ArtsetmétiersParisTech,151boulevarddel'hôpital,75013Paris,France bUniversitéParis-Est,LaboratoireNavier(UMR8205),CNRS,ENPC,IFSTTAR,77455Marne-la-Vallée,France
1.
Introduction
Pavement recycling has been increasingly gaining a great attentionintheasphaltindustry.Inadditiontotheecological benefits and the natural resources preservation, recycling allowsalso the reduction of energyconsumption resulting thereforeinamorecost-effectiveproductionandconsiderable economicalgains[1,2].InFrance,sincetheoilcrisisof1973,the bituminousmaterialshavebeenmoreandmorerecoveredfor recyclingand competitive effortsare made toinvolvehigh
amounts of RAP (reclaimed asphalt pavement). Nowadays, RAP rates up to 30% are commonly used in the road construction and rehabilitation projects making thereby necessary to control the properties of the final recycled product[3].
Inthiscontext,someresearchworkshavefocusedonthe investigation of the viscoelastic properties of the recycled asphalt fromdifferent pointof views.Eddhahak-Ouni etal.
[4,5] have presented an experimental laboratory protocol based on a progressive extraction combined with a FTIR spectroscopy analysisinorder tohighlighttherelationship Keywords:
Recycledasphalt Homogenization Blendedbinder Rheology
Shearcomplexmodulus
a
b
s
t
r
a
c
t
Thisresearchdealswithanexperimentalandmicromechanicalstudyofhighraterecycled asphaltwith70%ofRAP.Rheologicalmeasurementsofshearcomplexmodulusofvirgin, RAPandblendedbinderswereperformedatdifferenttemperaturesandfrequencies.Then,a micromechanical model based on the generalized self-consistent scheme (GSCS) was suggestedforthepredictionoftheeffectivemechanicalpropertiesoftherecycledasphalt. TheGSCS-based approachaimsto homogenizethe heterogeneousmaterialtaking into accounttheintergranularporosity,ontheonehand,andthepossibleinteractionsbetween its phases on the other one. The suggestedmethod was compared to a step-by-step (successive)homogenizationapproachandliteraturedatainelasticcasewereusedforthis purpose.TheresultshaveshownthattheGSCS-basedapproachpresentsagoodagreement with the data especially for higher volume fractions of aggregates. Furthermore, the significant influenceofthe blendhomogeneitylevelon theestimation oftheeffective mechanicalpropertiesoftherecycledasphaltwashighlighted.
* Correspondingauthor.Tel.:+33699247666.
betweenthemanufacturerecyclingprocessontheonehand andthelevelofhomogeneityoftheblendedbinderontheother one.Theyconcludedthatthetemperatureandthetimemixing parametersinvolvedintheasphaltmanufactureshallbewell calibratedinordertoreachthewishedblendinglevelofthefinal binderandtocontroltherebythemechanicalperformancesof therecycledasphalt.Otherresearchworkssuggestedempirical and analytical predictive models for the prediction of the viscoelasticpropertiesoftheasphalticconcretebaseduponthe rheologicalinformationofitsbituminousbinderandmixdesign
[6–9].Someofthesemodelshaveshownsomelimitationsfor specificsolicitationfrequencyandtemperatureranges[10]and othersoftenrequirecomplementaryexperimentaltestsonthe asphalticconcrete[11].
Although they are reliable, the experimental-based approaches are sometimes very long and cumbersome to perform.Furthermore,experiencescanbecostlyandtoomuch timeandenergy-consumingespeciallywhendealingwiththe investigation of thematerialresponses under different me-chanicalscenarios.Forthesereasons,innovativeapproaches based on the rational mechanics have been adopted in oppositionoftheso-calledempiricschool.Themicromechanics whichhasbeen developedfor decades is acomplementary approachwhosepurposeistoprovideefficientdeductivetools forthepredictionoftheeffectivepropertiesofthe heteroge-neousmaterialbasedontheknowledgeofthepropertiesofits constituents.Itwashighlightedwiththehelpoftheextensive literaturethatmicromechanicscanbeusedtounderstandthe mechanicalbehaviorofthebituminousmaterials.Buttlaretal.
[12]usedthegeneralizedself-consistentscheme(GSCS)model
[13,14]inordertoinvestigatetheasphaltmasticbehavior.Later, ShashidarandShenoy[15]suggestedamodifiedformofButtlar solutioninvolvingtheuseofanorderofmagnitudeanalysisfor simplificationofthecomplexsetsoftheGSCSequations.The researchofLachihab[16]consideredtheasphaltasabiphase compositeincludingspherical inclusionscoated bya bitu-menfilmandembeddedintheinfiniteequivalentmedium. Nevertheless,theintergranularporositywasnottakeninto accountinhiswork.Morerecently,AlamandHammoum[17]
presentedamicromechanicalanalysisbasedonasuccessive homogenizationprocessinwhichtheself-consistentmodel wasconsideredinthreemultiscalestepsforthepredictionof theviscoelasticpropertiesofHMAmixes.However,errorsup to50%wererecordedinsomecasesforthepredictionofthe mechanicalbehavioroftheasphalt.Fromtheabovestudies, thesuitabilityofthegeneralizedself-consistentmodelwas shownandits reliabilitytoprovidegoodpredictionsofthe equivalentpropertiesofthecomposite[18],butfurtherwork is still needed for a better prediction of the asphalt properties.
Inthisframework,theobjectivesofthepresentworkare: (i) Topresentanovelmicromechanicalapproachbasedon theGSCSmodelandillustratedonalaboratorycaseofa high rate recycled HMA including 70% of RAP. The intergranular porosity of the composite is taken into accountinthemodeling.
(ii) Toinvestigatethelevelofblendingofthefinalbinderand itsinfluenceinthepredictionoftheviscoelasticproperties ofthehighraterecycledHMA.
2.
Theoretical
development
Therecycledasphaltisaheterogeneousmaterialcomposedof typically95%(bymass)of aggregatesand5% ofbituminous binder.Thus,thiscompositecanbeassumedasamultiphase material reinforced by elastic inclusions (aggregates ‘‘a’’) whicharesurroundedbyaviscoelasticmatrix(mastic‘‘m’’) and weakenedbytheporosityphase(airvoids‘‘p’’)(Fig. 1). Accordingly, the micromechanical model which will be considered hereisbasedon thegeneralized self-consistent scheme (GSCS)[16].Thismodelwas initiallyformulatedby ChristensenandLo[13]fortwo-phasematerialsthrougha 3-phaseschemebyconsideringacompositesphericalinclusion coatedbyamatrixshellwhichisembeddedinanequivalent homogeneousinfinitemedium(EHM).Unliketheclassic self-consistentscheme,theGSCSisanimplicitmodelsincethe finalsolutionisexpressedasafunctionofthepropertiesofthe EHMwhichisunknowninitself.Forfurtherinformationon this micromechanicalscheme, thereader couldconsultthe detailedworkofHervéandZaoui[14].
Inaddition,forabetterandmoreaccuratedefinitionofthe multiphasecomposite,conclusionshavebeendrawnfromthe previous research of Navaro [19] through which it was highlightedthatinthecaseofagoodblendingofthevirgin binderintotheRAP-binder,theaggregateissurroundedbyjust onehomogeneousbinderlayer(Fig.2a).Accordingly,inthecase of a ‘‘good’’ blending configuration, one can suppose a monolayersphericalinclusionofa(2+1)-phase micromecha-nicalmodel[20].Whereas,inthecaseofa‘‘bad’’blendingof virginandRAPbinders,a2-layeredinclusionisconsidered.The firstlayerclosetotheaggregatecorrespondstotheRAPbinder andthesecondlayerisrelativetothevirginbinder(Fig.2b). Accordingly, a (3+1)-phase model will be adopted for the descriptionofthisconfiguration.Inthefollowing,the micro-mechanical development based on the (n+1)-phase model (n=2, 3) is presented in order to appreciate the effective homogeneous mechanical properties ofthe recycledasphalt compositeinbothconfigurations(goodandbadblendinglevels). Forthesakeofsimplicity,theproblemwillbefirsttractablein the caseoflinearelasticityandthesolutionisthenextendedtothe viscoelasticity.
2.1. TheGSCS-basedapproach(elasticitycase)
In this section, we will assume that the heterogeneous material is composed with purely elastic phases; thereby theassociatedrepresentativevolumeelement(RVE),denoted as ‘‘V’’, is described by a first elastic spherical inclusion (correspondingtotheaggregates)coatedbyanelasticphase and a second spherical inclusion (corresponding to the
porosity)whichareembeddedinanequivalenthomogeneous infinite phase subjected to uniform strain condition E at infinity.Eachphaseischaracterizedbyabulkmoduluski,a
Poisson'sratioyi,ashearmodulusmiandavolumefractionfi.
Notethatkiandmiareequaltozerofortheparticularcaseof
theporosity.
Thehomogenizationglobalproblemdenoted‘‘GP’’canthen be solved by considering simultaneously two secondary problemsnamed‘‘SP1’’and‘‘SP2’’,associatedrespectivelyto the rigid inclusion (aggregates) and the ‘‘weak’’ inclusion (porosity)andsubjectedtothesameauxiliarydeformationE0
atinfinity(Fig.3).
Asknown,thepredictionofthehomogeneousequivalent mechanical properties of a heterogeneous material by a multiscaleapproachrequiresthe knowledgeof theaverage of the localization tensors of each phase of the composite accordingtothefollowing:
complexes;hom¼hcomplexes;:Ai¼X
i
ficomplexes;i:hAii
(1) where complexes;hom and complexes;i are respectively the
effectiveandthe local stiffness4th-order tensors;Ai isthe
strainlocalization4th-ordertensorsofthephaseiassociated tothesecondaryproblem.Inthecaseoftheglobalproblem, theaveragestrainlocalizationtensor,denotedL,isdefinedas theparameterthatrelatesthemicro-strainfieldeofeachpoint ofmicro-coordinatexoftheRVEtothemacro-strainfieldE appliedatinfinity:
eiðxÞ¼hLiðxÞi:E (2)
Asamatterofthefact,thelocalizationproblemhastobe solvedatfirstforthederivationofthehomogenized proper-ties. Based onthe factthatthe average straineequals the macro-strainEontheonehandandtheaveragesumofthe strainsrelativetothedifferentphasesontheotherhand,itcan beeasilydemonstratedthatthestrainlocalizationtensorof theglobalproblemisrelatedtothatofthesecondaryproblems accordingtothefollowing:
hLiðxÞi¼hAii:
X
ifihAii
1
(3) By assuming that the material is isotropicandmaking useofthe spherical symmetryofthe problem, the micro-strainfieldofthesecondaryconfigurationscanbeobtained by the resolution of two elementary problems of simple shear and hydrostatic pressure applied at infinity. Note that in the case of linear isotropy, the stiffness tensor is givenby:
complexes;¼3kIvolþ2mIdev (4)
where k and m are respectively the elastic bulk and shear moduliwhereasIvolandIdevdenoterespectivelythespherical
anddeviatoricoperators.
By considering adisplacement approach withboundary displacements conditions, one can determine the average strainineach phaseforbothuniformanddeviatoricstrain conditionsas:
Fig.2–GSCSrepresentationsofrecycledasphalticconcretes.(a)Monolayercase(goodblending)and(b)bi-layercase(bad blending).
esph i D E ¼ Fi Fnþ11 E0¼A sph i E0 edev i D E ¼ 1 Anþ1 Ai 21 5 R5 i R5i1 ð12yiÞðR3i R3i1Þ Bi ! 1E0¼Adevi E0 8 > > > < > > > : (5) whereRidenotestheradiusofthephasei,thecoefficientsAi,Bi
andFiareconstantsdefinedinAppendixAandIistheidentity
tensorof4thorder.
Once the strain localization tensors of the secondary problems are known, the localization tensor of the global problemcanbederivedfromEq.(3).Then,theuseofEq.(1)
combinedtoEq.(4)allowsthecomputationof theeffective bulkandshearmodulioftheheterogeneousmaterial.Eq.(6)
presentsthe summarized form expressionsof the effective mechanicalproperties. khom¼ fakaAsaphþ fmkmAsmphfpkpAspph : faAsph a þ fmAsmphfpAspph 1
mhom¼ famaAdeva þ fmmmAdevm fpmpAdevp
: faAdeva þ fmAdevm fpAdevp 1 8 > > > > > > > > > < > > > > > > > > > : (6)
where Asph and Adev are the strain localization tensors of
respectivelythehydrostaticpressureandsimple shear pro-blems.
2.2. Successivehomogenizationapproach
In this section, we present a different approach for the predictionoftheeffectivemechanicalpropertiesoftheelastic compositebaseduponasuccessivehomogenizationmethod. Thelatterconsistsonastep-by-stephomogenizationprocess whichinvolvesdifferentREVsandhomogenizationschemes foreachstep.Theoutputeffectivesolutioncomputedinstep (i)isconsideredasaninputdataforthefollowingstep(i+1). Theadvantageofsuchanapproachisessentiallytotrimdown thecomplexproblemandtotrackitscomplexitiesindifferent stagesallowingthereforean‘‘easy’’andclassic homogeniza-tionoftheglobalproblem.Notethatthismodelwaspreviously usedbyAlamandHammoum[17]forthepredictionofthe viscoelasticpropertiesofHotMixAsphaltwhichisassumedas acomposite of fourphases(fines, bitumen,aggregatesand voids).Fig.4illustratesthedifferentstepsofthesuccessive homogenizationprocessconsideredherein.Thesteps corre-spondtoRVEs,eachofwhich,composedofthreephases,isa compositeinclusionembeddedinamatrix.
In the first step, bitumen and fines are homogenized together to obtain the mastic (EHM 1). Then, the latter is consideredasalayerfor theaggregatesinthesecondstep. Accordingly,thenewhomogenizedmediumisamixtureof aggregates and mastic (EHM 2). Finally, the porosity is taken into account in the thirdstep asan inclusion (void) surroundedbytheEHM2derivedfromthesecond homogeni-zationstep.
Since the inclusioniscomposite, weuseda generalized self-consistentschemeforeachstepofthehomogenization process;theeffectivebulkandshearmoduliofthecomposite can be predicted using a (n+1)-phase model (with n=2) accordingtothefollowing:
khom¼knþ ðR 3 n1=R3nÞðkn1knÞ 1þððR3 nR3n1Þ=R3nÞððkn1knÞ=ðknþ4=3mnÞÞ A mhom mn 2 þB mhom mn þC¼0 8 > > > < > > > : (7)
wheretheexpressionsofthecoefficientsA,BandCaregivenin
Appendix A. Note that in this particular caseof n=2, the equationsyieldtotheclassicalsolutionofthe3-phasemodel
[12].
2.3. Extensiontotheviscoelasticity
Thewell-knowncorrespondenceprincipleiswidelyusedfor viscoelastic composites in order to transpose the effective elasticsolutionsfromtheelasticitytotheviscoelasticity[21– 23].Thecorrespondenceisessentiallybasedonthesimilitude noticedbetweentheelasticconstitutivelaw,ontheonehand, and that of theviscoelasticity expressed inLaplace–Carson domain.Itwashighlighted,however,bymanyresearchesthat the determination of the viscoelastic effective creep and relaxationmoduliisnotoftenstraightforwardsinceitoften required rational fraction forms of the elastic effective solutionsoftheLaplaceCarsontransforminordertomake possiblethecomputationoftheinversetransform.Theworks ofHashin[24]establishedasimplecorrespondenceprinciple relatingtheeffectivecomplexmodulitoeffectiveelasticoneof composites. Accordingly, in order to obtain the effective complex moduli of a viscoelastic heterogeneous material, theelastici-phasemodulikiandmihavetobereplacedbythe
complex i-phase moduli kiðivÞ and miðivÞ. Accordingly, the
constitutivelawoftheviscoelasticcompositecanbewrittenas follows:
~sij¼ ~lðivÞ~ekkdijþ2~mðivÞ~eij (8)
whereiisthecomplexvariable(i2=1); ~land ~mare respec-tivelytheLamécoefficientandthecomplexshearmodulusof theviscoelasticcompositevisthesolicitationfrequency.
3.
Illustration
on
a
high
rate
recycled
HMA
3.1. MaterialsdescriptionTheGSCS-basedapproachisillustratedonalaboratorycaseof a hot recycled asphaltic concrete (BBSG 0/10) previously consideredbytheauthors[4].Therecycledmixtureincludes 70%ofRAPand30%ofvirginmaterials(bitumen+aggregates). TherichnessmodulusoftherecycledHMAisequalto3.6with a total bitumen content of 5.74% ppc (by weight of dry aggregates).TherecycledHMA isa20/30penetrationgrade bitumenwhereasthevirginandtheRAPbitumensare160/220 and10/20penetrationgradesrespectively.Itshallbe empha-sizedthatthe manufactureparameters,namelythe mixing temperatureandthemixingtime,werecalibratedsuchastwo experimentalconfigurations(goodblendingandbadblending) andtheycanbestudied.Table1summarizestheinputdataof theasphaltconstituents.
3.2. Rheologicalmeasurements
Rheologicaltestswereperformedonthreekindsofbitumen: thevirginbitumen,theRAPbitumenandthehomogeneous blendderivedfromtherecycledasphalticconcrete.TheRAP and homogeneous bituminous binders wereobtained by a conventionalextractionmethodusinganasphaltanalysator and then recovered by vacuum distillation using a rotary evaporatoraccordingtothestandardspecifications, respec-tively NF EN 12697-1 and NF EN 12697-3. Afterwards, the rheological experiments were carried out using a Bohlin DynamicShearRheometer(DSR)fromMalverninmodeplate and plate geometry (8mm, gap=1mm) within the linear viscoelastic(LVE)domaininthefrequencyrange0.1–10Hzand fortemperaturesvaryingbetween5and608C.
4.
Results
and
discussion
4.1. CaseofahomogeneousblendThesievescurvesofvirgin,RAPandrecycledaggregatesare presentedinFig.5.
4.1.1. Rheologicalresults(complexmodulus)
Theisothermsandisochronesofshearcomplexmodulusm*of thethreebituminousbindersaredepictedinFig.6.Itcanbe seen from these plots that the complex modulus of the three bindersexhibits the same trendswith regard tothe
temperatureontheonehandandthefrequencysolicitationon theotherone.Thehigherthetemperature,theloweristhe rigidity of the bitumen. On the other side, the higher the frequency,thehigheristhecomplexmodulus.Moreover,one cannoticethattheRAPbitumenpresentsthehigherrigidity whereastherecycledbitumenliesbetweenthevirginandthe RAPcurves.Thisisduetotheagingphenomenonexperienced
Table1–Summaryofthemechanicalpropertiesofasphaltconstituents.
Materials Stiffnessmodulus(GPa) Poisson'sratio Shearmodulus(GPa) Compressibilitymodulus(GPa)
Aggregates 105.500 0.25 42.233 70.053
Fines 55.200 0.15 2.400 26.285
Bitumen – 0.33 – –
Fig.5–Sievecurvesofvirgin,RAPandblendedaggregates.
Fig.6–Resultsoftheshearcomplexmodulus
measurements:isothermsatf=1.59Hz(uppercurves),and isochronesatT=108C(lowercurves).
bytheRAPbitumenduringitslifeservicethroughthedaily repetitivemechanicalandthermalloadswhichhaveaffected thehardnessofthebitumen.
4.1.2. Comparisonwiththesuccessivehomogenization (elasticitycase)
Fig.7depictstheresultsoftheeffectiveelasticshearmodulus mhomwithamonolayerrigidinclusionincomparisonwiththe predictionsgivenbythesuccessivehomogenizationapproach. Itcanbeoutlinedthatthetwomodelscorrelateverywellforan aggregatevolumefractionlessthan35%.However,whenthe inclusionconcentrationismoreimportant,ahigherdeviation canbenoticedbetweenthetwoapproaches.Thisfindingcould beexplainedbytheinteractioneffectbetweenthedifferent phases which is more highlighted when the aggregates volumefractionbecomesimportant.Infact,whenthevolume fractionislessthan35%,theaggregatesaresomewhatdiluted intheporousmatrix,sotheinteractioneffectisnotsufficiently strong and the different phases of the composite could therefore be assumed as independent ‘‘packages’’ which could be homogenized separately through a step-by-step method as preceded by the aforementioned successive
homogenizationapproach.However,whenthevolume frac-tionbecomesmoreimportant,somebondsandinteractions maybebuiltbetweenthedifferentphasesanditrequires,asa consequence,tobetakenintoaccountinthehomogenization processofthecomposite.
In order to arbitrate more objectively between the two approaches, some experimental results of effective shear moduli corresponding to a homogenized granular medium withtwophasesweregatheredfromtheliterature(Table2) andusedhereinfortheconfrontationofthehomogenization methodswiththeexperienceintheparticularelasticcase[25]. The data of the effective elastic shear modulus are presentedinFig.7byasteriskblackmarks.Itcanbenoticed that the tendency curve fitting the experimental resultsis closer to the GSCS-based approach than the successive homogenizationmodelespeciallyforhighaggregatesfraction volume(fa>55%).Althoughthesuccessivehomogenization methodhastheadvantagesofbeingmuchsimplerandless cumbersomethantheGSCS-basedmodel,itcanunfortunately provide inaccurate predictions of the apparentmechanical properties andperformancesofthe heterogeneousmaterial and yields thereforetounsafe roadpavement design.Note thatthesuccessivehomogenizationmethodunderestimates therigidityofthecompositeofabout20%whentheaggregates volumefractionisequalto80%.Onthebasisofthissimple casestudyofelasticheterogeneousmedium,theGSC-based approachisprovedtobethebestappropriatemethodforthe predictionofeffectivemechanicalpropertiesofthe heteroge-neousmaterial.Inthenextsection,thisapproachisadopted for the estimation of the equivalent shear modulus of the recycledasphalticconcrete.
4.2. Effectofthelevelofblending(caseofBBSGasphalt concrete)
PlotsofFig.8presenttheevolutionversusthefrequencyofthe equivalentshearmodulusoftherecycledasphalticconcrete base upontheGSCS-basedapproach.Thecalculations were performed by considering two configurations, namely the monolayerrigidinclusion(caseof‘‘good’’blending)andthe bi-layerrigidinclusion(caseof‘‘bad’’blending).Itcanbenoticed fromthisfigurethatthecaseofahomogeneousblendexhibits higher complex shear modulus of the recycled asphalt in comparisonwiththe‘‘bad’’recycledasphalt.Thisfindinglet deduce that the rigidity of the composite is significantly improvedwhenthevirginandRAPbindersarewellblended togetherduring themanufactureprocessyieldingtherebya uniquecontinuouslayeraroundtheaggregates.Nevertheless, whenthebindersarenotwellblended,therecycledbitumenis splitinto two differentlayerswithdifferentproperties and behaves therefore somewhat like two ‘‘unknown’’ and Fig.7–Effectiveelasticshearmoduluspredictedbythe
GSCS-basedapproachandthesuccessivehomogenization, anditscomparisonwiththeexperimentaldata.
Table2–Literaturedatausedforthecomparisonwiththe
homogenizationapproaches.
Matrix (epoxyresin)
Inclusion (normalizedSand)
Stiffnessmodulus(GPa) 2.518 73.08
Poisson'sratio 0.413 0.172
Source[25].
Fig.8–Effectivecomplexshearmodulusevolutionasa functionoffrequencyforT=108C,resultsforrecycled asphalt(caseofgoodandbadblending),andvirgin bitumen.
discontinuousmaterials.Thiscouldaffecttheshearmodulus oftheoverallcomposite.
5.
Conclusions
In this paper, a GSCS-based micromechanical model was presentedfortheinvestigationofthemechanicalbehaviorofa highraterecycledHMAmixture.Motivatedbytheknowledge of therheological propertiesof threedifferent binders,this studyaimstopredictbythemeansofamultiscalemodeling the equivalent mechanical properties of a recycled porous compositetakingintoaccountthemicrostructure configura-tionsresultingfromtheblendinglevelofthevirginbitumen intothe RAP binderduring the manufactureprocess. Some conclusionscanbedrawnattheendofthisresearch: - TheGSCS-basedmodel,basedonaone-step
homogeniza-tion,is shown to besuitable and efficient to predictthe effectiveshearmodulusof theheterogeneousmaterial. In particular,thecomparisonofthedevelopedapproachinthe elastic case has shown a good agreement with the homogenizationdatagatheredfromliterature.
- Although it has numerical complexity, the GSCS-based model was proved to be safer than the multi-steps homogenization since it allows taking into account the possibleinteractionswhichcouldoccurbetweenthe differ-entphasesofthecompositeespeciallywhentheinclusion volumefractionisnotnegligible.
- Theblendinglevelbetweenthenewandtheoldbituminous binderscanhavesignificanteffectsontheestimationofthe effectiveshearmodulusoftherecycledfinalproduct.The illustrationonthehighraterecycledHMAcasehasshown considerable differencesnoticed in the evaluation of the equivalent shear modulus between ‘‘good’’ and ‘‘bad’’ recycled asphaltic concretes. This finding highlights the strong relationship between the asphalt manufacture processontheonehandandtheviscoelasticperformances oftherecycledasphaltontheotherone.
However,severallimitationsofthisresearchareneededto bementionedsuchasthelackofexperimentaldataofshear complexmodulusofrecycledasphalttocomparewiththe GSCS-basedapproachresults.Furthermore,thesize distribu-tion and the random morphological characteristics of aggregateswerenottakenintoaccountinthepresentstudy. Accordingly, it is worthwhile to carry on complementary
experimental researches for the measurement of the mechanicalpropertiesoftherecycledasphalt,andtofurther investigate the aggregates properties and take them in considerationinthemicromechanicalmodel.
Acknowledgements
The rheological tests were performed at ESTP-France. The authorsthankgratefullythepreciouscollaborationofNavier laboratory (ENPC-France) for the scientific discussions and helpfulsuggestions.
Appendix
A.
Case
of
n-layered
composite
sphere
(average
strains
constants
and
effective
shear
modulus
solutions)
[14]
A.1. CoefficientsAi,Bi,Fi
SeeFig.A.1. Fi¼ 1 Qn 11 Fnþ1 (A.1) QðkÞ¼YðkÞ j¼1 NðjÞ (A.2) NðkÞ¼ 1 3Kkþ1þ4mkþ1 3Kkþ4mkþ1 R43 k ðmkþ1mkÞ 3ðKkþ1KkÞR3k 3Kkþ1þ4mkÞ 0 @ 1 A (A.3) Ai¼PðnÞ22 Anþ1 PðnÞ11PðnÞ22PðnÞ12PðnÞ21P ði1Þ; i¼½1;nþ1 (A.4) Bi¼PðnÞ21 Anþ1 PðnÞ11PðnÞ22 PðnÞ12PðnÞ21P ði1Þ; i¼½1;nþ1 (A.5) Pn¼Yn i¼1 Mi and MðkÞ¼ 1 5ð1vkþ1Þ Ck 3 R2 kð3bk7CkÞ 5ð12vkÞ 12ak R5 k 4ðfk27akÞ 15ð12vkÞR3k 0 ð12vkþ1Þbk 7ðð12vkÞ 20ðð12vkþ1Þak 7R7 k 12akð12vkþ1Þ 7ð12vkÞR5k R5 kak 2 R7 kð2akþ147akÞ 70ð12vkÞ dk 7 R2 k½105ð1vkþ1Þþ12akð710vkþ1Þ7ek 35ð12vkÞ 56ð12vkþ1ÞakR3k 7ð12vkþ1ÞakRk5 0 ekð12vkþ1Þ 3ð12vkÞ 0 B B B B B B B B B B B B @ 1 C C C C C C C C C C C C A (A.6)
with ak¼ð7þ5vkÞð710vkþ1Þ mk mkþ1ð710vkÞð7þ5vkþ1Þ bk¼4ð710vkÞþð7þ5vkÞ mk mkþ1 ck¼ð75vkþ1Þþ2ð4þ5vkþ1Þ mk mkþ1 dk¼ð75vkþ1Þþ4ð7þ10vkþ1Þ mk mkþ1 ek¼2ð45vkÞþð7þ5vkÞ mk mkþ1 fk¼ð45vkÞþð75vkþ1Þð45vkþ1Þð75vkÞ mk mkþ1 ak¼ mk mkþ11 (A.7)
A.2. Effectiveshearmodulussolutionaccordingtothe(n+ 1)-phasemodel A mhom mn 2 þB mhom mn þC¼0 (A.8) with A¼4R10 nð12nnÞð710nnÞZ12þ20R7nð712nnþ8n2nÞZ42 þ12R5nð12nnÞðZ147Z23Þþ20R3nð12nnÞ2Z13 þ16ð45nnÞð12nnÞZ43 B¼3R10 nð12nnÞð15nn7ÞZ12þ60R7nðnn3ÞnnZ42 24R5 nð12nnÞðZ147Z23Þ40R3nð12nnÞ2Z13 8ð45nnÞð12nnÞZ43 C¼R10 nð12nnÞð5nnþ7ÞZ12þ10R7nð7n2nÞZ42 þ12R5 nð12nnÞðZ147Z23Þþ20R3nð12nnÞ2Z13 8ð75nnÞð12nnÞZ43 (A.9) where Zab¼Pðn1Þa1 P ðn1Þ b2 P ðn1Þ b1 P ðn1Þ a2 ; a;b2½1;4 (A.10)
r
e
f
e
r
e
n
c
e
s
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