www.elsevier.com/locate/mechmat
Onthemechanicalbehaviorofasphalt
J.MuraliKrishnana,K.R.Rajagopal
abb,*DepartmentofCivilEngineering,IndianInstituteofTechnologyMadras,Chennai600036,IndiaDepartmentofMechanicalEngineering,TexasA&MUniversity,CollegeStation,TX77840,USA
Received15June2004;receivedinrevisedform29July2004
Abstract
Knowledgeofthemechanicalandthermodynamicbehaviorofstraightrunasphaltisdesirableinviewofitsuseasabinderforpavementapplications.Thedifferentcomplexprocessesundergonebyasphaltconcretemixturessuchasheal-ing,aging,etc.canbeunderstoodmoreclearly,ifonehasabetterunderstandingoftheconstitutivebehaviorofasphalt.Theneedispressingastheuseofpolymer-modifiedasphaltasabinderhasincreasedinrecentyears.
MostofthestudiescarriedoutonconstitutivemodelingofasphaltmodelpureasphalteitherasaNewtonianfluidorasalinearviscoelasticfluidoverawiderangeoftemperatures.Thecomplexityrelatedtothestudyoftheconstitutivebehaviorofasphaltiscompoundedbythefactthatasphaltisamixtureofdifferentchemicalspeciessomeofwhichareamorphousandsomeofwhicharecrystallineinnature.Therelaxationmechanismsofasphaltarediversewithdifferentrelaxationmechanismsatdifferenttemperatures.Inthisstudy,weuseathermodynamicframeworkfortheconstitutivemodelingofasphaltandwemodelasphaltasamaterialwithmultiplerelaxationmechanisms.Thisframeworkrecog-nizesthefactthatmaterialslikeasphaltcanexistinmorethanonenaturalconfiguration(forinstance,stressfreecon-figuration).Weusetheexperimentaldataavailableintheliterature(Lethersich,W.,1942.Themechanicalbehaviourofbitumen.JournaloftheSocietyofChemicalIndustry61,101–108;Cheung,C.Y.,Cebon,D.,1997.Experimentalstudyofputebitumensintension,compression,andshear.JournalofRheology41(1),45–73)forasphaltfromdifferentsourcesanddemonstratetheefficacyofthemodel.Ó2005ElsevierLtd.Allrightsreserved.
Keywords:Asphalt;Constitutivemodeling;Viscoelasticity;Multiplenaturalconfigurations;Relaxation
*Correspondingauthor.
E-mailaddresses:jmk@iitm.ac.in(J.M.Krishnan),krajagopal@mengr.tamu.edu(K.R.Rajagopal).
0167-6636/$-seefrontmatterÓ2005ElsevierLtd.Allrightsreserved.doi:10.1016/j.mechmat.2004.09.005
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1.Introduction
Asphaltisacomplexheterogeneousmixtureofhydrocarbonsusuallycollectedasabyproductoftherefiningprocessofcrudeoilinpetroleumrefin-eries.Asphalthasnumerousapplicationsandoneofitsmajorusesisasabinderforaggregatemate-rialsintheconstructionofhighwaysandrunways(MuraliKrishnanandRajagopal,2003).Themechanicalpropertiesofasphaltmixturesdependtoalargeextentonthetypeandquantityofas-phaltusedandhenceitisimperativethatonedevelopsabetterunderstandingofasphalt.Modi-fiersintheformofpolymer,crumbtirerubber,fill-ers,etc.arebeingaddedtoasphaltinanattempttoimproveitsmechanicalproperties.Aseachandeverymodifiercaninteractwithasphaltinawidelydifferentmanner,thecomplexityinmodel-ingtheconstitutivebehaviorofmodifiedasphaltisincreased.However,evenbeforeoneattemptsthemodelingofmodifiedasphalt,thebehaviorofstraight-runasphaltneedstobebetterunderstood.Inthisstudy,wefocusourattentiononmodelingthemechanicalbehaviorofasphaltforaspecifictemperaturerangeinwhichitsbehaviorispre-dominantlyviscoelastic.Thereasonformodelingthebehaviorofasphaltinthisspecificrangewillbecomeclearaswediscussindetailthenatureofasphaltandthevariouscomplexmanifestationsofitsbehavior.Severalexperimentalinvestigationshavebeencarriedoutonasphalt,however,mostoftheinformationthatisrequiredtomakereason-ableassumptionsconcerningthenatureofasphaltfortheentiretemperaturerangeofinterestisstilllacking.Forinstance,reliableinformationrelatedtotheglasstransitiontemperature,influenceoftemperatureandpressureontheapparentviscos-ity,influenceofdifferentsources/processingmeth-ods,etc.onthemechanicalandthermodynamicalbehaviorofasphaltarenotavailable.
Mostofthecurrentaswellearliermodelingat-temptswithregardtoasphalthavecharacterizedasphaltasalinearviscoelasticoraNewtonianfluidandhaveinvariablyusedassumptionssuchastime-temperaturesuperposition,andhavemod-eledthemasthermorheologicallysimplefluidswhichobeytheArrheniustyperateequations.Whilethesemodelshaveservedtheirpurposeincharacterizingdistressmeasurementsforasphaltconcretetoareasonabledegree,noneofthemcouldbeconsideredasrigorousconstitutivemod-elswiththeconstitutiveequationsreflectingthecomplexityofasphalt.Someoftheattemptsde-scribethebehaviorofasphaltinapiecemealfash-ionwithdifferentempiricalequationsdescribingthematerialbehaviorfordifferenttemperaturere-gimes.Asdetailsaboutthecomplexityofasphaltarebeingrevealedbytheuseoftoolssuchasgaschromatography,massspectrometry,differen-tialscanningcalorimetry,etc.,thereisaneedtodevelopconstitutivemodelswhichhavearigorousbasisandwhichhavetheabilitytotakeintoac-counttheinformationthatisgleanedasaconse-quenceofthesesophisticatedexperiments.
OneofthepopularmodelsfordescribingthemechanicalbehaviorofasphaltisduetoBurgers(1939).SaalandLabout(1940)usedamodifiedformofBurgersÕmodeltocharacterizeasphaltasamixedgel–solsandpredictedwithreasonableaccuracycertainexperimentalresults.Lethersich(1942)usedamodelbasedonamechanicalanalogconsistingoftwospringsandtwodashpotstocharacterizetheresponseofbitumenandobservedtheneedformorethanonerelaxationtimetodescribetheresponseofasphalt.Oneoftheearli-estattemptsinderivingafullythreedimensionaltheoryforratetypeviscoelasticfluidsisduetoFrohlichandSack(1946).Motivatedbytheexper-imentalresultsonbitumenofLethersich(1942),theydevelopedconstitutivemodelsforthevis-coelasticresponseofdispersions.VanderPoel(1954)assumedthatasphaltbehavedlikealinear-izedelasticmaterialatlowtemperaturesandforshortloadingtimesandbehavedlikeaNewtonianfluidforlongloadingtimesandsufficientlyhightemperatures.Whilehisnomographsusingpene-trationindexandsofteningpointmeasurementsofasphaltareverywidelyused,useofmeasureslikeÔstiffnessÕforcharacteringtheNewtonianresponseofasphaltisinappropriate.Brodnyanetal.(1960)andGaskinsetal.(1960)inaseriesofpapersinvestigatedtherheologicalcharacteris-ticsofasphalt.Theyusedtime-temperaturesuper-positionalongwiththeWilliam–Landel–Ferry
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equations.UsingdeWaele–OstwaldÕsempiricalequations,MajidzadehandSchweyertriedtocharacterizethenon-Newtonianbehaviorofas-phalts(MajidzadehandSchweyer,1965).ThestructuraldegradationofasphaltwasmodeledbythemusingaEyringrateprocesstheory.Mostofthelaterstudiesconcerningasphaltwerededicatedtothedevelopmentofsimplelinearviscoelasticmodelsforasphaltanddatareductionwithrespecttosuchmodelswhichledtothedeterminationofthecomplexmodulus,phaseangle,etc.,obtainedbycorrelatingwithtestsonasphaltindifferentkindsofrheometers.Forinstance,theydevelopedrelationsbetweenthedynamicmodulusandphaseangle(JongepierandKuilman,1969),mastercurvesforthecomplexshearmoduluswithfre-quencyandtemperature(Dobson,1969),linearviscoelasticmodelsforthinbituminousfilmsintensionandcompression(DickinsonandWitt,1974).Thelimitationsofusingtheasumptionoftime-temperaturesuperpositionathightempera-tureswerereportedinthestudyofLesueuretal.(1996).Crystallizationandthenatureofasphalt-enespresentinasphaltwereascribed,byLesueuretal.,asthereasonsforthefailureoftimetemper-aturesuperpositionfordescribingthebehaviorofasphalt.Assumingasphalttobeadispersionofasphalteneparticlespeptizedbyresins,abimodalmodelwasdevelopedbytheseauthors.CheungandCebon(1997)usedthe‘‘Eyringplasticitymodel’’attemperaturesbelowglasstransitionwithatemperaturedependenceoftheArrheniustypeattemperaturesaboveglasstransition,andtheyalsoassumedthatasphaltobeyedtime-temperaturesuperpositionathightemperatures.
Inthefollowingsections,weelaborateonsomeoftheissuesrelatedtothemodelingofasphalt.Wethendetailathermodynamicframeworkformod-elingasphalt.Thisframeworktakesintoaccountthefactthatmaterialslikeasphaltcanexistindif-ferentnatural(stress-free)configurationsandithasbeensuccessfullyusedformodelingdiversematerialbehavior.Thefinalsectionofthispaperisrelatedtotestingtheefficacyofthepredictionsofthemodelbymakingacomparisonofthepre-dictionwithsomeoftheexperimentaldatathatisavailableintheliterature.
2.Issuesrelatedtomodelingofasphalt2.1.‘Multi-constitutent’natureofasphalt
Itisnotsurprisingthatdefiniteconclusionsregardingthemechanicalresponseofdifferentchemicalspeciesthatconstituteasphaltarenotyetinplace.Thefactthatwearedealingwithanorganicmaterialprocessedbytheearthoverseveralthousandyearsandsubjectedtoseveralmillioncyclesoftemperatureandpressureover-whelmsus.Themulti-constituentnatureofasphalthasbeenexploredtodifferentdegreesofsophisti-cationandtheproceduresrelatedtothedifferentfractionationschemesarestillevolving.Oneoftheearlieststudieswhichaddressedthemulti-con-stituentnatureofasphaltisduetoBoussingault(1837)andheclassifiedthebitumenofBechel-bronnintopetroleneandasphaltene.Nellensteyn(1924)suggestedcharacterizingasphaltasacolloidalsystemandsignificantinvestigationscon-cerningthecolloidalnatureofasphaltwerecon-ductedbyPfeifferandSaal(1940).Inasimilarveintoearlierattemptsthatwereconcernedwithunderstandingthemicro-structureofasphalt,DickieandYen(1967)conceptualizedthatasphalt-enesandresinsarerepeatingelementsofsimilarcompositionwiththedifferenceintheirchemicalstructureascribabletosolubilityandaromaticity.Therehavebeeninnumerablestudiesrelatedtofractionatingasphaltintodifferentdistinctchemi-calspeciesandthefractionationschemesduetoRostlerandWhite(1962)andCorbett(1969)deservespecialmentionhere.Alltheseschemesconcerningfractionatingasphaltintodifferentspeciesuselowboilingpointhydrocarbonsolventsanditiswellknownthatdifferentsolventsyielddifferentquantitiesofthefractions.Henceanyconclusionbasedonaspecificfractionationschemewillbearbitraryandprematureatthisstage(Goodrichetal.,1986).Inthelightoftheabovedevelopments,itisinterestingtoobserveheretheresearchconductedaspartoftheStrategicHighwayResearchProgram(SHRP)intheUnitedStatesofAmerica.TheSHRPresearcheffortcrit-icallyexaminedtheavailablecolloidalmodelsforasphaltinthelightoftheadvancesmadeinthe
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fieldofcolloidchemistryandcametotheconclu-sionthat‘‘asphaltcementisarelativelyhomoge-neousandrandomlydistributedcollectionofmoleculesdifferinginpolarityandmolecularsize’’(Petersenetal.,1994)and‘‘asphaltisasinglephasemixtureofmanydifferentpolarandnon-polarmolecules,allofwhichinteractwithoneanother’’(YoutcheffandJones,1994).TheyalsousedIonExchangeChromatographytoseparateasphaltintostrongandweakacids,strongandweakbases,neutralsandamphoterics—com-poundswithbothacidandbasefunctionalities.Thesefractionsalongwiththeparentasphaltwerealsotestedinrheometersinanattempttolinkthephysicalpropertiesofasphaltwiththevariouschemicalfractions.
Basedontheabovecitedstudies,somegeneralconclusionsaboutthenatureofasphaltcanbereached.Wecanconcludewithoutanylossofgen-eralitythatasphaltisamixtureofdifferentchem-icalspeciesandthedifferentmanifestationsofthemechanicalbehaviorofasphaltdependsontherel-ativeproportionsofeachofthesespecies.Thechangeofbehaviorofasphaltovertime,suchasaging,internalstructurechange,etc.,areduetotheinterconversionofthedifferentchemicalspe-ciesconstitutingasphalt.Theproportionofthesedifferentconstituentsaswellasthepotentialforchemicalinterconversiondependstoalargeextentonthesourceofasphalt(crudesource),thepro-cessingmethod,etc.Acompleteandrigorouscon-stitutivemodelshouldthenbeabletotakeintoaccountthemulti-constituentnatureofasphalt,theabilitytointerconvertaswellastheinfluenceofthecrudesourceonthemechanicalbehaviorofasphalt.Toconstructsuchamodelisadifficultbutachievabletaskusingtheframeworkwehaveinhand,butduetothepaucityofexperimentalinformationsuchanattemptmaynotbefeasibleatthemoment.Hence,inthisinvestigationweignoretheinfluenceofthedifferentchemicalspe-ciesthatconstituteasphaltasfarasitsmechanicalbehaviorisconcerned.2.2.Asphalttransitions
Themechanicalandthermodynamicbehaviorofasphaltisverysensitivetotheinfluenceoftem-perature.Pavementengineersandpavingtechnol-ogistsspecifydifferenttemperaturerangesforthevarioustasksrelatedtolayinganasphaltconcretepavement,forinstance,thetemperatureofasphaltfortransportandstorage,thetemperatureduringmixing,layingandcompactionoperations,etc.Severalstandardsrelatedtotherequirementsofapparentviscosityforalltheseoperationsarewelldocumentedintheliterature(AsphaltInstitute,1994).Thefactthatasphaltexhibitsdifferentmechanicalbehaviorasthetemperatureisprogres-sivelyincreased/decreasedfromaspecifictempera-tureiswellknown.Forinstance,startingfromalowtemperatureofapproximatelyÀ40°Candheatingasphaltatauniformtemperatureratetoatemperatureof+100°C,thefollowingtransi-tionsareobserved:glassysolid)viscoelasticsolid)viscoelasticfluid)Newtonianfluid.Thespecifictemperaturerangesforeachtypeofas-phaltexhibitingthesedifferenttypesofbehaviorisdifficulttopinpoint.Thisisduetothefactthatthenatureofasphaltcontinuestochangeasitissubjectedtodifferentcyclesofheatingandcooling.Hence,aparticulargradeofasphaltwhichexhibitsviscoelasticfluidbehaviorinatemperaturerangeofsay,20–40°C,mayshowthissamebehavioratamuchhighertemperaturerangeafteranum-berofcyclesoftemperatureand/orloading.Themainreasonforsuchachangeinitsresponseisthechangeintheinternalstructureofasphaltwithrespecttotime.
Differenttemperatureregimeshavebeenassoci-atedwiththedifferenttypesofmechanicalre-sponseofasphalt.Forinstance,Schweyer(1973)classifiedasphaltbehaviorinthefollowingman-ner:hightemperature(>60°C)—Newtonianfluid,near-transitionregion(between0and60°C)—vis-coelasticandfar-transitionrange(betweenglasstransitiontemperatureand0°C)—elastic.MoreinsightintothetransitorynatureofasphalthasbeengainedrecentlyduetothestudiesbyStormetal.(1996).Usingasphaltsfromthreedifferentsourcesandtestingthemoverawiderangeoftemperatures,Stormetal.,concludedthatatthetemperaturerangeof65–150°C,theseasphaltsbehavedasNewtonianfluidsandinthetempera-turerangeof25–65°C,thebehaviorwasessen-tiallyviscoelastic.Stormetal.,hypothesizedthat
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thesolvationoftheasphalteneshellsbecomeslar-gerduringthistemperaturetransitionimpartinganewmicrostructureforasphalt.Thisisinadditiontothecrystallizationthatisgenerallyconsideredasbeingresponsibleforthistransitorybehavior.Henceasthetemperatureofanasphaltsampleisvariedfromaspecificvalue,theinternalstructurechangesgivingrisetodifferentmechanicalbehav-ior.Asdiscussedearlier,thisisduetothefactthatasphaltisinessenceamixtureofdifferentchemicalspecies,eachofthemexhibitingsolid-likeorfluid-likecharacteristicsasthetemperatureisvaried.Onecanalsoviewasphaltasamixtureofamor-phousandcrystallinephasesandtheinfluenceoftemperatureisinthemeltingofcrystallinephasesasthetemperatureisincreasedorintheformationofcrystallinephasesasthetemperatureisde-creased.Differentstudieshaveaddressedtheissuesrelatedtotheexistenceofamorphousandcrystal-linephasesinasphaltandtheirroleinthetransi-torynatureofasphalt.Mostofthesestudieshaveconcludedthatthelowtemperatureproper-tiesofasphaltbindersdependtoalargeextentontheamountofcrystallizablefractionsandtheglasstransitiontemperatureofasphalt.Forin-stance,increasedcrystallizedfractionsinasphaltleadtoreducedductility,reducedadhesiontothemineralaggregatesandincreasedbrittlenessatlowtemperatures.Unliketheliteratureconcerningpolymers,therehavebeenfewstudiesdevotedtounderstandingthecrystallizationkineticsofas-phalt.Someofthestudieswhichhaveconcernedthemselveswiththeroleofamorphousandcrystal-linefractionsofasphaltsincludeSmithetal.(1966),NoelandCorbett(1970),GiavariniandPochetti(1973),Albertetal.(1985),Claudyetal.(1992),Dalyetal.(1996),Netzel(1998),Michonetal.(1999),Massonetal.(2002),andEdwardsandRedelius(2003).
Thefactthatasphaltsexhibitmorethanoneglasstransitiontemperaturehasbeenrecordedindifferentexperimentalinvestigations(Chambrionetal.,1996;MassonandPolomark,2001;Massonetal.,2002).Fromaninitialcondition,differentcoolingratesinducedifferentinternalstructuralchangesinasphaltresultinginglasstransitionswhichareseveralordersapart.Massonandco-workers(2001,2002)investigatedthemicrostruc-tureofbitumenbymeansofmodulatedDSC(MDSC)andcametotheconclusionthatthede-greeofpolymerizationofasphaltsisoftheorder10andhenceasphaltcouldbeclassifiedasanoli-gomer.OnthebasisofMDSCofbitumenwithdif-ferentannealinghistory,thesestudiesconcludedthatasphalthadamesophasestructuresimilartothatencounteredinliquidcrystals.
Tosummarizetheabovediscussiononthetran-sitorynatureofasphalt,wecanconsiderasphaltasamixtureoftwocomplexamorphousphasesatroughly100°C.WehastentoaddherethatbyphasewemeanÔahomogeneous,physicallydistinctportionofmatterÕ.Asthetemperatureisreduced,onephaseofthismixturestartscrystallizingwhiletheotherremainsintheamorphousphase.Atthefieldservicetemperatureofinterest,asphaltcanbeassumedtoconsistofamorphousandcrystallinephases.Theviscoelasticbehaviorofasphaltthenisinfluencedbythevolume/massfractionsofthesedifferentphasesandthetendencyforthecrystal-linephasetoeitherdissolveorsolidifydependinguponwhetherthepavementtemperatureincreasesordecreases.Asofnow,therearenotenoughexperimentaldataavailablediscussingtheinter-conversionbetweenamorphousandcrystallinephasesasasphaltissubjectedtoawiderangeoftemperaturesandloading.Also,thecrystallizationkineticsofasphaltisnotwellunderstood.Forinstance,itisnotclearastowhatcausesthecrys-tallizationofsomefractionsofasphaltasthetem-peratureisreducedorwhattheinfluenceofthedifferentasphaltsourcesandprocessingmethodsareonthecrystallization.Wecanmakereasonableassumptionsontheconditionsfortheinitiationofcrystallizationandtheevolutionforthegrowthofthecrystallizedfractionsonlywhenwehaveanan-swertotheabovequestion.Inthisinvestigation,weassumethattheamorphousandcrystallinephasesgiverisetodifferentrelaxationmechanismsandhencewewillmodelasphaltwithmultiplerelaxationmechanisms.Weignoretheactualprocessofcrystallization/dissolutionofthecry-stallinephaseasthetemperatureisvariedandalsothepresenceofaninterfacialphasewhichcanbeconsideredtohavecharacteristicsthatliesinbetweenthatfortheamorphousandcrystallinephases.
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2.3.InternalstructuralchangeofasphaltwithtimeThecomplexresponseofasphaltisdirectlyre-latedtotheevolutionofitsinternalstructureintheabsenceofexternalforces.Theinternalstruc-tureofasphaltcandevelopandevolveifitisleftundisturbedataconstanttemperature.Thisinter-nalstructuredevelopsrapidlyatfirstandthenevolvesinanasymptoticmannerovertime.Essen-tially,thisevolutionofinternalstructurecancon-sistofareversibleandanirreversibleportion.Theirreversibleportioncanbeascribedtotheagingofasphaltthatresultsinthelossofchemicalspeciesduetotheevaporation/volatilizationanddependsonthetemperatureatwhichasphaltisheld.Italsodependsonthesourceofasphalt,theÔconsistencyÕofasphaltandthespecificfractionalcompositionofasphalt.ThemoreinterestingchangeintheinternalstructureisduetothereversibleportionandissimilartotheÔphysicalagingÕofpolymers(Hutchinson,1995).Thisspecificphenomenonisthechangeintheproperty(density,mechanicalre-sponsecharacteristics,dielectricproperties,etc.)ofasphaltwhenmaintainedataconstanttempera-tureforconsiderabletimeintheabsenceofanyexternalforcesandwithoutanyappreciablechangeinitschemicalcomposition.Thermaland/orexternalforcesactingonasphaltcanreverttheinternalstructuretotheoriginalconditioninwhichitexisted.Asofnow,twodifferentkindsofreversibleinternalstructuralchangehavebeenidentified.Thefirstoneoccursatroomtempera-tureandisanextremelyslowprocesstakingfromdaystoweekstoreachequilibriumconditions.Thesecondonewhichisobservedattemperaturesnearglasstransitionforasphaltismuchmorerapidandexperimentalinvestigationshavereportedthatittakesnormally1–2daysatthetemperaturerangeofÀ15toÀ35°C(LuandIsacsson,2000).HencethesimpleArrheniuskindofrelationshipbetweenratesandtemperaturesdonotapplyforasphalt,asaccordingtothisassumptiononewouldexpectfasterhardeningrateathighertemperatures(Mas-sonetal.,2002).Severalstudieshaveinvestigatedthereversibleinternalstructuralchangeofasphalt(seeforinstanceHubbardandReeve,1913;Hub-bardandPritchard,1916;TraxlerandSchweyer,1936;TraxlerandCoombs,1937;TraxlerandCoombs,1938;Pendleton,1943;Brownetal.,1957;Stinsky,1975;BahiaandAnderson,1992;Claudyetal.,1992;Huangetal.,1999;LuandIsacsson,2000).Thisreversiblechangeintheinter-nalstructurehasbeenidentifiedatroomtempera-tureasÔsterichardeningÕ(Brownetal.,1957)andatglasstransitiontemperatureasÔlowtemperaturephysicalhardeningÕ(BahiaandAnderson,1992).Theneedforsubjectingasphalttoidenticaltestconditionsbeforestartinganykindofexperimen-tationisduetothischangeininternalstructurethatoccurs,asanysuchdeviationmayresultinmeasurementofpropertiesofasphaltfromatran-sientconfigurationandnotfromitsnaturalconfig-uration.Forinstance,Traxlerandcoworkers(1936,1937)whilemeasuringtheviscosityofsev-eraldifferentasphaltsinfallingcoaxialcylinderviscometersnoticedthattheviscositiesofasphaltkeptintheviscometersforconsiderabletimeexhibitedincreasedviscosity.Traxleretal.,as-cribedthisstructureformationduetothetwo-phasenature(asphalteneandpetrolene)ofasphaltinwhichagradualisothermalsol–geltransforma-tionoccursasasphaltiskeptsteadyataspecifictemperature.SimilarresultswerereportedbyBrownetal.(1957)whochosetocallthisphenom-enonsterichardening.Brownetal.,attributedtheformationofinternalstructuretotheasphaltenefractionofasphalt.Thechangeintheinternalstructureofasphaltwhenheldforsufficienttimeneartheglasstransitiontemperaturehasbeencharacterizedasthatduetolowtemperaturephys-icalhardeningintheworkofBahiaandAnderson(1992).TheytriedtoexplainthisbehaviorduetothecollapseofÔfreevolumeÕasasphaltpassesthroughglasstransition(BahiaandAnderson,1992).Claudyetal.(1992)investigatedthelowtemperaturephysicalhardeningofasphaltandconcludedthatmolecularagglomerationsofcrys-tallinephasesatlowtemperaturecouldbeonerea-sonforthisbehavior.TheywerealsothefirsttoobservethatÔspinodaldecompositionÕaphenome-noninwhichaÔhomogeneousÕliquidseparatesintotwoliquidphasesasthematerialiscooled(Hilliard,1970)ischaracteristicofasphaltandisanotherpossiblereasonforthelowtemperaturephysicalhardeningofasphalt.Continuingontheselines,Massonetal.,ascribedafourstageinternal
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structuraldevelopmentprocessforasphalt(Mas-sonetal.,2002).Eachofthefractionsofasphaltinfluenceintheirownway,dependingontheirtendencyforcrystallization,intheformationofreversibleinternalstructure(seeMassonetal.,2002).
Withinthecontextofwhatwehavediscussedintheearliersections,itisclearthat(a)themechan-icalbehaviorofasphaltinthetemperatureregimeofinterestisquitecomplicatedandnotwellunder-stood,(b)acomprehensivetheoryformodelingasphaltevenwiththelimitedinformationthatisavailableislackingand(c)eachandeveryfacetofthemodelingofthemechanicalbehaviorofas-phaltrequiresfusingideasfromphysicsandchem-istry.Arigorousframeworkformodelingasphaltwiththeabilitytotakeintoaccountthemicro-structuraldetailsandreflecttheminamacroscopicsense,islacking.Suchaframeworkshouldtakeintoaccountthefactthattheinternalstructureofasphaltevolveswithtime(inthepresenceorotherwiseofexternalinfluencessuchasload,tem-perature,etc.).Inthisstudywetakeafirststepinthisdirection.Wevisualizeasphaltasamaterialpossessingmultiplerelaxationmechanisms.Thesedifferentrelaxationtimescouldbeassociatedwiththeamorphousandcrystallinephasesandthetypeandchangesintheinternalstructure,i.e.,theinter-actionbetweenthephases.Weassumethateachoftheserelaxationmechanismscanbemodeledbyaratetypeviscoelasticfluidmodelwithmultiplesetsofnaturalconfigurations.Thekeyelementoftheframeworkthatweuseisthatabodycanexiststressfreeinnumerousnaturalconfigurations.Theunderlyingnaturalconfigurationscanchangeduringanyprocesstowhichthebodyissubjectedtoandthechangeoftheinternalmicrostructureiscapturedbythisevolutionofthenaturalconfigu-ration.Theresponseofthebodyfromthesenatu-ralconfigurationsiselasticwhensubjectedtoexternalforces.Thisparticularframeworkhasbeenusedformodelingdifferenttypesofphenom-ena,forinstance,multi-networktheory(Raja-gopalandWineman,1992),plasticity(RajagopalandSrinivasa,1998a,b),crystallizationofpoly-mers(RaoandRajagopal,2000;RaoandRajag-opal,2001;RaoandRajagopal,2002),solidtosolidphasetransition(RajagopalandSrinivasa,
1999),viscoelasticliquids(RajagopalandSrini-vasa,2000),anisotropicfluids(RajagopalandSrinivasa,2001),andgrowthofbiologicalmateri-als(HumphreyandRajagopal,2002;Raoetal.,2003).WerefertheinterestedreadertoMuraliKrishnanandRajagopal(2003,2004a,b)withregardtotheapplicationofthistheorytoasphaltmixtures.
3.Modelingofasphalt3.1.Preliminaries
ConsiderabodyBinaconfigurationjRðBÞ.Weshall,fortheeaseofnotationrefertothecon-figurationasjR.LetXdenoteatypicalpositionofamaterialpointinjR.Letjtbetheconfigurationatatimet,thenthemotionvjaparticleRassignstoeachpar-ticleinconfigurationjRintheconfigura-tionjtattimet,i.e.,x¼vjRðX;tÞ.
ð1Þ
ThedeformationgradientFjRisdefinedthroughFovjjR
RoX
.ð2ÞTheleftandrightCauchy-GreenstretchtensorsBjRandCjRaredefinedthroughBjRFjTRFjR;ð3ÞCjRFTjRFjR.
ð4Þ
Nowthebalanceofmassisgivenbyq
_þqdivv¼0;ð5Þ
whereqisthedensityandvisthevelocity.Exper-imentalstudiesonthecompressibilityofasphalthavepointedoutthatthechangeinthedensityisoftheorderofonly1.5%undernormaltempera-turesandpressures(MehrotraandSvrcek,1987)andhenceinthisstudy,weassumeasphalttobeincompressible.Inthelightoftheassumptionofincompressibility,thebalanceofmassreducestodivv¼0.
ð6ÞTheqbalanceofov
ot
þðrvÞvlinearmomentumis¼divTþqg;ð7Þ
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whereTistheCauchystressandgistheaccelera-tionduetogravity.Foranincompressiblemate-rial,theCauchystresstensorTreducestoT¼ÀpIþTE;
ð8Þ
wherepistheLagrangemultiplierduetothecon-straintofincompressibilityandTEistheconstitu-tivelydeterminedextrastress.Thebalanceofangularmomentumforabodyintheabsenceofinternalcouplesimpliesthatthestresstensorissymmetric.Intheformulationoftheconstitutiveequationsderivedinthisstudy,thereduced-dissi-pationequationisusedandforisothermalcondi-tionsitisgivenbyGreenandNaghdi(1977),RajagopalandSrinivasa(1999)TÁLÀqw
_¼qhfnP0;ð9Þ
wherewistheHelmholtzpotential,fistherateof
entropyproductionandnistherateofdissipation.Inthepresentattemptweignoreradiation(MuraliKrishnanandRajagopal,2003).
3.2.Modelingasphaltwithmultiplerelaxationmechanisms
Wemodelasphaltasaratetypeviscoelasticfluid.WefollowthemethodologyofRajagopalandSrinivasa(2000)fordevelopingconstitutiverelationsforthestress.ReferringtoFig.1,jRisareferenceconfiguration,jc(t)istheconfigurationcurrentlyoccupiedbythematerialandjp(t)isthepreferrednaturalconfigurationoncethetractionsonjc(t)areremoved.Itispossibleforthematerialtopossessmorethanonenaturalconfigurationandweassumethatwecanassociatethenaturalconfigurationswithdifferentrelaxationmecha-nisms.Letusforthesakeofdiscussionassumethatwehaveanasphaltsampleat100°Candwecoolthesampletoroomtemperatureataratesuchthatthecrystallinephasedevelops.Ifweleavethissampleatroomtemperatureforsufficienttime,therelaxationofasphaltwillbeduetothetwodiffer-entphases,theamorphousandcrystallinephases.Duetothefactthattheinternalstructureofas-phaltevolvesoveraperiodoftime(Ôstericharden-ingÕ),therecanbeonemorerelaxationtimeassociatedwiththischangeandthisrelaxation
FκRκc(t)Fκp(t)κRGκp(t)Fig.1.Naturalconfigurationsassociatedwithasphalt.timewillbeoftheorderofdays.Hence,thecon-figurationjp(t)inthiscasecanbethoughtofbeingmadeupofabodywithwhichwecanassociatethreedifferentrelaxationtimes,onepertainingtotheamorphousphase,onetothecrystallinephaseandtheotherthatisrelatedtothetimescaleofthesterichardeningatroomtemperatures.Thesamematerialcanexhibitdifferentrelaxationmecha-nismsifitiscooledtoneartheglasstransitiontemperature.Totakeintoaccountallthesepossi-bilities,weassumethatasphaltpossessesmultiplerelaxationmechanisms,eachofthemtriggeredbyadifferentphysical/chemicalprocess.Wealsoas-sumethattheresponsefromeachofthesenaturalconfigurationsiselastic.ReferringtoFig.1,FjRdenotesthemappingbetweenthetangentspaceassociatedwithjR,atapointinthereferencecon-figurationtothetangentspaceassociatedwiththesamematerialpointinjc(t).FjpmappingbetweenthetangentspaceiðtÞreferstotheassociatedwiththeconfigurationjpiðtÞ,atamaterialpointtothetangentspaceassociatedwiththecurrentconfigurationjc(t)atthesamepoint.HeretheindexÔiÕrangesfromÔ1,...,nÕwhereÔnÕsignifiesthenumberofrelaxationmechanisms.WealsodefinethefollowingmappingGibetweentheappropriatetangentspacesofjRandjpiðtÞ,
J.M.Krishnan,K.R.Rajagopal/MechanicsofMaterials37(2005)1085–1100
1093
GiFjj1
R!pðtÞ¼FÀjpðtÞFjR;
i¼1;2;...;n.ð10Þ
i
WedefinethevelocitygradientLjpofLDiðtÞandthesym-metricpartjpiðtÞ;jpiðtÞasfollows:Ljp_iðtÞ¼GiGÀ1i
;D1jpiðtÞ¼
2
LjT
piðtÞþLjpðtÞ;i¼1;2;...;n.ð11Þ
iNow,onecanviewthetensorsBjpinformationaboutthedeformationsiðtÞascontainingoftheamor-phousandcrystallinephasesaswellasthespecificinternalstructuralchangesthattakeplaceasthematerialisunloaded.Also,astherecentstudiesofMassonandPolomark(2001),Massonetal.(2002)havesuggestedasphaltcouldhaveamicro-structuresimilartoliquidcrystals,onecouldalsomodelasphaltasananisotropicfluid.Inthatcase,howthestoredenergydependsonthetensorBjpinformationabouttheanisotropyofas-iðtÞ,containsphalt,ofcourseinamacroscopicsense.Nowitcanbeshownthat(seeRajagopalandSrinivasa,2000fordetails)
O
BjpiðtÞ
B_jpiðtÞÀLBjpiðtÞÀBjpiðtÞLT¼À2FjTpiðtÞDjpiðtÞFjpðtÞ;
i¼1;2;...;n;ð12Þ
i
wheretheinvertedtriangleistheÔupperconvectedÕ
Oldroydderivativeandthesuperposeddotsigni-fiesthematerialtimederivative.Sincewehaveas-sumedasphalttobeincompressible,themotionsassociatedwiththenaturalconfigurationsareiso-choricandhencetrðDjpiðtÞÞ¼0;
i¼1;2;...;n.
ð13Þ
WeassumethefollowingformfortheHelmholtzpotential:w¼wðIi;IIiÞ;i¼1;2;...;n;
ð14Þ
whereIi¼trðBjpiðtÞÞ;IIi¼
trðB2jpiðtÞÞ;
i¼1;2;...;n.
ð15Þ
Sincethematerialisassumedtobeisotropic,theconfigurationsjpiðtÞcanbechosensuchthatFjpiðtÞ¼VjpiðtÞ;
i¼1;2;...;n;
ð16Þ
whereVjpiðtÞ;i¼1;2aretherightstretchtensors
inthepolardecomposition.Wealsoassumethefollowingformfortherateofdissipation:n¼nðBjpiðtÞ;DjpiðtÞÞ;
i¼1;2;...;n.
ð17Þ
Also,wecanassumethatformotionswherethenaturalconfigurationsdonotchange,therateofdissipationiszeroandhence,n¼nðBjpiðtÞ;0Þ¼0;
i¼1;2;...;n.
ð18Þ
SubstitutingtheformfortheHelmholtzpotential(Eq.(14))intothereducedenergydissipationequation(Eq.(9)),andusingEqs.(12)and(16),we\"get,TÀXn2qow#i¼1
oIB2owjB2ipiðtÞþoIIijpiðtÞÁD
þ
Xn2qowBowj2piðtÞþ2Bjpi¼1
oIðtÞÁDjioIIiipiðtÞ¼nP0.
ð19Þ
Sincewearelookingatformssufficienttosatisfy
theaboveequation,itisreasonabletoassumethatthestressisgivenby
T¼Àp1þXn2qowBowj2
piðtÞþ2Bjpi¼1oIioIIiðtÞ.ð20Þ
iFromtheformsassumedfortheHelmholtzpoten-tialandtherateofdissipation,itisclearthat
X
nTiÁDjpiðtÞ¼nP0;ð21Þ
i¼1
whereTiisgivenby
Towowi¼2qoIBjiðtÞþ22
oIIBp;
ipijiðtÞ
i¼1;2;...;n.
ð22Þ
Eq.(21)placesrestrictionsonthetensors
DjpiðtÞ;i¼1;2;...;n.FollowingRajagopalandSrinivasa(2000),weassumethattheactualvaluesofDjpiðtÞ;i¼1;2;...;nchosensatisfytherestric-tionsgivenbyEqs.(21)and(13)alsocorrespondstothemaximumrateofdissipation.ThisiscarriedoutbyextremizingtherateofdissipationusingthemethodofLagrangemultiplierssubjecttothe
1094J.M.Krishnan,K.R.Rajagopal/MechanicsofMaterials37(2005)1085–1100
constraintsgivenbyEqs.(21)and(13)(fordetailsthereadersarereferredtoRajagopalandSrinivasa(2000)).Fromtheaboveprocedure,wegetTon
i¼c1i
oDÀc2i1¼0;i¼1;2;...;n;
ð23Þ
jpiðtÞ
wherec1iandc2iareLagrangemultipliers.Nowforthespecificproblemunderconsideration,weneedtoprescribeconstitutiveassumptionsfortheHelmholtzpotentialandtherateofdissipation.WeassumethattheHelmholtzpotentialisofthefollowingform:
w¼1Xn2qliðIiÀ3Þ;ð24Þ
i¼1whichimpliesthattheinstantaneouselasticre-sponseisthatofaneo-Hookeanelasticsolid.Wealsoassumethattherateofdissipationisofthefollowingform:
n¼
X
ngiDjpiðtÞÁBjpiðtÞDjpiðtÞþgiDÁD.ð25Þi¼1
Intheaboveequation,listheshearmodulusand
g,garetheviscosities.SubstitutingEqs.(24)and(25)intoEq.(9)andsimplifyingusingEqs.(16),(12),wegetthefollowingrepresentationforstress:
T¼Àp1þ
X
nlDiBjpiðtÞþgi;ð26Þi¼1
andthefollowingevolutionequationforBjpiðtÞ:1O\"#
2Bl3jpiðtÞ¼ig1ÀBjpitrðBÀ1iðtÞ;jpi
ðtÞ
Þi¼1;2;...;n.ð27Þ
Thiscompletesthedevelopmentofthemodelforasphalt.Intheremainingpart,weusethismodelandexaminetheefficacyofitspredictionsvis-a-vissomeoftheexperimentalresultsavailableintheliterature.4.Applications
Inthissection,weinvestigatetheapplicationtotwodifferentkindsofdeformations.Thefirstisre-latedtoaconstantextensionratetestandthesec-ondisrelatedtoaconstanttensilestresstest.
Assumingthedeformationtobehomogeneous,thekinematicsofdeformationincylindricalpolarcoordinatesystemforboththesecasesare,
r¼pffiffiffiffiffiffiffiffiffi11
KðtÞR;h¼pffiffiffiffiffiffiffiffiffiKðtÞH;z¼KðtÞZ.
ð28ÞHereapointinthereferenceconfigurationisde-notedby(R,H,Z)andthesamepointinthecur-rentconfigurationisdenotedby(r,h,z)andK(t)denotesthestretchalongtheZdirection.Thedeformation gradientforthismotion!
isgivenby
F11
jR¼diagpffiffiffiffiffiffiffiffiffiKðtÞ;pffiffiffiffiffiffiffiffiffiKðtÞ;KðtÞ;ð29Þ
andthusB¼diag
11j2
R
KðtÞ;KðtÞ;KðtÞ.ð30Þ
ThecomponentsofBjpðtÞareassumedtobe
diag1BðtÞ;1BðtÞ;B2ðtÞ.Thisassumptionisconsistentwiththestipulationthatthestress-freestateforthematerialisachievedviaamotionoftheformgivenbyEq.(28).
Forthecurrentproblem,weassumethatas-phalthasasinglerelaxationtime.Thisessentiallymeansthatthereisasinglenaturalconfigurationassociatedwithasphalt.Theconstitutivemodeldetailedintheearliersectionisverygeneralinnat-ureandtakesintoaccountthedifferentrelaxationmechanismspossibleforasphalt.However,sincethedetailsrelatedtothevolume/massfractionsofamorphousandcrystallinephasesaswellasthespecificinfluenceoftheinternalstructurechangesuchassterichardeningonthemechanicalbehaviorofasphaltarenotavailablewithregardtotheexperimentalstudiesthatweshalluseforcorrelation,itisproposedtosolvetheproblemre-latedtotheabovetwodeformationswithasinglerelaxationtime.Forthecasewithasinglerelaxa-tiontime,theconstitutiveequationisgivenbyT¼Àp1þlBjpðtÞþgD;
ð31Þ
andtheevolutionequationforthenaturalconfig-urationisgiven1Ol2by
3
3
2BjpðtÞ¼g4trB
À11ÀBjpðtÞ5.ð32ÞjpðtÞ
J.M.Krishnan,K.R.Rajagopal/MechanicsofMaterials37(2005)1085–11001095
Fortheproblemunderconsideration,thestressesinthelateraldirectionsarezeroandhencethestressinthezdirectionisgivenasfollows:gðDzzÀDrrÞ.Tzz¼lðBzzÀBrrÞþ
ð33Þ
Theevolutionequationforthenaturalconfigura-tioninthezdirectionisgivenby
oBzzoBzz2l2BzzðBrrÀBzzÞ
þvzz¼þ2LzzBzz.
g2BzzþBrrotoz
ð34Þ
4.1.Constantextensionrateexperiments
Intheconstantextensionrateexperiments,nor-mallyaÔdumb-bellÕshapedspecimenismounted
betweenthecross-headsandthecross-headsarethenpulledataconstantspeed.Insuchacase,thestretchisgivenasfollows:KðtÞ¼1þKt;
ð35Þ
whereKisaconstant.Thevelocitygradientforthismotionisgivenby
À1KÀ1KK
;;.L¼diag
21þKt21þKt1þKt
ð36Þ
Thevelocitygradient,LisdiagonalandhenceDthesymmetricpartofthevelocitygradientisthesameasL.Also,weassumethefollowingformfortheviscosities:
hmi
g¼g0NtrBjpðtÞÀ3þ1expðbÞ;ð37Þg¼g0.
ð38Þ
Theinitialconditionforthismotionisgivenasfollows:BjpðtÞ¼1;
fort¼0.
ð39Þ
Withthisinitialcondition,Eq.(34)issolvedandthestressintheasphaltspecimenisgivenby(33).ExperimentalstudiesonbitumenconductedbyCheungandCebon(1997)arecomparedwiththepredictionsofthemodelinFig.2anditisseenthatthemodelpredictstheexperimentalobserva-tionsreasonablywell.
Constant extension rate test for Bitumen at 10 °C2.00 X 1061.80 X 1061.60 X 1061.40 X 106ModelNominal Stress (Pa)1.20 X 1061.00 X 106Experiment0.2/s0.1/s0.05/s0.02/s0.01/s8.00 X 1056.00 X 1054.00 X 1052.00 X 100.00 X 105000.10.20.30.40.50.6Nominal StrainFig.2.Comparisonoftheconstantextensionratetestsat10°CofbitumenB1reportedinCheungandCebon(1997)withmodelpredictions.Themodelparametersusedarel=5MPa,g0=6.2MPas,g0¼0.75MPas,N=0.75,m=0.5,b=0.55.1096J.M.Krishnan,K.R.Rajagopal/MechanicsofMaterials37(2005)1085–1100
4.2.Tensilecreepexperiments
Intensilecreeptesting,atensileforceisappliedÔinstantaneouslyÕtothespecimenandmaintainedconstantuntiltheendofthetest.Onevariationofthistestisthatinwhichtheforceisreleasedattheendofaspecifictimeperiodandtherelaxa-tionofthematerialismeasured.Thevelocitygra-dientforthismotionisgivenby
__K_1K1K
L¼diagÀ;À;.ð40Þ
2K2KKDuetothefactthattheasphaltusedinthetensile
creepexperiments(blownbitumen)iscompletelydifferentwhencomparedwiththeasphaltusedintheconstantextensionratetesting(steamdistilledbitumen),theformfortheviscositygchosenisdif-ferentandisgivenas
hmi
ð41Þg¼g0NtrBjpðtÞÀ3þ1.
TheinitialconditionsforthismotionisBzz¼Kð0Þ;Brr¼Bhh¼
1
.Kð0Þ
2
ð42Þð43Þ
Theappropriateinitialconditionsherearearrivedatbyrecognizingthefactthataninstan-taneousapplicationofforceelicitsanelasticresponsefromthematerialwiththevalueofstretchattimet=0beingK(0).AsbeforeEq.(34)issolvedinconjunctionwithEq.(33)withthevelocitygradientgivenbyEq.(40).
Figs.3–5showthepredictionsofthemodelandthecomparisonoftheresultswiththeexperimen-talobservationsofLethersich(1942).ItistobepointedoutherethattheseclassicalexperimentalobservationsofLethersich(1942)werethemotiva-tionforthedevelopmentofafullythreedimen-sionaltheoryforratetypeviscoelasticfluidsbyFrohlichandSack(1946).
Extension/time curves for Bitumen25Temperature 30°CModel19.8 g/cm2 - Experiment35 g/cm2 - Experiment20Extension in % 151050010203040506070Time in daysFig.3.Comparisonofthetensilecreeptestsat30°Cofblownbitumenno.2reportedinLethersich(1942)withmodelpredictions.Themodelparametersusedarel=11.2MPa,g0=1.6416·106MPas,g0¼9.936Â104MPas,N=10,m=3.J.M.Krishnan,K.R.Rajagopal/MechanicsofMaterials37(2005)1085–11001097
Extension/time curves for Bitumen25Temperature 35°CModel12.8 g/cm2 - Experiment25 g/cm2 - Experiment20Extension in %151050010203040506070Time in daysFig.4.Comparisonofthetensilecreeptestsat35°Cofblownbitumenno.2reportedinLethersich(1942)withmodelpredictions.Theg0¼9.936Â104MPas,N=10,m=3.modelparametersusedarel=7.8MPa,g0=1.296·106MPas,Extension/time curves for Bitumen25Temperature 40°CModel8.45 g/cm2 - Experiment20.8 g/cm2 - Experiment20Extension in %151050010203040506070Time in daysFig.5.Comparisonofthetensilecreeptestsat40°Cofblownbitumenno.2reportedinLethersich(1942)withmodelpredictions.Themodelparametersusedarel=6.2MPa,g0=6.696·105MPas,g0¼9.936Â104MPas,N=10,m=3.1098J.M.Krishnan,K.R.Rajagopal/MechanicsofMaterials37(2005)1085–1100
5.Conclusions
Inthispaper,wehavedevelopedaframeworkfortheconstitutivemodelingofstraightrunas-phalt.Theconstitutivemodelisdevelopedbasedontheframeworkofevolvingnaturalconfigura-tions.BychoosingspecificformsfortheHelm-holtzpotential,rateofdissipationandusingthereducedenergy-dissipationequationandmaximi-zationoftherateofdissipation,wehaveobtainedconstitutiverelationsforthestressandtheevolu-tionequationsfortheunderlyingnaturalconfigu-rations.Wehaveassumedasphalttohavemultiplerelaxationmechanismsandincorporatedeachoftheserelaxationmechanismsintoaratetypevisco-elasticfluidmodel.Weillustratedtheefficacyofthemodelbycomparingitspredictionswithexper-imentalobservationsoftwodistinctasphalts.Therecentuseofmodifierstoasphaltintheformoffillers,polymers,etc.haveincreasedthecomplexityofthemechanicalandthermodynamicbehaviorofasphalt.Theinfluenceofthesemodifi-ersontheinitiationofcrystallization/dissolutionofdifferentfractionsinasphaltaswellasonthechangeinthetransitorybehaviorneedstobechar-acterizedclearly.Also,theagingofasphaltandtheaddedchangeinthebehaviorduetotheintercon-versionofdifferentchemicalspeciesthatconstituteasphaltneedstobetakenintoaccountifoneisinterestedinpredictingthelongtermbehaviorofasphaltpavements.Theframeworkthatwehaveusedinthisstudycomesinhandyformodelingsuchprocessesandthisstudyisthefirststepinthedirectionofdevelopingacomprehensivether-modynamicframeworkformodelingasphalt.
Acknowledgment
WethanktheNationalScienceFoundationforitssupportofthiswork.
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