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Review of the uses and modelingof bitumen from ancient to modern times

J Murali Krishnan and KR RajagopalDepartment of Mechanical Engineering, Texas A&M University, College Station TX 77845;[email protected], [email protected]

In this article, we provide a review of the uses of asphalt from ancient to modern times andwe discuss the main attempts at mathematically modeling its behavior. We start with a discus-sion of the various definitions for asphalt and bitumen and discuss briefly some of the manyfascinating uses to which they have been put from ancient to modern times. In Section 3 wesurvey the various attempts at unraveling the chemical structure of asphalt. The description ofits physical characteristics, the early constitutive models that have been employed to describeits mechanical response, and the numerous tests to assess its properties are dealt with in Sec-tion 4. The next section is devoted to a discussion of the aging of asphalt. In Section 6, after acursory discussion of the more classical models for asphalt concrete, we introduce the readerto a thermodynamic framework that has been recently put into place for bodies that have mul-tiple natural configurations and use this framework to develop a model for describing the me-chanical response of asphalt concrete. We use this model to describe various experiments thathave been carried out on asphalt concrete and we find that the predictions of the model agreewith experiments sufficiently well. This review article cites 580 references.@DOI: 10.1115/1.1529658#

1 INTRODUCTION 1

1.1 What are bitumen and asphalt?

The use of asphalt~bitumen!2 by the hunter-gatherer to fash-ion hunting implements out of stick and stone pre-dates re-corded civilization@1#. That bitumen’s widespread use datesback to the Palaeolithic times has been established beyond ashadow of doubt by the use of some of the most moderntools available to science: gas chromatography and massspectrometry@2–4#. During the Paleolithic era that extendedprior to 10,000 BC human societies were savages3 and eventhese savages recognized the usefulness of bitumen as anadhesive to fix flint onto wooden handles and to fashionother tools. According to Connan@2#, bitumen was the adhe-sive of choice to stick flint to wooden handles until theNeolithic times and he cites the unearthing of plentifulsamples of such tools in Syria, Israel, and the Northwesternfrontiers of the Indian subcontinent that date back to 8900BC. Much more striking is his reference to the use of suchtools from the Hummalian period around 180,000 BC. Ascraper and a Levallios flake were discovered in Syria thathave been conclusively dated by the use of gas chromatog-raphy and mass spectroscopy as having been used about40,000 BC@3,4#.

How did the savage recognize the adhesive nature oftumen? Marschner, Duffy, and Wright@6# offer a rather in-triguing explanation: ‘‘At hundreds of places on the facethe earth, bitumen seeps to the surface along with water fthe deep underground. Presumably the black viscous tareither petroleum in migration or something that would habecome petroleum if it had not escaped. Most of the seepwashes away, but some collects to thicken to pitch or sointo surrounding silt or sand and sets to natural asphalt. Sdeposits are hard in winter, but in the summer heat theymelt into hazardous sticky masses. Unwary animals attrato the water become caught in the bitumen and attract pretors, which are entrapped in turn. Perhaps, early mannessed such struggles, became intrigued by bitumen,gradually learned how to use it.’’~See also@7#.!

Amongst the earliest of construction materials, asphwas used aplenty in all the early civilizations: BabylonMesopotamia, Sumeria, Chaldea, Mohenjo-daro aHarappa, Phoenicia, China, and later in Greece and Roand unlike other construction materials its use has becoever more common. Finding ever-increasing use in thelikeliest of applications, asphalt has become indispensa

ASME Reprint No AMR344 $42.00Appl Mech Rev vol 56, no 2, March 2003 14

Transmitted by Associate Editor JN Reddy

1This article is a redacted version of a book that is to appear on this subject matter.2We shall use the term asphalt and bitumen interchangeably. We discuss in some detail the etymology and usage of these words later.3Childe@5# defines savages as those that ‘‘live exclusively on wild food by collecting, hunting, or fishing.’’ We shall use the term savage in the above sense, ascribing no derogatorymeaning to the term.

© 2003 American Society of Mechanical Engineers9

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150 Murali Krishnan and Rajagopal: Bitumen from ancient to modern times Appl Mech Rev vol 56, no 2, March 2003

as it is the material of choice for the construction of roaways and runways that serve as the circulatory system forworld economy.

Though the main emphasis of this review article concethe various attempts at describing the behavior of asphalcontinued use from prehistory demands a short discussiothe history of the use of asphalt. Before getting into a dcussion of such a history, it would serve us well to first gaan understanding of what we mean when we use the tasphalt.

According to the Oxford English Dictionary@8#, asphalt is‘‘A bituminous substance, found in many parts of the wora smooth, hard, brittle, black or brownish-black resinomineral, consisting of a mixture of different hydrocarboncalled also mineral pitch, Jew’s pitch and in the Old Tesment ‘slime’,’’ and according to the OED the first use of thword in English that they can ascribe to a specific persoduring the year 1325.4 The Oxford English Dictionary@8#defines Bitumen as, ‘‘Originally, a kind of mineral pitcfound in Palestine and Babylon, used as mortar, etc.same as asphalt, mineral pitch, Jew’s pitch, Bitumen Jucum.’’ It also provides the following more scientific defintion, ‘‘In modern scientific use, the generic name of certinflammable substances, native hydrocarbons more oroxygenated, liquid, semi-solid, and solid, including naphtpetroleum, asphalt,etcElastic Bitumen: Mineral Caoutchouor Elaterite.’’ The first use of the word that they cite is in thyear 1460.5 Today, both these words are used to describesame class of materials, asphalt being the word of choicthe United States and bitumen in the European and Eascountries~see also@9,10#!. Concerning the controversies surounding the usage of the word asphalt and bitumen inUnited States and Europe, we refer the reader to the booJaccard@11# and the discussion related to the usage of thwords in the review by Peckham@12#.

The Webster’s Dictionary@13# defines asphalt as, ‘‘Abrown or black, tarlike, bituminous substance that consmainly of hydrocarbons, found in large flat beds or maderefining petroleum’’ and bitumen as ‘‘1! asphalt found innatural state, 2! any of various black, combustible, solid tsemisolid mixtures of hydrocarbons that are usually obtaifrom the distillation of petroleum, used to make roofing mterials, sealants, paints, etc.’’

The above definitions do little to clearly delineate tclass of materials that are referred to as ‘‘bitumen.’’ The teseems to apply to a large class of hydrocarbons and eveearly as 1895 the use of the wordbitumenconfounded andconfused its users as the following remarks of Peckham@14#bear testimony: ‘‘The exact meaning of the word ‘bitumein modern scientific literature, has been a matter of perpity for many years. It has appeared to me impossible thatone unacquainted with the different substances includedthe makers of dictionaries and cyclopedias under theirscription and definitions, could form any clear idea of wh

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they are. For instance, in Genesis XI 3, a Hebrew wordcurs which designates the substance used in constructingwalls of the tower of Babel. In the Septuagint this wordtranslated asascaltos, and in the vulgate ‘bitumen.’ Inthe Bishop Bible of 1568, and subsequent translationsEnglish, the word is rendered as ‘‘Slime.’’ In the Douatranslation of 1600, it is ‘bitume.’ In the Protestant Frentranslation it is ‘bitume.’ In Luther’s German Bible it is‘thon’.’’ He sums up the early confusion most admirabthrough, ‘‘Since the confusion of tongues at Babel, no suconfusion of names and things and mixture of things wever witnessed as, in this case, confounded all forms oftumen, and all of the artificial products similar to or identicwith natural bitumens,’’ and advocates a clear and precdefinition for bitumen thus, ‘‘It therefore seems to me desable that the word bitumen should be once more, and cleadefined, as the generic name of that large class of substaoccurring in nature as minerals, and consisting chieflymixtures of compounds of carbon and hydrogen and nigen, sulphur and oxygen as more rare constituents. Unthis genus should be ranged, as species and sub-specieof the combustible gases, napthas, petroleum, malthasasphaltums.’’ Asbitumenrefers not to a substance but tovery large class of substances, the modeling of such a lclass of substances naturally presents exciting challenge

The Dictionary of Applied Chemistry compiled by Thorp@15# has the following definition for bitumen: ‘‘This termincludes a considerable number of inflammable mineral sstances consisting mainly of hydrocarbons. They are of vous consistence, from thin fluid to solid, but the solid bitmens are for the most part liquefiable at a moderate heat.purest kind of fluid bitumen, called naptha or rock oil, iscolorless liquid of sp gr 0.7–0.8, and with a bituminoodour. It often occurs in nature with asphalt and other sobitumens. Petroleum is a dark-coloured fluid variety containg much naptha. Maltha or mineral tar is a more viscvariety. The solid bitumens areasphalt(qv); mineral tallowor hatchetin; elastic bitumen; mineral caoutchouc or elateite; ozokerite, &c.’’

According to the Permanent International AssociationRoads Congress, the following are the definitions for Bimen, Asphaltic Bitumen, Asphalt, and Tar@16#:

Bitumens: Mixtures of natural or pyrogenous origin ocombination of both~frequently accompanied by their nonmetallic derivatives! which can be gaseous, liquid, semsolid, or solid, and which are completely soluble in carbdisulphide.

Asphaltic Bitumen: Natural or naturally occurring bitumenor bitumen prepared from natural hydrocarbons by distition or oxidation or cracking; solid or viscous, containinglow percentage of volatile products; possessing characteragglomerating properties, and substantially soluble in cardisulphide.

Asphalt: Natural or mechanical mixtures in which the aphaltic bitumen is associated with inert matter.

Tar: A bituminous product, viscous or liquid resultinfrom the destructive distillation of carbonaceous material

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4EE Allit PB 1038 ‘‘be spumande asphalton pat spyserez sellen.’’5CAPGRAVE Chron 30 ‘‘A vessel of wykyris, filled the joyntis with tow erde, clepebithumen.’’

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Appl Mech Rev vol 56, no 2, March 2003 Murali Krishnan and Rajagopal: Bitumen from ancient to modern times 151

The American Society for Testing and Materials~ASTM!in its standard ASTM D 8-97@17# has promulgated the following definitions for asphalt and bitumen:

Bituminous materials~Relating in general to bituminoumaterials!: a class of black or dark-colored~solid, semi-solid,or viscous! cementitious substances, natural or manuftured, composed principally of high molecular weight hydrcarbons, of which asphalts, tars, pitches, and asphaltitestypical.

Relating specifically to Petroleum or Asphalts: Asphaltdark brown to black cementitious material in which the pdominating constituents are bitumens which occur in nator are obtained in petroleum processing.

At this juncture, it is appropriate to quote selectively frothe deliberations of the ASTM the different revisions whiled to the above definitions. For instance, the first definitadopted in 1912 for bitumens is the following: ‘‘Bitumenare mixtures of native or pyrogeneous hydrocarbonstheir non-metallic derivatives, which may be gases, liquiviscous liquids, or solids, and which are soluble in carbdisulphide’’ @18#. This definition came under harsh criticisfrom Richardson who in his interesting minority report@19#questioned the validity of this definition as against the emological significance of the word since it suggested tcoal tar could be also classified as bitumen. It is also penent to point out that asphalt was not included in the listdefinitions.

The 1915 ASTM proceedings included a new set of dnitions for bitumen and asphalt based on the penetrationand the consensus reached in 1915 is given below@20#:

Solid Bituminous Materials: ‘‘Those having a penetrationat 25°C~77°F!, under a load of 100 g applied for 5 secondof not more than 10.’’

Liquid Bituminous Materials: ‘‘Those materials having apenetration at 25°C~77°F!, under a load of 50 g applied fo1 second, of not more than 350.’’

Semi-Solid Bituminous Materials: ‘‘Those having a pen-etration at 25°C~77°F!, under a load of 100 g applied forseconds, of more than 10 and a penetration at 25°C~77°F!,under a load of 50 g applied for 1 second, of not more th350.’’

Asphalt Cement: ‘‘A fluxed or unfluxed asphalt speciallyprepared as to quality and consistency for direct use inmanufacture of bituminous pavements, and having a penetion at 25°C~77°F!, of between 5 and 250, under a load100 g applied for 5 seconds.’’

The dissenting views of Mackenzie@21# with regard tothe above definitions based on the use of a Penetrometethe penetration value to classify bitumens as solid, sesolid, and liquid warrants mention here. The issues relatethe consistency of bitumen have been summed up byinventors of the penetration equipment in the following waBowen, the maker of the first penetration machine makesfollowing remarks about the aims and scope of the test,sharp polished needle, acting during a given short intervatime, under a given weight, and at a given temperature, petrates a homogenous, non-crystalline, semi-liquid bodydepth, apparently dependent chiefly on the strength of co

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sion of the body, or softness as commonly designated’’@22#.Dow who suggested some major revisions to the BowePenetrometer had the following observation about the naof asphalt, ‘‘The theory of asphalt construction is but littunderstood. This is because the proper physical laws areapplied in reasoning on this subject, which is not surpriswhen we consider that the cementing of other materialsgeneral use in engineering work are solids, and involveslaws governing the cohesion and adhesion of solids for eother, while in the case of asphalt construction we havetirely different principles involved—those governing the chesion of liquids and adhesion of liquids for solids, for aphalt paving cements are in every sense of the word liquIt is either because this fact is unknown, disregarded,thoughtlessly overlooked by analysts and engineers thatmajority of literature on the technical examination of aphalts is so lacking in value’’@23#. Elsewhere Dow refers toasphalt and solid bitumens as ‘‘hyperviscous liquids’’@24–26# within the context of the Penetrometer tests. This twarrants a much more detailed discussion and we deferto a later section.

The 1944 revision to the ASTM definition for bitumeand asphalt retained the 1915 classification of solid, sesolid, and liquid bituminous materials but had the followindefinitions for bitumens and asphalts@27#.

Bitumens: ‘‘Mixtures of hydrocarbons of natural or pyrogeneous origin, or combinations of both, frequently accopanied by their non-metallic derivatives, which may be geous, liquid, semisolid, or solid, and which are completsoluble in carbon disulphide.’’

Asphalts: ‘‘black to dark-brown solid or semisolid cementitious materials which gradually liquefy when heated,which the predominating constituents are bitumens allwhich occur in the solid or semisolid form in nature or aobtained by refining petroleum, or which are combinationsthe bitumens mentioned with each other or with petroleumderivatives thereof.’’

In all the above classifications, bitumen is clearly diffeentiated from asphalt. However, it is unclear what exactlyframers of the above definition meant by solid, liquid, aviscous. For instance, to classify a material as being eiviscous or liquid~or as in the latest ASTM definition fobitumen assolid, semisolid, or viscous! implies that a liquidor solid cannot be viscous! Clearly this is not a tenablesition. They do not seem to accept the possibility that marials could be viscoelastic solids or viscoelastic fluids athis is rather odd given the great strides made in viscoeticity by the time these definitions were put in place by tPermanent International Association of Road Congressewell as the current ASTM definitions. The above definitiowould leave the modern day student of mechanics nonewiser with respect to the true nature of the material bedescribed.

The recent tools of science such as gas chromatogramass spectrometry, and differential scanning calorimehave advanced the classification of asphalt and bitumensound footing and while the chemistry of asphalt and bimen have been re-examined by using these tools, the trlation of such information into modeling the thermomechacal response of these materials has not kept pace@28–33#.

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1.2 On the abundance of bitumen in nature

Bitumen was available in abundance naturally in seas, laponds, and rivers, on mountainsides, in coal pits and imines; practically in all forms of flora and fauna. This mait possible to put it to diverse use from times immemoria

Bitumen seems to have been available naturally apleAgricola @1# remarks, ‘‘Liquid bitumen, if there is muchfloating on springs, streams, and rivers, is drawn up in buets or other vessels; but, if there is little, it is collected wgoose wings, pieces of linen, ralla, shreds of reeds, and othings to which it easily adheres, and it is boiled in larbrass or iron pots by fire and condensed.’’ Nowadays, bmen is primarily obtained as a by-product during the procof distillation of petroleum. Lest Agricola’s remark concering the ubiquity of bitumen be ascribed to the opinion osolitary writer, we wish to cite a few others whose writinbear witness to the plentiful supply of bitumen. Two pubcations that appeared in the Philosophical Transaction1673, one by Paulo Boccone@34# and the other by Jesso@35#, comment on the natural occurrence of bitumen. PaBoccone@34# remarks that hillocks in Sicily are covered witbitumen, ready to be harvested, ‘‘A certain stony substanthat is fissile, and hath the scent ofbitumen, complicated andlaid together membrane-like, and found in theHybleanmountains ofSicily, nearMilelli , neighboring upon the townof Augusta, and the ancientMegara. Being burnt in a candlethe bituminous smell will soon be perceived; and is affirmthat this stony body, being recently sever’d from its mine abed, is flexible like paper; but being long exposed to Air aSun, becomes frangible. And the herbs, that grow onstone, do insinuate their fibers and roots between sevcoats of the same. It may deserve to have its uses examthere being found whole hillocks cover’d with it.’’ Jesso@35# talks about it as a mineral found in coal and iron minof England, uncommon in comparison to coal but commenough to be found in plenty: ‘‘And some of this fungosubstance is very soft and like gelly. In and about the msolid pieces~of which I have some, half a foot square! aremany big lumps ofbituminoussubstance. Thisbitumen isvery inflammable like resin; it is very light, it breaks firmand shines like good Aloes; and for color, it is not muunlike it, save it is more dark green color. We distilledparcel of it, which yielded us an Acidulous limpid watethen, a white liquor, which was, I guess, from some ofoily parts precipitate. And in the last place a copious yellOyl, not unlike that ofSuccinumor Pitch,’’ and ‘‘That Birch,~of which there is great plenty and hath vast woods all thmountanous parts ofEngland over! will yield bitumen, aslimpid as the sap is which runs from it by tapping...’’

Sources of water, the lifeblood of great civilizations, wealso sources of bitumen. The Babylonians found bitumenLake Asphaltitis. The great Roman architect and encyclodist Marcus Vitruvius Pollio, who died before the birth oChrist, remarks~see a translation of Vitruvius in@36#!, ‘‘InBabylon, a lake of very great extent, called Lake Asphaltihas liquid asphalt swimming on its surface, with which aphalt and with burnt brick Semiramis built the wall surrouning Babylon. At Jaffa in Syria and among the Nomads

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Arabia, are lakes of enormous size that yield very lamasses of asphalt, which are carried off by the inhabitathereabouts,’’ and Diodorus remarks~see Forbes@37#! ‘‘Is alarge sea which yields up much asphalt and from which ano means negligible revenue is drawn. The sea is aboutstadia in length and 60 stadia wide. The water stings anexceedingly bitter so that fish cannot live in it, nor do aother aquatic creatures occur in it. Although large riversvery fresh water flow in it, it remains bitter. Every yearlarge quantity of asphalt in pieces of more than three plethfloat in the middle of the sea, but often they are only twplethora in length. That is why the barbarians who live onshores of the sea call the large pieces ‘bull’ and the smapieces ‘calf.’ When the asphalt floats in the middle of twater, it looks like an Island to those standing on the shoThe advent of the asphalt is heralded 20 days before itsrival, for all around the lake the stench is wafted by the wiover many stadia and all the silver, gold, and copper inneighborhood becomes tarnished; but the tarnish disappagain when the asphalt rises to the surface. The district invicinity which is readily inflammable and which is pervadeby an unpleasant odour, makes the people’s bodies illthey die young.’’ And Strabo: ‘‘,Babylonia produces alsgreat quantities of asphalt, concerning which Eratosthestates, that the liquid kind, which is called naptha, is foundSusis, but the dry kind, which can be solidified, in Babylnia; and there is a fountain of this latter asphalt nearEuphrates river; and that when this river is at its flood attime of the melting of the snows, the fountain of asphaltalso finned and overflows into the river; and that there laclods of asphalt are formed which are suitable for buildinconstructed of baked bricks. Other writers say that the liqkind is also found in Babylonia.

‘‘Now writers state in particular the great usefulnessthe dry kind in the construction of buildings, but they salso that boats are woven with reeds and, when plastewith asphalt, are impervious to water.

The liquid kind, which they call naptha, is of a singulnature; for if it is brought near fire it catches fire; and if yosmear a body with it and bring it near to the fire, the bobursts into flame, It is impossible to quench these flamwith water ~for they burn more violently!, unless a greatamount is used, though they can be smothered and quenwith mud, vinegar, alum, and birdlime. It is said that Aleander, for an experiment, poured some naptha on a bobath and brought a lamp near him; and the boy, envelopeflames, would have been burned to death if the bystandhad not, by pouring on him a very great quantity of watprevaled over the fire and saved his life.’’

Poseidonius says of the springs of the naptha in Babythat some send forth white naptha and others black; andsome of the former consist of liquid sulphur~and it is thesethat attract the flames!, whereas the others send forth blanaptha, liquid asphalt, which is burnt in lamps insteadoil.’’

The Hebrews harvested the Dead Sea for bitumen. Acola @1# observes, ‘‘of this kind is the lake which the Hebrews call the Dead Sea, and which is quite full of bitum

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nous fluids.’’ The Greek Author Herodotus~see thetranslation of his work in@38#! records the existence of pooof Bitumen in the island of Cyrauis, ‘‘Off their coast~theCarthaginians say! lies an island called Cyrauis, twenty-fivmiles long and narrow across, accessible from the mainlait is full of olives and vines. It is said that there is a lakethis island from which the maidens of the country draw godust out of the mud on feathers smeared with pitch. I doknow whether this is true; I just write what is said. But athings are possible; for I myself saw pitch drawn from twater of a pool in Zacynthus. The pools there are numerothe greatest of them is seventy feet long and broad,twelve feet deep. Into this they drop a pole with a myrbranch fastened to its end, and bring up pitch on the mysmelling like asphalt, and for the rest better than the pitchPieria.’’ He later comments, ‘‘Asphalt and salt and oil adrawn from it in the following way: a windlass is used in thdrawing, with half a skin tied to it in place of a bucket; this dipped into the well and then poured into a tank.’’ Herdotus@38# is careful to distinguish that which he has heafrom that which he himself has witnessed and it seemssonable to assume that his account is accurate.

Bitumen was also found naturally in Assyria~see Forbes@37#!: ‘‘In Assyria there is asphalt in a lake called Sosingiin the bed of which the Tigris is absorbed, which flowsunderground to rise to the surface again ata a great distaway.

Naptha also occurs here; this material greatly resemasphalt and is very tough and sticky. Even if a small balights on it, it is drawn down, it can no longer fly anddisappears into the depths. Once this kind of moisture beto burn, human intelligence will find no other meansquenching it than by earth.

A big cleft will be seen in these districts from which motally fatal fumes rise up, the heavy odour of which will kiany living creatures coming within its reach.’’

The city of Hit was well known as a source of bitumenthe following quotation from Strabo bear witness~see Forbes@37#!: ‘‘There is another city, called Is, eight days journefrom Babylon, where a little river flows, also named Is,tributary stream of the river Euphrates; from the sourcethis river Is rise with the waters many gouts of bitumen afrom thence the bitumen was brought for the wall of Bablon.

‘‘These barks of the Tigris have no pumps in them, bcause of the great abundance of pitch, which they havpitch them withall: which pitch they have in abundance twdays journey from Babylon. Near unto the river Euphratthere is a city called Heit nere unto which city there is a grplain full of pitch, very marvellous to beholde, and a thinalmost incredible that out of a hole in the earth, which cotinually throweth out pitch into the aire with continuasmoke, this pitch is thrown with such a force that being hofalleth like as it were sprinckled over all the planie is alwafull of pitch, the Mores and the Arabians of that place sthat hole is the mouth of hell and in truth it is a thing venotable to be marked: and by this pitch the whole peohave their benefit to pich before their barks.’’

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It seems like bitumen was not found in abundance incontinent of Africa. However, mention is made in the wrings of Vitruvius concerning ‘‘oil seepages’’~see Forbes@37#!. Forbes also mentions that bitumen was imported frPalestine or Syria into Egypt and that in view of ‘‘The totlack of references to sources of bitumen in ancient EgypHowever, there are references that can be offered as evidof the natural availability of bitumen in other parts of Afric~see Forbes@37# quotations attributed to Vitruvius!: ‘‘Thereis also a lake of Ethiopia which anoints men who swimit... There is also a spring at Carthage on which floats anwith the perfume of cedar shavings and with this oil sheare usually dressed.’’

Bitumen was available in Palestine in great quantitieswas found in limestone deposits~see Forbes@37#!: ‘‘In No-mad Arabia are lakes of immense size producing much bmen which is gathered by neighboring tribes. This is nsurprising because there are many quarries of hard bituthere. When, therefore, a spring of water rushes throughbituminous land, it draws the bitumen with it, and passioutside the water separates and deposits the bitumen.’’

Asphalt was an important natural resource. If asphalt wnot available locally, it was imported. For instance, Edza@39# in the translation of the Sumerian Literature~‘‘TheBuilding of Ningirsu’s temple’’! mainly before the third dy-nasty of Ur observes, ‘‘For lord Nin-jirsu, Gudea had shiwith hanna dock there, and ships with gravel and dry bimen...bitumen, and gypsum from the hills of Madga, carglike boats bringing grain from the field.’’ For a discussionthe role of asphalt in the cross-cultural exchange in the eMesopotamian civilization and in proto-elamite Iran, we rfer the reader to the articles by Alden@40# and Algaze@41#~also see Woolley and Mallowan@42# for details on theUr excavations and the usage of bitumen in this ancicivilization!.

Peckham@43# discusses in great detail the geographidistributions of the various types of ‘‘semi-solid bitumenand bituminous rocks’’ in the United States. He mentions tthe different types of materials that go under the namebitumen are to be found in the valley of the Connecticriver, in the eastern portion of the state of New York, in NeJersey, in Ritchie county in West Virginia, in numerous couties in Texas~in the valleys of the Brazos in Brazos county!,and in the valley of the Red River in Montage county,Burnett county and Uvalde county, in the area of Upper Wlow Creek and white river in Colorado, in parts of Utaadjoining northwestern Colorado, Breckenridge, Hogan, aCarter Counties in Kentucky, the banks of the Washita Riand the Prairie adjoining the Arbuckle mountains in Okahoma, in the vicinity of Santa Crux, San Luis Obispo, LAlamos, Santa Barbara, Los Angeles, San Francisco, Carteria, and many other locations in California. In this informtive article by Peckham the interested reader can findextensive discussion of the various locations where bitumwas found in the United States~see also@44#!. It should beclear from the brief discussion above that bitumen was avable aplenty in the United States, but this abundance waslimited to the United States alone but true of Canada as w

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as the following remarks of Peckham@43# bear evidence: ‘‘InBritish North America there are enormous deposits of bminous sand in the valley of Athabasca river. In New Bruswick the well know deposit of Albertite, in the valley of thPeticodiac River in Albert county, was mined for manyears. The material is a clean, pure asphaltum, jet blaccolor, with a brittle, conchoidal fracture.’’

The remarks of Bousingault@45,46# that ‘‘immensemasses of pitch which are found on the banks of the RMagdalena at Payta, upon the coast of Peru and its availity on Lake Maricabo and in the state of Bermudez in Veezuela’’ attest to its abundance in South America. Laquantities of bitumen were also to be found in BarbadosTrinidad in the West Indies.

China was one of the first countries in the world to dcover natural gas and petroleum. In hisNotes on Mengxi,Shen Kuo~1031–1095 AD! remarks on the rock oil in Yanterritory ~see Yen@47# and Zhang and Huang@48#!. Bitumenwas also found, and put to good use, in various parts ofIndian subcontinent and the Far East.

For more information on the nature and availabilitybitumen, the reader is referred to the works of Abraham@49#,Akersten @50#, Barth @51#, Broome @52#, Bushong @53#,Clapp @54#, Davis @55#, Dowling et al @56#, Forbes@37,57#,Hackford et al @58#, Hartley and Brinegar@59#, Hatchett@60#, Hoffmannet al @61#, Hunt @62#, Jones@63#, Knox @64#,Lockhart@65#, Merriam@66#, Merriam and Stock@67#, Rich-ardson@68#, Traxler @69#, Tyler et al @70#, and Wolfard@71#.

Today we rarely if ever get our asphalt from natusources. It is available in large quantities at the end ofrefining process for petroleum. The total production of hmix asphalt in Europe and the USA was around 900 milltons for the year 2000@72#.

1.3 Historical references to bitumen

Bitumen seems to have been used in practically every clization that predates the birth of Christ. Let us start whistorical references in Sanskrit literature that mention bmen. There is not just one word for bitumen in Sanskrit. Tnumber of terms used to denote bitumen is truly astonishand clearly indicate the diverse sources for the bitumThey number over 20! That they made such subtle disttions based on the source of bitumen suggests that itavailable freely and used widely, different types of bitumbeing preferred for different applications. We list some of tmany terms used to denote Bitumen in Sanskrit:

Agaja—Bitumen produced on a mountain or from a tree~seeWilliams @73#, p 4!Asmottha—Bitumen produced from rock~@73#, p 114!Asmajatu—rock asphalt~@73#, p 123!Indrajatu—bitumen~@73#, p 166!Upa-rasa—a secondary mineral~as red chalk, bitumen!~@73#, p 205!Girisambhava~@73#, p 355!Jatu~Latin: bitumen, germ: Kitt! ~@73#, p 409!Dhatuja~@73#, p 514!Dhatuka~@73#, p 514!Silaniryasa—rock exuded bitumen~@73#, p 1073!

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Siladadru—rock eruption bitumen~@73#, p 1073!Silaprasuna, Silabhava—rock produced bitumen~@73#, p1073!Sila pushpa—rock efflorescence bitumen~@73#, p 1073!Silasveda—rock perspiration bitumen~@73#, p 1073!Silottha—produced from rock—bitumen~@73#, p 1073!Silajit—rock overpowering—bitumen~@73#, p 1073!Sila mala—rock impurity—bitumen~@73#, p 1073!Silavyadi—rock-ailment—bitumen~@73#, p 1073!Silasana—bitumen~@73#, p 1073!Silahva or sila hvaya—bitumen~@73#, p 1073!Silottha—bitumen~@73#, p 1073!Sileya—as hard as rock or stone, coming from rock, bitum~@73#, p 1073!Saila-ja, Silaja—bitumen~@73#, p 1090!Saileyaka—bitumen~@73#, p 1090!

The great Indian epic, The Mahabharatha,6 mentions a housebuilt with bitumen, among other construction materials, wthe specific intent of consigning the residents to fire by virof the combustible nature of the construction material.

In his three volume archeological study of Mohenjo-Dabetween 1922 and 1927, Sir John Marshall@77# providescopious evidence for the use of bitumen in a variety of wa‘‘The brick steps were lined with timber bedded in bitume...Backing this was an inch thick damp-proof course of bimen, which was kept in place and prevented from creepby another thin wall of burnt brick behind it.’’ While discussing the use of watertight backing in a tank he refers tothick layer of bitumen, averaging 1 inch in thickness, wapplied to the outside of the walls; and presumably to pvent the bitumen from creeping, a thin retaining wall, obrick or eleven inches in thickness was built against it.’’ Hgoes on to say, ‘‘traces of bitumen were found in mostthese holes, and it is probable, therefore, that the treads wcovered with wood, which was fixed into the sides of tstairways with a bituminous cement.’’

While bitumen was used in the above manner, its useMohenjo-Daro according to Marshall@77# was by no meanscommon, unlike its use in Babylonia. Marshall@77# ob-serves, ‘‘Bitumen up to the present has been found in oone building in Mohenjo-Daro,’’and he suggests this wprobably due to the unavailability of bitumen in India, ‘‘fobitumen is rare in India.’’

The comments of Marshall@77# concerning the rarity ofbitumen in India warrants some comment as it flies inface of the over 20 different terms in Sanskrit that were uto describe bitumen. The history and geography of a naand the cultural and religious proclivities of its populashape the languages of its people. Thus, it is highly unlikthat over 20 different terms were used to describe variforms of asphalt if it were not of common occurrence.conclude that bitumen was rare on the basis of the diggiat Mohenjo-Daro is a far stretch. Marshall@77# suggests that

6Various dates are ascribed to this epic. While the traditional date when the greadescribed in the epic took place is supposed to be 3102 BC, Majumdar@74# places theevent at around 1000 BC, Basham@75# around 800 BC and Zimmer@76# provides amuch more conservative estimate for the text that is current between 400 BC–40~The emphasis here being on the current version of the text of the great epic, posearlier versions would be consistent with the dates ascribed by Basham@75# or Ma-jumdar @74#!.

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the bitumen that was used in Mohenjo-Daro could hacome from the North-West Frontier Provinces. He obserthat ‘‘Bitumen is also found near Isa Khel on the right baof the Indus~North West Frontier Province! in India. Thequantity is small, but may have been greater in older timeIt is not our aim here to get into a lengthy discussion onavailability of bitumen in ancient times and, in particular, ginto a discussion of the accuracy of Marshall’s observatias we are grossly ill equipped for such a purpose.

Biblical references to bitumen could fill numerous pagand so we shall rest content citing but a few examples. Gesis 11@78# addresses the use of bitumen in constructi‘‘And they said to one another, ‘Come, let us make bricand burn them thoroughly.’And they had brick for stone, abitumen for mortar.’’ In Exodus 2@78#, the use of bitumen asan adhesive and a sealant is obvious from the remark, ‘‘Awhen she could hide him no longer she took for him a basmade of bulrushes, and daubed it with bitumen and pitand she put the child in it and placed it among the reedthe river’s brink.’’ In Isaiah 34@78# the combustible nature obitumen is made evident by the statement, ‘‘And the streaof Edom shall be turned into pitch, and her soil into brimstone; her land shall become burning pitch.’’ There aremerous other references to bitumen in the Old Testamenthe few above suffice to indicate its several uses in bibltimes.

The author of the authoritative tract on Greco-Perswars, Herodotus~see@38#!, who is supposed to have beeborn around 484 BC, has also made numerous referencbitumen and we have already come across one of them. Ewhere he discusses one of the sources of bitumen from Blon thus, ‘‘There is another city, called Is, eight days’ journfrom Babylon, where there is a little river, also named Istributary of the Euphrates river; from the source of this rivIs, many lumps of bitumen rise with the water; and frothere the bitumen was brought for the wall of Babylon.’’ Wshall discuss the construction of this wall of Babylon wbitumen and its role in the preservation of the city during odiscussion of the multifarious uses for bitumen.

The Philosopher Diodorous Cronus~see@79# for a trans-lation! who is supposed to have lived in the fourth centuBC points to the use of bitumen as a combustible matewhile describing the actions of a Syracusan general: ‘‘Whthis took place, Sicanus, the Syracusan general, straighfilling a merchant ship with faggots and pine-wood apitch, set fire to the ships which were wallowing in thshoals.’’

References to Bitumen are to be found in the writingsAristophanes~Acharnians, Frogsin @80,81#!, Flavius ~seeHistory of the Jewish War (75–79) and Against Apionin@82#!, Leon Battista Alberti~seeThe Ten Books of Architecture in @83#!, Pliny the Elder~seeThe second booke of thhistorie of natvre (Chap CIII, CIIII, CV) and the sevenbooke of the historie of natvre (Chap XV)in @84#!, PubliusOvidius Naso~seeMetamorphosesin @85#!, Strabo~seeTheGeography of Strabo, Book 7, Chap 5in @86#!, Xenophan~SeeAnabasis, Book 2, Chap 4 and Book 7, Chap 5in @87#!,and many others. There is little purpose to documenting

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such references here. Suffice it is to say that bitumenused extensively in the very earliest of civilizations. In tnext section we provide a very brief description of the mauses of bitumen.

2 DIVERSE USES OF BITUMEN

2.1 Uses of bitumen from prehistory to modern times

Few natural materials are used as diversely as bitumenthe introduction we touched upon some of the uses thattumen was put to in prehistory. We remarked earlier tclear evidence exists concerning the use of bitumen fromHummalian period from around 180,000 BC@3,4#. It wasused as the adhesive of choice to construct hunting imments out of stick and stone, the success of the hunt criticdepending on how well the sharpened flint stuck to the sthat was flung at a fleeing animal. We also came acrossuses as a sealant, as an adhesive, and filler in buildingstruction in the early civilizations of Babylonia, SumeriAssyria, Egypt, India, China, and Greece. We now consiits multifarious uses in the unlikeliest of places. The discsions do not follow a chronological order, rather we stwith a use, which probably due to the ignorance of thethors of this review, seems to be the most intriguing asurely the most unusual use of bitumen; a use which unmost other uses of bitumen is thankfully not being practic

Bitumen was used by the Egyptians for embalming.Greenhill@88# observes, ‘‘there are several kinds ofembalm-ing, viz, with asphalt or pissasphalt, with Oyl or Gum ofcedar, with aromatics and spices.’’ However, that asphalt winvariably one of the many ingredients is clear from the veuse of the word Mummy. The word Mummy is derived frothe Persian wordmumiya, the word for asphalt in Persia~see Pettigrew@89#!.7

Pettigrew@89# in his fascinating tome,A History of Egyp-tian Mummies, discusses various possible roots for the womummy and suggests that it may be derived from the Araword mumia that signifies a body ‘‘embalmed or aromatised.’’ He goes on to state, ‘‘The Persian wordmumiya,means bitumen or mineral pitch, the PISSASPHALTOwhich is generally found in the bodies embalmed by tancient Egyptians.’’ A few other references to the connectbetween mummy and bitumen that Pettigrew@89# providesare worth repeating: ‘‘Abd’Allatif, an Arabian physician whflourished in the twelfth century, describes mummy, propeso called, to be a substance that flows down from the topthe mountains, and which, mixing with the waters that cait down, coagulates like mineral pitch, and exhales an odresembling that of white~Burgundy! pitch and bitumen...Ibn

7A recent article in a popular science magazine~see Glausiusz@90#! questions whetherthe word mummy has its origins in the Persian word Mumiya or the Egyptian coword Mum that stands for beeswax. Such doubts are by no means new and in theno scholarly rationale for such a proposal is advanced and as most of the expertsfield are well aware of the Egyptian word it is highly unlikely that they have not givthis possibility much thought. The author also offers evidence that is far from comling against what currently seems to be inconvertible scientific evidence for the usasphalt in mummification. In truth, the article seems to be written by one that isfrom being an expert on these issues and in fairness to him he claims no such expIt is well known that Mummy is derived from the Arabic word for bitumen, which in ioriginal Persian form meant wax. The Persian and Arabic word for Mummy meabody preserved by wax or bitumen.

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Baitar says that...the Mumiya of the tombs, found in grquantities in Egypt, is nothing more than the amalgamciently used by the Greeks to preserve their dead bodies fputrefaction.’’ He says also that the term Mumia is given tkind of light black stone found near Sena in Yemen, awhich contains a ‘‘black fluid substance in a small cavityand finally, ‘‘The account given by Sir William Ouseley confirms many particulars that just quoted. Sir William visitethe Mummy Mountain~Kieh Mumiay! in the territory ofDarabgerd, Persia. He fancied that it presented a darkepearance than the mountains adjacent to it. He saysmummy is a blackish bituminous matter, which oozes frothe rock, and is considered by the Persians as far morecious than gold; for it heals cuts and bruises, as they affialmost immediately, and causes fractured bones to unitefew minutes, and, taken inwardly, is a sovereign remedymany diseases.’’

The most compelling evidence is offered by Pettigr@89# for the use of bitumen in mummification in the following lengthy discussion on the examination of a mummyM Rouelle: ‘‘The sixth and last examination made byRouelle was of balsamic matter found in a vessel preserin the chamber of the mummies. It was of a black colobrilliant, soft, and flexible. Upon handling it an agreeabodour diffused. Its nature was purely resinous, uniform insubstances, not bitter like aloe to the taste, and perfesoluble in spirit of wine. By distillation, a small quantity owater, slightly aromatic, was obtained, to which, uponincrease of heat, some drops of an acid, and a lightscarcely coloured, but quick and penetrating in its balsamatter was produced. The quantity of oil thus obtainamounted to almost one-fourth of the weight of balsammatter. This pitch, and acid accompanying this oil possesthe same odour. Thus it appeared that these entered intcomposition of this substance some matter of an aromand resinous nature, abounding with an essential oil—basis of the balsam in Jews’ pitch.

The Jews’ pitch or bitumen Judaicum called alsoasphal-tum, the name of the lake whence it has been obtained,solid friable substance of a brownish-black color, brilliantits fracture, and giving out a bituminous odour...It resufrom the whole of these analyses that M Rouelle has detethree modes of embalming, differing in the material usedWith the asphaltum or Jews’ pitch. 2. With the mixtureasphaltum with the liquour of cedar or the cedria. 3. Wthis mixture, to which were added some resinous and amatic ingredients.’’

Thus, according to Pettigrew@89#, irrespective of whichof the above methods of embalming was used, bitumena part of the embalming mixture, if not the whole.

In a later systematic inspection of many mummieMacalister@91# discusses the process of mummification thcomprised in the extraction of the brain through the nosethe filling of the cranial cavity with rags, the filling of thcavum oris and pharynx, and the use and the variety of linin the embalming procedure. Of particular relevance todiscussion is his following observation: ‘‘In about ten of mspecimens the mouth was filled with asphalte, in four wit

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powdery resinous aromatic material like powdered sandwood and cassia.’’ He also observes, ‘‘Some few of mmummies were apparently only salted, although carefubandaged; others were thoroughly impregnated withphalte, which was evidently melted when the bodies wimmersed in it. This heating was in one mummy carrialmost to the point of calcinations. One mummy in whithere was no trace of asphalte was calcined in like manThe asphalte was vitreous, or in some semi-fluid and sticapparently mixed with pitch from Byblus, and probably withe resin from Phoenicia and Coptos mentioned in the rituThe ‘oil of the black stone,’ probably some form of asphaor bitumen, was prescribed as an application to the feetlegs and thighs before they were enwrapped. I have fosuch a material in a thin layer soaking the deepest bandboth here and on the head. The saline mentioned founchiefly impure common salt, with some nitrates and carbates of sodium and potassium.’’

Despite the above evidence offered by Pettigrew, Macister, and several others, there was a considerable amoucontroversy concerning the use of bitumen in mummifiction. An investigation of Lucas@92,93# based on the solubil-ity tests prompted him to say, ‘‘asphalt was perhaps not uby the ancient Egyptians’’~see also Griffiths@94# and Spiel-mann@95#!. This controversy, however, has been laid to rby careful gas chromatographic studies aimed at determinthe presence of asphalt in mummies. The recent gas cmatographic study by Rullkotter and Nissenbaum@96# andNissenbaum@97# of Egyptian mummies provides incontrovertible evidence that bitumen was indeed used in the mmification. Rullkotter and Nissenbaum,@96# aver: ‘‘The useof asphalt in the mummification process, which was prticed in the ancient Egyptian dynasties from at leastFourth Dynasty~Circa 2600 BC! to the end of the RomanPeriod~fourth century AD! has been a matter of controversfor a long time. Several of the ancient historians, suchDiodorus of Sicily and Strabo~first century AD!, specificallystate that the Dead Sea asphalt was exported to Egyptused for embalming. Diodorus describes a major battletween Egyptians and Syrians to control the trade in Deadasphalt, which was considered a major source of revenCompared to this several archaeologists denied that aspwas used in the mummification. The widespread black marial in mummies, as for example in the mummy of Tutankhmun, was assumed to be a mixture of wood-derived tarresins...Based on the results of this molecular study, therno doubt that fossil asphalt has been used by the ancEgyptian for embalming, although admixtures of fresh plaresins or similar substances are likely’’~see also Hammond@98#, Proefke and Rinehart@99#!.

That bitumen was widely used in ancient times as a pservative is beyond contention. Whether it was due topresence of bitumen in the mummies or whether it wasother reasons, mummies were supposed to have greattive powers. That it was held to be a panacea, a gresought after drug, had rather unpleasant consequences.tigrew’s @89# following remarks make transparent the widspread use of bitumen as a drug: ‘‘In the sixteenth and par

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the seventeenth centuries, mummy formed one of the onary drugs, and was found to be in the shop of all apotheies, and considerable sums of money were expended inpurchase of it, principally from the Jews in the East. Tdemand, however, was not easily supplied; for the govement of Egypt was unwilling to permit the transportationthe bodies from their sepulchral habitation; too great a temtation was created to the commission of fraud.’’ The dreadconsequence of the popularity of parts of the mummydrugs led to abhorrent acts of murder of human beingstheir subsequent mummification. That such an inhumanwould even be contemplated, let alone carried through insixteenth and seventeenth centuries attests to the mesming power of mindless superstition, a sad human trait tpersists to this day.

There is some evidence to the bitumen in the mumbeing credited with great healing properties~see Pettigrew@89#!: ‘‘The asphalt and the bitumen it was contended cosolidated and healed the broken and lacerated veins, anpicquancy occasioning sickness, it was said to havepower of throwing off from the stomach collections of cogealed blood.’’

It would not be an exaggeration to say that the safetynations devolved around the use of bitumen, attested byfollowing narration by Flavius Josephus concerning the cquests of the legendary Babylonian King Nebuchadnezwho, after subduing the Jews, the Phoenicians, the Syriand the Egyptians, set about making his own Babylonpregnable: ‘‘He also rebuilt the old city, and added anotheit on the outside, and so far restored Babylon, that none wshould besiege it afterwards might have it in their powerdivert the river, so as to facilitate an entrance into it; and the did by building three walls about the inner city, and thrabout the outer. Some of these walls he built of burnt brand bitumen...’’ A reference to the use of bitumen in yanother strategic construction can be found in the writingsHerodotus@38# who mentions the construction of a moa‘‘Then using hot bitumen for cement and interposing layof wattled reeds at every thirtieth course of bricks, they bfirst the border of the moat and then the wall itself in tsame fashion.’’ These constructions with bitumen were mimposing and impressive, at times towering up to dizzyheights ‘‘and they built a tower, neither sparing any painor being in any degree negligent about the work: and,reason of the multitude of hands employed in it, it grew vehigh, sooner than any one could expect; but the thicknesit was so great, and it was so strongly built, that therebygreat height seemed, upon the view, to be less than it rewas. It was built of burnt brick, cemented together with mtar, made of bitumen, that it might not be liable to admwater’’ ~see Whiston@82#!. Practically all constructionshomes, bridges, walls, and arches used bitumen, in shmost imposing ancient constructions were all held togetby bitumen.

From antiquity to the present day, bitumen continues toused as waterproofing material. The terraces of the ‘‘HangGardens of Babylon’’ were waterproofed with bitumen, ain third world countries it is the waterproofing agent

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choice for flat roofs in view of its inexpensiveness. Aswaterproofing agent, it was used in antiquity for mats, bkets, jars, boats, water pipes, cisterns, and sarcophagi,this is borne out by samples unearthed during excavationnumerous locations in the Middle East~see Connan@2#!.

As to the use of bitumen in the construction of ships,an adhesive and sealant, numerous references are tfound. In Genesis 6~King James Version of the Bible@100#!,mention is made of the use of bitumen in the following maner: ‘‘Make thee an ark of gopher wood; rooms shalt thmake in the ark, and shalt pitch it within and without wipitch.’’ Aristophanes~see Henderson@80#! puts the followingwords in the mouth of one of his characters, Dicaeopolis:can’t stomach this. It smells of pitch and battleship constrtion,’’ highlighting the liberal use of pitch in ship buildingWe shall dispense with further discussion of the use of piin the construction of ships; whether it was Greece, EgyIndia, or China; bitumen was used liberally in the ship buiing industry.

Burning asphalt was used as a weapon; firebrands wburning asphalt were hurled at one’s enemies. We quotelow a couple of instances of bitumen used as weapon‘‘The inhabitants of Motye, now that the threat was at hangrips, were nevertheless not dismayed by the armamenDionysius, even though they had for the moment no allieshelp them. Surpassing the besiegers in thirst for glory, thethe first place raised up men in crow’s-nests resting on yaarms suspended from the highest possible masts, and tfrom their lofty positions hurled lighted fire-brands and buring tow with pitch on the enemies’ siege engines’’~see Old-father@79#!, and, ‘‘In a citie of Comagene, named Samosathere is a pond, yielding forth a kind of slimie mud~calledMaltha! which will burne cleare. When it meeteth with anything solide and hard, it sticketh to it like glew: also, if it betouched, it followeth them that flee from it. By this meanthe townesmen defended their walls, when Lucullus gaveassault, and his souldiours fried and burned in their owarmours. Cast water upon it, and yet it will burne’’~see Hol-land @84#!.

Bitumen was also used for decorative purposes andjewelry. Necklaces comprising of bitumen beads dating bto 500 BC have been found as also inlays of precious stoin bitumen~see Connan@2#!.

Another important use of bitumen was as a source ofergy. Bitumen was used as a fuel for lamps. In his paperthe possibility of producing perpetual sepulchral lamps, P@101# discusses the use of bitumen. In the process of makthis suggestion, he also echoes the sentiments that wepressed earlier of its plentiful supply: ‘‘Now that there mabe such aBitumenor inexhaustible oyl, I shall not need totrouble you withTestimoniesof old Authors, or carry youinto Italy, or other foreign parts: for will you but allow me ainconsumable weik, I will carry you no further thanPitchfordin Stropshire, where there isNaptha or liquid bitumen ~aspecimenwhere of I have to shew you! that constantly issuesforth with a Spring there, and floats upon the water:this Iwould have separated before it joynes with the water intductusof its own, and so conveyed to the place, thoug

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most convenient for such aLamp, into which it should per-petually distill, as it does now into thefountain; which Idoubt not but you will allow may be done without any grematter of difficulty: and if so we have anOyl aseverlastingas ourWeik.’’

Bitumen was in the ancient times considered an equacommon oils and thus a source of heat~energy in its thermalform! @102#.

The light sensitivity of bitumen was gainfully exploited imaking the world’s first heliograph. In the words of Spiemann @103#, ‘‘The remarkable recrudescence of interestthe changes that light produces in bitumen inevitably raienquiry as to how it originally began. Joseph Nice´phoreNiepce~1765–1833!, having interested himself in the practice of lithography, pondered on the problem of reproducthe colour of a picture. By good luck he had at hand a lesuitable for use in a camera, and obtained results in M1816, though not in colour, by a method undescribed. Amaking further experiments he employed a solution of ‘phalt’ dissolved in Dippel’s animal~bone! oil, a liquid al-ready known to be sensitive to light; but what led him tobitumen is impossible to tell. By this means he produced1824, a portrait on tin of Cardinal Georges d’Amboiswhich though requiring subsequent strengthening by enging, was undoubtedly the first ‘heliograph’ in the world@103#

As the preponderant use of bitumen is in the construcof roadways and runways, a brief discussion of the historyits use in the construction of roadways is warranted. Anthoritative and masterful account of the history of ancieroads and their construction has been given by Rose@104#and Forbes@57#. Several theories have been advanced fororigin and evolution of roads and the main viewpoints averbalized most succinctly by Forbes@57#. Was the concepof roads the product of man’s creativity, a means to facilitcommunication and exchange between distinct and disgroups, or they merely evolve from pathways establishedanimals, a discovery of an effective mode for contact thadeliberate invention of humankind’s creative genius? Thisan issue that is far from settled. As Forbes@57# observes, ‘‘Ithas also been suggested, eg, by Dressler, that roads arethan mankind, the theory being that the paths formed bydaily passage of wild animals to their drinking-pool or ttracks of herds of bison or wild horses became the first roawhich man used and improved upon. Plausible as this themay seem the ‘wild animal path’ origin has been violenattacked by various investigators, especially and perhapthe most efficient way by Roe. He found no connectiontween the bison tracks and the paths of the first tradernorthern America. He even denies the existence of spetracks along which wild beasts travel. According to him amals follow a different track every time and hardly evroam in the same direction during their wanderings. It seeto him impossible to up-hold the ‘wild animal path’ theoany longer. Gregory, Belloc, and many other writers still avance the ‘drinking-pool’ theory, though it has never beproved by explorers who have studied the behavior of wanimals in the jungle. In the majority of cases it was fouthat the track or path was only clearly visible in the neig

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bourhood of the stream or pool, but soon it became vaand was entirely lost in the ever encroaching vegetatiThese paths never attain a great length, because it takgreat amount of trouble to keep them open even if thereregular traffic.’’

It is impossible to settle these conflicting points of vieby appealing to ‘‘facts,’’ the thoughts of the first ‘‘road construction crew’’ that led them to the invention or discoverythe construction of roadways and thus we shall say no mabout the same. A concise discussion of an interesting relissue, the history of the wheel can be found in the workPiggott @105#.

There is incontrovertible evidence that by the Bronze athere was considerable trade between various human sements, and humankind had developed a network of trroutes that included roadways and waterways. Unlikelatter which were at that time exclusively carved out by Nture, the roadways were built by humans.

The first systematic construction of roadways is attributo the peoples living in Asia Minor and Mesopotamia btween 3500–3000 BC. The earliest of these were mud strand it took nearly a thousand years before bitumen wasused in the construction of roads.

The earliest long distance road seems to have beenAncient Royal Road that began~or ended! at Susa near thePersian Gulf and ran all the way to Ephesus on the wescoast through Asia Minor. Herodotus wrote about this Acient Royal Road. However, his description of the lay of troad has been called into question by later scholars~Calder@106#!.

Another ancient long distance road is the Silk Roadtrade route for traders in silk in central Asia to access theindustry that was founded in China by Si Ling Shi, wifethe Emperor Huang Ti, in 2640 BC.

Bitumen seems to have not been used in either of throads.

The Imperial roads of China were constructed aroundsame time as the Royal Persian Road and while these rwere magnificently constructed bitumen seems to havebeen used in their construction unlike the roads of the Invalley civilization that used bitumen.

Much later, the Romans built a remarkable networkroadways, the most famous of the roads being the first ldistance road whose construction started in 312 BC, thepian Way. Typically, the Roman roads consisted in concrmade of broken stones, sand, lime, and dross, that was swiched between broken stones in mortar and paving stoin mortar. Bitumen seems to have not been used in throads. The earliest long roads in Europe were construcbetween 1900 and 300 BC, and were put in place to facilitthe trade of Amber, a precious commodity. There were fprincipal roads~Schreiber@107#!:

‘‘The first ran southwards from where Hamburg is todto the Rhine, then up-river to Basle, westwards to the Rhoˆne,then down-river to the Mediterranean. There was an altertive route a little farther west, which crossed the LowRhine at Xanten and, passing through Metz, joined up w

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the first at Chalons.~The towns I have mentioned were nyet in existence; they are used here merely as locatpoints!.

The second, middle, road ran due south from the Bacoast near Lu¨beck through Magdeburg, Halle, Hof, Regenburg and Hallstatt to the Brenner Pass.

The third is perhaps the most interesting. It began inSamland on the East Prussian coast~where amber is still tobe found!, crossed the river Vistula, and reached the Danthrough the Moravian Gate: in the Etruscan period it pasto the west of Vienna, in Roman times to the east at Carmtum; from there it continued through Hungary to Aquileia~inthe northern corner of the Adriatic!.

The fourth road, the Balitc-Pontus amber road, followfor the most part the main eastern rivers, the Vistula, SSereth, Pruth, Bug, and Dnieper.’’

However none of these roads seem to have had bituas one of the ingredients of construction.

Slowly but inexorably the transition from roads of diffeent types to those that were predominantly made of aspand asphalt concrete took place in view of the comfort tsuch roads accorded the travelers. As Gregory@108# ob-serves: ‘‘Cobbled pavements provide the most jarring ofroad surfaces and form a contrast to those made of a unifsmooth sheet, which for light modern motor and cycle traforms the ideal roadway.

The first material largely used for this kind of road wasphalt, which is a limestone that contains bituminous main a sufficient proportion, usually 7–12%, to become plaswhen heated, and to cement the powdered limestone firwhen it sets on cooling.

Artificial asphalt is powdered limestone mixed with thsame proportions of some form of bitumen, such as namineral bitumen or one left as a residue during the distition of crude petroleum.

The amount of bitumen is not less than 7% or the aspwould be too brittle, and not over 12% or it would be tosoft, especially in hot weather.

The native asphalt most used in Europe was discovere1721: it is the bituminous limestone of the Val de Traversthe Jura Mountains near Neuchaˆtel. Numerous other bitumi-nous limestones are used. The material is crushed, and wheated the bitumen exudes and melts, and when it coolsthe powder, which can be rolled or pressed into a firm esurface. An asphalt road has the advantages of being smeasily cleaned, and comparatively noiseless; hence it isadapted for crowded city streets.

Asphalt paving was tried in Paris in 1838, but was nmuch used until 1854. It was introduced into London in 18and has since been extensively adopted.’’

Today, practically all of the highways are paved with aphalt concrete and it seems that this will be the materiachoice for some time to come. Bitumen was used in roconstruction as early as 2800 BC in Asia Minor, Mesopomia, and Persia, and there is evidence for the extensiveof bitumen and bitumen-joined bricks in the constructionroads in India around 2600 BC. Bitumen-grouted rubb

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bricks joined with bitumen, and bitumen mastic floors wecommonly used in Asia Minor and Babylonia from 1600 Bonwards.

In Mesopotamia, even at the time of Hammurabi~seeForbes@57#! there were no paved streets. By the timeNebuchadnezzar bitumen became a common construcmaterial ~see Forbes@57#!: ‘‘A very common pavement forfloors and courtyards consisted of a three-fold layerbricks, the undermost of which was always joined with bitmen; the top two layers showing either bituminous or gysum mortar.

‘‘The foundations of such floors were sometimes formof layers of rubble and crushed bricks mixed with loampenetrated with bituminous mortar. A construction of altnating layers of bricks and bituminous mortar was alreadyuse for masonry at an early date.’’

Being such a common construction material for pavments, it found its way into the laying of ceremonial roaand into the general construction of roadways.

As mentioned earlier, there is clear evidence of the usebitumen in the construction of roads by the peoples ofIndus Valley around 2600 BC~see Forbes@57#!: ‘‘Certainstreets and courtyards show that bitumen was also usedjoining grout, another example of technical skill.’’ The folowing table~Table 1! from Forbes@57# summarizes the his-tory of roads and the different construction methods used

2.2 Uses of bitumen in modern times

The Shell Bitumen Handbook@109# lists over 250 knowncurrent uses for bitumen in agriculture, construction, hydrlics, erosion control, automobile industry, electrical industrailways, paving industry, etc~see Table 2!, ~see also@110,111#!.

Bitumen is finding ever increasing use as an insulatorelectrical appliances such as transformers. It is also useinsulate cables, and the fact that it does not mix with wacould make it an insulator of choice for underwater cablespecially in virtue of its inexpensiveness. For more deton the use of asphalt in electrical appliances and the issrelated to it, the reader is referred to@112–118#. Bitumen isalso used to store and contain low and moderately radiotive wastes.

The most widespread use of asphalt today is in the cstruction of roadways and runways. For the first use of bmen in road construction, the reader is referred to the exlent monograph by Forbes@57#. ~See also the followingreferences for a history of the use of bitumen in road cstruction in the last two centuries: Richardson@26#, Boorman@119#, Langton@120#, Spielman and Hughes@121#, Ebbertsand Johnstone@122#, Hughes, Adam, and Chiaa@123#, RoadResearch Laboratory@124#, and Zakar@125#.! Today severalmillion tonnes of bitumen are used per annum just in roway and runway construction.

3 CHEMICAL ANALYSIS OF BITUMEN

The performance of bitumen in field applications such asconstruction of highways and runways or for that matter


Table 1. Chronological table of ancient roads and road building methods„from appendix 1 of †57‡…

Approx.Date

North WesternEurope Italy and Malta Crete and Greece

Egypt andPalestine

Asia Minor,Mesopotamia and

Persia India

B.C. 3500? Inventionof Wheel

3000 Earliest cretan floors 3200 PyramidCausewaysQuarry road toHatnub

Sumerian mudstreets

29002800 Slab pavements in

MaltaQuarry roads inSinai

Extensive use ofbitumen Track-system existing

Wheeled trafficoutside towns

27002600 Slab pavement Molfetta Sargonid slab

paved-roads nearNineveh

Bricks in mortarExtensive use ofbitumen

2500 Earliest roadengineers in army

Bitumen joinedbricks

2400 Probably earliest roads inCrete, later mostly rebuilt

Road throughWadi Hammamat

Bit. Mastic Floorsat Khafaje

Special care ofdraining2300 Ridge-ways England

2200 Alignments France ~Date stilluncertain, may beearlier,63000 BC!

2100

Track-ways England20001900 Artificial wheelruts

in MaltaRoads nearKnossos-Period of thegreat trade routes ofCrete slab pavement onconcrete foundation

Mud floors andstreets at Babylon

1800 Aichbuhl log roads

1700 Michelsberg logroadsfrom 1800 onwardsoldest Dutch logroads

1600 Slab pavements inAegean islands and Troy

Wheel in Commonuse

Brick pavementsBabylonBit. Mastic floorscome in moregeneral case

1500 Stone sets and cobbles~Sicily!

1400 Patrol roads at TellAmarna

1300 Well drained and pavedMycenaean roadsin Argolis etc

Map of the roads tothe goldminesRoads in delta~?!

Bricks on sand~Assur!1200 Terremare log roads

Several roadsmentioned

Decline of roadbuilding

1100 Ramses III buildsnew roads

Assyrian roadengineers in armyBitumen-groutedrubble

1000 Double lynchetroads~Engl.!Cobble pavements~Holland!

Greek tracks and forest pathsperhaps earliest paved roads

Solomon’s pavedroad?

Bricks1bitumenfor pavements ingeneral useBit. Mastic floors

900800 Clay1rubble burnt

in situ700 Earliest sacred roads

with artificial wheel ruts?‘‘Lithostratos’’ pavement?Processional road at Cyrene

Artif Wheelrutsat AssurCobble pavementat Khorsabad

600 Trackways nearGerman ‘‘limes’’

Processional roadof SanheribProcessional roadat Babylon

500 Younger Dutch logroads

Roman Track-ways andgravel roads312 Via Appia

Amphictyonic Lawon road repairs~380!

Bitumen ceases tobe usedPersian couriersystemTracks with posthousesMilestones~?!Persian Slabpaved roads?

Bricks on rammedrubble foundation

400

300 Log roads in Prussia New road Athens-PeirauesAth Building laws

Desert roads toFayum?

Traffic regulationsRoyal RoadMilestones erected

200 Hellenistic roads Hellenistic Roads Hellenistic roads Concrete forpavements

100 6150 Puzzolanodiscovered: Milestonesgeneral

Roman log roads Rapid progress of theRoman Highway

First attempts at drainingstreets

Sacred road toTaxilaLater glass tiled


Table 2. More than 250 known uses of bitumen†109‡

AgricultureDamp-proofing and water-proofing buildings,structures, Disinfectants, Fence post coating,Mulches, Mulching paper, Paved barn floors,barnyards, feed platforms,etc, Protection tanks,vats,etc, Protection for concrete structures, Treepaints, Water and moisture barriers,~above andbelow ground!,Wind and water erosion control, Weathermodification areas.BuildingsFloorsDamp-proofing and water proofing,Floor compositions, tiles, coverings,Insulating fabrics, papers, Step treads.RoofingBuilding papers, Built-up roof adhesives, feltsprimers, Caulking Compounds,Cement waterproofing compounds,Cleats for roofing,Glass wool compositions,Insulating fabrics, felts, papers,Joint filler compounds,Laminated roofing shingles,Liquid-roof coatings, Plastic cements,Shingles.Walls, siding, ceilingsAcoustical blocks, compositions, felts,Architectural decoration, Bricks,Brick siding, Building blocks, papers,Damp-proofing coatings, compositions,Insulating board, fabrics, felts, paper,Joint filler compounds, Masonry coatings,Plaster boards, Putty, Siding Compositions,Soundproofing, Stucco base, Wallboard.MiscellaneousAir drying paints, varnishes, Artificial timber,Ebonized timber, Insulating paints,Plumbing, pipes, Treated awnings.Hydraulic and erosion controlCanal linings, sealants, Catchments area, basins,Dam groutings, Dam linings, protection, Dikeprotection, Ditch linings,Drainage gutters, structures, Embankment protection,Groynes, Jetties, Levee protection,Mattresses for levee and bank protection,Membrane linings, waterproofing,Reservoir linings, Revetments,Sand dune stabilization, Sewage lagoons, oxidationponds, Swimming pools, Waste ponds, Waterbarriers.IndustrialAluminum foil compositions using bitumenBacked felts, Conduit insulation, lamination,Insulating Boards, Paint Compositions, Papers,Pipe wrappings, Roofing, Shingles.AutomotiveAcoustical compositions, felts, Brake linings,Clutch facings, Floor sound deadeners, Frictionelements, Insulating felts,Panel Boards, Shim strips, Tacking strips,Underseal.ElectricalArmature carbons, windings, Battery boxes, carbons,Electrical insulating compounds, papers, tapes, wirecoatings, Junction box compound, MouldedConduits.

CompositionsBlack grease, Buffing compounds, Cable splicingcompound, Coffin linings, Embalming, Etchingcompositions,Extenders, Explosives, Fire extinguishercompounds, Joint fillers, Lap cementLubricating grease, Pipe coatings, dips, jointseals, Plastic cements, Plasticisers, Preservatives,Printing inks, Well drilling fluid, Wooden caskliners.Impregnated, treated materialsArmoured bituminized fabrics, Burlapimpregnation, Canvas treating, Carpetingmedium, Deck cloth impregnation, Fabrics, felts,Mildew prevention, Packing papers, Pipes andpipe wrapping, Planks, Rugs, asphalt base,Sawdust, cork, asphalt composition, Treatedleather, Wrapping papers.Paints, Varnishes, etcAcid-proof enamels, mastics, varnishes,Acid-resistant coatings,Air-drying paints, varnishes,Anti-corrosive and anti-fouling paints,Anti-oxidants and solvents,Bases for solvent compositions,Baking and heat resistant enamels,Boat deck sealing compound,Lacquers, japans,Marine enamels.MiscellaneousBelting, Blasting fuses, Briquette binders,Burial vaults, Casting moulds, Clay articles,Clay pigeons, Depilatory, Expansion joints,Flower pots, Foundry cores,Friction tape, Fuel, Gaskets, Gramophone records, Mirrorbacking, Rubbers, moulded compositions, Shoe fillers,soles, Table tops.Paving~See also Hydraulics, Agriculture, Railways, Recreation!,Airport runways, taxiways, aprons,etc, Asphaltblocking, Brick fillers,Bridge deck surfacing, Crack fillers, Curbs, gutters,drainage ditches, Floors for buildings, ware houses,garages,etc, Highways, roads, streets, shoulders, Parkinglots, driveways,Portland cement concrete underseal, Roof-deck parking,Sidewalks, footpaths,Soil Stabilization.RailwaysBallast treatment, Curve lubricant,Dust laying, Paved ballast, sub-ballast,Paved crossings, freight yards, station platforms, Railfillers, Railway sleepers,Sleeper impregnating, stabilization.RecreationPaved surfaces for:Dance pavilions, Drive-in movies, Gymnasiums, sportarenas, Playground, school yards, Race tracks, Runningtracks, Skating tracks, Swimming and wading pools,Tennis courts, handball courts.Bases for:Synthetic playing field and running track surfaces.

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many of the innumerable applications, in which it is put touse, depends to a large extent on its initial chemical compo-sition, molecular structure, and the variation of its internal-structure with respect to time. Here, we wish to restrict ourattention to the specific use of bitumen in the applications

related to roadway and runway construction as the usebitumen for different application brings to the fore complissues related to its modeling.

The exact nature~structure, chemical composition! of bi-tumen is still clouded in controversy. Different crude sourc

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subjected to different refining processes give rise to differgrades of bitumen. Furthermore, the structure of bitumkeeps on changing from when it leaves the refinery tofinal mixing in the hot mix asphalt plant due to the repeaheating and cooling during its transport. As new toolsscience are being put to use in the exploration of the tnature of bitumen, different pictures arise that bring wthem attendant problems related to modeling. While thelowing section surveys the chemical and physical analysibitumen, we have restricted ourselves to the literature, whis of relevance to the phenomenological modeling of bimen. We make no attempt here to present a comprehensurvey of the literature related to the chemistry of asphaltasphaltenes nor are we competent to do so. The interereader is referred to the works of Chilingarian and Yen@126#,Bunger and Li@127#, and Yen and Chilingarian@128,129#where he can find detailed discussions of the same.

The pioneering chemical analysis of bitumensBoussingault@45,46# ~see also a translation of this woralong with a commentary in Chapter VIII of@43#! was sub-stantiated later by Bel and Muntz in@130# and by Bel@131#.Boussingault@45,46# did some of the earlier experiments othe composition of Bitumen and classified it into two groupnamely Petrolene and Asphaltene depending upon the typsolvents used to separate these hydrocarbons. Kayser in~see@43# for details! isolated asphalt into three componenand called them thea, b, andg asphalts. He also assignedifferent molecular formulas with an increasing percentaof sulphur. Thea asphalt was supposed to be oily, thebasphalt was a solid gummy substance melting at 60°Ctheg asphalt had the same consistency asb asphalt but melt-ing at 165°C.

Linton @132# conducted extensive chemical analyses onmany as 23 types of bitumen samples provided to herPeckham~Crude Trinidad Asphaltum, Cuban Asphalt, KubResiduum, Egyptian Asphalt, Seyssel Asphaltic Rock, Tulite, Kentucky Asphaltic Rock, California Maltha, Asphaltufrom three different sources, Asphaltic Sandstone, an Asptic mineral resembling Gilsonite, Grahamite, Hard ArtificiAsphalt, Soft Artificial Asphalt, Dubb’s Artificial AsphaltRoofing Pitch, Pitch, Asphaltic Rock, and two types of Aphaltic Pavement! to determine the amount of Petrolene, Aphaltene, water mineral matter, and other organic mattethe specimens. Sadtler@133,134# commenting about thework of Boussingault’s and Linton’s remarks: ‘‘Mention wamade in speaking of Boussingault’s studies of the Bechlbronn mineral tar, of the distinction made by him of ‘ptrolene’ and ‘asphaltene,’ and of the definitions given thoterms later by Kayser. We know now that there are no wcharacterized compounds to which these names can be gbut that the materials~obtained by fractional solution withdifferent solvents! to which these names have been givenmixtures. In general, the mixtures called ‘petrolene,’ awhich is extracted from these natural asphalts by meanpetroleum ether, acetone, or common ethylic ether, is touviscid, and of great cementing quality, although soft; whthat called ‘asphaltene,’ and which is extracted from the redue with hot turpentine, chloroform, or carbon disulphide

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dry, friable, and brittle; but the proportions of these extracfrom one and the same asphalt, as has recently beenclearly shown by Miss Laura A Linton...will vary quite notably according to the choice of the solvent.’’

Continuing on the ideas of classifying asphalts basedits solubility in various organic solvents, Frankforter@135#conducted studies on Obispo and Trinidad refined asphadiscussion of the distillation and other chemical tests pformed on bitumen, until 1909, can be found in the bookPeckham@43# and a discussion of the various analytical teand methods for asphalt in the book by Sadtler@136#. Rich-ardson@137# carried out a study of the composition of Trinidad asphalt and he systematically looked into the minematter and investigated its colloidal nature. Richardson inpaper in 1915@138# discussed the relation between the cloidal chemistry and what he calledperfect sheet asphaltinthat both rely on the studies of the behavior of surfacesfilms. He also recognized the increase of the surface endue to the increase in the number of colloidal particles athus the increase of the cementing power of bitumen. Tclassification of bitumen into its different constituents wcarried out by Marcusson in a systematic way@139–143#. Heclassified asphaltic bitumen into carboids, carbenes, aspenes, asphaltic resins, oily constituents, asphaltic acids,their anhydrides. The studies by Rosinger@144# and Errera@145# emphasized the colloidal nature of asphalt on the baof a variety of physical and chemical analysis. For instanthe following observation by Errera@145# is worth repeating~here he uses the terminology proposed by Kayser abouclassification of the constituents of asphalt into itsa, b, gasphalts!: ‘‘Modern theories seem to admit in sensitive aphalt the existence of three substances chemically definea, b, g asphalt. The latter is alone sensitive to light. Whstudying the question from a colloid point of view, it woulappear more correct to say that asphalt is a ‘polydispersothe sensitive part is the one which is found in a statecolloidal dispersion. When chemically analyzed, this colldal part proves to be the richer in sulphur, the valencieswhich are a factor in polymerization. According as their dgree of association is greater or smaller, we find in aspthe intermediate stages between molecular and colloidaphalt. Sunlight seems to have a coagulating action.’’ Ringer @144# earlier also came to the same conclusion abthe coagulating effect of sunlight. Lord@146# seems to haverecognized that the colloidal character of asphalt canmodified by adding inorganic solids or clay.

Nellensteyn@147# analyzed the constitution of asphalt anremarks on the fact that different bitumens behave differenas they congeal—some crystallize while others remain amphous. He says, ‘‘Asphalt when solidifying shows a vemarked increase in viscosity without crystallization. Othbitumens show either crystallization or a general increasviscosity.’’ In this widely quoted paper, he concludes th‘‘Asphalt contains elementary carbon in colloidal form anthat this colloidal form is the essential constituent of aphalt.’’ Observations of the Tyndall effect and Brownian mtion using ultramicroscopic examinations of asphaltenes

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the synthesis of asphalt from carbon and hydrocarbons hconfirmed his findings~see also the discussion of this papby Hackford@148#!.

Kirk and Reuerson@149# discuss the important studies uto 1925 in which the colloidal nature of asphalt has berecognized. The colloidal nature imparted to asphalt bypresence of inorganic matter~Richardson in 1915@138#!, bythe addition of either clay or certain inorganic salts~Lord in1919 @146#!, or the sheer nature of pure asphalt to existboth molecular and colloidal dispersion~Rosinger in 1914@144# and Errera in 1923@145#! were discussed by themThey showed that the influence of the addition of copsulfate on the colloidal nature of Trinidad asphalts is suthat it increased the number of the colloidal particlesdecreased their size. They also showed that by dispersicertain oil asphalt in carbon disulphide, carbon tetrachlorand acetone, there were dispersions of the inorganic matas well as the organic portion of the asphalt. Their remthat ‘‘It would seem probable from these experiments tthe solvents used is at least as important a factor as the cacter of the asphalt,’’ points to the difficulties inherentmodeling asphalt.

The principles of colloidal chemistry as it applies to tvarious processes related to the petroleum industry anparticular to the preparation of asphalt emulsions werecussed by Dunstan@150# and Morrell and Egloff@151#. Nel-lensteyn@152# classified asphaltic bitumens within the cotext of the colloidal chemistry into the following groups: 1!the medium, 2! a lyophile part: the protective bodies, and!a lyophobe part: the ultramicrons. According to his obsertions, the dispersed phase consists of the last two grothese being the constituents of the asphalt micelles. Alsotalks about the stability of this system: ‘‘The stability of thwhole system in the first place depends upon the relabetween the micelles and the medium. Changes in thisbility, which are known as flocculate and peptizing reactiogive rise to areversible flocculation. If, however, the micelleitself is destroyed, the micelles cannot be repeptized, at lnot directly. In this case we have anirreversible floccula-tion.’’ The observation takes on particular significanceview of the numerous practical applications related to pomer modified asphalt and asphalt emulsion. For instancas per Nellensteyn’s view point, asphalt can be considerea two-phase system, what really causes the precipitatiothe asphaltenes during emulsification? Is it due to the chaof the interfacial tension of the micelle-medium? This ismuch practical significance as summed up by Nellenst@152#: ‘‘... it is certainly very mysterious that the coagulatebitumen easily expels the water, while molten bitumenheres very poorly to moistened surfaces of mineral mattThe irreversible flocculation as described by Nellensteynconsiderable merit especially when it comes to describthe aging of asphalt. This also explains the anomaloushavior related to some ‘‘hard bitumens’’ having excellent svice life as opposed to ‘‘soft bitumens.’’

Kalichevsky and Fulton@153# summed up the importandevelopments in research related to the chemical comption of asphalt on the lines of the classification proposed

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Marcusson@139–143#. They proposed that asphalt could bconsidered as a three constituent mixture consisting ofphaltenes, asphaltic resins, and oils; asphaltenes impahardness and high melting point to the mixture, aspharesins responsible for the ductility and tensile strength ofmixture as well as playing the role as a stabilizer, andoily constituents acting as the dispersing phase~see alsoSchneider and Just@154#!.

Mack @155,156# continued along the lines of Marcusso@139–143# but considered only two constituents for the aphalt: asphaltenes as the dispersed phase and the mixtuasphaltic resins and oils as the dispersion medium. He uEinstein’s formula in trying to find the molecular weight othe asphaltenes by using the relation between relative visity ~ratio of the viscosity of suspension and the viscositythe dispersion medium! and concentration. However, his etimate of the molecular weight of the asphaltenes andgeneralization that asphaltenes have relatively low molecweight do not agree well with the recent estimates ofconstitution and makeup of asphaltenes.

Baskin @157# in his paper that was well ahead of its timdiscusses the chemistry of asphalt and the significance osource of asphalt on its properties. His pertinent observatrelating to the specific properties of asphalt and their depdence on the source, the oxidation of asphalt and onvarious empirical tests which were used during his tim~which are still in use now! can be of assistance in modelinasphalt or, for that matter, in writing out specifications fpractical engineering purposes. Various studies carried outhe chemistry of asphalt during and before the time of tpaper seem to have concluded that asphalt is a mixtHowever, the influence of the individual constituents on toverall behavior of asphalt depends to a large extent onsource, the manufacturing processes, etc, and this wasclearly enunciated by Baskin. His remarks, ‘‘Asphalt is insense a distinct group by itself, but rather a portion ofcrude petroleum containing several groups in form of a mture or mutual solution. The character and makeup of egroup to start with depends to a great extent on source.any given consistency, the oily constituents in a Californcrude residue are different in physical and chemical propties from those extracted from the same consistency Panresidual. This likewise, undoubtedly holds true of the resand the asphaltenes,’’ are right on the mark and the recuse of biomarkers in characterizing asphalts from a particcrude have proved that this is indeed true.

Hillman and Barnett@158# separated asphalt into differenconstituents by means of fractional precipitation and ccluded that the molecular weight of asphaltenes fromcracked residue was lower than that of the uncracked resiNellensteyn@159#, building on the work done by him on thcolloidal structure of bitumens, used microscopic and ultmicroscopic pictures of bitumen, natural and artificial aphalt, and tar and found that the colloidal structure of natubitumen is the same as the artificial bitumen mixed wmineral fillers~see also Nellensteyn and Kuipers@160# for adiscussion of the different asphaltenes!.

Identification and the classification of the bitumen by uing The Panchrometerwhich essentially uses the compariso


Table 3. Summary of composition and characterization as found intypical 85Õ100 straight reduced asphalt†169‡

ComponentWt %range

PhysicalNature

Density20Õ4°„Av…

MolecularWeight„Av…

Saturates 5–15 Colorlessliquid

0.87 650

Naphthene-Aromatics

30–45 Yellow toRed liquid

0.99 725

Polar-Aromatics

30–45 Blacksolid

1.07 1150

Asphaltenes 5–20 Brown toBlackSolid

1.15 3500

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and in his subsequent paper in 1969@169#, a systematic studyof the composition of asphalt was conducted leading to ware now called asCorbett fractions. Corbett and Swarbrick@168# used a new procedure for the separation of asphaltits different constituents~they separated asphalt into threconstituents such as paraffins plus napthalenes, aromaticand asphaltenes! and then used nuclear magnetic resonanelectron paramagnetic resonance, infrared radiation, mspectrometry, ultraviolet radiation, etc, to study the charteristics of each of these fractions. Using these techniqthey conducted the thin film oven test that is normally usfor simulating the short-term aging behavior of asphalt. Thfound that there was no change in the paraffin plus naplenes constituent but some portion of aromatic oils gets cverted into asphaltenes. Corbett, in his landmark pape1969 @169#, described a method for determining the compsition of asphalt based on fractionation into its four genecomponents. The following tables taken from this paper whelp us in understanding the composition of asphalt anddependence on the source of crude. For instance, Tabshows that asphalt is a mixture of four constituents, twothem are liquids and the other two are solids. Also, a peruof Table 4 makes it clear that the empirical tests usedcharacterizing asphalt such as penetration and ring andsoftening point test do not tell us anything about the contency of asphalt and, to the contrary, mislead one to belithat all the asphalts having either the same penetrationsoftening point are the same.

Within the context of modeling the behavior of asphalt,is clear from the above studies thata! asphalt is a mixture ofdifferent reacting and diffusing components,b! asphalt fromdifferent sources of crude have different amount of constents and possibly with different ability for reactions, andc!each and every manifestations of change in the behavioasphalt~such as aging, etc! is due to the inter-conversion oone type of constituent to the other type.

In a related development, Griffinet al @170# and Simpsonet al @171# looked into the influence of the chemical compsition of asphalt on various empirical measures such as petration, softening point, penetration index, and on somethe physical properties such as viscosity and viscostemperature susceptibility. Assuming asphalt to be a linviscoelastic material, they also looked into the influencedifferent constituents on the complex modulus. Sinceintent in this section is to discuss the developments inresearch related to the chemical composition of asphalt,desist from making observations about the assumptionspecific model~see the definitions proposed by Van der Po

Table 4. Effect of crude source on composition†169‡

Venezuelan USA Mexico Mid-East

Saturates 14.0% 10.5% 8.5% 8.0%Naphthene-Aromatics

34.5% 38.5% 29.8% 38.5%

Polar-Aromatics

36.3% 33.4% 42.6% 37.0%

Asphaltenes 14.1% 16.8% 28.3% 15.5%Penetration@77°F

90 92 88 85

Soft Point,°F

114 114 116 115

of the color of the bitumen being tested with that of a stadard sample was carried out by Attwool and Broome@161#.This method is capable of detecting the changes in the bmen during aging and the settlement of the straight run bmen, changes that are undetectable by routine tests suthe penetration test, viscosity test, softening point test,~See also Csanyi and Fung@162# for a discussion on thepossible use of ultraviolet light identification as a rapmeans for determining the constituents of asphalt.!

In a detailed study on the dependence of the rheologbehavior of the bitumen on molecular considerations sucthe molecular structure, weight, etc, Oliensis@163# summa-rized the complex response of the bitumen when subjectehigh temperatures: 1! there is an increase in the moleculweight at certain stages and decrease at certain other sand 2! the complexity of the internal structure is increasedcertain stages and simplified at certain other stages. Uthe more consistent parameters~as per Oliensis@163#! suchas dehydrogenation~increase of the carbon to hydrogen rtio! and increase of flocculability, Oliensis classified bitmens into three groups depending upon the heat treatmewhich they are subjected during their manufacturing procsuch as steam or vacuum refined asphalts, cracked aspand coal tars, and coal tar pitches produced by the destive distillation of bituminous coal~see also Darmois@164#on the influence of temperature in the change of the collostate of bitumen and Mariani@165# on the use of the methoof chemical attack in determining the constituents of natuasphalt and the influence of temperature on it!.

Hughes and Hardman@166# tested 20 different asphaltfrom different crude sources in an attempt to find some crelation between the chemical composition and the physproperties of asphalt. They considered only asphaltenespetrolenes in their study. Using the values of weight percasphaltenes, and viscosity and viscosity-gravity constanthe petrolenes, they obtained a working correlation betw77°F penetration, and with the degree of complexity offlow ‘‘n.’’ Similar correlations have been derived by Kinnaird @167# relating 77°F penetration and ring and ball soening point with the percentage of paraffinic constituentasphalt.

Significant contributions to the understanding of the coposition of asphalt and its change over time when subjecto specific tests or field applications have been made by Cbett. In their paper in 1958~Corbett and Swarbrick@168#!

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@172#! which have serious drawbacks that will become aparent from discussions given later. The importance ofwork by Griffin et al @170# and Simpsonet al @171# lies intheir suggestion concerning the need for the blending ofdifferent constituents to get the desired properties~thoughthe properties used by them to benchmark were empirmeasures which did not capture the complex makeup oftumen!.

At this stage, we need to review the early studies onphysical properties of asphalt as studies after the 1960specially those published within the realm of use by asphpavement technologists, which were mostly correlationschemical composition with the rheological behavior. Netypes of viscometers were developed but measurementsmade mostly within the linear response regime and thosewere carried out beyond such a regime were yet descrunder the presupposition that the material can be descrby a linear model~elastic or viscoelastic! for asphalt. Start-ing from the work of Boussingault in 1837 to the workCorbett in 1969, it had taken over 100 years to come to gwith an acceptable description for the compositional strture of asphalt~see @173# for the ASTM standard for tesmethods for separation of asphalt into four fractions!. Mostof the works after 1970 were carried out by geochemistspetroleum chemists interested in the chemical structureasphaltenes and with the advent of new tools for studyingsame, there have been numerous publications related toissues. For more details related to the use of sophisticequipments in unraveling the chemistry of asphalt, the reais referred to@174–195#, and for studies related to the relationship between the chemical constitution of asphalt apavement performance, the reader is referred to@196–208#.Our aim, as we emphasized earlier, in discussing the comsition of asphalt and the interaction mechanisms of its cstituents is to highlight the complexity that confronts a moeler and being aware of the variety of issues that influethe structure of asphalt, we would be in a better positiondescribe phenomena such as aging, fracturing, and heaand in turn we can describe better the many complexsponse characteristics exhibited by bitumen.

4 MODELING OF BITUMEN

4.1 Early studies on the physical analysisand modeling of bitumen

In this section, we review some of the important early devopments related to the physical analysis and modelingbitumen. As we discussed earlier, the applications of bitumare many. With respect to the issues related to the usbitumen as pavement material, there have been manywhich have been developed. Most of these tests do not bout the quintessential nature of bitumen related to its woing conditions. For instance, the penetration test widely uin characterizing bitumen is supposed to measure its ‘‘csistency,’’ the softening point test the transition of asphfrom its solid-like behavior to fluid-like behavior, etc. Eacof these tests leads to definitions of material parametersare ad hoc and leads to contradictory measures for descrthe behavior of bitumen under different working condition

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Also, most of the present models falling within the purvieof SUPERPAVE in the USA and similar initiatives in Europare for linearly viscoelastic materials. In the following setion we shall discuss issues relevant to the tests for chaterizing the nature of bitumen.

We start with a discussion of some early penetratingservations concerning the response of materials in whichbehavior of asphalt served a defining role. In his immorclassic titled theTheory of Heat, Maxwell @209# discussesthe quintessential features of solids and liquids. He drawsdistinction between notions such as asoft solidandviscousfluid thus: ‘‘Thus, a tallow candle is much softer than a stiof sealing wax; but if the candle and the stick of sealing ware laid horizontally between two supports, the sealing wwill in a few weeks in summer bend with its own weighwhile the candle remains straight. The candle is therefa soft solid, and the sealing wax a very viscous fluidMaxwell recognized the viscoelastic nature of bitumen, ahis comments concerning the same are worth documen‘‘What is required to alter the form of a soft solid is a sufficient force, and, this when applied produces its effectonce. In the case of viscous fluid it istimewhich is required,and if enough time is given, the very smallest force wproduce a sensible effect, such as would require a very laforce if suddenly applied. Thus a block of pitch may behard that you cannot make a dent in it by striking it with yoknuckles; and yet it will in course of time, flatten itself by iown weight, and glide down hill like a stream of waterHere, by pitch, Maxwell is referring to bitumen.

Maxwell is clearly making a case for understanding tnotion of time scales with regard to the response of bitumits short time or rather instantaneous response~being rappedby the knuckles! is that of a highly elastic solid like materiawhile its long time response is that of a viscous fluid limaterial.

A case could be made that even in the case of the canif we were to wait a sufficiently long time, the candle woueventually deform, that is it would not be able to sustainshape indefinitely and here by sustain8 we mean to remain inthe same position. Such a point of view would imply thatthe materials are fluids, there being no solids. Howevbased on the time scale of observation, if we can discerndeformation we could call such a response a solid-likesponse. Even given our ability to measure ever finer spascales there are objects that do not suffer discernable demation and these we would refer to as solids, the candlthe case of Maxwell. We also see in Maxwell’s descripti

8As we have observed earlier, we can at best talk about solid-like and fluid-like beior, the behavior being decided by a natural time scale associated with the maversus the time scale of the experiment. There is the point of view that no materiasustain indefinitely~forever! applied shear stresses, ie, all materials are essentifluid-like provided the experimental time scale is sufficiently large. However, wedefine, for mathematical purposes, a material that can sustain the applied shearIn this context we provide the sense in which we are using the term ‘‘sustain’’ amits various meanings~see @8#!: To cause to continue in a certain state, to keepmaintain at a proper level or standard, to preserve the status of, to endure wigiving way, to bear up against, withstand, to bear, support, to hold in position, toable to maintain a fixed position, to withstand weight.

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no confusion between solidity, fluidity, and viscosity thseem to plague the bitumen literature much after the writiof Maxwell.

Trouton @210# conducted torsional experiments on cyliders of pitch and found that the relation between ‘‘the rateflow of the material under shearing stress cannot be in simproportion to stress,’’ thereby establishing the noNewtonian nature of the material. In addition to these tsional experiments Trouton@210# also studied the uniaxiacompression of a cylinder of pitch, the free-flowing streaof the material, and the bending of a horizontal rod or bethat is supported at its end, due to its own weight. In 19Frankforter @135# studied the effect of temperature on thproperties of asphalt. Predictably, he found that as the tperature increased, in its solid-like response regime, theterial became less brittle~more ductile!.

In his landmark paper on ‘‘Fluidity and Plasticity,’’ Bingham @211# appeals to experiments on Trinidad asphalt toto dispel some longstanding prejudices of his time thoughclassification would hardly be found clear or acceptableday: ‘‘Perhaps no one any longer regards viscosity as bean additive property in true solutions, yet, due presumablforce of habit, data on flow are still expressed in termsviscosities instead of more useful fluidities. The differenceparticularly striking in the study of colloids, and may billustrated by a wide variety of suspensions or emulsions.will take as our example the dispersion of Trinidad asphalbenzene studied by C Richardson. The first addition ofphalt to benzene are of very slight effect on the viscosity,as the concentration increases the effect is greater and gruntil finally the viscosity becomes infinite. This behaviorquite general and remained unexplained until the fluiditwere plotted against the concentration, when it was obsein this and many other cases the fluidity-concentration cuis linear. The mystery as to why the viscosity varies mumore rapidly in the more concentrated solutions is therefexplained by the mathematical requirements of the cwhich may be summed up in the following law. Whether tsolution be a true solution or merely a colloidal disperseach successive equal increase in the concentration ofsolute decreases the fluidity by a constant amount untilfluidity reaches that of the pure solute, ie, zero in the casmost colloids.’’ We would be hard pressed to find many whare trained today in mechanics to accept the concept ofidity as advanced by Bingham. His comments seem to imthat viscosity becoming infinite leads to lack of fluidity~bythis he has to mean a rigid body that is a fluid, and nosolid!. If we were to take a Navier-Stokes fluid and let its tviscosity tend to infinity would we get a solid or a fluimodel with infinite stresses? In fact, in allowing the viscoties to tend to infinity, if we can indeed model a solid~andthis we cannot! do we get a deformable elastic solid materor a rigid solid? The fluidity or solidity of a material is related to the time scales for the response and the explanaof Bingham are way off the mark. Also, it depends on whone means by a fluid and a solid, and fluidity and solidalso is one to construe lack of fluidity as solidity? As wobserved earlier, if one takes the point of view that ‘‘eve

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thing flows’’ given sufficient time, and that which is capabof flow is a fluid, then there would be no solid. However,one takes such a point of view, we cannot at the same ttalk about a Young’s modulus or a Shear modulus associwith a solid, all that we have is a fluid with a sufficientllarge viscosity. A solid is not a fluid with a sufficiently largviscosity, and this point cannot be overemphasized, butis an error which is all too commonly, in fact invariablmade even in the present day. Characterizing a materialpends on the time scales for the experiments and obsetions. We cannot get into these issues here and they arecussed at great length elsewhere~see Rajagopal@212#!.Bingham also gives a somewhat narrow view of thesponses of liquids and gases, a view which is quite untennow: ‘‘When viscous substance, either a liquid or a gassubjected to shearing stress, a continuous deformation rewhich is, within certain restrictions, directly proportionalthe shearing stress. This fundamental law of viscous flow.’’ Of course, his caveat ‘‘within certain restrictions’’ coulallow him to wiggle out of such afundamental lawfor allgases and liquids, but given that he only discusses linviscous and plastic responses~and his notion of plasticity isnot what is meant by that term with regard to the responsmetals or polymeric solids! suggests that it was not nonlinearities in response but some other restrictions that he hamind.

Bingham recognizes that liquids could have elasticity,idea in keeping with the earlier ideas of Maxwell@213# thathad led Maxwell in developing his model for a viscoelasfluid. However, his work does not have the clarity or depthMaxwell’s.

Bingham uses pitch and clay to differentiate between mterials with and without a ‘‘yield’’ stress. The following remarks by him suggest that pitch, if its deformatioin is viewin a sufficiently small time scale, would exhibit a yield streif the magnitude of stress is sufficiently small, but given timwould flow under the slightest of stress. He states, ‘‘Theresome flow at every shearing stress, but at the smallest shing stresses the time required for a given amount of flowvery great, hence pitch, which would require great stresseshear rapidly, nevertheless moves continuously so thaloses shape and takes the shape of the containing vessethis context, it is worthwhile to comment here on the pitdrop experiment, perhaps one of the longest running expments in the history of science. This experiment was setuthe foyer of the Department of Physics, University of Queesland in Brisbane by Professor Thomas Parnell in 1927.experimental setup in the words of Edgeworthet al @214# isas follows: ‘‘The pitch was warmed and poured into a glafunnel, with the bottom of the stem sealed. Three years wallowed for the pitch to consolidate, and in 1930 the seastem was cut. From that date, the pitch has been alloweflow out of the funnel and a record kept of the dates whdrops fell...The pitch in its funnel is not kept under any spcial conditions, so its rate of flow varies with normal, sesonal changes in temperatures.’’ The following figure~Fig. 1!

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shows the experimental setup@215#. Figure 2 shows the pitchat room temperature, before and after being hit with a hamer @215#.

The table~Table 5! from Edgeworthet al @214# showingthe record of pitch drops is given below.9

Thus, any one walking into the Queensland laboratand gazing upon the pitch drop experiment might concluthat pitch is a material that does not flow, or that it is a sorather than that pitch is a fluid that flows ever so slowunder those conditions. Thus, to ascribe a ‘‘yield’’ stresspitch might or might not be accurate; it really depends ontime scales of interest. However, from a philosophical stapoint, the assumptions that a material is a fluid~and here bya fluid we mean a body that cannot sustain a shear stress! andthat it has a yield stress are contradictory. All that one csay is that during the time interval of interest, no deformatwas detectable with the instruments at hand. It is quite psible, in fact most likely, that sophisticated measuremtechniques might show minute deformations~see also Chapter XI of @216# and @217#!.

Mack @156# studied the response of asphalt at variotemperatures. He measured the viscosity by consideringflow of asphalt between two concentric cylinders due toinner cylinder moving axially, and found that viscosity dcreases with increasing temperature. He also developemethod for determining the molecular weight of the solufrom viscosity measurements as there is a one to one colation between the viscosity, the molecular weight of the sute and the concentration in weight percent of the solute

The importance of recognizing the source of asphaltthe method by means of which it is processed, in the choof an appropriate model, was recognized by Pittman

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Traxler @218#. Pittman and Traxler@218# recognized that asphalt from the same source, if treated differently, could leto products whose response could vary dramatically: ‘‘Fther, asphalts processed in the same manner from petrofrom different sources rarely have the same physical propties; in fact, those prepared from the same petroleum canmade to differ in properties by changes in refining proceduFor these reasons, in any investigation dealing with asphthe source, method of processing, and some commonly msured properties should be given to aid in identifying tmaterial...’’ Traxleret al @219# found that the simple formulaproposed earlier by Saal and Koens@220# for determining theviscosity of asphalt based on the penetration depth had tmodified for asphalt from different sources, etc: ‘‘Asphaobtained from different sources or those prepared or pcessed differently require distinctly characteristic constain the equation proposed by Saal and Koens.’’ Saal aKoens@220# had proposed the formula

h55.133109

P1.93 , (1)

whereh denotes the viscosity andP denotes the penetratioby a 100 gram load applied for 5 seconds. In a later stuSaal and Labout@221# criticized the earlier study of Traxleet al @219# for having overlooked elastic effects, which theclaimed was the reason for the results of Traxleret al @219#disagreeing with the formula~1!.

A very detailed investigation of the ‘‘plastic’’ properties oasphaltic bitumen was carried out by Saal and Koens@220#.By ‘‘plastic’’ they mean that the viscosity of asphaltic bitumen was dependent on the shear stress, and this is notis currently understood as theplastic response of metallic orpolymeric solids. They also studied the relation between

Table 5. Record of pitch drops†214‡

Year Event

1930 The stem was cut1938 ~Dec! 1st drop fell1947 ~Feb! 2nd drop fell1954 ~Apr! 3rd drop fell1962 ~May! 4th drop fell1970 ~Aug! 5th drop fell1979 ~Apr! 6th drop fell

9The seventh drop fell on July 1988 and the eighth drop fell on November 2000@215#.

Fig. 1 Pitch drop experiment@215#
Fig. 2 Pitch, before and after being hit with a hammer@215#

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notion ofplasticityand the composition of asphaltic bitumeThey seem to have recognized that the viscosity of asphbitumen depended on the pressure. Thus, viscosity, accorto them depended on both the shear components andnormal components of the stress and thus they had aimplicit model in mind for the material. The paper was denitely pioneering for its time. However, that the viscositya fluid could depend on the normal stress was recognizeStokes in 1845@222#. Saal and Koens@220#, however, as-sumed that the viscosity could depend on all the componof the stress tensor.

In their study of the fluidity of materials such as resabiotic acid, and asphalt, Bingham and Stephens@223# inves-tigated a very interesting aspect of rheology, namely thefect of pressure on the fluidity of the material. It might bworthwhile at this juncture to deviate from our discussionthe modeling of asphalt and take a short detour concernthe relevance of pressure on the fluidity of bodies. As mtioned earlier, Stokes@222# recognized that the viscosity oliquids could depend upon the pressure and in his semarticle he derived the general Stokesian fluid model~thatsatisfies the requirements of frame indifference and matesymmetry!. In fact, he felt the need to discuss and justify tsituations when viscosity could be considered a constantstates, ‘‘Let us now consider in what cases it is allowablesuppose thatm to be independent of the pressure. It has beconcluded by Dubaat from his experiments on the motionwater in pipes and canals, that the total retardation ofvelocity due to friction is not increased by increasing tpressure...I shall therefore suppose that for water, by anafor other incompressible fluids,m is independent of the pressure.’’ However, Stokes recognized that the viscosity codepend on pressure in other situations. A rather interesmistake was made by Coulomb in 1785@224# in the differ-entiation between a fluid and solid based on whetherfrictional properties depended on pressure~see Rajagopa@212# for a discussion of the same!.

There are many physical situations where the pressare so high that significant changes can take place invalue of the viscosity, so much so that glass transitionsupposed to take place in the case of liquids~see Szeri@225#!. This is thought to be the case in elastohydrodynamlubrication. The pressure dependence on the viscosityflows at exceedingly high pressure necessitates a carefuevaluation of the classical no slip boundary condition at ipermeable surfaces. At such high pressures, it is possiblethe fluid stick-slips at the boundary. A detailed discussionthe relevant issues can be found in Rajagopal@212#.

A great deal of work has been carried out concerningvariation of the viscosity with pressure and much of the wotill the 1930s can be found in the book by Bridgman@226#~see also@227#! devoted exclusively to the physics of matrials at high pressure. Interestingly, some of this work dewith different types of asphalt. Andrade@228# suggested thathe viscosity is related to the pressure and temperature infollowing manner

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There is a significant body of experimental literature sinthe work of Andrade and Bridgman that document the vation of the viscosity with the pressure for a variety of fluid~Cutler et al @229#, Griestet al @230#, Johnson and Camero@231#, Johnson and Greenwood@232#, and Johnson andTevaarwerk@233#!. Recently Hron, Ma´lek, and Rajagopal@234# have studied the flows of fluids with pressure depedent viscosities in special geometries. Rigorous mathemcal results concerning the existence and uniqueness of flof fluids whose viscosity depends both on the pressurethe shear rate can be found in Ma´lek, Necas, and Rajagopa@235#, and numerical results pertinent to special flowssuch fluids can be found in Hronet al @236#.

In the case of asphalt, the increase in the viscosity duan increase in pressure is much greater as the molecweight of the asphalt increases. This is because the freeume of higher molecular weight polymers is smaller thanlower molecular weight polymers ‘‘so that at equal compressibility the decrease of this volume caused by pressurelatively greater in the former case’’~see Saal and Labou@237#!. Sample data depicting the influence the pressurethe ‘‘stiffness’’ of asphalt is tabulated in Table 6.

It is well known that asphalts are extremely sensitivetemperature. Given that there are a great variety of asphit is not surprising that a single relationship cannot be estlished between the viscosity of asphalt and temperatSince asphalts are generally viscoelastic, it is not sufficienknow how the viscosity changes with temperature. We aneed to know how the elastic response changes with tperature. Schweyer, Coombs, and Traxler@238# have shownthat for a class of asphalts the viscosity-temperature relatship is linear in a logarithmic plot, with the viscosity decreasing with temperature. This sort of behavior is in keepwith that of most liquids. They also studied the effectadding mineral powder to asphalt and then varying the teperature. Their experiments, for the types of materials twere considered, indicated that the percentage changes iviscosity, for a given change in the temperature, were neidentical in both cases. For viscosity-temperature relatiships for a variety of asphalt found in the earlier literatuthe reader is referred to@239–241#.

Pfeiffer and Van Doormaal@242# investigated the thermaproperties of asphalt and developed a temperature suscbility index that provided a measure for the departureasphalt from purely viscous behavior. Their work containreference to an unpublished experiment of Adriani and Liburg that clearly brings out the viscoelastic nature of asph

Asphalt is an inexpensive insulator and is used extsively in insulating transformers and as an impregnating mterial for condensers. Given that it does not mix with watit is currently considered as a possible source of insulatiounderwater cables. Notwithstanding such important appltions, there are not that many studies concerning the elecal properties of asphalt. One of the few studies concernthe electrical properties of asphalt that provides some inmation regarding its dielectric strength, electrical conduct

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ity, dielectric loss, and dielectric constant is that by Saal,Meinema, and Blokker@243#. They measured the dielectricstrength~‘‘the minimum value of the electric field intensityat which a disruptive discharge occurs in the material underspecified conditions’’! in between parallel plates and spheri-cal electrodes for a variety of bitumens, at different tempera-tures. They also measured the dielectric losses due to heatingand the dielectric constant at different temperatures. Theyfound that the heating losses are negligible if the frequencyof the alternating current is low, and further, the losses in-crease with temperature. Given the importance of asphalt asan insulator, there is a great need for characterizing its elec-trical properties and this in an area that is crying out forfurther research. We now turn to a more detailed discussionof the mechanical response characteristics of asphalt.

Around the same time as the studies by Traxler, Saal, andPfeiffer and others, Houwink@244,245# tried to obtain a cor-relation between the elasticity~Young’s modulus! of asphalt~this of course presupposes that asphalt in linearized elasticmaterial which we now know is a totally untenable sup-position10! and the interatomic forces. He also provided aformula for the computation of the rupture energy, carryingout the calculation based on just a pair of atoms and the forcebetween them. Houwink suggested that the higher the energycontent of the internal bonds, the higher the elasticity~seealso Houwink@246#!.

Another early study that presumed that asphalt can bemodeled as a linearized elastic solid was that by Grime andEaton @247# wherein they determined the Young’s modulusof asphalt by considering flexural vibrations. We shall notdiscuss this or other studies that assumed that asphalt was alinearized elastic solid as such an assumption is unrealistic,the aforementioned studies being mentioned merely to indi-cate that early studies made such an assumption.

When asphalts are left to themselves, they develop inter-nal structure thereby leading to a change in theconsistencyof the material. Traxler and Coombs in 1937@248# found that

the development of internal structure inhibits the flow of tmaterial. They also found that the structural developmenmuch more rapid in air blown asphalt than in asphalt thasteam or vacuum refined. They attribute the formationstructure to sol-gel transformation that takes place as theterial ages. Stress relaxation can occur in viscoelastic mrials that do not age. Also, stress relaxation can occur in epurely elastic materials that age. In the case of viscoelamaterials that can age, stress relaxation can have disorigins. Recently, Rajagopal and Wineman@249# have triedto delineate the distinction between the stress relaxationto aging and that which is not related to aging. As aging isimportant issue in asphalts, we discuss this aspect in sdetail in Section 5.

In an attempt at delineating the physical constants ofphaltic bitumens used in a wide variety of applications, Set al @250# performed a variety of experiments and detemined the specific gravity, coefficient of expansion, specheat, thermal conductivity, permeability to water vapor, sface tension, and total surface energy of a particular clasasphaltic bitumen. Of the many findings, of particular inteest is their recognition of the dependence of the crude soon the specific gravity of asphalt, the correlation betweenincrease in the coefficient of expansion with the hardnesthe bitumen~measured by its penetration value!, and theirperspicacity in recognizing the limitations in using Ficklaw for describing diffusion of water vapor through aspha

Broome@251#, in a review of the measurement of the floproperties of bitumens, cautioned against the use of thepirical tests on bitumen such as the penetration test, ringball softening point test, etc, as they do not test bitumens,the temperature at which they will exist in the road.’’ Hcriticism that tests need to be carried out close to the opating conditions are well taken as asphalt is extremely ssitive to changes in temperature as well as deformatWhile he was on the mark in his observations concerning

Table 6. Influence of pressure on the stiffness of different asphalts at temperatures between 20°C and 60°C„from 381 in †237‡…

Asphalt

Measuring temp, °C

Pressure P, kgÕcm2

Origin Type Pen./25°C

TempR&B,

°C

P.I. Asphaltenes,% wt

0 100 200 300 400 500

RatioStiffness at pressure P

Stiffness at pressure PÄ0

Borneo Newton-iantype

47 47 22.2 1.2 25.0 1 4.0 13 45 167 59740.0 1 - - 20 59 17049.0 1 2.2 5.8 15 37 90

California Sol type 54 48 21.8 5.1 20.0 1 2.5 6.6 17 44 11540.0 1 2.1 4.5 9.2 19 4049.4 1 1.8 3.7 7.1 14 27

Venezuela Sol type 44 55 20.3 15.5 20.9 1 2.2 4.9 11 24 5430.0 1 2.0 4.2 8.5 18 3640.0 1 1.9 3.6 7.0 14 2650.0 1 1.8 3.1 5.5 10 19

Venezuela~blown!

Geltype

35 86.5 14.4 28.9 60.0 1 1.7 3.0 5.2 9.1 16

itsthens

in-

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need for characterizing the properties of asphalt nearworking conditions, he muddied issues by characterizingflow properties of bitumen by introducing imprecise notiosuch asdegree of plasticityandthixotropyand describing thechange in the behavior of bitumen when subjected to

mea-rial.able.ng’s

10Unfortunately, even today in 2002, there are numerous studies that supposedlysure the Young’s modulus of asphalt for different intervals of response of the matEven if asphalt were an elastic solid, such an approximation would be unaccepGiven the nonlinear viscoelastic nature of asphalt, such measurements of Yomodulus are devoid of meaning in that they cannot characterize the body.

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crease in temperature as transition fromsolid—plasticsolid—liquid, andplastic—liquid. His classifications do littleto help understand the behavior of asphalt.

The importance of making measurements which reflthe fundamental properties of bitumen on the one handthe use of these results in developing specifications relatethe actual behavior of bitumen in the field conditions onother hand were succinctly expressed by Lee and Wa@252#. They faulted the use of the empirical measurememade on bitumen using the penetration test, softening ptest, ductility test, etc, and went on to make measuremendifferent varieties of bitumen using a coni-cylindrical vicometer. However their use of the terminology ‘‘plastic liuid’’ for a certain type of non-Newtonian response of asphconfuses rather than clarifies the type of material that aspis. Their point that most earlier descriptions of asphaltinadequate, in one manner or another, is right on targetwhile their following remark from the vantage point of ouunderstanding of mechanics over six decades later msound unnecessarily confusing it nonetheless pinpointssalient difficulties associated with characterizing asphalt:attempting to define plasticity for purposes of physical msurements the idea has become prevalent that a plastic mrial is one for which a yield value can be determined. Tvalue represents the stress above which continuous defotion occurs, and below which there is no continuous demation. No satisfactory term, however, has been generadopted for describing a liquid~a material having zero yieldvalue! for which the rate of shear is not directly proportionto the shearing stress. Asphaltic bitumens generallywithin this category. Such terms as non-Newtonian flopseudo-plasticity, quasi-viscosity, and structural viscosityinconvenient when used to compare materials that diffetheir degree of deviation from ideal flow. It has consequenbeen found desirable to refer to the plastic flow of such mterials even though they have a zero yield value. It is csidered that if such an artificial division is necessary the teplastic solidshould be applied to materials that have a denite yield value and the termplastic-liquid to which havenot.’’ The problem lies with their not recognizing that this scalled yield is a question of time scales. We repeat onagain, in absolute terms, given sufficient time a fluid canhave the notion of ‘‘yield’’ associated with its responsHowever, depending in the time scale of observation, it mseem to exhibit ‘‘yield.’’ Their attaching more significancethe implications of the load-deformation tests for bitumrather than purely concentrating on the temperature-viscocoefficients for characterizing bituminous layers in field coditions is noteworthy.

Ford and Arabian@253# proposed a simplified form of aconi-cylindrical viscosimeter for measuring the deformatiproperties of asphalts. They plotted the changes in ratdeformation with deformation of asphalt when measureding this equipment and they suggested the use of threestants such as coefficient of viscosity, coefficient of elasticin shear, and coefficient of internal slip to describe the coplex behavior of asphalt~see also Poole@254,255#!. Such

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classifications do little by way of helping describe asphalt,for that matter make little sense from our current perspecof mechanics.

Some studies pertaining to the modeling of gunnedphalt were published, notably by Broome and Bilm@256,257# in 1941. Gunned asphalt essentially consists inblend of bitumen with mineral matter and instead of applyiit in the molten state, the mixture is pulverized and spraythrough a flame-gun on the surface to be treated. This pcess mimicked the ‘‘metallising’’ process prevalent at thtime. The end product of such a complex process leadmaterial response that is quite different from asphalt pduced in other processes thereby leading to further clenges in modeling. The relevant response characteristicgunned asphalt that need to be taken into account in its meling is discussed at length by Broome and Bilmes@257#.They used Burger’s model@258# and the Scott Blair andCoppen@259,260# equation for modeling the rheological response of gunned asphalt and found that the equationposed by Scott Blair and Coppen (h5Ss21t1 for a truefluid, n5Ss21t0 for a solid andc5Ss21tk for an interme-diate material whereS is the shearing stress,s is the shear,tis the time,h is the viscosity,n is the modulus of rigidity,andc andk are constant. Presumablyc andk are supposedto measure ‘‘firmness’’ and the ‘‘coefficient of dissipation’!fitted their experimental data reasonably. From our stapoint, such a model leaves much to be desired. Their dission related to the notions of fluidity and solidity, andviscosity and plasticity is worth recording here: ‘‘For eample, a road~or other bituminous structure! will deformunder the slightest stress if it is made with a bitumen whis a true fluid even if its viscosity~firmness! is very high,whereas it will deform much less easily if the bitumenplastic and, therefore, not a true fluid. Since the value ofk is1 for a true fluid and 0 for a true solid, this implies that thvalue of k for the mixture employed should be as lowpossible other things being equal.’’ The point that they atrying to make but not with much clarity, is that one couhave a fluid that is highly elastic and one could have a sothat is soft. In this regard we wish to bring the readeattention to the comments of Maxwell which we quoted elier and which is much more to the point and recognizaccurately the nature of a viscoelastic material.

There have been a lot of studies faulting the use of epirical measurements such as the penetration test, ringball softening point test, ductility test, etc~these are mostlyrelated to customer specifications in insuring uniformitythe product being used! in characterizing the mechanicaproperties of asphalt. We also share the need for develoa rational theory for describing the response of asphaltthe need for less reliance on such empirical correlatioThus, we shall discuss some of the important studies thataway from empirical correlations and pave the way for tdevelopment of a proper theory for the response of asp~see for instance the extensive summary of 325 sampleasphalt from 105 refineries in the study by Welborn and Hstead@261#!. Some of the investigations were pioneering ftheir time in the sense that most of the issues discussed

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relevant even today as numerous correlations are curreused between experimental measurements and the paeters used for describing the material, and in the analysisdesign of asphalt concrete pavements. As an example ofrelations in vogue, we shall look at the penetration tesSection 4.2.

Traxler et al @262# tried to interpret the penetration tesring and ball softening point test, and the ductility test withthe purview of the rheological characterization prevalenthat time and concluded that apart from the softening potest, none of the other tests correlated with what they defias ‘‘consistency.’’ Much of the terminology that they usewell as many of the concepts that they introduce are nouse today. For instance, they defined a Newtonian liquidbe a simple fluid and a non-Newtonian liquid to be a coplex fluid, but such terms by themselves do little by waymaking the modeling any clearer. They also introduced tesuch as ‘‘pliability’’ ~reciprocal of consistency! and ‘‘flaccid-ity’’ ~reciprocal of elasticity!, concepts of little or no use incharacterizing a viscoelastic or a viscoplastic material. Tuse the term ‘‘consistency’’ as defined by Bingham@263#.

Let us turn our attention to asphalt concrete for a momeas understanding asphalt has a bearing on understandinphalt concrete. The response of asphalt concrete pavedepends to a large extent on the type and percentage ophalt used and there were studies relating the type of aspwith the deformation characteristics of asphalt concrete mtures. Mack@264# discussed the type of loads which alikely to act on the pavement and commented on the physand chemical properties a binder should possess for optperformance of the pavement. This was a step in the rdirection in the sense that while modeling of bitumen andcharacterization of its physical and chemical propertiesneeded, it is the complete response of the asphalt concmixture which is of interest. Mack defined a parameter caldegree of plasticitythrough the following equation:

h5Fn/R (3)

whereh is the viscosity,F is the shear stress,R is the rate ofshear, andn is the degree of plasticity, and concluded that tstrength of the pavement structure increased with increasn. Some of his observations about the rheology of bitumnous mixtures and on what constitutes elastic and plabehavior reflect the prevailing confusion in the asphalt ccrete literature of the day regarding such ideas as cangleaned from the following remarks: ‘‘The term strain is usally applied to a recoverable change of the shape of a band deformation to a non-recoverable change; ...specimebehaves like a solid, and the stress and the elastic recoremains constant, ie, no stress is dissipated.’’ It is puzzthat while the basic ideas of viscoelasticity were firmlyplace at that time, no mention is made of the viscoelanature of asphalt concrete, rather he concluded asphaltcrete to be a mixture of elastic and plastic elements,‘‘plastic’’ nature due to the type and amount of asphalt~it isby no means clear that he was thinking of what wouldconsidered a viscoplastic material today as ‘‘plastic’’ to t

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practitioners in asphalt rheology meant, and unfortunatcontinues to mean, a fluid with a threshold for flow!.

Lethersich in 1942@265# succinctly expressed the complcated response characteristics of asphalt; ‘‘It is shownthe behavior of bitumen may be expressed in terms omechanical model containing two elasticities and two vcosities. The two viscosities increase with frequency...It mbe further deduced that a penetration test is largely inenced by the variation of the viscosity with strain, and mindeed serve as a measure of this variation.’’ He providemechanical analog and derived a one-dimensional~1D!model based on his observation@265#. While his ideas andhis analysis are commendable for his time, and in relationthe research that was current at that time, he unfortunadid not put into place in his modeling the facts about tresponse he highlights, for example, while he states thatviscosity is a function of strain~in this he is in error, but heobviously means strain rate or to be more precise the smetric part of the velocity gradient as the strain rate isproperly frame invariant! his modeling assumes that the vicosity is a constant. All such issues not withstanding, wein his paper, for the first time, the need for more than orelaxation time, to describe the response of asphalt.

Frohlick and Sack in 1945@266# developed rate typemodels for the viscoelastic response of dispersions~solid mi-celles in a viscous fluid!, and here we clearly see the sparof true understanding of viscoelastic response. Their studmotivated by the experiments on asphalt of Lethersich@265#that clearly indicate the viscoelastic character of asphThey model the dispersion as equal linearized elastic sphdispersed in a Newtonian fluid, and take into accountincompressibility and isotropy of the dispersion to obtainrate type model, although this model is not properly framindifferent.

In their assessment of the state of the rheology of asphTraxler@267# and Romberg and Traxler@268# suggest the useof ‘‘thermodynamic methods’’ in characterizing asphalt athe infusion of this new idea into the field was noteworthHowever, the confusion between plastic deformation andcoelasticity persists in their work.

In his paper, ‘‘Elasticity beyond the elastic limit,’’ Reine@269# comments on the viscoelastic nature of asphalt: ‘addition one has to consider elastic materials such as bmen or cement stone showing creep: their elastic potengradually disappear through relaxation.’’ Of course, hewrong in thinking of bitumen as an elastic material that ccreep, for an elastic material by definition cannot crerather he should think of it as a material that can exhibit ban elastic and a viscous response, namely a viscoelasticterial. Reiner’s work is, however, not concerned with tdevelopment of a proper viscoelastic model for asphalt.

Thelen in 1949 @270# suggested correlations betweewhat he called the five basic rheological properties: yivalue, mobility, elasticity, work of adhesion, and vibratioresistance, and 15 working properties of asphalt. Unfonately, he failed to recognize the nonlinear behavior of bimen and also appealed to testing methods such as penetrtest, etc, which are empirical in nature. Similar attempts w

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made by Corbett and Vokac@271# in the sense that theselected eight basic types of tests~extrude, rebound, impresshammer, fold, squeeze, pull, and twist! and tried to relatethem to the ‘‘firmness’’ and ‘‘yielding’’ of asphalt.

While Reiner, Rigden, and Thrower@272# ~see also@273#!recognized that asphalt is viscoelastic and that voluchanges take place in simple shear flows~experimentallyshown by Lee and Marwick@274#!, they made the interestinobservation that even a simple compression of asphaltlead to volume increase. They adopt certain ideas of Resuch as a viscous Poisson’s ratio and deformational Poissratio to describe the volume changes of asphalt specim~their asphalt consists in a mixture of 14 parts by weightasphalt and 86 parts by weight of limestone! subject to ten-sion. However, while the notion of a Poisson’s ratio has revance to the response of elastic bodies in the linearrange and can be extended to linear viscoelasticity, the ccept has no meaning in finite deformations of elastic bodor in nonlinear viscoelasticity. Given that the responseasphalt is in general quite nonlinear, it is clear that thattempts to describe the behavior of such a material witnotion of more than one Poisson’s ratio, as well as otconceptual, ideas is flawed~see also Lee, Reiner, and Rigde@275#!.

The brittle fracture of tars and bitumens were studiedRigden and Lee@276# and they tried to interpret the empircal Frass brittle test mathematically. However, their assution that bitumen is an incompressible Newtonian fluidinconsistent with the notion of a material that can be frtured, at least within the context of their study.

There is not much point in discussing these early studon asphalt in any greater detail for all of them are flaweda fundamental way. Any data reduction depends onmodel used for the purpose. Thus, all these early corrtions, which did not recognize the complex rheological chacteristics of asphalt, while they served some practical csiderations have little to say about the response of asph

It is not meaningful to provide plots of stress versus strfor viscoelastic bodies, whether they be solid-like or flulike, or for viscoplastic bodies. Nor is there much senseplotting the ratio of the stress and strain, in a purelysetting, versus time. However, it is common to see plotsthe ‘‘stiffness’’ S that is defined throughS5s/«, wheresdenotes the stress in one dimension and« the strain in onedimension, versus time~see Saal and Labout@237#!. Firstly,such a quantity cannot be defined in a meaningful fashionnonlinear materials. Secondly, there is no physical meanto such a ratio when the stress depends on both the strainthe strain rate. In viscoelasticity, for 1D response it onmakes sense to plot stress versus time and strain versusA stress versus strain plot makes no sense because an innumber of them can be plotted for the same response. Wat a fixed time, we can find the ratio of the value of stressthat of the strain, the ratio has no physical meaning. Extralating from linearized elasticity to call that ratio the ‘‘stiffness’’ does not lead to a meaningful description of the marial.

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It is important to recognize that Frohlick and Sack@266#developed a fully 3D theory and derived a rate type flumodel that takes the form:

S1lS5m1D11m2D2, (4)

where the superposed dot denotes the usual materialderivative and

D151

2 S ]v

]x1S ]v

]xDTD , D25

1

2 S ]a

]x1S ]a

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wherev anda denote the velocity and acceleration, respetively. Neither S nor D2 is frame-indifferent. However, wenote that the model~4! is nonlinear by virtue ofS beingnonlinear. The model~4! was one of the early nonlineamodels for viscoelasticity.

At this juncture, it would be appropriate to point out ththe influential work of Oldroyd@277# draws greatly from thework of Frohlick and Sack@266#. Oldroyd recognized thathe usual material time derivative is not frame-indiffereand introduced time derivatives that are frame invariant.replacing the usual material time derivatives in~4! by theupper convected time derivative, we obtain

S1lS¹

5mA11A¹

1, (6)

where

B¹

5B2LB2BLT, (7)

and

L5gradv. (8)

The expressionS denotes the constitutively determined extstress, the Cauchy StressT being given by

T52p11S. (9)

The constitutively indeterminate part of the stress2p1 is aconsequence of the constraint of incompressibility.

While asphalt is fluid-like at temperatures above its rinball softening point, approximately below 20°C it behavlike a viscoelastic solid. As its response as a viscoelasolid tends to display an instantaneous elastic responsecannot model its behavior with a standard viscoelastic smodel such as a Kelvin-Voigt model as such a model canexhibit instantaneous elastic response. A variety of viscoetic solid models can be used and here we shall discusssimplest of them, a model usually referred to as thestandardgeneral linear viscoelastic model~see Wineman and Rajagopal @278#!. The constitutive relation for the standard genelinear viscoelastic model can be expressed in two ways:

T5E0

t

G~ t2s!«ds, (10)

or

«5Eo

t

J~ t2s!Tds, (11)

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whereT is the shear stress and« the linearized strain,G andJ are the relaxation and creep compliance fourth ordersors. The above representations for the stress and strainbe expressed in a simple form in terms of the Stieltjes ingral representation~see Appendix D of Wineman and Rajagopal @278#!:

T5G* d«, (12)

«5J* dT. (13)

It is important to recognize that the models~12! and~13! arenot frame-indifferent, as the linearized strain is not framindifferent. In our conversations with asphalt rheologists,are struck by the fact that most of them are firstly unawarethe reason for requiring frame-indifferent models, seconthat the linearized strain is not frame-indifferent, and thirdthat in a rigid body motion the linear strain does not vanidentically. Of course, frame-indifference is anassumptionthat is expected to be met in a certain class of processbody is subjected to and not aprinciple that holds for allbodies in all processes. A dogmatic belief in such aprinciplecan be as harmful as not recognizing that such an assumought to hold for certain processes~see Rajagopal@212# for adetailed discussion of the same!. For instance, when concerned with processes where relativistic effects cannotignored one finds that a different space-time invarianceplains the processes better. Similarly, if we are concerwith processes for which Maxwell’s equations hold, we awell aware that we cannot demand frame-indifference,these equations are not frame-indifferent. What we are adcating here is not an extreme form of skepticism. Tocontrary, our point is that there is no substitute for gophysics, and that uncommon of all senses, common seneeds to be used in good measure, it being the responsibof the modeler to ensure that assumptions that are consiwith that which is being modeled are put in place. Unfornately, standard texts devoted to linearized elasticity dostart from a nonlinear perspective and then delineate thetent and scope of the linearized theory.

It might be appropriate to develop the basic kinematiconcepts at this juncture. However, we shall defer it till Stion 6 as the material that follows does not require a clgrasp of these concepts. An interested reader can skipward to Section 6.2 where a nonlinear theory is develofor describing the response of asphalt concrete. To followmaterial given below, an understanding of classical lineized elasticity and classical linear viscoelasticity suffices.

We recognize the similarity in structure of the constitutirelations ~12! and ~13!, and those for linearized elasticityThe fact that both these theories are linear leads to a mused correspondence principle that allows us to write stions to problems in linear viscoelasticity from those thatknown for linearized elasticity. In fact, Wineman and Ragopal @278# have shown that such correspondence canestablished between many linear constitutive theories andlinearized theory of elasticity. However, such a correspdence principle does not apply between linearized elasticids and nonlinear elastic or viscoelastic solids. Unfortunat

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however, such a correspondence is used frequently inliterature pertaining to the response of asphalt which cleaexhibits nonlinear response. If the viscoelastic materiaisotropic, then the tensorsG andJ can be expressed in termof two material functions that depend on time.

Let us consider the uniaxial extension of a linear vcoelastic body~we follow the discussion of Wineman anRajagopal@278# in the material presented below!, ie, thestressT takes the following matrix representation in a Catesian coordinate system:

@T#5F T11~ t ! 0 0

0 0 0

0 0 0G . (14)

When subject to such a state of stress, a body will undetransverse strains«22(t) and «33(t), in addition to the elon-gational strain«11(t). Thus the strain« has the followingmatrix representation

@«#5F «11~ t! 0 0

0 «22~ t! 0

0 0 «33~ t!G . (15)

It is important to realize that, in general, if the materialanisotropic we could produce shear strains due to the apcation of an uniaxial stress. While asphalt can be considean isotropic material, there is compelling reason to suppthat asphalt concrete is anisotropic.

A few words of explanation are in order here as momodels for asphalt concrete are fluid models and fluidstacitly assumed to be isotropic. This need not necessarilythe case, for it is perfectly possible for a fluid to be anistropic. Recently, Rajagopal and Srinivasa@279# have pro-vided a thermodynamic framework for developing constitive relations for anisotropic fluids based on the notion thabody can have many stress-free configurations~modulo rigidmotions!. Moreover, such fluids can exhibit more than otype of anisotropic response, one associated with the fluelastic response and the other associated with the dissipresponse. In Section 6.2 we introduce the framework of bies with multiple natural configurations and develop 3models for asphalt concrete.

It would be more appropriate to call this 3D modelviscoelastic model that is linear in terms of the linearizstrain. In fact, if the strains as measured from the stress-state of the body to which it would go in an instantaneoelastic response is sufficiently small, then it can be lineized. To be more precise, the strain in the elastic respowith respect to the natural configuration of the material atcurrent configuration is ‘‘sufficiently small.’’ These notionare discussed in Section 6.2 in detail. In keeping withnomenclature in viscoelasticity, we shall refer to it as lineviscoelastic material.

It might be a worthwhile digression to discuss briefly tmeaning of Poisson’s ratio for linear viscoelastic materiagiven its wide use in asphalt and asphalt concrete mecha

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To motivate the notion of Poisson’s ratio for a certain claof viscoelastic materials, let us consider the linear viscoetic model introduced earlier. Now

T11~ t !5«11~0!G~ t !1E0

t

G~ t2s!«11~ t !ds, (16)

5E02

t

G~ t2s!«11~s!ds5G* d«11, (17)

5E02

t

«11~ t2s!G~s!ds5«11* dG, (18)

where* denotes the Stieltjes Convolution~see Wineman andRajagopal@278#!, « the linearized strain, andG the relaxationmodulus.

Alternatively, we can express the model for a linear vcoelastic material through

«11~ t !5T11~0!J~ t !1E0

t

J~ t2s!T11~s!ds, (19)

5E02

t

J~ t2s!T11~s!ds5J* dT11, (20)

5E02

t

T11~ t2s!J~s!ds5T11* dJ. (21)

Now suppose that the specimen is subject to an extensistep-strain history

«11~ t !5«110 H~ t ! (22)

whereH(t) is the usual Heaviside step function. We defithe Poisson’s ratio in linear viscoelasticity through a functn(t) as follows:

«22

«110 52n~ t !, t>0, (23)

«33

«110 52n~ t !, t>0. (24)

It is important to recognize that the transverse strainsvary with time even though the extensional strain is hfixed. We also note that the Poisson’s ratio is a functiontime. If the material were anisotropic, then it is possible tthere exist more than one Poisson’s ratio. Suppose thamaterial is orthotropic, then we could define two Poissoratio through

«22

«110 52n22~ t !, t>0, (25)

ssas-

is-

nal

eon

anldofatthe

n’s

«33

«110 52n33~ t !, t>0. (26)

We have been constantly referring to the viscoelasticsponse albeit linear response of asphalt. However, giventhe term asphalt or bitumen does not refer to a clearly idtifiable substance but to a variety of substances, it issurprising that there is a wide range of responses assocwith asphalt. Depending on the temperature and the typloading which certain asphalts are subject to they canspond like elastic-plastic, viscoplastic, or even brittle-elasmaterials. We shall discuss later a model that is capabledescribing the behavior of asphalt over the whole rangeresponse characteristics.

When dealing with mixtures of asphalt with mineral agregates we might need to take into account the inhomoneity as well as the anisotropy of the material, especiawhen the temperature is sufficiently low when the mixturethe asphalt and the mineral aggregate is solid-like. At suciently low temperatures, such mixtures are supposed tospond like a ‘‘Bingham plastic’’ material exhibiting yieldNijboer @280# discusses the ‘‘Bingham plastic’’ responsesuch mixtures, especially the need for taking such respointo account in developing a fully comprehensive designroads. The‘‘Bingham plastic’’-like response is a consequeof the time scale of the experiment or observation not besufficiently large. As we have remarked earlier, if the marial is fluid-like it cannot resist shear and thus there cannoa threshold stress, it is just that the flow is so small that inot observable.

A variety of spring-dashpot models have been proposedescribe the 1D constitutive response of asphalt. SaalLabout @237# proposed the use of a Maxwell model andVoigt model ~Figs. 3 and 4! ~see also@281#! and Lethersich@265# suggested the use of a model due to Burgers, whmechanical analog is shown in Fig. 5.

Reiner et al @272# are critical of the above model anstate, ‘‘It has become evident that generally both shear-fland volume-flow will occur as a parallel phenomena ingranular material such as asphalt and concrete and therect procedure is to resolve the deformation into thesecomponents and to consider each one separately.’’ Howeit is hard to find any merit to the notion of Reineret al thatthe deformation should be split into two components ashould be considered separately. It smacks of thinking ablinear response~most asphalts respond nonlinearly! and, fur-thermore, it is totally unclear what they mean by shear-fland volume-flow being parallel phenomena.

Fig. 4 Voigt Model
Fig. 3 Maxwell Model

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It is worthwhile to discuss in passing another approachmodeling asphalt that draws upon ideas introduced inunlikeliest of areas, psychology. Blair and Veinoglou, in claboration with Caffyn@282# seem to have been the firresearchers to systematically use fractional derivative mofor viscoelastic fluids motivated by the work of Gema@283,284# which is in turn based on operational calculus dveloped by Heaviside@285#. The work of Blair et al @282#introduces the notion of quasi-properties and appealsprinciple called thePrinciple of Intermediacyused in psy-chology at that time. It ascribes to these ‘‘quasi-propertiethe meaning of ‘‘a measure of a process than a state,’’ ie,measurements of certain quantities such asfirmnessare notnecessarily performed at equilibrium but at various statequasi-equilibrium. Unfortunately, an infinite number of suprocess-dependent quasi-properties can be defined, asthemselves acknowledge. They also remark that such acedure does not lead to a constitutive characterization ofmaterial but involves the process in question~see also Blairand Coppen@259,260#!. On the other hand, one can righttake the point of view that it is impossible to provide constutive relations for a body that hold for all processes and twe can at best describe as how a body behaves when suto a class of processes. In fact, it is perfectly reasonablsuppose that the same body can have different constitucharacterizations in different classes of processes and theffect, is the basis for the framework for the study of tthermomechanics of materials with multiple natural configrations.

We shall not get into a detailed discussion of fractionderivative models but shall merely present one that has bused to describe the response of asphalt. As we shalllater, a material like asphalt which exhibits viscoelastic flulike behavior above a certain temperature, exhibitsviscoelastic-solid-like behavior in a certain range, and whbehaves like a brittle-elastic-solid below a certain tempeture needs a much more complex constitutive prescripthan any of the popular viscoelasticity models in voguwhether they be differential type, rate type, integral type,fractional derivative model. We shall discuss a reasonageneral model for asphalt in Section 6.2. We now turnattention to a fractional rate type model that has been usedescribe the response of asphalt.

Before introducing the fractional derivative model, wshall first define what is meant by a fractional derivativethese derivatives are not commonly used in mechanThese ideas have a long history. The notions of fractiodifferentiation and fractional integration were recognizedLeibniz in 1695@286# and can be traced back to the work

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Euler in 1730@287#. It was given a firm footing by Liouvillein 1832@288# and Riemann in 1876@289#. An excellent andsuccinct review of the history of fractional calculus canfound in Oldham and Spanier@290# which also contains anannotated chronological bibliography on the subject by Btram Ross~see pp 3–15 of@290#!.

Fractional calculus has been used in a wide varietyareas and early use of it was made in the field of rheologyBlair et al @282#, Blair @291,292#, and Blair and Caffyn@293#. A review of the use of fractional calculus in solimechanics can be found in Rossikin and Shitikova@294#.

The derivative or integral of orderq of a function is givenby ~we follow the notation used in Oldham and Span@290#!

dqf

@d~x2a!#q 5 limN→`

H Fx2a

N G2q

G~2q! (j 50

N21G~ j 2q!

G~ j 11!

3 f S x2 j Fx2a

N G D J , (27)

wherea denotes the lower limit of the integration,q is arbi-trary, andG denotes the gamma function defined through

G~x!ª limN→`

F N!Nx

x~x11!~x12!1¯~x1N!G . (28)

The definition~27! holds for both fractional derivatives anintegrals. We shall now restrict ourselves to the notionfractional derivatives and introduce a couple of models thave been used to study the response of viscoelastic maals. For the purpose of illustration, we shall restrict ourselto 1D models.

Let us recall the 1D Kelvin-Voigt model:

s5E«1md«

dt, (29)

whereE is the Young’s modulus,m is the viscosity,s is thestress, and« is the strain. A generalization of the same usifractional calculus would be

s5E«1mdr«

dtr, (30)

where the derivative on the right hand side is a fractioderivative. Similarly, the three parameter solid model is

1

ms1

1

E

ds

dt5

E1

m«1S 11

E1

E D d«

dt, (31)

Fig. 5 Burger’s Model

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whereE and E1 are appropriate Young’s moduli~or springconstants for the spring-dashpot analogs!. The above modecan be generalized to

1

ms1

1

E

drs

dtr5

E1

m«1S 11

E1

E D dr«

dtr, (32)

Many generalizations of the Maxwell fluid and the KelviVoigt solid are possible. For instance, ifN Maxwell elementsare put together, then the constitutive relation for the strcan be expressed as

P~D !s5Q~D !«, (33)

whereD denotes the usual differential operator and

P~D !5S D

E11

1

m1D S D

E21

1

m2D¯S D

EN1

1

mND , (34)

and

Q~D !5~q1D1q2D21¯1qNDN!. (35)

We can immediately generalize such a model by introducfractional derivatives. Bagley and Torvik@295# ~based onBagley @296#, see also@297#! provide a 3D constitutivemodel based on fractional derivatives:

S 11 (k51

K

akDbkD S 11 (

p51

P

bpDgpDT

5F S 11 (p51

P

bpDgpD S l01(j 51

J

l jDa j D ~ tr«!G1

12S 11 (k51

K

akDbkD S m01 (

m51

M

mmDdmD «. (36)

Unfortunately, in view of the linearized strain that is usedthe constitutive representation for the material, we havmodel that is not frame-indifferent, a feature this modshares with classical linear viscoelasticity. There are mpapers that are devoted to the study of the response ofcoelastic materials that appeal to fractional calculus~see@298# and the many references listed in@294#!. As asphalt isviscoelastic, we recognize that such models have also bused to study the response of asphalt~see@299#!. We shallnot get into a more detailed discussion of such studies. Wa great deal of progress has been made with regard to elish a rigorous framework for fractional calculus, much rmains to be done.

Depending on the type of the asphalt concrete thathas, we would have to develop the model that can capthe material’s response and this has to be rooted in senthermodynamics, but before doing that we need to discother developments in the modeling of asphalt and aspconcrete.

4.2 Common engineering tests for characterizingbitumen—An evaluation of the penetration test

There is no point to providing details of the methodologfor characterizing bitumen in the numerous tests that are,that have been, used. We shall discuss briefly just one

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them, the penetration test, a test that we have referrenumerous times in the course of this review. Perhapsother test has been as widely used to classify and charaize the large class of materials that are called bitumen aspenetration test. The literature on the different kinds of petration tests that were practiced till a consensus was reaand a standard put in place would be voluminous~ASTMD5-97, 2000@300#!. This test is presumed to be capabledetermining properties of bitumen that can characterizeresponse over a wide range of conditions. Needless to shas been widely used, misused, and abused. While mostof this type have given way to tests aimed at determinmaterial moduli that appear in the constitutive representafor the material, this test is being put to use in areas wheven the originators of this test did not envision. We wbriefly describe this test and discuss some of the ramifitions of this test. We make no attempt to review all theerature pertinent to the test. Our main aim here is to recits usefulness but caution the user as to its limitationscharacterizing bitumen. A test of this kind is very usefulthe laboratory of a hot mix plant to check whether the bimen that is to be used has the quality as specified, butyond this, it offers little in the way of providing informationfor the constitutive modeling, or for that matter even categrizing the physical and chemical characteristics of bitumThe penetration test is considered as one of the basic rlogical tests for characterizing asphalt and thus our interesdiscussing it in some detail below.

The penetration test is defined by the ASTM~ASTM D5-97, 2000@300#! as follows: ‘‘consistency of a bituminoumaterial expressed as the distance in tenth of a millimethat a standard needle vertically penetrates a sample omaterial under known conditions of loading, time, and teperature,’’ and the ASTM further makes the following rmarks about the significance of the test: ‘‘The penetrattest is used as a measure of consistency. Higher valuepenetration indicate softer consistency.’’ This particular tmethod is used to determine the penetration in a solidsemisolid bituminous material. We have discussed in deelsewhere in this article the confusion caused by the usthe terms such as ‘‘consistency.’’ Consistency as definwithin the context of asphalt can have different meaninFor instance, the earliest methods of testing the consisteor degree of softening were bychewingand this is detailedby Halstead and Welborn@301# as follows: ‘‘If the materialcrumbled on chewing, it was too hard. If it stuck to the teeit was too soft. If it was pliable like chewing gum, the cosistency was correct. Even after the invention of the penetion machine, the chewing method—crude as it may appto the uninitiated—served as a valuable check. An aspman prided himself on the fact that by chewing the matehe could very closely estimate the results obtained bymachine.’’

The originator of the penetration test, Bowen@22#, con-ceptualized this test as performing the following task: ‘paving cement should be both adherent and coherentdegree which experience alone can determine. Hencegreat need of being as exact, respecting this degree, as

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sible. Should a certain observed degree of softness of ament be found associated with a certain good pavemenfollows, other things being equal, that a superintendent wsuch experience would be better able to perform continugood work. An effort to bring this quality, softness, into eact measurement has resulted in the following apparatuspendent on the following consideration: A sharp polishneedle, acting during a short interval of time, under a givweight, and at a given temperature, penetrates a homneous, non-crystalline, semi-liquid body to a depth, appently dependent chiefly on the degree of strength of coheof the body, or softness as commonly designated.’’ Dow@24#who made extensive modifications to the original Bowepenetration test described the penetration value as relatethe susceptibility to change in temperature. RichardsonForrest @25# clearly delineate the following regarding thscope of the test: ‘‘It is intended only as a convenient meof determining, in laboratories where control is exercisover bituminous cements and at plants where they are inwith some degree of uniformity and accuracy, whether thare of the proper consistency or that which has been specand recording the consistency in figures. The true viscositnot the matter at issue but whether the person who prepsuch cements is doing so in a regular manner.’’ From thdiscussions, it is clear that the penetration value wastended as a quality control test but as of now it is usedwidely different areas of modeling the physical, chemicand rheological behavior of asphalt.

Correlations between the penetration value and the phcal and chemical properties of asphalt and their changenumerous. For instance, some of the correlations whichcommonly used are~we provide below the types of correlations used and references to the literature wherein suchrelations are criticized!: the hysteresis effects on the penettion value ~Woog et al @302#!; elastic behavior of bitumenand the penetration value~Saal and Labout@221#!; surfaceconsistency characteristics of asphalt and penetra~Knowles and McCoy@303#!; relationship between penetration and temperature in the form of susceptibility facto~Holmeset al @304#! and criticism of these susceptibility factors ~Schweyeret al @238#, Traxler and Schweyer@305#!; ra-tios of penetrations with different weights and susceptibiindex ~Bencowitz and Boe@306#! and criticism of this index~Broome @251#, Hillman @307#, Pfeiffer and van Doorma@242#!; penetrations at different temperature and a differsusceptibility index based on the log penetration-temperacurve ~Lewis and Welborn@308#! and criticism of these in-dices~Grant and Hoiberg@309#, Neppe@310#!; empirical re-lationship between penetration and absolute viscosity~Saaland Koens@220#!; flow properties of bitumens made by sucessive penetration tests~Thelen @311#, Rhodes and Volk-mann @312#! ~it also provides a criticism of the empiricarelationship between viscosity and penetration of SaalKoens @220# and the successive penetration test of The@311# by Traxler and Pittman@313# and Traxler et al@219,262#!; on relation between plastic flow and penetrati~Mack @314#!; on relation between the results from the cocal viscometer and penetration test~Pendleton@315#!; on

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‘‘true penetration’’ values based on a regular cylindricshaped thin-walled Penetrometer needle which offers no ftional resistance or end effects~Neppe@310,316–319#!; PVNnumber~penetration at 25°C–viscosity at 135°C number! asa measure of susceptibility to temperature~McLeod @320#!;relations of fatigue life and permanent deformation withspect to penetration~Rogge et al @321#!; relationship be-tween chemical constituents and penetration~Stock and But-ton @322#!; correlation of penetration index with thasphaltenes/~wax 1 polar-aromatics ratio! ~Porier and Sa-watzky @323#, Hagenet al @324#!; and the list of correlationsgoes on and on!

Determining the aging of asphalt using indices basedthe penetration values of aged and unaged asphalt havebeen proposed. For instance, empirical relations betwpenetration and hardness were proposed by Vokac@325#based on some arbitrary definitions such as, ‘‘Hard—not eily penetrated, or separated into parts; not yielding to prsure and opposed to soft; Soft—easily impressible; eayielding to pressure; the contrary to hard.’’ We discussmore detail the different aging indices in the section on ag~also see Shiauet al @326# and Noureldin and Wood@327# forrelations between penetration value and aging testsBrown et al @328# for change of penetration value with timand its correlation with aging!.

The whole intent of the penetration test, not withstandthe empirical correlations discussed above, is to put iplace a robust test method requiring minimum skill andsimple test equipment. The entire test process takes lesstwo hours for most of specimens and, as quoted earlier,intent is to ensure that the quality of bitumen used is ‘‘cosistent’’ enough and wide disparities are not encountereding the construction process. The penetration test is thufield test for asphalt quality control and attempts at correing the test results with viscosity have been rightfully inored in the SUPERPAVE specifications. However, studrelated to the determination of the penetration value usnear infrared spectroscopy have been reported in the litture very recently. Blancoet al @329# reported correlations openetration test with spectral data based on the assumpthat, ‘‘The penetration value of a bitumen is not dependon any specific individual component, but rather on the tocomposition of the sample and on interactions between cponents. One can therefore reasonably assume that theetration value may be acceptably estimated by correlawith spectral data.’’ This correlation with spectral data wouhave significance if, in the first place, tests such as the petration test had a significant role to play in characterizthe materials. They do not. All that they do is provide a ruof thumb which an asphalt technician in the hot mix labotory can turn to in order to get a feel for the bitumen thatis using. As Bingham@330# aptly observes, such tests caand do prove to be impediments to obtaining a clear undstanding of the behavior of the material: ‘‘If one to-day dsires to know the consistency of a material, one mustmeasure the consistency directly. He must first analyzematerial. If the material turns out to be an oil, he must thproceed to get the latitude and longitude of the laborato

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and if he finds himself in the USA, he selects the Sayboltpossibly the Saybolt Furol, but in England the Engler; ifmetal, he directs his thoughts towards one of several hness machines; if a cement, he goes to the slump test; ba clay, he may exercise considerable individuality of taand measure shrinkage on drying, absorption of a dye, ormechanical strength. When it comes to plastics, the contency may be measured~1! as a temperature in the ring anball test ~2! as a solubility, or~3! as a distance which asample of let us say, a pitch will fall in a given time. This bno means exhausts the lists of instruments in commonwhich are not based on any sort of theory of the fundamenature of flow, and which give results which are qualitatiin character, and more than any other thing preventsprogress of the art.’’

4.3 Recent approaches to the modeling of bitumen

The earlier approaches related to the modeling of bitumdepended heavily on the works of Bingham and ReinTerms such as plasticity, consistency, fluidity, pliability, stiness, etc, were freely used without adequate explanationwithout any consistent physical basis. The complex naturasphalt and its relation to the performance of asphalt ccrete pavement perplexed many investigators and they reheavily on the use of empirical tests to circumvent the diculty related to the modeling issues. In the meantime, escially after the Second World War, there was an increasethe vehicular traffic in the highways and the loads onhighways increased considerably. The need to understanphalt and design pavements that can withstand the incretraffic loads prompted some immediate simple solutions. Tuse of Marshall and Hveem mix design methods for desiing asphalt concrete mixtures and the use of Boussine@331# and Burmister’s@332,333# theories of layered structures for structural designing of pavements were the methologies that were most popular. Most of the pavement desmanuals developed during the later half of the last centwere based on the series of papers by Van der P@172,334–336# in which he advocated the assumption oflinear viscoelastic model for asphalt. The assumption othermo-rheologically simple material behavior for asphand the validity of the time-temperature superpositionsumption have dominated most of the studies concerningphalt during the last 50 years. These assumptions wwidely criticized in different studies but due to the lack ofproper alternate approach for modeling asphalt, these mods are widely followed even today. The far-reaching devopments in unraveling the chemistry of asphalt have not btranslated into its constitutive modeling. Our main attempthis section is to summarize the various modeling attemstarting with the work of Van der Poel and critically examisome of their deficiencies.

Van der Poel@172,334–336# proposed a simple systemfor finding out what he defined as ‘‘stiffness’’ of asphalt uing the ring and ball softening point and the penetrationdex. He defined stiffness in the following way:

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od-ignuryoela

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s-in-

S5s

«, (37)

whereS is the stiffness modulus,s is the tensile stress, and«is the total strain. His comments concerning the stiffnemodulus are as follows~Van der Poel@172#!: ‘‘The mostsimple rheological magnitude used in applied mechanicYoung’s modulus, which, however, is defined only when tmaterial is a purely elastic solid, ie, when there is a linerelationship between stress and strain, and this shouldindependent of the time of loading. Without modification thmodulus cannot be used for visco-elastic materials suchbitumens. Provided we confine ourselves to such conditiof stress and deformation under which a linear relationsbetween the two exists, a simple extension of the concepYoung’s modulus can be given that also applies to viselastic materials. ...This stiffness modulus will, in generdepend on:~a! loading procedure,~b! time of loading orfrequency, and~c! temperature.’’ He also advocated the uof a superposition principle which allows inter-conversiondata from the creep test to a dynamic test. The justificatfor the assumption of a linear behavior was made onassumption that at low temperatures and short times of loing asphalt behaves linearly, while for long loading time asufficiently high temperature asphalt seems to behave likNewtonian fluid. There is no mention of how an expressof the above type can model a Newtonian fluid. The folloing remarks: ‘‘Only in the range of moderate temperatuand loading times can nonlinear effects occur, and thesecome more marked with increase of deformation. ...Thlarge deformations, however, are far greater than thosemally met in practice, where generally the object is to avoexcessive deformations.’’ were used to justify the neglectthe nonlinear behavior of asphalt concrete. This assumpthat large deformations are rarely met in practice are btantly wrong, especially when one is concerned with defmations suffered due to the motion of large tractor trailersaircraft during takeoff and landing. The notion of modelinasphalt concrete as a linear elastic material is quite wroheaded. Using such ideas and measuring the penetratiodex and ring and ball softening point for 47 samples frodifferent parts of the world, a nomograph was proposedfinding the stiffness of bitumen. This nomograph was subquently used in all the pavement design codes and simnomographs for finding the stiffness of asphalt concrknowing the stiffness of asphalt were proposed~Shell Pave-ment Design Manual@337#, Asphalt Institute Pavement Design Manual@338#, British Standards Specification@339#!.Unfortunately, the assumption of a linear, small strain behior coupled with the use of empirical measures such as petration index, etc, cannot capture the complex behavioasphalt concrete and asphalt, and such studies set bacmodeling efforts substantially.

Correlation between the properties of constituents of dferent kinds of asphalt~straight run, thermal, blown! and thepenetration and other parameters were attempted by Co@340#. Assuming that asphalt is a mixture of just two costituents, asphaltenes and petrolenes, Corbett tried to r

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the viscosity of the individual constituents with the penettion value, softening point, and the pentane insolubles. Whe could find some sort of a correlation in the case of strarun asphalts, he could not show any pronounced correlafor the other types of asphalt.

In a review of the mechanical testing of asphaltic bimen, Saal@341# commented on the accuracy of the nomgraph proposed by Van der Poel and remarked that bitumshowing the same viscoelastic character may still difsomewhat in the empirical penetration index and attributhese differences to the different crude sources or methomanufacture.

Wood and Goetz in their study@342# tried to find a rela-tionship between the unconfined compressive strengthbituminous mixture and the viscosity of the binder. Thused four different kinds of asphalt in their study and cocluded that ‘‘it is obvious that some factor other than tviscosity of the binder affects the strength of the mixturesThey pointed to the need to take into account variables sas steric hardening, surface-chemical reactions, and mollar orientation of asphalt to provide a more accurate desction of asphalt. In a very interesting study on the crackcharacteristics of asphalt, Doyle@343# used two differenttypes of asphalts possessing different brittle points~a point atwhich asphalt is no longerbendableat a given temperatureie, it shows a transition from behaving as a viscoelastic sto responding like a brittle elastic solid! and suggested amodification of the ductility test prevalent at that time. Hremarks concerning the necessity for taking into accountbehavior of asphalt as a thin film and the drawbacks relato the oven weathering test to simulate the aging behavioasphalt are quite thought provoking.

Using direct electrical means, Csanyi and Bassi@344#separated asphalt into its different constituents. They useelectrolytic cell and studied the different constituents depited on the electrodes by ultraviolet light. They reportsome unusual results normally not captured in the rouasphalt tests. For instance, two asphalts manufacturedtwo different sources but having the same penetration vasoftening point, and specific gravity behaved in a totally dferent manner showing totally different affinities to the pasage of electricity.

In a series of three papers~Brodnyan@345#, Gaskinset al@346#, and Brodnyanet al @347#! concerning the rheology oasphalt, results pertaining to its mechanical characteriswere discussed. Ten different types of asphalts characteras sol, medium, and gel were used. Also, two different tyof instruments were used for testing asphalt; a cocylindrical viscometer~rotational viscometer! and a high-pressure capillary tube viscometer. Their main conclusiare that asphalts are viscoelastic bodies which respondNewtonian fluids at high temperatures. At the temperatuwhere the transition to viscoelastic behavior occurred,viscosity was found to be of the order of 1023 poise. Anotherinteresting observation that they made is that related tonon-Newtonian response of the material at the temperarange of 130–250°C for a medium type asphalt with viscity varying from 1 million to 10 poise at a constant she

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stress of 3000 dyne/cm2. They also concluded that the notioof time-temperature superposition was valid for the materthat they tested for their range of deformations and they vfied the appropriateness of the empirical Williams-LandFerry @348# relation for asphalt albeit to within an error o150–200% at high temperature and low frequencies. Thesumption of time-temperature superposition, while wideprevalent even today, is not based on reliable studiesdoes not definitely hold for the general nonlinear responsmaterials such as asphalt.

The influence of the internal structure on the rheologiproperties of asphalt, specifically related to its noNewtonian behavior was discussed by Schweyer and hisworkers in a series of papers~Majidzadeh and Schweye@349,350# and Schweyer and Busot@351#!.

In their paper in 1965 Majidzadeh and Schweyer@349#,using a power-law model based on the de Waele-Ostwaequation attempted to relate the empirical parameter inpower-law model~called by them as a non-Newtonian costant! with the testing variables. They also proposed a strtural index to correlate the structural degradation and triedexplain the complex non-Newtonian behavior of asphwithin the context of the Eyring rate process theory. Dispeing with the Eyring equation in their later study@350#, theytried to use the concepts of ‘‘free volume’’ or ‘‘free space~see Batchinski@352# and Doolittle@353,354#! and proposeda viscosity-temperature equation capable of describingeight asphalts they tested within the range of 25–100°C.worth pointing out that they observed a breakdown inviscosity-temperature relationships when these conceptsused at exactly the softening point temperature~as we dis-cussed earlier, this is the temperature at which theretransition from a viscoelastic solid-like behavior to a vicoelastic fluid-like behavior for asphalt!. Using the com-pressibility of asphalt when tested in a confined moSchweyer and Busot@351# proposed a rheological equatioof state for asphalt. They recognized that memory effemay be of consequence in the description of asphalt,they were among the earliest to do so. Their followingmarks are worth repeating: ‘‘A Stokesian fluid is one whethe stress caused by deformation is only a function of theof deformation; it may be Newtonian or non-Newtonian. Aphaltic cements used in paving have memories at ambtemperatures that will be between the limits for these textremes because they are viscoelastic.’’ While Schweand Busot@351# are correct insofar as the ability of thStokesian fluid model to capture certain non-Newtonian ftures such as shear thinning, shear-thickening, or nonlincreep, they are misleading as the remark seems to imviscoelastic response is something in between Newtonand non-Newtonian fluid response captured by the Stokefluid. A Stokesian fluid cannot stress-relax, it cannot exhthe type of instantaneous elasticity Maxwell commentedearlier.

Most of the later studies on asphalts presumed that aspcould be described as a linear viscoelastic material andmainly concentrated on investigations of the linear viscoetic behavior of different types of bitumens in terms of th

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dynamic modulus and phase angle curves~Jongepier andKuilman @355#!, development of master curves for the complex shear modulus with frequency and temperature~Dobson@356#!, linear viscoelastic behavior of thin films of bitumein tension and compression and loading in direction normto the plane~Dickinson and Witt@357#, Dickinson @358#!,behavior in steady and transient shear flow~Attane et al@359#!, physicochemical analysis of different kinds of bitmen~Buisineet al @360#!. Other studies concerned the chaacterization of asphalt using a Generalized Maxwell mo~Stastnaet al @361,362#!, a bi-model description of asphastructure and the limitations of the time-temperature supposition ~Lesueur et al @363#!, use of Eyring Plasticitymodel, time-temperature superposition for describing asp~Cheung and Cebon@364,365#!, use of fractional model fordescribing the ‘‘mechanical and dielectric transitio~Stastna and Zanzotto@366#; see also@367,368#!, linear andnonlinear domains for asphalt and asphalt concrete~Croixand Di Benedetto@369#!, etc. For more complete reviews othe rheology of asphalt pertinent to its linear response,reader is referred to Schweyer@370# and Welborn@371#.

5 AGING OF BITUMEN

5.1 Current approaches

Let us start with a discussion of what is meant by agingthe asphalt mechanics community. A very clear descriptof this process which is used even now has been provideDow @23#. He remarks, ‘‘All bitumens undergo a more oless rapid change with aging that appears to be due to twpossibly more causes. Two distinct changes manifest thselves. One is the surface hardening, which is likely dueoxidation, and possibly to the volatilization of some ligoils. It begins at the surface and gradually extends intobitumen. The other is a hardening of the entire mass,dently due to condensation of molecules. Both these chantake place in all bitumens, but one or the other may predonate. The former is much the less objectionable, as it mabut slow progress into the mass.’’ As the constitution ofphalt became more clearer during the 1950s, it was possto discuss the influence of the different constituents ofphalt with regard to its response characteristics. For instain the case of asphalt that is a mixture of asphaltenes,and resins, Brown@372# sums up the entire process of aginas follows: ‘‘When asphalt ages in the pavement, some ofoils and resins are preferentially absorbed into the porethe aggregate, but the chief aging process is that of oxtion. Qualitatively, the oxidation changes experienced inair-blowing still, in the hot mixer, and in the pavement areof the same general nature. Over the course of the oxidachanges, the non-polar oils remain substantially constantresins decrease, and the asphaltenes increase. This indconversion of the smaller resin molecules to the largerphaltene molecules. Undoubtedly, also, asphaltenes furcombine with resins or themselves to still larger asphaltenThis size growth of the reactive molecules proceeds with lof hydrogen, which is converted into water. Because agggate is preferentially hydrophilic, such formation of water

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the body of the road probably adversely affects the adheof the asphalt. Also, because of increase in size, per sgreater degree of physical entanglement of these comspecies must result with increased resistance to slippaglayers at the molecular level. This decreased freedommovement shows as an increase of consistency signalehigher softening point and lower penetration.’’ The admrable clarity with which the mechanics of aging is explainabove is unfortunately not transferred to delineating macscopic variables which can be used as proper measuredescribing the process of aging. But this is no easy taskremains an open problem to this day. In Brown’s study, ois confronted with terms likeconsistencyor relations pertain-ing to one empirical test or another, which do little to quatify the process of aging. An interesting aspect of agingdiscussed by Brown is the manner in which exposure to liaffects the aging process.

Blokker and Hoorn@373# classified the manifestations oaging of bitumen as physical hardening and chemical haening. Physical hardening was supposed to be due to thein the adsorption equilibria due to the colloidal nature of tbitumen, crystallization of paraffin wax, and the evaporatof volatile substances. Oxidation of bitumen due to the enronmental conditions was considered as the main causchemical hardening. That oxidation proceeds at a mfaster rate in the presence of light is well known and caninferred from the experimental studies by Blokker and Hoo@373#. For instance, the oxidation rate varies by as much50 times when measurements on the percentage of bituwere made for 2.75 days in the dark and when exposelight. Also, in the case of oxidation in the presence of ligexperiments have shown that only layers of up to 4mmthickness are oxidized. While this may hold good under ctrolled laboratory conditions, the field conditions are qudifferent in the sense that the oxidized asphalt may cruminto a powder on the surface and this may further expfresh material for reacting with the atmospheric oxygen. Tmain difference between oxidation when exposed to liand in the absence of light is the way in which the reactoxygen is bound~Knotnerus@374#!.

Traxler @375# suggested that there are 15 different effeinduced by time, heat, oxygen, sunlight, and beta and gamrays leading to bitumen’s durability problems. These effewere oxidation~in dark!, photooxidation under direct sunlight, volatilization, photooxidation under reflected lighphotochemical action of direct light, photochemical actionreflected light, polymerization, age hardening, exudationoil, changes by nuclear energy, action of water, absorptionoil by solid, absorption of asphaltic components at a sosurface, chemical reactions or catalytic effects, and micbiological deterioration.

Kinnaird @376# suggested that there can be two hardenprocesses that can take place in a bituminous mixtureoxidative hardening and a non-oxidative hardening and pposed empirical measures such as a characterizing fact~apower law relation between penetration and softening po!and an activity coefficient~the ratio of the characterizingfactor and the percentage of paraffinic constituents!. How-

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ever, later studies showed that the non-oxidative hardeninessentially related to low temperature oxidation.

While there have been studies related to the modelingthe aging process of asphalt using very simplified schemthe majority of the studies in the asphalt concrete literatwere related to devising test methods which can mimicaging process in the laboratory and the development ofgression equations to correlate it with field studies. The eliest known tests to measure the loss on heating and haring of asphalt are due to Dow@23#. Mackenzir@377# studiedthe weathering of natural asphalt based on the studiescarbenes~the constituent of bitumen that are soluble in cbon disulphide but insoluble in carbon tetrachloride anamed so by Richardson@26#!. On examining Texas and Durango pitch, Mackenzir came to the conclusion that light aing upon a solution of bitumen in tetrachloride causesbitumen to decompose and this decomposition is directlylated to the amount of overheating involved. However,all the bitumens were found to show this behavior~for in-stance, the Bermudez asphalt which was also tested!. Mack-enzir @377# called the carbenes formed during the influenof light as ‘‘pseudo-carbenes’’ to differentiate them from tcarbenes that were formed in dark conditions. Spielm@103# ascribed this phenomenon to ‘‘destructive heatinThe amount of carbenes determined the reliability ofpavement and Broome and Child@378# suggested that thepresence of more than 0.5% of carbenes makes the bituunsuitable for road construction~see also Broome and Chil@378# for details concerning a typical weathering test!.

Commenting on the influence of heat on bitumen, Spman@103# remarked: ‘‘The changes caused by heat are prably three-fold: 1! a physical change, after which the matrial tends to revert to its original condition, but whichprobably never reaches, owing to the temperatures, veties, and conditions of hardening of the numerous constitusubstances; 2! a specific chemical change caused by heand 3! chemical changes which would take place slowlyordinary temperatures, but which are accelerated by heFurther, he concluded, ‘‘The changes that occur after bmen has been removed from its source may be negligiblgreat. Albertite and its relatives suffer no change; Trinidasphalt rapidly hardens on the surface of the lake, whresidual bitumens undergo a considerable and prolonchange after suffering the heating necessary for its prodtion. This change is two-fold—a slow hardening due to taction of light and air on the residual oils remaining in tbitumen, and a slow and mysterious ‘settling down,’ ofnature that can only, and unsatisfactorily, be termed ‘intermolecular arrangement.’ That this vague phrase is actuallthe direction of correct interpretation is seen by the increof the asphaltenes and melting-point, with the time.’’ Usithe penetration test as an indicator of the aging susceptibof bitumen, Spielman@103# found that there were differencein the penetration value measured at the surface ofsample and at 3 mm below the surface and increase ophaltene content. Dow@23# and Errera@145# had recorded

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earlier that the increase in asphaltene content due to thefluence of heating is due to oxidation, action of light, apolymerization.

Ebberts@379# devised a method for measuring the oxidtion of asphalt in thin films by using acid potassium permaganate as the indicator. Some of the interesting observatare relations pertaining to the oxidation rate of asphalt mafactured using different types of processing methodsfrom different crude sources. Ebberts also suggested sempirical indices for oxidation rate in terms of the consumtion of potassium permanganate.

Pavement engineers requiring immediate and dependsolutions to the problem of aging of asphalt used data frreal life pavements to construct empirical correlations whcould be used for designing and constructing pavemeGallaway @380–382# reported results of extensive testincarried out on the durability characteristics of asphalt,data for which was gathered over a period of four years frasphalt concrete pavements in service. Some of the impofindings of this study are related to the change of the checal composition of asphalt when subjected to field contions. In the span of four years, the percentage of asphaltincreased with the percentages of resins and oils showincorresponding decrease. Using a modified Ebberts@379#method to measure the oxidation value of asphalt, it wfound that it decreased with service life and, as expecthere was a marked change in the viscosity with time.

There have been few attempts at modeling the agingasphalt even within the realm of linearized viscoelasticHere, we review some of the important literature pertainto the aging of asphalt and finally we present a recent stof Wineman and Rajagopal@249# concerning modeling theaging of a viscoelastic material in general. Their study wohave to be greatly modified to take into account the vmany factors that influence the aging of asphalt. It mightbest provide a staring point, if even that, for such andeavor. We have discussed here some of the issues relatthe aging of asphalt. There is a great body of literaturelated to the modeling of the aging of asphalts and thedura-bility of asphalt concrete layers. We make no attempt to pvide an extensive review of the same. We shall rest condiscussing some of the issues and citing some of the relereferences related to the modeling of these issues.

Most of the current approaches to the modeling of agof asphalt fall into two categories, one related to that prticed by asphalt paving technologists in devising experimtal procedures that can simulate the aging behavior of aspduring mixing and laying operations and during the servperiod in highways and runways and the other relatedasphalt chemists trying to understand the changes taplace at the molecular level in asphalt when it ages usingmost modern tools available to the scientist such as gas cmatography, mass spectrometry, etc. Now, a distinctneeds to be made between the aging of asphalt as a sconstituent and aging of asphalt as part of the asphalt ccrete mixture. With what confidence can one use the expmental data on the aging of a pure asphalt sample and uto interpret the aging of an asphalt concrete layer? In the c

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of asphalt concrete, one needs to take into account facsuch as percentage of air voids, amount of asphalt, thickof the asphalt coating, type of aggregates, etc, all of whaffect and are affected by the aging process. The situabecomes even more complex if one is concerned withscribing the aging of modified asphalts~eg, polymer modi-fied asphalts!.

Oort @383,384# experimentally studied the diffusion ooxygen using the classical diffusion equation and useddiffusivity and penetration of oxygen into asphalt filmscorrelate with the extent of aging. While this may be callthe first attempt in trying to mathematically model the agiprocess, the constitutive properties of asphalt were not tainto account. He idealized the process of aging by meantwo mechanisms; the diffusion of oxygen and the oxygconsumption of asphalt by reaction. He also assumed thdark conditions, ie, in the absence of light, conditions ptaining to slow diffusion and shallow penetration existeWith these assumptions, for instance, for a layer thicknes5 mm, a diffusion coefficient of 5.5310216 m2/sec wasfound. However, studies by Blokker and Hoorn@373# con-tradicted the findings of Oort@383,384# in the sense that alarger diffusion coefficient and penetration were found frosimilar experiments. The reasons for the contradiction wexplained by Blokker and Hoorn@373# as stemming from thelack of recognition of the different reactivities of the compnents and an assumption of a constant coefficient of dision. Using the notion ofstiffness~Van der Poel@172#!,Blokker and Hoorn@373# defined an aging index and camea conclusion that ‘‘between most bitumen, there are nopreciable differences in type of ageing.’’ This is not the caand we will discuss it later.

Majidzadeh@385,386# used laboratory aged asphalt asand asphalt to study their creep response. He foundthere was a significant difference in the creep responssand asphalt made with aged asphalt and the sand asmixture aged as a whole. This is of practical significancein the field the aging of asphalt takes place in the presencaggregates and this finding also shows that studies on clation of the aging of asphalt with aging of asphalt concrneeds to be done with great care. He also found that ccompliance decreased in both types of mixtures as the ‘gree’’ of aging is increased. Anaging indexdescribed as theratio of the viscosity of the aged and unaged asphalt@386#was introduced. At low temperatures, the aging indtemperature relation was highly nonlinear and Majidzad@386# ascribed this to theactivation energyof asphalt at lowtemperature. By assuming a simple linear viscoelastic mofor capturing the creep response of sand asphalt, the dedence of the parameters that characterize the model ontype of asphalt, degree of aging, and the test temperawere determined. Majidzadeh and Schweyer@387# used alinear viscoelastic model to describe the aging processproposed adynamic aging index~given as the ratio of lossmodulus of the aged and unaged asphalt! since they reasonethat the earlier proposition of anaging indexdoes not takeninto account the time variation related to the aging phenoena of asphalt.

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Brown et al @388# coined the termsteric hardeningtoillustrate the reversible hardening resulting from reorientions taking place at the atomic, molecular, and micelle lels in the absence ofdimensional deformation. This is differ-ent from the ‘‘strain hardening’’ which is due to the internorientations caused by continued deformation or straSteric hardening has not received much attention in the fias reflected by the remarks of Brownet al @388#, ‘‘This phe-nomenon is not a new discovery, but has received little ephasis,’’ and the comments of Petersen@389#, ‘‘The tendencyfor an asphalt to harden from structuring, which may bemajor contributor to loss of durability and pavement failuris being virtually ignored in pavement performance conserations by present pavement technologists.’’ From the fipublication on the hardening of asphalt due to structuringHubbard and Pritchard@390# to the experiments of Traxleand Schweyer@305# and Traxler and Coombs@248#, andthose by Brownet al @328,388#, nothing much has been donin terms of experimental investigations or the developmof a theoretical framework to explain this aspect to thehavior of asphalt and its incorporation within the asphconcrete life estimation studies. This characteristic of asphas important practical implications and these are sumrized by Neppe@391,392# in the following manner: ‘‘Slowchanges in the internal physical structure attributed to thitropic phenomena are recognized as contributory factorthe natural hardening of bitumens particularly in an undturbed state. This fact largely explains the relatively rapdead-end street cracking sometimes encountered in prawhere there is a lack of traffic, whilst a similar road structusubjected to much compaction by traffic may remain inperfectly satisfactory condition for a long time, other condtions being equal. At the dead-end, the relatively undisturmat is allowed to undergo thixotropic hardening, and a rigtype of pavement results. Cracking then occurs owingcontractions and expansion caused by changes in tempture. On the other hand, a road of similar construction whis subject to the continual kneading action of traffic is molikely to retain its original flexibility.’’ Here, we wish to pointout that while the terminologies used by Neppe@391,392#and Brownet al @388# are different, the phenomenon theare referring to is the same. Also, this type of behaviordifficult to observe experimentally since any mechaniworking or any increase of temperature can alter the intestructure of bitumen.

Hubbard and Pritchard@390#, in a study aimed at estimating the effect of different variables on the asphalt penetrattest, reported for the first time the reversible hardeningfect. By changing the initial conditions before the start of tpenetration test such as the time of cooling in air, water,ice, Hubbard and Pritchard@390# found that samples kept fomore time in air showed a gradual hardening or loweringthe penetration value. The reversible nature of this phenenon was shown by Hubbard and Pritchard@390# when thesesamples were then heated and tested again by cooling tin water. The fact that these samples showed no sign ofearlier hardening had nothing to do with the aging of asphwhen they are exposed to prolonged heating, and thispointed out by Hubbard and Pritchard@390#. Traxler andSchweyer @305# called a similar phenomenon as ‘‘timehardening’’ in which there was an increase in the viscosityan asphalt specimen when it was allowed to remain un

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turbed. They also defined an asphalt aging index given byslope of the log-viscosity versus log-time curve assumintime scale of 100 hours as the basis of comparison. Addiof limestone filler did not change the time hardening~alsocalled as ‘‘age-hardening’’ by Traxler and Schweyer@305#!but recent studies on the addition of limestone fillers hashown that there is a significant change in the hardenbehavior of asphalt. However, it is to be pointed out thmost of the recent studies are related to the general hardebehavior of asphalt and they do not necessarily talk abouttime hardening behavior as considered in the study by Tler and Schweyer@305#. Continuing on their studies on thage-hardening behavior of asphalt, Traxler and Coom@248# conducted studies on two different types of asphclassified by them asessentially viscous asphaltsand non-viscous asphaltsand also on lime-asphalt mixtures. The aghardening effect was more apparent in thenon-viscous as-phalt and this led them to conclude that ‘‘...the rateincrease of consistency with time becomes greater withdegree of anomalous flow exhibited by an asphalt leadthe conclusion that age hardening is directly dependenthe gradual formation of some kind of internal structure.’’

In the field, where asphalt coexists with aggregates,amount of asphalt used to bind the aggregates together pa major role in the aging process. It is well known that taging of asphalt as a thin film and as a bulk materialdifferent. Recent studies on the steric hardening effect ofasphalt films in the presence of aggregate surfaces~Huanget al @393#! have shown that current performance-gradsystem of SUPERPAVE may not actually capture the diffence in the behavior of bitumen graded alike, especiwhen related to hardening behavior.

Thurston and coworkers in a series of articles@394–396#studied the influence of the various constituents onweathering nature of asphalts. That the oxidation in the invidual constituents of asphalt does not take place in an idtical manner as compared with asphalt as a whole was ontheir main findings. Most of the oxidation related problemoccur on the surface of the asphalt concrete layers, thepho-tooxidation as defined by Thurston and Knowles@395# re-lates to the oxidation triggered due to exposure to sunligThe dependence of the oxidation on the crude source asas on different constituents and their makeup is one fawhich needs to be taken into account to understand theface oxidation of asphalt concrete layers. For instance, Tston and Knowles@395# found that Gulf Coastal residuumand its constituents are more resistant to oxidation duesunlight than the Mexican or Mid-Continent residua. Obsvations related to the influence of the proportions of resand asphaltenes on recombined asphalt on weatheringalso reported by Thurston@396#.

Griffin et al @397# studied the influence of different constituents of asphalts on their durability characteristics. Baon the assumption that laboratory aging in the presencnitrogen alone can be attributed to the loss of volatiwhereas aging in air takes into account the hardening duoxidation as well as loss of volatiles, they studied four dferent types of asphalts. They also assumed that fract

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having molecular weight less than 400 contribute only to loof volatiles in the entire aging process. Using the ratio ofair aging index to nitrogen aging index, they calculatedhardening susceptibility due to oxidation of different kindsasphalt.

Corbett and Swarbrick@398#, in a very important study,addressed the issues related to the hardening of bitumerelation to its distinct hydrocarbon types. Using chromatoraphy and nuclear magnetic resonance, they separatedphalt into asphaltenes and four other hydrocarbon ty(paraffins1napthenes, single ring aromatics, light multirinaromatics, heavy multiring aromatics!. Using the increase oviscosity as a measure to describe the aging of whole aspthey found that the aging process essentially consists of cversion of heavy multiring aromatics into asphaltenes. Aanalyzing the asphaltene content of asphalt aged in serconditions for three years, they confirmed their finding thincrease of asphaltenes contribute to the aging of asphalt~seealso Kuhn and Rigden@399# for details regarding the use oa ball viscometer to follow the increase of viscosity relatto the aging of asphalt!.

Burgh et al @400# conducted extensive experimental stuies on the aging of bitumen. Their findings conclude thaging of bitumen and bitumen mixed with fillers proceedtotally different manner and that an ‘‘initially soft bitumencan be artificially aged so that its susceptibility to furthaging can be tailored to proceed at a retarded rate underconditions. However, in view of the absence of a modelexplain the entire mechanism, they were not able to many valid conclusions. A similar finding concerning the tedency of soft bitumen to take more time to harden andmethods for measuring the same were detailed by Andeet al @401#. However, the assumptions related to the tendeof soft bitumen to harden at a much slower rate may notalways correct as recent studies related to the use of biarkers have shown. The aging tendencies of typical asphdo depend on the source of the crude oil and the conditiunder which it has matured.

In a significant study on the short-term and long-termfects of aging, Lee@402# designed experiments to mimic thaging process of asphalt during the mixing and laying opetions and the long-term aging in the field during service coditions. Different types of asphalt were subjected to a tfilm oven test and in a pressure aging vessel and the chain viscosity and percentage of asphaltene were carefmeasured. One of the sample measurements made on tyasphalt used in highway construction is shown in TableThe dependence of oxygen pressure on viscosity and thecrease of asphaltene content clearly show the change oternal structure during the aging process. Elsewhere inarticle we have discussed the pressure dependence of viity and some of the recent attempts at modeling the sa~see also Lee@403#!.

The aging behavior of asphalt film as related to its tensstrength was studied by Sisko@404# for 12 different kinds ofasphalt from road samples which were aged in the road11 years. The tensile strength of asphalt, in general, is relits cohesive and adhesive bonding and some of the con

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Table 7. Effects of pressure on the variation of asphalt properties„Table 4 of †402‡…

Condition

Asphalt 10

Viscosity, 77 F~megapoises!

Asphaltenes~%!

Original 1.70 12.9Residue, TFOT 6.15 15.9Vacuum, 24 hr 5.70 15.9

N2 , 1 atm, 24 hr 6.80 15.5Air, 1 atm, 24 hr 7.20 16.3O2 , 1 atm, 24 hr 10.0 16.9

O2 , 10 psig, 24 hr 10.5 16.4O2 , 3 psig, 24 hr 11.0 17.6O2 , 60 psig, 24 hr 12.5 17.7O2 , 90 psig, 24 hr 14.0 18.0O2 , 106 psig, 24 hr - -O2 , 132 psig, 24 hr 14.5 19.2O2 , 154 psig, 24 hr 15.4 18.5O2 , 200 psig, 24 hr 17.8 19.4

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can go a lot further; we know such models are inapproprfor the material in question. Thus, all these results will hato be reinterpreted in the light of a proper nonlinear mofor asphalt and only those experiments that present rawcan be used with some amount of reliability. Even so, maof these studies at least go towards describing the comnature of asphalt.!

Dickinson@408#, describing the hardening of Middle Eapetroleum asphalts, suggested the use of treating bituwith liquid propane or lime. While the rolling thin film oventest~ASTM! does take into account the aging occurring ding the mixing operation, it does not take into account tinteraction of hot asphalt films with aggregate surfacesthe subsequent adsorption of volatiles into the aggregateface or the influence of the energy associated with the mixprocess on the aging. There are not enough experimestudies devoted to this issue. Ensley, in a series of articstudied the asphalt-aggregate interaction mechanismpostulated mechanisms for the asphalt-aggregate interaduring the mixing process@409–411#. He observes that alarge amount of energy is released in asphalt-aggregatemersion as compared with the immersion of ‘‘pure solidinto pure liquids. He also observes that the reaction timethe asphalt and aggregate is large. None of these phenomhave been modeled to any reasonable extent in the asconcrete literature.

Almost all the procedures for predicting the aging charteristics of asphalt depend to a large extent on the laboramethods of accelerated aging and then correlating it withbehavior under operating conditions. Most of these testsvolve testing asphalt at a specific temperature and for acific period of time, and empirical indices either based onchange in viscosity or change in the penetration valuesdeveloped. In the earlier sections concerning the modelinasphalt, we discussed in detail the complicated makeupasphalt and why linear viscoelastic models do not accuracapture its behavior. The same also holds true for aging sies. Ishai@412# pointed out the problems in attemptingcorrelate the aging of asphalt based on what he called aonepoint evaluation. He suggested that any aging study shoinvolve the variation of asphalt’s characteristics over timand temperature change. In this context, he suggested aphalt durability aging index which had the linear combintion of the influences of time as well as temperature onaging tendency of asphalt. However, this type of linear cobinations of time and temperature on the aging susceptibof asphalt may not hold good for a material like asphwhose internal structure changes in a non-linear mannertime. Similar problems with using single point temperatuoxidation studies on the aging mechanisms of asphaltbeen demonstrated by Herrington and Ball@413# in theirstudies on the temperature dependence of the oxidamechanism using gel permeation chromatography profile

The aging behavior of asphalt in asphalt concrete depeto a very large extent on the internal structure of asphconcrete characterized by means of air voids, connectedoccluded. Since aging is influenced to a large extent byamount of asphalt that is exposed to the atmosphere,

sions related to the independence of the crude source ophalt on its tensile strength paints a very simplistic picturethe entire structure and chemistry of asphalt.

Traxler @405# conducted studies on the chemical compsition of asphalt and its aging behavior. Using a correlatbetween thecoefficient of dispersion~ratio of resins and cy-clics to asphaltenes and saturates! to the hardening index~ratio of the viscosity of the aged asphalt to the unaged o!,he concluded that there exists a relation between agingdegree of dispersion of asphalt.

The simplistic notions adopted with regard to the predtion of the hardening behavior of asphalt, and quantifyingin terms of its constituents such as asphaltenes and macan lead to serious problems as enunciated by AltgeltHarle @406# in their study on the effect of asphaltenesasphalt viscosity. It is the property of aggregation of asphenes in the presence of maltenes which influences thehardening behavior. Also, the irregularities related toblending of different kinds of asphalt to produce a typicasphalt may not be all that straightforward as the followremarks by Altgelt and Harle@406# make evident: ‘‘For ex-ample, two asphalts of the same viscosity may yield a blof different viscosity. More generally, asphalts often fto obey the empirical formulas for predicting blend viscoties which have been found useful for oils and other viscliquids.’’

Specific field tests to monitor the aging of asphalt habeen attempted in different locations for various periodstime. For instance, Corbett and Merz@407# describe theMichigan test road after 18 years of service. Studies oftype have given us a wealth of information relating to thardening of asphalt and the changes in the physicalchemical composition of asphalt; changes in the asphaltecontent and aromatics, changes in the viscosity, penetraand ductility values, etc.~Of course, all these quantities aobtained on the basis of data reduction that presupposspecific model and we cannot overlook the fact that tmodel might be inappropriate for the material. In fact, w

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more porous the asphalt concrete mixture, the more istendency toward aging. Studies relating the amount ofvoids to the tendency towards aging have been reportethe literature by Kumar and Goetz@414#. Using Fick’s law ofdiffusion, attempts to model the diffusion of oxygen into tasphalt films and thus induce aging have been carried ouOort @383,384#, Dickinson@408#, Blokker and Hoorn@373#,and Tuffour and Ishai@415#. However, the process of diffusion of oxygen is much more complicated and the simplifiassumptions which ensure the appropriateness of Fick’sare not necessarily valid under normal operating conditio

The constitution of asphalt and the separation into thgroups depending on their molecular size has been achiusing high pressure gel permeation chromatography studLaboratory aging of these asphalts showed a clear shift ofmedium and small molecular sized groups to the large mlecular sized ones@416#.

Some of the important physical and chemical parameassumed as significant in the aging processes are the dgen content, temperature, time, film thickness, and finallythe basis of recent studies, the origin of the crude oil.oxidation is the only aging mechanism assumed, in termthe chemistry of asphalt, it points to the increase of thephaltenes content. Using a new approach to characteriztumen chemical functions based on29Si NMR spectroscopyon different types of bitumens~with some of them aged inthe laboratory!, a correlation between the acidic characand the aging of bitumen has been established@417#.

Use of hydrated lime as an additive to bitumen to improits aging properties has been studied in detail by many woers in the field. However, contradictory results about the beficial effects of the additives are reported.~See Lesueur andLittle @418# and Johanssonet al @419# for details.! Similarly,the use of polymer modified bitumen in place of straight rbitumen to have better aging properties has been advocin the literature~see, for instance, Lu and Isacsson@420,421#and many references listed there!. However, we do not wishto enter into a discussion of the polymer modified bitumand its rheological properties in this article.

The use of a laboratory aging procedure and comparof its results with the field aging of asphalt is fraught wiinconsistencies. For instance, the pressure aging vesselin the laboratory to mimic the long term aging of asphuses high pressure in an attempt to accelerate the agingcess. In the earlier sections, we have discussed the isrelated to the dependence of viscosity on pressure andthe modeling attempts can become quite involved. Usesome empirical indices either related to the ratios of thephaltenes weights of the aged and unaged asphalts or rof the viscosities of the aged and unaged asphalts doclearly capture the aging process. In fact, some studies heven shown that the pressure aging vessel~eg, Siddiqui andAli @422#! has a more detrimental effect on asphalt thanrolling thin film oven test which is used to mimic the aginoccurring during the mixing process.

Use of parameters such as the complex modulus or pangle from the assumption that asphalt behaves as a linized viscoelastic fluid has been used in the modeling of

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ing. Relations between the complex modulus and shearconducted at different temperatures have been discussethe literature by Mastrofini and Scarsella@423#.

The issues related to the comparison of the field and laratory aging of asphalt has been discussed in detail by Vhasselt and Choquet@424,425#. They used a kinetic theoryapproach using the classical kinetic theory and assumin1D diffusion model for capturing oxidation. Using asphaenes content, ring and ball softening point, and the reciproof penetration as some of the indicators for oxidation, thsimulated aging of asphalt in the laboratory. One importaspect of aging which was studied in detail by them is relato the correlation of aging data from laboratory to the fie

In a significant study Petersen@426# and Petersenet al@427# reinvestigated the problem of oxidative aging with rspect to laboratory testing. They provided an explanationthe difference in the oxidative aging between laboratory afield conditions and they developed a new model for oxidtive aging. Petersen’s remarks concerning the aging proare worth repeating@426#: ‘‘Although viscosity increaseswithin a given asphalt are related to the amount of oxidatwhich takes place, the sensitivity of the asphalt to viscosincrease from oxidation varies widely from one asphsource~composition! to another. For example, two asphaforming identical amounts of oxidation products duringgiven aging exposure might have viscosity increases thatorders of magnitude different from each other... It is poslated that the change in the physical properties of asphaltoxidation are dominated more by the sensitivity of thephalt to the oxygen-containing functional groups formed~itsresponse to physicochemical interactions! than by the abso-lute level of oxidation which occurs. Thus, the physical proerty response to oxidation appears to be governed largelthe state of dispersion of the asphalt components: ie, hwell the solvent phase of the asphalt keeps the polar comnents dispersed during oxidation, thus reducing their effon viscosity increase.’’ Earlier studies by Lee@402,403# hadshown that the functional relationship between the rateviscosity change with time is hyperbolic. Petersen@426# alsoremarks: ‘‘In general, ...high temperature aging produplots of log viscosity versus oxidation time that deviate onmoderately from a linear relationship, while low temperatuaging produces hyperbolic plots in which the rate of viscity increase progressively decreases at a significant rateaging time.’’ In an attempt to explain the difference in aginbetween high temperature testing as practiced and the ataking place at representative temperatures in the field,tersen@426# proposed a hypothetical model related to tchange of microstructure and its influence on the agingasphalt. Assuming that the asphalt microstructure mobilition is largely dependent on the temperature of testing,tersenet al @427# were able to explain different types of experiments~see also Harnsbergeret al @428# and Branthaver@429# for relations between oxidation of asphalt and the labratory testing procedures, and Wanget al @430# and Chater-goonet al @431,432# on the chemistry of asphalt oxidation!.Drawing upon ideas related to many of the naturally occring phenomena which change with time, such as bact

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growth, population dynamics, and chemical reactionsmodel to describe the aging of asphalt over time was pposed by Chen and Huang@433#.

Asphalt chemists have recently used very sophisticaequipment to monitor the changes taking place at thelecular level during the process of oxidation. For instanthe dependence of pressure on the aging susceptibilitdifferent asphalts~Domkeet al @434,435#!, the relevance ofthe use of pressure aging vessel pertaining to the dependof temperature for modified asphalts~Chipps et al @436#!,correlations pertaining to carbonyl content and oxygen ctent for different SHRP asphalts~Liu et al @437#!, air blow-ing of asphalts and the increase of their temperature sustibility and their aging characteristics~Vassilievet al @438#!,asphalt recycling agents and their influence on the aginghavior of subsequently recycled asphalts~Chaffin et al@439#!, and the role of using a model dihydroaromatic copound and petroleum bitumen in studies related to the obvation of the oxidation process~Herrington@440#! are someof the recent studies. The results of these studies need tdigested and the information used in the development oproper model for the aging of asphalt.

5.2 Simple mechanical framework for modelingthe aging of a viscoelastic material withpossible application to bitumen

We now turn to some recent work by Rajagopal and Wiman @249# that provides a means for distinguishing thelaxation of stresses in viscoelastic materials that do notfrom the relaxation of stress due to aging. Here, by aging,mean any structural changes that take place with time leing to an increase or decrease of stresses. In classicalcoelastic materials, stresses relax without any irreversstructural changes in the material.

Let us consider the class of viscoelastic materials thave an instantaneous elastic response, and let us supthat the natural configurations of the body can be reacheinstantaneously unloading to the stress free state. We ssuppose that a material that has been strained for longeriods of time ages more rapidly than a material that has bstrained by the same amount for shorter periods of time. Tis, specimens subject to larger strains age faster than tsubject to smaller strains, other things being equal. Interingly, asphalt’s properties can change for the better or wowith time depending upon the parent crude oil’s thermmaturity @31#.

Let us illustrate the difference between stress relaxatiin viscoelastic bodies that do not age with those that dothe case of a viscoelastic material that can age, we introdthe fourth order tensorE that characterizes the material rsponse:

E:R3R→L, (38)

where

L5$EuE:Lin Sym~V,V!→Lin Sym~V,V!%, (39)

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ence

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where Lin Sym(V,V) represents the symmetric linear tranformations from a vector spaceV into itself. In the case of a1D response, we shall assume a very simple form forE:

E~ t1 ,t2!5E0 expF2l1t1

tveGexpF2l2a~ t2!

tagG , (40)

wherea is a function oft2 and E0.0, l1.0, l2.0, tve

.0, andtag.0, andE0 , l1 , l2 , tve , andtag are constants.It is possible to consider situations whentag is a function oftime, but we shall not consider such a possibility.

We would like the material to respond like a purely vicoelastic material for timest,ta , ta being the instant whenaging starts. This would be possible if we were to picka(t2)to be given by,

a~ t2!5bH~ t22ta!, (41)

wheret2 is the running variable,b is a constant, andH is theHeaviside step function given by

H~ t2 t !5H 0 t, t

1 t> t. (42)

Thus, E reduces to the classical expression for linear vcoelasticity. Of course, we notice that whentag→`, themodel reduces to the model of classical viscoelasticity.

Let us suppose that the stress in a viscoelastic matethat can age is given by

T~ t !5E~0,t !«~ t !1E0

t

E~s,t !«~ t2s!ds. (43)

To clearly illustrate our ideas, let us consider a simpleresponse~the theory is equally valid for a general 3D response!

T~ t !5E~0,t !«~ t !1E0

t

«~ t2s!]E~s,t !

]sds, (44)

which can be expressed as

T~ t !5E~ t,t !«~0!1E0

t

E~ t2s,t !d«~s!

dsds, (45)

where the lower limit of integration corresponds to when tmaterial was created and the relaxation function E depeon both real time and anaging time. The above model suffersfrom the same lack of frame indifference as the classlinear viscoelastic model.

If «(t)5«0H(t2a), where H denotes the Heaviside stefunction, it immediately follows that

T~ t !5«0E~ t2a,t !. (46)

We could either have the material start activating at sotime ta or it could start aging from the time of creationt50. For the purpose of illustration, let us suppose that agstarts att5ta . For all timest,ta , as the material has nostarted aging, the stress will be given by that for a viscoetic material that does not age.

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In general, even in a homogeneous body, different mrial points will start aging at different instants of timevirtue of their being subject to different deformation histoThe stress can be expressed as

T~x,t !5«~0!E~ t,t !1E0

taE~ t2s,t2s!

d«~s!

dsds

1«~ ta!E~ t2ta ,t2ta!

1Eta

t

E~ t2s,t2ta!d«~s!

dsds, (47)

5TVE~x,t !1«~ ta!E~ t2ta ,t2ta!

1Eta

t

E~ t2s,t2ta!d«~s!

dsds. (48)

In the above representation,TVE(x,t) denotes the contribution due to purely viscoelastic response before aging andrest of the terms the contribution after aging has started.

To illustrate the effect of aging, let us consider the prolem of the torsion of a right circular cylinder.

Torsional deformation is described through a mapping

r 5R, u5Q1CZ, z5Z, (49)

from a referential frame expressed in a cylindrical polarordinate system (R,Q,Z), to a current deformed state reprsented in a cylindrical coordinate system by (r ,u,z). Thedeformation under consideration is inhomogeneous.11 Wepurposely pick an inhomogeneous deformation as it willsure that different points in the body will start aging at dferent instants of time. However, since we are interestedsmall strains we do not need to make a distinction betwthe Eulerian and Lagrangian strain.

Let us suppose aging is activated when the condition

f~Rc~ t !,t !50 (50)

is met, wherec(t) is the angle of twist per unit length of thcylinder. Now, the material will first activate at the boundaof the cylinder as that is where the strain is the largest~recallour criteria for aging!, say R0 . Let ta(R) denote the timewhen aging is activated at the radiusR. Thus, ta(R),ta(R0), if R,R0 .

Now

f~Rc~ ta~R!!,ta~R!!50. (51)

If t,ta(R0), no aging will be taking place in the cylindeThus, the stress will be given by

T~R,t !5RFc~0!E~ t !1E0

t

E~ t2s!dc

dsdsG , (52)

and thus the applied momentM (t) is given by

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ry

.

M ~ t !52pEo

R0T~R,t !R2dR

5JH c~0!E~ t !1E0

t

E~ t2s!dc

dsdsJ . (53)

We note that the ratioM (t)/J is independent of the size othe cylinder~hereJ5pRo

4/2 is the polar moment of inertia!.Aging will start at R5R0 , and as we holdc fixed, the

aging will proceed towards the center. Let us consider whaging is first initiated atR corresponding to a radial locatioR. The time at which aging is initiated ista(R).ta(R0). Thecylinder is divided into two regions; in the domainR<R,R0 the material is aging while in the domain 0,R,R thematerial has not started to age. Thus,

T~R,ta~R!!5RH c~0!E~ ta~R!!

1Eo

ta~R!E~ ta~R!2s!

dc

dsdsJ

5RT~ ta~R!!,;0,R,R, (54)

where

T~ ta~R!!5c~0!Eta~R!)1*0ta~R!E~ ta~R!2s!

dc

dsds.

Next,

T~R,ta~R!!5TVE~R,ta~R!!1Rc~ ta~R!!E~ ta~R!

2ta~R!,ta~R!2ta~R!!1Eta~R!

ta~R!E~ ta~R!

2s,ta~R!2ta~R!!]~Rc~s!!

]sds,

;R,R,R0, (55)

which can be expressed as

T~R,ta~R!!5R@ T~ ta~R!!#1RTa~R,ta~R!!, (56)

where

RTa~R,ta~R!!5Rc~ ta~R!!E~ ta~R!2ta~R!,

ta~R!2ta~R!!

1Eta~R!

ta~R!@E~ ta~R!2s,ta~R!2ta~R!!#

]~Rc~s!!

]sds. (57)

We can compute the moment att5ta(R),

rma-n.neousght

11A motion is said to be homogenous if in a Cartesian coordinate system the defotion gradientF ~see Section 6.2! has constant entries in its matrix representatioDeformations that are not homogeneous are called inhomogeneous. In a homogedeformation, material lines that are straight before deformation will remain straafter deformation.

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M ~ ta~R!!52pE0

R0T~R,ta~R!!R2dR,

52pE0

R@RT~ ta~R!!R2dR#

12pER

R

@RT~ ta~R!!1RTa~R,ta~R!!#R2dR,

(58)

52pE0

R0T~ ta~R!!R2dR

12pER

R

@Ta~R,ta~R!!#R3dR. (59)

Thus,

M ~ ta~R!!5JTa~ ta~R!!12pER

R0@Ta~R,ta~R!!#R3dR,

(60)

which immediately leads to

M ~ ta~R!!

J5Fc~0!ES ta~R! D1E

0

ta~R!E~ ta~R!2s!

dc

dsdsG

11

J ER

R0@Ta~R,ta~R!!#R3dR. (61)

We recognize that the first term is the contribution dueviscoelasticity in the absence of aging, while the second tis the contribution due to aging. While the contributionviscoelasticity does not depend on the size of the specimthe second term,

1

J ER

R0@Ta~R,ta~R!!#R3dR

52

p ER/R0

1

@Ta~R0R,ta~R!!#R3dR, (62)

clearly depends onR0 and thus the sample size.The model above is a purely phenomenological mo

that does not explicitly take into account oxidation and seral other variables that influence aging. In order to do so,would have to introduce parameters which reflect thesefects, and the tensorE has to be generalized to take it inaccount. For instance,tag could depend on the extent ooxidation. However, such a generalization would stop shof developing a proper thermodynamic framework for taging of asphalt.

While the development of such a thermodynamic theis quite daunting, it is by no means beyond the realmspossibility. In the next section we describe a general thermdynamic framework for describing the response of dissitive bodies. It is possible to include the effects of oxidatiwithin the purview of such a model, provided we can dscribe the entropy production associated with the oxidaprocess.

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6 MODELING OF ASPHALT CONCRETE

6.1 Classical models for asphalt concrete

In this section, we give a brief overview of the differeattempts at modeling asphalt concrete till recently and follit up with a thermodynamic framework that has been justin place for modeling asphalt concrete. Our main emphathroughout this article has been to identify the various factthat play a role in determining the behavior of asphathereby setting the stage for its constitutive modeling. Hoever, since the main use of asphalt~here we are merely referring to the quantity of asphalt used as we well know ofmyriad uses! is in the construction of highways and runwaywe feel it is necessary to look at some of the importantpects related to the modeling of asphalt concrete.

It is important to distinguish asphalt from asphalt cocrete. In marked contrast to our discussions concerningvarious experimental studies on asphalt, we make no atteto review all the literature related to the experimental studand field observations on asphalt concrete. We straightaget to the current literature related to the constitutive moding of asphalt concrete. The recent attempts at the modeof asphalt concrete have culminated in the developmenthe SUPERPAVE system@441# for arriving at the mix andstructural design in the United States. There have been slar attempts in Europe and elsewhere@442#. However, allthese guidelines share a similar philosophy with SUPEPAVE and hence we will restrict ourselves to a discussionSUPERPAVE. SUPERPAVE, as we shall see, is not a contutive specification for asphalt concrete, rather it is rulethumb design manual that serves its purpose, more orand is itself based on a very rudimentary constitutive thefor asphalt concrete.

In the following sections, we discuss what is meantasphalt concrete, its different failure modes, and the differtypes of constitutive models that have been proposed.discuss in detail the failure due to fatigue as it brings ointeresting issues related to the modeling of the crackinghealing of asphalt concrete. Here we use the term aspconcrete in a generic sense: it includes any mixture of aggates of different sizes with asphalt acting as the binder.

6.1.1 What is asphalt concrete?Asphalt concrete consists in a mixture of aggregates contously graded from a maximum size, typically less thanmm, through fine filler that is smaller than 0.075 mm. Sufcient asphalt is added to the mix so that the compacted m‘‘effectively impervious’’ and will have acceptable dissipative and elastic properties. Due to the use of different siand fractions of the aggregate, air voids are present inmix during the manufacture of asphalt concrete. It is necsary that a minimum amount of air void be maintainedallow for the deformation and recovery of the material witout serious structural damage~usually incorrectly referred toas plastic flow!.

There is a progressive change in the material propertiethe asphalt concrete during its lifetime. This is due to tinfluence of different micro-mechanical activities that are

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play during its deformation. The micro-mechanical activitiare due to interactions at the constituent interface or witthe constituents themselves. This, of course, depends laon the nature of the aggregate matrix and the temperarelated response of the asphalt cement~such as transitionfrom solid-like to fluid-like behavior, hardening, etc!. Micro-mechanical activities include effects such as reduction ofvoids due to densification in virtue of the loads imposedthe traffic, debonding and possible rebonding of aggregcontact points during the application of dynamic trafloads, and asphalt absorption by the aggregate partoccurring at the constituent interface. Changes in the rhlogical and micro-structural properties of effective asphcontent and degradation of aggregate particles are sof the micro-mechanical activities occurring within thconstituents.

In the lifetime of an asphalt concrete layer, there iscontinuous reduction of the air voids~and hence, increase ithe density of the asphalt concrete layers! due to the loadsdue to the traffic. This consolidation in asphalt concretecurs via two mechanisms: 1D densification and consolidain the mineral aggregates which is identified asplastic flowin the mineral aggregates. When reduction in air voids taplace without significant deformation of asphalt, it is referrto as densification. However, when significant deformatof asphalt takes place and itflows into the air voids, it iscalledplastic flow~the mechanism underlying what is meaby plastic flowhas nothing in common with the mechanisunderlying theplastic flowin metals!. These two are separatphenomena and depend to a large extent on the constitproperties of asphalt. The contribution of each to overall csolidation varies. Pavement performance studies@443# haveshown that when the air voids are less than 3% the consdation mechanism predominates. The reason for this isbuildup of high pore pressure in the reduced air void spaThe result is that some amount of asphalt is forced to flThe flow of asphalt into the voids and the reduction in tasphalt film thickness ultimately results in the reductionthe relative distance between the aggregate particles. Pareorientation may result from the flow of asphalt into tvoids. This relocation of particles can only occur after tfriction among the particles has been overcome. In fasome mixtures that have adequate internal friction and cosion at the time of construction areover-lubricated~througha reduction in air voids! due to additional consolidation. Duto this over-lubrication, the strength and shear propertiesthe mixture are greatly reduced. While the reduction invoids below 3% may lead to excessive consolidation duethe mechanisms discussed above, air voids greater than a7% result in high air and water permeability that leaddurability problems.

The development of asphalt concrete mixtures andstructural design of pavements were initially trial and erprocesses, with the criteria depending greatly on the expence of the materials engineer. During the last few decaempirical methods for designing the mixture and for tstructural design have evolved and have been standardbut these two aspects, the design of the mix and that of

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structures are still considered independent functions. Hethe structural design of the pavements is based on propeassumed for the material, and the question whether theterial used in the construction of the pavement meetsinitial assumptions that were made during the design stremains largely unanswered. One important aspect instructural design of the pavement that leaves much todesired is the absence of the consideration of the varmicro-mechanical activities that come into play during tperformance of asphalt concrete in the field. The influenof these micro-mechanical activities are generally accounfor by ad-hoc corrections that can sometimes be contratory.

The current approach to flexible pavement design isrelate the engineering properties of asphalt mixtures to pament distress and these are mostly distress prediction moand not constitutive models as usually understood in ctinuum mechanics. In these models, some measure oftress~number of cycles to failure, amount of permanent dformation, etc! is related to material properties that prsupposes the asphalt mixtures to be either linear viscoelaor linear elastic. The critical distress parameter thus assuis related to distresses such as rutting, fatigue cracking,temperature cracking, and moisture-induced damage,combination of these@441,443–445#. Mixtures are tested forcertaincritical propertiesand based on whether thecriticalpropertyof the mix satisfies the assumption initially madethe structural design, the mix is accepted or rejected.impetus for a more mechanistic approach instead of theditional empirical approach was provided by the Asphalt Agregate Mixture Analysis System~AAMAS ! developed aspart of the National Cooperative Highway Research Progrbetween 1986 and 1991@443#. The transition from totallyempirical to an empirical-mechanistic approach culminain the development of the SUPERPAVE mix design methcomprised of performance based material specificatiomechanistic test methods, design and analysis procedpractices for materials selection, accelerated performatesting, and mixture design for asphalt concrete@441#. Theconstitutive theory for asphalt plays a minor role in the grascheme of things, and the appropriateness or otherwise oconstitutive theory is not exposed to a direct test as it is,a small cog in a huge device which has built into it varioad hoc factors. Our aim here is to provide a constituttheory for asphalt concrete that can be either falsified or croborated to the extent that it conforms to some experimetest~as no experiment can prove a theory correct!. We do thisin Section 6.2.

The link between the testing protocol and pavement pformance prediction in the SUPERPAVE system is baseda theory that often presupposes that the asphalt respondsa linear viscoelastic material, and in some cases as a lielastic material. However, traffic loads that ultimately caudistress within the asphalt layers lead to a response thaoften outside the linear viscoelastic regime. This is especitrue of the material within the upper 25 mm of the asphlayer, which is subjected to tire induced shearing stresThus, when laboratory determined distress parameters

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compared with corresponding field data, the predictionspoor. There are several reasons why this approach hasbeen as successful as hoped for. One reason is the chanthe property of asphalt concrete over time. This is causedthe combined effects of the traffic and the environment. Treasons for the property change of asphalt concrete madue to a change in air voids, aging of asphalt, seepagmoisture, or a combination of all these effects. As thechanges occur over time, the engineering properties ofphalt concrete mixtures also change. Mixture propertiesdefined by the practitioners such as stiffness, stability, dubility, fatigue resistance, and permeability change due toconcomitant change in the structure of the mixture. Forstance, fatigue resistance of the mixture generally increawith a decrease in air voids. However, oxidation of asphleading to its aging makes the mix brittle, resulting in pofatigue performance and added susceptibility to low tempeture cracking. Asphalt undergoes micro-structural and rhlogical changes due to the traffic and environmental effe

During its lifetime, an asphalt concrete layer is subjecto several manifestations of distress: rutting, fatigue craing, temperature cracking, and moisture induced dam~Yoder and Witczak@444#!. All these distresses are interrelated. For instance, moisture induced damage can acceldistress in other modes. The damage due to moisture mfests itself in the form of stripping of the asphalt coatifrom the aggregate surface and loss of cohesion resultinreduced stiffness and stability. Studies of stripping duewater damage have indicated that the water trapped inthe air voids can reach up to a pressure of 138 KPa~20 psi!,exceeding the adhesive strength of the binder-aggregateface ~see Acott@446#, Hallberg @447#, and Khandal@448#!.Furthermore, the presence of moisture in the intersticethe asphalt concrete layers accelerates the age hardeningcess of asphalt and thus further reduces the life of the pment @51,449#.

Two methods historically used for mix design are tMarshall and Hveem methods@450#. These empirical procedures were designed five decades ago and are being usof today. This, in itself, is not necessarily bad if the metholead to reliable and predictable outcomes. Unfortunately tdo not. Although concepts such as design limits for stabilamount of air voids, asphalt adsorption, voids in mineaggregate, etc, seem reasonable, the premature distremost of the pavements constructed using these methoddicate that these cannot be used for designing the mixtuThere are a number of correlations which exist in the liteture such as the modeling of the Marshall test using a pticity model based on the Drucker-Prager yield criteri@451# or the prediction of engineering modulus of asphconcrete from the Marshall load-deformation values@452#.However, as with any empirical correlation, they are too liited and restrictive in nature@445#. More importantly, theyprovide no understanding of the true behavior of asphconcrete.

In the process of the characterization of asphalt concrthere are a number of material parameters that are emcally linked to performance. For instance, the creep modu

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is related to rutting; fatigue resistance is related to tenstrength, etc. One important parameter which has beenlated to rutting, fatigue cracking, and low temperature craing is stiffness. We saw that Van der Poel@172# introducedthe termstiffnessfor asphalt concrete based on an extensof the Young’s modulus used in the theory of linearized elticity. Two approaches for finding stiffness have been uswidely. One is based on laboratory tests and the other onomographic solution@338,453#. Stiffness~according to Vander Poel@172#! generally depends upon the type of asphcement and the aggregate volumetric concentration.volumetric concentration is indirectly related to the air voicontent in the stiffness model. Stiffness generally increawith a decrease in air voids~increased degree of compactio!and asphalt hardness@454#. This indicates that the stiffnesshould be a function of air voids and the change in the stness with respect to the change in air voids needs toincorporated in the material models. The majority of distremodels are phenomenological and, hence, the influence ovoids, if considered at all, is through some correction facor by the proposition of some upper limit for air void@455,456#.

6.1.2 Constitutive models for asphalt concreteThe motivation for most of the studies on asphalt concrbehavior has been to develop models to cater to some odistresses to which an asphalt concrete pavement is subjeto and determining some experimental variables based onanalysis of such distress models. Thus, unlike asphaltwhich models that describe the behavior of viscoelasticids have been developed, with regard to asphalt concrconstitutive specifications are primarily, though not excsively, related to empirical correlations for different typesdistresses. However, studies assuming asphalt concretea viscoelastic material have also been carried out. Thstudies assume that the macroscopic mechanical behaviasphalt concrete is viscoelastic and use either a sprdashpot analogy in the form of a Burger’s model~eg, Bur-gers @258#, Lee et al @274#, Saal and Labout@237#, Saal@457#, Poel@335#, Reiner@273#, Monismith and Secor@458#,Secor and Monismith@459#, Pagen@460#, Monismith et al@461#, and Huschek@462#!, or some simple form of vis-coelastic constitutive equation~eg, Daviset al @463#, Huang@464#, Moavenzadeh and Soussou@465#, Perlet al @466#, andKim and Little @467#!.

There are quite a few models for asphalt concrete whconsider the microstructure but as with most of the phenoenological models, they neglect the evolution of the micstructure during the life time or consider it, for instance,means of some ‘‘shift factors.’’ To illustrate our point, wreview some of the pertinent literature in this regard. Micrstructural modeling of asphalt concrete using the analogysoil mechanics was carried out by Nijboer@280#. He postu-lated that the entire deformation resistance of bituminomixes can be explained in terms of initial resistance, interfriction, and viscous resistance.

Huschek@462# built upon the work of Nijboer and usedthree phase system consisting of regions characterized

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viscosity, modulus of elasticity, and a modulus of plasticiVan der Poel@336# modeled the behavior of asphalt mixescalculating the rigidity of concentrated solutions of elasspheres in an elastic medium, using a method developeddilute dispersions by Frolich and Sack@266#. Hills @468# de-veloped models for the long time creep behavior of aspmixes by characterizing the internal structure of the mixmeans of the asphalt film thickness. Evolution equationsthis internal variable as a function of the macroscopic straing of the material were assumed. Recently, Cheunget al@469# and Deshpande and Cebon@470# have developed models for asphalt concrete mixes using isolated contact moand shear box models. Boutin and Auriault@471#, using theanalogy of a porous media saturated by a viscoelastic flclassified the macroscopic behavior of asphalt concretebiphasic, elastic, and viscoelastic depending upon the ratithe dimension of the pores to the macroscopic wave lenUse of a viscoplastic potential for developing an elasviscoplastic model for bituminous concrete has also bproposed in the literature by Florea@472,473#.

The deformation resistance of asphalt concrete is maderived from the aggregate matrix and the viscous aspmastic. Due to the change in the microstructure~either due tomechanical changes such as reduction of the air voidschemical changes such as aging of asphalt!, the response othe aggregate matrix and the asphalt mastic to the traloading changes with time. Also, the ability of the pavemeto stress relax upon load removal changes as the microsture is continuously modified. This change in microstructuand changes in the loading conditions and the environmeconditions, and hence the response, are the cause fornomena such as rutting, fatigue cracking, low temperacracking, and moisture induced damage. For instance,well known that the distress due to rutting is due to t‘‘accumulation of deformation’’ under repeated traffic loaing resulting in the development of ruts longitudinally alothe pavement. This ‘‘accumulation of deformation’’ depento a large extent on the 1D densification due to the air voreduction and the plastic flow of the asphalt mastic. Whmost of the studies have not considered the mechanismdensification assuming that the pavement will be compacwell during construction@474,475#, the plastic flow of theasphalt concrete has been assumed to be dependent onperature, loading rate, and the loading time interval. Ading to the complexity is the dilation of some sand asphmixes @464# under constant compressive stress for loloading times similar to that observed in the responsegeomaterials.

Asphalt concrete can be considered to be a thrconstituent system of aggregate particle skeleton~solid!, vis-cous mastic~fluid!, and the air filled voids~gaseous!. Thistype of analogy has been used in modeling the air voreduction of asphalt concrete using the continuum theorymixtures and the theory of linear elastic material with vo@476–478#. During the mixing process of the aggregate pticles with asphalt, it would be reasonable to assume thataggregate particles are coated closely by the asphalt. Duthe compaction process, this coating is penetrated so

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some direct particle-to-particle contact is developed@462#.The smaller mineral particles which are not part of the agregate mineral skeleton form the asphalt mastic along wthe binder. Both constituents resist the external loadiThough the air voids do not possess mechanical strentheir distribution is very critical in deciding whether a hydrstatic pressure condition can exist in the constituentswhether the micro-cracks present in the asphalt masticcoalesce to become macro-cracks due to the presenceair void in the neighborhood. Also, the air voids serve asort of a buffer into which the asphalt mastic is squeezwhen acted upon by a compressive force. In this respect,pertinent to address the concept ofrefusal air voids contentused in the asphalt concrete literature. It is the air void ctent beyond which there is no change in the air voidsasphalt concrete for any subsequent loading. So, frominitial virgin state of 5%–8% air voids in an asphalt concrepavement, there is a progressive change in the micstructure of the material due to traffic loading. These chanmay be due to the readjustment of the aggregate matrixthe asphalt mastic, thus resulting in the reduction ofvoids. Also, changes occur in the internal structure dueoxidation and aging of asphalt in the asphalt mastic, ahence the ease with which the asphalt mastic can moveaffected. In the case of the aggregate matrix, this resultobtaining an optimum interlock position of the contapoints, thus involving greater resistance to displacementthis context, a remark concerning some interesting obsetions made in the experimental studies on sand-asphalt aorder as this will reinforce the importance of taking the evlution of the micro-structure into account. In an experimenstudy on sand-asphalt, it was observed that there wasincrease in volume under constant compressive st~Huang@464#!. A similar study on sand-asphalt suggesteduse of amodulus of recoveryandmixture viscosityfor char-acterizing asphalt mixes~Wood and Goetz@479#!. Comparedto asphalt concrete mixtures, sand-asphalt consists offine aggregate particles and the stiff skeleton developeding the compaction process in a normal asphalt concspecimen are absent. Hence, this results in the dilation ofspecimen during normal compressive loading as there isstiff aggregate matrix to resist the movement of the saasphalt mastic mixture in the lateral direction.

To summarize, the traditional methods of modeling aphalt concrete have not considered in detail the change inmicro-mechanisms as the material deforms, and assumethe Cauchy stress tensor depends upon the deformationdient measured from a single reference configuration. Wthis may be true for materials which fall within the purvieof classical elasticity such as rubber, for a material likephalt concrete whose stress-strain response is not onlyand temperature dependent but also dependent on the inal structure and its change, this would not be correct.materials in which the microstructure evolves, thereby alloing the body to have an evolving set of natural stress fconfigurations, the stress would depend on kinematical msures from this evolving set of natural configurations. Tobservation is reinforced by the variety of shift factors whiare used when comparing laboratory tested specimensthe data collected from the field. Before we discuss theplication of the framework in simulating the response of aphalt concrete for different types of loading conditions, a fe

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remarks concerning the use of different types of testsmodels prevalent in the literature are in order. It is clear tthe properties of the constituents of the mixture have anfluence on the mechanical behavior of the mixture. The meral aggregates of various sizes are held together by thephalt binder. A properly compacted asphalt concrete mixtproduces a structure whose mechanical properties depenthe interlocking of the aggregate matrix and the cohesivenof the asphalt mastic. The introduction of large volumesaggregate of different size into asphalt give rise to two dferent mixtures within asphalt concrete. The first one isaggregate matrix which is bonded in the contact pointasphalt and the second is the asphalt mastic in whichfillers float in the asphalt. This type of reasoning has beadopted in the past to evaluate the partial volumes of aspneeded for the aggregate matrix and the asphalt mastiusing a so calledmastic theory@480#.

The influence of the type of test method employedelicit the mechanical behavior of asphalt concrete dependa large extent on the response of the aggregate matrix anasphalt mastic. In the case of compressive creep testsaggregate matrix interlocks and the increase in rate of demation ceases after the load has been applied for someWhen the percentage of aggregate matrix is considerless, as in a sheet asphalt mixture, the increase in the radeformation ceases after a considerably longer time@479#.This is due to the absence of the development of the aggate interlock which occurs in a normal asphalt concrlayer. Also, in asphalt concrete mixtures at high temperatthe asphalt viscosity decreases and the aggregate particleable to move more easily past one another. As the phygoverning relaxation is different from that regarding crefor asphalt concrete, it is incorrect to try and infer strerelaxation response from creep response or vice versa. Athere are inherent differences in the tensile creep test acompressive creep test. In the compressive creep test, themore likelihood of a sharing of stress between the particand the asphalt with the aggregates transmitting some sto the asphalt. In the case of a tensile creep test, the majof the stress is transmitted through the asphalt mastic win turn transmits it to the aggregate mastic.

To sum up, the difference between the mechanical behior of asphalt concrete when subjected to different typestests is influenced by the way in which the individual costituents, ie, aggregate matrix and asphalt mortar matrix,have. This is articulated well by Daviset al @463#: ‘‘Al-though the asphalt/aggregate composite retains some oproperties of the viscoelastic cementing agent, the differebetween the viscoelastic response of the asphalt alonethat of the composite seems to be a strong function oftype of test employed. The more aggregate displacemenvolved, the greater is the difference between the viscoelaproperties of the composite and of the asphalt.’’ Significstrides have been made in the use of tomography for chaterizing the internal structure of asphalt concrete. Recentvances, both technology wise and in the application of maematics in developing various inversion methods thatneeded for different modes of data collection, have made

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tool very attractive for many engineering applications. Tmathematical essence of tomography is in the reconstrucof a distribution from samples of its Radon transform~seeHermanet al @481#, Helgason@482#, and Radon@483#!. Mostof these applications on tomography are essentially aimemeasuring the dimensions of the internal structure, interflaws of the materials, etc. Use of tomography in imagiasphalt concrete has been mostly in characterizing vstructure and aggregate interlock@484,485#, determining therelationship between the size of the sample to its gloproperties@486#, quantifying the effect of compaction on thinternal structure@487,488#, study of crack propagation@489#, aggregate angularity@490#, etc. Differing from theabove studies, the use of tomography in estimating thetions from a sequence of images has been carried outasphalt concrete and other materials and is detailed inet al @491#, Zhouet al @492#, and Synolakis@493,494#. Whilethe use of computer tomography to study the 3D interdisplacement field of a composite material such as aspconcrete would help in identifying the local material propeties and validating the constitutive assumptions, these stu@492–494# have stopped with detailing the kinematics of dplacement and have not gone further in using the resultthe development of constitutive theories.

6.1.3 Fatigue life prediction modelsOne of the important distress mechanisms in asphalt concpavements that is least understood is the fatigue cracbehavior. In the design process followed as part of the SPERPAVE and other methods, the tensile strain at the botof the asphalt concrete layer is assumed to be the paramcontrolling the fatigue cracking behavior. Thus, most of tpavement design process is then directed towards miniming the horizontal tensile strain so that failure by fatigcracking is prevented. Experiments are conducted atlaboratory for the characterization of the fatigue resistaproperties of the chosen asphalt concrete mixture under ristic loading and environmental conditions. It is well reconized that the number of cycles of a standard axle to cafailure by fatigue cracking in service is far greater than tpredictions made from laboratory testing. Hence shift factof the order of 10–20 are used to scale up the predictionthe laboratory testing. Here, failure is defined as the poinwhich the stiffness of the mix is reduced to 50% of its initistiffness@441#.

The fatigue behavior of asphalt concrete within the claof tests performed by experimentalists working with asphconcrete depends to a large extent on the characteristicthe mix ~asphalt content, type of asphalt, air voids, etc!, thespecific test being conducted, and mode of loading~whetherthe stress or strain is being controlled, the frequency, theperiods, the temperature at which testing is carried out, e!.

Several experimental tests are available for characterizthe fatigue behavior of asphalt concrete. Simple flexubeam fatigue test in three-point loading of rectangular spmens@495#, cantilever type loading of trapezoidal specime~Bonnot @496#!, and diametrical loading in indirect tensilmode @497# are those that are most often used. Similar

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different types of loading patterns such as sinusoidal, hasine, triangular, and square shaped wave forms are usedinstance, the SHRP test M-009@441# proposes a four poinbending of rectangular specimens subjected to sinusoloading with constant strain loading for determining its ftigue characteristics. There have been a lot of discrepanin the estimation of the fatigue life when displacementcontrolled as opposed to that when force is controlled. Owould expect that the estimation of a parameter of a matesuch as asphalt concrete should not depend on testing aratus provided the material is not being tested in disparegimes of its response. However, this is not the case, atwithin the protocol of testing methods employed to descrthe behavior of asphalt concrete. Two types of testing meods are used, one in which the displacement is maintaconstant and the force required to maintain this displacemis measured, and the other in which the magnitude offorce is kept constant and the displacement is measureboth the cases, the failure is deemed to have occurred wthe ‘‘stiffness’’ is reduced to half the ‘‘initial stiffness.’There are many reasons proposed in the literature fordifferences in the fatigue life estimation when these two dferent types of testing methods are used. For instance,numbers of cycles to failure when the displacement is ctrolled is supposed to include the number of cycles to crpropagation~Pell @498#! whereas it is not the case in thforce controlled tests. Practitioners in the field use controforce testing for layers of thicknesses exceeding 100 mmcontrolled displacement testing for layer thicknessesthan 100 mm~Bazin and Saunier@499#, Epps and Monismith@495#, Pell and Cooper@500#!. While it may be reasonable texpect minor differences in the life expectancy of asphconcrete in experiments in which the force is controlled athat in which displacement is controlled when an approprforce history is used to engender a displacement andcorresponding displacement history used to engenderforce, radical departures should not be observed. Inevent, there is a need to recognize the nonlinear respcharacteristics of asphalt, its tendency to age resultingchange in its microstructure, and the recognition that it imulti-constituent mixture, and factor these ideas into devoping an appropriate constitutive framework within whiits fatigue characteristics can be studied. This is easierthan done as fatigue is ill understood even for much simpmaterials.

To realistically simulate the field traffic conditions in thlaboratory, fatigue testing with rest periods between loadcycles has been carried out. The rest period may be betweach loading cycle or after some cycles over a certain peof time. As expected, the rest periods do have beneficiafect on asphalt concrete as it allows the specimen to relaxaccumulated stresses and heal. Different ratios are avairelating the number of cycles to failure with rest periodthat without rest periods and these ranges from 5 to 10emphasized earlier, the use of a parameter like ‘‘stiffnesscharacterize asphalt concrete and thence its failure sevehandicaps the understanding of the complex structural re

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There has been a lot of work related to the shift factand how they depend on various parameters related tolife asphalt concrete pavements such as loading, envimental, material conditions, etc. For details relating touse of different shift factors for parameters such as tempture @501#, rest periods@502–507#, healing@337,508–510#,aging@511#, differences in axle loads@512#, etc, the reader isreferred to the relevant literature cited here and also toreview articles on asphalt fatigue@513–515#.

The analysis of the laboratory fatigue test data haslowed three different approaches: 1! a phenomenological approach in which some form of empirical relationship is asumed between initial strain or stress, mix stiffness, binproperties, air voids, voids in mineral aggregate, etc, andfatigue life ~eg, Monismith and Epps@516#, Pell @498#, Pelland Cooper@500#, Finn et al @517#, Shell @337#, Bonnaureet al @518#, Asphalt Institute@338#, Crauset al @512#, Mo-nismith et al @454#, Baladi @497#, De La Roche and Riviere@519#, SUPERPAVE @441#, De La Rocheet al @520#, DiBenedetto and De la Roche@521#, Harvey and Tsai@511#!; 2!a fracture mechanics approach in which the fatigue damaccumulation is related to the propagation of crack usconcepts borrowed from the fracture of a linearized elabodies @522–525# or using the concepts proposed bSchapery@526–529# for crack growth in viscoelastic medi@530–532#. Once again, even in single constituent materiafracture mechanics is largely more an art than a scienNotions that are applied to conservative systems are appwith impunity to materials undergoing dissipative proces~for example, the results of Noether@533# which apply toconservative systems and thus to elastic bodies are apwithin the context of inelastic materials. Even in the caseconservative elastic systems, the strains at the crack tipsuch that sufficiently near the crack tip the material canbe described as an elastic material and the results of Noe@533# do not apply!. However, in the spirit of engineeringapproximations such studies have value but the sense oapproximation ought not to be forgotten; finally 3! a dissi-pated energy concept in which the fatigue damage is relato the energy dissipated in the specimen during testing~eg,Van Dijk @534#, Van Dijk and Visser@535#, Shell @337#,Pronk @510#!. Such an approach is also not without its limtations as a proper characterization of the dissipation is nessary~see also@536,537#!. It cannot just be based on somad hoc measurements of heat generated, as it is differendifferent specimens subject to the same working.

Before we proceed further, the following remarks relatto the fatigue and fracture characterization of asphalt ccrete are in order. The ability to model the evolution ainteraction of the void structure~air voids as well as micro-cracks! within the asphalt mixture is critical to the ability tpredict fatigue performance as the void structure affectsstress distribution within the mixture and affects the permability of the mixture to both air and water which, in turnaffects the rate and level of oxidative aging and moistdamage. In fact, without an understanding of the evolution

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the microstructure we are extremely limited in our undstanding of the fatigue damage process. Also, the methoogy discussed above related to the dissipated energy conor the studies based on Schapery’s work assumes thastrain energy introduced into the system is balanced byincrease in a system’s surface free energy density due toformation of cracks@526–529#. However, the premises owhich these models have been developed need to befully reevaluated, as there is sufficient evidence to conclthat many of them are untenable. All of these studies onfracturing of viscoelastic solids appeal to a correspondeprinciple between elasticity and linear viscoelasticity undcertain conditions~in fact, such a correspondence canestablished for any linear material! @538#. Firstly, asphaltconcrete is not a linear viscoelastic material; its responshighly nonlinear. Secondly, in the fracture of certain solidthere is no dissipation of energy except in a small plazone adjacent to the crack tip and it is assumed thatcould be neglected in comparison to the energy supplhowever, in viscoelastic materials, there is dissipation inbulk of the material in addition to that near the crack tip athe total dissipation is not necessarily negligible. Thirdly, tcorrespondence principle does not apply to problems sucfracture where dissipation, and thus irreversibility, cleahave a role to play, which entails a fully thermodynamtheory in which the equations are not necessarily similastructure. Thus, with regard to asphalt concrete, the consions drawn concerning fatigue and fracture response netotal reevaluation. However, this is no mean task and mtake a long time to resolve, and until then we might haverest content with certain ad hoc procedures that are in plHowever, it is possible to minimize the ‘‘ad hocness’’ of thprocedure and take some of the relevant issues into accoFor instance, we might start by giving up on the correspdence principle, especially when dissipation occurs inbulk, and study the problem of fracture from a more reasable perspective.

6.2 Thermodynamic frameworkfor modeling asphalt concrete

After having discussed in some detail the classicalproaches to modeling asphalt concrete, we discuss in detthermodynamic framework that has been recently put iplace which can be used for the constitutive descriptionasphalt concrete. This framework has a reasonably genstructure within which a host of dissipative processes candescribed, and we shall begin with a brief description ofgeneral framework and then turn our attention to a specmodel for asphalt concrete that seems to describeuniaxial compressive and tensile creep behavior of aspconcrete as documented in the extensive experimentalby Monismith and Secor@458#. This is not to say that themodel is only able to capture the 1D response—it is the lof experimental data for more general deformations thathibits our determining the efficacy of the model for mogeneral deformations.

Traditional models for asphalt concrete do not considedetail how the changes that take place at the mic

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mechanical level, ie, the structural reorganization of thegregate particle skeleton, the viscous mortar matrix, andair filled voids, affect the stress response of the asphalt ccrete mixture. From a continuum stand point, we haveintroduce field variables that can, in some homogenizsense, describe the evolution of the microstructure and sparameters are usually referred to as ‘‘internal state vables.’’ These ‘‘internal variables’’ should, however, hasome clear physical meaning. The framework we shall adto model asphalt concrete recognizes the change in thecrostructure of the material through the changes in thenatu-ral configurationsof the body. For our purpose here, we cunderstand bynatural configurationa stress free configuration ~see Rajagopal@212,539# for a lengthy exposition on therole of ‘‘natural configurations’’ in mechanics!. Correspond-ing to any current configuration of the body, we can haseveral associated natural configurations, possibly onewhich the body would go to when the tractions are removpreferably a stress free state. The specific natural configtion the body would go to depends on the process. Forstance, it is possible that if the stresses are relieved inadiabatic process it would go to a natural configurationk (B)

ad

while if stresses are relieved in an isothermal process it cogo to a different natural configurationk (B)

is . Perfectly elasticmaterials have a unique stress free configuration, modrigid body motions, and during the deformation the undering microstructural mechanism does not change and the breturns to the same stress free state on the removal oftraction. In a classical viscoelastic material capable of insttaneous elastic response, we could associate the stressstate obtained by an instantaneous elastic unloading respas a natural configuration. This, in effect, is the stress fstate reached in an adiabatic process, as there is not suffitime for heat to be transferred during an instantaneouscess. On the other hand, we could associate the fully strrelaxed state in an isothermal process as the natural conration. Here, we shall consider the one reached throughinstantaneous elastic response as the relevant naturalfiguration. In bodies which suffer fundamental micrstructural changes as they deform, as do many metallic sothat undergoplastic deformation12 and as do polymeric solids, removal of traction does not necessarily lead to the satraction free configuration as the deformation progressThus, different underlying microstructures are associawith different natural configurations. As the material de-forms and as the microstructure evolves, the natural confiration associated with the current microstructure evolvThe stress response of the body changes as the microsture changes and the stress depends on kinematical meafrom an evolving set ofnatural configurationsin addition toother quantities such as temperature, etc. Thus, in matewith evolving microstructures, in order to characterize tmaterial we have not just a single constitutive responsethe stress based on kinematical measures from a single

12The terminologyplastic deformationleaves much to be desired. However, we shnot get into a discussion concerning the inappropriateness of the terminology heruse it as is customary in the literature related to plasticity.

rft

si

s

s

t

t

ea

r

esendnd

mcanfor-

are

ors

gu-n-ionatu-ra-eess

at ifosero-

even


erence configuration, but a multiparameter family ofsponse functions that depend on kinematical measuresan evolving set of natural configurations. This is indeedsituation in the case of bodies that exhibit a variety of dispative responses such as plasticity due to slip@540,541#,twinning @542,543#, solid to solid phase transition@544#, andinelastic response of multinetwork polymers@545–547#, vis-coelastic response of materials@279,548#, and the responseof anisotropic fluids such as liquid crystals@549#, crystalli-zation of polymers@550–552#, biological materials such aligaments and tendons@553#, and rate-independent inelastbehavior@554#. Recently, theories have been developedgrowth which appeal to the fact that thenatural configura-tions of the body evolve as such processes take place~seeHumphrey and Rajagopal@555#, Rao Humphrey, and Rajagopal @556#, Rajagopal@212#!.

We shall not get into a detailed discussion of the thermdynamic framework here, but refer the reader to the numous papers citied above for the details. Here, we preseredacted discussion of the framework so that the discusthat follows is reasonably self-contained and clear.

The response of many materials during certain proce~such as those described above, ie, twinning, etc! can bemodeled as elastic responses from an evolving set of naconfigurations. These elastic responses can be differentmost importantly, they can have different material symmtries with respect to the appropriate natural configuratioWithin a fully thermodynamic framework we need to assciate a class of stored energy functions, rate of dissipafunctions, rate of entropy production functions, latent healatent energies~latent energy is not to be confused with thconcept of latent heat, it is a measure of the difference ininternal energy as the configurations evolve and playsimportant role in phenomenon such as phase transition!, etc,with the body so that we may characterize its response.now describe the mathematical framework.

Let kR(B) denote a reference configuration of a bodyand let XPkR(B) denote a typical position of a materiaparticle PPB. Let k t(B) denote the configuration of thbody B at time t. By a motion of the body, we meanone-to-one assignment fromkR(B) to k t(B), that is suffi-ciently smooth, for each timet, ie,

x5xkR~X,t !. (63)

We have, for the purpose of ease of notation, chosen toresentkR(B), k t(B), askR andk t , respectively.

Since the motion is invertible, at eacht, we can express

X5xkR

21~x,t !. (64)

Now, a quantityf can be defined over the body as

f5f~X,t !5f~x,t !. (65)

We shall use the convention,

df

dt5

]f

]t, ¹f5

]f

]X, (66)

e-romhesi-

cfor

-

o-er-

nt aion

ses

uraland,e-ns.o-ionts,ethean

We

Bl

ep-

]f

]t5

]f

]t, gradf5

]f

]x. (67)

The derivatives~66! are referred to as Lagrangian derivativand those defined by~67! as Eulerian derivatives, though thLagrangian derivative was introduced earlier by Euler athe Eulerian derivative was used earlier by D’Alembert aD Bernoulli.

We shall keep our kinematical definitions to the minimupossible so that we do not lose clarity and coherence. Wedefine a measure of the deformation of the body, the demation gradientFkR

, through

FkRª

]xkR

]X, (68)

and we shall assume that

detFkR.0. (69)

Next, the left and right Cauchy-Green stretch tensorsintroduced through

BkR5FkR

FkR

T , (70)

and

CkR5FkR

T FkR. (71)

The Green-St Venant and the Almansi-Hamel strain tensEkR

andekRare defined respectively through

EkR5 1

2 @CkR21#, (72)

and

ekR5 1

2 @12BkR

21#. (73)

The velocityv is defined through

v5]xkR

~X,t !

]t, (74)

and the velocity gradientL (x,t) is defined through

L5]v

]x. (75)

The symmetric part of the velocity gradient is denoted byD,ie,

D5 12 @L1LT#. (76)

We can define such kinematical measures from any confiration k. Now, corresponding to the current deformed cofigurationk t , we associate a stress free natural configuratkp(t) . As mentioned earlier, there can be more than one nral configuration that corresponds to the current configution k t , which are not related by rigid body motion. Thnatural configuration of relevance depends on the procclass under consideration. However, it does not mean thwe consider isothermal processes, we only need to chothe natural configuration achieved through an isothermal pcess. For materials with instantaneous elastic response,

c

ms

,t

a

e

t

g

altus

s

s ine

pro-

s,ithun-that

sralnta-nse,t ofof a

re-d isor-sso-

m-in-


in isothermal processes, it is best to choose the naturalfiguration reached through an adiabatic process. The chof the natural configuration is based on what renderscalculations the simplest. Here, we shall consider isotherprocesses and use the instantaneously unloaded stresconfiguration as the natural configuration. For the classmaterials that we are studying, namely asphalt concreteshall suppose that the material has instantaneous elasand thus we suppose thatk t can be instantaneously unloadeto a natural configurationkp(t) . In general, if the body un-dergoes an inhomogeneous deformation, we cannot unloaa geometrically compatible~ie, a body without holes, a contiguous body! stress free state by the removal of surface trtion in Euclidean space. However, it is possible to placebody in a stress free geometrically compatible configuratin a non-Euclidean space~see Eckart@557#, and Rajagopaland Srinivasa@540#!. If the deformation is homogeneous, wcan reach a natural configurationkp(t) that is geometricallycompatible, and we can define a deformation gradient

Fkp~ t !5

]x

]Xkp~ t !

(77)

For general deformations, we can define a mapping fromtangent space atXkp(t)

to the tangent space atx ~see Rajago-pal and Srinivasa@540# for a detailed discussion for thsame!. As asphalt concrete can be thought of as a mixturethree constituents, an aggregate matrix that is a solid, acous mortar that is a fluid, and air voids, at the very leasis necessary to associate natural configurations with thegregate matrix and the asphalt mortar matrix. The aggre

s ae-the

Ra-

rethe

rel-

, wefromthef

e

e-r

on-oiceour

alfreeofweicityd

d to-c-

theion

e

the

ofvis-, itag-ate

matrix has a very small relaxation time while the asphmortar matrix has a relatively large relaxation time. We thassociate with the current configurationk t two natural con-figurationskp1(t) and kp2(t) ~see Fig. 6!. We can define themappingsFkp1(t)

andFkp2(t)which, in general, are mapping

from the tangent space of the appropriate material pointthe configurationskp1(t) andkp2(t) to the tangent space of thcorresponding material point atk t . In the case of homog-enous deformations, these mappings are gradients of appriate motions.

We feel that the choice of the two natural configurationand modeling asphalt concrete as a mixture of materials wthese two natural configurations has very sound physicalderpinnings because experiments unmistakably indicateasphalt concrete exhibits instantaneous elastic response~re-bound! as well as exhibiting flow like that of a fluid that haa long relaxation time. Thus, we can think of the two natuconfigurations as those which correspond to the instaneous elastic response and a very slow long-term resporespectively. In the asphalt concrete literature, this aspecmaterial response had been modeled using the notionmodulus of elasticityand mixture viscosityto describe itsfluid-like response~Wood and Goetz@479#!. There are othertypes of material response where there are typically twolaxation times—an immediate example that comes to minthe problem of crystallization in polymers where the amphous and crystalline phases have distinct time scales aciated with them~see Rao and Rajagopal@550–552#!.

We shall model asphalt concrete as a mixture of two coponents which have their own natural configurations andternal energies, entropies, etc. We shall model them aco-existent mixture and not allow for a relative motion btween the constituents and thus it is not truly a mixture intrue sense of mixture theory~see Rajagopal and Tao@558#!but a constrained mixture in the sense of the studies byjagopal and Srinivasa@542#. Let the mappingsGi , i 51, 2 bedefined through~see Fig. 6!

Gi5Fkpi~ t !

21 FkR, i 51,2, (78)

and let

Lkpi~ t !5GiG

21, (79)

Dkpi~ t !5 1

2 ~Lkpi~ t !1Lkpi~ t !

T !. (80)

In general,Gi will not be the gradient of a mapping, they aappropriate mappings from the tangent space at a point inreference configuration to the corresponding point in theevant natural configuration.

In order to describe the response of asphalt concretehave to characterize its instantaneous elastic responsean evolving set of natural configurations and we describeevolution of the natural configurations implicitly in terms ohow the stretchBkpi(t)

varies with time whereBkp(t)

5Fkp(t)Fkp(t)

T . We shall discuss this issue later within th

context of the modeling. Here, we establish a purely kinete
Fig. 6 Natural configuration associated with asphalt conc
mixture

o

o

,

etd

tv

,

.

m

o-isallary

ingbi-ne

toificb-

theics.

ar-lawonec-

ntwtedoaved a

ig-i-

halteci-

for

theand

uc-en-ofm.n

in


matical relationship that will prove useful later. It is straighforward to show~see Rajagopal and Srinivasa@279#! that

B¹

kpi~ t !5Bkpi~ t !

2LBkpi~ t !2Bkpi~ t !

LT,

522Fkpi~ t !Dkpi~ t !

Fkpi~ t !

T , i 51,2. (81)

where the triangle denotes the upper convected Oldrderivative.

Since we shall be modeling asphalt concrete as an incpressible material, we shall require that

tr D50, trDkpi~ t !50. (82)

Because asphalt concrete consists of three constituentsaggregate matrix, asphalt mortar matrix, and air voidsmight seem inappropriate in view of the presence of air voto model asphalt concrete as an incompressible mateHowever, as the void percentage is quite small, such ansumption may be a reasonable first approximation. Whilis necessary to study the problem within the context ofbody being compressible, given the complexity of the mothat is being proposed, we shall dispense with the samenow.

It is necessary to know how these two natural configutions evolve as the body deforms, and this is determinedthermodynamic grounds. However, before getting into a dcussion of these issues, we record the basic balance lawthe sake of completeness and clarity. We state the balalaws for a single continuum. The balance of mass inLagrangian and Eulerian form are expressed respectithrough

rkR5rk t

~detFkR!, (83)

where the symbol det denotes the determinant, and

]rk t

]t1div~rk t

v!50, (84)

where]/]t denotes the usual Eulerian partial time derivativand div denotes the Eulerian divergence.

As we shall be interested in an incompressible materwe shall require that the following constraint be satisfied

div v50. (85)

Next, the balance of linear momentum in the Eulerian fois

div T1rb5rdv

dt, (86)

whereT is the Cauchy stress andb is the specific body forceSince the body is incompressible, the Cauchy stressT isdetermined only to within a spherical part, ie,

T52p11Te, (87)

whereTe is the constitutively determined extra stress. In tabsence of body couples, the balance of angular momenrequires that Cauchy stress be symmetric.

The local form for the balance of energy takes the for

t-

yd

m-

, theit

idsrial.as-it

heelfor

ra-onis-s fornceheely

e,

ial,

rm

hetum

r«1div q5T"L1rr , (88)

where« denotes the specific internal energy,q denotes theheat flux vector, andr is the radiant energy.

Finally, we need to document the second law of thermdynamics. One commonly used form for the second lawthe Clausius-Duhem inequality that is supposed to hold inprocesses that the body is subject to. In fact, it is customto obtain restrictions on constitutive equations by requirthat the body meet the Clausius-Duhem inequality in artrary motions. A difficulty with such an approach is that ocannot expect a particular form of a constitutive equationhold in arbitrary motions. In fact, we expect that a specform of the constitutive equation will hold in a certain suclass of processes.

There is a more serious shortcoming concerning howradiant heating is treated in continuum thermodynamUsually, an expression for the radiationr is substituted interms of the internal energy« and the heat flux vectorq intothe second law, thereby eliminating completely the appeance of the radiation in the second law. Thus, the seconddoes not, in this way of thinking, place any restrictionsthe radiation. It is then assumed to take whatever form nessary to render the first law~the balance of energy! mean-ingful. Unfortunately, radiation cannot be whatever we wait to be. In fact, the key to our very life on this planet is hothe electromagnetic radiation that we receive is converinto energy in various forms, including its thermal form. Twdifferent bodies exposed to the same amount of microwradiation have distinct temperature fields. Thus, we neeconstitutive specification for radiation~see Rajagopal andTao @559# for a detailed discussion of this issue!. Here, un-fortunately, we shall have to rest content by completelynoring radiation. However, in light of the recent use of mcrowave heating in performing accelerated aging of aspin the laboratory, the issue related to the constitutive spfication for radiation needs to be considered~see@560,561#for literature related to the use of microwave heatingcharacterizing asphalt!.

We thus prefer to adopt a different approach towardsuse of the second law that has been used by GreenNaghdi @562# and Rajagopal and Srinivasa@540,541#. Wechoose a specific constitutive form for the entropy prodtion that automatically meets the requirements that thetropy production be nonnegative. The choice of the formentropy production is dictated by the physics of the probleLet z be the entropy, andu the absolute temperature. Thewe define

rz1divS q

u D2rr

u5rf, (89)

wheref is the rate of entropy production. We shall choosef,ie, a constitutive choice is made so that

f>0. (90)

Details of the modeling of asphalt concrete can be foundMurali Krishnan and Rajagopal@548#. Here, we provide anabbreviated version of the same.

o

nh

ht

dt

t

e

et

all

asticpo-

ll

thema-

ve

inl

xtra


We shall suppose that the internal energy and entropypend on the temperatureu andFkpi (t)

, ie,

« i5«kpi~u,Fkpi ~ t !

!, i 51,2, (91)

z i5zkpi~u,Fkpi ~ t !

!, i 51,2, (92)

However, it is important to recognize that the form of theresponse functions can change as the preferred naturalfiguration changes.

Then, the Helmholtz potentialc i will also depend onuandFkpi (t)

, ie,

c i5c i~u,Fkpi ~ t !!, i 51,2, (93)

wherec is defined throughc i5« i2uz i . Here, for the sakeof simplicity, we shall denote the suffixkpi

as justi.In making the above choices, we have essentially

sumed that both the aggregate matrix and the asphalt mmatrix have a common temperatureu and we have also assumed that each component’s Helmholtz’s potential depeonly on the deformation of that constituent. In general, itpossible that there are effects due to interactions betweenconstituents during the deformation, ie,

c i5c i~u,Fkp1~ t !,Fkp2~ t !

!. (94)

For the sake of simplicity, we shall ignore such interactioWe shall assume, as a first approximation, that asp

concrete is isotropic. Then it follows from the requirementframe indifference and isotropy that

c i5c i~u,I i ,II i !, i 51,2, (95)

whereI i5tr(Bkpi (t)), II i5tr(Bkpi (t)

2 ), i 51,2.

Finally, we introduce the constitutive assumption for trate of entropy production. Here, we shall just considerentropy production due to the conversion of the working inenergy in the thermal form, ie, what is usually referred toheat. Since the experimental data that is only availableisothermal problems, the theory that is being developwould then be tested against these experiments. Thisnot mean that we are ignoring the variation of the properof asphalt concrete with temperature. To the contrary,take such variations into account. It is only that we are inested in isothermal processes and can consider suchcesses at different temperatures.

Following the study of Rajagopal and Srinivasa@279# ofrate type differential models for viscoelastic fluids, we shsuppose that the rate of dissipationj i has the form

j i5j i~u,Bkpi ~ t !,Dkpi ~ t !

!, i 51,2. (96)

We shall require that

j i~•,•,0!50, i 51,2, (97)

that is, there is no dissipation ifDkpi (t)is zero, the respons

being purely elastic from an appropriate natural configution. It is possible that there is dissipation during a proceven while the natural configurations do not change, but

de-

secon-

as-rtar

-ndsisthe

s.alt

of

ehetoasforedoesiesweer-pro-

all

ra-sshis

concerns a totally different class of materials and we shnot consider such a possibility here~see Rajagopal@212#!.

We shall make special choices for« i andz i :

« i5Cu1Ai , i 51,2, (98)

z i5C ln u1Bi2mi(Ii23), i 51,2, whereA, B, C, andm i areconstants. The models essentially presume that the elresponses from the natural configuration for the two comnents are that of a neo-Hookean solid. Also,C represents theconstant specific heat andm i is the shear modulus. We shaassume that

j i5h i~u,Bkpi ~ t !!Dkpi ~ t !

•Bkpi ~ t !Dkpi ~ t !

, (99)

h i being the viscosities of the components. Notice thatviscosity depends on both the temperature and the defortion of the body. We shall further choose the specific form

h15h1@N~ trBkp1~ t !23!m11#, (100)

and

h25h2, (101)

whereh1 andh2 are material constants andm is a constant.Requiring that the rate of dissipation be maximized~see

Ziegler @563,564#, Ziegler and Wehrli@565#, Rajagopal andSrinivasa@540,541#!, subject to~82! and ~97! as constraintsleads to~see Murali Krishnan and Rajagopal@548#! the fol-lowing constitutive model for asphalt concrete:

T52p11m1Bkp1~ t !1m2Bkp2~ t !

, (102)

with

1

2B¹

kpi ~ t !5

m i

h i F 3

tr~Bkpi ~ t !

21 !12Bkpi ~ t !G , i 51,2. (103)

Rajagopal and Srinivasa@540,541# provide a rationale formaximizing the rate of dissipation to make constitutichoices and we will not repeat it here.

The above constitutive representation can be rewrittenthe following manner~see Murali Krishnan and Rajagopa@548#!:

T52p11S, (104)

Si1h i

2m iS¹

5l i1 i 51,2, (105)

where

S5S11S2, (106)

Si5m i~Bkpi ~ t !!, i 51,2, (107)

l i5S 3

tr~Bkpi ~ t !

21 !D , i 51,2. (108)

We can express the above system in terms of the total estressS as:

ener

ti

o

th

F

ns,

t alas-at

oadistan-ion

sl,

nsi-

ela-pletion.

o-rs,

crib-eri-than

con-tlyin

uresuid-likebe

s ofd a


S1aS¹

1b S¹¹

5l1¹

1lb 1¹¹

, (109)

where

a5h1

2m11

h2

2m2, (110)

b5h1h2

4m1m2. (111)

The model~109!–~111! is a fully 3D model that satisfies thinvariance of frame indifference. It is a nonlinear model aunlike the popular Maxwell model and the Oldroyd-B modetc, it involves the stress being differentiated twice withspect to the Oldroyd upper convected derivative.

We can linearize the above model by requiring thatelastic responses be sufficiently small from the approprnatural configuration, namely that

iBkpi ~ t !21i5O~d!,d!1. (112)

Linearizing the model and restricting it to one dimensileads to

s1p1s1p2s5q1«1q2«, (113)

wheres is the stress,« is the strain,p1 , p2 , q1 , andq2 arematerial constants, and the model corresponds to the mproposed by Burger in 1935~Burger @258#!. In deriving

~113!, we have also used the fact 1¹

522D.We are now in a position to compare the predictions of

model~102!–~103! with the experimental data of Monismitand Secor@458# for uniaxial creep problems. The appropriakinematics is given by a motion of the form

r 51

AL~ t !R,u5

1

AL~ t !Q,z5L~ t !Z, (114)

where (R,Q,Z) and (r ,u,z) denote a typical material particle in a cylindrical polar coordinate system in the referenconfiguration and the current configuration, respectively.such a deformation,

FkR5diagS 1

AL,

1

AL,L D , (115)

where diag(a,b,c) denotes a diagonal matrix witha, b, andcbeing its entries. The velocity gradient for this motion takthe form

L5diagS 2L

2AL,

2L

2AL,L

L D . (116)

The velocity gradientL is diagonal and hence the symmetrpart of the velocity gradientD is the same asL . Also, allother kinematical tensors, namelyBkpi (t)

, i 51,2, are diago-

nal tensors. The constitutive equation can be now written

Trr 52p11m1B1rr1m2B2rr

, (117)

Tuu52p11m1B1uu1m2B2uu

, (118)

d,l,e-

heate

n

odel

he

te

-ceor

es

ic

as

and

Tzz52p11m1B1zz1m2B2zz

. (119)

For a uniaxial loading, the stresses in the lateral directioTrr andTuu , are zero and hence

Tzz5m1~B1zz2B1rr

!1m2~B2zz2B2rr

!, (120)

with the initial conditions given by

B1zz5B2zz

5L~0!2 (121)

B1rr5B2rr

5B1uu5B2uu

51

L~0!. (122)

Here, we obtain the initial conditions by recognizing thasudden application of the force elicits an instantaneous etic response from the material with the value of stretchtime t50 beingL~0!.

Also, the same conditions are reapplied at the end of lremoval to monitor the relaxation of the mixture. Thisnecessary as a sudden removal of the force elicits an instaneous elastic rebound of the material. Now, the evolutequations forBkpi (t)

are given by Eq.~103!.

This can be written in the following form along thezdirection as

]Bi zz

]t1n i zz

]Bi zz

]z22Li zz

Bi zz

52m i

h iF2Bi zz

~Bi rr2Bi zz

!

2Bi zz1Bi rr

G , i 51,2. (123)

Ignoring the inertial terms, Eq.~120! in conjunction with theEq. ~123! are solved numerically with the initial condition~Eqs. ~121! and ~122!! and the conditions at load reversausing Gear’s method for stiff ordinary differential equatio~Gear@566#!. The model predictions along with the expermental data are shown in Figs. 7–9.

While the model~106!–~111! qualitatively captures theessential features, it does not provide quantitative corrtion, but this is not surprising as we have made rather simassumptions for the stored energy and the rate of dissipaIn fact, the rationale Murali Krishnan and Rajagopal@548#had for developing the model~106!–~111! was primarily toobtain a generalization of Burger’s model within the thermdynamic framework developed by Rajagopal and coworkeas Burger’s model has been reasonably successful in desing the behavior of asphalt concrete. To better fit the expmental data, it might be necessary to use a model otherBurger’s model.

The model considered thus far presumes that asphaltcrete is fluid-like; however, if the temperature is sufficienlow, the material behaves in a solid-like manner. Thus,order to model asphalt concrete over a range of temperatit might be necessary to develop a model that reflects a fllike behavior above a certain temperature and a solid-response below a certain temperature. In fact, it mightnecessary to allow asphalt concrete to exhibit three typeresponse, a fluid-like response, a solid-like response, an

typewesticrechand

l-


transitional behavior in which the material behaves as thoit is a mixture of a solid and a fluid. Once again the theorymultiple natural configurations provides an ideal vehicledescribing such behavior. Before we get into a discussiothe mathematical description of the model, it is first necsary to decide on what types of responses accurately desthat of asphalt concrete. We shall suppose that at sufficie

ughof

forn ofes-cribently

high temperature, asphalt concrete behaves like a rateviscoelastic liquid described earlier. In its solid-like state,would assume that it is either an elastic solid or a viscoelasolid, while in the transitional regime it behaves as a mixtuof a viscoelastic fluid and a solid. In regard to modeling sua response, we can use the methodology adopted by RaoRajagopal@550–552# for describing the behavior of crysta

Fig. 7 Unconfined compression creep test: Comparison of experimental values of Monismith and Secor@458# with the model predictions

Fig. 8 Simple tension test creep test: Comparison of experimental values of Monismith and Secor@458# with the model predictions

l

i

h

i

e

olidtial

e

n-n thebe-con-onn-lt

mete

kesthe

d


lizing of polymers. Since we can assume that asphalt ccrete in its solid-like state is isotropic, we could possibadopt the framework presented in Kannanet al @567# tostudy the solidification of polymer melts to amorphous pomers. This framework, however, might yet be inadequateKannanet al @567# assume that the material transitions froa viscoelastic liquid to a mixture of an elastic solid andviscoelastic liquid to finally an isotropic~rubber like! solid,as the temperature decreases. However, asphalt conmight behave as a viscoelastic solid rather than an elasolid in an appropriate range of temperature, finatransforming into a brittle elastic solid below a certatemperature.

We shall describe a model for asphalt based on a tration from a viscoelastic fluid to an isotropic elastic solid va mixture~see Kannanet al @567# for details!. The method-ology is essentially the same for the case of when the mtransforms into a viscoelastic solid, the only difference bethe structure of the final model.

The model for the liquid-like asphalt concrete is identicto that outlined earlier. Now, as the temperature is lowerat a critical temperature, the response of the fluid-like aspconcrete changes to that of an elastic solid-like responsefor a certain range of temperatures we have a mixture oviscoelastic fluid and an elastic solid.

We now discuss the model for the mixture of the vcoelastic fluid and the elastic solid, in the transitional regimAs the temperature reduces, at certain critical ranges fortemperature, the response of asphalt concrete is assum

on-ly

y-as

ma

cretesticllyin

nsi-ia

eltng

aled,altandf a

s-e.thed to

be an isotropic homogenous elastic material. For the sthat is forming, we shall assume that its Helmholtz potenis given through

cS5CSFu2uS2u lnu

uSG1

mS

2r~ I kr

23!, (124)

wherems is assumed to be a constant,Cs is the specific heatthat is assumed to be constant, andI kr

is the trace of the

Cauchy-Green stretchBkrwhere k r denotes the referenc

configuration at which solidification starts in the asphalt cocrete. We are assuming that as the temperature reduces ifield and as asphalt concrete starts showing a solid-likehavior, the response is that measured from a stress freefiguration. It is imperative to recognize that the configuratikR , the reference configuration for the fluid-like asphalt cocrete, is different fromk r , the configuration at which asphaconcrete starts showing elastic solid-like response~see Fig.10!!.

We are now going to assume that in a transitional regiwe have a mixture of the original fluid-like asphalt concreand the solid-like asphalt concrete. As the transition taplace over a range of temperatures, we shall supposeHelmholtz potential for the mixture of the fluid-like ansolid-like asphalt concrete is

nd Secor
Fig. 9 Triaxial creep test at temperature 140°F and zero lateral pressure: Comparison of experimental values of Monismith a@458# with the model predictions

s.ys-f it-

rehalten-olidthatis a

e

ltoolidav-

taof akehat


c5H~u2~ug2d!!

@11exp~2s~u2ug!!#c f

11

@11exp~2s~u2ug!!#cS . (125)

Here,ug denotes the transition temperature which is takenbe the mean value of the temperature at which the solid-behavior of asphalt concrete starts and that at which it es is a constant~which is the larger in value the shorter thinterval for transition!, d is a constant that is sufficientlsmall ~that depends on the transition interval!, andH denotesthe usual Heaviside function. We note that the coefficiemultiplying c f and cs , namely 1/(11exp(2s(u2ug))) and1/(11exp(s(u2ug))) add up to unity and they can be intepreted as the mass fraction of the asphalt concrete havifluid-like behavior and solid-like behavior, respectively.

The rate of dissipationj for the mixture of asphalt concrete having solid-like behavior and fluid-like behaviorassumed to be

j5j f5(i 51

2

H~u2~ug2d!/11exp~2s~u2ug!

3@2v if~u,Bkpi ~ t !

!Dkpi ~ t !•Bkpi ~ t !

Dkpi ~ t !#,

where the material moduliv if , i 51, 2 are such thatv i

f→v i

as one approaches the initiation point andv im→vs( i 51,2)

when one approaches the pure elastic regime.It is possible to think of a more general form for th

entropy production that includes entropy production duedissipation as well as that due to conduction and phase tsition. The maximization of the entropy production subjectthe appropriate constraints leads not only to the constiturelations for the stress, but it also provides the kineticssolidification for the change of the mass fraction of the flu

he

tolikends,e

y

nts

r-ng a

-is

eto

ran-to

tiveof

id-like material to a solid-like material. Such an idea has been

gainfully exploited to study crystallization of polymer meltEquations that govern the kinetics of the growth of the crtals such as the Avrami equation and generalization ofollow logically from such a procedure~see Rao and Rajagopal@550–552# and Kannanet al @567#!. A procedure simi-lar to that used by Rao and Rajagopal@552# leads to thestress

T52p11H~u2~ug2d!!

11exp~2s~u2ug!!T f

11

11exp~s~u2ug!!TS, (126)

whereT f5( i 512 @m f i

(u f)u/u f i#Bkpi (t)

andTs5msBkr, T f be-

ing the stress in the fluid component of the mixture andTs

that in the solid component.The modeling is complete when the model for the pu

solid-like behavior is specified. We shall assume that aspconcrete in its solid state is described by a Helmholtz pottial cs and there is no dissipation as we assume that the sis elastic. In the above formulation, we have assumedthere is no entropy production due to phase transition. Itsimple matter to include this~see Rao and Rajagopal@552#,Kannanet al @567#!. It might be reasonable to ignore thentropy production due to conduction~as we have done inthe above derivation!, though incorporating it into the modeprovides no difficulty. It is a reasonably simple mattergeneralize the above model to lead to a viscoelastic smodel below the temperature at which the solid-like behior starts~see Kannanet al @567#, Murali Krishnan and Ra-jagopal@548#!. Unfortunately, there is no experimental dafor the stress response in place wherein the transitionparticular sample of asphalt takes place from its liquid-listate all the way through its solid-like state. We hope tsuch experiments will be carried out in the future to put t

Fig. 10 Natural configurations associated with the mixture of solid-like asphalt concrete and fluid-like asphalt concrete

n

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ieg

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tv

s

d

u

re

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ntopelop

ul-ntribectri-is

pedabledis-bitu-nity,eningeralmap-

e inralmo--y-

ini-

d ton is

w-mi-theds toersand

na-ioningheingthech-reeenentw

or


above model to test. We are, however, confident thatgeneral framework should be capable of yielding the reqsite model based on our past experience for a varietyproblems such as film blowing and fiber spinning involvipolymer crystallization.

The practical implication of the above discussion wregard to modeling asphalt concrete as exhibiting differkinds of material behavior has wider ramifications. Thisespecially so when it comes to modeling the compactprocess involved in the laying of asphalt concrete pavemeAs rightly pointed by Geller, ‘‘Quantifying the behavior oan asphalt mix during the compaction process has desolution because there are so many complexities and vables involved’’@568#. And this is one area in which verlittle modeling or quantitative results are available. Mostthe mix design methods used nowadays are semiempiand none of these methods indicate the level of resistaoffered by the mix to the compaction process@280, 568#.

The hot mix asphalt concrete mix when initially brougfrom the mixing plant has a temperature of 275–325°F aafter the final rolling is completed, the temperature is arou100–150°F. During this temperature drop, the characterisof the mix also change from a viscoelastic fluid to a vcoelastic solid. This transition can be ascribed mostly duthe increase in the viscosity of the asphalt binding the aggate particles due to the temperature decrease. There isanother frequent transition of this solid-like behaviorfluid-like behavior and vice versa, and this occurs almevery day during the time when the asphalt concrete pament is in service. While the former transition occurs durithe compaction process within one to two hours of time,latter is a process happening for the full lifetime of the pament ~see McLeod@569#, Finn and Epps@570#, and otherreferences listed therein!.

The literature on the influence of compaction of asphconcrete on its physical properties are enormous, howethey are mostly related to the experimental and field obvations of the entire process. Due to the number of variabinvolved during this process such as the type of asphalt, tof mix, the type of compaction equipment, the field contions, etc, even reasonable laboratory observationsfraught with inaccuracies. Even more compelling is the nefor a systematic study since most of the compacting eqment and the field procedures developed and used nowtain mostly to compacting granular layers, a mixturegranular materials, water, and air voids, as compared tophalt concrete, a mixture of aggregate particles, asphalt,air voids. Clearly, the role of asphalt in the compaction pcess of asphalt concrete is not equivalent to that of the rolwater in the compaction process of granular layers. Recstudies on the development of new compacting equipmattest to this fact~see Paganiet al @571#, Rickards et al@572,573#, and Halimet al @574#!.

Thus, developing a thermodynamic framework for ttransition of asphalt concrete from the state in which it is lthrough its working life is no mean task to achieve in viewthe numerous factors that play a role in the process. Hever, it should be possible to put a reasonable theory w

theui-of

g

thntis

ionnts.ffiedari-

oficalnce

tndndticss-to

re-also

tostve-nghee-

altverer-les

ypei-areedip-per-ofas-ando-of

entent

eid

ofw-ith

some predictive capability in place. This is the importachallenge facing the research worker in asphalt and we hthe reader will rise to the challenge and attempt to devesuch a model.

7 EPILOGUE

Though bitumen has been used in a variety of forms in mtifarious applications from times immemorial to the preseday, there is no theory in place that can accurately descits response when subjected to mechanical, thermal, elecal, and other external stimuli. A large part of the problemdue to the fact that large classes of materials are lumtogether as bitumen, the members of the class being capof diverse response characteristics. Thus, modeling theparate response of the various materials that are calledmen presents a great challenge and a wonderful opportufor the development of novel models, to the theorists. Givsuch diversity of response, a theory capable of describsuch materials needs, by necessity, to have a fairly genframework. We would also like the theory to have a firthermodynamic basis. One possible framework that can cture the distinct and different responses that are possiblbitumen is the theory for materials that have multiple natuconfigurations. The framework has been able to accomdate within its fold viscoelasticity, traditional plasticity, twinning, solid-to-solid phase transition, crystallization of polmer melts, anisotropic liquids, and growth and adaptationbiological materials. Given its ability to describe such dverse response of materials, we felt that it could be usemodel asphalt concrete and it seems that our assumptiopartly justified by the theory presented in this review. Hoever, much remains to be done. A model that takes thecrostructural details, the change in the air voids, aging,size, shape, and distributions of the aggregates, etc, neebe put into place. We hope that this review would spur othto get interested in modeling the response of asphaltasphalt concrete.

NOTE ADDED IN PROOF

The more we delve into the early investigations into theture of bitumen, the more we are drawn to the conclusthat, that which we think was the earliest study concernany aspect of the behavior of bitumen is definitely not tearliest investigation into this matter, such a study havbeen carried out earlier. We were totally unaware ofground breaking and profound early investigations of Hatett @575,576#. Equally embarrassing is our becoming awaof unexpected, but important uses to which bitumen has band is being put to. An example in point is the developmof bituminization processes for conditioning and storing loand medium level radioactive wastes@577–580#.

NOMENCLATURE

h - Viscosity ~Eq. ~1!!P - Penetration by a 100 grams load applied f

5 seconds ~Eq. ~1!!r - Density ~Eq. ~2!!

i

a

a

nen

s-

H,e-

n,

-

,

ew

u

r

-

d

to

t-der


A, r, s - Constants ~Eq. ~2!!u - Temperature ~Eq. ~2!!p - Pressure ~Eq. ~2!!F - Shear stress ~Eq. ~3!!R - Rate of shear ~Eq. ~3!!n - Degree of plasticity ~Eq. ~3!!S - Constitutively determined extra stress

~Eq. ~4!!D - Symmetric part of the velocity gradient

~Eq. ~4!!v - Velocity ~Eq. ~5!!a - Acceleration ~Eq. ~5!!T - Cauchy Stress ~Eq. ~9!!« - Linearized strain ~Eq. ~10!!G - Relaxation function ~Eq. ~10!!J - Creep compliance ~Eq. ~11!!

* - Stieltjes Convolution ~Eq. ~17!!H(t) - Heaviside step function ~Eq. ~22!!v(t) - Poisson’s ratio within the context of linea

vis coelasticity ~Eq. ~23!!G(x) - Gamma function ~Eq. ~28!!E - Young’s modulus ~Eq. ~29!!D - Differential operator ~Eq. ~33!!S - Stiffness modulus ~Eq. ~37!!V - Vector space ~Eq. ~39!!R, Q, Z - Referential coordinates in a cylindrical coo

dinate system ~Eq. ~49!!r, u, z - Current coordinates in a cylindrical coord

nate system ~Eq. ~49!!c(t) - Angle of twist per unit length of a cylin-

der ~Eq. ~50!!M (t) - Applied moment ~Eq. ~53!!J - Polar moment of inertia ~Eq. ~53!!kR(B) - Reference configuration of a body Bk t(B) - Configuration of the body B at timetFkR - Deformation gradient ~Eq. ~68!!BkR - Left Cauchy-Green stretch tensor~Eq.

~70!!CkR - Right Cauchy-Green stretch tensor~Eq.

~71!!EkR - Green-St Venant strain tensor~Eq. ~72!!ekR - Almansi-Hamel strain tensors~Eq. ~72!!L (x,t) - Velocity gradient ~Eq. ~75!!kp(t) - Stress-free natural configurationkp1(t) - Stress-free natural configuration of aggreg

matrixkp2(t) - Stress-free natural configuration of asph

mortar matrixGi - Mappings from the tangent space at a poi

in the reference configuration to the corrsponding point in the relevant natural cofiguration ~Eq. ~78!!

b - Body force ~Eq. ~86!!Te - Constitutively determined extra stress~Eq.

~87!!« - Specific internal energy ~Eq. ~88!!q - Heat flux vector ~Eq. ~88!!

r

r-

-

te

lt

t--

r - Radiant energy ~Eq. ~88!!z - Entropy ~Eq. ~89!!u - Absolute temperature ~Eq. ~89!!f - Rate of entropy production ~Eq. ~89!!c - Helmholtz potential ~Eq. ~93!!j - Rate of dissipation ~Eq. ~96!!diag~a,b,c! - Diagonal matrix with a, b, and c being its

entries¹ - Upper convected Oldroyd derivative~Eq.

~81!!tr - The trace operatordet - DeterminantLynSym - Symmetric linear transformation]/]t - Eulerian partial time derivativediv - Eulerian divergence

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J Murali Krishnan is an Engineering Research Associate in the Department of Mechanical Eneering, Texas A&M University. He earned his Master of Engineering Degree in TransportEngineering from Regional Engineering College, Tiruchirappalli, India in 1993, and a PhD dein Civil Engineering from the Indian Institute of Technology, Madras, India in 1999. His minterest in research includes asphalt and asphalt concrete. He has been an Associate MembeInstitution of Engineers (India) since 1990 and an Associate Member of the American SocCivil Engineers since 2002.

KR Rajagopalholds the Forsyth Chair in Engineering and is a professor of mathematics andmedical engineering at Texas A&M University. He is a Fellow of ASME and a past PresidentSociety for Natural Philosophy. He has authored or coauthored over 200 archival papersvariety of subfields in continuum mechanics that includes among them non-Newtonian fluid mics, finite elasticity, viscoelasticity, turbulence, mixture theory, mechanics of granular material,trorheology, and continuum thermodynamics. He has coauthored three books and coeditedothers in the field of mechanics. He currently serves on the editorial advisory boards of 30 arjournals and book series.

Documents

Krishnan Rajagopal 2003