Nonlinear thermal expansion and contraction ofasphalt concrete

Menglan Zeng and Donald H. Shields

Abstract: Asphalt concrete for paving roads is a viscoelastic material. In the prediction of thermal stress in asphaltpavements, the thermal expansion–contraction property of the material is required. In current practice, thermalexpansion–contraction is assumed to be a linear function of temperature, and a constant thermal coefficient is used.The fact that a viscoelastic material may have a glass transition temperature and the thermal property may have adiscontinuity at the glass transition temperature has not been considered. This study investigates the thermalnonlinearity of asphalt concrete. In this research, the thermal expansion–contraction was continuously measured on asingle type of asphalt concrete in the temperature range from +40°C to –40°C. It was found that the thermalexpansion–contraction was a continuous nonlinear function of temperature, resulting in a variable thermal coefficient.Evaluations of the effect of the nonlinearity indicated that the assumption of thermal linearity can result in moderateerrors in stress prediction in asphalt pavements.

Key words: asphalt concrete, thermal expansion–contraction, thermal coefficient, nonlinear thermal behaviour, asphaltpavement, thermal stress.

Résumé: Le béton bitumineux pour le pavage de routes est un matériau visco-élastique. Dans la prédiction de lacontrainte thermique dans les chaussées bitumineuses, la propriété de l’expansion–contraction thermique du matériauest nécessaire. Selon la pratique actuelle, l’expansion–contraction thermique est supposée une fonction linéaire de latempérature, et un coefficient thermique constant est utilisé. Le fait qu’un matériau visco-élastique puisse avoir unetempérature de transition vitreuse et que la propriété thermique puisse avoir une discontinuité à la température detransition vitreuse, n’a pas été considéré. Cette étude examine la non-linéarité thermique du béton bitumineux. Danscette recherche, l’expansion–contraction thermique a continuellement été mesurée sur un seul type de béton bitumineuxpour une variation de température de +40°C à –40°C. Il a été trouvé que l’expansion–contraction thermique était unefonction non-linéaire continue de la température, aboutissant à un coefficient thermique variable. Des évaluations del’effet de la non-linéarité ont indiqué que la supposition de linéarité thermique peut mener à des erreurs modérées dansla prédiction de la contrainte dans les chaussées bitumineuses.

Mots clés: béton bitumineux, expansion–contraction thermique, coefficient thermique, comportement thermiquenon-linéaire, chaussée bitumineuse, contrainte thermique.

[Traduit par la Rédaction] Zeng and Shields 34

Asphalt concrete is widely used as a paving material. It isaccepted that asphalt concrete is a viscoelastic materialwith complex material properties. In the prediction of ther-mally induced stress in asphalt pavements, the propertyof thermally induced expansion–contraction of the materialis required. In the development of SUPERPAVE, the state-of-the-art specifications and procedures for the design ofasphalt pavements (Lytton et al. 1993), the frequency of

thermal cracking wascalculated on the basis of predicted ther-mal stresses forknown temperature histories. When predict-ing the thermal stress, a constant thermal coefficient wasused, assuming that the thermal expansion–contraction ofthe asphalt concrete in a pavement is a linear function oftemperature (Lytton et al. 1993).

Engineering materials expand when they are subjectedto an increase in temperature, and they contract when sub-jected to a decrease in temperature, under restriction-freeconditions. Traditionally, the amount of thermal deforma-tion (expansion–contraction) of a material is calculated usinga thermal coefficient, denoted byα. In general mechanics,αis termed the coefficient of thermal expansion. In pavementproblems, since the concern is the stress built up at low tem-peratures,α is often called the coefficient of thermal con-traction. The thermal coefficient describes the relationshipbetween the change in temperature and the change in ther-mally induced strain. A general definition is given by

[1] α ε= dd

T( )TT

Can. J. Civ. Eng.26: 26–34 (1999) © 1999 NRC Canada

26

Received October 28, 1997.Revised manuscript accepted July 21, 1998.

M. Zeng. Department of Civil and Geological Engineering,The University of Manitoba, Winnipeg, MB R3T 5V6,Canada.D.H. Shields.Faculty of Engineering, The University ofManitoba, Winnipeg, MB R3T 5V6, Canada.

Written discussion of this article is welcomed and will bereceived by the Editor until July 31, 1999 (address insidefront cover).

whereεT is thermal strain, andT is temperature (Collieu andPowney 1973, p. 38). When the relationship betweenT andεT is linear, α becomes a constant. In reverse, the generalequation for the calculation of thermal strain is given by

[2] ε α αT

i

dd

dd= ′ ′ = ′ ′

′′∫ ∫T

tT T T t

T tt

tT

( ) [ ( )]( )

0

whereTi is the initial temperature at zero time,T′ is the tem-perature integral variable, andt′ is the time integral variable.If α is a constant, the equation reduces to the conventionalform of εT = α(T – Ti).

The thermal coefficient for a particular material is usuallytemperature dependent over a wide range of temperature. Ingeneral, the thermal coefficient decreases with temperature(Collieu and Powney 1973). For most materials, represent-ing the thermal coefficient by a constant is reasonable withinthe temperature range of interest. In some circumstances,making the thermal coefficient a function of temperatureshould be considered. This is particularly true for visco-elastic materials if the glass transition temperature lieswithin the temperature range of interest (Christensen 1982).In viscoelastic theory, the thermal deformation property mayhave a discontinuity at the transition temperature.

Many researchers have studied the thermal expansion–contraction of asphalt concrete, for example, Domaschuk etal. (1964), Monismith et al. (1965), Littlefield (1967), Joneset al. (1968), Haas and Phang (1988), Seddik and Haas(1995), and Stoffels and Kwanda (1996). In some of thesestudies, the thermal deformation of asphalt concrete wasfound, or was implied to be, a nonlinear function of temper-ature. In the discussions following the work by Stoffels andKwanda (1996), Anderson described the phenomenon ashaving curvilinear effects. However, the thermal coefficientwas treated, nevertheless, as a constant by averaging theslope of measured thermal strain versus temperature. The

study by Bahia and Anderson (1993) provides importantknowledge about the determination of the glass transitiontemperature and its variation and significance for asphalt ce-ment. For asphalt concrete, such information has not beenfound. This study attempts to deal with the nonlinearity ofthermal expansion–contraction, or the variability1 of thethermal coefficient, of asphalt concrete, and its effects onstress prediction, under uniaxial-stress conditions.

A single type of hot-mixed, dense-graded asphalt con-crete, as a representative of the class of material, was usedin the laboratory testing. The cement was from ELF,Donges, France. The aggregate was a crushed igneous rockcomprising basalt, diorite, and granite. Results of a sieveanalysis of the aggregate are presented in Fig. 1 and Table 1,and the physical characteristics of the asphalt concrete aresummarized in Table 2. The specimens were prepared at theLaboratoire Central des Ponts et Chaussées (LCPC), Centrede Nantes, France. Slabs of the asphalt concrete were madeusing a pneumatic-tired rolling-wheel compactor, fromwhich the specimens were cut to a standard size with a dia-mond blade. The standard specimens had a dimension of50 mm square in cross section and 170 mm in length. Theywere shipped to the University of Manitoba for testing boththe mechanical and thermal properties of the material. Pre-cise details concerning the preparation of the specimens canbe found in Zeng (1997).

The setup of the test for determining thermal expansion–contraction is illustrated in Fig. 2. An environmental cham-ber was fabricated for the test; the chamber was made ofplywood with hard insulation lining the inside. In the cham-ber, an electric heater, a solenoid valve, and a fan were pro-vided. Heating was achieved by means of the electric heater,and cooling by injection of liquid carbon dioxide under pres-sure through the solenoid valve. The fan circulated the in-side air to maintain the air temperature in the chamberuniform. A thermocouple was placed in the chamber as atemperature sensor. An IBM-compatible computer with a

© 1999 NRC Canada

Zeng and Shields 27

Sieve size(mm)

Percentretained

Percentpassing

(0) 7.5 00.08 7.5 7.50.315 10 151 8 252 12 334 11 456.3 18 56

10 22 7414 4 9620 0 100

Table 1. Results of a sieve analysis of theaggregate in the asphalt concrete under study.

Fig. 1. Results of a sieve analysis of the aggregate in the asphaltconcrete under study.

1A temperature-dependent thermal coefficient is sometimes called a nonlinear thermal coefficient (Seddik and Haas 1995). In this study, theterm “variable thermal coefficient” is used. This is because the term “linear” as applied to the thermal coefficient usually refers to a one-dimensional property, constant or variable, to distinguish it from areal and cubic properties. The use of nonlinear thermal coefficient mightcause confusion if one has to consider multiple-dimensional properties.

data acquisition and control card was used for both data ac-quisition and control purposes. A long test specimen was ob-tained by gluing two standard specimens together withepoxy (LePage 502). This specimen, measuring 340 mm inlength and 50 mm square in cross section, was used in thetest to improve the accuracy of measurement. The specimenwas placed on a steel support in the chamber. Interjected be-tween the support and the specimen was a sheet of teflon tominimize friction. Two invar rods were glued to the ends ofthe specimen as LVDT targets. Two LVDTs were used tomeasure thermal deformation; they were fixed to a steelframe outside the chamber. In order to facilitate temperaturemeasurements without risking damage to the test specimenand alteration to the material, a dummy specimen, measur-ing 50 mm square in cross section and 85 mm in length, wasplaced in the chamber to replicate the temperature of the testspecimen. Two thermocouples were glued to the dummyspecimen, one on the surface and the other in the centre. The

average of the readings from the two thermocouples wastaken as the specimen temperature.

In the test, the thermal deformation was continuouslymeasured in a temperature range from +40°C to –40°C for acycle of cooling and heating at an ambient rate of 10°C perhour. This temperature rate was slow enough to ensure uni-form rates of changes of temperature throughout the speci-men while at the same time fast enough to make the durationof the test convenient. A rate of 10°C per hour is also speci-fied in the Thermal Stress Restrained Specimen Test, the so-called TSRST (Jung and Vinson 1993). The specimen waskept for 1 h at thestarting temperature (40°C), 1 h at thelowest temperature (–40°C), and 1 h at theterminating tem-perature (40°C). The temperature history was preset in thesoftware. A margin of ±0.5°C was used. When the ambienttemperature in the environmental chamber was lower thanthe desired value by 0.5°C, the heater was activated to bringup the temperature; when the temperature was higher than

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28 Can. J. Civ. Eng. Vol. 26, 1999

Characteristic Test method

Original asphalt cementPenetration (25°C, 100 g, 5 s), 0.1 mm 61 ASTM D5Softening point (ring and ball) (°C) 50.8 ASTM D36Penetration index –0.52 Calculated

Recovered asphalt cementPenetration (25°C, 100 g, 5 s), 0.1 mm 32 ASTM D5Softening point (ring and ball) (°C) 58.0 ASTM D36Penetration index –0.38 Calculated

Asphalt concreteRelative density of asphalt cement 1.030 ASTM D70Relative density of aggregate 2.84 ASTM C127/C128Relative density of concrete (bulk) 2.480 ASTM D2726Content of asphalt by total mass (%) 5.1 CalculatedAir voids by total volume (%) 4.81 Calculated

Table 2. Characteristics of the asphalt concrete under study.

Fig. 2. Illustration of setup for the thermal expansion–contraction test.

the desired value by 0.5°C, the solenoid valve was opened toallow the injection of the liquid carbon dioxide into thechamber to bring the temperature down. Temperature andthermal-deformation data were logged every one-half hour.Figure 3 shows the temperatures measured on the surfaceand in the centre of the (dummy) specimen, the averagespecimen temperature, and the difference in temperature be-tween the surface and the centre of the specimen.

Figure 4 shows the results of the thermal expansion–con-traction test in terms of thermal strain versus specimen tem-perature measured in both cooling and heating. The thermalstrain is arbitrarily defined as zero at the beginning of thetest at 40°C. The relationships between temperature andthermal strain for cooling and heating are nearly coincident

for the single cycle of temperature change. To be specific,the following considers the data from the cooling process.

Measured data for cooling are shown in Fig. 5. The rela-tionship between thermal strain and temperature is repre-sented by a smooth curve. The slopes are steeper at highertemperatures and more horizontal at lower temperatures. Nodistinct discontinuity can be observed. In order to pheno-menologically describe the variation of the thermal strainwith temperature, the data underwent a regression analysisassuming various functions. The best fit with the leastsquares criterion is found to be a nonlinear, sigmoidal func-tion of the form

[3] εT = ++ − −

aaT a a

01

2 31 exp[ ( ) )]y

where the thermal strainεT is given in percent; andai are thecoefficients for the regression analysis, witha0 = –0.194,a1= 0.239,a2 = 9.77°C, anda3 = 20.6°C. The fitted curve isplotted in Fig. 5.

In order to better understand the nature of theεT versusTcurve, the curvature,k = (εT ′′) /[1 + (εT)′2]3/2, of the nonlinearfunction is derived as shown in Fig. 6. In Fig. 6 there aretwo (absolute) maximum values of curvature, one atT =37°C and the other atT = –17°C. The significance of themaximum curvature at 37°C is not apparent and is not of in-terest in this discussion. The 37°C maximum curvature oc-curs at a temperature that is too high for most thermal stressproblems, and in addition, at this high temperature, the as-phalt concrete is soft and the measurements are prone to er-ror. The fact that theεT versus T curve has a markedcurvature atT = –17°C is of greater engineering interest. Inviscoelastic theory, the temperature at which the relationshipof thermal deformation versus temperature exhibits a discon-tinuity is termed the glass transition temperature, denoted byTg. Although there is not a distinct discontinuity, it is proba-ble thatTg = –17°C for the asphalt concrete under study.

In principle, the curve ofεT versusT with a glass transi-tion temperature can be idealized by two straight lines.

© 1999 NRC Canada

Zeng and Shields 29

Fig. 3. Measured temperatures on the surface and in the centreof the specimen, the averaged specimen temperature, and thetemperature difference between the surface (Tsur) and the centre(Tcen) in the thermal expansion–contraction test.

Fig. 4. Results of the thermal expansion–contraction test for acycle of cooling and heating.

Fig. 5. Nonlinear (sigmoidal), bilinear, and linear approximationsof thermal strain versus temperature.Tg, glass transitiontemperature.

Bilinear regression analyses are performed on the test data.In the analyses, the data are divided into two groups, oneabove and one below some arbitrary temperature. Eachgroup undergoes a regression analysis. The error sums ofsquares (SSE) from the regressions at the chosen tempera-ture are summed as a global value. Figure 6 shows the SSEvalues based on SSE =Σi

n=1(Yi – $Yi)

2, whereYi is the ob-served value and$Yi is the fitted value. SSE reaches a mini-mum at about –17°C or a little lower. This agrees with theglass transition temperature determined using the maximumcurvature method. The best fit, bilinear equation is given by

[4] εT C)

C)=

++

> − °< − °

a a T

a a T

T

T0 1

3 4

17

17

(

(

with a0 = –0.101,a1 = 0.00262/°C,a2 = –0.122, anda3 =0.00135/°C. The fitted curve is also plotted in Fig. 5.

In accordance with tradition, the test data further undergoa straight-line regression. The best-fit equation is

[5] εT C)= + + > > − °a a T T0 1 40 40(

with a0 = –0.0954 anda1 = 0.00233/°C, which is plotted inFig. 5.

Equations [3], [4], and [5] are different ways to approxi-mate the test data. Figure 7 shows the absolute errors bythese approximations in fitted value minus observed valueversus temperature. Also indicated in Fig. 7 are the coeffi-cients of determination (r2) from the regression analyses.The nonlinear approximation, eq. [3], is very close to thetest data, and can be considered exact for engineering pur-poses.

For eqs. [3], [4], and [5], the first derivatives ofεT withrespect toT give, respectively, the continuously variable ther-mal coefficient as

[6] α = − −+ − −

× °−aa

T a aT a a

1

3

2 3

2 32

2

110

exp[ ( ) ]{ exp[ ( ) )]}

yy

y C

(+ > > − °40 40T C)

in which a0 = –0.194,a1 = 0.239, a2 = 9.77°C, anda3 =20.6°C; the bi-constant thermal coefficient as

[7] α =× °

× °

> − °< − °

−

−

2 62 10

1 35 10

17

17

5

5

.

.

(

(

y

y

C

C

C)

C)

T

T

and the single-constant thermal coefficient as

[8] α = × ° + > > − °−2 33 10 40 405. (y C C)T

In eq. [7] the thermal coefficient at high temperatures is al-most twice the value at low temperatures.

Equations [6], [7], and [8] are three ways to represent thethermal coefficient. Each of the equations exhibits a reduceddegree of accuracy in comparison with the preceding equa-tion. The continuously variable coefficient in eq. [6] is con-sidered to be the most accurate, the bi-constant coefficient ineq. [7] is of intermediate accuracy, and the single-constant

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30 Can. J. Civ. Eng. Vol. 26, 1999

Fig. 6. Curvature of the nonlinear approximation, and error sumof squares (SSE) of bilinear regression, of thermal strain versustemperature.

Fig. 7. Absolute errors of various approximations of theobserved value of thermal strain (εfit

T – εobsT ) versus temperature,

and coefficients of determination from the regression analyses.

Fig. 8. Three ways to represent thermal coefficient resulted fromvarious approximations of thermal strain versus temperature.

coefficient in eq. [8] is the most approximate. Figure 8graphically illustrates these representations.

In the evaluation of the effect of thermal nonlinearity onstress prediction, a continuous function of thermal strain ver-sus temperature, in place of the discrete test data, is requiredin the stress computation. In view of the precision of thenonlinear approximation of thermal strain versus tempera-ture (Fig. 7) and the necessity of such a function, the evalua-tion will be based on the variable thermal coefficient, whichis derived from the nonlinear approximation, in the follow-ing section.

In the prediction of thermal stress in asphalt pavements, astress–strain constitutive relationship for the asphalt concreteis required, in addition to the knowledge of thermal expan-sion–contraction. The stress–strain relationship of the as-phalt concrete under study has been investigated by Zeng(1997). A series of direct tensile relaxation tests was run onthe standard size (50 mm × 50 mm × 170 mm) specimens.The tests were conducted for temperatures from +40°C to–40°C, at strains up to 0.8%. Results of the tests showed thatthe material was strongly nonlinear in mechanical properties.Schapery’s (1969) theory for nonlinear viscoelastic materialswas able to characterize the mechanical nonlinearity:

[9] σ ε ρ ρτ

ε τ τ( ) ( ) ( ) [ ( )]t h E t h E ht

= + − ′∫e edd

d1 0 2∆

whereρ is the reduced time defined as

[10] ρεε

( )[ ( )] [ ( )]

tt

a T t a t

t= ′

′ ′∫ d0

In eq. [9] and the associated eq. [10],t is present time;τ ispast time, a time integral variable (0≤ ≤τ t); ρ ρ τ′ = ( ) is areduced time variable;E(t) is the relaxation modulus;Ee =E(∞) is the (equilibrium) long-term value of the modulus;∆E(t) = E(t) – Ee is the transient component of the modulus;he, h1, h2, and aε are nonlinear parameters, functions ofstrain as well as temperature; andaT is the temperature shiftfactor. Linear viscoelastic theory is recovered whenhe = h1= h2 = aε = 1, a condition when strains tend to zero. Ineq. [9], ε is referred as to the (stress-associated) mechanicalstrain. When transient temperature conditions are involved,ε ε ε= −t T, whereεt is the (observed) total strain andεT isthe (stress-free) thermal strain.

The effect of the nonlinearity of thermal expansion–con-traction, or the variability of thermal coefficient, on stressprediction is evaluated through two specific temperature his-tories of interest to engineers, a TSRST case and a sinusoi-dal temperature variation case. The first case requireslaboratory data. The second case is an idealized temperaturehistory. Schapery’s (1969) theory for nonlinear viscoelasticmaterials is used to relate strain to stress. Due to the unavail-ability of analytical solutions, numerical procedures weredeveloped and used to solve eq. [9] in conjunction witheq. [10] (Zeng 1997, Appendix 3). In the numerical method,the real time was divided into small increments and the inte-gral was approximated with a summation.

In the field, the total strain in intact pavement remainszero with temperature changes. The TSRST simulates thisimportant zero total strain condition. In a TSRST, the (ambi-ent) temperature drops at a rate of 10°C per hour. During thecourse of temperature change, the length of the specimen iskept essentially constant, in other words, the total strainεt iszero over time. Therefore, the mechanical strain is equal tothe thermal strain in magnitude with a reverse sign,ε ε= − T.Some researchers have used the TSRST to evaluate the re-sistance of asphalt concrete to thermal cracking (Jung andVinson 1993; Partl et al. 1995; Ponniah and Hesp 1996). Theconcern in this study is the stress developed as the tempera-ture drops.

In addition to the thermal expansion–contraction test, aTSRST was performed in this study for evaluation purposes.In the test, the (ambient) temperature dropped at a rate of10°C per hour. The procedures of the test abided byAASHTO Designation TP10-93 (First Edition). The startingtemperature in the TSRST was chosen to be 40°C instead ofthe usual 0°C. Although little stress might develop when thetemperature dropped from +40 to 0°C, a certain amount ofstrain accumulated. This strain, under the condition of mate-rial nonlinearity, would affect the magnitude of stress whenthe temperature was lowered farther. Figure 9 shows the re-sults of the TSRST. It is interesting to note that there is littleto no measurable buildup of stress until the temperature hasbeen lowered to about +10°C. At high temperatures, the ma-terial is soft and prone to viscous flow.

In order to evaluate the thermal nonlinearity, the stressesin the TSRST case are predicted using the three differentways to represent the thermal coefficient, and these predic-tions are compared with the test data, also shown in Fig. 9.When either the continuously variable representation(eq. [6]) or the bi-constant representation (eq. [7]) of thethermal coefficient is used, the predicted stress is, in general,in good agreement with the test data. On the other hand, thesingle-constant representation (eq. [8]) gives a stress predic-tion which is always higher than the test data. Figure 10shows the relative errors in stress resulting from the bi-constant and single-constant representations on the basis ofthe continuously variable thermal coefficient. Data at tem-peratures higher than 0°C are omitted, as at these tempera-tures little stress is developed. The error from bi-constantrepresentation is usually within ± 10%. Using the single-constant representation, the error is higher. The lower thetemperature, the greater is the error. At failure of the speci-men (–26°C), the single-constant error is about 30%. If thespecimen failed at a lower temperature for some reason, forexample if a lower temperature reduction rate were used, theoverestimate in stress would be even more significant. At–40°C, the error is expected to exceed 60%.

The sinusoidal temperature variation originally proposedby Monismith et al. (1965) was considered to be representa-tive (at the pavement surface) of a severe condition thatmight occur in north-central United States and south-centralCanada during the winter months. In the variation, the tem-perature drops from an initial value of –17.8°C to –40°C andreturns to –17.8°C over a 24 h period as shown in Fig. 11. Ina real situation, a diurnal temperature variation such as thiswould be superimposed on a longer wavelength, seasonaltemperature variation. There would exist some amount of

© 1999 NRC Canada

Zeng and Shields 31

stress in a pavement at the beginning of the temperaturevariation. The predicted stress for the Monismith et al. tem-perature history should be understood as an increment whichis superimposed onto the existing (unknown) stress at zerotime. (Strictly speaking, the simple superposition principle isnot applicable when material nonlinearity is involved.)

Figure 11 shows the stresses predicted using the variable,bi-constant, and single-constant representations of the ther-mal coefficient for the sinusoidal temperature variation case.Since mechanical nonlinearity is involved, the state of strainof the material should be identified. It is assumed that theasphalt concrete is in a stress–strain-free state on completionof construction of the pavement at 120°C. For the variableand bi-constant representations, the thermal coefficient is as-sumed to be a constant value ofα = 2.62 × 10–5/°C for T >Tg = –17°C. Figure 12 presents the relative errors in stressresulting from using the bi-constant and single-constant rep-

resentations instead of the continuously variable thermal co-efficient. Data before the 4th hour and after the 16th hourare omitted, as at these times little stress is developed. Theerror from bi-constant representation is within ±10% at mosttimes, as was found in the TSRST case. Using the single-constant representation, the error is much higher. At thepeak stress, the error is close to 90%.

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32 Can. J. Civ. Eng. Vol. 26, 1999

Fig. 10. Relative errors in stress resulting from bi-constant andsingle-constant representations on the basis of the variablethermal coefficient in the TSRST case.

Fig. 9. Observed stress in the TSRST and the predicted stresses using various representations of the thermal coefficient.

Fig. 11. Temperature versus time and predicted stresses usingvariable, bi-constant, and constant representations of the thermalcoefficient in the sinusoidal temperature variation case.

Since thermal expansion–contraction is a continuousnonlinear function of temperature, the thermal property forasphalt concrete should be measured over the range of tem-perature which a pavement may experience in the field. Al-though a clearly defined glass transition temperature was notfound, the value ofTg can be determined for a particular ma-terial by either a maximum curvature method or a bilinearregression method. WhenTg is within the temperature rangeof engineering interest, it is satisfactory to represent thethermal coefficient by two constants, one for above and onefor below Tg. The bi-constant thermal coefficient has the ad-vantage of simplicity of application, as compared to the con-tinuously variable representation. It is noted that, althoughthe sigmoidal function fits the test data best and facilitatesthe determination of the glass transition temperature, the re-sulting thermal coefficient in eq. [6] approaches zero whenTtends to ±∞, which is not logical.

The foregoing attempt to treat thermal nonlinearity isbased on a simplified approach. Thermally induced deforma-tion of viscoelastic materials is complex. For some materi-als, thermal expansion–contraction may exhibit instability:after a sudden temperature change, a viscoelastic materialmay continue to expand or contract in apparent contradictionto the normal response of an elastic material to a tempera-ture change (Findley et al. 1976). For asphalt concrete, therecan be an effect of densification: after a number of cycles ofcooling and heating, the material tends to reduce in volume(Littlefield 1967). Generally, the thermal deformation of aviscoelastic material depends not only upon the current valueof temperature, but also on the entire temperature history.Additionally, the thermal deformation can be dependent onthe inherited level of strain or stress (Schapery 1969; Findleyet al. 1976). These effects are not considered in the study.Only the nonlinearity of thermal expansion–contraction, or

the variability of the thermal coefficient, under clearly pre-scribed conditions is considered here.

At low temperatures, the development of thermal strain inasphalt concrete is slowed considerably with temperaturechanges. Ignoring this fact may result in moderate errors instress prediction of pavements. For a single-constant thermalcoefficient, the stress predicted using linear viscoelastic the-ory is directly proportional to the value of the coefficient.When either thermal or mechanical nonlinearity is involved,the simple proportional relationship no longer exits. The im-portance of the thermal nonlinearity should be examinedcase by case in conjunction with the consideration of me-chanical nonlinearity. The sinusoidal temperature variationexamined earlier in this study is an idealized temperaturehistory. Measured air and pavement temperature data areavailable (Young et al. 1969; Deme and Young 1987). Forthese data, Christison et al. (1972) made predictions of ther-mal stress, assuming the material to be linear, both thermallyand mechanically. The authors further related the results tocracking performance of pavements. It would be of interestto further examine whether there would be improvement instress prediction and in crack propagation prediction if boththermal and mechanical nonlinearities were considered.

The thermal expansion–contraction of a single type of as-phalt concrete was continuously measured over a tempera-ture range from +40 to –40°C. A comprehensive analysis ofthe test data reveals the following:

(1) The thermal expansion–contraction of asphalt concreteis a continuous nonlinear function of temperature. Such afunction results in a variable thermal coefficient. Ignoringthe continuous variability of the thermal coefficient may re-sult in moderate errors in stress predictions of pavements.

(2) There may exist a glass transition temperature for as-phalt concrete within the temperature range of engineeringinterest. The coefficient of thermal expansion–contractionfor asphalt concrete is markedly different below and abovethe transition temperature.

(3) A bi-constant thermal coefficient is a satisfactory rep-resentation of the continuously variable thermal coefficient.

(4) Pavement engineers are urged to test for and collectthermal expansion–contraction data for asphalt concrete. Mod-ern computers make it possible to use sophisticated tech-niques of regression analysis to properly characterize thethermal expansion–contraction of this complex material.

The authors would like to acknowledge the financialsupport from the Natural Sciences and Engineering ResearchCouncil of Canada. Thanks are due to all those who contrib-uted to this work, in articular Professor K.O. Anderson ofthe University of Alberta and Mr. R. Kwok of the Univer-sity of Manitoba. A special recognition is addressed toDr. J.-F. Corté and his colleagues at Laboratoire Central desPonts et Chaussées (LCPC) in France for their assistance inpreparing the test specimens.

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Zeng and Shields 33

Fig. 12. Relative errors in stress resulting from bi-constant andsingle-constant representations of the thermal coefficient in thesinusoidal temperature variation case (as compared to a variablerepresentation).

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34 Can. J. Civ. Eng. Vol. 26, 1999

Bahia, H.U., and Anderson, D.A. 1993. Glass transition behaviorand physical hardening of asphalt binders. Asphalt Paving Tech-nology, Journal of the Association of Asphalt Paving Technolo-gists,62: 93–129.

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ai (i = 0, 1, 2, ...): coefficients for regression analysisaT: temperature shift factorE(t): relaxation modulusEe: (equilibrium) long-term value of relaxation modulus∆E(t): transient component of relaxation modulushe, h1, h2, aε: nonlinear parameters in Schapery’s (1969) the-

ory for nonlinear viscoelastic materialsk: curvature ofεT = εT(T)r 2: coefficient of determinationSSE: error sum of squarest: real timet′: time integral variableT: temperatureT′: temperature integral variableTcen: centre temperature of specimenTg: glass transition temperatureTi: initial temperatureTsur: surface temperature of specimenY i: observed value$Yi: fitted valueα: coefficient of thermal expansion–contractionε: (mechanical) strainεt : total strainεT: thermal strainεfit

T : fitted thermal strainεobs

T : observed thermal strainρ: reduced timeρ′: reduced time variableτ: time integral variableσ: stress