60 ■ Transportation Research Record 1767Paper No. 01-3086

Microdamage accumulation due to fatigue loading, healing due to restperiods, and hysteresis heating during fatigue testing of asphalt mixturesusing a dynamic mechanical analyzer were investigated. Specially fabri-cated sand-asphalt specimens were tested under a repetitive, controlled-strain torsional mode at 25°C and 10 Hz. Fatigue performance was evalu-ated using hysteresis loops of stress-pseudostrain and a simple parameter,pseudostiffness, based on the elastic-viscoelastic correspondence prin-ciple. Heat generation and dissipation during fatigue testing were alsoinvestigated by measuring temperature changes of a tiny thermo-couple inside the asphalt mixtures. The results demonstrate that micro-damage healing due to rest periods results in an increase of fatigue life.It was observed that the effects of hysteresis heating on the changes ofstiffness during the fatigue test were not significant. Successful develop-ment of this testing method is suggested as a potential specification-typetest method because of its rapidity, repeatability, and accuracy.

Fatigue cracking in asphalt concrete pavements is considered one offour primary distresses, along with rutting, low-temperature crack-ing, and moisture damage. The fatigue cracks initiate as microcracksand are followed by a crack propagation process including coales-cence of the microcracks. The finding that microcracks that occur inasphalt mixtures heal due to rest periods has been verified in severalstudies (1–4). Quantitative analysis of both damage accumulationand healing in asphalt mixtures can be performed using the extendedelastic-viscoelastic correspondence principle developed by Schapery(5). Using the correspondence principle, mechanical behavior in bothlinear and nonlinear viscoelastic materials can be solved in the ab-sence of time-dependent effects. The elimination of time dependencyreduces the several potential energy dissipation factors to plasticdeformation, heat, and microstructural changes such as fracture andhealing (6). It is, therefore, required to know how mechanical energyapplied to the asphalt mixtures is elastically and viscoelasticallystored, and how it is lost to plastic strain, fracture, molecular re-arrangements, distributed heating, and localized cracktip or surfaceheating.

Since crack initiation and propagation are governed by stress-strain relationships, accurate evaluation of fatigue behavior can beachieved with appropriate constitutive models. Lee and Kim (7, 8)Lee et al. (9) developed a successful constitutive model using con-tinuum damage mechanics. The model accurately predicts mechan-ical behavior of asphalt concrete under uniaxial repeated cyclicloading with and without rest periods. The successful studies of Lee

et al. suggest a relatively easy way to understand the potential of theasphalt mixture to fracture and to heal, on the basis of tests per-formed in a reasonable period of time with repeatable and accurateresults and with acceptable precision.

In this study, a dynamic mechanical analyzer (DMA) was used tocharacterize fatigue damage and healing potential in asphalt mixturesunder controlled-strain, torsional testing. According to a study byReese (10), the torsional loading is a better simulation of damage thanbending loads when considering traffic movements. The impetus ofthe DMA testing was from work performed by Goodrich (11), Smithand Hesp (12), Christensen and Anderson (13), and Shashidharand Romero (14). Specifically, Smith and Hesp demonstrated thatcontrolled-strain testing of mastics in a DMA leads to a controlledrate of microcrack development and growth, whereas controlled-stress testing would lead to rapid and uncontrolled crack growth.Smith and Hesp demonstrated the effect of crack-pinning caused byfiller material and the rate and degree of microcrack growth. Heatgeneration and dissipation during the fatigue testing were measuredusing a tiny thermocouple wire and a digital thermometer. Threemain objectives of this paper are

1. To verify microdamage healing due to rest periods using theDMA on the basis of the elastic-viscoelastic correspondence principle,

2. To develop a simple and accurate method to evaluate fatiguedamage potential and microdamage healing potential of asphaltmixtures, and

3. To evaluate the effects of heat generation and dissipation onthe stiffness of asphalt mixtures.

BACKGROUND

Evaluation of Microdamage Healing Using Correspondence Principle

Viscoelastic materials generally produce hysteresis stress-strainloops. The area within these loops represents dissipated strain energyconsisting of heat dissipation, molecular restructuring, immeasur-able entropy, fracture, healing, and so forth. Therefore, evaluationof fracture and healing in viscoelastic materials is not a simpleprocess. Schapery’s elastic-viscoelastic correspondence principle,which is applicable to both linear and nonlinear viscoelastic ma-terials, allows one to evaluate damage growth and healing in visco-elastic materials (5). Schapery stated that constitutive equations forcertain viscoelastic media are identical to those for the elasticcases, but stresses and strains are not necessarily physical quanti-ties in the viscoelastic body. Instead, they are pseudovariables. In

Evaluation of Microdamage, Healing, andHeat Dissipation of Asphalt Mixtures,Using a Dynamic Mechanical Analyzer

Yong-Rak Kim, Dallas N. Little, and Robert L. Lytton

Y.-R. Kim, 501J CE/TTI Building; D. N. Little, 601 CE/TTI Building; R. L. Lytton,508G CE/TTI Building, Texas Transportation Institute, Texas A&M University,College Station, TX 77843-3135.

the case of a growing traction boundary surface, such as crackgrowth, the viscoelastic problem can be reduced to an elastic caseusing physical stresses and pseudostrains (4). Uniaxial pseudostrainis defined as

where

�R = pseudostrain,ER = reference modulus that is an arbitrary constant,

E(t) = relaxation modulus,� = time-dependent strain, andξ = time variable of integration.

For linear viscoelastic materials, a uniaxial stress-strain relationshipis established by the constitutive equation for linear viscoelasticmaterials as follows:

where σLVE(t) is the calculated linear viscoelastic stress.With the use of the definition of pseudostrain in Equation 1,

Equation 2 can be rewritten as

When the measured stress, σ(t), and the calculated linear visco-elastic stress, σLVE(t), are the same, no damage occurs to the samplebeing tested. It is, therefore, noted that measured stress-pseudostrainbehavior is linear viscoelastic without damage. Specimens which ex-perience significant damage form hysteretic loops when plotted in thestress-pseudostrain domain. Damage accumulation can be demon-strated by observing changes in the loop area and slope from fatiguetesting with any loading condition such as repetitive controlled stressor controlled strain.

In the case of torsional shear loading, the pseudostrain can bedefined in the same way as the uniaxial tensile or compressive loading.

where

γR = pseudostrain in shear mode,GR = reference shear modulus that is an arbitrary constant,

G(t) = shear relaxation modulus, andγ = time-dependent shear strain.

For linear viscoelastic materials, a torsional shear stress–shear strainrelationship can be expressed as

where τLVE(t) is the calculated linear viscoelastic stress.With the use of the definition of pseudostrain in Equation 4,

Equation 5 can be rewritten as follows:

τ γLVE( ) ( )t GRR= 6

τ ξ γξ

ξLVE( ) ( ) ( )t G t dt

= − ∂∂∫0

5

γ ξ γξ

ξR

R

t

GG t d= − ∂

∂∫14

0( ) ( )

σLVE( ) ( )t ERR= � 3

σ ξξ

ξLVE( ) ( ) ( )t E t dt

= − ∂∂∫0

2�

��R

R

t

EE t d= − ∂

∂∫11

0( ) ( )ξ

ξξ

Kim et al. Paper No. 01-3086 61

On the basis of the relationship between the pseudostrain andphysical stress, Kim (1) and Kim et al. (15) introduced a simpleparameter, secant pseudostiffness (SR, simply pseudostiffness), thattogether with the rate of change of pseudostrain energy can measurequantitatively the occurrence of microdamage and healing. Pseu-dostiffness represents the change in the slope of loops betweenphysical stress and pseudostrain. Pseudostiffness is expressed as follows:

where

SR = pseudostiffness,γm

R = peak pseudostrain in each physical stress–pseudostraincycle, and

τm = physical stress corresponding to γmR.

The change in pseudostiffness indicates damage growth fromrepeated cyclic loading. The pseudostiffness decreases with increas-ing number of cyclic loading and is then recovered after the restperiods due to microdamage healing of mixtures. An ideal trend forpseudostiffness versus the number of cycles before and after a restperiod is shown in Figure 1 (15). In Figure 1, the curve OBCD rep-resents the reduction in the pseudostiffness due to damage growthwithout a rest period, and the curve AB′D′ depicts the reduction inthe pseudostiffness due to damage growth after the rest period. Thepseudostiffness increased from Point B to Point A after the restperiod due to the microdamage healing, and it decreased as the load-ing continued after the rest. Therefore, it can be concluded that therest periods and corresponding microdamage healing contributed toan increase in fatigue life by an amount equal to ∆Nf .

Thermo-Viscoelastic Properties

The question must be addressed of how much of this reduction ofpseudostiffness is due to heating within the test sample. The answeris provided by measuring the change of temperature in the test sam-ple and using the principle of time-temperature superposition todetermine the expected amount of change of pseudostiffness withthe measured change of temperature. Because the asphalt concreteis commonly classified as a thermorheologically simple material,

SR m

mR

= τγ

( )7

FIGURE 1 Ideal trend, which is conceptual not real, before andafter rest period (15 ).

the time-temperature superposition concept can be applied success-fully to predict creep and stress relaxation behavior at fast rates ofloading—corresponding to the behavior at low temperatures—andslow rates of loading—corresponding to the behavior at high tem-peratures (16 ). Using individual stress relaxation curves at differ-ent temperatures, a stress relaxation master curve for a designatedtemperature can be constructed by horizontally shifting the stressrelaxation curves at various temperatures.

Several power laws, such as the pure power law, generalizedpower law, and modified power law, can be used to represent ana-lytically the master curve. The pure power law is commonly used inrepresenting time-dependent behavior of asphalt mixtures becauseof its simplicity. A mathematical derivation was performed usingthe pure power law to investigate change of pseudostiffness due toheating in the test sample.

where

SR(ζ) = master pseudostiffness at a certain reference temperature,ζ = reduced time,

S1 = coefficient of ζ−m, andm = exponent for relaxation modulus-time relationship.

Taking derivative of Equation 8 with respect to ζ and dividingboth sides by SR(ζ) yields

The reduced time can be defined by the time-temperature shiftfactor and loading time as follows:

where

t = loading time, which is approximately equal to 0.1/f;aT = time-temperature shift factor; and

f = frequency of loading in a dynamic loading condition.

Various laboratory tests on asphalt mixes have shown that a plot ofln aT versus temperature results in a straight line with a slope, β (17).

where

β = slope of straight line;∆T = T −T0, the change of temperature during cyclic shear testing;

T = temperature of the heated sample; andT0 = original temperature of the sample.

When the change of temperature, ∆T, is positive, Equation 11 isillustrated as follows:

Substituting Equation 12 for Equation 10 and taking the derivativeof ζ with respect to ∆T gives the following equation:

d

d Tt T

ζ β β( )

exp ( ) ( )∆

∆= [ ] 13

ln ( ) ( )a Tt = −β ∆ 12

ln ( ) ( )a Tt = β ∆ 11

ζ = t

aT

( )10

dS

Sm d

R

R

m( )

( )( ) ( )

ζζ

ζ ζ= − − 9

S SR m( ) ( )ζ ζ= −1 8

62 Paper No. 01-3086 Transportation Research Record 1767

Substituting Equation 13 for Equation 9 yields a final relationshiprepresenting the rate of change of pseudostiffness with respect totemperature change as follows:

Based on Equation 14, the rate of change of pseudostiffness dueto the temperature change at any time can be determined by knownvalues of β and m from an experimental test. The results of applyingthis methodology are presented in the section entitled Description ofLaboratory Tests and Results.

DEVELOPMENT OF METHODOLOGY

Fatigue performance was evaluated by applying a constant torsionalstrain to a specimen in a dynamic rheometer. In this study, the DMAwas used to characterize fatigue performance. Test data were collectedby a data acquisition system with a 16-bit multichannel board.

Testing Method and Materials

Fatigue testing was performed in a DMA using a rectangular torsiontest fixture. Constraint (controlled) strain amplitude, repetitive tor-sional oscillations were applied to the specimen. Figure 2 shows therectangular-bar configuration, imposed strain, and stress response.A Strategic Highway Research Program (SHRP)-classified binder,AAD-1, was mixed with Ottawa sand to form a sand-asphalt mix-ture capable of resisting plastic flow at an intermediate temperature(25°C). The Ottawa sand was selected, as it is a uniformly graded,clean aggregate that maximizes homogeneity in the specimen.

After specimen fabrication, each specimen was mounted in therheometer, and the chamber was closed and allowed to equilibrateto the desired testing temperature (25°C). The fatigue test was begunafter a 20-min period of equilibration at the test temperature. Allspecimens were tested at 10 Hz.

Specimen Fabrication

Sand-asphalt samples were prepared by mixing Ottawa sand with8% asphalt binder by weight of dry sand at the mixing temperature.The 8% asphalt content was selected as a reasonable arbitrary value toprovide an average “film thickness” of approximately 10 µm. The term“film thickness” is used with the understanding that it is a controver-sial term. It is used to establish an asphalt-cement distribution rea-sonably simulative of a mixture. Specimen compaction was per-formed at a predetermined compaction temperature. The mixing andcompaction temperatures were determined according to AASHTOT209 and were 164°C and 145°C, respectively. Each specimen wascompacted in a specially fabricated mold, as shown in Figure 3. Theloose sand-asphalt mixture was evenly distributed into the mold andcompacted by applying static pressure. A fine bare wire thermo-couple (0.13-mm diameter) was buried in the sand-asphalt specimenduring compaction in order to measure temperature changes due toheat generation and dissipation. The 8.5 g of loose sand-asphalt wasdetermined by trial and error to provide the mass necessary to pro-duce a specimen geometry of 60 mm long, 12 mm wide, and 6 mmthick. The allowable dimensional tolerance was with ±0.3 mm. Anyspecimen exceeding the geometry tolerance was excluded.

dS

Sm t T m d T

R

R

m( )

( )exp ( )( ) ( ) ( )

ζζ

β β= − −[ ]−1 1 14∆ ∆

DESCRIPTION OF LABORATORY TESTS AND RESULTS

A torsional shear stress relaxation test was performed to determinethe strain level causing nonlinear viscoelasticity and to obtain astress relaxation curve representing time-dependent properties of themixture. Based on results from the stress relaxation test, controlled-shear strain cyclic tests with and without rest periods were per-formed at a frequency of 10 Hz to investigate fatigue performanceand characteristics of microdamage healing due to the applied restperiods. Temperature measurements with the thermocouple inside

Kim et al. Paper No. 01-3086 63

the specimen provided evidence of heat storage and dissipation dur-ing the fatigue testing.

Stress Relaxation Test

Torsional stress relaxation tests were performed to determine thestrain level causing nonlinear viscoelasticity and to acquire a desiredstress relaxation curve to calculate pseudostrain. In linear viscoelas-tic materials, the ratio of stress to any constant strain is independentof the strain magnitude in the stress relaxation test. This property is

FIGURE 2 Rectangular-bar configuration, imposed strain, and stress response.

FIGURE 3 Aluminum mold assembly for compaction of specimen.

called homogeneity. Based on this concept, four different strains(0.10, 0.35, 1.05, and 3.00 percent) were used, with test resultsshown in Figure 4. In Figure 4, stress relaxation behaviors at 0.10and 0.35 percent were similar, while the stress relaxations at 1.05and 3.00 percent shifted downward significantly. It is inferred thatnonlinearity in the asphalt mixture starts at about 0.35 percent strain.

On the basis of the above results, a stress relaxation curve wasobtained at 0.20 percent strain that was considered not to induce anydamage in the asphalt mixture during testing. Four different tem-peratures, −10°C, 5°C, 25°C, and 40°C, were used to obtain a mas-ter stress relaxation curve at 25°C using the time-temperature super-position concept. A minimum of 40 min of recovery time was allowedbetween each test. An additional test at 25°C was performed afterthe series of tests to ensure negligible damage accumulation duringthe entire testing period. As seen in Figure 5, the first relaxationbehavior at 25°C is the same as the second one, indicating that nodamage was accumulated in the specimen during the test sequence.Time-temperature shift factors at different temperatures were ob-tained and their dependency on temperature is illustrated in Figure 6.As expected, the relationship between shift factors and temperaturesis linear on log-linear scales, and the linear regression slope, β, isdetermined as 0.27.

Figure 7 illustrates the master stress relaxation curve obtained byapplying the 0.2 percent strain. The experimental data were fit by a

64 Paper No. 01-3086 Transportation Research Record 1767

Prony series representation to calculate pseudostrain. As can beseen in Figure 7, the Prony series fit the measured data smoothly onlog-log scales.

Controlled Shear Strain Cyclic Test

In an attempt to validate the applicability of the correspondence prin-ciple to cyclic loading conditions, torsional shear cyclic tests withand without rest period were conducted on rectangular-bar speci-mens under a constant strain amplitude of 0.50 percent. Relativelylow strain (0.50 percent) was selected to avoid abrupt macrocrackinitiation due to thermocouple wire inclusion. Four different restperiods (1, 2, 1, and 4 min) were introduced after 600, 6,000, 12,000,and 24,000 cycles, respectively. A total of 30,000 load cycles wereapplied.

The cyclic stress-strain behavior with rest periods is presented inFigure 8. Stress-strain loops shifted downward with loading, whichcan be noted in the 20th and 600th cycle. The area inside the stress-strain loop was determined as dissipated energy. The stress-strainloop of the 601st cycle shifted upward after a rest period due tomicrodamage healing and viscoelastic relaxation.

FIGURE 4 Stress relaxation test results to determine strain levelcausing nonlinear viscoelasticity.

FIGURE 5 Shear stress relaxation curves at various temperatures.

FIGURE 6 Time-temperature shift factors.

FIGURE 7 Master shear stress relaxation (25�C, 0.2 percentstrain) and curve fitting.

The effect of the rest period on a damaged specimen was alsoinvestigated using the pseudostrain, as described in Figure 9. Changesof the pseudostrain-stress hysteresis loop indicate that microdamagehas occurred, as mentioned earlier. The changes in area and slope ofthe hysteresis loop during the rest periods, as observed in Figure 9,reflect the fracture healing of microcracks. The propensity for micro-crack healing in different mixtures can be determined by measuringthe difference in areas under the stress-pseudostrain curves and chordslopes of the stress-pseudostrain plots before and after a rest period.This method has been used successfully in differentiating the healingpotentials of various binders.

To investigate the effects of microdamage healing and the result-ing fatigue life increase due to rest periods, a simple parameter,pseudostiffness, was used to represent the changes in the slope ofhysteresis loops. In an attempt to remove sample-to-sample vari-ability, each pseudostiffness was divided by the initial pseudostiff-ness, I. This approach is demonstrated in several studies by Lee andKim (7, 8). Figure 10 presents the changes of Pseudostiffness I ofeach specimen with and without rest period at selected loadingcycles. It was observed that Pseudostiffness I increased after the restperiod and this resulted in an increase of fatigue life. Because theeffects of the relaxation phenomenon have been removed by the

Kim et al. Paper No. 01-3086 65

use of the pseudostrain concept, any recovered pseudostiffness afterthe rest period can be considered to be due to the microdamage heal-ing. Based on these results, it is verified that fatigue life of asphaltmixtures increases with rest periods. Extensive testing shows that ifrest periods are not continually introduced, the fatigue curve withrest periods will eventually merge the curve without rest periods.

Heat storage and dissipation was recorded using a fine-wire ther-mocouple and a digital thermometer to determine the impact of theeffect of heating and to be able to separate out damage due to crackdevelopment and that due to loss of stiffness from heating. Temper-ature changes during the fatigue testing with rest periods were mea-sured, and the results are plotted in Figure 11. As seen in Figure 11,some amount of heat was stored during the repeated cyclic loadingand dissipated during the rest periods. The trend of heating historieswas the same as the one of pseudostiffness, inferring probable effectsof heating on the stiffness loss and microdamage healing.

The effects of heating on stiffness loss can be investigated quanti-tatively using Equation 14, which was illustrated earlier. Equation 14shows that the rate of change of stiffness due to any change of tem-perature is dependent on an exponent (m) for the pseudostiffness-time relationship and a slope (β) between a time-temperature shift

FIGURE 8 Hysteresis loops from torsional shear cyclic loading.

FIGURE 9 Hysteresis loops of stress versus pseudostrain before and after rest period.

FIGURE 10 Pseudostiffness I versus number of loading cycles with and without rest period.

FIGURE 11 Temperature measurements in fatigue testing withrest periods.

factor and amount of temperature change. Since the maximum dif-ference of temperatures during testing was approximately 0.3°C, thepercentage of pseudostiffness reduction is computed to be 0.62 per-cent based on the values of m(0.56) and β(0.27), determined fromthe stress relaxation test at various temperatures. It may be inferredthat the reduced amount of 0.62 percent in pseudostiffness is negli-gible; therefore, heat generation and dissipation during the fatiguetesting have no significant effect on stiffness changes.

CONCLUDING REMARKS

Microdamage and fracture healing from rest periods were evaluatedsuccessfully using the DMA for one SHRP-classified binder, AAD-1,in this study. Changes in slope and area of stress-strain or stress-pseudostrain loops in asphalt mixtures demonstrated microdamageaccumulation and healing due to rest periods during the torsional load-ing fatigue test. Microdamage healing and the resulting fatigue lifeincrease due to rest periods were investigated using a simple parame-ter, pseudostiffness. It was observed that the pseudostiffness increasedafter the rest period and resulted in increased fatigue life. However,the fatigue tests were limited to 30,000 cycles, which is an acceptablelevel to develop microdamage but not complete fatigue failure of the specimens. It is recommended that future tests be performed tocomplete failure of the specimens and to compare the fatigue liveswithout rest period with the fatigue lives with rest periods.

The shift in fatigue life based on this test can define the healing prop-erties of a binder or a mastic that promote extended fatigue life due tohealing. Therefore, the specification test has the potential not only toestablish resistance to fatigue damage but also to promote healing.Since the two processes work hand-in-hand in the damage process,knowledge of both is critical. Successful development of this protocolcould result in a potential specification-type test method because of itsrapidity, repeatability, and accuracy. This simple technique would beuseful in binder selection and specification based on the observedmechanical behaviors including fatigue resistance and healingpotential. Future research will extend this approach to other SHRP-classified binders, particularly fabricated binders, and modified bindersto investigate their fatigue fracture and healing potential. The effectsof fillers on resistance of microcrack propagation and healing char-acteristics can also be estimated using this simple technique. Sand-asphalt specimens mixed with fillers or asphalt mastics are suggested.

Measurements of heat generation and dissipation inside of asphaltmixtures during the fatigue testing demonstrate that the amount ofheat generation from repetitive loading and dissipation from restperiods was not significant in the change of stiffness. Thermal imag-ing techniques may be used to further investigate temperature effectsand DMA results.

ACKNOWLEDGMENTS

The authors would like to acknowledge the Western Research Insti-tute and the Federal Highway Administration for their financial sup-

66 Paper No. 01-3086 Transportation Research Record 1767

port. Particular thanks are due to Naga Shashidhar of FHWA, whosediscussion of the testing method and data acquisition was invaluable.

REFERENCES

1. Kim, Y. R. Evaluation of Healing and Constitutive Modeling of As-phalt Concrete by Means of the Theory of Nonlinear Viscoelasticityand Damage Mechanics. Ph.D. dissertation. Texas A&M University,College Station, 1988.

2. Kim, Y. R., and D. N. Little. One-Dimensional Constitutive Modeling ofAsphalt Concrete. Journal of Engineering Mechanics, Vol. 116, No. 4,April 1990, pp. 751–772.

3. Kim, Y. R., S. L. Whitmoyer, and D. N. Little. Healing in Asphalt Con-crete Pavements: Is It Real? In Transportation Research Record 1454,TRB, National Research Council, Washington, D.C., 1994, pp. 89–96.

4. Kim, Y. R., Y. C. Lee, and H. J. Lee. Correspondence Principle forCharacterization of Asphalt Concrete. Journal of Materials in CivilEngineering, Vol. 7, No. 1, Feb. 1995, pp. 59–68.

5. Schapery, R. A. Correspondence Principles and a Generalized J-Integralfor Large Deformation and Fracture Analysis of Viscoelastic Media.International Journal of Fracture, Vol. 25, 1984, pp. 195–223.

6. Williams, D. A. Microdamage Healing in Asphalt Concretes: RelatingBinder Composition and Surface Energy to Healing Rate. Ph.D. disser-tation. Texas A&M University, College Station, 1998.

7. Lee, H. J., and Y. R. Kim. Viscoelastic Constitutive Model for AsphaltConcrete Under Cyclic Loading. Journal of Engineering Mechanics,Vol. 124, No. 1, Jan. 1998, pp. 32–40.

8. Lee, H. J., and Y. R. Kim. Viscoelastic Continuum Damage Model ofAsphalt Concrete with Healing. Journal of Engineering Mechanics,Vol. 124, No. 11, Nov. 1998, pp. 1224–1232.

9. Lee, H. J., J. S. Daniel, and Y. R. Kim. Continuum Damage Mechanics-Based Fatigue Model of Asphalt Concrete. Journal of Materials in CivilEngineering, Vol. 12, No. 2, May 2000, pp. 105–112.

10. Reese, R. Properties of Aged Asphalt Binder Related to Asphalt Con-crete Fatigue Life. Journal of the Association of Asphalt Paving Tech-nologists, Vol. 66, 1997, pp. 604–632.

11. Goodrich, J. L. Asphalt and Polymer Modified Asphalt PropertiesRelated to the Performance of Asphalt Concrete Mixes. Asphalt PavingTechnology, Vol. 57, 1988, pp. 116–175.

12. Smith, B. J., and S. A. M. Hesp. Crack Pinning in Asphalt Mastic andConcrete: Regular Fatigue Studies. In Transportation Research Record:Journal of the Transportation Research Board, No. 1728, TRB, NationalResearch Council, Washington, D.C., 2000, pp. 75–81.

13. Christensen, D. W., and D. A. Anderson. Interpretation of DynamicMechanical Test Data for Paving Grade Asphalt Cements. Journal of theAssociation of Asphalt Paving Technologists, Vol. 61, 1992, pp. 67–116.

14. Shashidhar, N., and P. Romero. Factors Affecting the Stiffening Poten-tial of Mineral Fillers. In Transportation Research Record 1638, TRB,National Research Council, Washington, D.C., 1998, pp. 94–100.

15. Kim, Y. R., H. J. Lee, and D. N. Little. Fundamental Properties of As-phalts and Modified Asphalts—Task K: Microdamage Healing inAsphalt and Asphalt Concrete, Vol. 4. Report DTFH61-92-C-00170.FHWA, U.S. Department of Transportation, Feb. 1998.

16. Lee, H. J. Uniaxial Constitutive Modeling of Asphalt Concrete UsingViscoelasticity and Continuum Damage Theory. Ph.D. dissertation.North Carolina State University, Raleigh, 1996.

17. Huang, Y. H. Pavement Analysis and Design. Prentice Hall, EnglewoodCliffs, N.J., 1993.

Publication of this paper sponsored by Committee on Characteristics of Bituminous Paving Mixtures to Meet Structural Requirements.