1990s there has been increased focus on challenging these assump- tions, particularly for modified asphalts that have better fatigue resistance and show a nonlinear response to loading. Currently, there is a considerable amount of effort placed on the develop- ment of an asphalt binder test procedure that can accurately deter- mine binder contribution to asphalt mixture fatigue by means of damage characterization. Multiple procedures, such as the time sweep, have been proposed to improve the current fatigue specification. The time sweep test method consists of applying repeated cyclic loading at a fixed load amplitude to a binder specimen by using the dynamic shear rheome- ter (DSR). The time sweep was developed during NCHRP Project 9-10 to solve the deficiencies of the current specification (1). The test is based on the definition of fatigue damage: degradation of material integrity under repeated loading. The procedure allows for selection of load amplitude, thus allowing for consideration of pave- ment structure and traffic loading. However, time sweep tests are time consuming and often unrepeatable. Recently, an effort has been placed on developing an accelerated asphalt binder fatigue test to replace the time sweep. Accelerated pro- cedures that systematically increase the load amplitude have shown great potential for fatigue resistance characterization (3). The focus of the present study was to modify the test procedure originally developed by Johnson and Bahia as a surrogate for the time sweep (4). This accel- erated method, called the linear amplitude sweep (LAS), consists of a series of cyclic loads at systematically linearly increasing strain ampli- tudes at a constant frequency of 10 Hz. Similar to the time sweep, the LAS test is run in the DSR and uses standard 8-mm parallel plate geometry. Figure 1 provides a schematic of the amplitude sweep load- ing. Loading begins with 100 cycles of sinusoidal loading at 0.1%. Each successive loading step consists of 100 cycles at a rate of increase of 1% applied strain. Johnson and Bahia recommended using load steps between 1% and 20% applied strain to evaluate damage evolu- tion (4). The procedure also includes a frequency sweep test at a very low strain amplitude of 0.1% to obtain undamaged material properties. The amplitude sweep can be run directly after the frequency sweep as no damage is induced during this stage. Note the combination of the frequency and amplitude sweep tests takes approximately 10 min plus conditioning time. LAS test results can be analyzed using viscoelastic continuum damage (VECD), which has been used extensively to model the complex fatigue behavior of asphalt binders and mixtures (5–8). The primary benefit of using VECD is that results from a single test run at a specific set of conditions can be used to predict the behavior of that material under any variety of alternate conditions. Application of VECD follows Schapery’s theory of work potential to model Modification and Validation of Linear Amplitude Sweep Test for Binder Fatigue Specification Cassie Hintz, Raul Velasquez, Carl Johnson, and Hussain Bahia 99 Current asphalt binder specifications lack the ability to characterize asphalt binder damage resistance to fatigue loading. Multiple acceler- ated testing procedures that attempt to efficiently and accurately char- acterize the contribution of asphalt binders to mixture fatigue are under investigation. One of these tests, which has received significant accep- tance by experts and has been submitted as a draft AASHTO standard, is the linear amplitude sweep (LAS) test. This procedure uses viscoelas- tic continuum damage mechanics to predict binder fatigue life as a func- tion of strain in the pavement. The LAS test uses cyclic loading with systematically increasing load amplitudes to accelerate damage and pro- vides sufficient data for analysis in less than 30 min. Although results of the current LAS testing protocol are promising, the time and the com- plex numerical procedures required for the analysis have raised con- cern. In addition, insufficient damage accumulation was observed when the strain amplitudes proposed in the LAS test were used for a set of polymer-modified binders. This paper presents simplifica- tions of the current analysis procedures and evaluates the ability of extended strain levels to cause sufficient damage for better calculation of the binder fatigue law parameters. The effectiveness of the modified procedure was validated by comparison of the results with the fatigue performance recorded by the Long-Term Pavement Performance pro- gram with consideration of the pavement structure. The fair correla- tions showed the potential for effective use of the modified method for binder specifications. The current PG specification to evaluate asphalt fatigue resistance is based on linear viscoelastic properties (i.e., G*sin δ). This approach lacks the ability to characterize actual damage resis- tance (1, 2). Furthermore, this specification does not account for pavement structure or traffic loading as the measurement is made at one strain level and for a very few cycles of loading. Develop- ers of the PG fatigue parameter were aware of this limitation, but they speculated binder in pavements functions mostly in the lin- ear viscoelastic range, and thus strain is not likely to affect its properties. They also assumed energy estimated in this range is a good indicator of binders’ resistance to fatigue. Since the late C. Hintz, R. Velasquez, and H. Bahia, Department of Civil and Environmental Engineering, University of Wisconsin–Madison, WI 53706. C. Johnson, Stark Asphalt, Division of Northwest Asphalt Products, Inc., Milwaukee, WI 53225. Corresponding author: C. Hintz, [email protected]. Transportation Research Record: Journal of the Transportation Research Board, No. 2207, Transportation Research Board of the National Academies, Washington, D.C., 2011, pp. 99–106. DOI: 10.3141/2207-13
Transcript
1990s there has been increased focus on challenging these assump-tions, particularly for modified asphalts that have better fatigueresistance and show a nonlinear response to loading. Currently,there is a considerable amount of effort placed on the develop-ment of an asphalt binder test procedure that can accurately deter-mine binder contribution to asphalt mixture fatigue by means ofdamage characterization.
Multiple procedures, such as the time sweep, have been proposedto improve the current fatigue specification. The time sweep testmethod consists of applying repeated cyclic loading at a fixed loadamplitude to a binder specimen by using the dynamic shear rheome-ter (DSR). The time sweep was developed during NCHRP Project9-10 to solve the deficiencies of the current specification (1). Thetest is based on the definition of fatigue damage: degradation ofmaterial integrity under repeated loading. The procedure allows forselection of load amplitude, thus allowing for consideration of pave-ment structure and traffic loading. However, time sweep tests aretime consuming and often unrepeatable.
Recently, an effort has been placed on developing an acceleratedasphalt binder fatigue test to replace the time sweep. Accelerated pro-cedures that systematically increase the load amplitude have showngreat potential for fatigue resistance characterization (3). The focus ofthe present study was to modify the test procedure originally developedby Johnson and Bahia as a surrogate for the time sweep (4). This accel-erated method, called the linear amplitude sweep (LAS), consists of aseries of cyclic loads at systematically linearly increasing strain ampli-tudes at a constant frequency of 10 Hz. Similar to the time sweep, theLAS test is run in the DSR and uses standard 8-mm parallel plategeometry. Figure 1 provides a schematic of the amplitude sweep load-ing. Loading begins with 100 cycles of sinusoidal loading at 0.1%.Each successive loading step consists of 100 cycles at a rate of increaseof 1% applied strain. Johnson and Bahia recommended using loadsteps between 1% and 20% applied strain to evaluate damage evolu-tion (4). The procedure also includes a frequency sweep test at a verylow strain amplitude of 0.1% to obtain undamaged material properties.The amplitude sweep can be run directly after the frequency sweep asno damage is induced during this stage. Note the combination of thefrequency and amplitude sweep tests takes approximately 10 min plusconditioning time.
LAS test results can be analyzed using viscoelastic continuumdamage (VECD), which has been used extensively to model thecomplex fatigue behavior of asphalt binders and mixtures (5–8). Theprimary benefit of using VECD is that results from a single test runat a specific set of conditions can be used to predict the behavior ofthat material under any variety of alternate conditions. Applicationof VECD follows Schapery’s theory of work potential to model
Modification and Validation of LinearAmplitude Sweep Test for Binder Fatigue Specification
Cassie Hintz, Raul Velasquez, Carl Johnson, and Hussain Bahia
99
Current asphalt binder specifications lack the ability to characterizeasphalt binder damage resistance to fatigue loading. Multiple acceler-ated testing procedures that attempt to efficiently and accurately char-acterize the contribution of asphalt binders to mixture fatigue are underinvestigation. One of these tests, which has received significant accep-tance by experts and has been submitted as a draft AASHTO standard,is the linear amplitude sweep (LAS) test. This procedure uses viscoelas-tic continuum damage mechanics to predict binder fatigue life as a func-tion of strain in the pavement. The LAS test uses cyclic loading withsystematically increasing load amplitudes to accelerate damage and pro-vides sufficient data for analysis in less than 30 min. Although results ofthe current LAS testing protocol are promising, the time and the com-plex numerical procedures required for the analysis have raised con-cern. In addition, insufficient damage accumulation was observedwhen the strain amplitudes proposed in the LAS test were used for a set of polymer-modified binders. This paper presents simplifica-tions of the current analysis procedures and evaluates the ability ofextended strain levels to cause sufficient damage for better calculationof the binder fatigue law parameters. The effectiveness of the modifiedprocedure was validated by comparison of the results with the fatigueperformance recorded by the Long-Term Pavement Performance pro-gram with consideration of the pavement structure. The fair correla-tions showed the potential for effective use of the modified method forbinder specifications.
The current PG specification to evaluate asphalt fatigue resistanceis based on linear viscoelastic properties (i.e., �G*� � sin δ). Thisapproach lacks the ability to characterize actual damage resis-tance (1, 2). Furthermore, this specification does not account forpavement structure or traffic loading as the measurement is madeat one strain level and for a very few cycles of loading. Develop-ers of the PG fatigue parameter were aware of this limitation, butthey speculated binder in pavements functions mostly in the lin-ear viscoelastic range, and thus strain is not likely to affect itsproperties. They also assumed energy estimated in this range is agood indicator of binders’ resistance to fatigue. Since the late
C. Hintz, R. Velasquez, and H. Bahia, Department of Civil and Environmental Engineering, University of Wisconsin–Madison, WI 53706. C. Johnson, StarkAsphalt, Division of Northwest Asphalt Products, Inc., Milwaukee, WI 53225. Corresponding author: C. Hintz, [email protected].
Transportation Research Record: Journal of the Transportation Research Board,No. 2207, Transportation Research Board of the National Academies, Washington,D.C., 2011, pp. 99–106.DOI: 10.3141/2207-13
damage growth (9). Work is related to damage by using Schapery’swork potential theory:
where
t = time,W = work performed,D = damage intensity, andα = material constant related to the rate at which damage
progresses.
The parameter α is taken to be 1 + 1/−m, where m is the slope ofa log–log plot of relaxation modulus versus time. Johnson and Bahia(4) proposed calculating m by converting frequency sweep data torelaxation modulus by using approximate interconversion methodspresented by Schapery and Park (10). The relationship proposed forthe calculation of α was developed for crack propagation in Mode I,but testing in this study is conducted in shear.
Quantification of work performed using dissipated energy followsKim et al. (11). Dissipated energy under strain-controlled loading iscalculated with
where
W = dissipated energy,γ0 = shear strain,
�G*� = complex modulus, andδ = phase angle.
Equation 2 can then be substituted in Equation 1 and numericallyintegrated to arrive at the following equation to allow for calculationof damage intensity (D):
where ID is the initial undamaged value of �G*�. Johnson and Bahia(4) proposed using �G*� � sin δ as the material integrity parameter anda power law to model �G*� � sin δ versus damage:
D t I G G t tD i i i i( ) ≅ −( )⎡⎣ ⎤⎦ −−+π γ δ δα
α02
11* sin * sin −−
+
=( )∑ 1
11
1
3α
i
N
( )
W G= π γ δi i02 2* sin ( )
dD
dt
W
D= ∂
∂⎛⎝⎜
⎞⎠⎟
α
( )1
100 Transportation Research Record 2207
where C0, C1, and C2 are model coefficients. C0 can be taken as theaverage value of �G*� � sin δ during the 0.1% strain amplitude loadstep, and C1 and C2 are fitted to experimental data. Determination ofmodel coefficients in Equation 4 requires use of an optimization toolsuch as Microsoft Excel’s SOLVER. After the parameters C1 and C2
are known, the derivative of Equation 4 with respect to D can bedetermined and substituted into Equation 1. Equation 1 can then beintegrated to obtain the following closed-form relation betweennumber of cycles to failure (Nf) and strain amplitude for a definedfailure criterion (3):
Simplification of Equation 5 can be accomplished to arrive at thebasic form shown below:
where
and B = −2α. The number of cycles to a given damage intensity canbe calculated at any strain level by using Equation 6. Thus, one canaccount for pavement structure and traffic loading by adjusting thestrain level.
While results of the LAS test are promising, there are importantconcerns about the current testing and analysis protocols. In the pro-posed procedure, strain amplitudes from 0.1% to 20% are used. How-ever, some asphalt binders exhibit little damage under this procedure.Figure 2 presents a typical plot of shear stress versus strain for fourbinders tested at the intermediate PG temperature. The PG 76-10binder exhibits significant damage, as evident by significant decreasesin stress response at high strains. However, the PG 52-40 and PG 58-34 binders demonstrate less damage as the shear stress responsedoes not degrade significantly even at 20% strain. It is unknownwhether VECD is able to accurately characterize the damage resistanceof these materials under the current testing protocol.
Additionally, calculation of the α parameter using frequency sweepresults is difficult. A model is fit to the frequency sweep data, and thatmodel is used to convert from the frequency domain to the time (relax-ation) domain. Furthermore, fitting of the model parameters in Equa-tion 4 by using an optimization tool such as Excel SOLVER is highlydependent on initial guesses for C1 and C2. A poor initial guess canlead to a poor model fit compared with that of a good initial conditionas illustrated in Figure 3. Thus, it is desirable to have a method to
Af D
kIG
C C
Gf
k
D
=( )
⎛⎝⎜
⎞⎠⎟
−
πα
α
*
* ( )
1 2
7
N Af
B= ( )γ max ( )6
Nf D
kIG
C C
Gff
k
D
=( )
⎛⎝⎜
⎞⎠⎟
( )− −
π
γα
α α
*
* ( )max
1 2
25
G C C DC
* sin ( )δ = − ( )0 12 4
0 500 1,000 1,500 2,000 2,500
Loading Cycles
0
5
10
15
20
25A
pp
lied
Str
ain
(%
)
FIGURE 1 Strain sweep loading scheme proposed by Johnson andBahia (4 ).
determine model coefficients that is independent of prior knowledgeof typical values for C1 and C2.
OBJECTIVES
The goal of this study was to address the concerns raised about thecurrent LAS testing and analysis methods. The following pointssummarize the main objectives of this study:
1. Propose a simplified method for determining the parameter α.2. Implement a method to determine the model parameters in
Equation 6 that does not require the use of standard least squaresoptimization tools.
3. Evaluate the effect of extending the strain amplitudes to 30%to determine if higher strains are necessary to fully characterize binderfatigue resistance.
4. Compare the modified LAS method to field performanceobtained from the Long-Term Pavement Performance (LTPP)program.
MATERIALS AND TESTING
Eight LTPP binders were used in this study. The LTPP program mon-itors selected pavement sections for various distresses. The asphaltbinder testing was conducted with an Anton Paar SmartPave DSR. All
Hintz, Velasquez, Johnson, and Bahia 101
tests were conducted at the intermediate-temperature PG of the asphaltbinder after rolling thin film oven aging. Table 1 provides a summaryof the asphalt binders used. Determination of the undamaged proper-ties of the binders was obtained by means of frequency sweep tests,which were conducted using a 0.1% strain. Frequencies ranged from0.1 to 30 Hz. After the frequency sweep test, the binders were loadedusing the strain amplitude sweep. In the amplitude sweep, 100 cycleswere initially applied at 0.1% strain. After this step, each successiveload step consisted of 100 cycles at a rate of increase of 1% appliedstrain per step for 30 steps, starting at 1% and ending at 30% appliedstrain. Generally, two replicates were run for each binder. If results var-ied by more than 15%, a third replicate was run. Analysis of the resultswas used to evaluate the proposed simplifications and changes to theLAS test.
SIMPLIFIED METHOD TO CALCULATE �
The damage exponent (α) is calculated from undamaged rheolog-ical properties by using the slope of a log–log plot of relaxationmodulus [G(t)] versus time. This calculation can be accomplishedin two ways: (a) by direct measurement of relaxation modulusfrom a stress relaxation test and (b) by converting frequencysweep test data to relaxation by using approximate interconver-sion methods developed by Schapery and Park (10). All standardDSRs are capable of conducting frequency sweep tests. Thispaper proposes a method to obtain α directly from frequencysweep data.
The original method converts the data from the frequency to thetime domain. Storage moduli [i.e., G′(ω)] at each angular frequency(ω) must be calculated using complex modulus and phase angle asfollows:
The slope (n) of a log–log plot of G′(ω) versus ω must be calculatedat each frequency as follows:
nGi t
i t
=′( ) − ′( )
−−
−
log log
log log( )
G ω ωω ω
1
1
9
′( ) = ( ) ×G Gω ω δ* cos ( )8
0 5 10 15 20 25
Shear Strain (%)
0
50
100
150
200
300
350
400
450
500PG 76-10
PG 58-34
PG 52-40
PG 64-22
250
Sh
ear
Str
ess
(kP
a)
FIGURE 2 Stress versus strain curve from LAS test.
Data
Fit: Good Initial Guess
Fit: Bad Initial Guess
0 2,0001,000 3,000 4,000
Damage Intensity
0
1
2
3
4
5
6
|G*|
sin
δδ (
MP
a)
FIGURE 3 Example of effect of initial guess on model fit.
The parameter λ′ can then be calculated for each n by using
where Γ is the gamma function and Γ(x) = (x − 1)!. After λ′ is cal-culated for each frequency, complex moduli as a function of fre-quency can be converted to relaxation moduli as a function of timeby using Equation 11. The time corresponding to a given frequencyis calculated as 1/ω.
Adding to the complexity of the conversion, a model must be fit tofrequency sweep data before conversion to allow for prediction of theresponse at frequencies outside of the testing range. Frequencies inthe range of 0.1 to 30 Hz correspond to relaxation times ranging fromapproximately 0.005 to 1.59 s using the presented interconversionmethod. Hence, direct measurements only allow for calculation of asmall range of relaxation times, and therefore predictions of complexmoduli and phase angles at additional frequencies are advantageousin order to obtain relaxation moduli over a reasonable time span.
The conversion of frequency sweep data to relaxation modulus islargely based on the relationship between storage modulus and fre-quency. Johnson demonstrated that the slope of a log–log plot of stor-age modulus versus frequency can be used to calculate α (12). Unlikethe slope of the log of relaxation modulus and log of time, the slopeof a log–log plot of storage modulus versus frequency is positive.However, the magnitude of this slope is nearly identical to the log–log plot of the relaxation modulus versus time. Thus, α can be cal-culated as 1 + 1/m, where m is the slope of the log storage modulusversus log frequency curve. Johnson concluded on the basis ofan analysis of variance that for four binders, the method for calcu-lating α was statistically insignificant on the resulting fatigue lawparameter A from Equation 6 (12). In this study, a different approachwas taken to determine if the α values calculated from relaxationmodulus versus time and storage modulus versus frequency arestatistically equivalent.
For the eight binders tested, α was determined using both methods.Two tests were run on each binder. The α values calculated using logof relaxation modulus versus log of time and log of storage modulus
G t G( ) =′
⎛⎝⎜
⎞⎠⎟
′( )111
λω ( )
λ π= −( ) × ⎛⎝⎜
⎞⎠⎟
Γ 12
10nn
cos ( )
102 Transportation Research Record 2207
versus log of frequency are shown in Table 2. The maximum discrep-ancy in α between the two methods is 2.24%. Thus, observation ofindividual values provides evidence that the two calculation methodsresult in very similar α values.
To further evaluate if the two methods result in the same α values,a test of hypothesis of the means of α estimated using the two meth-ods was conducted. A t-statistic was used to test the hypothesis thatthe two α means are statistically equal, that is, H0: μ1 = μ2 versus H1:μ1 ≠ μ2. The underlying assumption of conducting the hypothesis testis that the samples are normally distributed. Normality plots of α forboth computation methods indicated that the samples are normallydistributed. The resulting test statistic (t0) computed using the twosample distributions was 0.195, and the t-statistic using a significancelevel of 95% was 2.042. Hence, the null hypothesis cannot be rejected,and therefore the α’s computed using the two methods are statisticallyequivalent.
ELIMINATION OF NEED FOR STANDARDOPTIMIZATION TOOLS
Use of optimization tools to solve for best-fit model parameters isproblematic if one does not have a reasonable initial guess for theseparameters. Optimization tools use an iterative approach to find bestpossible values to minimize the error of a model fit. That is, the ini-tial values entered by a user are altered iteratively until the values thatprovide the smallest error between the measurements and model pre-dicted values are reached. If the initial guesses for model coefficientsare highly erroneous, the best model parameters obtained correspondto a local instead of global minima of the error function. Thus, amethod to determine model coefficients without initial guessing isdesirable.
To eliminate the need for iterative optimization tools, Equation 4can be linearized using the logarithmic transformation, as shown in
TABLE 1 LTPP Binder Summary
Test Pavement FatigueTemperature Thickness Cracking
Sample PG (°C) (in.) (m2)
04-B901 76-10 37 10.7 328.0
09-0902 64-28 22 7.2 0.01a
34-0961 78-28 28 6.4 178.8
37-0962 76-22 31 8.5 0.01a
09-0961 58-34 16 6.9 2.1
34-0901 64-22 25 5.6 49.5
89-A902 52-40 10 4.9 6.7
35-0902 64-22 25 10.8 32.0
aMeasured distress is 0 but is listed as 0.01 for inclusion on logarithmic plot.SOURCE: Data are from LTPP database at http://www.datapave.com.
TABLE 2 Comparison of α Values Calculated with Log(RelaxationModulus) Versus Log(Time) and Log(Storage Modulus) VersusLog(Frequency)
α, Storageα, Relaxation ModulusModulus Versus Difference
Sample PG Versus Time Frequency (%)
04-B901_1 76-10 2.291 2.240 2.24
04-B901_2 76-10 2.293 2.247 2.02
09-0902_1 64-28 2.443 2.405 1.57
09-0902_2 64-28 2.444 2.399 1.82
34-0961_1 78-28 2.609 2.572 1.42
34-0961_2 78-28 2.619 2.561 2.21
37-0962_1 76-22 2.391 2.345 1.91
37-0962_2 76-22 2.368 2.340 1.17
09-0961_1 58-34 2.741 2.689 1.91
09-0961_2 58-34 2.729 2.701 1.03
34-0901_1 64-22 2.399 2.356 1.81
34-0901_2 64-22 2.404 2.353 2.13
89-A902_1 52-40 2.809 2.818 0.33
89-A902_2 52-40 2.926 2.882 1.51
35-0902_1 64-22 2.609 2.580 1.13
35-0902_2 64-22 2.618 2.585 1.29
Equation 12, to allow for easy calculation of model coefficients C1
and C2. Recall that C0 is taken to be the average value of �G*� � sin δmeasured in the 0.1% strain step of the amplitude sweep. In otherwords, C0 represents the undamaged �G*� � sin δ and does not need tobe determined through model fitting.
Equation 12 is the common form of a linear equation, y = b + m � x,where m is the slope and b is the intercept. Hence, log(C1) is the inter-cept of a line formed as log(C0 − �G*� � sin δ) versus log(D), and C2 isthe slope of this line. Solving for the slope and intercept is easilyaccomplished with no initial guesses for C1 and C2.
If Equation 4 is an accurate representation of experimental data,the plot of log(C0 − �G*� � sin δ) versus log(D) should be perfectly lin-ear. However, at very low damage levels, experimental data exhibitnonlinear trends. The plot of log(C0 − �G*� � sin δ) versus log(D) for
log * sin log log ( )C G C C D0 1 2 12−( ) = ( ) + ( ) ( )i iδ
Hintz, Velasquez, Johnson, and Bahia 103
a typical binder is shown in Figure 4. If data at damages below 100are ignored, the linear fit is greatly improved, as evident in Figure 5.Thus, if C1 and C2 are determined using data corresponding todamage levels above 100, an adequate model fit is attained.
Both Excel SOLVER and the linearization method were used todetermine C1 and C2 for the asphalt binders tested using all strains(i.e., up to 30%). The fatigue law parameter A in Equation 6 wasobtained using the two methods. For this analysis, A was chosento be defined using the damage intensity corresponding to a 35%decrease from the initial �G*� � sin δ as recommended by Johnsonand Bahia (4). Therefore, in the following analyses, A is denotedas A35. The fatigue law parameter B only depends on α and there-fore is independent of the amplitude sweep results. The values ofA35 computed using optimization and linearization for all 16 testscompleted (i.e., two tests for each binder) are shown in Table 3.The results in Table 3 indicate that values of A35 computed by usingthe two methods are very similar.
R2 = 0.9721
log(Damage)-2.0
-1.5
-1.0
-0.5
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.5
1.0
1.5
log
(C0-
|G*|·
sin
δδ)
FIGURE 4 Linear fit of log(C0 � �G*� � sin �) versus log(D), including allexperimental data.
R2 = 0.9984
log(Damage)-0.2
1.0
0.0
1.5 2 2.5 3 3.5 4
0.2
0.4
0.6
0.8
log
(C0-
|G*|·
sin
δδ)
FIGURE 5 Linear fit of log(C0 � �G*� � sin �) versus log(D), excluding damagesbelow 100.
104 Transportation Research Record 2207
used in place of optimization to solve for model coefficients inEquation 4.
EFFECT OF USING HIGHER STRAINS ON LAS RESULTS
Figure 7 shows stress–strain curves for the binders tested using theLAS procedure. Some of the binders (e.g., 09-0961, 37-0962, and35-0902) did not exhibit significant decreases in shear stress at 20%strain, which could potentially result in poor prediction of damage(Figure 7). However, all binders displayed a considerable reductionin stress when higher amplitudes were applied.
A comparison between results using maximum strain amplitudesof 20% and 30% was performed. This was conducted by comparingA35 values computed by using all strains (0.1% to 30%) with valuescomputed by using only the strain levels up to 20%. That is, compu-tation of A35 by using a maximum strain of 20% was accomplishedby using the test data excluding strain levels above 20%. Table 4shows the comparison of A35 values computed by using maximumstrain amplitudes of 30% and 20%. Overall, using strain amplitudesabove 20% in the LAS test did not drastically affect A35 values. Withthe exception of binder 09-0902_1, A35 values did not differ by morethan 20%. The modified binders, which generally exhibit less dam-age at 20% applied strain, showed higher difference in A35 valueswith 20% versus 30% maximum strain. If a binder does not exhibitsufficient damage by 20% applied strain, fitting by including databeyond 20% is clearly affecting results. Hence, it is necessary thatA35 values be estimated by using strain amplitudes up to 30%.
A linear fit without intercept was applied to A35 values from testswith a maximum strain amplitude of 30% versus values from testswith a maximum strain of 20% (Figure 8). The slope of the best fit lineis 1.056 with an R2 of .96, indicating that using a maximum strain of30% increases the computed A35 parameter by an average of 5.6%.
Whereas using additional higher strains does not appear to havesubstantially affected VECD results, it is recommended that strainsup to 30% are used in the LAS test. Increasing the maximum strainused in the LAS test from 20% to 30% requires only an additional100 s. Testing the binders up to a level of γ = 30% leads to a signif-icant degradation of material integrity and is thought to be the bestrepresentation of fatigue damage behavior.
TABLE 3 A35 Computed with Optimization andLinearization
A35
Sample PG Optimization Linearization
04-B901_1 76-10 7.43 E+06 8.44 E+06
04-B901_2 76-10 6.95 E+06 7.57 E+06
09-0902_1 64-28 9.27 E+06 1.04 E+07
09-0902_2 64-28 8.28 E+06 9.50 E+06
34-0961_1 78-28 2.78 E+07 3.33 E+07
34-0961_2 78-28 2.93 E+07 3.47 E+07
37-0962_1 76-22 5.58 E+07 6.29 E+07
37-0962_2 76-22 5.31 E+07 5.97 E+07
09-0961_1 58-34 3.62 E+07 4.01 E+07
09-0961_2 58-34 3.64 E+07 4.08 E+07
34-0901_1 64-22 8.81 E+06 9.77 E+06
34-0901_2 64-22 8.15 E+06 9.37 E+06
89-A902_1 52-40 6.26 E+07 8.80 E+07
89-A902_2 52-40 7.29 E+07 9.70 E+07
35-0902_1 64-22 4.86 E+07 4.41 E+07
35-0902_2 64-22 4.69 E+07 4.30 E+07
0.0E+00
2.0E+07
4.0E+07
6.0E+07
8.0E+07
1.0E+08
0.0E+00 2.0E+07 4.0E+07 6.0E+07 8.0E+07 1.0E+08
Lin
eari
zati
on
A35
Optimization A35
Line of Equality
R2 = 0.95
89-A902
FIGURE 6 Comparison of A35 computed with different methods to derivepower law coefficients C1 and C2.
A plot of A35 computed by using linearization and optimizationmethods is provided in Figure 6. A line of equality on the plotreveals the results from optimization and linearization are similarfor all binders with the exception of binder 89-A902, which is aheavily modified binder of a PG 52-40 grade. This binder consistsof a very soft base with heavy polymer modification to stiffen it.The combination of the soft base and heavy modification leads toa highly elastic binder with a unique relationship between dam-age and loss modulus. Hence, the power law relationship betweendamage and loss modulus does not provide as accurate a fit com-pared with the other binders tested. An R2 of .95 is achieved when comparing the A35 values determined from the two methods.The results indicate the simplified linearization procedure can be
CORRELATION OF LAS RESULTS WITH FATIGUEPERFORMANCE OF LTPP SECTIONS
The U.S. LTPP program maintains records of the pavement distresson selected sections of Interstate highways. The asphalt binders usedin this study are included in several of these pavement sections. The
Hintz, Velasquez, Johnson, and Bahia 105
pavement distress indicator used by LTPP for fatigue cracking iscracked area per 500 m of pavement section. LAS results from thisstudy were compared with measured fatigue cracking. Because pave-ment thickness greatly affects the strains at the bottom of pavements,where fatigue cracking initiates, LTPP cracked area measurementswere normalized by pavement thicknesses. Cracked areas normalizedby pavement thickness were plotted against number of cycles to fail-ure predictions at 4% strain using LAS results. With the exception ofbinder 09-0902, the LAS results correlated well with field measure-ments (Figure 9). This outcome provides promising evidence that theLAS test is capable of measuring asphalt binder contribution tomixture fatigue.
CONCLUSIONS
On the basis of the results presented, the following conclusions canbe drawn:
• A simplified method that avoids the use of interconversionmethods for the calculation of the α parameter needed in the VECDanalysis was successfully implemented.
• VECD coefficients can be easily obtained by applying a logarith-mic transformation to the damage curves. The use of this linearizationtechnique eliminates the need of iterative optimization tools thatrequire initial guesses to estimate the model coefficients.
• It is recommended that the current LAS procedure be modifiedto include strains ranging from 0.1% to 30% because significantmaterial degradation is achieved for strain levels above 20%. More-over, testing time only increases by 100 s with respect to the currentprocedure.
0
50,000
100,000
150,000
200,000
250,000
300,000
350,000
400,000
450,000
500,000
0 5 10 15 20 25 30 35
Sh
ear
Str
ess
(Pa)
Shear Strain (%)
04-B901
09-0902
34-0961
37-0962
09-0961
34-0901
89-A902
35-0902
FIGURE 7 LAS results.
TABLE 4 Comparison of A35 Values Computed with MaximumStrain Amplitudes of 20% and 30%
• VECD analysis of LAS test results yields promising correlationsbetween accelerated binder fatigue life and measured cracking inactual pavements constructed as part of the LTPP program. It isbelieved that this method could be further improved by investigatinga method to separate nonlinearity from damage accumulation to moreaccurately predict fatigue lives. Accounting for nonlinearity could dif-ferentiate between results from tests using strain amplitudes rangingfrom 0.1% to 20% and tests including strains exceeding 20%.
ACKNOWLEDGMENTS
This research was sponsored by the Asphalt Research Consortium,which is funded by the Federal Highway Administration. This supportis greatly appreciated.
REFERENCES
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The Characteristics of Asphalt Materials Committee peer-reviewed this paper.
