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Road Materials and Pavement Design
ISSN: 1468-0629 (Print) 2164-7402 (Online) Journal homepage: http://www.tandfonline.com/loi/trmp20
Prediction of phase angles from dynamic modulusdata and implications for cracking performanceevaluation
Mirkat Oshone, Eshan Dave, Jo Sias Daniel & Geoffrey M. Rowe
To cite this article: Mirkat Oshone, Eshan Dave, Jo Sias Daniel & Geoffrey M. Rowe (2017):Prediction of phase angles from dynamic modulus data and implications for cracking performanceevaluation, Road Materials and Pavement Design, DOI: 10.1080/14680629.2017.1389086
To link to this article: http://dx.doi.org/10.1080/14680629.2017.1389086
Published online: 30 Oct 2017.
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Road Materials and Pavement Design, 2017https://doi.org/10.1080/14680629.2017.1389086
Prediction of phase angles from dynamic modulus data and implicationsfor cracking performance evaluation
Mirkat Oshonea, Eshan Dave a∗, Jo Sias Daniela and Geoffrey M. Roweb
aDepartment of Civil Engineering, University of New Hampshire, Durham, NH, USA; bAbatech, Inc.,Blooming Glen, PA, USA
(Received 15 August 2016; accepted 25 October 2016 )
The need for a viscoelastic characterisation of hot mix asphalt is increasing as advancedtesting and modelling is incorporated through mechanistic-empirical pavement design andperformance-based specifications. Viscoelastic characterisation includes measurement of themixture stiffness and relative proportion of elastic and viscous response. The most commonmethod is to measure the complex modulus, where dynamic modulus represents the stiffnessand the phase angle represents the relative extent of elastic and viscous response. Deter-mination of phase angle from temperature and frequency sweep tests has been challenging,unreliable and prone to error due to a high degree of variability and sensitivity to signal noise.There are also large amounts of historical dynamic modulus data that are either missing phaseangle measurements or have poorly measured phase angle data that inhibit their use in furtherevaluation. This paper evaluates the robustness of phase angle estimation from stiffness datafor asphalt mixtures. The objectives of the study are to: (1) evaluate the procedure of esti-mating phase angle from the slope of log-log stiffness master curve fitted with a generalisedlogistic sigmoidal curve and compare it with lab measurements and the Hirsch model; (2)assess the effect of measured and predicted phase angles on a mixture Black Space diagram;(3) evaluate the effect of using predicted phase angles on SVECD fatigue analysis particularlyregarding damage characteristics curves and fatigue coefficients and (4) evaluate the impacton layered viscoelastic pavement analysis for critical distresses (LVECD) pavement fatigueperformance evaluation due to the use of predicted phase angles. Three sets of independentmixtures were evaluated in this study comprising a wide range of mixture conditions. Theresults indicate good agreement between measured and predicted phase angle values in termsof shape and peak master curve values. In terms of magnitude, the values from both matchedvery well for certain sets of mixtures and subsequently manifested in similar performance pre-dictions. However, for other sets of mixtures, a considerable difference was observed betweenmeasured and predicted phase angle values as well as SVECD and LVECD results. The differ-ences may be attributed to the use of different types of linear variable displacement transducers(loose core versus spring loaded). Another possible explanation for the difference could be thecontribution of plastic strain, which may create a difference in phase angles of 1–2°.
Keywords: asphalt mixtures; slope of dynamic modulus master curve; phase angle mastercurve; rheological indices; fatigue performance
Introduction and backgroundAsphalt binders exhibit aspects of both elastic and viscous behaviours, and therefore they are con-sidered as viscoelastic materials. Viscoelastic materials are some of the most common materialsthat we encounter and are frequently used for many engineering applications. The mechanical
*Corresponding author. Email: [email protected]
© 2017 Informa UK Limited, trading as Taylor & Francis Group
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properties of viscoelastic materials are temperature, frequency, loading history and being time-dependent. Most often the creep and flow behaviour is small and can be neglected in engineeringcomputations. However, asphalt binders and mixtures need to be fully characterised to capturethe viscoelastic behaviour to understand the performance of pavement structures. Viscoelasticmaterials exhibit both elastic and viscous characteristics during load application. The elasticcomponent of the response is described by the storage modulus and the viscous componentof the response is the loss modulus. It is important to accurately measure both components ofthe response; however, the complex nature of the mechanical behaviour presents experimentaldifficulties and uncertainties during material characterisation.
Asphalt mixtures manifest a more complex viscoelastic behaviour due to the combination ofthe viscoelastic asphalt binders and the aggregate skeleton. Researchers have shown that asphaltmixtures demonstrate linear viscoelastic properties within a small strain level ( < 100 micro-strain) and a limited number of cycles (Airey & Rahimzadeh, 2004; Gardner & Skok, 1964). Yet,some studies argue that nonlinear viscoelastic behaviour can appear at strain levels as low as 40micro-strain (Sayegh, 1967). For materials that exhibit linear viscoelastic behaviour, the relation-ship between stress and strain depends on loading frequency and temperature and can be fullydescribed using a complex modulus (dynamic modulus and phase angle); this test is straight-forward and easy to adopt for the characterisation of asphalt mixes in the small strain region.Moreover, when asphalt mixtures are in the linear viscoelastic range, they generally exhibit ther-morheologically simple properties. The time–temperature superposition principle can then beemployed to horizontally shift results measured at different temperatures along the time or fre-quency axis to construct a master curve for the full characterisation of material behaviour (Vander Poel, 1955). The amount of time or frequency shift is called the time–temperature shift factor.By combining the master curve with the shift factor, it is possible to predict the linear viscoelasticbehaviour of materials over a wide range of frequency and temperature conditions.
The complex modulus test has been one of the methods in use for linear viscoelastic charac-terisation of asphalt mixtures in undamaged states since the 1950s (Heukelom & Klomp, 1964;Van der Poel, 1955). This is achieved by determining two fundamental viscoelastic proper-ties, namely, dynamic modulus, |E*| and phase angle. Based on comparative studies, Elseifi,Al-Qadi, and Yoo (2006) concluded that inclusion of a viscoelastic constitutive model intopavement design methods leads to improved accuracy. Currently, different structural and per-formance mechanistic models use dynamic modulus and phase angle master curves for linearviscoelastic characterisation of asphalt mixtures at a required range of temperature, strain ratesand stress states. Specific applications include the determination of various parameters includingbinder or mix rheological parameters – such as R-value (Christensen & Anderson, 1992), theGlover–Rowe (G–R) parameter (King, Anderson, Hanson, & Blankenship, 2012; Rowe, King,& Anderson, 2013), inflection point frequency, mixture Black Space plots (Mensching, Rowe, &Daniel, 2017), the C1 and C2 parameters in the Williams Landel Ferry (WLF) equation (Rowe,Baumgardner, & Sharrock, 2009; Kaelble, 1985), and lower and upper asymptote of mix mas-ter curves with a sigmoidal form. The application also extends to fatigue characterisation andperformance prediction models such as simplified viscoelastic continuum damage (SVECD) andlayered viscoelastic pavement analysis for critical distresses (LVECD).
The mechanistic analysis of pavements greatly depends on the material characterisationmethod and its accuracy. In recent years, significant advances have been made in specimenfabrication and testing equipment resulting in increased precision of the test data and lower vari-ability associated with |E*| measurements. Moreover, well-developed and robust |E*| predictionequations such as the Witczak model (Andrei, Witczak, & Mirza, 1999), Hirsch model (Chris-tensen, Pellinen, & Bonaquist, 2003) and others are available and have been successfully used byresearchers. The long-term pavement performance (LTPP) programme has also employed these
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Road Materials and Pavement Design 3
models to determine |E*|. While it is known that the accurate measurement of phase angle isvery important for determining the elastic and viscous components, the measurement of phaseangle in the laboratory still remains a challenge due to the need to accurately capture time-baseddata with existing measurement technology. The variability mostly arises from the large amountof inherent noise in the deformation measurement signal. In addition, the calibration aspect isalso complex. Generally, a testing device is evaluated using a solid fixture for a zero-phase lagresponse. However, standards of materials with a known stiffness and intermediate values ofphase angle (typical of that found in asphalt mixtures) are not available. The accuracy furtherdepends upon many other factors (e.g. adjustments for equipment compliance that have beenmade, details of displacement transducer design). Moreover, large amounts of historic data existwith |E*| measurements but with no accompanying phase angle measurements. For example,the LTPP database is populated with measured and predicted |E*| data but lacks phase angle (φ)data. This prohibits the use of these data for rheological and performance evaluation. Further, itinhibits their use for verification of rheological parameters and linkage of historic data to fieldperformance data.
Several researchers have developed relationships between phase angle and modulus for asphaltmixtures. Bonnaure, Gest, Gravois, and Uge (1977) developed a relationship that was limited tobinder stiffness (Sb) values greater than 5 MPa and less than 2 GPa (when Sb is greater than 2 GPathe mixture phase angle (φm) is taken to be zero). The relationship used the binder stiffness (Sb)volume of binder (Vb) in the prediction and is as follows:
φm = 16.36V0.352b exp
[log10Sb − log105 × 106
log10Sb − log102 × 109 × 0.974V−0.172b
]. (1)
During the SHRP project, Tayebali, Tsai, and Monismith (1994) developed several relationshipslinking phase angle to mix stiffness, an example of which is shown below:
φo = 260.096 − 17.172Ln(So), (2)
where φ0 is the mixture phase angle and S0 is the mixture stiffness. This relationship was devel-oped from a study of fatigue properties. The subscript to the parameters denotes that the initialcondition is used. The preceding two relationships for bituminous mixtures are empirical innature with derived constants from regression analysis.
The Hirsh model was originally developed in the late 1960s based on the modified law ofmixtures. The law states that the property of a composite material can be treated as a combinationof the properties of its components assuming the influence of each component is proportional toits volume fraction. Christensen et al. (2003) developed a calibrated phenomenological modelbased on the Hirsch model that links phase angle to binder properties and mixture volumetrics,as follows:
|E∗|mix = Pc·[
4, 200, 000·(
1 − VMA100
)+ 3·|G∗|binder·
(VFA·VMA
10, 000
)]
+ (1 − Pc)·[
1 − VMA100
4, 200, 000+ VMA
3·VFA·|G∗|binder
]−1
, (3)
where the contact area Pc is defined as follows:
Pc =(
20 + VFA·3·|G∗|binder
VMA
)0.58
650 +(
VFA·3·|G∗|binder
VMA
)0.58 , (4)
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4 M. Oshone et al.
and
φ = −21(log Pc)2 − 55logPc, (5)
where VFA is voids filled with asphalt, VMA is percent voids in the mineral aggregate and Gb*is the complex shear modulus of the binder.
It is generally accepted that asphalt mixtures are best modelled with a viscoelastic solid modelrepresented by a sigmoidal shape. The only model format that describes this shape to some extentis that of the Hirsch model whereas the other equations are clearly limited.
The concept of predicting phase angle from the slope of the complex modulus versus fre-quency was first suggested by Booij and Thoone (1982). Christensen and Anderson (1992) usedthe same concept in the development of the Christensen Anderson binder model phase angle cal-culation. Rowe (2009) presented equations using a similar basis for sigmoidal forms that can beapplied to asphalt mixture analysis, as follows (Equations (6–9)). The equations can be applied toeither the analysis of master curves that contain |E*| and phase angle data or just |E*| data alone.The works conducted by Booij and Thoone (1982), Christensen and Anderson (1992) and morerecently by Rowe (2009) demonstrate that the phase angle response can be determined from amathematical understanding of the dependency on the stiffness (either |G*| or |E*|) versus thefrequency (ω) using regression parameters for a sigmoidal model. Each of the regression coef-ficients (δ, α, β, λ) is related to the shape of the sigmoid fit to the master curve as detailed byMensching et al. (2017). This results in a method to determine phase angle from just dynamicmodulus vs. frequency data, when it is available.
Standard logistic
log |E∗| = δ + α
(1 + e[β+γ (logw)]), (6)
Standard logistic
φ = 90 × dlogE∗
dlogw= −90 × αγ
e[β+γ (logw)]
[1 + e[β+γ (logw)]]2 , (7)
Generalised logistic
log |E∗| = δ + α
(1 + λe[β+γ (logw)]1/λ, (8)
Generalised logistic
φ = 90 × dlogE∗
dlogw= −90 × αγ
e[β+γ (logw)]
[1 + λe[β+γ (logw)]1+1/λ ]. (9)
This paper evaluates a fundamental relationship approach to determine phase angle via theslope of the log–log of the stiffness curve (from now on referred to as the slope method). Thestudy by Rowe (2009) investigated this relationship and concluded the validity of the relationshipto a large set of modified and unmodified binders, asphalt mixes and some other polymers. Thispaper extends the previous research to a larger set of asphalt mixtures and evaluates the reliabilityof the relationship to asphalt mixtures. Phase angles predicted using the slope method are com-pared to laboratory measurements to assess the validity of the relationship for various mixtures.Further comparisons are made with the Hirsh model, the slope method and laboratory-measuredphase angle values. Finally, the implication for rheological parameters and pavement fatigue per-formance predictions due to the use of predicted phase angle values as opposed to lab-measuredvalues is assessed.
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Road Materials and Pavement Design 5
Research approach and materialsMaterials and testingThree sets of independent data were used for this study; two from specimens fabricated and testedby the research team as part of ongoing projects and the third from the LTPP database. The firstset of specimens includes two 9.5 mm nominal maximum aggregate size (NMAS) (C-9.5 mmand L-9.5 mm) and one 12.5 mm NMAS (C-12.5 mm) laboratory-produced mixtures. The aggre-gates and binders were obtained from Rhode Island Department of Transportation (RIDOT) andspecimens were fabricated to replicate a range of acceptable as-built field conditions in termsof asphalt binder and air void content. The low, optimum and high levels of asphalt and airvoid content combinations for the three mixtures resulted in 27 specimen conditions overall.The second set test specimens were plant-produced materials obtained from New HampshireDepartment of Transportation (NHDOT) projects. These included 12.5 and 19 mm NMAS mix-tures containing various amounts of RAP and RAS and a virgin mixture. The last set includesthree mixtures from Vermont, New Jersey and North Carolina LTPP sections. Overall, the wideselection of mixtures used in this study covers a range in as-built conditions, NMAS, laboratoryversus plant production, virgin and modified binder (V), and % RAP and RAS; this providesa platform for comparing the potential effects of these parameters on phase angle predictions.Mixture information is summarised in Table 1.
Specimens fabricated and tested by the research team were compacted using a Superpavegyratory compactor and then cut and cored to test specimen dimensions of 100 × 150 mmand 100 × 130 mm for dynamic modulus and fatigue testing, respectively. Complex modulus(E*) testing was performed following the test procedure provided in AASHTO T342 in load-controlled uniaxial compression mode. Testing was done at three temperatures (4.4°C, 21.1°C
Table 1. Mixture parametric information.
Rhode Island (RI) mixtures
Mixture label AC level AV level NMAS, mm Binder
L-9.5mm 5.9, 6.3, 6.8% 4, 7, 9% 9.5 PG 64-28C-9.5mm 5.9, 6.3, 6.8% 4, 7, 9% 9.5 PG 64-28VC-12.5mm 5.2, 5.6, 6.1% 4, 7, 9% 12.5 PG 64-28V
New Hampshire (NH) mixtures
Mixture Label%Total binder replacement
(%RAP/%RAS) AV level NMAS, mm Binder
NH-12.5 mm, 20%RAP 18.9 (18.9/0) 7.7 12.5 PG 58-28NH-12.5 mm, 30%RAP 28.3 (28.3/0) 6.8 12.5NH-12.5 mm, RAP/RAS 18.5 (7.4/11.1) 7.4 12.5NH-19 mm, 20%RAP 20.8 (20.8/0) 6.1 19NH-19 mm, 30%RAP 31.3 (31.3/0) 6.3 19NH-19 mm, RAP/RAS 20.4 (8.2/12.2) 6.0 19Virgin mixture 0 (5.9% ac content) 4.4 12.5 PG 58-28
Mixtures from LTPP
Mixture label SHRP ID |E*| link VFA VMA
NC mixture 1992 807 55.1 21.4NJ mixture 1033 715 76.4 14.4VT mixture 1682 1110 72.4 14.5
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6 M. Oshone et al.
and 37.8°C) and six loading frequencies (25, 10, 5, 1, 0.5 and 0.1 Hz) on three replicate speci-mens using an Asphalt Mixture Performance Tester (AMPT). Axial deformation was measuredusing four linear variable displacement transducers (LVDTs) with a 70 mm gauge length. Strainamplitudes during tests were limited to 50–75 micro-strain to ensure the test response remains inthe linear viscoelastic range. The resulting stress and strain of the final six cycle of each loadingseries were used to calculate the dynamic modulus and phase angle of the mixtures. Dynamicmodulus is determined by dividing peak-to-peak specimen stress to strain (Equation (10)), andphase angle is obtained from the time lag between peak to peak strain and stress (Equation 11):
|E∗| = σamp
εamp, (10)
φ = 2π f �t, (11)
where σampis the amplitude of applied sinusoidal stress, εampis the amplitude of sinusoidal strainresponse, f is the stress and strain frequency and �t is the average time lag between stress peakand strain peak at a given frequency and temperature.
Cyclic fatigue testing using AMPT was carried out on specimens following the test procedurein AASHTO TP 107. Four specimens were tested at four different peak-to-peak on-specimenstrains to cover the appropriate range of number of cycles to failure.
Dynamic modulus and phase angle master curve construction from measured dataIn this study, the RHEA™ software (Rowe & Sharrock, 2000) is used to construct |E*| and φ
master curves from measured |E*| and φ data points using the time–temperature superpositionprinciple. For tests performed in the linear viscoelastic range, the time–temperature superpositionprinciple allows test isotherms to be shifted to a required temperature at a reduced frequency. Theshifting in the RHEA™ software is done following the work done by Gordon and Shaw (1994).The storage modulus, representing the elastic behaviour, and loss modulus, representing the vis-cous behaviour, are shifted separately. Then the shifting from the two components is averaged toproduce the final shift factor. In the absence of phase angle data, the shifting in the programmeis done only once using the dynamic modulus component. The same shift factors are applied tothe corresponding phase angle measured points to produce phase angle master curves. The five-parameter, generalised sigmoidal model (Richards curve) was used to fit the dynamic modulusand phase angle master curves (Equations 8 and 9).
1.E+01
1.E+02
1.E+03
1.E+04
1.E-03 1.E-01 1.E+01 1.E+03
Dyn
amic
Mod
ulus
(M
Pa)
Reduced Frequency (Hz)
Measured 4.4°CMeasured 21.1°CMeasured 37.8°CMeasured Low & High Range
Ref. Temp = 21.1ºC
Figure 1. Measured |E*| raw data.
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Road Materials and Pavement Design 7
-4
-3
-2
-1
0
1
2
3
0 10 20 30 40 50
Log
shi
ft f
acto
r a(
T)
Temperature, ºC
(b)Shift Factor4.4°C21.1°C37.8°C
0
10
20
30
40
1.E-03 1.E-01 1.E+01 1.E+03
Pha
se A
ngle
(D
eg.)
Reduced Frequency (Hz)
(c)
Measured FittedMeasured Shifted
Ref. Temp = 21.1 ºC
3.E+01
3.E+02
3.E+03
3.E+04
1.E-03 1.E-01 1.E+01 1.E+03
Dyn
amic
Mod
ulus
(M
Pa)
Reduced Frequency (Hz)
(a)
Measured FittedMeasured ShiftedMeasured Low & High Range
Ref. Temp = 21.1ºC
Figure 2. (a) |E*| master curves, (b) time–temperature shift factors, (c) phase angle master curves.
The |E*| and φ master curve construction process is illustrated in Figures 1 and 2. Figure 1shows lab-measured |E*| isotherms at test temperatures of 4.4°C, 21.1°C and 37.8°C and loadingfrequencies of 25, 10, 5, 1, 0.5 and 0.1 Hz. The isotherms are shifted horizontally to a reduced fre-quency to construct the |E*| master curve at a reference temperature of 21.1°C (Figure 2(a)). Theshifted points are fitted with a generalised logistic equation (Figure 2(a)). The time–temperatureshift factors are shown in Figure 2(b). The same shift factors (Figure 2(b)) are applied to mea-sured phase angle points to construct phase angle master curves (Figure 2(c)). The measuredphase angle master curve is fitted with a generalised logistic equation (Figure 2(c)).
To determine mixture phase angle using the slope method, first the |E*| master curve is con-structed using average measured |E*| data from replicates and |E*| values deviating from the
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8 M. Oshone et al.
mean by one standard deviation. This is done to account for some degree of variability thatexists during |E*| measurement. The sample standard deviation for each replicate at each testtemperature and frequency was calculated to obtain the high and low range of the measured |E*|.For each set of measured data (average, average + 1 standard deviation and average − 1 standarddeviation), independent time–temperature shifting was conducted and this yielded three mastercurves for |E*|. These are labelled as “Measured”, “Measured High Range” and “Measured LowRange” (Figure 2(a)).
Phase angle prediction from the slope of the log–log |E*| master curveUsing the fitting sigmoidal logistic function, 28 |E*| master curve points were computed at5 equally spaced points per decade on a logarithmic scale over a frequency range of 0.001–10,000 Hz. Figure 3(a) shows the points determined in this manner for average |E*|, low andhigh range |E*|. The unfitted master curve points which correspond to the unfitted phase anglepoints in Figure 2(b) are also shown in Figure 3(a). The phase angle points (Figure 3(b)) aredetermined from the slope of the log–log |E*| master curve of Figure 2(a) using Equation (9).Figure 3(b) shows the predicted phase angle from average, low and high ranges. The measuredunfitted and fitted curves are shown to allow visual comparison between measured and predictedphase angle points (Figure 3(b)).
Phase angle Prediction using the Hirsch ModelIn this portion of the study, the Hirsch model is used to determine the phase angle from bindershear modulus |G*| and mix volumetrics (Equation (5)) to allow comparison to the slope method
0
10
20
30
40
1.E-03 1.E-01 1.E+01 1.E+03
).geD(
elgnA
esahP
Reduced Frequency (Hz)
(b)
Measured FittedMeasured ShiftedPredicted AveragePredicted Low & High Range
1.E+02
1.E+03
1.E+04
1.E-03 1.E-01 1.E+01 1.E+03
Dyn
amic
Mod
ulus
(M
Pa)
Reduced Frequency (Hz)
(a)
Measured FittedMeasured ShiftedMeasured Low & High Range
Figure 3. (a) |E*| master curves, (b) measured and predicted phase angle master curves.
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Road Materials and Pavement Design 9
prediction and lab measurements. Lab-measured |G*| binder values were used for the NH virginmixture. However, the |G*| values back calculated using Equations (3) and (4) are used for theLTPP mixtures because the binder shear modulus values were not available. The back-calculatedshear modulus is then used in Equation (5) to determine the phase angle. It should be notedthat for the phase angle prediction done in this manner, the compounding error from each modelcould contribute to the differences or similarities observed with the Hirsch method, slope methodand lab measurement comparison for the LTPP mixtures.
Statistical analysisStatistical analysis is performed to examine the accuracy of the phase angle estimation from theslope of the log–log |E*| master curve. First, the Pearson linear correlation coefficient is usedto measure the strength of linear association between the 28 measured and predicted data pointscorresponding to each mixture condition. Next, a linear relationship is fitted to the points and theequation of the fitted line is generated to allow comparison on a unity plot (x = y). Further statis-tical analysis is employed by calculating the root mean square error (RMSE) between measuredand predicted phase angle points. RMSE complements the correlation coefficient because it pro-vides the difference in measurements between each measured and predicted points. The RMSEfor individual mixtures is normalised by dividing the total RMSE by the number of points.
Impact on pavement performance evaluationMixture Black Space diagramMixture Black Space diagrams assess the stiffness and relaxation capability of mixtures froma plot of |E*| versus phase angle. A recent study by Mensching et al. (2017) evaluated thecorrelation between certain Black Space points and low temperature cracking performance basedon the Glover–Rowe (G–R) binder cracking parameter. The G–R parameter correlates binderBlack Space points to non-load-associated cracking (King et al., 2012; Rowe et al., 2013). Thetemperature and frequency combinations evaluated for mixture Black Space points were 15°Cand 0.005, 5 and 500 rad/s. For this study, the Black Space points are generated using measuredand predicted phase angle data to evaluate the relative location of the points.
Pavement fatigue life evaluation, SVECD and LVECDIn recent studies the SVECD approach (Underwood, Baek, & Kim, 2012) has been used to eval-uate the fatigue performance of various mixtures using uniaxial cyclic fatigue testing. In thisapproach, two relationships are developed to characterise the fatigue performance of the mix-tures and are used in subsequent modelling and pavement performance prediction. The damagecharacteristic curve (C versus S) is represented by the reduction in the pseudo secant modulus(material integrity) and accumulated damage during cyclic loading. The curve is fitted with anexponential function (Equation (12)). This relationship shows how the stiffness, or integrity, ofthe material changes as micro cracks grow during continued load applications.
The fundamental energy-based failure criterion, GR (the rate of change of the average releasedpseudo strain energy), is plotted versus the number of cycles to failure and demonstrates that ifdamage is accumulating faster, material failure will occur sooner. The curve from this relation-ship is fitted with Equation (13). The fitting parameters a, b, γ and s in Equations (12 and 13) arereferred to as damage model coefficients. Both relationships require dynamic modulus and phaseangle master curves for the linear viscoelastic characterisation of the asphalt mixtures:
C = e−aSb, (12)
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Nf = (GR)s, (13)
where a, b, γ and s are fitting parametersThe LVECD programme employs a finite element structural analysis for pavement response
computation and pavement fatigue performance prediction based on the SVECD approach(Eslaminia, Thirunavukkarasu, Guddati, & Kim, 2012). The fatigue coefficients from SVECDanalysis are used in the model to determine the fatigue performance of the study mixtures in apavement structure in terms of number of failure points. Failure of an element is defined when theratio of applied loading cycles to the failure loading cycle is equal to 1 (N/Nf = 1), in which casethe asphalt element is considered completely cracked. A spatial distribution plot from LVECDanalysis displays the ratio of N/Nf for the matrix of 11 by 101 finite element nodes producing atotal of 1111 nodes along the pavement cross-section. For this study, the number of failure pointswas tallied to compare relative fatigue performance.
For this study, SVECD analysis and LVECD pavement simulations were performed using bothmeasured and predicted phase angles. The SVECD approach was used to analyse the fatigue testresults and determine the damage characteristics curves (C versus S and GR versus Nf ) for thestudy mixtures. One of the required inputs to the SVECD analysis is phase angle measurementof mixtures at three temperatures and six frequencies for linear viscoelastic characterisation ofmixtures. The phase angle prediction from the stiffness data was used to obtain these values. Thefatigue coefficients obtained from SVECD analysis using predicted and measured phase anglevalues are used in LVECD to compute the fatigue performance of the mixtures in a pavementstructure. A typical pavement structure and traffic were used in the analysis. Fatigue crackingperformance evaluation in terms of C versus S and GR versus Nf as well as number of failurepoints determined using the predicted and measured phase angle values for the mixtures werecompared. This is done to assess the potential effect of predicted phase angle on the damagecharacteristic curves and fatigue performance prediction and further gauge the accuracy of theprediction approach in the context of pavement performance estimates.
Results and discussionThe results from statistical analysis, rheological indices and pavement performance evaluationare discussed in this section. Results for all of the mixtures evaluated are summarised in tabularform and example figures for two cases were chosen for illustration. The two cases represent thebest and worst examples, from the perspective of match between measured and predicated phaseangles.
Figure 4 demonstrates the |E*| master curve fitted with the generalised logistic equationfor Measured (Mean), measured high range (Mean + 1 Standard deviation) and measured lowrange (Mean – 1 Standard deviation). Comparing all Rhode Island and New Hampshire Mix-tures, the L-9.5 mm 6.8AC 7AV mixture exhibited the lowest variability (Figure 4(a)) whereasthe C-12.5 mm 6.1AC 7AV showed the highest variability (Figure 4(b)) in |E*| measure-ment among replicates. Considering highest variability observed, it can be noted that deviationin dynamic modulus measurements among replicates is slight (CV = 11%). Therefore, theeffect on phase angle prediction due to |E*| measurement variation is also expected to beminimal.
Comparison of measured and predicted phase anglesFigure 5 shows the different measured and predicted phase angle master curves. The actual mea-sured phase angle values are shown along with the logistic curve fitted to the measured points
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1.E+02
1.E+03
1.E+04
1.E-03 1.E-01 1.E+01 1.E+03
Dyn
amic
Mod
ulus
(M
Pa)
Reduced Frequency (Hz)
(b) C-12.5mm 6.1AC 7AV
Measured FittedMeasured ShiftedMeasured Low & High Range
1.E+02
1.E+03
1.E+04
1.E-03 1.E-01 1.E+01 1.E+03
Dyn
amic
Mod
ulus
(M
Pa)
Reduced Frequency (Hz)
(a) L-9.5mm 6.8AC 7AV
Measured FittedMeasured ShiftedMeasured Low & High Range
Figure 4. |E*| master curves: (a) L-9.5 mm 6.8AC 7AV, (b) C-12.5 mm 6.1AC 7AV.
to illustrate the variability in phase angle measurements and its influence on the shape of thefitted logistic curve. Throughout the study, the statistical and other comparisons are done usingfitted measured phase angle values as opposed to predicted values. So, it should be noted that thefitting might magnify or reduce the differences between measured and predicted values.
The phase angle master curve predicted from the measured |E*| curve is shown in Figure 5.The mixtures presented represent the best and worst matches between measured and predictedvalues. The L-9.5 mm 5.9AC 7AV and NH-12.5 mm, 30% RAP mixtures had the lowest dif-ferences among the Rhode Island and New Hampshire mixtures, respectively (Figure 5(a,b)).The C-12.5 mm 6.1AC 7AV and NH-19 mm, RAP/RAS mixtures exhibited the largest differ-ences between measured and predicted values (Figure 5(c,d)). The difference between measuredand predicted values appears to be higher at the peak phase angle, which corresponds to highertesting temperature or lower loading frequency. As it can be seen from actual phase angle mea-surements, the variability in lab measurements is usually higher at this location as well. Thisraises the question about the accuracy of phase angle measurements and how to incorporate theerror that is present during model verification. It has to be noted that in both best and worst sce-narios, the measured and predicted phase angle master curves have comparable shape and peaksat similar frequencies. This is further quantified by computing the inflection point frequency forboth curves, shown in Figure 6. Overall, the inflection points match very well which is indicatedby the high R2 value (0.98) and low average RMSE (0.02 Hz).
The statistical analysis for Rhode Island and New Hampshire mixtures is presented in Table 2,and the relationship is shown graphically in Figure 6. A strong association between measured andpredicted values is observed for the L-9.5 mm and C-9.5 mm mixture whereas the C-12.5 mm and
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0
10
20
30
40
1.E-03 1.E-01 1.E+01 1.E+03
Pha
se A
ngle
(D
eg.)
Reduced Frequency (Hz)
(a) L-9.5mm 5.9AC 7AV
Measured FittedMeasured Shifted Predicted Average
0
10
20
30
40
1.E-03 1.E-01 1.E+01 1.E+03
Pha
se A
ngle
(D
eg.)
Reduced Frequency (Hz)
(b) NH-12.5 mm, 30% RAP
Measured FittedMeasured ShiftedPredicted Average
0
10
20
30
40
1.E-03 1.E-01 1.E+01 1.E+03
Pha
se A
ngle
(D
eg.)
Reduced Frequency (Hz)
(d) NH-19 mm, RAPRAS
Measured FittedMeasured ShiftedPredicted Average
0
10
20
30
40
1.E-03 1.E-01 1.E+01 1.E+03
Pha
se A
ngle
(D
eg.)
Reduced Frequency (Hz)
(c) C-12.5mm 6.1AC 7AV
Measured FittedMeasured Shifted Predicted Average
Figure 5. Measured and predicted phase angle master curves: (a) L-9.5 mm 5.9AC 7AV, (b)NH-12.5 mm,30% RAP, (c) C-12.5 mm 6.1AC 7AV, (d) NH-19 mm, RAPRAS.
New Hampshire mixtures appear to have the least agreement between measured and predictedvalues. Overall, the prediction appears to generate lower phase angle values as compared tomeasured values. This is indicated by the consistent deviation of the points to the lower side ofthe equality line (Figure 7).
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Road Materials and Pavement Design 13
y = 1.03xR² = 0.98
0.0
0.5
1.0
1.5
2.0
0.0 0.5 1.0 1.5 2.0Measured Inflection Freq., Hz
RI and NH Mixtures
RMSE = 0.02
Pre
dict
ed I
nfle
ctio
n F
req.
, Hz
Figure 6. Line of equality plot for measured and predicted inflection point frequencies.
Generally, the predicted and measured phase angles are in good agreement in terms of shape ofthe master curve and magnitude for Rhode Island L-9.5 mm and C-9.5 mm mixtures with 7% and8% deviation from the equality line and 1.34 and 1.62 RMSE, respectively. For the 18 mixtureconditions under the 2 sets of mixtures, the average difference between measured and predictedvalues is less than 2°. However, a larger difference is observed in New Hampshire mixes fol-lowed by the Rhode Island C-12.5 mm with RMSE 4.24 and 4.96 and 16% and 17% deviationfrom the equality line, respectively. From the observation, it appears that the differences betweenmeasured and predicted values are consistently lower for certain sets of mixtures and are higherfor other sets of mixtures despite the same lab measurement procedure and prediction methodemployed. Particularly, the composition in terms of aggregate and binder used for the C-12.5 mmmix was similar to the C-9.5 mm mix and both mixtures used identical production procedure andequipment, environmental chamber and AMPT. Hence, the higher difference observed in theC-12.5 mm could not be explained with respect to any of these parameters. However, for theC-9.5 mm and L-9.5 mm mixtures, for which measured and predicated phase angles are in betteragreement (both with low average RMSE of 1.34 and 1.62), spring-loaded linear variable dis-placement transducers (LVDTs) were used to measure specimen deformation. However, for theC-12.5 mm and New Hampshire mixtures where the differences are higher, loose core LVDTswere used.
For all study mixtures, the same AMPT device was used. There were no differences in cali-bration, machine compliance, algorithm and PID parameters used. The only difference as statedabove was the type of LVDT that was used for deformation measurements. A comparative studydone by Lacroix (2013) showed that the measurements from spring-loaded LVDTs are less vari-able (within the tolerance given by the manufacturer) as compared to loose core LVDTs in asituation where both are calibrated. The study also suggested that the alignment between LVDTand the specimen causes a higher discrepancy in measurement of displacement and phase anglefor loose core models as compared to spring-loaded ones. Based on the results presented hereas well as from observations by others, the type of LVDTs used could be one of the reasons forhigher differences between measured and predicted phase angle values observed.
The prediction of phase angle from dynamic modulus relies upon the assumption of linearviscoelastic behaviour in the small strain region and that materials can be considered thermorhe-ologically simple and time–temperature supposition is valid. However, it has been known formany years that a true representation of asphalt mixtures over the entire range of temperature
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Table 2. Statistical evaluation.
RI mixtures AC–AV R2 Trend line equation RMSE
L-9.5mm 5.9–4 1.00 y = 0.93x 1.435.9–7 1.00 y = 0.96x 0.855.9–9 1.00 y = 0.94x 1.156.3–4 1.00 y = 0.93x 1.306.3–7 0.99 y = 0.94x 1.696.3–9 0.98 y = 0.91x 2.286.8–4 1.00 y = 0.95x 1.096.8–7 1.00 y = 0.94x 1.166.8–9 1.00 y = 0.95x 1.14
Average 94% (6%) 1.34C-9.5mm 5.9–4 0.99 y = 0.94x 1.26
5.9–7 0.96 y = 0.93x 1.315.9–9 0.95 y = 0.92x 1.616.3–4 0.97 y = 0.93x 1.516.3–7 0.97 y = 0.93x 1.46
6.3––9 0.94 y = 0.92x 1.626.8–4 0.95 y = 0.93x 1.736.8–7 0.98 y = 0.91x 1.956.8–9 0.86 y = 0.91x 2.15
Average 92% (8%) 1.62C-12.5mm 5.2–3 0.98 y = 0.86x 3.57
5.2–7 0.96 y = 0.82x 4.845.2–9 0.89 y = 0.87x 2.915.6–5 0.99 y = 0.87x 3.415.6–7 0.93 y = 0.82x 5.985.6–9 0.97 y = 0.81x 4.996.1–4 0.83 y = 0.82x 5.596.1–7 0.77 y = 0.73x 8.656.1–9 0.95 y = 0.84x 4.73
Average 83% (17%) 4.96NH mixtures NMAS R2 Trend line equation RMSENH 12.5 mm, 20%RAP 12.5 0.86 y = 0.88x 3.05NH 12.5 mm, 30%RAP 12.5 0.99 y = 0.89x 2.40NH 12.5 mm, RAP/RAS 12.5 0.99 y = 0.82x 4.36NH 19 mm, 20%RAP 19 0.99 y = 0.83x 4.45NH 19 mm, 30%RAP 19 0.94 y = 0.83x 5.14NH 19 mm, RAP/RAS 19 0.97 y = 0.78x 6.03Average 84% (16%) 4.24
and frequency requires the use of viscoelastic-plastic models. The plastic strain that occurs islargely a function of movement within the aggregate skeleton and represents the complex natureof a mixture rather than a bituminous binder. Work conducted on mixture visco-plastic behavioursuggests that the plastic deformation occurs at very low stress and most likely can be consid-ered as a stress-dependent zero yield plasticity (Darabi, Al-Rub, Masad, Huang, & Little, 2011;Drescher, Kim, & Newcomb, 1993; Rowe & Pellinen, 2004). If the response to load is consid-ered a simple model – zero yield stress-dependent linear plastic strain – the strain produced bya sinusoidal stress application will be completely out of phase with the load response in a sim-ilar manner to a viscous strain. The stress imposed upon the aggregate skeleton will be higherwhen the binder is less stiff, associated with lower frequencies or higher temperatures. Thus,
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0
10
20
30
40
50
0 10 20 30 40 50
Measured Phase Angle, Degrees
RI Mixtures
L-9.5mmC-9.5mmC-12.5mm
0
10
20
30
40
50
0 10 20 30 40 50
Measured Phase Angle, Degrees
NH Mixtures
NH-12.5mm, 20%RAPNH-12.5mm, 30%RAPNH-12.5mm, RAP/RASNH-19mm, 20%RAPNH-19mm, 30%RAPNH-19mm, RAP/RAS
Pred
icte
d Ph
ase
Ang
le, D
egre
esPr
edic
ted
Phas
e A
ngle
, Deg
rees
Figure 7. Line of equality plot for measured and predicted phase angles.
the measured phase angle of a mixture (which includes a plastic response) is always theoreti-cally going to be greater than that of a mixture represented only by viscoelastic behaviour. Thus,when a viscoelastic consideration is applied to the calculation of phase angle from the complexmodulus of a mixture a bias in the data is expected.
It should be noted that the calibration of the phase angle measurement is complex. Phase angleis not a direct measurement but rather a calculation from collected data. The system in use wasverified to deliver a zero-phase angle with an aluminium specimen fixed in the device. However,at higher frequencies some bias in measurement may exist for reasons not immediately intuitiveto the researchers. While magnitudes of these effects are not known, they may exist and arementioned in this paper for completeness of this discussion. If these effects are insignificant itshould be possible to assess the plastic strain effects compared to the various models that exist.
Comparison of the slope-based prediction method with the Hirsh model predictionA virgin mixture is used to compare phase angle prediction among the slope-based method,Hirsch model and lab measurement. A virgin mixture is used for this comparison because thebinder modulus is a direct representation of the material in the mixture; this is not possible with
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0
10
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40
1.E-03 1.E-01 1.E+01 1.E+03
Pha
se A
ngle
(D
eg.)
Reduced Frequency (Hz)
(a)
Measured FittedMeasured ShiftedPredicted SlopePredicted Hirsch
1.E+01
1.E+03
1.E+05
0 10 20 30 40
Dyn
amic
Mod
ulus
, MPa
Phase Angle (degrees)
(b)
Measured FittedMeasured ShiftedPredicted SlopePredicted Hirsch
RMSE=1.97R² = 0.94
RMSE=3.17R² = 0.83
0
10
20
30
40
50
0 10 20 30 40 50
Pred
icte
d Ph
ase
Ang
le, D
egre
es
Measured Phase Angle, Degrees
(c)
× Hirsch Model
SlopeMethod
Figure 8. (a) Phase angle master curves; (b) Black Space diagram; (c) line of equality plot for phase anglepredicted using the slope method and the Hirsch model.
RAP mixtures due to unknown degree of blending between the RAP and virgin binders in themixture. Phase angle master curves from the slope method, Hirsch model and lab measurementand the relationship between them are shown graphically in Figure 8. The lab-measured andpredicted phase angles with the slope method are in better agreement in terms of shape andmagnitude than those from the Hirsch model which is indicated by the lower RMSE (Figure8(c)). The Black Space diagram (Figure 8(b)) also supports the above observation. While theHirsch model has been found useful, it tends to follow a certain defined shape (as seen in thefigure) and does not show the same level of flexibility that is observed with the slope-basedmethod.
The mixture data from the LTPP sections are used to produce phase angle master curvesusing the slope method and the Hirsch model. Although the predictions cannot be verified
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0
10
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40
1.E-03 1.E-01 1.E+01 1.E+03
Pha
se A
ngle
(D
eg.)
Reduced Frequency (Hz)
(a) NC
Predicted SlopePredicted Hirsch
0
10
20
30
40
1.E-03 1.E-01 1.E+01 1.E+03
Pha
se A
ngle
(D
eg.)
Reduced Frequency (Hz)
(c) NJ
Predicted SlopePredicted Hirsch
0
10
20
30
40
1.E-03 1.E-01 1.E+01 1.E+03
Pha
se A
ngle
(D
eg.)
Reduced Frequency (Hz)
(b) VT
Predicted SlopePredicted Hirsch
Figure 9. Phase angle predicted using the slope method and the Hirsch model: (a) NC, (b) VT, (c) NJ.
against lab measurements, the presented plots show some of the possible differences that couldbe encountered due to the use of one method over the other. For the North Carolina mixture,(Figure 9(a)) the master curve peaks do not match and larger differences are observed in termsof magnitude. The Vermont mixture master curves from both methods are relatively similarin terms of shape and peak location but a larger difference is observed in the magnitude ofthe peak point (Figure 9(b)). The New Jersey mixture master curves follow distinctly differ-ent shapes and the peak points do not match (Figure 9(c)). It should be noted here that the Hirschmodel predictions here utilised binder |G*| values that were predicted from |E*| values. Onceagain due to lack of measured phase angle data, it is difficult to comment as to which method
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0.E+00
2.E+03
4.E+03
6.E+03
8.E+03
1.E+04
0 10 20 30 40Phase Angle (degrees)
(a) L-9.5mm 5.9AC 7AV
Measured FittedPredicted from |E*|
5 rad/s
0.005 rad/s
15ºC
500 rad/s
0.E+00
2.E+03
4.E+03
6.E+03
8.E+03
0 10 20 30 40
|E*|
, MPa
|E*|
, MPa
Phase Angle (degrees)
(b) L-9.5mm 6.3AC 9AV
Measured FittedPredicted with |E*|
5 rad/s
0.005 rad/s
15ºC
500 rad/s
Figure 10. Black Space points determined using predicted and measured phase angles: (a) L-9.5 mm5.9AC 7AV, (b) L-9.5 mm 6.3AC 9AV.
provides more accurate predictions and the observations made here are purely comparativein nature.
Impact on pavement performance evaluationMixture Black Space diagramThe mixture Black Space diagrams produced using measured and predicted phase angles areshown in Figure 10. These plots correspond to the best and worst matches between lab mea-surement and slope prediction among C-9.5 mm and L-9.5 mm mixtures. Since the |E*| valuesare the same, the difference due to the use of measured versus predicted phase angles is a shiftto the right or the left. For the best fit the measured Black Space points are shifted to the leftvery slightly. For the poorest fit, there is a larger difference between the points, with the largestdifference at 500 rad/s.
Pavement fatigue life evaluation, SVECD and LVECDThe damage characteristic curves (C versus S) and the fatigue failure criterion curves (GR versusNf ) are generated using measured and predicted phase angle values and are shown in Figures 11and 12, respectively. The L-9.5 mm 5.9AC 7AV and C-12.5 mm 6.1AC 7AV mixtures presentedhere represent the best and worst matches. The damage characteristic and failure criterion curvesare almost identical for the best match case. The worst match case shows slightly different dam-age characteristic curves and a shifted failure criterion curve. The values of the model parameters(a, b, r and s) show how similar the best curves are and the difference in the worst fit.
Figure 13 shows the predicted number of failure points in the pavement over 20 years of ser-vice for the L-9.5 mm 5.9AC 7AV (best match among phase angle values) and C-12.5 mm 6.1AC
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0
0.2
0.4
0.6
0.8
1
1.2
0.E+00 1.E+05 2.E+05
C (
PSe
udo
Stif
fnes
s)
S (Damage Parameter)
C-12.5mm 6.1AC 7AV
Measured Phase AnglePredicted Phase Angle
0
0.2
0.4
0.6
0.8
1
1.2
0.E+00 1.E+05 2.E+05
C (
PSe
udo
Stif
fnes
s)
S (Damage Parameter)
L-9.5mm 5.9AV 7AV
Measured Phase AnglePredicted Phase Angle
Figure 11. Damage characteristics curves (C versus S) with measured and predicted phase angles.
7AV (worst match among phase angle values) mixtures from the LVECD pavement evaluation.As expected, the pavement simulations for the mixture with the least difference between mea-sured and predicted values generated nearly identical curves. The mixture with the worst matchbetween measured and predicted phase angles shows a difference of approximately 25 failurepoints at the end of the analysis period (20 years); this translates to a decrease of approximately25% in the number of failure points with the predicted values. However, when this is translatedto the percentage of failed points in the overall structure, the difference is only 2% (7% using thepredicted values and 9% using the measured values). Since the LVECD model is not calibratedwith field performance data, it is difficult to translate the differences to actual field cracking andevaluate whether this magnitude of difference would change any decisions that would be madewith respect to using this mixture or design.
Generally, mixtures where the spring-loaded LVDT is used exhibited less variation amongmeasured and predicted values and, hence, with missing phase angle data, the phase angleobtained from the slope-based method can be used for linear viscoelastic characterisation ofmixes in the SVECD and LVECD models to obtain a comparable fatigue performance prediction.
Summary and conclusionIn this study, a fundamental relationship to determine the phase angle from the slope of thelog–log of the |E*| master curve is evaluated using three sets of independent mixtures. The
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1.E+00
1.E+01
1.E+02
1.E+03
5.E+03 5.E+04
GR
Nf
L-9.5mm 5.9AV 7AV
Measured Phase AnglePredicted Phase Angle
1.E+00
1.E+01
1.E+02
1.E+03
5.E+03 5.E+04
GR
Nf
C-12.5mm 6.1AC 7AV
Measured Phase AnglePredicted Phase Angle
Figure 12. Damage characteristics curves (GR versus Nf ) with measured and predicted phase angles.
validity of the method is investigated by comparing predicted values with lab measurements.Further comparison is performed between lab measurements, the slope method and Hirschmodel values. Statistical quantities are calculated to examine the accuracy of the phase angleestimations. Rheological parameters are generated using measured and predicted values andcompared. Finally, fatigue characterisation using the SVECD approach and pavement evaluationusing the LVECD model is conducted using measured and predicted phase angle values to assessthe implication of using one over the other in fatigue performance prediction. The followingconclusions are drawn based on the observations:
• Overall, the variability in |E*| measurement between replicates was low. Due to this thepredicted phase angle values from average |E*|, low and high ranges |E*| were very close.
• For specimens where the lab measurements were done using spring-loaded LVDTs, mea-sured phase angle values match very well with values predicted using the slope methodwith an average difference of less than 2°. However, a larger difference (up to an averagedifference of 5°) is observed for specimens that used loose core LVDTs for measurement.The highest differences are observed around the peak values where the phase measure-ments are also more variable due to high test temperatures or low test frequencies. Furtherstudy is needed to determine the exact attribution of measurement inaccuracy on theobserved differences due to the type of LVDT and other factors.
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0
10
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60
70
0 50 100 150 200 250Months
L-9.5mm 5.9AC 7AV
Measured Phase AnglePredicted Phase Angle
0
20
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60
80
100
120
0 50 100 150 200 250
Num
ber
of F
ailu
re P
oint
sN
umbe
r of
Fai
lure
Poi
nts
Months
C-12.5mm 6.1AC 7AV
Measured Phase AnglePredicted Phase Angle
Figure 13. LVECD pavement simulation with predicted and measured phase angles.
• Generally, the predicted phase angles from the slope method are consistently lower thanlab-measured values. Two hypotheses were presented as to why this might be observedfor the majority of mixtures. The first is that neglecting the plastic response that might bepresent at lower frequencies and higher temperatures may cause the measured phase angleto be higher than it actually is. The second reason could be attributed to the complexity inphase angle calibration. The calibration performed at zero-phase angle might cause biasduring lab measurement of phase angle at different temperatures and frequencies.
• The phase angle master curves constructed from lab measurements and predicted using theslope method follow a comparable shape and exhibit very similar inflection points.
• Though extensive study has not been done for comparing lab measurement, the Hirschmodel and the slope method, it is observed that measured and slope method values agreevery well in terms of master curve shape and magnitude as compared to the Hirsch model.The same interpretation applies for the Black Space diagram generated from the threemethods. The comparison between the Hirsch and slope methods using LTPP data showedthe possible differences that arise due to the use of one method over the other. At differentinstances the phase angle curves were different in magnitude, shape and peak point loca-tion. Generally, it is observed that the Hirsch model lacks flexibility in terms of shape dueto the underlying functional form of the model.
• The mixture Black Space points generated using both measured and predicted phase anglesfrom the slope method are comparable.
• For the two set of mixtures corresponding to best match among predicted and measuredphase angle values, the damage characteristics curves from SVECD and number of failurepoints from LVECD were similar, indicating the phase angle predicted from the slope
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method can be used when no phase angle data are present for the linear viscoelasticcharacterisation of mixtures without affecting the results.
The advancement in pavement performance mechanistic models calls for better accuracy inmaterial characterisation. The estimation of phase angle from measured stiffness using the slopemethod is viable due to the availability and growing reliability of stiffness measurements withadvancements in equipment as compared to phase angle measurements. With the availability ofsuch a simple and robust method, the phase angle can be computed from |E*| master curves formixtures for which phase angle measurements are missing. This also largely applies to LTPPdata. Subsequently, rheological indices which require phase angle can be generated from historicdata and can be calibrated with available field performance data. The study also gives insightinto some bias that exists during lab measurements.
Since the validation of the prediction of phase angle from the slope of the log–log curveis done by comparing with lab-measured values, any bias on lab measurement presents thesame bias on the validation. Future work is needed to identify the magnitude of bias. More-over, the binder grades evaluated in this study are limited to PG 58-28 and PG 64-28 bindersthat are extensively used in the Northeastern part of the United States. Future study is rec-ommended to evaluate the influence of binder grade on the prediction capability of theproposed method. Furthermore, future study should look at the potential effect of permanentstrain that could be encountered during complex modulus testing (acceptable up to 1500 μ
according to AASHTO T342). Finally, the validity of the method to polymer-modified andaged mixtures should be investigated by employing the method described in this study todetermine if the modification and aging alter the relationship between dynamic modulus andphase angle.
AcknowledgmentsThe authors would like to acknowledge New Hampshire and Rhode Island Departments of Transporta-tion for providing the materials for this study. Acknowledgement is also extended to Katie Haslett for herassistance with specimen preparation at the University of New Hampshire.
Disclosure statementNo potential conflict of interest was reported by the authors.
ORCIDEshan Dave http://orcid.org/0000-0001-9788-2246
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