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Construction of triaxial dynamic modulus master curve for asphalt mixtures
Construction and Building Materials 37 (2012) 21–26
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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Construction of triaxial dynamic modulus master curve for asphalt mixtures Yanqing Zhao ⇑, Jimin Tang, Hui Liu School of Transportation Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China
h i g h l i g h t s " The effect of confining pressure on dynamic modulus was investigated. " The effect of confining pressure on time–temperature shift factor can be ignored. " A model for vertical shift factor was proposed. " A model for triaxial dynamic modulus master curve was proposed.
a r t i c l e
i n f o
Article history: Received 14 February 2012 Received in revised form 19 June 2012 Accepted 22 June 2012 Available online 24 August 2012 Keywords: Master curve Triaxial dynamic modulus Confining pressure Vertical shift factor
a b s t r a c t The dynamic modulus is considered as the most important factor of asphalt concrete influencing the field performance of asphalt pavements. The newly developed Mechanistic-Empirical Pavement Design Guide (MEPDG) used the uniaxial dynamic modulus master curve to characterize the temperature and time dependent behavior of asphalt concrete. It is evident that the triaxial dynamic modulus can better characterize the in situ mechanical behavior of the material. This study proposed a model for the construction of triaxial dynamic modulus master curve. A key feature of the proposed model is that it employs the vertical shifting technique to take into account the effect of confining pressure on the dynamic modulus. The vertical shift factor is modeled as a function of the reduced frequency and confining pressure. The triaxial master curve model was evaluated using dynamic modulus test results of three asphalt mixtures obtained at various temperatures, loading frequencies and confining pressures. The predictive accuracy of the proposed model is about the same as that of uniaxial and isobaric master curves, indicating the proposed model is accurate and effective in constructing triaxial dynamic modulus master curves. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Pavement design is moving towards more mechanistic-based methodologies. The new Mechanistic-Empirical Pavement Design Guide (MEPDG), developed under the National Cooperative Highway Research Program (NCHRP) Project 1–37A, represents a major step in this direction [1]. In mechanistic-empirical (M-E) design procedures, characterization of mechanical properties of paving materials plays a vital role in the determination of pavement structure responses. In the MEPDG, the dynamic modulus master curve is used to characterize the temperature and frequency dependent behavior of asphalt concrete [1]. The master curve can be constructed for an arbitrarily selected reference temperature by horizontally shifting the test results obtained at various temperatures and loading frequencies according to the time–temperature superposition principle (TTSP). There exists a concern that the dynamic modulus required in the MEPDG is measured from uniaxial testing, while it is well ⇑ Corresponding author. Tel.: +86 15804269202; fax: +86 041184707761. E-mail address: [email protected] (Y. Zhao). 0950-0618/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2012.06.067
known that asphalt concrete behaves in multiaxial stress states in the field. It is apparent that the triaxial dynamic modulus can better characterize the in situ behavior of asphalt concrete. Also, the triaxial dynamic modulus is recommended in the NCHRP Project 9–19 as one of the simple performance tests for permanent deformation, showing its importance in the area of asphalt mixture design and evaluation [2]. Researchers have shown that the confining pressure affects the dynamic modulus of asphalt concrete significantly, especially at high temperatures [3–6]. Therefore, there is an immediate need to develop a model of triaxial dynamic modulus master curve that can characterize not only the temperature and time dependent behavior but also the pressure dependent behavior of asphalt concrete. Such a model can be used to construct a triaxial master curve for an arbitrarily selected reference temperature and reference confining pressure by shifting dynamic modulus test results obtained at various temperatures, frequencies and confining pressures to form a single smooth curve [7]. Although the triaxial master curve can provide a better characterization of the mechanical behavior of asphalt concrete, limited research has been carried out to this end. Pellinen and Witczak
Y. Zhao et al. / Construction and Building Materials 37 (2012) 21–26
2. Experimental program This study used three types of hot-mix asphalt (HMA) mixtures, a Stone Mastic Asphalt (SMA) with a nominal maximum aggregate size (NMAS) of 12.5 mm and two Superpave mixtures with 19 and 25 mm NMAS, respectively. The three mixtures were used in the construction of Huning Expressway in Jiangsu, China. For brevity, the three mixtures are referred to as SMA 12.5, Superpave 19 and Superpave 25, respectively. The SMA 12.5 and Superpave 19 mixtures used a SBS modified PG76–22 binder, while the Superpave 25 mixture used a neat PG64–22 binder. Detailed information on aggregate gradations, binder and aggregate properties, and mixture designs can be found elsewhere [10]. Triaxial dynamic modulus tests were performed on the three mixtures following the guidelines specified in AASHTO TP 62–07 [11]. All specimens were compacted using a Superpave gyratory compactor to dimensions of 178 mm in height and 150 mm in diameter. These specimens were then cored and cut to a diameter of 100 mm and a height of 150 mm for testing. The dynamic modulus tests were initially conducted for evaluating the performance of Huning Expressway using the MEPDG software. Therefore, the target air voids of the 100 mm by 150 mm specimens were chosen to be the as-built air voids which were 4.1, 4.1 and 7.0% for the SMA 12.5, Superpave 19 and Superpave 25 mixtures, respectively. Specimens with air voids that differed by more than 0.5% from the target air voids were rejected. Dynamic modulus tests were conduced in a stress-controlled compressive loading mode using a closed-loop servohydraulic testing system. Each specimen was tested at temperatures of 4, 15, 25, 40 and 55 °C and at confining pressures of 0, 100, 200 and 250 kPa. At each combination of temperature and confining pressure, dynamic moduli were measured at frequencies of 0.1, 0.5, 1, 5, 10, 20 and 25 Hz. The applied stress was adjusted for each testing condition through trial and error so that the strain response was kept within 50–80 le. Three replicates were tested for each mixture and the results were averaged. Two quantities, the dynamic modulus, |E⁄|, and phase angle, /, were obtained from the tests using the standard technique given in AASHTO TP 62–07 [11]. The test results measured at temperatures of 4 and 55 °C and at confining pressures of 0 and 250 kPa are presented in Table 1 for the SMA 12.5 mixture. Detailed information on the test procedure and test results can be found elsewhere [10].
3. Modeling of triaxial dynamic modulus master curve
curve model. The dynamic modulus test results for the SMA 12.5 mixture at low (4 °C), median (25 °C) and high (55 °C) temperatures are presented in Fig. 1a–c. It is seen that at 4 °C the confining pressure has almost no effect on the dynamic modulus. However, at 55 °C the effect of confining pressure is significant, and the dynamic modulus measured at a confining pressure of 250 kPa could be up to eight times larger than the unconfined value. At 25 °C the dynamic modulus increases with increasing confining pressure but at a lower rate than that observed at 55 °C. Similar observations have been reported by other researchers [3–6]. It is also shown in Fig. 1 that the effect of confining pressure on the dynamic modulus is dependent not only on the testing temperature but also on the loading frequency. The dependency on the
(a) Dynamic Modulus (MPa)
[8] proposed a triaxial master curve model which related the dynamic modulus to the reduced frequency through a sigmoidal function similar to the one used for uniaxial master curve in the MEPDG. In the model the minimum dynamic modulus was no longer a constant. Instead, it was modeled as a function of stress state using the k1k3 model proposed by Witczak and Uzan for unbound materials [9]. However, the overall effects of confining pressure on dynamic modulus may not be accurately characterized by only changing the minimum value of dynamic modulus. This study aims to propose a new model for the construction of triaxial dynamic modulus master curve.
20000
15000 25Hz 20Hz 10000
10Hz 5Hz 1Hz
5000
0.5Hz 0.1Hz 0
0
50
100
150
200
250
300
Confining Pressure (KPa)
(b) Dynamic Modulus (MPa)
22
8000 25Hz 6000
20Hz 10Hz 5Hz
4000
1Hz 0.5Hz 0.1Hz
2000
3.1. Effect of confining pressure on dynamic modulus 0
An investigation of the effect of confining pressure on the dynamic modulus is an essential step for developing a triaxial master
0
50
100
150
200
250
300
Confining Pressure (KPa)
(c)
1500
Confining pressure (kPa)
0
Temperature (°C)
Frequency(Hz)
|E⁄| (MPa)
/ (°)
|E⁄| (MPa)
250 / (°)
4 4 4 4 4 4 4 55 55 55 55 55 55 55
25 20 10 5 1 0.5 0.1 25 20 10 5 1 0.5 0.1
17,870 17,476 16,200 14,948 11,900 10,636 7747 631 590 433 317 152 137 101
9.0 9.4 10.5 11.7 14.8 16.3 20.2 32.0 31.2 29.3 28.9 27.1 24.9 20.7
18,202 17,780 16,476 15,172 12,136 10,818 7982 1359 1283 1122 1013 871 842 822
9.0 9.4 10.5 11.8 14.9 16.3 20.2 19.1 18.0 15.9 13.8 10.5 9.4 8.0
Dynamic Modulus (MPa)
Table 1 Test results of SMA 12.5 mixture.
1000 25Hz 20Hz 10Hz 500
5Hz 1Hz 0.5Hz 0.1Hz
0
0
50
100
150
200
250
300
Confining Pressure (KPa) Fig. 1. Variation of dynamic modulus with confining pressure for SMA 12.5 mixture. (a) 4 °C; (b) 25 °C; and (c) 55 °C.
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Y. Zhao et al. / Construction and Building Materials 37 (2012) 21–26
100000
Dynamic Modulus (MPa)
loading frequency is more evident at high temperatures. At 55 °C, the dynamic modulus increases by 8.1 times when the confining pressure increases from 0 kPa to 250 kPa at a loading frequency of 0.1 Hz, while it only increases by 2.1 times at 25 Hz. At 25 °C, the dynamic modulus increases by 58% and 5% for 0.1 and 25 Hz, respectively. Similar observations can also be found for the Superpave 19 and Superpave 25 mixtures. The dynamic modulus test results show that asphalt concrete exhibits stress hardening behavior, especially at high temperatures and low frequencies. This kind of nonlinearity may be related to the dilation properties of the material [4].
10000
4 15 25 40 55 Master Curve
1000
100 0.00001
3.2. Isobaric dynamic modulus master curve
0.001
0.1
10
1000
100000
Reduced Frequency (Hz) The uniaxial master curve model adopted in the MEPDG has the form of a sigmoidal function as shown below:
ða dÞ 1 þ ebcflogðfr Þg
ð1Þ
3
Log (time-temperature shift factor)
logðjE jÞ ¼ d þ
Fig. 2. Schematic representation of horizontal shift (SMA 12.5, 100 kpa).
where |E⁄| is the dynamic modulus; fr is the reduced frequency; a and d are logarithms of maximum and minimum values of |E⁄|, respectively; b and c are shape parameters; The reduced frequency is defined as follows:
fr ¼ f aT
or
log fr ¼ log f þ log aT
ð2Þ
where f is the loading frequency; aT is the time–temperature shift factor. The Williams–Landel–Ferry (WLF) and Arrhenius equations are the most often used models for aT of asphalt concrete [12,13]. In this study the WLF model was used as shown in the following equation
C 1 ðT T 0 Þ log aT ¼ C 2 þ ðT T 0 Þ
0KPa 100KPa
2
200KPa 1
250KPa Same shift factor
0 -1 -2 -3
0
10
20
30
40
50
60
Temperature ( )
ð3Þ
Fig. 3. Time–temperature shift factors for Isobaric master curves.
100000
Dynamic Modulus (MPa)
where C1 and C2 are model coefficients; T0 is the reference temperature. Isobaric master curves can be constructed by horizontally shifting dynamic modulus test results obtained at a given confining pressure and various temperatures and loading frequencies. The horizontal shift process is schematically shown in Fig. 2. The amount of horizontal shift is the time–temperature shift factor. The master curve, combined with the time–temperature shift factor, allows for the prediction of mechanical properties of asphalt concrete over a wide range of conditions. For the purposes of this study, it is necessary to investigate the effect of confining pressure on the time–temperature shift factor. If the effect is significant, the shift factor function, as shown in Eq. (3), needs to be modified to include the confining pressure as a variable. The shift factors were solved simultaneously with the sigmoidal function coefficients using non-linear optimization and the Solver function in the Excel spreadsheet. The shift factors obtained for various isobaric master curves are presented in Fig. 3 for the SMA 12.5 mixture. It is seen from the figure that the shift factor varies slightly with the confining pressure. By assuming that the shift factor is independent of the confining pressure, isobaric master curves for various confining pressures were also constructed using the same set of shift factors. The shift factors obtained are also presented in Fig. 3. The corresponding isobaric master curves are presented in Fig. 4 in which the label A1 means each isobaric master curve has its own set of shift factors, while A2 means all isobaric master curves use the same set of shift factors. It is seen from Fig. 4 that the two approaches produce almost the same master curves. A major function of the master curve is to predict the dynamic moduli of asphalt concrete at various conditions. The accuracy and effectiveness of the master curve can be evaluated by comparing the predicted and measured values. The dynamic moduli predicted from isobaric master curves obtained using different sets of shift
10000 0KPa,A1 0KPa,A2 100KPa,A1 100KPa,A2 200KPa,A1 200KPa,A2 250KPa,A1 250KPa,A2
1000
100
10 0.00001
0.001
0.1
10
1000
100000
Reduced Frequency (Hz) Fig. 4. Isobaric master curves.
factors and the same set of shift factors for various confining levels were compared to the measured values, and the prediction errors were normalized using the following equation:
NE ¼
jEmea Epre j 100% Emea
ð4Þ
where NE is the normalized error; Epre and Emea are predicted and measured dynamic moduli, respectively. The averaged normalized errors for the three mixtures are listed in Table 2. It can be seen that using the same set of shift factors for various confining levels has almost no adverse effect on the prediction accuracy. Therefore, the shift factor was assumed to be independent of the confining
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Y. Zhao et al. / Construction and Building Materials 37 (2012) 21–26
Table 2 Averaged normalized errors of predicted dynamic moduli (%). Confining pressure (kPa) 0
100
200
250
Use different shift factors SMA 12.5 4.82 Superpave 19 4.27 Superpave 25 7.25
5.01 3.36 4.43
4.43 3.13 3.21
4.61 2.92 2.71
Use the same shift factors SMA 12.5 4.90 Superpave 19 4.73 Superpave 25 7.48
5.08 3.59 4.52
4.45 3.15 3.24
4.66 2.93 2.76
Dynamic Modulus (MPa)
Mix type
100000
10000
4 4 4 4 55 55 55 55
1000
100
10 0.01
1
0.1
10
,0KPa ,100KPa ,200KPa ,250KPa ,0KPa ,100KPa ,200KPa ,250KPa
100
Frequency (Hz)
pressure and Eq. (3) was used to determine the time–temperature shift factor in the subsequent analyses.
Fig. 5. Dynamic modulus test results of SMA 12.5 mixture.
3.3. Vertical shift factor and triaxial dynamic modulus master curve A triaxial master curve is constructed for an arbitrarily selected reference temperature and reference confining pressure by shifting test results measured at other temperatures and confining pressures to form a smooth curve. Researchers have successfully constructed master curves for polymers, asphalt binders and other materials using the free volume theory to take into account the effects of temperature, loading time and pressure [7,14–17]. The general steps adopted in these studies include first horizontally shifting the test results using the TTSP to get the isobaric master curves and then horizontally shifting the isobaric master curves according to the time–pressure superposition principle (TPSP) to get the final triaxial master curve, or the two horizontal shifts can be completed in one step by using the time–temperature-pressure superposition principle (TTPSP). However, an examination of Fig. 4 indicates that it is very difficult to obtain a smooth triaxial master curve for asphalt concrete by only performing horizontally shifting. The vertical shifting technique has also been used by researchers for constructing master curves [18–22]. By applying the vertical shifting, a master curve has the following general form:
kw ¼ Fðfr Þ
ð5Þ
where k is the vertical shift factor; w is a material property, such as the relaxation modulus or creep compliance, etc. The quantity on the left side of Eq. (5) is called a ‘‘reduced variable’’ [19]. In this study w is the dynamic modulus and kw is the reduced dynamic modulus. A master curve can be obtained by plotting the reduced dynamic modulus against the reduce frequency on a log–log scale [18,19]. Different models were proposed for vertical shift factors of various materials. Dealy and Plazek [19] suggested that the vertical shift factor was related to material’s density based on the Bueche-Rouse theories of linear viscoelasticity of polymer solutions. Plazek and Chelko [20] estimated the vertical shift factor from the steady-state compliance for amorphous polymers. Schapery [21] used the vertical shift factor to describe the strain dependency of nonlinear viscoelastic materials. Masad et al. [22] characterized the nonlinear behavior of asphalt binders by vertically shifting the master curves at different stress levels. In this study, the vertical shift factor was used to model the effect of confining pressure on the dynamic modulus for the construction of triaxial master curve. In other words, the dynamic moduli measured at various confining pressures were shifted vertically to the reference pressure to produce a smooth master curve. The amount of shift is the vertical shift factor. To formulate an appropriate model for the vertical shift factor, it is necessary to identify the factors influencing the shift factor and
investigate how the shift factor varies with these influencing factors. The dynamic moduli of the SMA 12.5 mixture measured at 4 and 55 °C are presented in Fig. 5. It is seen from the figure that at 4 °C the shift factor is close to zero since the test results at various confining levels are almost the same, while at high temperatures test results measured at confining pressures higher than the reference pressure need to be shifted downward and results at lower confining pressures need to be shifted upward. The amount of shift is dependent not only on the difference between the testing pressure and the reference pressure but also on the testing pressure itself. For example, for a reference pressure of 100 kPa the amount of vertical shift required for 0 kPa is significantly larger than that for 200 kPa at 55 °C. It is also shown in Fig. 5 that besides the temperature and confining pressure, the loading frequency significantly affects the vertical shift factor. For example, in order to shift the data measured at 0 kPa to a reference pressure of 100 kPa at 55 °C, the amount of vertical shift required at 0.1 Hz is much larger than that at 25 Hz. The above observations lead to a conclusion that the amount of vertical shift required for a condition at which asphalt concrete is soft (high temperature, low frequency and low confining pressure) is larger than that for a condition at which the material is stiff (low temperature, high frequency and high confining pressure). Since it is well known that the effects of temperature and loading frequency can be characterized using a single variable, the reduced frequency as shown in Eq. (2), the vertical shift factor can be modeled as a function of the reduced frequency and confining pressure. Based on the above analyses, a vertical shift factor model was proposed in this study as follows:
logðkÞ ¼
ðP P0 Þ eC3 þC 4 logðfr Þ
þ C 5 ðP þ Pa ÞC 6
ð6Þ
where P is the confining pressure; P0 is the reference pressure; Pa is the atmospheric pressure and Pa = 101.3 kPa; C3–C6 are regression coefficients. At the reference pressure, the vertical shift factor is equal to 1 and the triaxial master curve is reduced to an isobaric master curve as shown in Eq. (1). Therefore, the sigmoidal function shown in Eq. (1) was used in this study to relate the reduced dynamic modulus to the reduced frequency. By putting together all the components discussed above, a model was proposed for the construction of triaxial dynamic modulus master curve as follows:
logðjE jr Þ ¼ d þ
ða dÞ 1 þ ebcflogðfr Þg
log fr ¼ log f þ log aT ¼ log f þ
ð7aÞ C 1 ðT T 0 Þ C 2 þ ðT T 0 Þ
ð7bÞ
25
Y. Zhao et al. / Construction and Building Materials 37 (2012) 21–26 Table 3 Coefficients for triaxial dynamic modulus master curves. Mix type
a
b
c
d
C1
C2
C3
C4
C5
C6
SMA12.5 Superpave 19 Superpave 25
4.359 4.369 4.356
0.0044 0.0051 0.0050
0.687 0.747 0.704
2.379 2.640 2.455
16.772 11.053 8.268
150.998 104.097 82.171
1.214 1.341 1.280
1.062 1.292 0.415
0.845 1.849 0.990
0.761 0.994 1.050
ð7cÞ
eC3 þC4 logðfr Þ þ C 5 ðP þ Pa ÞC 6
where |E⁄|r is the reduced dynamic modulus. 4. Analytical results and discussion
Reduced Dynamic Modulus (MPa)
100000
10000
0KPa 100KPa 200KPa 250KPa Master Curve
1000
100
10 0.00001
0.001
0.1
10
1000
100000
Reduced Frequency (Hz) Fig. 6. Triaxial dynamic modulus master curve of SMA 12.5 mixture.
100000
10000
1000 0KPa, 55 250KPa, 4 Master Curve
100
10 0.000001 0.0001
0.01
1
100
10000 1000000
Reduced Frequency (Hz) Fig. 7. Schematic illustration of horizontal and vertical shifting.
0.6 0.5
Log (Vertical Shift Factor)
The triaxial master curve model shown in Eq. (7) has 4 sigmoidal function coefficients, 2 WLF equation coefficients and 4 vertical shift factor coefficients. Christensen et al. [23] and Bonaquist [24] reported that the uniaxial maximum dynamic modulus of asphalt concrete can be predicted reasonably well using the Hirsch model. As discussed earlier the effect of confining pressure on the dynamic modulus can be ignored at low temperatures and high frequencies indicating that the maximum dynamic modulus does not vary with the confining pressure. Thus, the Hirsch model was used to estimate the maximum dynamic modulus of triaxial master curve. There are a total of nine unknown model coefficients in Eq. (7) which can be solved simultaneously by performing non-linear least squares regression. The model coefficients determined at a 25 °C reference temperature and a 100 kPa reference pressure for various mixtures are listed in Table 3. The confining pressure values used in the analyses were in a unit of MPa. The triaxial master curve of the SMA 12.5 mixture is shown in Fig. 6. It is seen from Fig. 6 that a smooth master curve can be constructed using dynamic modulus test results measured at various temperatures, loading frequencies and pressures by performing both horizontal and vertical shifting. A schematic illustration of the horizontal and vertical shifting for the aforementioned reference condition is presented in Fig. 7. It is seen that at 4 °C and 250 kPa almost no vertical shifting is required, while at 55 °C and 0 kPa the amount of vertical shifting is significant. The vertical shift factor was modeled in this study as a function of the reduced frequency and confining pressure. A three-dimensional surface plot is presented in Fig. 8 showing how the vertical shift factor varies with the reduced frequency and confining pressure for the SMA 12.5 mixture. The accuracy and effectiveness of the proposed triaxial master curve model were evaluated by comparing the predicted and measured dynamic moduli. The reduced dynamic moduli were first
0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 5 4 3 2 1 0 Log -1 -2 (Red -3 -4 uced -5 F
requ
ency
0
)
50
150
100
200
ressure
gP Confinin
100000
10000
SMA 12.5 Superpave 19 Superpave 25
1000
100
10 10
100
1000
10000
250
(KPa)
Fig. 8. Three-dimensional surface plot of vertical shift factor.
Predicted Dynamic Modulus (MPa)
¼ logðjE jÞ þ
ðP P0 Þ
Reduced Dynamic Modulus (MPa)
logðjE jr Þ ¼ logðjE jÞ þ logðkÞ
100000
Measured Dynamic Modulus (MPa) Fig. 9. Comparison of predicted and measured dynamic moduli.
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Y. Zhao et al. / Construction and Building Materials 37 (2012) 21–26
computed using Eq. (7a) and dynamic moduli were then predicted using Eq. (7c). The dynamic moduli measured at various testing conditions for the three mixtures are plotted against the predicted values in Fig. 9. A very good agreement is observed between the measured and predicted values. The prediction errors were evaluated using Eq. (4) and the averaged normalized errors are 4.96%, 4.34% and 4.74% for the SMA 12.5, Superpave 19 and Superpave 25 mixtures, respectively. A comparison of these normalized errors to the values listed in Table 2 shows that the predictive accuracy of the triaxial master curve is about the same as that of the uniaxial or isobaric master curve, indicating the proposed model is accurate and effective in constructing triaxial dynamic modulus master curves. 5. Conclusions A model for the construction of triaxial dynamic modulus master curve was proposed in this study and the accuracy and effectiveness of the model were evaluated using the dynamic modulus test results of three asphalt mixtures measured at various temperatures, loading frequencies and confining pressures. A key feature of the proposed model is that it employs vertical shifting to take into account the effect of confining pressure on the dynamic modulus. Based on the analyses and results presented, the following observations and conclusions can be made: 1. At low temperatures the confining pressure has little effect on the dynamic modulus, while at high temperatures the effect of confining pressure is significant. The effect of confining pressure is also dependent on the loading frequency, especially at high temperatures. 2. The effect of confining pressure on the time–temperature shift factor can be ignored for the construction of triaxial dynamic modulus master curve. 3. The amount of vertical shift required for the construction of triaxial master curves is dependent on temperature, loading frequency and pressure. A larger amount of vertical shift is needed for a condition at which asphalt concrete is soft (high temperature, low loading frequency and low confining pressure) compared to a condition at which the material is stiff. The vertical shift factor can be modeled as a function of the reduced frequency and confining pressure. 4. The triaxial master curve model uses a sigmoidal function to relate the reduced dynamic modulus to the reduced frequency by performing both horizontal and vertical shifting on dynamic modulus test results. The predictive accuracy of the proposed model is about the same as that of uniaxial and isobaric master curves, indicating the proposed model is accurate and effective in constructing triaxial dynamic modulus master curves.
Acknowledgments This research was sponsored by the China Western Transportation Research Program (#2009318773054) and Specialized
Research Fund for the Doctoral Program of Higher Education (#20100041120005). The supports are gratefully acknowledged. References [1] Applied research associates (ARA). Guide for mechanistic-empirical design of new and rehabilitated pavement structures. Final report, national cooperative highway research program (NCHRP) Project 1–37A, New Mexico: Albuquerque; 2004. [2] Witczak MW, Kaloush K, Pellinen T, El-Basyouny M, Von Quintus H. Simple performance test for superpave mix design, NCHRP report 465, transportation research board, National Research Council, Washington, DC; 2002. [3] Zeiada W, Kaloush K, Biligiri K, Reed J, Stempihar J. Significance of confined dynamic modulus laboratory testing for asphalt concrete: conventional, gapgraded, and open-graded mixtures, Transportation research record no. 2210, National Research Council, Washington, DC; 2011. pp. 9–19. [4] Kim Y, Guddati M, Underwood B, Yun T, Savadatti V. Development of a multiaxial viscoelastoplastic continuum damage model for asphalt mixtures, FHWA publication no. DTFH61-05-H-00019; Federal Highway Administration, McLean, VA; 2009. [5] Underwood B, Yun T, Kim Y. Experimental investigations of the viscoelastic and damage behaviors of hot-mix asphalt in compression. J Mater Civil Eng ASCE 2011;23(4):459–66. [6] Zhu H, Sun L, Yang J, Chen Z, Gu W. Developing master curves and predicting dynamic modulus of polymer-modified asphalt mixtures. J Mater Civil Eng ASCE 2011;23(2):131–7. [7] Ferry JD. Viscoelastic properties of polymers. 3rd ed. New York: John Willey & Sons; 1980. [8] Pellinen TK, Witczak MW. Stress dependent master curve construction for dynamic (complex) modulus. J Assoc Asphalt Paving Technol 2002;71:281–309. [9] Witczak MW, Uzan J. The universal airport pavement design system – report i of iv: granular material characterization. Department of Civil Engineering, University of Maryland at College Park, MD; 1988. [10] Zhao Y, Bai Q. Performance evaluation of pavement structures with asphalt treated bases used in huning expressway, Jiangsu DOT report no. 04Y50; Nanjing, China; 2007. [11] American association of state highway and transportation officials (AASHTO). Standard method of test for determining dynamic modulus of hot-mix asphalt (HMA). AASHTO TP 62–07, Washington, DC; 2007. [12] Pellinen T, Zofka A, Marasteanu M, Funk N. Asphalt mixture stiffness predictive models. J Assoc Asphalt Paving Technol 2007;76:p575–625. [13] Anderson D, Christensen D, Bahia H. Physical properties of asphalt cement and the development of performance related specifications. J Assoc Asphalt Paving Technol 1991;60:437–75. [14] Bhuvanesh Y, Gupta V. Long-term prediction of creep in textile fibers. Polymer 1994;35(10):2226–8. [15] Jazouli S, Luo W, Bremand F, Vu-Khanh T. Nonlinear creep behavior of viscoelastic polycarbonate. J Mater Sci 2006;41(2):531–6. [16] Martin-Alfonso M, Martinez-Boza F, Navarro F, Fernandez M, Gallegos C. Pressure–temperature–viscosity relationship for heavy petroleum fractions. Fuel 2007;86:227–33. [17] Dastoorian F, Tajvidi M, Ebrahimi G. Evaluation of time dependent behavior of a wood flour/high density polyethylene composite. J Reinf Plast Comp 2010;29(1):132–43. [18] Van Gurp M, Palmen J. Time–temperature superposition for polymeric blends. Rheol Bull 1998;67:5–8. [19] Dealy J, Plazek D. Time–temperature superposition – a users guide. Rheol Bull 2009;78:16–31. [20] Plazek D, Chelko AJ. Temperature dependence of the steady state recoverable compliance of amorphous polymers 1977;18(1):15–8. [21] Schapery R. On the characterization of non-linear viscoelastic materials. Polymer Eng Sci 1969;9(4):295–310. [22] Masad E, Huang C, Airey G, Muliana A. Nonlinear viscoelastic analysis of unaged and aged asphalt binders. Constr Build Mater 2008;22(11):p2170–2179. [23] Christensen D, Pellinen T, Bonaquist R. Hirsch model for estimating the modulus of asphalt concrete. J Assoc Asphalt Paving Technol 2003;72:97–121. [24] Bonaquist R. Refining the simple performance tester for use in routine practice, NCHRP report 614, Transportation Research Board, Washington, DC; 2008.
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Construction of triaxial dynamic modulus master curve for asphalt mixtures Yanqing Zhao ⇑, Jimin Tang, Hui Liu School of Transportation Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China
h i g h l i g h t s " The effect of confining pressure on dynamic modulus was investigated. " The effect of confining pressure on time–temperature shift factor can be ignored. " A model for vertical shift factor was proposed. " A model for triaxial dynamic modulus master curve was proposed.
a r t i c l e
i n f o
Article history: Received 14 February 2012 Received in revised form 19 June 2012 Accepted 22 June 2012 Available online 24 August 2012 Keywords: Master curve Triaxial dynamic modulus Confining pressure Vertical shift factor
a b s t r a c t The dynamic modulus is considered as the most important factor of asphalt concrete influencing the field performance of asphalt pavements. The newly developed Mechanistic-Empirical Pavement Design Guide (MEPDG) used the uniaxial dynamic modulus master curve to characterize the temperature and time dependent behavior of asphalt concrete. It is evident that the triaxial dynamic modulus can better characterize the in situ mechanical behavior of the material. This study proposed a model for the construction of triaxial dynamic modulus master curve. A key feature of the proposed model is that it employs the vertical shifting technique to take into account the effect of confining pressure on the dynamic modulus. The vertical shift factor is modeled as a function of the reduced frequency and confining pressure. The triaxial master curve model was evaluated using dynamic modulus test results of three asphalt mixtures obtained at various temperatures, loading frequencies and confining pressures. The predictive accuracy of the proposed model is about the same as that of uniaxial and isobaric master curves, indicating the proposed model is accurate and effective in constructing triaxial dynamic modulus master curves. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Pavement design is moving towards more mechanistic-based methodologies. The new Mechanistic-Empirical Pavement Design Guide (MEPDG), developed under the National Cooperative Highway Research Program (NCHRP) Project 1–37A, represents a major step in this direction [1]. In mechanistic-empirical (M-E) design procedures, characterization of mechanical properties of paving materials plays a vital role in the determination of pavement structure responses. In the MEPDG, the dynamic modulus master curve is used to characterize the temperature and frequency dependent behavior of asphalt concrete [1]. The master curve can be constructed for an arbitrarily selected reference temperature by horizontally shifting the test results obtained at various temperatures and loading frequencies according to the time–temperature superposition principle (TTSP). There exists a concern that the dynamic modulus required in the MEPDG is measured from uniaxial testing, while it is well ⇑ Corresponding author. Tel.: +86 15804269202; fax: +86 041184707761. E-mail address: [email protected] (Y. Zhao). 0950-0618/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2012.06.067
known that asphalt concrete behaves in multiaxial stress states in the field. It is apparent that the triaxial dynamic modulus can better characterize the in situ behavior of asphalt concrete. Also, the triaxial dynamic modulus is recommended in the NCHRP Project 9–19 as one of the simple performance tests for permanent deformation, showing its importance in the area of asphalt mixture design and evaluation [2]. Researchers have shown that the confining pressure affects the dynamic modulus of asphalt concrete significantly, especially at high temperatures [3–6]. Therefore, there is an immediate need to develop a model of triaxial dynamic modulus master curve that can characterize not only the temperature and time dependent behavior but also the pressure dependent behavior of asphalt concrete. Such a model can be used to construct a triaxial master curve for an arbitrarily selected reference temperature and reference confining pressure by shifting dynamic modulus test results obtained at various temperatures, frequencies and confining pressures to form a single smooth curve [7]. Although the triaxial master curve can provide a better characterization of the mechanical behavior of asphalt concrete, limited research has been carried out to this end. Pellinen and Witczak
Y. Zhao et al. / Construction and Building Materials 37 (2012) 21–26
2. Experimental program This study used three types of hot-mix asphalt (HMA) mixtures, a Stone Mastic Asphalt (SMA) with a nominal maximum aggregate size (NMAS) of 12.5 mm and two Superpave mixtures with 19 and 25 mm NMAS, respectively. The three mixtures were used in the construction of Huning Expressway in Jiangsu, China. For brevity, the three mixtures are referred to as SMA 12.5, Superpave 19 and Superpave 25, respectively. The SMA 12.5 and Superpave 19 mixtures used a SBS modified PG76–22 binder, while the Superpave 25 mixture used a neat PG64–22 binder. Detailed information on aggregate gradations, binder and aggregate properties, and mixture designs can be found elsewhere [10]. Triaxial dynamic modulus tests were performed on the three mixtures following the guidelines specified in AASHTO TP 62–07 [11]. All specimens were compacted using a Superpave gyratory compactor to dimensions of 178 mm in height and 150 mm in diameter. These specimens were then cored and cut to a diameter of 100 mm and a height of 150 mm for testing. The dynamic modulus tests were initially conducted for evaluating the performance of Huning Expressway using the MEPDG software. Therefore, the target air voids of the 100 mm by 150 mm specimens were chosen to be the as-built air voids which were 4.1, 4.1 and 7.0% for the SMA 12.5, Superpave 19 and Superpave 25 mixtures, respectively. Specimens with air voids that differed by more than 0.5% from the target air voids were rejected. Dynamic modulus tests were conduced in a stress-controlled compressive loading mode using a closed-loop servohydraulic testing system. Each specimen was tested at temperatures of 4, 15, 25, 40 and 55 °C and at confining pressures of 0, 100, 200 and 250 kPa. At each combination of temperature and confining pressure, dynamic moduli were measured at frequencies of 0.1, 0.5, 1, 5, 10, 20 and 25 Hz. The applied stress was adjusted for each testing condition through trial and error so that the strain response was kept within 50–80 le. Three replicates were tested for each mixture and the results were averaged. Two quantities, the dynamic modulus, |E⁄|, and phase angle, /, were obtained from the tests using the standard technique given in AASHTO TP 62–07 [11]. The test results measured at temperatures of 4 and 55 °C and at confining pressures of 0 and 250 kPa are presented in Table 1 for the SMA 12.5 mixture. Detailed information on the test procedure and test results can be found elsewhere [10].
3. Modeling of triaxial dynamic modulus master curve
curve model. The dynamic modulus test results for the SMA 12.5 mixture at low (4 °C), median (25 °C) and high (55 °C) temperatures are presented in Fig. 1a–c. It is seen that at 4 °C the confining pressure has almost no effect on the dynamic modulus. However, at 55 °C the effect of confining pressure is significant, and the dynamic modulus measured at a confining pressure of 250 kPa could be up to eight times larger than the unconfined value. At 25 °C the dynamic modulus increases with increasing confining pressure but at a lower rate than that observed at 55 °C. Similar observations have been reported by other researchers [3–6]. It is also shown in Fig. 1 that the effect of confining pressure on the dynamic modulus is dependent not only on the testing temperature but also on the loading frequency. The dependency on the
(a) Dynamic Modulus (MPa)
[8] proposed a triaxial master curve model which related the dynamic modulus to the reduced frequency through a sigmoidal function similar to the one used for uniaxial master curve in the MEPDG. In the model the minimum dynamic modulus was no longer a constant. Instead, it was modeled as a function of stress state using the k1k3 model proposed by Witczak and Uzan for unbound materials [9]. However, the overall effects of confining pressure on dynamic modulus may not be accurately characterized by only changing the minimum value of dynamic modulus. This study aims to propose a new model for the construction of triaxial dynamic modulus master curve.
20000
15000 25Hz 20Hz 10000
10Hz 5Hz 1Hz
5000
0.5Hz 0.1Hz 0
0
50
100
150
200
250
300
Confining Pressure (KPa)
(b) Dynamic Modulus (MPa)
22
8000 25Hz 6000
20Hz 10Hz 5Hz
4000
1Hz 0.5Hz 0.1Hz
2000
3.1. Effect of confining pressure on dynamic modulus 0
An investigation of the effect of confining pressure on the dynamic modulus is an essential step for developing a triaxial master
0
50
100
150
200
250
300
Confining Pressure (KPa)
(c)
1500
Confining pressure (kPa)
0
Temperature (°C)
Frequency(Hz)
|E⁄| (MPa)
/ (°)
|E⁄| (MPa)
250 / (°)
4 4 4 4 4 4 4 55 55 55 55 55 55 55
25 20 10 5 1 0.5 0.1 25 20 10 5 1 0.5 0.1
17,870 17,476 16,200 14,948 11,900 10,636 7747 631 590 433 317 152 137 101
9.0 9.4 10.5 11.7 14.8 16.3 20.2 32.0 31.2 29.3 28.9 27.1 24.9 20.7
18,202 17,780 16,476 15,172 12,136 10,818 7982 1359 1283 1122 1013 871 842 822
9.0 9.4 10.5 11.8 14.9 16.3 20.2 19.1 18.0 15.9 13.8 10.5 9.4 8.0
Dynamic Modulus (MPa)
Table 1 Test results of SMA 12.5 mixture.
1000 25Hz 20Hz 10Hz 500
5Hz 1Hz 0.5Hz 0.1Hz
0
0
50
100
150
200
250
300
Confining Pressure (KPa) Fig. 1. Variation of dynamic modulus with confining pressure for SMA 12.5 mixture. (a) 4 °C; (b) 25 °C; and (c) 55 °C.
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Y. Zhao et al. / Construction and Building Materials 37 (2012) 21–26
100000
Dynamic Modulus (MPa)
loading frequency is more evident at high temperatures. At 55 °C, the dynamic modulus increases by 8.1 times when the confining pressure increases from 0 kPa to 250 kPa at a loading frequency of 0.1 Hz, while it only increases by 2.1 times at 25 Hz. At 25 °C, the dynamic modulus increases by 58% and 5% for 0.1 and 25 Hz, respectively. Similar observations can also be found for the Superpave 19 and Superpave 25 mixtures. The dynamic modulus test results show that asphalt concrete exhibits stress hardening behavior, especially at high temperatures and low frequencies. This kind of nonlinearity may be related to the dilation properties of the material [4].
10000
4 15 25 40 55 Master Curve
1000
100 0.00001
3.2. Isobaric dynamic modulus master curve
0.001
0.1
10
1000
100000
Reduced Frequency (Hz) The uniaxial master curve model adopted in the MEPDG has the form of a sigmoidal function as shown below:
ða dÞ 1 þ ebcflogðfr Þg
ð1Þ
3
Log (time-temperature shift factor)
logðjE jÞ ¼ d þ
Fig. 2. Schematic representation of horizontal shift (SMA 12.5, 100 kpa).
where |E⁄| is the dynamic modulus; fr is the reduced frequency; a and d are logarithms of maximum and minimum values of |E⁄|, respectively; b and c are shape parameters; The reduced frequency is defined as follows:
fr ¼ f aT
or
log fr ¼ log f þ log aT
ð2Þ
where f is the loading frequency; aT is the time–temperature shift factor. The Williams–Landel–Ferry (WLF) and Arrhenius equations are the most often used models for aT of asphalt concrete [12,13]. In this study the WLF model was used as shown in the following equation
C 1 ðT T 0 Þ log aT ¼ C 2 þ ðT T 0 Þ
0KPa 100KPa
2
200KPa 1
250KPa Same shift factor
0 -1 -2 -3
0
10
20
30
40
50
60
Temperature ( )
ð3Þ
Fig. 3. Time–temperature shift factors for Isobaric master curves.
100000
Dynamic Modulus (MPa)
where C1 and C2 are model coefficients; T0 is the reference temperature. Isobaric master curves can be constructed by horizontally shifting dynamic modulus test results obtained at a given confining pressure and various temperatures and loading frequencies. The horizontal shift process is schematically shown in Fig. 2. The amount of horizontal shift is the time–temperature shift factor. The master curve, combined with the time–temperature shift factor, allows for the prediction of mechanical properties of asphalt concrete over a wide range of conditions. For the purposes of this study, it is necessary to investigate the effect of confining pressure on the time–temperature shift factor. If the effect is significant, the shift factor function, as shown in Eq. (3), needs to be modified to include the confining pressure as a variable. The shift factors were solved simultaneously with the sigmoidal function coefficients using non-linear optimization and the Solver function in the Excel spreadsheet. The shift factors obtained for various isobaric master curves are presented in Fig. 3 for the SMA 12.5 mixture. It is seen from the figure that the shift factor varies slightly with the confining pressure. By assuming that the shift factor is independent of the confining pressure, isobaric master curves for various confining pressures were also constructed using the same set of shift factors. The shift factors obtained are also presented in Fig. 3. The corresponding isobaric master curves are presented in Fig. 4 in which the label A1 means each isobaric master curve has its own set of shift factors, while A2 means all isobaric master curves use the same set of shift factors. It is seen from Fig. 4 that the two approaches produce almost the same master curves. A major function of the master curve is to predict the dynamic moduli of asphalt concrete at various conditions. The accuracy and effectiveness of the master curve can be evaluated by comparing the predicted and measured values. The dynamic moduli predicted from isobaric master curves obtained using different sets of shift
10000 0KPa,A1 0KPa,A2 100KPa,A1 100KPa,A2 200KPa,A1 200KPa,A2 250KPa,A1 250KPa,A2
1000
100
10 0.00001
0.001
0.1
10
1000
100000
Reduced Frequency (Hz) Fig. 4. Isobaric master curves.
factors and the same set of shift factors for various confining levels were compared to the measured values, and the prediction errors were normalized using the following equation:
NE ¼
jEmea Epre j 100% Emea
ð4Þ
where NE is the normalized error; Epre and Emea are predicted and measured dynamic moduli, respectively. The averaged normalized errors for the three mixtures are listed in Table 2. It can be seen that using the same set of shift factors for various confining levels has almost no adverse effect on the prediction accuracy. Therefore, the shift factor was assumed to be independent of the confining
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Y. Zhao et al. / Construction and Building Materials 37 (2012) 21–26
Table 2 Averaged normalized errors of predicted dynamic moduli (%). Confining pressure (kPa) 0
100
200
250
Use different shift factors SMA 12.5 4.82 Superpave 19 4.27 Superpave 25 7.25
5.01 3.36 4.43
4.43 3.13 3.21
4.61 2.92 2.71
Use the same shift factors SMA 12.5 4.90 Superpave 19 4.73 Superpave 25 7.48
5.08 3.59 4.52
4.45 3.15 3.24
4.66 2.93 2.76
Dynamic Modulus (MPa)
Mix type
100000
10000
4 4 4 4 55 55 55 55
1000
100
10 0.01
1
0.1
10
,0KPa ,100KPa ,200KPa ,250KPa ,0KPa ,100KPa ,200KPa ,250KPa
100
Frequency (Hz)
pressure and Eq. (3) was used to determine the time–temperature shift factor in the subsequent analyses.
Fig. 5. Dynamic modulus test results of SMA 12.5 mixture.
3.3. Vertical shift factor and triaxial dynamic modulus master curve A triaxial master curve is constructed for an arbitrarily selected reference temperature and reference confining pressure by shifting test results measured at other temperatures and confining pressures to form a smooth curve. Researchers have successfully constructed master curves for polymers, asphalt binders and other materials using the free volume theory to take into account the effects of temperature, loading time and pressure [7,14–17]. The general steps adopted in these studies include first horizontally shifting the test results using the TTSP to get the isobaric master curves and then horizontally shifting the isobaric master curves according to the time–pressure superposition principle (TPSP) to get the final triaxial master curve, or the two horizontal shifts can be completed in one step by using the time–temperature-pressure superposition principle (TTPSP). However, an examination of Fig. 4 indicates that it is very difficult to obtain a smooth triaxial master curve for asphalt concrete by only performing horizontally shifting. The vertical shifting technique has also been used by researchers for constructing master curves [18–22]. By applying the vertical shifting, a master curve has the following general form:
kw ¼ Fðfr Þ
ð5Þ
where k is the vertical shift factor; w is a material property, such as the relaxation modulus or creep compliance, etc. The quantity on the left side of Eq. (5) is called a ‘‘reduced variable’’ [19]. In this study w is the dynamic modulus and kw is the reduced dynamic modulus. A master curve can be obtained by plotting the reduced dynamic modulus against the reduce frequency on a log–log scale [18,19]. Different models were proposed for vertical shift factors of various materials. Dealy and Plazek [19] suggested that the vertical shift factor was related to material’s density based on the Bueche-Rouse theories of linear viscoelasticity of polymer solutions. Plazek and Chelko [20] estimated the vertical shift factor from the steady-state compliance for amorphous polymers. Schapery [21] used the vertical shift factor to describe the strain dependency of nonlinear viscoelastic materials. Masad et al. [22] characterized the nonlinear behavior of asphalt binders by vertically shifting the master curves at different stress levels. In this study, the vertical shift factor was used to model the effect of confining pressure on the dynamic modulus for the construction of triaxial master curve. In other words, the dynamic moduli measured at various confining pressures were shifted vertically to the reference pressure to produce a smooth master curve. The amount of shift is the vertical shift factor. To formulate an appropriate model for the vertical shift factor, it is necessary to identify the factors influencing the shift factor and
investigate how the shift factor varies with these influencing factors. The dynamic moduli of the SMA 12.5 mixture measured at 4 and 55 °C are presented in Fig. 5. It is seen from the figure that at 4 °C the shift factor is close to zero since the test results at various confining levels are almost the same, while at high temperatures test results measured at confining pressures higher than the reference pressure need to be shifted downward and results at lower confining pressures need to be shifted upward. The amount of shift is dependent not only on the difference between the testing pressure and the reference pressure but also on the testing pressure itself. For example, for a reference pressure of 100 kPa the amount of vertical shift required for 0 kPa is significantly larger than that for 200 kPa at 55 °C. It is also shown in Fig. 5 that besides the temperature and confining pressure, the loading frequency significantly affects the vertical shift factor. For example, in order to shift the data measured at 0 kPa to a reference pressure of 100 kPa at 55 °C, the amount of vertical shift required at 0.1 Hz is much larger than that at 25 Hz. The above observations lead to a conclusion that the amount of vertical shift required for a condition at which asphalt concrete is soft (high temperature, low frequency and low confining pressure) is larger than that for a condition at which the material is stiff (low temperature, high frequency and high confining pressure). Since it is well known that the effects of temperature and loading frequency can be characterized using a single variable, the reduced frequency as shown in Eq. (2), the vertical shift factor can be modeled as a function of the reduced frequency and confining pressure. Based on the above analyses, a vertical shift factor model was proposed in this study as follows:
logðkÞ ¼
ðP P0 Þ eC3 þC 4 logðfr Þ
þ C 5 ðP þ Pa ÞC 6
ð6Þ
where P is the confining pressure; P0 is the reference pressure; Pa is the atmospheric pressure and Pa = 101.3 kPa; C3–C6 are regression coefficients. At the reference pressure, the vertical shift factor is equal to 1 and the triaxial master curve is reduced to an isobaric master curve as shown in Eq. (1). Therefore, the sigmoidal function shown in Eq. (1) was used in this study to relate the reduced dynamic modulus to the reduced frequency. By putting together all the components discussed above, a model was proposed for the construction of triaxial dynamic modulus master curve as follows:
logðjE jr Þ ¼ d þ
ða dÞ 1 þ ebcflogðfr Þg
log fr ¼ log f þ log aT ¼ log f þ
ð7aÞ C 1 ðT T 0 Þ C 2 þ ðT T 0 Þ
ð7bÞ
25
Y. Zhao et al. / Construction and Building Materials 37 (2012) 21–26 Table 3 Coefficients for triaxial dynamic modulus master curves. Mix type
a
b
c
d
C1
C2
C3
C4
C5
C6
SMA12.5 Superpave 19 Superpave 25
4.359 4.369 4.356
0.0044 0.0051 0.0050
0.687 0.747 0.704
2.379 2.640 2.455
16.772 11.053 8.268
150.998 104.097 82.171
1.214 1.341 1.280
1.062 1.292 0.415
0.845 1.849 0.990
0.761 0.994 1.050
ð7cÞ
eC3 þC4 logðfr Þ þ C 5 ðP þ Pa ÞC 6
where |E⁄|r is the reduced dynamic modulus. 4. Analytical results and discussion
Reduced Dynamic Modulus (MPa)
100000
10000
0KPa 100KPa 200KPa 250KPa Master Curve
1000
100
10 0.00001
0.001
0.1
10
1000
100000
Reduced Frequency (Hz) Fig. 6. Triaxial dynamic modulus master curve of SMA 12.5 mixture.
100000
10000
1000 0KPa, 55 250KPa, 4 Master Curve
100
10 0.000001 0.0001
0.01
1
100
10000 1000000
Reduced Frequency (Hz) Fig. 7. Schematic illustration of horizontal and vertical shifting.
0.6 0.5
Log (Vertical Shift Factor)
The triaxial master curve model shown in Eq. (7) has 4 sigmoidal function coefficients, 2 WLF equation coefficients and 4 vertical shift factor coefficients. Christensen et al. [23] and Bonaquist [24] reported that the uniaxial maximum dynamic modulus of asphalt concrete can be predicted reasonably well using the Hirsch model. As discussed earlier the effect of confining pressure on the dynamic modulus can be ignored at low temperatures and high frequencies indicating that the maximum dynamic modulus does not vary with the confining pressure. Thus, the Hirsch model was used to estimate the maximum dynamic modulus of triaxial master curve. There are a total of nine unknown model coefficients in Eq. (7) which can be solved simultaneously by performing non-linear least squares regression. The model coefficients determined at a 25 °C reference temperature and a 100 kPa reference pressure for various mixtures are listed in Table 3. The confining pressure values used in the analyses were in a unit of MPa. The triaxial master curve of the SMA 12.5 mixture is shown in Fig. 6. It is seen from Fig. 6 that a smooth master curve can be constructed using dynamic modulus test results measured at various temperatures, loading frequencies and pressures by performing both horizontal and vertical shifting. A schematic illustration of the horizontal and vertical shifting for the aforementioned reference condition is presented in Fig. 7. It is seen that at 4 °C and 250 kPa almost no vertical shifting is required, while at 55 °C and 0 kPa the amount of vertical shifting is significant. The vertical shift factor was modeled in this study as a function of the reduced frequency and confining pressure. A three-dimensional surface plot is presented in Fig. 8 showing how the vertical shift factor varies with the reduced frequency and confining pressure for the SMA 12.5 mixture. The accuracy and effectiveness of the proposed triaxial master curve model were evaluated by comparing the predicted and measured dynamic moduli. The reduced dynamic moduli were first
0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 5 4 3 2 1 0 Log -1 -2 (Red -3 -4 uced -5 F
requ
ency
0
)
50
150
100
200
ressure
gP Confinin
100000
10000
SMA 12.5 Superpave 19 Superpave 25
1000
100
10 10
100
1000
10000
250
(KPa)
Fig. 8. Three-dimensional surface plot of vertical shift factor.
Predicted Dynamic Modulus (MPa)
¼ logðjE jÞ þ
ðP P0 Þ
Reduced Dynamic Modulus (MPa)
logðjE jr Þ ¼ logðjE jÞ þ logðkÞ
100000
Measured Dynamic Modulus (MPa) Fig. 9. Comparison of predicted and measured dynamic moduli.
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Y. Zhao et al. / Construction and Building Materials 37 (2012) 21–26
computed using Eq. (7a) and dynamic moduli were then predicted using Eq. (7c). The dynamic moduli measured at various testing conditions for the three mixtures are plotted against the predicted values in Fig. 9. A very good agreement is observed between the measured and predicted values. The prediction errors were evaluated using Eq. (4) and the averaged normalized errors are 4.96%, 4.34% and 4.74% for the SMA 12.5, Superpave 19 and Superpave 25 mixtures, respectively. A comparison of these normalized errors to the values listed in Table 2 shows that the predictive accuracy of the triaxial master curve is about the same as that of the uniaxial or isobaric master curve, indicating the proposed model is accurate and effective in constructing triaxial dynamic modulus master curves. 5. Conclusions A model for the construction of triaxial dynamic modulus master curve was proposed in this study and the accuracy and effectiveness of the model were evaluated using the dynamic modulus test results of three asphalt mixtures measured at various temperatures, loading frequencies and confining pressures. A key feature of the proposed model is that it employs vertical shifting to take into account the effect of confining pressure on the dynamic modulus. Based on the analyses and results presented, the following observations and conclusions can be made: 1. At low temperatures the confining pressure has little effect on the dynamic modulus, while at high temperatures the effect of confining pressure is significant. The effect of confining pressure is also dependent on the loading frequency, especially at high temperatures. 2. The effect of confining pressure on the time–temperature shift factor can be ignored for the construction of triaxial dynamic modulus master curve. 3. The amount of vertical shift required for the construction of triaxial master curves is dependent on temperature, loading frequency and pressure. A larger amount of vertical shift is needed for a condition at which asphalt concrete is soft (high temperature, low loading frequency and low confining pressure) compared to a condition at which the material is stiff. The vertical shift factor can be modeled as a function of the reduced frequency and confining pressure. 4. The triaxial master curve model uses a sigmoidal function to relate the reduced dynamic modulus to the reduced frequency by performing both horizontal and vertical shifting on dynamic modulus test results. The predictive accuracy of the proposed model is about the same as that of uniaxial and isobaric master curves, indicating the proposed model is accurate and effective in constructing triaxial dynamic modulus master curves.
Acknowledgments This research was sponsored by the China Western Transportation Research Program (#2009318773054) and Specialized
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