## 5 Dynamic Modulus Test  
To verify the FWD back-calculation with temperature correction, core drilling was performed at the test site. The cored specimens were 10 cm in diameter to satisfy the size requirements of the test specimens in dynamic modulus test. The drilling depth was approximately 35 cm to ensure that both DGAC and BTB were contained. Subsequently, the cored specimens were cut to separate DGAC and BTB, producing DGAC and BTB test specimens of 15 cm in height. However, the surface layers (PAC, OGAC, and SMA) were too thin to meet the height requirement for dynamic modulus testing. Consequently, dynamic modulus tests were only conducted on the DGAC and BTB specimens.  
Dynamic modulus tests are widely used to evaluate the influence of temperature and frequency on the properties of asphalt materials. In this study, a material test system was used to perform the dynamic modulus tests at different temperatures and frequencies. The tests were performed at 15 \({}^{\circ}\)C, 25 \({}^{\circ}\)C, 35 \({}^{\circ}\)C, 45 \({}^{\circ}\)C, and 55 \({}^{\circ}\)C, and frequencies of 25, 10, 5, 1, 0.5, and 0.1 Hz were used for the tests. Figure 9 presents the dynamic modulus test results of DGAC and BTB layers at different temperatures and frequencies. The frequency-temperature superposition principle was employed to form the master curve by horizontal shifting (Figure 9) to the reference temperature 35 \({}^{\circ}\)C in this study. A sigmoidal function as shown in Equation (4) was employed to create a master curve of dynamic modulus. In Equation (4), \(\omega_{r}=\frac{\omega}{a_{T}}\) is the reduced frequency, \(a_{T}\) is the frequency-temperature shift factor, \(\omega\) is the frequency, \(\delta\) is the minimum logarithmic value of \(|E^{*}|\), \(\delta+\alpha\) is the maximum logarithmic value of \(|E^{*}|\), \(\beta\) and \(\gamma\) are parameters describing the shape of the sigmoidal function.  
\[\text{log}|E^{*}|=\delta+\frac{\alpha}{1+e^{\beta+\gamma(log\omega_{r})}} \tag{4}\]  
Table 5 presents the sigmoidal function coefficients for the DGAC and BTB specimens obtained by using the least squares method. Figure 10 illustrates the relationship between the frequency-temperature shift factor and temperature of the DGAC and BTB specimens, whereas Figure 11 displays the master curve of dynamic modulus for the DGAC and BTB specimens at 35 \({}^{\circ}\)C. Then, the relationship between  
Figure 8: The comparison between measured and evaluated temperature.  
frequency-temperature shifted factor and temperature can be formulated as shown in Equations (5) and (6) for DGAC and BTB, respectively. In Equations (5) and (6), \(T\) is the pavement temperature estimated by the temperature-evaluation model (Equation (3)).  
\[a_{T}=0.0004*Exp(0.2244*T) \tag{5}\]  
\[a_{T}=0.0006*Exp(0.2023*T) \tag{6}\]  
The frequency-temperature shift factor \(a_{T}\) calculated by Equations (5) and (6) is employed to conduct the temperature correction of FWD back-calculation at different temperatures and then the corrected modulus will be compared with master curve (Figure 11) to verify the effectiveness of the temperature correction of back-calculation using the frequency-temperature superposition principle.  
Figure 10: Cont.  
Figure 9: Results of dynamic modulus tests.  
Figure 11: Cont.  
Figure 10: Relationship between temperature and shift factor.