## 3 Results and Discussion
### Effects of Loading Frequency and Testing Temperature on the Dynamic Modulus

Table 6: The test results of the dynamic modulus and phase angle at 50 \({}^{\circ}\)C.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**G-S**} & \multirow{2}{*}{**M-P**} & \multirow{2}{*}{**S-P**} & \multicolumn{8}{c}{**Loading Frequency (Hz)**} \\ \cline{3-11}  & & & **25** & **20** & **10** & **5** & **2** & **1** & **0.5** & **0.2** & **0.1** \\ \hline \multirow{8}{*}{1} & \multirow{4}{*}{(MPa)} & \(|\overline{E^{\ast}}|\) & 4296 & 4012 & 3215 & 2461 & 1748 & 1351 & 1041 & 736 & 556 \\  & & \(\sigma\) & 139.0 & 243.3 & 106.4 & 68.2 & 79.2 & 74.9 & 49.0 & 47.9 & 23.6 \\  & & \(C_{\rm p}\) & 3.2 & 6.1 & 3.3 & 2.8 & 4.5 & 5.5 & 4.7 & 6.5 & 4.2 \\  & & \(\overline{\delta}\) & 30.9 & 32.1 & 33.3 & 34.0 & 34.5 & 34.0 & 33.2 & 32.0 & 30.6 \\  & & \(\sigma\) & 1.1 & 1.4 & 1.0 & 1.0 & 1.1 & 0.8 & 1.4 & 1.5 & 1.3 \\  & & \(C_{\rm p}\) & 3.5 & 4.5 & 3.0 & 3.0 & 3.1 & 2.4 & 4.2 & 4.7 & 4.3 \\ \hline \multirow{8}{*}{2} & \multirow{4}{*}{(MPa)} & \(|\overline{E^{\ast}}|\) & 3637 & 3374 & 2693 & 2120 & 1481 & 1145 & 892 & 646 & 490 \\  & & \(\sigma\) & 173.7 & 70.9 & 113.3 & 119.9 & 68.2 & 73.8 & 37.8 & 16.0 & 35.6 \\  & & \(C_{\rm p}\) & 4.8 & 2.1 & 4.2 & 5.7 & 4.6 & 6.4 & 4.2 & 2.5 & 7.3 \\  & & \(\overline{\delta}\) & 31.5 & 32.1 & 32.3 & 33.0 & 33.2 & 32.6 & 31.7 & 30.2 & 29.0 \\  & & \(\sigma\) & 0.6 & 1.1 & 1.5 & 0.9 & 1.4 & 0.4 & 1.2 & 0.6 & 1.5 \\  & & \(C_{\rm p}\) & 2.1 & 3.4 & 4.8 & 2.8 & 4.1 & 1.3 & 3.7 & 2.0 & 5.2 \\ \hline \multirow{8}{*}{3} & \multirow{4}{*}{(MPa)} & \(|\overline{E^{\ast}}|\) & 3384 & 3207 & 2546 & 1996 & 1410 & 1089 & 833 & 585 & 439 \\  & & \(\sigma\) & 112.9 & 134.3 & 77.2 & 90.3 & 98.0 & 87.0 & 38.3 & 29.3 & 36.3 \\ \cline{1-1}  & & \(\overline{\delta}\) & 3.3 & 4.2 & 3.0 & 4.5 & 7.0 & 8.0 & 4.6 & 5.0 & 8.3 \\ \cline{1-1}  & & \(\overline{\delta}\) & 31.2 & 32.0 & 32.8 & 33.1 & 33.1 & 32.4 & 31.5 & 30.1 & 29.0 \\ \cline{1-1}  & & \(\sigma\) & 1.2 & 0.9 & 1.7 & 0.6 & 1.4 & 1.8 & 1.1 & 1.4 & 0.8 \\ \cline{1-1}  & & \(C_{\rm p}\) & 3.7 & 2.8 & 5.2 & 1.8 & 4.3 & 5.6 & 3.3 & 4.8 & 2.8 \\ \hline \multirow{8}{*}{4} & \multirow{4}{*}{(MPa)} & \(|\overline{E^{\ast}}|\) & 3284 & 3106 & 2468 & 1934 & 1364 & 1057 & 815 & 577 & 443 \\  & & \(\sigma\) & 195.0 & 89.0 & 85.4 & 100.5 & 60.4 & 65.8 & 24.6 & 26.3 & 37.5 \\ \clFor the construction of master curves, the shift factors should be obtained firstly. In this paper, the Williams-Landel-Ferry (WLF) equation was adopted to calculate the shift factors (Equation (3)):  
\[\text{lg}\alpha_{T}=\frac{-C_{1}(T-T_{r})}{C_{2}+(T-T_{r})} \tag{3}\]  
where \(\alpha_{T}\) is the shift factor; \(C_{1}\) and \(C_{2}\) are constants depending on the reference temperature; \(T_{r}\) is the reference temperature in Kelvin; \(T\) is the test temperature in Kelvin. The WLF equation is widely used in the calculation of shift factors and has a limitation of applicable temperature range of \(\text{T}_{\text{g}}<\text{T}<\text{T}_{\text{g}}+100\)\({}^{\circ}\)C (\(\text{T}_{\text{g}}\): glass transition temperature), in which range, the equation applicable results satisfy the test data better [33]. With shift factors, the curves at other temperature can be shifted to the reference temperature to form the master curve. The Sigmoidal function is considered to fit the master curves of the dynamic modulus of asphalt mixtures well by shift factors [34]. In this paper, the Sigmoidal function (shown in Equation (4)) is adopted to determine the predicted dynamic modulus values by nonlinear least square method fitting in Origin software (OriginPro 2016, OriginLab Corporation). \[\text{lg}|E^{*}|=\delta+\frac{\alpha}{1+e^{\beta+\gamma(\text{lg}/r)}} \tag{4}\]  
\[f_{r}=\text{f}\times\alpha_{T} \tag{5}\]  
where \(\delta\) is the minimum value of \(|E^{*}|\); \(\delta+\alpha\) is the maximum value of \(|E^{*}|\); \(\beta\) and \(\gamma\) are parameters describing the shape of the Sigmoidal function; \(f_{r}\) is the reduced frequency at reference temperature (Hz); \(f\) is the loading frequency (Hz).