## 3 Results and Discussion
### Effects of Loading Frequency and Testing Temperature on the Dynamic Modulus
The shift factors at different reference temperature are shown in Table 7. The fitting parameters values of Sigmoidal function by Origin nonlinear least square fitting are shown in Table 8. \begin{table}
\begin{tabular}{c 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{4-13}  & & & **25** & **20** & **10** & **5** & **2** & **1** & **0.5** & **0.2** & **0.1** \\ \hline \multirow{4}{*}{4} & \(|\text{E}^{*}|\) & \(|\overline{E}^{*}|\) & 917 & 775 & 585 & 439 & 309 & 247 & 200 & 155 & 127 \\  & (MPa) & \(\sigma\) & 45.4 & 39.7 & 44.4 & 19.1 & 11.6 & 13.3 & 9.5 & 9.5 & 5.9 \\  & \(C_{\text{g}}\) & 5.0 & 5.1 & 7.6 & 4.3 & 3.8 & 5.4 & 4.8 & 6.1 & 4.7 \\  & \(\delta\) & \(\overline{\delta}\) & 37.3 & 36.1 & 34.5 & 33.0 & 30.7 & 29.1 & 27.6 & 26.0 & 25.0 \\  & (deg) & \(\sigma\) & 1.5 & 2.1 & 1.8 & 2.0 & 1.0 & 1.3 & 1.9 & 2.1 & 1.4 \\  & (C\({}_{\text{g}}\) & 4.1 & 5.7 & 5.1 & 6.1 & 3.2 & 4.4 & 6.7 & 8.0 & 5.5 \\ \hline \multirow{4}{*}{5} & \(|\text{E}^{*}|\) & \(|\overline{E}^{*}|\) & 803 & 701 & 533 & 402 & 283 & 228 & 186 & 143 & 121 \\  & (MPa) & \(\sigma\) & 72.5 & 46.2 & 27.9 & 21.9 & 13.8 & 12.8 & 8.2 & 9.6 & 9.0 \\  & \(C_{\text{g}}\) & 9.0 & 6.6 & 5.2 & 5.5 & 4.9 & 5.6 & 4.4 & 6.7 & 7.5 \\  & \(\delta\) & \(\overline{\delta}\) & 38.1 & 37.3 & 35.4 & 33.8 & 31.8 & 30.2 & 28.9 & 27.5 & 26.7 \\  & (deg) & \(\sigma\) & 1.6 & 2.6 & 1.6 & 2.0 & 1.7 & 2.2 & 1.6 & 2.3 & 1.5 \\  & \(C_{\text{g}}\) & 4.2 & 6.9 & 4.4 & 5.9 & 5.2 & 7.2 & 5.5 & 8.4 & 5.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: _Cont._ \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Reference Temperature (\({}^{\circ}\)C)**} & \multicolumn{3}{c}{**Values of Shift Factors (\(\text{lg}\alpha_{T}\))**} \\ \cline{3-5}  & **20\({}^{\circ}\)C** & **35\({}^{\circ}\)C** & **50\({}^{\circ}\)C** \\ \hline
20 & 0 & \(-\)1.1398 & \(-\)2.0198 \\
35 & 1.5346 & 0 & \(-\)1.1398 \\
50 & 3.7123 & 1.5346 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The values of shift factors at different reference temperature. Figure 5 presents the correlation between the dynamic modulus ([E*]) and loading frequency of five groups of asphalt mixture samples at different testing temperatures. Figure 6 shows the master curves of dynamic modulus test results. Figure 7 shows the master curves of dynamic modulus of five groups of samples. Figure 5: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Group of** \\ **Samples** \\ \end{tabular} } & \multicolumn{2}{c}{**Asphalt Contents (\%)**} & \multicolumn{2}{c}{**Values of Fitting Parameters**} & \multicolumn{2}{c}{**Correlation**} \\ \cline{2-7}  & **Upper** & **Lower** & \multirow{2}{*}{\(\delta\)} & \multirow{2}{*}{\(\alpha\)} & \multirow{2}{*}{\(\beta\)} & \multirow{2}{*}{\(\gamma\)} & \multirow{2}{*}{**R-Square (R\({}^{2}\))**} \\  & **Half Layer** & **Half Layer** & & & & & \\ \hline
1 & 5.0 & 5.0 & \(-\)16.5297 & 21.1705 & \(-\)3.0329 & \(-\)0.3107 & 0.99994 \\
2 & 5.3 & 4.7 & \(-\)34.4801 & 39.0201 & \(-\)3.7386 & \(-\)0.3357 & 0.99987 \\
3 & 5.6 & 4.4 & \(-\)30.1671 & 34.7170 & \(-\)3.5301 & \(-\)0.3352 & 0.99997 \\
4 & 5.9 & 4.1 & \(-\)0.5430 & 4.9250 & \(-\)1.6089 & \(-\)0.4177 & 0.99972 \\
5 & 6.2 & 3.8 & \(-\)38.9270 & 43.6006 & \(-\)3.5857 & \(-\)0.2801 & 0.99988 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The values of fitting parameters of Sigmoidal function by Origin (reference temperature: 20 \({}^{\circ}\)C). Figure 5: The correlation between the dynamic modulus and loading frequency (ACU: asphalt content in the upper half layer, ACL: asphalt content in the lower half layer): (**a**) 20 \({}^{\circ}\)C; (**b**) 35 \({}^{\circ}\)C; (**c**) 50 \({}^{\circ}\)C.  
Figure 6: _Cont._  
As expected, from Figures 5 and 6, the dynamic modulus of asphalt mixtures increase monotonously with the increase of loading frequency while they decrease with the increase of testing temperature.