# Dynamic Characteristic Master Curve and Parameters of Different Asphalt Mixtures
## 3 Characteristic Parameters of Master Curves
### Modulus Master Curve Parameters  
Figure 2A shows that the modulus master curve of the asphalt mixture tends to a stable modulus value--the minimum modulus value (\(10^{\text{E}\circ}\)) and the maximum modulus value (\(10^{\text{E}\circ+\alpha}\)) under the condition of extremely low frequency and extremely high frequency. This shows that the dynamic modulus of the asphalt mixture at very low frequency (or high temperature) and very high frequency (low temperature) is not affected by load frequency and only shows obvious frequency sensitivity in the general range of \(1\times 10^{-5}\) to \(1\times 10^{5}\) Hz. According to the related research, the ultimate modulus of the asphalt mixture is only related to the mixture gradation, which depends on the mixture volume parameters and presents uniqueness [9; 11]. Due to the use of the same gradation, the volume indexes of the three kinds of asphalt mixtures are almost the same. The ultimate modulus is also close to each other, and the difference is small. The parameters \(\alpha\) and \(\gamma\) of the principal curve of modulus are also stable. The linear correlation analysis was carried out by using the master curve parameters of three kinds of mixture modulus (E\({}_{0}\), \(\alpha\), \(\gamma\)) and the volume index of the mixture. Figure 3 shows that the correlation coefficients of the aggregate void ratio (VMA), mixture density (\(\gamma_{\text{f}}\)), and asphalt content (Pb%) are the highest, while the effects of the asphalt saturation (VFA) and void percentage (Vv%) are not significant. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Type**} & \multirow{2}{*}{
\begin{tabular}{c} **Temp-** \\ **rature**, \({}^{\circ}\)C \\ \end{tabular} } & \multicolumn{4}{c}{**Fitting Parameters for E* Master Curve**} & \multicolumn{4}{c}{**Fitting Parameters for \(\delta\) Master Curve**} \\ \cline{3-13}  & & \(\mathbf{E_{0}}\) & \(\mathbf{\alpha}\) & \(\mathbf{\beta}\) & \(\gamma\) & \(\mathbf{R^{2}}\) & \(\delta_{0}\) & \(\mathbf{A}\) & \(\omega_{c}\) & \(\omega\) & \(\mathbf{R^{2}}\) \\ \hline \multirow{4}{*}{ZH50-AC20} & 20 & \multirow{4}{*}{1.4522} & \multirow{4}{*}{2.844} & \(-1.537\) & \multirow{4}{*}{\(-0.664\)} & \multirow{4}{*}{0.997} & \multirow{4}{*}{4.436} & \(-2.084\) & \multirow{4}{*}{2.961} & \(-0.889\) & \multirow{4}{*}{2.893} & \multirow{4}{*}{0.989} \\  & 30 & & & & \(-0.743\) & & & & & \(0.161\) & \\  & 40 & & & & \(0.561\) & & & & & \(1.074\) & \\  & 20 & & & \(-1.339\) & & & & & & \(-1.607\) & \\  & 30 & & & \(-0.607\) & & & & & & \(-0.551\) & \\  & 40 & & & \(0.149\) & & & & & & \(0.536\) & \\  & 50 & & & & \(0.924\) & & & & & & \(1.654\) & \\  & 20 & & & \(-0.998\) & & & & & & \(-1.216\) & \\  & 30 & & & \(-0.461\) & & & & & & \(-0.364\) & \\  & 40 & & & \(0.156\) & & & & & & \(0.624\) & \\  & 50 & & & & \(0.846\) & & & & & & \(1.725\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Master curve parameters for three asphalt mixtures. Figure 2: Master curves of three kinds of asphalt mixtures. (**A**) Dynamic modulus master curve, (**B**) Phase angle master curve. However, the parameter \(\beta\) of the master curve increases monotonously with the increase of the reference temperature and presents the linear relationship shown in Figure 4. This phenomenon is related to the shift factor of the principal curve and is essentially related to the temperature sensitivity of asphalt. A good power-law relationship can be established by using the principal curve parameter \(\beta\) at 20, 30, 40, and 50 \({}^{\circ}\)C and the complex modulus (G*) of asphalt at the same temperature (test conditions: stress 100 Pa, frequency 5 rad/s). Figure 5 shows that the shape parameter \(\beta\) of the main curve of mixture modulus decreases with the increase of complex modulus G* of asphalt (when the temperature decreases or the asphalt hardens). Under the same gradation condition, a low \(\beta\) value is more likely to lead to a higher modulus.