## 3 Numerical Examples
### Experimental Data Processing  
As designed in the experiments, the micro-strain of specimen under different frequency is around 100 \(\mu\epsilon\). Moreover, the measured dynamic modulus and phase angle is constant under changeable loading. Thus, the experiments can be regarded as non-destructive ones that can maintain the specimen within linear viscoelastic range [26]. According to the time-temperature superposition principle, dynamic modulus and phase angle under different temperatures and frequencies can be translated to an integrated curve, called master curve [27]. Therefore, to obtain a dynamic modulus master curve, the curves under different temperatures are required to be horizontally shifted to one curve at a given temperature [28]. In this case, the long-term mechanical behavior of the asphalt material can be predicted without having to perform experiments for a long duration of testing time. In addition, the mechanical behavior of the material is at very short loading times (or very high frequencies) which are difficult to obtain from the experiments can be determined using the master curve [29]. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Sieve Size (mm)** & 19 & 12.5 & 9.5 & 4.75 & 2.36 & 1.18 & 0.6 & 0.3 & 0.15 & 0.075 \\ \hline
**Passing Rate (\%)** & 100 & 93.0 & 74.8 & 50.3 & 30.3 & 24.6 & 19.4 & 15.0 & 10.3 & 6.5 \\ \hline \end{tabular}
\end{table} 
Table 1: Aggregate Gradation of Hot Mix Asphalt AC-13. Sigmoid functions as Equation (16) can be employed to pre-smooth the discrete experimental data of dynamic modulus. Similarly, Equation (17) can be used to pre-smooth the discrete experimental data of phase angle [30]. The function of calculating the reduced frequency can be obtained through Equation (18). The reduced frequency is defined as the computed frequency at the reference temperature, equivalent to the actual loading frequency at the test temperature. All of these fitted parameters can be determined by fitting the functions to experimental data and the summary of the fitted results are shown in Table 2.  
\[\log\lvert E^{*}\rvert=\delta+\frac{\alpha}{\left(1+e^{\beta+\gamma\log\left(f_ {r}\right)}\right)} \tag{16}\]  
\[\delta(f_{r})=\xi_{1}+\xi_{2}\log(f_{r})-\xi_{3}\frac{\pi}{2}\alpha\gamma\frac {e^{\beta+\gamma\log\left(f_{r}\right)}}{\left(1+e^{\beta+\gamma\log\left(f_{ r}\right)}\right)^{2}} \tag{17}\]  
\[\log(f_{r})=\log(f)+\log(\alpha_{T}) \tag{18}\]  
where, \(f_{r}\) is the reduced frequency; \(f\) is the loading frequency; \(\delta\) is the minimum modulus; \(\alpha\) is the parameter of vertical span of dynamic modulus; \(\beta\), \(\gamma\) are the shape parameters of the material; \(\xi_{1},\xi_{2},\xi_{3}\) are all fitting parameters of experimental data; \(\alpha_{T}\) is the time-temperature shifting factor. The determination of \(\alpha_{T}\) is based on Williams-Landel-Ferry (WLF) model [28] in Equation (19). \[\log(\alpha_{T})=\frac{-C_{1}(T-T_{0})}{C_{2}+(T-T_{0})} \tag{19}\]  
where \(C_{1}\) and \(C_{2}\) are the WLF coefficients; \(T\) is the testing temperature; \(T_{0}\) is the reference temperature, which indicates that temperature at which the master curve is constructed. Thus, after the processing of experimental data, dynamic modulus and phase angle master curves at 21.1 \({}^{\circ}\)C Reference Temperature of asphalt mixtures can be obtained as presented in Figure 2.  
Similar to the process of fitting master curves of dynamic modulus and phase angle, the creep compliance master curve can be found as in Figure 3. Considering the requirement of comparison between dynamic modulus and creep compliance interconversions, the reference temperature is required to be the same.