# Accuracy of Dynamic Modulus Models of Asphalt Mixtures Containing Reclaimed Asphalt (RA)
## 5 Parameter Acquisition
### Generalized Huet-Sayegh Model (2S2P1D)  
For the second model, the 2S2P1D model, the regression coefficients \(\alpha\), \(\beta\), \(\mathbf{k}\), and \(\mathbf{h}\) were determined to plot the master curve of the mixtures. These coefficients, predefined as 0, were obtained using the nonlinear regression method with the SOLVER function in Microsoft Excel. The coefficients for each asphalt mix are provided in Table 7, with the same reference temperature, \(T_{R}=20\)\({}^{\circ}\)C.  
The k parameter affected the left asymmetry of the Cole-Cole curve and the height of the Black space curve, while the h parameter affected the radius of the Black space and Cole-Cole curves. The \(\alpha\) parameter affected the height of the Cole-Cole curve and the right asymmetry of the Black curve, and the \(\beta\) typically represented the distribution of relaxation times in the material, describing the overall shape of the relaxation spectrum. Additionally, the Cole-Cole diagram and Black space diagram helped to obtain the dynamic modulus limits, i.e., E\({}_{0}\) and E\({}_{\text{inf}}\). The Black space diagram is the result of plotting the dynamic modulus \(\mid\)E\({}^{*}\)\(\mid\) versus the phase angle \(\Phi\) (Figure 16a). The Cole-Cole diagram (Figure 16b) illustrates the storage modulus E\({}_{1}\) and the loss modulus E\({}_{2}\). These two diagrams are considered valuable tools since they evaluate the materials independently of the time-temperature superposition principle. It is evident that all the mixtures shared almost the same value for the parameter \(\beta\). It is also clear that the model parameters for the mixtures with 30% RA and 40% RA were close to each other, which could have been due to the slight difference in the reclaimed asphalt compared to the other mixtures. The \(\mid\)E\({}^{*}\)\(\mid\) master curves for the tested asphalt mixes are presented as a function of frequency for the reference temperature of 20 \({}^{\circ}\)C in Figures 17-20. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**RA (\%)**} & \multicolumn{6}{c}{**Generalized Huet–Sayegh Model Parameters**} \\ \cline{2-7}  & **E\({}_{0}\)** & **E\({}_{\text{inf}}\)** & **k** & **h** & \(\alpha\)** & \(\beta\) \\ \hline
0 & 760 & 13,966 & 0.157 & 0.539 & 1.469 & 2,259,955 \\
30 & 1154 & 27,717 & 0.215 & 0.749 & 2.728 & 2,259,953 \\
40 & 214 & 26,105 & 0.242 & 0.725 & 3.559 & 2,259,961 \\
50 & 16 & 23,935 & 0.195 & 0.670 & 4.644 & 2,259,953 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Generalized Huet–Sayegh model parameters. Figure 15: Comparison between measured \(\mid\)E\({}^{*}\)\(\mid\)_exp and calculated \(\mid\)E\({}^{*}\)\(\mid\)_Sigmoid model for the asphalt mixture with 50% RA (the red line represents the identity function \(y=x\)). Figure 16: Graphical representation of Huet-Sayegh model parameters in (**a**) Cole–Cole diagram and (**b**) Black diagram [30]. Figure 17: Comparison between the master curves of the experimental data and those of the Generalized Huet-Sayegh (2S2P1D) model for asphalt mixture with 0% RA at \(T_{R}\) = 20 \({}^{\circ}\)C.  
Figure 18: Comparison between the master curves of the experimental data and those of the Generalized Huet-Sayegh (2S2P1D) model for asphalt mixture with 30% RA at \(T_{R}\) = 20 \({}^{\circ}\)C.  
The evaluation of the master curves showed that the dynamic modulus estimated by the Generalized Huet-Sayegh model differed from the actual laboratory measurements, particularly in the range of low and high frequencies. However, these differences varied from one mixture to another. This suggests that the 2S2P1D model did not represent the linear behaviour of asphalt mixes at low and high frequencies, and modifications to the model are required. Figures 21-24 show the calculated \(|E^{*}|\) as a function of the measured \(|E^{*}|\) for all the asphalt mixes tested.