## 3 Results and Discussion
### Relaxation Characteristics of Asphal Mixtures
#### 3.3.2 Fitting Analysis of Relaxation Modulus  
To analyze the relaxation properties of the eight groups of asphalt mixtures, the second-order extensive Maxwell model and the Scott-Blair model were applied to analyze the relaxation moduli \(E(t)\) of the eight types of asphalt mixtures with time. The expressions of the second-order extensive Maxwell model [31; 32; 33; 34] and the Scott-Blair model [35; 36] are shown in Equations (16) and (17), respectively. \[E(t)=E_{1}e^{-t/\rho_{1}}+E_{2}e^{-t/\rho_{2}}, \tag{15}\]  
where \(E_{1}\) and \(E_{2}\) are the strengths to characterize the stress relaxation \(\rho_{1}\) and \(\rho_{2}\) are the relaxation times used to characterize the stress relaxation ability of asphalt mixtures. \[E(t)=\eta\frac{t^{-\alpha}}{\int\limits_{0}^{1-\alpha}e^{-\tau}\tau^{-\alpha} d\tau}, \tag{16}\]  
where \(\eta\) is the viscous damping coefficient, MPa\(\cdot\)s, and \(\alpha\) is a fractional order, which can be used to analyze the stress relaxation process of asphalt mixtures. The larger the value is, the closer the material property is to the rheological property of fluid, and the smaller the value is, the closer it is to the elastic property of solid. The unknown parameters of Equations (16) and (17) are solved by EXCEL software. The solution results of the two model parameters are shown in Table 8. From the correlation coefficient R\({}^{2}\) in Table 8 and the comparison diagram in Figure 9, we can see that the fitting curves of Scott-Blair model and second-order extended Maxwell model are in good agreement with the calculated results. This means that both the Scott-Blair model and the second-order extended Maxwell model can better describe the relaxation trend of eight groups of asphalt mixtures. From the parameter calculation results of second-order extensive Maxwell model, the addition of rubber particle and diatomite can reduce the relaxation time of asphalt mixture, make the internal stress of asphalt mixture rapidly relax due to temperature change or external load, and then improve the stress relaxation ability of asphalt mixture. According to the results of the Scott-Blair model, it can be found that the addition of rubber particle and diatomite can reduce the fractional order \(\alpha\) and increase the solid elastic properties of asphalt mixture, which means that the addition of rubber particles and diatomite can improve the elastic properties of asphalt mixture, the deformation can be recovered quickly after the load is removed, and the residual irrecoverable deformation will be reduced. Comparing the test results of different modified bitumen, the improvement of rubber particle and diatomite on the relaxation modulus of asphalt mixture is weaker than that of SBS modified asphalt, and the joint use of diatomite, rubber particles, and SBS modified asphalt will make the asphalt mixture have the best relaxation characteristics. From the point of view of gradation, it can \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{8}{c}{**Asphalt Mixture Types**} \\ \cline{2-9}  & **AC** & **DRAC** & **SAC** & **SDRAC** & **SMA** & **DRSMA** & **SSMA** & **SDRSMA** \\ \hline \(E_{1}\) (MPa) & 30.237 & 39.483 & 44.639 & 53.825 & 32.043 & 35.408 & 46.865 & 50.913 \\ \(E_{2}\) (MPa) & 9.407 & 8.863 & 9.921 & 14.024 & 10.511 & 10.324 & 11.335 & 11.385 \\ \(\rho_{1}\) (s) & 36.717 & 63.395 & 53.442 & 64.322 & 30.117 & 49.098 & 56.630 & 56.201 \\ \(\rho_{2}\) (s) & 53.847 & 39.806 & 46.885 & 35.649 & 65.801 & 36.844 & 41.031 & 45.090 \\ R\({}^{2}\) & 0.9773 & 0.9609 & 0.9715 & 0.9608 & 0.9779 & 0.9689 & 0.9617 & 0.9702 \\ \hline \(\eta\) & 42.631 & 49.023 & 56.532 & 67.768 & 47.683 & 46.236 & 59.382 & 64.181 \\ \(\alpha\) & 0.04967 & 0.03117 & 0.03428 & 0.03291 & 0.05769 & 0.03865 & 0.03416 & 0.03350 \\ R\({}^{2}\) & 0.9833 & 0.9397 & 0.9688 & 0.9153 & 0.9947 & 0.9384 & 0.9450 & 0.9627 \\ \hline \hline \end{tabular}
\end{table}