## 4 Viscoelastic Damage Model Considering the Damage Effect
### Fitting Results of Viscoelastic Damage Model  
The viscoelastic properties of two kinds of BRA compound-modified asphalt are analyzed by the viscoelastic damage model established in this paper. This Levenberg-Marquardt method was implemented to fit the creep compliance of BRA-modified asphalt in \(-6\)\({}^{\circ}\)C, \(-12\)\({}^{\circ}\)C, and \(-18\)\({}^{\circ}\)C. The parameters of creep compliance of the viscoelastic damage model are shown in Table 7. The columnar diagram viscoelastic parameters (\(E_{1}\), \(E_{2}\), \(\eta_{1}\) and \(\eta_{2}\)) of the model changed with temperature are shown in Figure 11.  
* The coefficient of determination \(R^{2}\) between the model and test data shown in Table 7 indicated that the fitting degree of test data and creep compliance of viscoelastic damage model is very precise. So, considering the effect of microdefects during the loading process can improve the precision in the prediction of viscoelastic parameters of compound-modified asphalt. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(T\) (\({}^{\circ}\)C) & **Asphal Type** & \(m\) & \(n\) & \(E_{1}\) & \(E_{2}\) & \(\eta_{1}\) & \(\eta_{2}\) & \(\tau\) & \\  & & & **(MPa)** & **(MPa)** & **(MPa-s)** & **(MPa-s)** & **(s)** & **\(R^{2}\)** \\ \hline  & BRA & 0.123 & 321.869 & 546.8 & 302.0 & 33,912.3 & 8225.6 & 27.24 & 0.9989 \\  & +3\% SBR & 0.100 & 442.900 & 423.1 & 209.7 & 21,979.7 & 6223.6 & 29.68 & 0.9990 \\ \(-6\) & +5\% SBR & 0.010 & 2634.839 & 357.9 & 127.0 & 9948.6 & 3930.3 & 30.95 & 0.9994 \\  & +5\% nano-CaCO\({}_{3}\) & 0.093 & 452.713 & 446.0 & 277.2 & 23,538.8 & 8207.3 & 29.61 & 0.9985 \\  & +10\% nano-CaCO\({}_{3}\) & 0.086 & 513.592 & 480.0 & 182.9 & 22,868.5 & 5118.1 & 27.98 & 0.9988 \\ \hline  & BRA & 0.159 & 321.821 & 717.6 & 680.1 & 66,678.5 & 16,874.3 & 24.81 & 0.9983 \\ \(-12\) & +3\% SBR & 0.168 & 275.811 & 706.5 & 631.0 & 66,919.3 & 16,043.5 & 25.43 & 0.9979 \\ \(-12\) & +5\% SBR & 0.066 & 854.2 & 684.2 & 326.3 & 27,941.3 & 9520.7 & 29.18 & 0.9993 \\  & +5\% nano-CaCO\({}_{3}\) & 0.190 & 167.516 & 759.9 & 590.2 & 78,495.5 & 13,824.1 & 23.42 & 0.9976 \\  & +10\% nano-CaCO\({}_{3}\) & 0.152 & 392.156 & 691.7 & 678.2 & 64,479.4 & 15,774.8 & 23.26 & 0.9985 \\ \hline  & BRA & 0.215 & 192.268 & 1220.6 & 998.7 & 121,513.3 & 27,984.9 & 28.02 & 0.9988 \\  & +3\% SBR & 0.272 & 142.491 & 981.0 & 937.4 & 132,872.4 & 24,969.3 & 26.64 & 0.9985 \\ \(-18\) & +5\% SBR & 0.148 & 1955.665 & 790.8 & 966.2 & 55,410.8 & 30,105.7 & 31.159 & 0.9999 \\  & +5\% nano-CaCO\({}_{3}\) & \(-\)20.539 & 0.005 & 1150.8 & 148.8 & 68,578.3 & 55,528.0 & 30.03 & 0.9999 \\  & +10\% nano-CaCO\({}_{3}\) & \(-\)10.058 & 0.431 & 1128.3 & 1975.7 & 61,545.1 & 59,415.5 & 30.07 & 0.9999 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Fitting results of viscoelastic damage model and Burgers’ model for different asphalts. Figure 11: The comparison among viscoelastic damage model’s parameters of creep compliance: (**a**) the viscoelastic parameters (\(E_{1}\)) of the model changed with temperature; (**b**) the viscoelastic parameters (\(E_{2}\)) of the model changed with temperature; (**c**) the viscoelastic parameters (\(\eta_{1}\)) of the model changed with temperature; (**d**) the viscoelastic parameters (\(\eta_{2}\)) of the model changed with temperature. 2. As shown in Figure 11a,b,d, the viscoelastic parameters (\(E_{1}\), \(E_{2}\) and \(\eta_{2}\)) of BRA compound-modified asphalt increased as the temperature decreased, which indicated that the low-temperature performance of the material decreased. Among the viscoelastic parameters, \(E_{2}\) and \(\eta_{2}\) had the same change trend as the temperature decreased, so both have good correlation with each other. 3. In three test temperatures, the viscoelastic parameters (\(E_{1}\), \(E_{2}\), and \(\eta_{1}\)) of BRA compound-modified asphalt, in which 5% SBR was the smallest among the test conditions, indicated that SBR could improve the flexibility and low-temperature performance of BRA-modified asphalt.