## 4 Results and Discussion
### Effect of freeze-soak-scour Cycles on Properties of NHISS-AC
\[\frac{d^{2}v(t)}{d^{2}t}=\frac{Kab^{3}e^{-bt}\Big{(}1-4 abe^{-bt}+a^{2}e^{-2 bt} \Big{)}}{\big{(}1+ae^{-bt}\big{)}^{4}}=0 \tag{7}\]  
We can get that  
\[t_{1}=\frac{\ln a-1.317}{b} \tag{8}\]  
\[t_{2}=\frac{\ln a+1.317}{b} \tag{9}\]  
Re-number the three key points  
\[t_{1}=\frac{\ln a-1.317}{b} \tag{10}\]  
\[\mathrm{y}=\frac{K}{1+ae^{-bt}} \tag{3}\]0  
\[\mathrm{y}=\frac{K}{1+ae^{-bt}} \tag{3}\]1  
\(t_{1}\) is the initial time point of the growth peak of the study, \(t_{2}\) is the peak, \(t_{3}\) is the end. \(\varphi_{\mathrm{i}}\) is used as the damage coefficient of the performance of the mixtures after the freeze-soak-scour cycle, then  
\[\mathrm{y}=\frac{K}{1+ae^{-bt}} \tag{3}\]2  
In the formula, \(\theta_{\mathrm{i}}\) is the test index of specimens after i cycles. According to the tests, the voids content is used to evaluate the durability of mixtures, the Marshall stability is used to evaluate the high-temperature mechanical property of mixtures, \(-10\)\({}^{\circ}\)C splitting tensile strength and the destruction tensile strain are used to evaluate the low-temperature mechanical property, the freeze-soak-scour splitting tensile ratio is used to evaluate the water stability. If two indexes are needed to evaluate the damage of a certain mechanical property of mixtures, the damage coefficient takes the mean of the damage coefficient of two indexes  
For the convenience of analysis, the damage process represented by the logistic curve equation is transformed into  
\[\mathrm{y}=\frac{K}{1+ae^{-bt}} \tag{3}\]3  
The regression equation of the damage model and the corresponding value of each model is shown in \[\mathrm{y}=\frac{K}{1+ae^{-bt}} \tag{3}\]4 The freeze-thaw-scour damage model is effective, because all R\({}^{2}\) of the regression equation are above 0.95. In order to compare the model parameters and analyze physical meaning, the contrast figure was given in \[\mathrm{y}=\frac{K}{1+ae^{-bt}} \tag{3}\]5 \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Mixture** & **Pavement Property** & **Logistic Damage Model** & **SS\({}_{\mathbf{e}}\)** & **R\({}^{2}\)** & **RMSE** & **x\({}_{\mathbf{0}}\)** & **x\({}_{\mathbf{1}}\)** & **x\({}_{\mathbf{2}}\)** \\ \hline \multirow{4}{*}{AC} & Durability & \(\varphi=-\frac{0.34}{1+\left(\frac{\alpha_{1}}{\Delta}\right)^{2}}+0.34\) & \(1.5\times 10^{-4}\) & 0.9976 & 0.0063 & 4.3 & 0.35 & 7.3 \\  & High temperature & \(\varphi=-\frac{0.19}{1+\left(\frac{\alpha_{1}}{\Delta}\right)^{2}}+0.19\) & \(6.7\times 10^{-8}\) & 0.9999 & 0.0011 & 5.5 & 0.51 & 7.9 \\  & Water stability & \(\varphi=-\frac{0.33}{1+\left(\frac{\alpha_{1}}{\Delta}\right)^{2}}+0.33\) & \(1.9\times 10^{-4}\) & 0.9964 & 0.0069 & 0.7 & 0.03 & 3.4 \\  & Low temperature & \(\varphi=-\frac{0.13}{1+\left(\frac{\alpha_{1}}{\Delta}\right)^{2}}+0.13\) & \(1.3\times 10^{-4}\) & 0.9796 & 0.0057 & 4.8 & 0.21 & 10.1 \\  & mechanical property & & & & & & & & \\ \hline \multirow{4}{*}{NHSS-AC} & Durability & \(\varphi=-\frac{0.10}{1+\left(\frac{\alpha_{1}}{\Delta}\right)^{2}}+0.10\) & \(9\times 10^{-6}\) & 0.9982 & 0.0015 & 3.7 & 0.26 & 7.1 \\  & High temperature & \(\varphi=-\frac{0.15}{1+\left(\frac{\alpha_{1}}{\Delta}\right)^{2}}+0.15\) & \(8\times 10^{-5}\) & 0.9941 & 0.0045 & 4.6 & 0.43 & 7.1 \\  & Mechanical property & \(\varphi=-\frac{0.25}{1+\left(\frac{\alpha_{1}}{\Delta}\right)^{2}}+0.25\) & \(2\times 10^{-4}\) & 0.9940 & 0.0070 & 2.0 & 0.15 & 4.8 \\  & Low temperature & \(\varphi=-\frac{0.11}{1+\left(\frac{\alpha_{1}}{\Delta}\right)^{1}}+0.11\) & \(1.7\times 10^{-4}\) & 0.9547 & 0.0066 & 2.8 & 0.08 & 9.1 \\ \hline \end{tabular}
\end{table} 
Table 6: Fitting result of the logistic damage model. Figure 10: Damage degree result of the various properties of the AC and NHSS-AC. From Figure 10(a), it can be seen that after 20 freeze-soak-scour cycles, the durability and water stability are the most damaged of the four properties of the AC, reaching 0.34 and 0.33 respectively.