## 3 Failure Criterions
### Failure Criterion after Freeze-Thaw Cycles  
According to the establishment of engineering failure criterion and its testing method, the failure criterion of AC-13 and SMA-13 asphalt mixtures following freeze-thaw cycles can be established by conducting uniaxial compressive, uniaxial tensile, and the conventional triaxial tests. The average value of three effective test results of failure strength after 0, 1, 3, 5, 10, 15, and 20 freeze-thaw cycles are listed in Table 7.  
The three-dimensional failure criteria after freeze-thaw cycles can be represented by Tensile meridian:  
\[\frac{\tau_{oct}^{t}}{f_{c}}=a_{2}+b_{2}\frac{\sigma_{oct}}{f_{c}}-c_{2}N,\ R_{AC}^{2}=0.85;R_{ SMA}^{2}=0.83 \tag{13}\]  
Compressive meridian:  
\[\frac{\tau_{oct}^{c}}{f_{c}}=k_{1}\bigg{(}a_{2}+b_{2}\frac{\sigma_{oct}}{f_{c }}-c_{2}N\bigg{)},\ R_{AC}^{2}=0.98;R_{SMA}^{2}=0.98 \tag{14}\]  
Strength envelope curve:  
\[\tau_{oct}(\theta)=\tau_{oct}^{t}-(\tau_{oct}^{t}-\tau_{oct}^{c})3\theta/\pi \tag{15}\]  
where \(N\) is the number of freeze-thaw cycles; \(a_{2}\), \(b_{2}\)\(c_{2}\), and \(k_{1}\) are model parameters as shown in Table 8.  
Figure 11: Linear tensile and compressive meridians. Figure 10: Linear strength envelope curves in the \(\pi\)-plane. The failure characteristics of AC-13 and SMA-13 asphalt mixtures after freeze-thaw cycles are similar to the previous failure modes. The specimens experienced shear failure and tensile failure and there are a few loose grains on the failure surfaces. Seeing from Figures 12 and 13, there are multivariate linear relationships between the octahedral shear strength and the octahedral normal strength, as well as the freeze-thaw numbers of AC-13 and SMA-13 asphalt mixtures, the failure envelope curves also shrink during the freeze-thaw process. This criterion provides the testing and theoretical reference for material and structure design of asphalt pavement in three-dimensional stress states under freeze-thaw conditions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{3}{*}{\
\begin{tabular}{c} **Mixture Type** \\ \end{tabular} } & \multirow{3}{*}{
\begin{tabular}{c} **Freeze-Thaw** \\ **Cycles (N)** \\ \end{tabular} } & \multicolumn{2}{c}{**Uniaxial Compressive Strength**} & \multicolumn{2}{c}{**Uniaxial Tensile Strength**} \\ \cline{3-6}  & & \(f_{\text{c}}\) (MPa) & \(\sigma_{1}\) (MPa) & \(\sigma_{2}\) (MPa) & \(\sigma_{3}\) (MPa) & \(f_{\text{t}}\) (MPa) \\ \hline \multirow{6}{*}{AC-13} & 0 & \(-8.477\) & \(-0.2\) & \(-0.2\) & \(-9.372\) & \(1.110\) \\  & 1 & \(-8.201\) & \(-0.2\) & \(-0.2\) & \(-9.194\) & \(1.072\) \\  & 3 & \(-7.913\) & \(-0.2\) & \(-0.2\) & \(-8.765\) & \(0.994\) \\  & 5 & \(-7.460\) & \(-0.2\) & \(-0.2\) & \(-8.451\) & \(0.936\) \\  & 10 & \(-6.866\) & \(-0.2\) & \(-0.2\) & \(-7.835\) & \(0.818\) \\  & 15 & \(-6.358\) & \(-0.2\) & \(-0.2\) & \(-7.423\) & \(0.733\) \\  & 20 & \(-6.104\) & \(-0.2\) & \(-0.2\) & \(-7.301\) & \(0.692\) \\ \hline \multirow{6}{*}{SMA-13} & 0 & \(-9.240\) & \(-0.2\) & \(-0.2\) & \(-10.506\) & \(1.258\) \\  & 1 & \(-9.067\) & \(-0.2\) & \(-0.2\) & \(-10.287\) & \(1.217\) \\ \cline{1-1}  & 3 & \(-8.634\) & \(-0.2\) & \(-0.2\) & \(-9.862\) & \(1.109\) \\ \cline{1-1}  & 5 & \(-8.271\) & \(-0.2\) & \(-0.2\) & \(-9.643\) & \(1.043\) \\ \cline{1-1}  & 10 & \(-7.596\) & \(-0.2\) & \(-0.2\) & \(-8.970\) & \(0.925\) \\ \cline{1-1}  & 15 & \(-7.128\) & \(-0.2\) & \(-0.2\) & \(-8.553\) & \(0.844\) \\ \cline{1-1}  & 20 & \(-6.714\) & \(-0.2\) & \(-0.2\) & \(-8.302\) & \(0.796\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Failure strength of asphalt mixtures after freeze-thaw cycles. Figure 12: Tensile and compressive meridians of asphalt mixtures after freeze-thaw cycles. **Note:** The meridians from top to bottom represent the tensile and compressive meridians after 0, 1, 3, 5, 10, 15, and 20 freeze-thaw cycles, respectively.