## 3 Failure Criterions  
### Nonlinear Failure Criterion  
The average value of three effective test results of triaxial compressive/tensile tests, the plane tensile, and compressive/axial tensile tests are presented in Table 3. Based on the test results, the octahedral normal stress, \(\sigma_{oct}\), octahedral shear stress, \(\tau_{oct}\), and lode angle, \(\theta\) can be obtained by the following formulas[29]:  
\[\sigma_{oct}=\sigma_{m}=(\sigma_{1}+\sigma_{2}+\sigma_{3})/3 \tag{1}\]  
\[\tau_{oct}=\sqrt{\left(\sigma_{1}-\sigma_{2}\right)^{2}+\left(\sigma_{2}- \sigma_{3}\right)^{2}+\left(\sigma_{3}-\sigma_{1}\right)^{2}}/3 \tag{2}\]  
\[\theta=\arccos{\frac{2\sigma_{1}-\sigma_{2}-\sigma_{3}}{3\sqrt{2}\tau_{oct}}} \tag{3}\]  
Figure 4: Stress path of tensile and compressive meridians. Figure 5: Stress path of strength envelope curve. For triaxial compressive tests, both AC-13 and SMA-13 specimens are mainly represented as shear failure shown in Figure 6. For the triaxial tensile test, the plane tensile, and compressive/axial tensile test, the specimen failures are shown in Figure 7.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Test** & **Mixture Type** & \(\mathbf{\sigma_{1}}\) **(MPa)** & \(\mathbf{\sigma_{2}}\) **(MPa)** & \(\mathbf{\sigma_{3}}\) **(MPa)** & \(\frac{\sigma_{\text{sat}}}{f_{\epsilon}}\) & \(\frac{\tau_{\text{sat}}}{f_{\epsilon}}\) & \(\theta^{\circ}\) \\ \hline \multirow{8}{*}{ \begin{tabular}{c} triaxial compressive \\ \end{tabular}  } & \(-\) & \(0.000\) & \(-8.477\) & \(-0.333\) & \(0.471\) & \(60\) \\ \cline{3-6}  & \(-0.200\) & \(-0.200\) & \(-9.372\) & \(-  
Based on the triaxial test results, the SMA-13 asphalt mixture has a higher strength in three-dimensional stress states (especially in triaxial tensile stress states) compared with AC-13 asphalt mixture. The three-dimensional nonlinear failure criterions of AC-13 and SMA-13 asphalt mixtures were established as follows:  
Tensile meridian:  
\[\frac{\tau_{oct}^{t}}{f_{c}}=a+b\frac{\sigma_{oct}}{f_{c}}+c\left(\frac{\sigma _{oct}}{f_{c}}\right)^{2},\ R_{AC}^{2}=0.98;R_{SMA}^{2}=0.98 \tag{4}\]  
Compressive meridian:  
\[\frac{\tau_{oct}^{c}}{f_{c}}=d+e\frac{\sigma_{oct}}{f_{c}}+f\left(\frac{ \sigma_{oct}}{f_{c}}\right)^{2},\ R_{AC}^{2}=0.99;R_{SMA}^{2}=0.98 \tag{5}\]  
Strength envelope curve:  
\[\tau_{oct}(\theta)=\tau_{oct}^{t}-\left(\tau_{oct}^{t}-\tau_{oct}^{c}\right) \sin^{n}1.5\theta,\ R_{AC}^{2}=0.96;R_{SMA}^{2}=0.97 \tag{6}\]  
where \(f_{c}\) is the uniaxial compressive strength, \(a\), \(b\), \(c\), \(d\), \(e\), \(f\), and \(n\) are model parameters  
It is shown that these failure criteria are in good agreement with the test results and reflect the difference between the tensile and compressive strength of the asphalt mixtures under complex stress states, as well as the synergistic failure effect of each stress component in the pavement structures. However, these criteria are complicated and have many fitting parameters. In the process of regression, the tensile meridian can be approximated with a quadratic polynomial. The tensile and compressive meridians must intersect at the triaxial equal tensile point when \(\tau_{oct}\) = 0. The tensile meridian and compressive meridian were set at a proportion relationship to simplify the meridians. Furthermore, it is assumed that the strength envelope in the region of 0\({}^{\circ}\)\(\sim\)60\({}^{\circ}\) is interpolated with the sine function [29; 30].