# Shear Properties of Asphalt Mixtures under Triaxial Compression
## 3 Results and Discussion
### Nonlinear Mohr-Coulomb Failure Criterion  
Based on the triaxial results, the Mohr-Coulomb circles can be drawn, and all the tangent points can be obtained by two Mohr-Coulomb circles tangent to each other. Considering that the Mohr-Coulomb criterion is one of the most popular strength theories in highway engineering, the triaxial compressive test results are plotted in the \(\sigma-\tau\) space to investigate the shear failure properties of asphalt mixtures, and the results are shown in Figures 8-10 (take the results of 60 \({}^{\circ}\)C as example). The data is fitted with linear function and power function respectively, and the nonlinear function is shown in Equation (3). \[\tau=a\sigma^{b}+c \tag{3}\]  
where \(\sigma\) represents the normal stress in the shear failure plane, \(\tau\) is the shear stress in the shear failure plane, \(a\), \(b\) and \(c\) are constants. In fact, \(c\) is cohesion strength when \(\sigma=0\). By curve fitting of the test data, all constants obtained are listed in Table 6. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Mixture Type**} & \multicolumn{4}{c}{**40 \({}^{\circ}\)C**} & \multicolumn{4}{c}{**50 \({}^{\circ}\)C**} & \multicolumn{4}{c}{**60 \({}^{\circ}\)C**} \\ \cline{2-11}  & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{c}\) & \(\mathbf{R^{2}}\) & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{c}\) & \(\mathbf{R^{2}}\) & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{c}\) & \(\mathbf{R^{2}}\) \\ \hline SMA-13 & 0.80 & 0.80 & 0.38 & 0.99 & 0.74 & 0.79 & 0.26 & 0.97 & 0.66 & 0.78 & 0.16 & 0.97 \\ AC-13 & 0.72 & 0.77 & 0.32 & 0.98 & 0.64 & 0.76 & 0.23 & 0.98 & 0.58 & 0.74 & 0.15 & 0.98 \\ OGFC-13 & 0.63 & 0.74 & 0.25 & 0.98 & 0.57 & 0.72 & 0.19 & 0.99 & 0.53 & 0.70 & 0.12 & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters of failure envelope in \(\sigma-\tau\) space. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Mixture Type**} & \multicolumn{2}{c}{**40 \({}^{\circ}\)C**} & \multicolumn{4}{c}{**50 \({}^{\circ}\)C**} & \multicolumn{4}{c}{**60 \({}^{\circ}\)C**} \\ \cline{2-11}  & \(\mathbf{A}\) & \(\mathbf{n}\) & \(\mathbf{R^{2}}\) & \(\mathbf{A}\) & \(\mathbf{n}\) & \(\mathbf{R^{2}}\) & \(\mathbf{A}\) & \(\mathbf{n}\) & \(\mathbf{R^{2}}\) \\ \hline SMA-13 & 3.319 & 0.919 & 0.98 & 3.043 & 0.913 & 0.98 & 2.656 & 0.905 & 0.97 \\ AC-13 & 2.883 & 0.894 & 0.97 & 2.550 & 0.885 & 0.99 & 2.338 & 0.876 & 0.98 \\ OGFC-13 & 2.440 & 0.847 & 0.98 & 2.363 & 0.835 & 0.99 & 2.113 & 0.824 & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of the triaxial strength model of asphalt mixtures. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mixture Type** & \(\mathbf{a_{0}}\) & \(\mathbf{b_{0}}\) & \(\mathbf{a_{1}}\) & \(\mathbf{b_{1}}\) & \(\mathbf{R^{2}}\) \\ \hline SMA-13 & \(-0.033\) & 4.664 & \(-0.70\times 10^{-3}\) & 0.947 & 0.97 \\ AC-13 & \(-0.027\) & 3.953 & \(-0.90\times 10^{-3}\) & 0.930 & 0.99 \\ OGFC-13 & \(-0.016\) & 3.123 & \(-1.15\times 10^{-3}\) & 0.893 & 0.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters of the triaxial strength model considering the influence of temperature and confining pressure. As can be seen from Figures 8-10 that the linear Mohr-Coulomb criterion is not suitable to describe the shear failure behavior of asphalt mixtures, the failure envelopes of asphalt mixtures can be interpolated by a power function. With the increase of temperature, the parameters \(a\), \(b\), and \(c\) decrease. The failure envelopes of these asphalt mixtures bend down when the confining pressure is close to 0.6 MPa. It is clear that the failure strength of asphalt mixtures is highly dependent on the confining pressure and temperature. Moreover, this power functional Mohr-Coulomb failure envelope under the loading speed of 1 mm/min can be validated by the nonlinear failure envelope in the \(I_{1}-\sqrt{J_{2}}\) space proposed by [PERSON] et al.