# Shear Properties of Asphalt Mixtures under Triaxial Compression
## 3 Results and Discussion
### Nonlinear Mohr-Coulomb Failure Criterion
[15] (where \(I_{1}\) represents the first invariant of stress tensor, \(J_{2}\) denotes the second invariant of stress deviator). It is also reasonable to predict the failure envelope of asphalt mixtures in \(\sigma-\tau\) space considering the influence of temperature and confining pressure as following:  
\[\tau_{t}=(m_{0}t+n_{0})\sigma^{(m_{1}t+n_{1})}+(m_{2}t+n_{2}) \tag{4}\]  
Figure 8: Failure envelope of SMA-13 asphalt mixture in \(\sigma-\tau\) space at 60 \({}^{\circ}\)C.  
Figure 10: Failure envelope of OGFC-13 asphalt mixture in \(\sigma-\tau\) space at 60 \({}^{\circ}\)C.  
Figure 9: Failure envelope of AC-13 asphalt mixture in \(\sigma-\tau\) space at 60 \({}^{\circ}\)C.  
where \(\sigma\) represents the normal stress in the shear failure plane; \(\tau_{l}\) is the shear stress in the shear failure plane at different temperatures; \(t\) is the temperature; \(m_{0}\), \(n_{0}\), \(m_{1}\), \(n_{1}\), \(m_{2}\), and \(n_{2}\) are constants. All the parameters of this model are listed in Table 7.  
Consider linear case, i.e., \(b=1\) and \(a=\tan\varphi\), where \(\varphi\) is defined as friction angle, linear Mohr-Coulomb failure criterion is given in Equation (5) [27; 28]. Furthermore, the strength parameters that are cohesion strength and friction angle obtained by linear Mohr-Coulomb theory are shown in Table 8.  
\[\tau=\sigma\tan\varphi+c \tag{5}\]  
Table 8 and Figure 11 show that according to the linear Mohr-Coulomb theory, the cohesion strength of asphalt mixtures decreases with the increase of temperature; the friction angle slightly decreases with temperature. Both cohesion strength and friction angle of asphalt mixture do not change with confining pressure. In the Mohr-Coulomb criterion, there are two important constants included, i.e., the cohesion strength \(c\) and friction angle \(\varphi\). The expression of friction angle \(\varphi\) for asphalt mixtures can be obtained as follows. \[\tan\varphi=\tau^{\prime}=\frac{\partial\tau}{\partial\sigma}=ab\sigma^{b-1} \tag{6}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Mixture Type** & \(\mathbf{m_{0}}\) & \(\mathbf{n_{0}}\) & \(\mathbf{m_{1}}\) & \(\mathbf{n_{1}}\) & \(\mathbf{m_{2}}\) & \(\mathbf{n_{2}}\) & \(\mathbf{R^{2}}\) \\ \hline SMA-13 & \(-7.0\times 10^{-3}\) & 1.083 & \(-0.10\times 10^{-2}\) & 0.840 & \(-1.10\times 10^{-2}\) & 0.817 & 0.99 \\ AC-13 & \(-6.9\times 10^{-3}\) & 0.990 & \(-0.15\times 10^{-2}\) & 0.832 & \(-0.85\times 10^{-2}\) & 0.658 & 0.98 \\ OGFC-13 & \(-5.0\times 10^{-3}\) & 0.827 & \(-0.20\times 10^{-2}\) & 0.820 & \(-0.65\times 10^{-2}\) & 0.512 & 0.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Parameters of the \(\sigma-\tau\) model considering the influence of temperature and confining pressure. Figure 11: Cohesion strength and friction angle by linear failure envelope. (**a**) The friction angle by linear failure envelope; (**b**) The cohesion strength by linear failure envelope. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Mixture Type**} & \multicolumn{2}{c}{**40 \({}^{\circ}\)C**} & \multicolumn{2}{c}{**50 \({}^{\circ}\)C**} & \multicolumn{2}{c}{**60 \({}^{\circ}\)C**} \\ \cline{2-7}  & \(\mathbf{c}\) & \(\mathbf{\varphi}\) & \(\mathbf{c}\) & \(\mathbf{\varphi}\) & \(\mathbf{c}\) & \(\mathbf{\varphi}\) \\ \hline SMA-13 & 0.518 & 35.030 & 0.389 & 32.129 & 0.273 & 30.795 \\ AC-13 & 0.475 & 31.048 & 0.323 & 29.249 & 0.253 & 27.022 \\ OGFC-13 & 0.391 & 27.474 & 0.293 & 25.641 & 0.218 & 24.228 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Strength parameters of asphalt mixtures by linear failure envelope.