# Shear Properties of Asphalt Mixtures under Triaxial Compression
## 3 Results and Discussion
### Nonlinear Mohr-Coulomb Failure Criterion
Change the partial differential to ordinary differential, and then  
\[\varphi=\arctan(ab\sigma^{b-1}) \tag{7}\]  
In addition, the tangent equation passing through the point \((\sigma_{m},\tau_{m})\) can be derived from Equation (3)  
\[\tau=\tau_{m}+\tau^{\prime}(\sigma-\sigma_{m})=\tau_{m}+ab(\sigma_{m})^{b-1}( \sigma-\sigma_{m}) \tag{8}\]  
Let \(\sigma=0\), then Equation (8) gives  
\[\tau_{\sigma=0}=\tau_{m}-ab\sigma_{m}{}^{b} \tag{9}\]  
Finally, denote \(\tau_{\sigma=0}\) as \(C\), and substitute Equation (3) into Equation (9), then one can obtain the expression for cohesion strength as  
\[C=a(1-b)\sigma_{m}{}^{b}+c \tag{10}\]  
where \(C\) is cohesion strength under different confining pressure, \(a,b,c\) are parameters in Equation (3). As a result, the nonlinear evolutional laws of Mohr-Coulomb strength parameters cohesion strength \(C\) and friction angle \(\varphi\) are obtained and shown in Table 9.  
In Figures 12-14, the dotted lines from top to bottom represent the friction angle and cohesion strength of SMA-13, AC-13, and OGFC-13 obtained by linear Mohr-Coulomb theory, respectively. It can be seen from Figures 12-14 that the linear Mohr-Coulomb theory cannot reflect the nonlinear variation of the cohesion strength and friction angle with testing conditions. Besides this, the linear Mohr-Coulomb theory in current asphalt pavement design produces a relatively large deviation in the calculation of cohesion strength and friction angle, which makes the design of pavement materials not accurate. Therefore, it is more appropriate to use the nonlinear failure criterion in the design of the asphalt pavement materials and structures. As can be seen from Table 9, the increase of temperature degrades the triaxial strength of asphalt mixtures. Compared with the continuous-graded AC-13 asphalt mixture and the open-graded OGFC-13 asphalt mixture, the gap-graded SMA-13 asphalt mixture exhibits better performance on the resistance to shear damage under complex stress conditions. It is mainly due to the framework dense structure of SMA-13 asphalt mixture was formed by coarse aggregates interlocked with each other and fine aggregates filled with voids. In addition, there are some fibers in the SMA-13 asphalt mixture which improves the shear strength [34; 35]. Considering the overloading phenomenon that generally \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Mixture Type** & **Confining Pressure** & \multicolumn{2}{c}{**40 \({}^{\circ}\)C**} & \multicolumn{2}{c}{**50 \({}^{\circ}\)C**} & \multicolumn{2}{c}{**60 \({}^{\circ}\)C**} \\ \cline{3-8}  & 0.2 & 0.516 & 33.666 & 0.372 & 32.516 & 0.249 & 30.565 \\  & 0.4 & 0.552 & 32.163 & 0.413 & 30.431 & 0.287 & 28.119 \\ SMA-13 & 0.6 & 0.594 & 30.769 & 0.448 & 29.052 & 0.320 & 26.621 \\  & 0.8 & 0.630 & 29.782 & 0.483 & 27.959 & 0.352 & 25.459 \\  & 1.0 & 0.664 & 29.004 & 0.517 & 27.093 & 0.381 & 24.564 \\ \hline  & 0.2 & 0.453 & 30.606 & 0.334 & 28.848 & 0.241 & 27.096 \\  & 0.4 & 0.491 & 28.776 & 0.374 & 26.384 & 0.279 & 24.408 \\ AC-13 & 0.6 & 0.525 & 27.466 & 0.405 & 25.003 & 0.310 & 22.782 \\  & 0.8 & 0.561 & 26.376 & 0.437 & 23.892 & 0.342 & 21.521 \\  & 1.0 & 0.595 & 25.487 & 0.467 & 22.994 & 0.370 & 20.600 \\ \hline  & 0.2 & 0.370 & 27.445 & 0.296 & 25.660 & 0.217 & 24.595 \\  & 0.4 & 0.414 & 24.982 & 0.334 & 23.151 & 0.254 & 21.776 \\ OGFC-13 & 0.6 & 0.447 & 23.596 & 0.366 & 21.540 & 0.288 & 19.937 \\  & 0.8 & 0.478 & 22.546 & 0.402 & 20.187 & 0.316 & 18.753 \\  & 1.0 & 0.507 & 21.707 & 0.423 & 19.494 & 0.341 & 17.844 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Strength parameters of asphalt mixtures by nonlinear failure envelope.