# Shear Properties of Asphalt Mixtures under Triaxial Compression
## 3 Results and Discussion  
### Triaxial Strength at 40 \({}^{\circ}\)C, 50 \({}^{\circ}\)C, and 60 \({}^{\circ}\)C  
The average values of three effective test results of triaxial strength are presented in Table 3. In the triaxial test, the specimens were mainly represented as shear failure, and some diagonal cracks appeared on the surface of the specimen as shown in Figure 4.  
Figures 5-7 show the triaxial compressive strength \(\sigma_{3}\) of asphalt mixtures under various confining pressure at 40 \({}^{\circ}\)C, 50 \({}^{\circ}\)C, and 60 \({}^{\circ}\)C; the sequence of triaxial strength of asphalt mixtures is SMA-13 \(>\) AC-13 \(>\) OGFC-13. It can be seen from those test results that the triaxial strength is highly influenced by confining pressure and temperature. The triaxial strength increases with the increase of confining pressure, and it increases with the decrease of temperature. All the test results were fitted well with a power function as following:  
\[\sigma_{3}=A\sigma_{1}^{n}+f_{c} \tag{1}\]  
Figure 4: Shear failure of specimen.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Mixture Type**} & \multirow{2}{*}{**Confining Pressure \(\sigma_{1}\) (MPa)**} & \multicolumn{3}{c}{**Triaxial Strength \(\sigma_{3}\) (MPa)**} \\ \cline{3-5}  & & **40 \({}^{\circ}\)C** & **50 \({}^{\circ}\)C** & **60 \({}^{\circ}\)C** \\ \hline \multirow{5}{*}{SMA-13} & 0 & 1.968 & 1.342 & 0.837 \\  & 0.2 & 2.822 & 2.079 & 1.456 \\  & 0.4 & 3.364 & 2.688 & 2.018 \\  & 0.6 & 4.139 & 3.248 & 2.509 \\  & 0.8 & 4.690 & 3.844 & 3.047 \\  & 1.0 & 5.284 & 4.350 & 3.506 \\ \hline \multirow{5}{*}{AC-13} & 0 & 1.613 & 1.142 & 0.750 \\  & 0.2 & 2.339 & 1.651 & 1.269 \\  & 0.4 & 2.856 & 2.252 & 1.783 \\  & 0.6 & 3.398 & 2.686 & 2.238 \\  & 0.8 & 4.024 & 3.229 & 2.737 \\  & 1.0 & 4.567 & 3.743 & 3.105 \\ \hline \multirow{5}{*}{OGFC-13} & 0 & 1.183 & 0.842 & 0.657 \\  & 0.2 & 1.835 & 1.465 & 1.153 \\ \cline{1-1}  & 0.4 & 2.397 & 1.941 & 1.605 \\ \cline{1-1}  & 0.6 & 2.796 & 2.388 & 2.059 \\ \cline{1-1}  & 0.8 & 3.207 & 2.801 & 2.407 \\ \cline{1-1}  & 1.0 & 3.599 & 3.243 & 2.766 \\ \hline \hline \end{tabular} Note: \(\sigma_{3}\) and \(\sigma_{1}\) are sorted by absolute values of compressive stresses.  
\end{table}
Table 3: Triaxial results of asphalt mixtures.  
where \(\sigma_{3}\) denotes the triaxial compressive strength, MPa, \(\sigma_{1}\) is the confining pressure, MPa, \(f_{c}\) is the uniaxial compressive strength, MPa. \(A\) and \(n\) are parameters to be determined by least square method. The parameters of this model are listed in Table 4.  
Figure 5: Triaxial results of SMA-13 asphalt mixture.  
Figure 6: Triaxial results of AC-13 asphalt mixture.  
Figure 7: Triaxial results of OGFC-13 asphalt mixture.  
It can be seen from Table 4 that with the increase of temperature, the parameter \(A\) and \(n\) decrease, and the \(\sigma_{3}-\sigma_{1}\) curves bend down. The increase in amplitude of triaxial strength with confining pressure became larger and larger. Based on the results, a regression/prediction model of triaxial strength considering the effects of temperature and confining pressure for asphalt mixtures is shown in Equation (2).  
\[\sigma_{3t}=(a_{0}t+b_{0})\sigma_{1}^{(a_{1}t+b_{1})}+f_{ct} \tag{2}\]  
where \(\sigma_{3t}\) denotes triaxial compressive strength at different temperatures, \(\sigma_{1}\) is the confining pressure, \(f_{ct}\) is the uniaxial compressive strength at different temperatures, \(t\) is the temperature. The parameters of this triaxial strength model are shown in Table 5.