## 3 Results and Discussion  
### Strength and Failure Modes  
Figure 4 shows the peak strength of the 3D nonpersistent-jointed specimen obtained from numerical simulation. For the specimen with inclinations \(\theta\) = 0\({}^{\circ}\) and 90\({}^{\circ}\), when the joint persistence varies from 0.42 to 0.94, the peak strength is basically stable, and the strength of the specimen is closer to the initial strength value (peak strength of the intact specimen). However, when the inclination angle \(\theta\) = 45\({}^{\circ}\) to 75\({}^{\circ}\), as the joint persistence varies from 0.42 to 0.94, the peak strength of the specimen decreases obviously. This also indicates that the joint persistence plays an important role in the peak strength for the specimens with joint inclinations of 45\({}^{\circ}\) to 75\({}^{\circ}\). With the change in \(\theta\), the failure characteristics of jointed specimens will also change greatly. For the specimens with \(\theta\) = 45\({}^{\circ}\) to 75\({}^{\circ}\), due to the influence of \(\theta\), the specimens mainly failed due to slippage along the joint plane. Thus, with the increase in joint persistence (K), the smaller the proportion of rock bridges on the joint plane, the weaker the resistance to sliding along the joint surface. Therefore, the joint persistence has a great effect on the strength of the specimens that experience slippage failure. However, when \(\theta\)\(\leq\) 45\({}^{\circ}\) or \(\theta\) = 90\({}^{\circ}\), the failure behaviour of the specimens is closer to the split failure, and almost no slippage occurs along the joint plane. Thus, the joint persistence seems to play a minor role in the strength of the specimens that experience splitting failure. Figure 5 shows the peak strength comparisons between the test and numerical results. In general, the numerical results show great agreement with those obtained from the laboratory tests. For all joint persistence conditions, the strength of the specimen with \(\theta\) = 0\({}^{\circ}\) and 90\({}^{\circ}\) is closer to the initial strength value, and it is obviously different from the strength of specimens with inclinations of 45\({}^{\circ}\) to 90\({}^{\circ}\). However, it should be noted that when the joint inclination of the specimen is 30\({}^{\circ}\), its failure mode is the combination of splitting and shear. Thus, its strength is also in the transition position, and the change in joint penetration has a medium impact on it. Figure 4: Variation in peak strength with joint persistence. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Normal stiffness \(\overline{k}_{n}\) (\(\mathrm{GPa}\times\mathrm{m}^{-1}\)) & 85 \\ Shear stiffness \(\overline{k}_{s}\) (\(\mathrm{GPa}\times\mathrm{m}^{-1}\)) & 20 \\ Friction coefficient \(\mu\) & 0.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Microparameters of the smooth-joint (S-J) model. Figure 6 shows the failure patterns for jointed specimens with different inclinations and persistence. Figure 6 shows that for the specimen with inclination \(\theta\) = 0\({}^{\circ}\), all specimen failures belong to splitting failure, and the splitting plane in the specimen originates near the 3D nonpersistent joint plane. When the inclination angle \(\theta\) = 45\({}^{\circ}\) to 75\({}^{\circ}\), there is no splitting failure plane in the specimen; instead, there is an obvious slippage fracture plane along the joint plane. This also indicates that the specimen fails from the shear crack coalescence between the rock bridges. It is not difficult to find that the failure mode of the specimen with \(\theta\) = 30\({}^{\circ}\) is in the transition region between splitting failure and shear failure. There are not only shear fracture planes but also partial splitting failure bands in the specimen, especially for the specimen with joint persistence values of 0.42 and 0.61.