## 2 Model Setup and Microparameter Calibration  
In practical rock engineering, most of the joints in natural rock masses are discontinuous joints. As shown in Figure 2, there are rock bridges with different shapes in the three-dimensional discontinuous joints. In previous studies, few papers have focused on the failure mechanism of 3D nonpersistent jointed rock masses. In the present paper, the fracture behaviour of a three-dimensional nonpersistent jointed rock mass was studied and analysed based on DEM. To study the effect of the inclination and persistence of 3D nonpersistent joints on the failure behaviour of rock masses, different inclination (\(\theta\) = 0\({}^{\circ}\), 30\({}^{\circ}\), 45\({}^{\circ}\), 60\({}^{\circ}\), 75\({}^{\circ}\), and 90\({}^{\circ}\)), and persistence (K1 = 0.42, K2 = 0.61, K3 = 0.75, K4 = 0.88 and K5 = 0.94) values are considered in this paper. The specimen number is used by S-\(\theta\)-K, where S indicates the specimen, \(\theta\) indicates the inclination, and K is the persistence. The dimensions of the experimental specimens are also shown in Figure 2.  
Figure 1: Joints with different inclinations in pillars [19]. For the joint persistence, as there are rock bridges with different shapes in the three-dimensional discontinuous joints, it is calculated based on the area ratio. Thus, the joint persistence of the 3D nonpersistent joint is:  
\[K=\sum\biggl{[}\frac{S_{I}-S_{RBi}}{S_{J}}\biggr{]} \tag{1}\]  
where \(S_{RBi}\) is the total area of the rock bridges and \(S_{I}\) is the area of the complete joint [21]. Figure 3 shows the PFC3D numerical model for 3D nonpersistent joint specimens, and the external dimension of the PFC model is the same as that of the experimental specimen (70 mm in length and width, 140 mm in height). The particles belonging to the individual joint parts are identified separately, and the lower strength parameters are set to form a 3D nonpersistent joint surface. In previous studies, PFC has been widely adopted to simulate the whole failure process of jointed rocks [22; 23; 24]. As we know, there are several kinds of contact models in PFC. In this paper, the linear parallel bond model and smooth-joint model have been adopted to simulate the rock-like material and joint interface, respectively. For the linear parallel bond model, since the bond follows the Coulomb criterion, as the corresponding strength limitations are breached, the bond undergoes tensile or shear failure. Previously, the joints of rock masses were modelled by eliminating particles or weakening strength parameters. However, this also leads to the problem of the high strength of the rock mass in the simulation results [14; 25; 26]. As the smooth-joint model allows the particles intersecting with S-J to overlap and pass through each other (Figure 3) and follows the Coulomb sliding model, this problem was also successfully resolved. The experimental and simulated values during the parameter calibration are shown in Table 1. The microparameters of the linear parallel bond model are calibrated (by the \"trail-and-error\" method) based on the macro strength and deformation parameters (including the uniaxial compression strength, elastic modulus and Poisson's ratio), which were acquired from mechanical tests of rock-like materials, and the calibration results are shown in Table 2. At the same time, the microparameters of the smooth-joint model were calibrated by the shear parameters (including normal stiffness, shear stiffness and friction angle) obtained from laboratory direct shear tests of specimens with intact planar joint planes, as shown in Table 3.  
Figure 2: 3D nonpersistent joint and jointed specimen [20].