## 4 Verification of the Established Rock Constitutive Model
### Effect of the Number of Points  
In order to investigate the effect of the number of joints on the transmission coefficient of stress waves under different biaxial static loads, numerical calculations were conducted on the cases of jointed rock masses with vertical incidence (i.e., the angle of joints was 90\({}^{\circ}\)) and oblique incidence (the angle of joints was 60\({}^{\circ}\)) of stress waves propagation. Based on this, the numerical models of rock masses containing a different number of 90\({}^{\circ}\) and 60\({}^{\circ}\) joints were established. In the numerical calculation, six different number of joints were selected, which were 1, 2, 3, 5, 7, and 9, respectively. At the same time, according to the joint spacing in the three-joints test sample T3 in the model test, the joint spacing selected in the numerical model was also 50 mm. According to the number of joints, the size of the numerical model established was 1300 mm (length) \(\times\) 1300 mm (width), and the numerical models including five joints with angles of 90\({}^{\circ}\) and 60\({}^{\circ}\) are shown in Figure 13.  
The applied biaxial static loads in the numerical simulation were somewhat different from those in Section 4.1, which were 0 MPa, 0.75 MPa, 1.5 MPa, 2.25 MPa, 4.5 MPa, and 6.75 MPa, for a total of six conditions, while the applied stress wave load \(P(t)\) remained the same as in Section 4.1. To eliminate the effects of the geometric and physical attenuation of the stress waves, the selected measurement points in the numerical model were node A located on the measurement line near the left side of the first joint and node B located on the measurement line near the right side of the ninth joint when the number of joints was nine, as shown in Figure 13 above, which was slightly different from the arrangement of the measurement points in the T2 test sample in the model test. The variation of the transmission coefficient with the number of joints for vertical and oblique incidence of stress waves under different biaxial static loads was sorted out as shown in Figure 14.  
Figure 13: Numerical calculation model of the rock mass containing five joints with different angles (unit: mm): (**a**) joint angle of 90\({}^{\circ}\); (**b**) joint angle of 60\({}^{\circ}\).  
As can be seen from Figure 14 above, the transmission coefficients of both the 90\({}^{\circ}\) and 60\({}^{\circ}\) joint rock masses showed a decreasing trend with an increase in the number of joints under the same static load, and the decreasing increment gradually became smaller, especially after the number of joints reached a certain number (more than five), the attenuation effect of the increase in the number of joints on the stress wave propagation gradually weakened.  
At the same time, the reduction in the transmission coefficient for the 90\({}^{\circ}\) joint rock mass was 70.5%, 47.1%, 34.2%, 32.6%, 29.1%, and 26.8% when the number of joints was increased from 1 to 9 with biaxial static loads of 0 MPa, 0.75 MPa, 1.5 MPa, 2.25 MPa, 4.5 MPa, and 6.75 MPa, respectively, and the corresponding reduction in the transmission coefficient for the 60\({}^{\circ}\) joint rock mass was 36.3%, 34.4%, 31.7%, 30.0%, 25.7%, and 22.6%, respectively. The above results showed that when the static load was small, such as 0 MPa and 1.5 MPa, the decrease in the transmission coefficient of the 90\({}^{\circ}\) jointed rock masses caused by increasing the number of joints was significantly larger than that of the 60\({}^{\circ}\) jointed rock masses, and the larger the static load was, the smaller the decrease in the stress waves transmission coefficient caused by increasing the number of joints for both 90\({}^{\circ}\) and 60\({}^{\circ}\) jointed rock masses, indicating that the attenuation effect of the number of joints on stress wave propagation became weaker as the static load increased.