## 4 Verification of the Established Rock Constitutive Model
### Numerical Model and Calculation Procedure  
According to the three test samples designed in the model test as shown in Figure 2, the corresponding discrete element numerical models based on UDEC were established as shown in Figure 7. The dimensions of the numerical models were identical to the model test samples, whose length and width were 1600 mm and 1300 mm, respectively, while the lengths and spatial locations of the joints in the different numerical models were consistent with the model test samples. Due to the high mechanical strength of the seamless steel pipe, its deformation under static loads was approximately negligible, so it was not necessary to consider the quick-drying material inside the seamless steel pipe in the numerical model. The interior of the steel pipe was blank and the equivalent blast loads were applied directly to the inner wall of the steel pipes. Figure 6: Calculation flow of the Mohr-Coulomb elasto-plastic constitutive model of the rock considering the initial damage. After meshing, the established numerical models all contained intact rock, joint, and seamless steel pipe elements, and the number of degrees of freedom of the element was 3, including the translation in the x and y directions, and the rotation in the x-y plane. According to the statistics, the number of seamless steel pipe elements in the T1, T2, and T3 test blocks was 40, while the number of intact rock and joint elements were 25,510 and 84, 36,752 and 178, and 32,536 and 148, respectively. The static loading and constraints of the numerical models were the same as that of the model test samples. The fixed constraint was applied at the lower end of the model, and the uniformly distributed load was applied on the other three sides. The magnitude of the static loads applied on the numerical models was the same as the biaxial static loads in the model test. In reference to the physical and mechanical parameters of the rock and the joint simulated materials in the model test, the corresponding mechanical parameters of the rock, joint, and seamless steel pipe in the numerical simulation are shown in Table 2.  
Only the elastic modulus and Poisson's ratio of the simulated rock material were derived in the model tests, while the mechanical parameters used in the established rock constitutive model were the bulk modulus \(K\) and shear modulus \(G\), which can be converted by the following equation. \[\left\{ \begin{array}{l}K=\frac{E}{3(1-2\mu)}\\ G=\frac{E}{2(1+\mu)}\end{array} \right. \tag{7}\]  
Figure 7: Numerical model of the different model test samples (unit: mm): (**a**) T1 sample; (**b**) T2 sample; (**c**) T3 sample. In the numerical model, the developed Mohr-Coulomb elasto-plastic constitutive model considering the initial damage was used to simulate the intact rock. According to the results of the model test, the nonlinear BB model could express the mechanical response of the joints under dynamic loading in the presence of the initial static loading, so the joints were simulated in the numerical simulation using the BB model in UDEC. Meanwhile, the strength of the seamless steel pipe was relatively higher compared to the rock and joints simulation materials as seen in Table 2, so the linear elastic model in UDEC was chosen for the simulation. The detailed numerical calculation process was as follows:  
1. The initial static loads were first applied on the numerical model, where the applied static loads in the horizontal and vertical directions were of equal magnitude. To compensate for the insufficient static load conditions in the model test and considering the biaxial compressive strength of the rock material of 7.826 MPa, the number of static load conditions was increased in the numerical simulation, and there were 11 different static load conditions, namely 0, 0.75 MPa, 1.5 MPa, 2.25 MPa, 3 MPa, 3.75 MPa, 4.5 MPa, 5.25 MPa, 6 MPa, 6.75 MPa, and 7.5 MPa. 2. After the static loads were applied, an equivalent blast load curve was applied to the inner wall of the seamless steel tubes in the numerical model.