## 4 Numerical Simulation
### Numerical Results and Discussions
#### 4.4.2 Cavern-Existing Sample  
The step-stress-crack relationship of the numerical cavern-existing sample is shown in Figure 13. Five main steps, including tensile crack initiation, shear crack initiation, intermediate rock failure, start of splitting failure, and final peak stress status, were selected because of the prefabricated cavern; the appearance times of tensile, shear, and splitting failure are relatively earlier than the corresponding results of the cavern-free sample (Figure 10). Figure 14 shows the progressive crack locations and displacement and velocity trends, and the main failure characteristics are as follows:  
(1) When the tensile cracks initiate (Figure 14a), the displacement trend field presents a typical centripetal distribution on both sides of the sample (like the results of the cavern-free sample), while the velocity trends in the sidewalls present an opposite direction to the displacement vectors (outward). (2) Shear cracks begin to gather at the bottom of the right sidewall (Figure 14b), and some shear cracks distribute in the intermediate rock between the tunnel and cavern. Displacement vectors mainly show the vertical direction in the left sidewall and present a slight horizontal displacement component in the right sidewall. The velocity trend field in the left sidewall maintains a stable distribution, while some particles (intermediate rock, marked by the small arrow) present a centripetal distribution. (3) Because the velocity trends always show an outward direction, particles' horizontal displacement components approach zero in the left sidewall, and displacement trends in the right sidewall present an opposite direction (the inner rock shows outward movement, and the outer rock illustrates inward movement in the horizontal direction). Tensile and shear cracks densely distribute in the intermediate rock, and there is almost no crack in the right sidewall. In the above three stages, particles above the cavern continue to show vertical displacement and velocity, and particles outside the cavern show stable right-bottom moving direction. (4) When the splitting failures are formed (Figure 14d), the splitting boundary in the left sidewall is clear (illustrated by the displacement and velocity trends) and secondary cracks quickly appear in the inner layer of the right sidewall. Many tensile cracks wrap the cavern, and the shear cracks only occur in the intermediate rock and the right surrounding rock. Furthermore, the velocity trends show a disorganized distribution in some regions (red frames), such as the bottom of the left sidewall, the whole right sidewall, and the top-right surrounding rock of the cavern. Figure 12: Distribution of the 3D principal stress axes at the initial and final states of the cavern-free sample (semi-figures (**a**,**b**) are initial state, semi-figures (**a1**,**b1**) are final state). (5) The above disorganized velocity trends disappear at the peak stress state, and both sides of the splitting failures remain as unchanged distribution areas (the corresponding crack density gradually increases). However, particles in the intermediate rock present several velocity and displacement directions, which follows the moving characteristics of the rock block in the experimental test. Distributions of the principal stress axes fields are illustrated in Figure 15. Because the prefabricated cavern is located at the right vault of the tunnel, the measuring region of the principal stress axes is thus broadly rightward for 20 mm. Like the results of the cavern-free sample, the maximum principal stress axes are mainly vertical to the boundaries of the tunnel and cavern in both the initial and final states. In addition, some maximum principal stress axes (in the tunnel vault and the bottom and upper surrounding rock of the cavern) approach zero at the final state (Figure 15(a1)); the prefabricated cavern shows almost no influence on the maximum principal stress axes in the right sidewall.