## 4 Numerical Simulation
### Numerical Results and Discussions  
#### 4.4.1 Cavern-Free Sample  
Figure 10 presents the step-stress-crack relationship during the numerical test, the maximum z-axis strength of the numerical cavern-free sample is 22.38 MPa (slightly higher than the corresponding result), and the initiation of the tensile crack is earlier than the shear crack. Four special states of the sample were selected, which include the initiations of tensile and shear cracks, the construction of splitting failure, and the final peak stress state; the corresponding progressive displacement and velocity trends of the cavern-free sample are shown in Figure 11:(1) The displacement trend field presents a typical centripetal distribution before the shear failure occurs (Figure 11a), while the velocity trend field shows an opposite moving direction in the sidewall (moving outwards).  
(2) Sporadic tensile cracks distribute near the tunnel free face. When the secondary shear cracks start to gather at the bottom of sidewall (Figure 10b), particles in the sidewall only show vertical deformation because the corresponding velocity direction is outward.  
(3) Surrounding rock adjacent to the tunnel face starts to show an inward displacement trend as the splitting failure forms, and the corresponding velocity trend illustrates a haphazard distribution in this region (Figure 10c); the splitting failure mainly distributes in the tunnel sidewall, and the top vault and bottom boundaries of the tunnel suffer slight tensile failure (tensile crack bands formed).  
(4) The displacement and velocity trends of particles present the same distribution characteristic at the peak stress state (Figure 10d), and the majority of secondary failures appear in the tunnel sidewall, which shows a perfect agreement with the experimental results. Although the thickness of secondary cracks is large, the typical splitting failure only covers about 3 mm thickness (based on the displacement and velocity trends).  
To summarize, the particle displacement trend field demonstrates a gradual change during the true triaxial compression test, which is different from the uniaxial compression test we have carried out before [25]. The application of the velocity trend field helps to capture the real change characteristics of the surrounding rock.  
Distributions of the initial and final principal stress axes fields are illustrated in Figure 12. After the prestress loading is completed, the principal stress directions in the surrounding rock remain basically unchanged from the beginning to the end of the test. Moreover, the maximum principal stress axes are vertical to the tunnel boundary; the minimum principal stress is almost zero near the tunnel free face and gradually increases as the measuring position moves outward.  
On the other hand, the main difference between the initial and final state is the relationship between the magnitude of the intermediate and minimum stress values in different regions. The sidewall shows relatively smaller principal stress values (red frame in Figure 12b) than those obtained from other positions in the initial state; however, the relatively high principal stress distributes in the sidewall in the final state (red frame in Figure 12(b1)), and the outermost layer of the principal stress axes is essentially the same as the external load characteristics (x-axial loading is larger than y-axial loading in the experimental test).