## 2 Numerical Modeling
### Materials and Parameters  
In this study, the numerical simulation assumes that the physical and mechanical properties of the rock mass and fault fracture zones are highly homogeneous and that joints in the surrounding rock are not considered. The intact rock mass and fault fracture zones are modeled using the Mohr-Coulomb criterion rock damage model. The physical and mechanical parameters of the rock mass are determined based on on-site geological investigation data and the Standard for engineering classification of rock masses (GB/T50218-2014) [24], as summarized in Table 1.  
To enhance the characterization of the model's dynamics, Rayleigh damping is applied to the surrounding rock, with the damping coefficient calculated using Equation (1):  
\[C=\alpha_{1}M+\beta_{1}K \tag{1}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Density/(kg/m\({}^{3}\))** & \
\begin{tabular}{c} **Elastic** \\ **Modulus/GPa** \\ \end{tabular} & **Poisson Ratio** &
\begin{tabular}{c} **Internal Friction** \\ **Angle/(\({}^{\circ}\))** \\ \end{tabular} & **Cohesion/(MPa)** \\ \hline The complete rock mass & 2100 & 3.0 & 0.33 & 31 & 0.40 \\ Fault fracture zone & 1700 & 1.1 & 0.41 & 21 & 0.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Rock mechanical parameters [25]. Figure 2: Three-dimensional numerical model. Here, \(\alpha_{1}\) and \(M\) represent the mass coefficients and mass matrices, respectively, while \(\beta_{1}\) and \(K\) represent the stiffness coefficients and stiffness matrices, respectively. \(\alpha_{1}\) and \(M\) are calculated using Equation (2):  
\[\alpha_{1}=\frac{2\xi\omega_{i}\omega_{j}}{\omega_{i}+\omega_{j}},\beta_{1}= \frac{2\xi}{\omega_{i}+\omega_{j}} \tag{2}\]  
Here, \(\omega_{i}\) and \(\omega_{j}\) represent the \(i\)-th and \(j\)-th self-oscillation frequencies of the model, respectively, selected as the 1 st and 3 rd orders in the simulation. \(\xi\) denotes the damping ratio of the surrounding rock, set to 0.05 in the simulation [26]. Typically, to realistically simulate the mechanical behavior of tunnels under fault dislocation, it is crucial to account for the nonlinear characteristics of the tunnel lining materials. Given that tunnel damage is associated with concrete cracking and crushing, the tunnel structure in the simulation is modeled using the Concrete Damaged Plasticity model (CDP model) integrated into ABAQUS. The CDP model can simulate the strain softening and stiffness degradation of concrete under loading. It divides concrete damage into tensile and compressive damage, and material stiffness degradation is depicted using two separate normalization indices: tensile damage (\(d_{\text{t}}\)) and compressive damage (\(d_{\text{c}}\)). Figure 3 illustrates the tensile and compressive stress-strain relationship of the lining material. Here, \(E\) represents the initial modulus of elasticity,\(f_{\text{c}}\) and\(f_{\text{t}}\) are the stresses at the material's failure points, and \(\varepsilon\) is the equivalent plastic strain. When the material is damaged, the slope of its unloading curve is given by (\(1-d_{\text{ct}}\))\(E\). The stress-strain relationship of the lining material conforms to the design standards for Code for the Design of Concrete Structures (GB50010-2010) [27]. The specific parameters are listed in Table 2.  
In tunnel construction, the primary support can effectively integrate with the surrounding rock; thus, the binding constraint in ABAQUS is applied to model the interaction between the primary support and the surrounding rock. Additionally, considering the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Density/(kg/m3)** & **Elastic** & **Poisson Ratio** & **Compressive** & **Tensile Yield** \\  & **Modulus/MPa** & & **Stress/MPa** & **Stress/MPa** & **Stress/MPa** \\ \hline Primary support & 2400 & 30,000 & 0.2 & 20.1 & 2.01 \\ Secondary lining & 2500 & 31,500 & 0.2 & 23.4 & 2.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical parameters of the lining.