## 4 Numerical Model and Materials
### Soil-Tunnel System
#### 4.1.2 Material Properties  
During the calculations, the concrete damage plasticity (CDP) model is adopted to describe the irreparable damage of the concrete under the seismic action. Based on the CDP model, the damage factor ranges from 0 to 1, representing no damage and complete failure of concrete, respectively. As C60 high performance concrete is used for the segmental lining, the corresponding elasticity modulus, Poisson's ratio, shear expansion angle, flow potential offset value, density, cohesion coefficient, and eccentricity are 3.55 \(\times\) 10\({}^{4}\) MPa, 0.2, 0.67, 2450 kg/m\({}^{3}\), 35\({}^{\circ}\), 0.001, and 0.1, respectively. Additionally, the ratio of biaxial ultimate compressive strength to uniaxial ultimate compressive strength is set to be 1.16. All the above parameters were determined as [PERSON] et al. reported [41]. More detailed governing parameters of the CDP model for C60 concrete are listed in Table 1. The damage parameters of the concrete can be obtained by the following equation.  
\[d_{\mathrm{k}}=\frac{(1-\beta)\epsilon_{\mathrm{k}}^{\mathrm{in}}E_{0}}{ \sigma_{\mathrm{k}}+(1-\beta)\epsilon_{\mathrm{k}}^{\mathrm{in}}E_{0}},( \mathrm{k=t,c}) \tag{1}\]  
in which, \(d_{\mathrm{k}}\) denotes the damage parameter, while \(\mathrm{k=t}\) means tension and \(\mathrm{t=c}\) means compression; \(E_{0}\) is the initial elasticity modulus of the concrete; \(\epsilon_{\mathrm{k}}^{\mathrm{in}}\) is the strain in the inelastic stage; \(\sigma_{\mathrm{k}}\) is the stress; \(\beta\) is the proportional coefficient of plastic strain to inelastic strain.  
On the other hand, the elastoplastic model is employed to describe the plastic behavior of the bending bolt joint. For the bending bolt with a strength grade of 8.8, the corresponding elastic modulus, yield stress, and Poisson's ratio are set to be 210 GPa, 640 MPa, and 0.25, respectively [40]. While for the anchor joint, multidirectional springs with bilinear behavior are used, including the tensile stiffness and shear stiffness. This assumption was used in this paper based on the reports published in [42]. The initial tensile stiffness of the anchor joint is set to be 350 MN/m, while that will be decreased to 0 if the joint opening exceeds 1.5 mm. Similarly, the initial shear stiffness of the anchor joint is 80 MN/m, while that will be increased to 230 MN/m if the joint offset exceeds 2 mm.  
Figure 5: 3D numerical model of the soil-tunnel system and local details.  
During the calculations of wave propagation in soft soil and rock, it is important to capture the soil nonlinearity. Consequently, the equivalent linear approach is selected to define the approximate nonlinear behavior of soil through a continuous update of the shear modulus and damping ratio according to the calculated soil strain, which has been widely used in this field. The maximum shear modulus of the soil can be calculated from the following equation. More detailed geological properties of different soil layers can be found in Table 2.  
\[G_{\max}=\rho V_{\mathrm{s}}^{2} \tag{2}\]  
where, \(G_{\max}\) is the maximum shear modulus; \(\rho\) is the density; \(V_{\mathrm{s}}\) is the shear wave velocity.