## 3 Establishing the Numerical Model
### Constitutive Model  
The modified [PERSON] viscoelastic dynamic constitutive model was used to describe the dynamic characteristics of soil and rock [35], through the following expressions:  
\[\tau-\tau_{\text{c}}=G_{\text{max}}\cdot(\gamma-\gamma_{\text{c}})\cdot\bigg{[}1- H\bigg{(}\frac{|\gamma-\gamma_{\text{c}}|}{2n_{a}}\bigg{)}\bigg{]}, \tag{2}\]  
\[(2n_{a}\gamma_{0})^{2\text{B}}=\left(\gamma_{\text{ex}}\pm\gamma_{\text{c}} \right)^{2\text{B}}\cdot\bigg{(}\frac{1-R}{R}\bigg{)}, \tag{3}\]  
\[R=\bigg{(}1-\frac{\tau_{\text{ex}}\pm\tau_{\text{c}}}{G_{\text{max}}\cdot( \gamma_{\text{ex}}\pm\gamma_{\text{c}})}\bigg{)}^{\frac{1}{\alpha}}, \tag{4}\]  
where \(\tau\) and \(\gamma\) are the shear stress and the strain, respectively; \(G_{\text{max}}\) is the initial shear modulus; \(A\), \(B\), and \(\gamma_{0}\) are the dimensionless fitting parameters; \(n_{a}\) is the coefficient controlling the scale of hysteresis loop; \(\tau_{\text{c}}\) and \(\gamma_{\text{c}}\) denote the shear stress and the strain at the last stress reversal point, respectively; and the \(\tau_{\text{ex}}\) and \(\gamma_{\text{ex}}\) are the shear stress and strain at the last extreme value point, respectively. The parameters of this model for soil and rock are shown in Table 2.  
The tunnel lining's concrete was assumed to be C30, and, to describe the compressive and tensile damage and degrading of the concrete, the concrete damage-plasticity constitutive model proposed by [PERSON] and [PERSON] [36] was adopted. The corresponding material property parameters for the tunnel's concrete lining are shown in Table 3. The physico-mechanical parameters of the fluid in the present analysis are described in Table 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**No.**} & \multicolumn{1}{c}{**Model**} & \multicolumn{1}{c}{**Incidence Direction**} & \multicolumn{1}{c}{**Wave Type**} & \multicolumn{1}{c}{**Incidence Angles**} \\ \hline \multicolumn{1}{c}{} & 1 & & & & 33.69\({}^{\circ}\) \\ \multicolumn{1}{c}{} & 2 & & Left-side & & 26.57\({}^{\circ}\) \\ \multicolumn{1}{c}{} & 3 & & incidence & & 18.43\({}^{\circ}\) \\ \multicolumn{1}{c}{} & 4 & One-sided mountain & & & 11.31\({}^{\circ}\) \\ \multicolumn{1}{c}{} & 5 & tunnel model & & & 33.69\({}^{\circ}\) \\ \multicolumn{1}{c}{} & 6 & Right-side & & & 26.57\({}^{\circ}\) \\ \multicolumn{1}{c}{} & 7 & & incidence & & 18.43\({}^{\circ}\) \\ \multicolumn{1}{c}{} & 8 & & & & 11.31\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sets of numerical simulation. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Soil Types**} & **Density** & **Poisson’s** & **Shear Wave** & **Compressive** & **Initial Shear** & \multicolumn{2}{c}{**Fitting Parameters**} \\ \cline{3-8}  & **(kg/m\({}^{3}\))** & **Ratio** & **Velocity** & **Wave Velocity** & **Modulus** & \multicolumn{1}{c}{\(A\)} & \multicolumn{1}{c}{\(B\)} & \multicolumn{1}{c}{\(\gamma_{0}\) (\(\times 10^{-4}\))} \\ \hline Medium- & & & & & & & \\ coarse sand & & & & & & & \\ Bed rock & 2522 & 0.2 & 899.82 & 1469.40 & 2042 & 1.30 & 0.40 & 10.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of the fitting parameters of the modified Davidenkov model. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & **Description** & **Value** \\ \hline \(\rho\) & Density & 2450 kg/m\({}^{3}\) \\ \(E_{0}\) & Elastic modulus & 30,000 MPa \\ \(\mu\) & Poisson’s ratio & 0.18 \\ \(\overline{\
u}\) & Dilation angle & 36.31\({}^{\circ}\) \\ \(\varepsilon_{0}\) & Flow potential eccentricity & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Material property of the tunnel’s concrete lining.