## 3 Model Description for Water Conveyance Tunnel
### Material Nonlinearity and Constitutive Model  
The concrete model of the tunnel is assumed as the linear elastic material. Its mass density is 2500 kg/m\({}^{3}\). Its elastic modulus is 35.5 GPa and its Poisson ratio is 0.2. The widely used rubber model of Mooney-Rivlin [25; 32] is adopted for describing the deformation joints. The law of Mooney-Rivlin for the incompressible rubber is based on a strain energy function, which can be described as:  
\[W=C_{1}\left(I_{1}-3\right)+C_{2}\left(I_{2}-3\right)+\left(0.5C_{1}+C_{2} \right)\left(I_{3}{}^{-2}-1\right)+\frac{C_{1}\left(5\upsilon-2\right)+C_{2} \left(11\upsilon-5\right)}{2\left(1-2\upsilon\right)}\left(I_{3}-1\right)^{2} \tag{11}\]  
in which the material parameters derived from the uniaxial test data are represented by \(C_{1}\) and \(C_{2}\); the Cauchy-Green tensor invariants are described by \(I_{1}\), \(I_{2}\) and, respectively; the Poisson ratio is represented by \(\upsilon\). The parameters of this material used in this study include mass density of 1300 kg/m\({}^{3}\), a Poisson ratio of 0.4999, constant \(C_{1}\) of \(7.0\times 10^{5}\) Pa, and constant \(C_{2}\) of \(3.5\times 10^{5}\) Pa.  
Because of the slight compressibility of the Newtonian fluid (inner water) used in this study, the viscous stress caused by volumetric change is nearly zero. So, the constitutive equation is given by:  
\[\sigma_{ij}=-p\delta_{ij}+\tau_{ij} \tag{12}\]  
where \(p\) is pressure; \(\delta_{ij}\) is Kronecker-\(\delta\); \(\tau_{ij}\) signifies the deviatoric stress tensor. On the right side of Equation 13, the first item is defined with EOS (equation of state) in simulation; the second item is the deviatoric stress and is obtained through the material constitutive model.  
The null material model is used to compute the deviatoric stress of the inner water. The EOS of inner water can be described by the Gruneisen equation with appropriately chosen parameters. The Gruneisen equation can be given as:  
\[p=\frac{\left[\left(1-\frac{\gamma_{0}}{2}\right)\varphi+1-\frac{\alpha\varphi ^{2}}{2}\right]\rho_{0}\varphi K^{2}}{\left[1-\frac{C_{2}\varphi^{2}}{\varphi+ 1}-\frac{C_{3}\varphi^{3}}{\left(\varphi+1\right)^{2}}-\left(C_{1}-1\right) \varphi\right]^{2}}+E\left(\gamma_{0}+\alpha\varphi\right) \tag{13}\]  
in which \(C_{1}\), \(C_{2}\) and \(C_{3}\) represent the slope factors for the curve of \(\upsilon_{s}-\upsilon_{p}\); \(K\) signifies the intercept for the curve of \(\upsilon_{s}-\upsilon_{p}\); \(\gamma_{0}\), \(\varphi\), \(E\) and \(\alpha\) represents Gruneisen gamma, kinematic viscosity, initial internal energy and a volume correction of \(\gamma_{0}\), respectively. In this simulation, the variable parameters \(K\) and \(\varphi\) are set as 1400 m/s and \(8.684\times 10^{-4}\), respectively; the constant parameters \(\gamma_{0}\)\(\alpha\), \(C_{1}\), \(C_{2}\) and \(C_{3}\) are 0.35, 0, 1.192, \(-0.92\) and 0, respectively.  
in which  
\[C_{R}\left(t\right)=C\left[1+\frac{2\uppi}{\pi}E_{1}\left(\beta t\right)\right] \tag{15}\]  
where \(C\) is the elastic isotropic constitutive tensor; \(\beta=2\pi f_{d}/10\), where \(f_{d}\) means dominant excitation frequency with the unit of Hz; \(\uppi\) means the hysteretic damping factor and is described as \(\uppi=\pi\mathrm{atan}(10)=2.14\varsigma\), where \(\varsigma\) is the prescribed damping ratio. The function \(E_{1}\) can be given by the Prony series as:  
\[E_{1}\left(s\right)\approx\sum\limits_{k=1}^{5}b_{k}e^{a_{k}s} \tag{16}\]  
where \(a_{k}\) and \(b_{k}\) are appropriate constants to be determined and the pertinent value can be found in the literature [33].  
The soil surrounding a tunnel mainly affects its behavior during seismic excitation, and Table 1 shows the mechanical properties of the soil layers surrounding this specific tunnel. The soil can be modeled by the Type 232 material model in LS-DYNA.