## 2 Development of the Numerical Model
### Modeling of the Saturated Soils  
The saturated soil can have one layer or multi-layers in the simulation model. The one-layered soil model is efficient for carrying out parametric studies and obtaining the general influences of parameters. And the multi-layered soils are actually common in practice. Both one-layered and multi-layered soil models are developed in this study. To avoid repetition, the development of a multi-layered model is elaborated, and the one-layered soil model has the same characteristics. \begin{table}
\begin{tabular}{l c} \hline \hline
**Properties** & **Values** \\ \hline
**CRH380 EMU Train [42]** & \\ Mass of the car body (kg) & 40,000 \\ Mass of the bogie (kg) & 3200 \\ Mass of the wheelset (kg) & 2400 \\ Inertia of pitch motion of the car body(kg\(\cdot\)m\({}^{2}\)) & \(5.47\times 10^{5}\) \\ Inertia of pitch motion of the bogie(kg\(\cdot\)m\({}^{2}\)) & 6800 \\ Primary suspension stiffness (N/m) & \(1.04\times 10^{6}\) \\ Primary suspension damping (N\(\cdot\)s/m) & \(5\times 10^{3}\) \\ Secondary suspension stiffness (N/m) & \(4\times 10^{5}\) \\ Secondary suspension damping (N\(\cdot\)s/m) & \(6\times 10^{3}\) \\
**CRTS II slab track [19; 37; 38]** & \\ Mass density of the rail (kg/m\({}^{3}\)) & 7830 \\ Modulus of elasticity of the rail (Pa) & \(2.059\times 10^{11}\) \\ Poisson’s ratio of the rail & 0.3 \\ Stiffness of the rail pads (N/m) & \(5.0\times 10^{7}\) (dynamic stiffness) \\ Damping of the rail pads (N\(\cdot\)s/m) & \(7.5\times 10^{4}\) \\ Mass density of the concrete slab (kg/m\({}^{3}\)) & 2500 \\ Modulus of elasticity of the concrete slab (Pa) & \(3.6\times 10^{10}\) (reference static value, strain-rate dependent) \\ Poisson’s ratio of the concrete slab & 0.2 \\ Mass density of the CA mortar (kg/m\({}^{3}\)) & 1900 \\ Modulus of elasticity of the CA mortar (Pa) & \(7\times 10^{9}\)(reference static value, strain-rate dependent) \\ Poisson’s ratio of the CA mortar & 0.2 \\ Mass density of the concrete base (kg/m\({}^{3}\)) & 2400 \\ Modulus of elasticity of the concrete base (Pa) & \(2.55\times 10^{10}\)(reference static value, strain-rate dependent) \\ Poisson’s ratio of the concrete base & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the vehicle and slab track; CRH380 EMU = China Railway High-speed (CRH) 380 Electric Multiple Unit; CRTS = China Railway Track System; CA = cement asphalt. Figure 2: Track irregularity (a) track irregularity with distance and (b) power spectrum density (PSD) with wavelength. The multi-layered soils are developed based on a typical cross-section in the Beijing-Shanghai high-speed railway in China [32]. The subgrade consists of three layers: Surface layer, bottom layer, and subgrade body. Furthermore, the ground consists of two layers of clay and three layers of rock. The subgrade and clay are simulated as solid elements based on the FEM theory in LS-DYNA. In order to improve the calculation efficiency, the rock is not developed in the model, and the fixed boundary is set under the bottom of the second layer of ground instead. Although LS-DYNA provides the infinite boundary options for soils such as Perfectly Matched Layers (PML) and non-reflecting boundary, they are mainly applicable for normal-dry soils without pore waters. When the soils are saturated, the infinite boundary can induce some deformations at the edge of the model in LS-DYNA. Therefore, the normal translational and rotational constraint boundary is still used, but the dimension of the model is set as \(100\) m \(\times\) 110 m \(\times\) 15.5 m to avoid the influence of reflected waves on the vibration responses of soils in the focus-area, which is the soil underneath the slab track, as shown in Figure 3.  
In order to simulate the saturated soils, LS-DYNA uses effective stress theory to model materials with pore water pressure. The pore fluid and soil skeleton are assumed to carry loads in parallel.