# Dynamic Response of Railway Subgrade Under Train Load and Freeze-Thaw Action
## 2 Three-Dimensional Finite Element Model of Railway Subgrade
### Parameter Selection and Meshing
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & **Parameters** & \multicolumn{4}{c}{**Temperature/\({}^{\circ}\)C**} \\ \cline{3-6}  & & **5** & **0** & \(-\)**5** & \(-\)**15** \\ \hline Surface layer of subgrade & \(E\)/MPa & 21 & 21 & 25 & 29.6 \\ /subgrade fill & \(\mu\) & 0.24 & 0.24 & 0.13 & 0.13 \\ /subgrade fill & \(c\)/MPa & 0.001 & 0.001 & 0.5 & 0.75 \\  & \(\varphi\)/\(\circ\) & 28 & 28 & 40 & 40 \\ \hline \hline \multirow{4}{*}{Active layer} & \(E\)/MPa & 34 & 34 & 50.8 & 67 \\  & \(\mu\) & 0.45 & 0.45 & 0.13 & 0.13 \\  & \(c\)/MPa & 0.003 & 0.003 & 0.6 & 0.6 \\  & \(\varphi\)/\(\circ\) & 30 & 30 & 34 & 34 \\ \hline \hline \multirow{4}{*}{Ice-rich frozen soil} & \(E\)/MPa & 2 & 2 & 176 & 520 \\  & \(\mu\) & 0.4 & 0.4 & 0.18 & 0.15 \\ /MPa & 0.007 & 0.007 & 1.3 & 1.3 \\  & \(\varphi\)/\(\circ\) & 12 & 12 & 25 & 25 \\ \hline \hline \multirow{4}{*}{Ice-poor frozen soil} & \(E\)/MPa & 500 & 500 & 730 & 835 \\  & \(\mu\) & 0.25 & 0.25 & 0.25 & 0.25 \\ /MPa & 32 & 32 & 38 & 38 \\ /MPa & \(\varphi\)/\(\circ\) & 14 & 14 & 28 & 28 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical parameters of permafrost subgrade materials. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & **W/\%** & \(\lambda\)**(**W**-**m\({}^{-1,\circ}\)C\({}^{-1}\))** & \(C\)**/**(**kg**-**\({}^{-1,\circ}\)C\({}^{-1}\))** & \(L\)**/**(**J**-**kg**-**\({}^{-1}\))** & \(k\)**/(m**-**s\({}^{-1}\))** \\ \cline{2-6}  & \(\lambda_{f}\) & \(\lambda_{s}\) & \(C_{f}\) & \(C_{u}\) & & **Thawed State** & **Frozen State** \\ \hline Surface layer of subgrade & 6 & 1.4 & 1.15 & 706.6 & 861.7 & 10,461.5 & \(1.0\times 10^{-7}\) & \(1.0\times 10^{-19}\) \\ Subgrade fill & 6 & 1.4 & 1.15 & 706.6 & 861.7 & 10,736.8 & \(1.0\times 10^{-7}\) & \(1.0\times 10^{-19}\) \\ Active layer & 15 & 2.414 & 1.837 & 851.8 & 1099.5 & 37,687.5 & \(1.3\times 10^{-7}\) & \(1.3\times 10^{-19}\) \\ Ice-rich frozen soil & 30 & 2.12 & 1.42 & 1222 & 1608 & 52,527 & \(5.0\times 10^{-9}\) & \(5.0\times 10^{-21}\) \\ Ice-poor frozen soil & 15 & 1.86 & 1.6 & 981.8 & 1272 & 20,944.4 & \(1.6\times 10^{-9}\) & \(1.6\times 10^{-21}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Thermophysical parameters and permeability coefficients of permafrost subgrade materials. The \(L_{e}\) is maximum element size and \(\lambda_{s}\) is shear wavelength. In this paper, the value of \(\lambda_{s}\) is 7.9 m, it indicates that the maximum element size \(L_{e}\) is 1.32 m. To ensure precise analysis of the dynamic response of the model, the phenomenon of stress concentration in the region beneath the track structure has been considered. Consequently, the solid element size has been specified at 0.14 m to capture the intricate stress distributions in this critical area. The load location is further away, the element size is larger, the solid element size is gradually increased to 0.5 m at the boundary of the foundation. The railway track, subgrade and foundation are adopted eight-node hexahedral solid units. To maintain the continuity of interface deformation, a tie connection is employed to guarantee interlayer contact beneath the slab. The foundation base is rendered immobile, while the periphery is encircled by a semi-uniform viscoelastic artificial boundary to simulate the edge conditions. The observation points of the non-ballasted track-subgrade coupling model are distributed within the subgrade cross-section as shown in Figure 3, the coordinate point A (\(-\)2.5,0) is positioned on the subgrade surface directly beneath the left track of the left railway line. The coordinate point B (0,0) marks the central axis of the subgrade. Meanwhile, the coordinate point C (2.5,0) is located on the subgrade surface below the right track of the right railway line.