## 5 Results and Discussions  
To present the results more clearly, the coordinates were constructed as illustrated in Figure 3 on the ground surface. The Y-axis is aligned with the longitudinal direction of the tunnel, while the X-axis is set as the transverse direction of the tunnel. In addition, the origin is right above the center of the cross passage. Development of Frost Heave and Thaw Settlement and How the Thickness of the Freezing Wall Affected Them  
Figure 4 illustrates the frost heaves (the vertical displacements arising from the freezing process) and thaw settlements (the vertical displacements caused by the thawing process) at the origin O in Figure 3 induced by freezing walls of different thicknesses with respect to time. As illustrated in Figure 4a, there is a lag in the frost heave of the ground surface during the freezing process after the heat-absorbing source is added to the freezing wall. This lag corresponds to a slight settlement of the surface ground (the shrinkage stage), \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Material** & **Temperature (\({}^{\circ}\)C)** & **Density (kg/m\({}^{\circ}\))** & **Bulk Modulus (MPa)** & **Shear Modulus (MPa)** & **Constitutive Model** & **Friction Angle (\({}^{\circ}\))** & **Cohesion (kPa)** \\ \hline \multirow{3}{*}{Soil} & 20 (Original soil) & 1880 & 16.67 & 7.69 & Mohr-Coulomb & 17 \({}^{\ast}\) & 22 \\  & \(-10\) & 1880 & 106 & 57.9 & Mohr-Coulomb & 17 \({}^{\ast}\) & 330 \\  & 20 (Thawed soil) & 1880 & 40 & 18.46 & Mohr-Coulomb & 17 \({}^{\ast}\) & 5.5 \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} Liner of \\ the tunnel \\ \end{tabular} } & - & 2500 & \(2\times 10^{4}\) & \(1.5\times 10^{4}\) & Elastic & - & - \\ \hline \multirow{3}{*}{
\begin{tabular}{c} Liner of the \\ cross passage \\ \end{tabular} } & - & 2400 & \(1.56\times 10^{4}\) & \(1.17\times 10^{4}\) & Elastic & - & - \\ \hline \hline \end{tabular}  
* When the temperature is above the freezing temperature, the thermal expansion coefficient of soils is positive and constant.  
\end{table} 
Table 2: Mechanical properties of the materials. Figure 3: The schematic diagram of the coordinate used in the presentation of the results. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & **Temperature** & **Thermal** & **Specific Heat** & **Thermal** & & \\  & (\({}^{\circ}\)C) & **Conductivity** & **Capacity** & **Expansion** & & **Heave Rate (\({}^{\circ}\)C\({}^{\ast}\))** & **Shear Settlement** \\  & (\({}^{\circ}\)C) & (W/m\({}^{\circ}\)C) & (J/kg\({}^{\circ}\)C) & **Coefficient (\({}^{\circ}\)C\({}^{\ast}\))** & & **Rate (\({}^{\circ}\)C\({}^{\ast}\))** \\ \hline \multirow{3}{*}{\begin{tabular}{c} Soil \\ \end{tabular} } & 20 & 1.23 & 1040 & \(1\times 10^{-5}\) * & - & - \\  & \(-10\) & 1.38 & 1110 & - & \(-9.64\times 10^{-4}\) & \(-1.03\times 10^{-3}\) \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} Liner of the tunnel \\ cross passage \\ \end{tabular} } & - & 0.5 & 1000 & \(1\times 10^{-5}\) & - & - \\ \hline \multirow{2}{*}{
\begin{tabular}{c} Air in the tunnel \\ \end{tabular} } & - & -.* & -.* & \(1\times 10^{-5}\) & - & - \\ \hline \hline \end{tabular}  
* When the temperature is above the freezing temperature, the thermal expansion coefficient of soils is positive and constant.  
\end{table} 
Table 3: Thermal properties of the materials. which is most likely caused by a small shrinkage in the soil before the temperature is below the freezing temperature of the soil. After passing the shrinkage stage, the frost heave begins to develop approximately linearly with respect to time (the fast-freezing stage). In addition, as shown in Figure 4b, the thawing process, the reversed process of the frost heave, shows 'inverted' curves compared with their counterparts of the frost heaving process. Similar to the the frost heave curve having a fast-freezing stage, the thaw settlement curve has a 'fast-thawing stage' just after the removal of the heat-absorbing source.