## 2 Establishment of Three-Dimensional Pipeline-Sooil Thermo-Mechanical Coupling Model
### Pipeline-Soil Model Parameters
#### 2.1.2 Pipe Material Parameters  
According to the working conditions, both pipeline and thermal insulation materials are within the standard operating temperature range. Compared with the sensitivity of soil to temperature, the small changes in the temperature-dependent material properties of pipeline and thermal insulation materials can be ignored. The pipe material is X65 steel, the elastic modulus is 2.06 \(\times\) 10\({}^{5}\) MPa, the Poisson's ratio is 0.3, the density is 7850 kg/m\({}^{3}\), the thermal conductivity is 60.5 w/(m\({}^{\circ}\)C), the specific heat capacity is 434 J/(kg.\({}^{\circ}\)C), the yield strength is 448 MPa, and the tensile strength is 672 MPa. The thermal insulation material is rigid polyurethane foam material, the elastic modulus is 200 MPa, the Poisson's ratio is 0.39, the density is 60 kg/m\({}^{3}\), the thermal conductivity is 0.047 w/(m\({}^{\circ}\)C), the specific heat capacity is 1200 J/(kg.\({}^{\circ}\)C).  
Due to the differential frost heaving of soil, the physical and mechanical characteristics of X65 pipeline material show nonlinear performance (material nonlinearity) and the displacement of the pipeline structure system makes the mechanics of the system change significantly (geometric nonlinearity). Therefore, the Ramberg-Osgood model selected for the constitutive relationship of X65 pipeline in this paper can fully consider the influence of soil on the elastoplastic properties and large deformation of pipeline [28; 29]. Furthermore, the true stress-strain curve of the material was calculated as shown in Figure 2.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{8}{c}{**Temperature/\({}^{\circ}\)C**} \\ \cline{2-9}  & \(-\)**30** & \(-\)**10** & \(-\)**5** & \(-\)**2** & \(-\)**1** & \(-\)**0.5** & **0** & **15** & **30** \\ \hline \multicolumn{9}{c}{**Weathered**} & **Bedrock** & **Resolute** (\(\omega\) = 2.59\%) \\ \hline Thermal conductivity \(\lambda\) (W/m\({}^{\circ}\)C) & 2.12 & 2.12 & 2.12 & 2.12 & 2.12 & 1.42 & 1.42 & 1.42 \\ Specific heat C (J/kg.\({}^{\circ}\)C) & 1500 & 1500 & 1500 & 1500 & 1500 & 1900 & 1900 & 1900 \\ Soil expansion coefficient \(\alpha\) (10\({}^{-4}\)) & 0.259 & 0.777 & 1.554 & 3.885 & 7.770 & 15.54 & 0 & 0 & 0 \\ \hline \end{tabular}
\end{table}
Table 1: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Temperature/\({}^{\circ}\)C**} \\ \cline{2-7}  & \(-\)**20** & \(-\)**10** & \(-\)**5** & \(-\)**2** & **0** & **20** \\ \hline \multicolumn{7}{c}{**Silty Clay**} \\ \hline Density \(\rho\) (kg/m\({}^{3}\)) & 1920 & 1920 & 1920 & 1920 & 1920 & 1920 & 1920 \\ Poisson’s ratio \(\
u\) & 0.32 & 0.32 & 0.32 & 0.32 & 0.32 & 0.35 & 0.35 \\ Internal friction angle \(\varphi\) (\({}^{\circ}\)) & 26 & 26 & 26 & 26 & 24 & 24 \\ Cohesion c (MPa) & 0.6 & 0.6 & 0.6 & 0.57 & 0.15 & 0.15 \\ Elastic modulus E (MPa) & 200 & 100 & 50 & 23.4 & 6 & 6 \\ \hline \multicolumn{7}{c}{**Fine Sand and Gravel**} \\ \hline Density \(\rho\) (kg/m\({}^{3}\)) & 1834 & 1834 & 1834 & 1834 & 1834 & 1834 \\ Poisson’s ratio \(\
u\) & 0.15 & 0.15 & 0.15 & 0.15 & 0.2 & 0.2 \\ Internal friction angle \(\varphi\) (\({}^{\circ}\)) & 20 & 20 & 20 & 20 & 18 & 18 \\ Cohesion c (MPa) & 1.3 & 1.3 & 1.3 & 1.3 & 0.1 & 0.2 \\ Elastic modulus E (MPa) & 500 & 300 & 100 & 70 & 3 & 3 \\ \hline \multicolumn{7}{c}{**Weathered bedrock residue**} \\ \hline Density \(\rho\) (kg/m\({}^{3}\)) & 2330 & 2330 & 2330 & 2330 & 2330 & 2330 \\ Poisson’s ratio \(\
u\) & 0.3 & 0.3 & 0.3 & 0.3 & 0.35 & 0.35 \\ Internal friction angle \(\varphi\) (\({}^{\circ}\)) & 42 & 42 & 42 & 42 & 33.32 & 33.32 \\ Cohesion c (MPa) & 25.2 & 25.2 & 25.2 & 25.2 & 12.6 & 12.6 \\ Elastic modulus E (MPa) & 1400 & 1400 & 1400 & 1400 & 700 & 700 \\ \hline \end{tabular}
\end{table}
Table 2: Soil mechanical parameters.