## 3 Thermo-Mechanical Coupling Mathematical Model
### Coupling of Stress Field to Temperature Field  
For the plane strain problem, the equilibrium equation of the stress field can be expressed as  
\[\left\{ \begin{array}{c}\frac{\partial\sigma_{x}}{\partial x}+\frac{\partial \tau_{xy}}{\partial y}=0\\ \frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\sigma_{y}}{\partial y}=- \gamma\end{array} \right. \tag{18}\]  
where \(\gamma\) is the gravity of soil. The geometric equation can be expressed as  
\[\left\{ \begin{array}{c}\varepsilon_{x}=\frac{\partial t}{\partial x}\\ \varepsilon_{x}=\frac{\partial t}{\partial x}\\ \varepsilon_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \end{array} \right. \tag{19}\]  
Due to the frozen soil being defined as a linear elastomer, the mechanical properties of frozen soil change with temperature, considering its orthotropic characteristics. So, the thermo-mechanical coupling mathematical model can be expressed as  
\[\left\{ \begin{array}{c}\varepsilon_{x}=\frac{1-\mu(T)^{2}}{E(T)}\Big{[} \sigma_{x}-\frac{\mu(T)}{1-\mu(T)}\sigma_{y}\Big{]}+\sigma_{x}^{T}\\ \varepsilon_{y}=\frac{1-\mu(T)^{2}}{E(T)}\Big{[}\sigma_{y}-\frac{\mu(T)}{1- \mu(T)}\sigma_{x}\Big{]}+\sigma_{y}^{T}\\ \varepsilon_{xy}=\frac{2[1-\mu(T)]}{E(T)}\tau_{xy}+\varepsilon_{xy}^{T}\end{array} \right. \tag{20}\]  
where \(\varepsilon_{x}^{T},\varepsilon_{y}^{T}\) and \(\varepsilon_{xy}^{T}\) are the instantaneous strain component caused by temperature. To further effectively discuss the orthotropic deformation characteristics of frozen soil, generally, the frost heaving rate \(\varepsilon_{f/r}^{T}\) is used to measure the strength of frozen soil and the parameter is defined as the instantaneous volumetric strain related to temperature. As shown in Figure 2, the local coordinate system is established in the direction of temperature gradient in three directions (1, 2 and 3). According to the linear elastic theory of frozen soil, it can be known that the instantaneous volumetric strain \(\varepsilon_{f}^{T}\) and elastic strain \(\varepsilon^{\theta}\) constitute the total strain \(\varepsilon\) of frozen soil  
Figure 2: Local coordinate system. when the soil begins to freeze and the instantaneous thermal strain component generated by temperature in the local coordinate system is expressed as  
\[ \begin{cases}\varepsilon_{11}^{T}&\varepsilon_{12}^{T}&\varepsilon_{13}^{T}\\ \varepsilon_{21}^{T}&\varepsilon_{22}^{T}&\varepsilon_{23}^{T}\\ \varepsilon_{31}^{T}&\varepsilon_{32}^{T}&\varepsilon_{33}^{T}\end{cases}  = \begin{bmatrix}\eta&0&0\\ 0&(1-\eta)/2&0\\ 0&0&(1-\eta)/2\end{bmatrix} \varepsilon_{f}^{T} \tag{21}\]  
where \(\eta\) is an orthotropic parameter. When \(\eta=1\), the frozen soil exhibits unidirectional deformation characteristics (i.e., in the direction of the temperature gradient); when \(\eta=1/3\), the frozen soil exhibits isotropic deformation characterized. Therefore, whether unidirectional deformation or isotropic deformation, frozen soil has characteristics of orthotropic deformation. Employing [PERSON]'s law, the bulk strain component of frozen soil can be expressed as  
\[\left\{ \begin{array}{c}\varepsilon_{11}=\frac{1}{E}[\sigma_{11}-\mu(\sigma _{22}+\sigma_{33})]+\eta\varepsilon_{f}^{T}\\ \varepsilon_{22}=\frac{1}{E}[\sigma_{22}-\mu(\sigma_{11}+\sigma_{33})]+ \frac{1}{2}(1-\eta)\varepsilon_{f}^{T}\\ \varepsilon_{33}=\frac{1}{E}[\sigma_{33}-\mu(\sigma_{11}+\sigma_{22})]+ \frac{1}{2}(1-\eta)\varepsilon_{f}^{T}\\ \varepsilon_{12}=\frac{2(1+\mu)}{E}\tau_{12}\\ \varepsilon_{23}=\frac{2(1+\mu)}{E}\tau_{23}\\ \varepsilon_{13}=\frac{2(1+\mu)}{E}\tau_{13}\end{array} \right. \tag{22}\]  
where \(E\) is the elastic modulus of frozen soil and \(\mu\) is Poisson's ratio of frozen soil. For the plane strain problem, \(\varepsilon_{33}=\varepsilon_{13}=\varepsilon_{23}=0\),  
\[\sigma_{33}=\mu(\sigma_{11}+\sigma_{22})-\frac{E}{2}(1-\eta)\varepsilon_{f}^{T} \tag{23}\]  
Combined with Equations (22) and (23), bulk strain component can be expressed as  
\[\left\{