## 3 Coupling Mathematical Model for Tunnel Freezing-Frost Heave
### Coupling Mathematical Model for Freezing-Frost Heave  
Thermal elastic-plastic analysis is performed with the features of the frozen soil curtain in mind [32; 33]. Assuming frozen soil is an isotropic linear elastic material, its total strain \(\left\{\epsilon\right\}\) may be thought of as a combination of elastic strain increment \(\left\{\epsilon\right\}\)\({}_{\text{e}}\) induced by load and instantaneous bulk strain increment \(\left\{\epsilon\right\}\)\({}_{\text{v}}\) caused by temperature change [34; 35], namely:  
\[\left\{\epsilon\right\}=\left\{\epsilon\right\}_{e}+\left\{\epsilon\right\}_{v} \tag{13}\]  
The elastic strain increment \(\left\{\epsilon\right\}\)\({}_{\text{e}}\) can be expressed:  
\[\left\{\epsilon\right\}_{e}=\left[D\right]^{-1}\{\sigma\} \tag{14}\]where [\(D\)] is the elastic matrix, and its expression is as follows:  
\[[D]=\frac{E(1-\mu)}{(1+u)(1-2\mu)} \begin{Bmatrix}1&\mu/(1-\mu)&0\\ \mu/(1-\mu)&1&0\\ 0&0&(1-2\mu)/[2(1-\mu)]\end{Bmatrix}  \tag{15}\]  
The thermal strain component produced by temperature change may be described in the local coordinate system as follows:  
\[ \begin{Bmatrix}\epsilon_{11}^{T}&\epsilon_{12}^{T}\\ \epsilon_{21}^{T}&\epsilon_{22}^{T}\end{Bmatrix} = \begin{bmatrix}1/3&0\\ 0&1/3\end{bmatrix} \{\epsilon\}_{v} \tag{16}\]  
[PERSON]'s elastic theorem states that the total strain component of frozen soil in the local coordinate system is expressed as follows:  
\[\left\{ \begin{array}{l}\epsilon_{11}=\frac{1}{\Gamma}[\sigma_{11}-\mu( \sigma_{22}+\sigma_{33})]+\frac{1}{3}\{\epsilon\}_{v}\\ \epsilon_{22}=\frac{1}{\Gamma}[\sigma_{22}-\mu(\sigma_{11}+\sigma_{33})]+ \frac{1}{3}\{\epsilon\}_{v}\\ \epsilon_{12}=\epsilon_{21}=\frac{2(1+\mu)}{\Gamma}\tau_{12}\end{array} \right. \tag{17}\]  
For the planar strain issue, with \(\epsilon_{13}=\epsilon_{23}=\epsilon_{33}=0\), it can be obtained that:  
\[\sigma_{33}=\mu(\sigma_{11}+\sigma_{22})-\frac{E}{3}\{\epsilon\}_{v} \tag{18}\]  
The formula for the total strain of frozen soil in the local coordinate system is found by inserting Equation (18) into Equation (17):  
\[\left\{ \begin{array}{l}\epsilon_{11}=\frac{1-\mu^{2}}{E}\Big{(}\sigma_{11}- \frac{\mu}{1-\mu}\sigma_{22}\Big{)}+\frac{1+\mu}{3}\{\epsilon\}_{v}\\ \epsilon_{22}=\frac{1-\mu^{2}}{E}\Big{(}\sigma_{22}-\frac{\mu}{1-\mu}\sigma_ {11}\Big{)}+\frac{1+\mu}{3}\{\epsilon\}_{v}\\ \epsilon_{12}=\epsilon_{21}=\frac{2(1+\mu)}{\Gamma}\tau_{12}\end{array} \right. \tag{19}\]  
The stress field equilibrium equation in the global coordinate system for the planar strain issue is as follows:  
\[\left\{ \begin{array}{l}\frac{\partial x_{x}}{\partial x}+\frac{\partial\tau _{xy}}{\partial y}=0\\ \frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\tau_{xy}}{\partial y}=- \gamma\end{array} \right. \tag{20}\]  
where \(\gamma\) is the soil gravity, and the geometric equation is  
\[\left\{ \begin{array}{l}\epsilon_{x}=\frac{\partial u}{\partial x}\\ \epsilon_{y}=\frac{\partial v}{\partial y}\\ \epsilon_{xy}=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} \end{array} \right. \tag{21}\]  
The displacement boundary conditions:  
\[\left\{ \begin{array}{l}u=\overline{u}(x,y);\text{Horizontal displacement}\\ v=\overline{v}(x,y);\text{ Vertical displacement}\end{array} \right. \tag{22}\]  
Given that mechanical properties of frozen soil--such as elastic modulus \(E\) and Poisson's ratio \(\mu\)--fluctuate with temperature, the thermodynamic coupling constitutive equation can be found by rewriting Equation (19):  
\[\left\{ \begin{array}{l}\epsilon_{x}=\frac{1-\mu(T)^{2}}{E(T)}\Big{[}\sigma_ {x}-\frac{\mu(T)}{1-\mu(T)}\sigma_{y}\Big{]}+\frac{1+\mu(T)}{3}\{\epsilon\}_{v} \\ \epsilon_{y}=\frac{1-\mu(T)^{2}}{E(T)}\Big{[}\sigma_{y}-\frac{\mu(T)}{1-\mu(T) }\sigma_{x}\Big{]}+\frac{1+\mu(T)}{3}\{\epsilon\}_{v}\\ \epsilon_{xy}=\frac{21+\mu(T)}{\Gamma}\tau_{xy}\end{array} \right. \tag{23}\]According to the incremental theory [36; 37; 38], the stress-strain relationship for the assumed isotropic elastic-plastic unfrozen soil is as follows [39]:  
\[\left\{