## 3 Establishing of Mechanical Model  
Figure 5 shows a sketch of the mechanical model of the surrounding rock and lining structure of a cold-region tunnel. Zone I represents the lining structure. Zone II and III represent the frozen and unfrozen rock mass, respectively. Zone II is further subdivided into II-1 and II-2, which represent the plastic and elastic rock mass, respectively. As zone II freezes and expands, additional pressure is exerted on the lining structure due to the limitations of zone I and III, which is referred to as the frost heave force [10; 12; 13]. As shown in Figure 5, \(R\_0\) and \(R\_1\) represent the inner and outer radius of the lining structure, respectively. R_2 and represent the outer radius of the frozen rock mass. \(R_{\rm P}\) represents the radius of the yield range of frozen rock mass. In addition, \(P\_0\) represents the far-field pressure in the tunnel situ. \(F\_0\) represents external pressure, such as aerodynamic force [8]. \(F\_1\) and \(F\_2\) represent internal pressure induced by freezes and expands of the frozen rock mass. \(R_{\rm P}\) represents internal pressure acting on the contact surface between plastic and elastic rock mass. The following basic assumptions were adopted to solve the mechanical model: (1) The cross-sectional shape of a deeply buried tunnel is circular. (2) Zone I and zone III  
Figure 4: Schematic of rock damage caused by freeze–thaw exposure. are isotropic, homogeneous, and elastic media, and zone II is a transversely isotropic homogeneous elasto-plastic medium. (3) Zone II satisfies the unified strength theory [24; 25]. (4) The plane strain condition is employed. (5) The body forces of the rock mass and structure are ignored. In the circumferential and radial stresses of zones I, II, and III, the equilibrium equation satisfied can be given as follows:  
\[\frac{d\sigma_{r}^{i}}{dr}+\frac{\sigma_{r}^{i}-\sigma_{\theta}^{i}}{r}=0 \tag{3}\]  
where \(\sigma\) represents the stress. Superscript \(i\) represents the zone number (e.g., \(I\) = I, II, II-1, II-2, and III). The geometric equations satisfied by the circumferential and radial strains of zones I, II, and III can be given as follows:  
\[\varepsilon_{r}^{i}=\frac{du_{r}^{i}}{dr}\ \ \varepsilon_{\theta}^{i}=\frac{u_{r}^ {i}}{r} \tag{4}\]  
where \(u\) represents displacement. The constitutive equations satisfied by zone I can be given as follows:  
\[\left\{ \begin{array}{l}\varepsilon_{r}^{\text{I}}=\frac{1-\mu_{\theta}^{2}} {E_{\text{s}}}(\sigma_{r}^{\text{I}}-\frac{\mu_{\theta}}{1-\mu_{\theta}} \sigma_{\theta}^{1})\\ \varepsilon_{\theta}^{\text{I}}=\frac{1-\mu_{\theta}^{2}}{E_{\text{s}}}( \sigma_{\theta}^{1}-\frac{\mu_{\theta}}{1-\mu_{\theta}}\sigma_{r}^{1})\\ \end{array} \right. \tag{5}\]  
where \(\mu\) represents [PERSON]'s ratio and subscript s represents support zone I. The constitutive equations [5; 8] satisfied by zone II can be given as follows:  
\[\left\{ \begin{array}{l}\varepsilon_{r}^{\text{I}e}=\frac{1-\mu_{\theta}^{2} }{E_{\text{s}}(r)}((\sigma_{r}^{\text{I}\text{I}e}-P\_0)-\frac{\mu_{\theta}}{ 1-\mu_{\theta}}(\sigma_{\theta}^{\text{I}\text{I}e}-P\_0))-C_{1}\\ \varepsilon_{\theta}^{\text{I}\text{I}e}=\frac{1-\mu_{\theta}^{2}}{E_{\text{ s}}(r)}((\sigma_{\theta}^{\text{I}\text{I}e}-P\_0)-\frac{\mu_{\theta}}{1-\mu_{ \theta}}(\sigma_{r}^{\text{I}\text{I}e}-P\_0))-C_{2}\\ C_{1}=(\frac{\mu_{\theta}}{k_{\text{T}}+2}\varepsilon_{\theta}+\mu_{\theta} \frac{1}{k_{\text{T}}+2}\varepsilon_{\theta})\\ C_{2}=(\frac{1}{k_{\text{T}}+2}\varepsilon_{\theta}+\mu_{\theta}\frac{k_{ \text{T}}}{k_{\text{T}}+2}\varepsilon_{\theta})\\ \end{array} \right. \tag{6}\]  
where superscript e represents the elasticity in zone II. The constitutive equations [7; 22] satisfied by zone III can be given as follows:  
\[\left\{