## 2 Dual Transverse Isotropic Model of Frozen Rock Mass  
A plane in which the properties of the rock in different directions are identical is called an isotropic plane. When the properties of the rock perpendicular to the isotropic plane are different from those within the isotropic plane, this rock is referred to as transversely isotropic [19; 20], as shown in Figure 2. Accordingly, such rock is an isotropic material.  
In cold regions, the longitudinal and circumferential temperature gradient is largely invisible, and the radial temperature gradient is prominent in tunnel engineering. On-site monitoring data from cold-region tunnels [21] also demonstrated that the measured temperature gradient is positively correlated with the radial direction of the tunnel, as shown in Figure 3. Hence, the frozen rock mass can be regarded as a transversely isotropic material. Subsequently, the strain was used as an indicator of the transverse isotropy of the frozen rock mass, and the corresponding expressions [6] are  
\[\left\{\begin{array}{l}\varepsilon_{r}=\frac{k}{k+2}\varepsilon_{v}\\ \varepsilon_{\theta}=\varepsilon_{l}=\frac{1}{k+2}\varepsilon_{v}\end{array}\right. \tag{1}\]  
where \(\varepsilon\) represents strain. The subscripts \(r\), \(\theta\), and \(l\) represent the radial, circumferential, and longitudinal directions, respectively. Parameters \(k\) and \(\varepsilon_{v}\) represent the anisotropic frost heave coefficients and volumetric strain, respectively. If \(k=1\) is assumed, the expressions can be simplified as the isotropy case.  
Figure 3: Temperature distribution along the radial direction.  
Figure 2: Schematic of transversely isotropic material.  
The periodic freezing and thawing caused by the alternation of winter and summer significantly weaken the bearing capacity of the rock mass, which is composed of solid, liquid, and gas materials. Weakening is mainly the result of the additional expansion pressure exerted by the increases in volume as water freezes, which internally damages the rock mass, as shown in Figure 4. Typically, a decrease in elastic modulus can indicate that the surrounding rock has been damaged. The larger the temperature difference, the greater the elastic modulus decreases. Therefore, similar to the temperature gradient, the elastic modulus increases with the increase in the radial depth of the surrounding rock. The elastic modulus becomes constant when the depth reaches the unfrozen rock mass boundary. The increase in the elastic modulus with depth along the radial direction is assumed to be a power-law function [22; 23], and the formula is as follows:  
\[E_{\rm f}(r)=E_{\rm uf}(\frac{r}{R_{\rm uf}})^{\alpha_{\rm f}} \tag{2}\]  
where \(E\) represents elastic modulus; subscripts f and uf represent the frozen and unfrozen rock mass, respectively; \(r\) represents the distance from the tunnel center in the radial direction; \(R_{\rm uf}\) represents the depth of the frozen rock mass; and \(\alpha_{\rm f}\) represents the radial gradient influence coefficient of the frozen rock mass. If \(\alpha_{\rm f}\) = 0 is assumed, the expression can also be simplified as an isotropy case.  
In summary, a dual transverse isotropic model of frozen rock mass was proposed based on the strain and elastic modulus parameters. If the parameter values are reasonable (e.g., \(k\) = 1 and \(\alpha_{\rm f}\) = 0), the dual transverse isotropic model can be simplified as an isotropy case. Subsequently, the mechanical model of the cold-region tunnel is established and further solved by considering the dual transverse isotropy of frozen rock mass.