The Distribution Characteristics of Frost Heaving Forces on Tunnels in Cold Regions Based on Thermo-Mechanical Coupling

Abstract

To address the freezing damage to tunnel lining caused by frost heaving of the surrounding rock in water-rich tunnels in cold regions, a numerical thermo-mechanical coupling model for tunnel-surrounding rock that considers the anisotropy of frost heave deformation was established by examining overall frost heaves in a freeze–thaw cycle. Using a COMSOL Multiphysics 6.0 platform and the sequential coupling method, the temperature field evolution of tunnel-surrounding rock, freezing cycle development, and distribution characteristics of the frost heaving force of a tunnel lining under different minimum temperatures, numbers of negative temperature days, frost heave ratios, and anisotropy coefficients of frost heave deformation were systematically simulated. The results revealed that the response of the temperature field of tunnel-surrounding rock to the external temperature varies spatially with time lags, the shallow surface temperatures and the area around the lining fluctuate with the climate, and the temperature of the deep surrounding rock is dominated by the geothermal gradient. The extent of the freezing cycle and the frost heaving force increase significantly when lowering the minimum temperature. The maximum frost heaving force usually occurs in the region of the side wall and the spring line, and tensile stress is prone to be generated at the spring line; the influence of slight fluctuations in the minimum temperature or the short shift in the coldest day on the frost heaving force is limited. A substantial increase in frost heaving force is observed with higher frost heave ratios; for example, an increase from 0.25% to 2.0% results in a 116% rise at the sidewall. Although the increase in the anisotropy coefficient of frost heave deformation does not change the overall distribution pattern of frost heaving force, it can exacerbate the directional concentration of frost heave strain, which can increase the frost heaving force at the periphery of the top arch of the lining. This study revealed the distribution pattern and key influencing factors of the freezing cycle and frost heaving force for tunnels, providing a theoretical basis and data reference for the frost resistance design of tunnels in cold regions.

1. Introduction

Tunnels are important parts of transportation networks. According to public statistics, by the end of 2024, there were 28,724 road tunnels in China, with a total mileage of 32,596.6 km [1], and 18,997 railway tunnels, with a total mileage of 24,246 km [2], with gradual expansion to high altitudes and alpine regions. However, the ambient meteorological and geological conditions faced by cold regions are extremely complex, posing a great challenge to the design and construction of tunnel projects. Under the effect of continuous low temperatures in winter, the fissure water in the surrounding rock at the entrance of a water-rich tunnel can freeze, leading to an overall volume expansion of the surrounding rock and the generation of frost heave stress. This stress is transferred to the lining structure through the surrounding rock–lining interface, resulting in freezing damage such as cracking, spalling and icing in the lining concrete [3,4,5,6,7], thus threatening traffic safety. To address this outcome, experts and scholars have conducted extensive studies on calculating frost heave mechanisms and frost heaving forces in tunnels and have proposed corresponding frost heave calculation models. Frost heave models can be divided into three main categories: accumulated water frost heave models [8], weathered layer frost heave models [9], and freeze–thaw cycle overall frost heave models [10]. Currently the most widely used is the more applicable freeze–thaw cycle frost heave model [10,11,12,13,14,15], whereas the remaining two types of models have many assumptions, and their application is limited [16,17].

To determine the frost heaving force, methods such as field monitoring, indoor testing, theoretical analysis, and numerical simulation are used. Zhang et al. [18] simultaneously measured the surrounding rock temperature and the lining pressure to carry out field measurements of frost heaving force in the Jichoushan Tunnel and analysed the variation pattern of the frost heaving force. He et al. [19] conducted a series of unidirectional freezing experiments on clay in an open system and obtained the evolution patterns of the freezing depth, frost heaving force, and water content versus time. Zheng et al. [20] considered the depth of water behind the lining and carried out a three-dimensional (3D) geotechnical model test of the frost heaving force of a lining; their results showed that the model can well predict the evolution of frost heaving force caused by water accumulation. Zhao et al. [21] established a frost heave model based on the principles of ice–water phase change and water–heat coupling and discussed effects of factors such as geometric parameters. Yang et al. [22] proposed an analytical solution for the frost heaving force in a circular tunnel based on the displacement method, performed parameter sensitivity analysis, and noted that lining thickness should be a central concern when designing tunnels for frost resistance. Based on an overall frost heave model of a broken freeze–thaw lithosphere, Cui et al. [15] proposed a calculation method for the overall frost heave pressure of a horseshoe-shaped tunnel with broken surrounding rock in a seasonally frozen area. Shen et al. [23] established a 3D thermo-mechanical coupling finite element model for fractured rocks to analyse the effects of fracture size, number of fractures, and fracture inclination on frost heaves. These studies used different methods and different assumptions to calculate the frost heaving force, and these methods have their own characteristics in terms of applicability, calculation accuracy, parameter requirements, and calculation efficiency, thus reflecting the lack of unified standards and specifications for frost heaving force calculations. In addition, most studies have shown that anisotropic frost heave deformations [24,25,26] and the multiphysics field coupling effect [27,28] significantly affect the frost heaving force.

In summary, the distribution of frost heaving forces in tunnels is affected by the geometry, multifield coupling effect, and anisotropy of frost heave deformation. To describe the evolution pattern of the freezing depth and frost heaving force of the rock surrounding a tunnel under the action of different factors, this study adopted the freeze–thaw cycle overall frost heave model and established a thermo-mechanical coupling model for frost heaves in the surrounding rock of tunnel by considering the anisotropy of frost heave deformation via the COMSOL Multiphysics finite element simulation platform. Sequential coupling was used to perform thermo-mechanical coupling calculations under different minimum temperatures, numbers of negative temperature days, frost heave ratios, and anisotropy coefficients of frost heave deformation to obtain the distribution characteristics of the freezing cycle and the frost heaving force. Finally, the effects of different factors on the frost heaving force of the lining were analysed. The results indicate that the frost heave ratio and the minimum air temperature have significant effects on the frost heaving force. An increase in the anisotropy coefficient leads to a directional concentration of frost heave strain, which in turn amplifies the frost heaving force. In contrast, the shift in the coldest day has a negligible impact on the frost heaving force. This study is important for revealing the size of the freezing cycle, the distribution characteristics of the frost heaving force and the key influencing factors of tunnels in cold regions.

2. Materials and Methods

2.1. Thermo-Mechanical Coupling Governing Equations

2.1.1. Governing Equation of the Temperature Field

Assuming that the phase change interval of rock and soil masses is [T_d, T_r], the governing equation of the temperature field of the 2D tunnel model can be written as [29]

$$c^{*}{\displaystyle \frac{\partial T}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left({\lambda }^{*}{\displaystyle \frac{\partial T}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left({\lambda }^{*}{\displaystyle \frac{\partial T}{\partial y}}\right)$$

(1)

With

$$c^{*}=\left\{\begin{array}{cc}{c}_f& T<T_d\\ {\displaystyle \frac{c_f+c_u}{2}}+{\displaystyle \frac{L}{T_r-T_d}}& T_d\le T\le T_r\\ c_u& T>T_r\end{array}\right.$$

(2)

and

$${\lambda }^{*}=\left\{\begin{array}{cc}{\lambda }_f& T<T_d\\ {\lambda }_f+{\displaystyle \frac{{\lambda }_u-{\lambda }_f}{T_r-T_d}}(T-T_d)& T_d\le T\le T_r\\ {\lambda }_u& T>T_r\end{array}\right.$$

(3)

where $T_d$ is the freezing temperature of the soil (°C); $T_r$ is the soil thawing temperature (°C); $c^{*}$ is the effective volumetric specific heat of the soil (J·m⁻³·°C⁻¹); $c_f$ is the volumetric specific heat of the frozen soil (J·m⁻³·°C⁻¹); $c_u$ is the volumetric specific heat of the thawed (unfrozen) soil (J·m⁻³·°C⁻¹); ${\lambda }^{*}$ is the equivalent thermal conductivity of the soil (J·m⁻¹·s⁻¹·°C); ${\lambda }_f$ and ${\lambda }_u$ are the thermal conductivity of the frozen soil and thawed soil (unfrozen soil) (J·m⁻¹·s⁻¹·°C); and L is the latent heat of freezing per unit volume of soil (J·m⁻³).

2.1.2. Governing Equations of the Stress–Strain Field

To better describe the orthotropic anisotropic deformation of frozen soil, a local coordinate system was established, with directions 1, 2, and 3 representing the three principal axes. In the local coordinate system, the relationship between the thermal strain component caused by temperature and the freeze–thaw volume deformation is as follows [29]:

$$\left[\begin{array}{ccc}{\epsilon }_{11}^T& {\epsilon }_{12}^T& {\epsilon }_{13}^T\\ {\epsilon }_{21}^T& {\epsilon }_{22}^T& {\epsilon }_{23}^T\\ {\epsilon }_{31}^T& {\epsilon }_{32}^T& {\epsilon }_{33}^T\end{array}\right]=\left[\begin{array}{ccc}\eta & 0& 0\\ 0& (1-\eta )/2& 0\\ 0& 0& (1-\eta )/2\end{array}\right]{\epsilon }_v$$

(4)

Assuming that frozen soil is a linear elastic body, when the soil reaches the freezing or thawing temperature, the total strain of frozen soil is composed of two parts—the elastic strain caused by the load and the volume deformation caused by the freezing–thawing—which can be expressed as

$$\left\{\begin{array}{l}{\epsilon }_{11}={\displaystyle \frac{1}{E}}[{\sigma }_{11}-\mu ({\sigma }_{22}+{\sigma }_{33})]+\eta {\epsilon }_v\\ {\epsilon }_{22}={\displaystyle \frac{1}{E}}[{\sigma }_{22}-\mu ({\sigma }_{11}+{\sigma }_{33})]+{\displaystyle \frac{1}{2}}(1-\eta ){\epsilon }_v\\ {\epsilon }_{33}={\displaystyle \frac{1}{E}}[{\sigma }_{33}-\mu ({\sigma }_{11}+{\sigma }_{22})]+{\displaystyle \frac{1}{2}}(1-\eta ){\epsilon }_v\\ {\epsilon }_{12}={\displaystyle \frac{2}{E}}(1+\mu ){\tau }_{12}\\ {\epsilon }_{23}={\displaystyle \frac{2}{E}}(1+\mu ){\tau }_{23}\\ {\epsilon }_{31}={\displaystyle \frac{2}{E}}(1+\mu ){\tau }_{31}\end{array}\right.$$

(5)

where E is the elastic modulus of the frozen soil, μ is the Poisson’s ratio of the frozen soil, and η is the deformation characteristic coefficient, which is a dimensionless quantity. When η = 1/3, the deformation characteristic of frozen soil is isotropic; ${\epsilon }_v$ is the volume deformation of soil caused by frost heave or thaw settlement due to temperature changes.

For the plane strain problem, given by ${\epsilon }_{33}={\epsilon }_{23}={\epsilon }_{31}=0$, the following equation can be used:

$$\left\{\begin{array}{l}{\epsilon }_{11}={\displaystyle \frac{1-{\mu }^2}{E}}\left({\sigma }_{11}-{\displaystyle \frac{\mu }{1-\mu }}{\sigma }_{22}\right)+\left[\eta +{\displaystyle \frac{\mu }{2}}(1-\eta )\right]{\epsilon }_v\hfill \\ {\epsilon }_{22}={\displaystyle \frac{1-{\mu }^2}{E}}\left({\sigma }_{22}-{\displaystyle \frac{\mu }{1-\mu }}{\sigma }_{11}\right)+{\displaystyle \frac{1}{2}}(1-\eta )(1+\mu ){\epsilon }_v\hfill \\ {\epsilon }_{12}={\displaystyle \frac{2}{E}}(1+\mu ){\tau }_{12}\hfill \end{array}\right.$$

(6)

Suppose that the angle between heat flow direction 1 and the x-axis of the global coordinate system is $\theta$; then, the thermal strain component caused by the temperature in the global coordinate system is

$$\left\{\begin{array}{l}{\epsilon }_x^T=\left\{\left[\eta +{\displaystyle \frac{\mu }{2}}(1-\eta )\right]{\cos }^2\theta +\left[{\displaystyle \frac{1}{2}}(1-\eta )(1+\mu )\right]{\sin }^2\theta \right\}{\epsilon }_v\hfill \\ {\epsilon }_y^T=\left\{\left[\eta +{\displaystyle \frac{\mu }{2}}(1-\eta )\right]{\sin }^2\theta +\left[{\displaystyle \frac{1}{2}}(1-\eta )(1+\mu )\right]{\cos }^2\theta \right\}{\epsilon }_v\hfill \\ {\epsilon }_{xy}^T={\displaystyle \frac{1}{2}}(3\eta -1)\sin \theta \cos \theta {\epsilon }_v\hfill \end{array}\right.$$

(7)

In summary, the governing equation for the thermo-mechanical coupling stress–strain field of the 2D tunnel cross section is as follows:

$$\left\{\begin{array}{l}{\epsilon }_x={\displaystyle \frac{1-\mu {(T)}^2}{E(T)}}\left({\sigma }_x-{\displaystyle \frac{\mu (T)}{1-\mu (T)}}{\sigma }_y\right)+{\epsilon }_x^T\hfill \\ {\epsilon }_y={\displaystyle \frac{1-\mu {(T)}^2}{E(T)}}\left({\sigma }_y-{\displaystyle \frac{\mu (T)}{1-\mu (T)}}{\sigma }_x\right)+{\epsilon }_y^T\hfill \\ {\epsilon }_{xy}={\displaystyle \frac{2}{E(T)}}\left[1+\mu (T)\right]{\tau }_{xy}+{\epsilon }_{xy}^T\hfill \end{array}\right.$$

(8)

2.2. Numerical Model and Meshing

The location of the tunnel studied in this paper is shown in Figure 1. Ling et al. [30] revealed that considering the geometric construction of noncircular linings could make the distribution of frost heaving force of a tunnel more consistent with the actual situation; therefore, this study used a horseshoe structure that was consistent with the actual situation. The centre of the top arch of the tunnel lining is taken as the origin to establish the model, and the perpendicular distance between the model boundary and the origin is 35 m. The model includes the surrounding rock, primary lining, secondary lining, and pavement. The tunnel net area is 63.23 m², the equivalent diameter of the tunnel clearance cross section is 8.2 m, the thickness of the secondary lining is 0.45 m, and the thickness of the primary lining is 0.25 m. The numerical calculation model and meshing of the tunnel are shown in Figure 2.

Table 1. Thermodynamic parameters of the materials.

Materials	Density (kg·m−3)	Thermal Conductivity (W·m−1·°C−1)	Specific Heat Capacity (kJ·kg−1·°C−1)	Elastic Modulus (GPa)	Poisson’s Ratio (-)
Surrounding rock	2056	1.5 (unfrozen)/ 1.6 (frozen)	1.2 (unfrozen)/ 1.05 (frozen)	1.5 (unfrozen)/ 2.0 (frozen)	0.3 (unfrozen)/ 0.2 (frozen)
Primary lining	2400	1.3	0.9	20	0.2
Secondary lining	2400	1.6	0.9	30	0.2

lists the thermodynamic parameters required for numerical simulation.

2.3. Initial Field and Boundary Conditions

2.3.1. Initial Stress Field and Initial Temperature Field

When the initial geostress field is calculated, only the action of gravity is considered to limit the horizontal displacement of the left and right boundaries and the vertical displacement of the lower boundary. After the steady-state calculation is completed, stress is applied to the model.

Using the monitoring data provided by the scientific meteorological observation station near the tunnel site, the variation in temperature over time, from 1-1-2020 to 1-1-2022, was analysed, as shown in Figure 3. During the monitoring period, the highest daily average temperature is 21.3 °C in mid-August and the lowest temperature is −21.8 °C in mid-January.

When calculating the initial temperature, boundary layer theory is combined with the temperature fitting equation to establish the upper boundary temperature as

$$T=1.15-a\times 12.37\sin (2\pi t/365+1.25)$$

(9)

where T is the temperature (°C), t is time (d), and a is the amplitude adjustment coefficient, where a = 1.0 without adjustment.

The lower boundary of the model is set to be the heat flow boundary with a heat flux density of 0.06 W/m², and the left and right boundaries are set as adiabatic boundaries. These temperature boundary conditions are applied for long-term transient calculations. The calculated surrounding rock temperature field in the 100th year is used as the initial temperature field of the calculation domain.

2.3.2. Boundary Conditions for Transient Calculation

The settings of the upper, lower, left, and right displacement and temperature boundaries of the model are the same as those in Section 3.1. The left and right boundaries of the model do not move in the horizontal direction; that is, U_x = 0. The lower boundary of the model does not move in the vertical direction; that is, U_y = 0. The ground surface and the lining surface are free boundaries. The temperature fitting function is set on the upper surface boundary, the interface between the lining and air is set according to the actual temperature in the tunnel, the lower surface is set as the heat flow boundary, and the heat flux density is 0.06 W/m². The remaining boundaries are set to be adiabatic. The calculation time is 3650 d.

3. Results and Discussion

3.1. Effect of the Minimum Temperature

From the tunnel entrance to a certain depth, the temperature amplitude inside the tunnel gradually decreases. Based on the fitting function of the temperature near the tunnel site in Equation (9), the freezing cycle and distribution of frost heaving force are calculated under the four minimum temperatures of −8 °C, −10 °C, −12 °C, and −14 °C, with changing the temperature amplitude, and the frost heave ratio is 0.2%. Figure 4 shows the distribution characteristics of the tunnel temperature field on the coldest day under different minimum negative temperatures. The distribution characteristics of the temperature near the tunnel lining are basically the same, i.e., as the magnitude increases, the temperature of the lining and the surrounding rock on the coldest day decreases; but, simultaneously, the temperature of the surrounding rock around the lining is slightly higher due to the influence of the maximum temperature in the warm season.

The method for calculating the freezing cycle and the frost heaving force is as follows. First, the initial stress field is calculated, and the transient temperature is calculated with the temperature boundary condition to determine the thickness of the freezing cycle. Then, the given thermal expansion coefficient of the freezing cycle is comparable to the frost heave ratio, thus enabling the simulation of the frost heaving force. In thermo-mechanical coupling, sequential coupling is adopted; that is, the temperature field affects the stress field, and the stress field calculation is realised based on the temperature field. To explore the changes in the freezing cycle, the distribution of the freezing cycle on the coldest day extracted under different minimum temperatures is shown in Figure 5. The distribution of the freezing cycles around the tunnel lining is uneven; therefore, the inhomogeneity of the thickness distribution of the freezing cycle should be considered when utilising the overall frost heave model of the freeze–thaw cycle to reasonably determine the freezing range. Among them, the freezing cycle of the top arch is relatively uniform and has the largest thickness, while the freezing cycle of the inverted arch is uneven. From the smallest minimum temperature to the largest minimum temperature, the freezing depth of the tunnel vault is 0.95 m, 1.14 m, 1.25 m, and 1.39 m, and the freezing depth of the side wall is 0.79 m, 0.98 m, 1.09 m, and 1.23 m. As the temperature continues to decrease, the scope of the freezing cycle gradually expands, and from −8 °C to −14 °C, the average increase in the freezing depth of the arch top is 0.147 m/°C.

The temperatures at the tunnel vault position along the outer normal direction are extracted under each minimum air temperature, and the results are shown in Figure 6. As the minimum air temperature decreases, the temperature at the same position near the lining gradually decreases, but this trend becomes increasingly less obvious as the depth increases; the temperatures are basically the same after approximately 3 m of depth. This indicates that the effect of external temperature changes on the surrounding rock temperature is limited. After a certain depth, the surrounding rock temperature is mainly controlled by the stable geothermal gradient and is no longer disturbed by external temperature fluctuations, thus remaining relatively constant.

The frost heaving force acting on the periphery of the tunnel lining can be divided into two components: the normal stress and the tangential stress. However, in general, the normal stress plays an important role in the frost heaving force [31,32]. Therefore, this study focused on the normal stress component and extracted it via the following equation:

$${\sigma }_N={\sigma }_x{\cos }^2\alpha +{\sigma }_y{\sin }^2\alpha +2{\sigma }_{xy}\sin \alpha \cos \alpha$$

(10)

where α is the angle between the outer normal of the lining and the x-axis and ${\sigma }_N$ is the normal stress on the lining periphery.

Figure 7 shows the distribution of the frost heaving force at the periphery of the primary lining under different minimum temperatures. The distribution characteristics of the frost heaving force under different minimum temperatures are basically the same; the frost heaving force gradually increases from the vault downwards to the sidewall, and the frost heaving force gradually decreases from the sidewall to the arch bottom. The frost heaving force is the greatest from the sidewall to the spring line, and the frost heaving force is the least at the arch bottom. The maximum frost heaving force under each working condition is 0.245 MPa, 0.252 MPa, 0.264 MPa, and 0.273 MPa (in ascending order), with the maximum increment of the frost heaving force being 0.028 MPa, but the overall change was not significant. Under general natural working conditions that assume that there are no extremely low temperatures, fluctuations in the minimum temperature cannot cause drastic changes in the frost heaving force.

3.2. Effect of the Number of Days with Negative Temperatures

To investigate the effects of the duration of negative temperature on the temperature field distribution around the lining, in this study, amplitude modulation is performed according to Equation (9) to obtain a temperature change function with a minimum temperature of −10 °C. The coldest day of this function is used as the cut-off point to advance or delay the coldest day, the temperature amplitude remains unchanged, and the angular frequency is varied to generate five working conditions (∆T = ±15 d, ±30 d, and the original function). The positive sign indicates that the coldest day is delayed, the period of the temperature function becomes larger, and the number of days with negative temperatures decreases in that year; the negative sign indicates that the coldest day is advanced, the period decreases, and the number of days with negative temperatures increases in that year. The frost heave ratio is the same as that in Section 3.1. Figure 8 shows the freezing depth of the tunnel vault and sidewall at 365 d under different negative temperature days. The freezing depth of the vault and sidewall decreases with increasing ∆T, and the freezing depth becomes smaller when the coldest day is delayed, which is due to the fact that the temperature is approaching the minimum and has not yet reached the minimum. The freezing depth becomes greater when the coldest day advances. As the number of negative temperature days increases, the coldest day is not the day when the freezing depth is the greatest; the thermal response of the surrounding rock has a certain lag, and the peak freezing depth lags behind the minimum temperature.

Figure 9 shows the distribution of frost heave pressure under different numbers of negative temperature days, and the distribution characteristics are similar to those in Section 3.1. The distribution characteristics of the frost heaving force under different conditions are basically the same, with the frost heaving force from the sidewall to the spring line being the greatest and the frost heaving force being the least at the arch bottom. The maximum frost heaving force under each working condition in descending order of ∆T (+30 d, +15 d, 0 d, −15 d, and −30 d) is 0.239 MPa, 0.247 MPa, 0.252 MPa, 0.256 MPa, and 0.251 MPa, respectively, and the absolute maximum increment in the frost heaving force is 0.013 MPa. The advance or delay of the coldest day (i.e., the change in angular frequency) has an insignificant effect on the frost heaving force. The time accumulation effect is not considered because the angular frequency adjustment would destroy the interannual periodicity of the temperature data, resulting in significant differences in the temperature between a certain day and the same day a few years later, which is inconsistent with the general pattern.

3.3. Effect of the Frost Heave Ratio

The frost heave ratio range of rock masses under different frost heave grades is shown in

Table 2. Frost heave ratio ε_v (%) of rock masses under different frost heave grades [33].

Non-Frost Heave	Weakly Frost Heave	Frost Heave	Strongly Frost Heave	Extremely Strong Frost Heave
εv ≤ 0.13	0.13 < εv ≤ 0.47	0.47 < εv ≤ 0.80	0.80 < εv ≤ 1.6	1.6 < εv

. The frost heave ratio is determined by referring to the method of Xia et al. [33], who proposed a method to value the frost heave ratio of the rock mass in the cold region tunnel and carried out the corresponding classification of frost heave susceptibility (5 grades: non-frost heave susceptible rocks, weakly frost heave susceptible rocks, frost heave susceptible rocks, strongly frost heave susceptible rocks, and extremely strong frost heave susceptible rocks). In this study, four working conditions with frost heave ratios of 0.25%, 0.5%, 1.0%, and 2.0% are designed to calculate the frost heaving force of the lining; the values of the frost heave ratio cover a range from weakly frost heave-susceptible rocks to extremely strong frost heave-susceptible rocks, and the minimum temperature is −10 °C. Figure 10 shows the distribution of frost heave pressure under different frost heave ratios. The frost heave pressure from the vault to the sidewall gradually increases, the frost heave pressure from the sidewall to the vault bottom gradually decreases, and the frost heave pressure from the inverted arch to the spring line is the maximum. As the frost heave ratio increases, the frost heave pressure around the lining increases accordingly. The maximum frost heaving force is located in the sidewall section, in descending order of the frost heave ratio, which is 0.261 MPa, 0.305 MPa, 0.391 MPa, and 0.565 MPa. The largest increase in the frost heaving force occurs on the sidewall, i.e., when the frost heave ratio increases from 0.25% to 2.0%, the frost heaving force increases by 116%. The frost heave ratio of the surrounding rock has a significant effect on the frost heaving force.

Figure 11 shows the distributions of the maximum principal stresses of the initial lining under the four frost heave ratios. Figure 10 shows that when the frost heave ratio is 0.25%, the stresses are all negative, and the lining is completely under compression; when the frost heave ratios are 0.5%, 1.0%, and 2.0%, tensile stresses appears inside the lining, with maximum values of 0.05 MPa, 0.17 MPa, and 0.54 MPa, respectively, and the maximum tensile stress occurs at the spring line. The possible reason is as follows. The radius of curvature of the arch is relatively large; under the action of the frost heaving force of the surrounding rock, the backward bending of the inverted arch occurs to a certain extent, resulting in local tension at the spring line. Considering that the tensile strength of concrete is low, the strength of the arch can be improved by adding reinforcement or laying an insulation layer at the spring line [34].

3.4. Effect of the Anisotropy Coefficient of Frost Heave Deformation

Under actual working conditions, the frost heave deformation of the rock and soil masses around the tunnel mainly occurs in the direction of heat flow, but at the same time, the corresponding frost heave deformation also occurs in the direction orthogonal to heat flow, i.e., the volume deformation is spatially distributed along and perpendicular to the heat flow direction [24,25]. The introduction of the frost heave anisotropy coefficient of the surrounding rock can clarify the calculation and analysis results of the frost heaving force of the tunnel. To investigate the effect of the anisotropy coefficient on the frost heaving force of the tunnel lining, the analysis is performed under four working conditions (the anisotropy coefficient η = 0.7, 0.8, 0.9, and 1.0), with the minimum temperature being −10 °C and the frost heave ratio being 0.2%.

Figure 12 shows the distribution of the frost heaving force of the tunnel under different anisotropy coefficients of frost heave deformation. The distribution patterns of the frost heaving force at the periphery of the lining under different frost heave deformation anisotropy coefficients are basically the same; the overall frost heaving force follows the order of top arch > spring line > inverted arch, and the frost heaving force is located between the arch waist and the sidewall. As the frost heave deformation anisotropy coefficient increases, a greater proportion of frost heave deformation is concentrated along the direction of heat flow. This shift leads to a corresponding enhancement in the frost heaving force exerted perpendicular to the tunnel lining. As illustrated in Figure 11b, this effect is most pronounced at the vault (top arch), where the increase in stress is greatest. The arch waist experiences a moderate rise in frost heaving force, while the inverted arch shows only a slight increase. Specifically, in comparison to the isotropic case (η = 1/3), when η reaches 1.0, the frost heaving force increases by 0.121 MPa at point C (arch waist), by 0.107 MPa at point A (vault), and by just 0.01 MPa at point E (inverted arch). These results are similar to those in previous research [24,25,35], which showed that directional deformation and stress concentration often occur around tunnel linings. They found that the uneven thermal and mechanical behaviour of frozen ground can greatly influence how frost forces are distributed.

4. Conclusions

To study the distribution characteristics of the freezing cycle and frost heaving force of a tunnel in cold regions, the coupling effect of the temperature field and the stress field during the freezing process of the surrounding rock was considered, a thermo-mechanical coupling model of tunnel-surrounding rock considering the frost heave deformation anisotropy was established, and numerical simulations of the frost heaving force under different influencing factors were carried out. Based on the simulation results, the following conclusions are obtained:

(1) The response of the tunnel-surrounding rock temperature field to the external temperature fluctuations reveals significant spatial differences. The temperature of the shallow surface and the surrounding area of the lining changes with the external climate; on the other hand, the temperature distribution of the surrounding rock outside the affected area is mainly controlled by the geothermal gradient and remains relatively stable.

(2) As the minimum temperature decreases, the extent of tunnel freezing gradually increases. The frost heaving force distribution along the lining is closely influenced by the size of the freezing zone. Specifically, when the minimum temperatures are set to −8 °C, −10 °C, −12 °C, and −14 °C, the corresponding freezing depths at the tunnel vault are 0.95 m, 1.14 m, 1.25 m, and 1.39 m, respectively. As a result, a deeper frost penetration corresponds to a greater frost heaving pressure. The locations with the maximum frost heaving force are usually located at the sidewall and spring line sections. Owing to the influence of geometrical construction, tensile stress may be generated at the spring line of the lining, so measures such as adding reinforcement or laying an insulation layer can be used to prevent premature lining failure.

(3) When the coldest day is advanced or delayed within a 30-day range, the absolute maximum increment in the frost heaving force is only 0.013 MPa. This indicates that such small temporal shifts do not substantially influence the mechanical response of the tunnel lining. The frost heave ratio of the surrounding rock has a significant effect on the frost heaving force at the periphery of the lining. The frost heaving force at various locations on the lining increases with increasing frost heave ratio. When the frost heave ratio increases from 0.25% to 2.0%, the frost heaving force at the sidewall increases by 116% from 0.261 MPa to 0.565 MPa, the largest increase. The distribution patterns of the frost heaving force at the periphery of the lining under different frost heave deformation anisotropy coefficients are basically the same. With the increase in the frost heave deformation anisotropy coefficient, frost heave deformation becomes more concentrated along the heat flow direction, resulting in an enhanced frost heaving force perpendicular to the lining. Specifically, as the coefficient increases from 1/3 to 1.0, the frost heaving force increases by 0.121 MPa at the arch waist, 0.107 MPa at the vault, and only 0.01 MPa at the inverted arch, indicating that the most significant increase occurs at the vault.