## 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 \(-8\)\({}^{\circ}\)C, \(-10\)\({}^{\circ}\)C, \(-12\)\({}^{\circ}\)C, and \(-14\)\({}^{\circ}\)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.  
Conceptualization, L.P. and Q.L.; methodology, Y.S.; software, Y.S.; validation, Y.S. and L.P.; formal analysis, Y.S.; investigation, Y.S.; resources, L.P.; data curation, Y.S.; writing--original draft preparation, Y.S.; writing--review and editing, L.P.; visualization, Y.S. and L.P.; supervision, Q.L.; project administration, L.P. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
Conflicts of Interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.