## 1 Introduction  
Following the implementation of the \"Strong Transportation Country\" and \"West Development\" strategies, the density of the transportation network in China is expected to increase. Hence, new highway and railway tunnels will be designed, constructed, and operated in the cold regions of western and northeastern China. Cold-region tunnels are primarily characterized by the frost-heaving force, which differs from that in typical tunnels. On-site surveys have shown that the frost heave force is one of the important external factors causing damage to tunnel structures in cold regions [1, 2], as shown in Figure 1. Therefore, studying the frost-heaving force has important scientific value and engineering significance in cold-region tunnels. Based on the elastic solution of the frost-heaving force, a visco-elastic solution was obtained by combining the corresponding principle and the Laplace transform. The analytical results indicated that the frost-heaving force was an important influencing factor. If the frost-heaving force was considered, the stress of the lining structure was higher than that when the frost-heaving force was ignored [1]. The Mohr-Coulomb yield criterion, as one of the classic yield criteria, is widely used in the derivation of plastic solutions. The elasto-plastic solution meets the Mohr-Coulomb yield criterion, revealing that the frost heave was also an important factor during the development of plastic zones, and the frost heave will promote the expansion of the plastic zone range [3,4]. Subsequently, the non-uniform frost heave was further considered, and the analytical solution was established. The analysis results demonstrated that the stress difference in the support structure increases when frost heave is considered [5]. For the saturated rock, the frost heave behavior was systematically studied by conducting a series of model experiments. The results demonstrated that the frost heave behavior is more significant in the parallel freezing direction than in the vertical direction. A transversely isotropic frost-heaving model was then presented based on the experimental results, and the anisotropic frost heave coefficient was proposed and defined [6]. Additionally, by combining different yield criteria and non-associated flow rules, the plastic behavior of the frost heave that considers transversely isotropic frost heave was explored [7, 8, 9]. The analytical results revealed that the frost-heaving force increases with the increase in the frost heave coefficient, which indicates that the effect of transverse isotropy of the freezing rock mass on the frost-heaving force is significant [10]. In addition, mechanical models for composite lining structures [11] and dual non-uniform frost-heaving for the case of frozen rock mass [12] were proposed, solved, and analyzed. Compared to hydrostatic stress, non-hydrostatic stress more realistically reflects engineering practice. Accordingly, elastic analytical solutions of circumferential and radial stresses and radial displacement for rock mass and structure have been proposed [13], and the influence of the construction time of the structure on the stress and displacement has been explored [14]. The results showed that the influence of the lateral pressure coefficient on the vertical direction was slighter. However, existing solutions mainly consider the transversely isotropic characteristic of the frozen rock mass but neglect the impact of the radial temperature gradient on the mechanical properties of the frozen rock mass. The literature indicates that the lowest temperature zone is located on the inner surface of the frozen rock mass, and the temperature increases with the increase in the radial depth of the surrounding rock [15,16]. Cyclic freeze-thaw action increasingly damages surrounding rock with the increase in the temperature difference. Hence, the mechanical properties of frozen rock mass deteriorate with increasing depth along the radial direction [17,18].