## 1 Introduction  
The AGF (artificial ground freezing method) has been widely used in underground engineering construction in water-rich strata because the resultant frozen wall has a good isolation effect on the groundwater [1, 2, 3, 4], the principle of AGF is shown in Figure 1. In the construction of shaft freezing method, the depth of the frozen wall depends on the buried depth of mineral resources. For example, the freezing depth of Polby potash salt mine in the UK reached 930 m, and the freezing depth of the auxiliary shaft of Hutaoyu coal mine in China's Gansu Province reached 950 m [5]. The frozen soil undergoes frost heaving during the formation freezing, which is caused by the migration of unfrozen water and the expansion of frozen soil. The deformation of stratum is the main cause of the damage of an underground building structure [6, 7, 8], and the frost heaving is a major cause of the fracture of freezing pipes and the deformation of surrounding strata of underground buildings [9, 10]. Currently, research on frost heave of stratum has been mainly focused on two aspects: the frost heave of frozen surrounding rock of tunnels in cold areas and the frost heave caused by the freezing of artificial formations. In the theoretical calculation of the frost heave force of tunnels in cold regions, [PERSON] et al. [11, 12, 13] used the unified strength theory as the basis for determining tunnels, considering the comprehensive influence of the intermediate principal stress and transverse isotropic frost heave on the strength of the surrounding rock. They established the unified plastic analytical solution of the stress and displacement of tunnels in cold areas. [PERSON] et al. (2014); [PERSON] et al. (2014) found an analytical solution of the frost heave force of tunnels in cold areas. In this solution, the anisotropic frost heave coefficient \(k\) describes the transverse isotropic frost heave of the surrounding rock. The rationality of the solution was verified by comparing its results with field data. [PERSON] (2014) established a model for determining the elasticity and plasticity of the surrounding rock of tunnels in cold zones. [PERSON] (2014) divided the frozen surrounding rock into partitions according to support strength and support time, established a calculation model that considers the influence of frost heave, and examined the stress and deformation of tunnel lining and surrounding rock in cold areas through engineering cases. [PERSON] (2014) established a three-dimensional temperature field model based on the test results of the temperature field of a tunnel in a cold region, obtaining the variation rule of the freezing depth of surrounding rock using [PERSON]'s formula. In addition, the author established a new frost heave model based on the freezing depth of surrounding rock, combined with the frost heave model of a freeze-thaw zone and the frost heave model of regolith. [PERSON] et al. [PERSON] et al. (2014) established an elastic-plastic damage mechanics calculation model, considering the characteristics of the uneven frost heave of surrounding rock, verifying the theoretical solutions and analyzing the parameters. Unlike the frozen surrounding rock of a tunnel in a cold zone, the artificial frozen wall is generated by freezing pipes. The heat conduction distance of freezing pipes and the direction of the temperature gradient influence the temperature within the frozen wall, which changes with the distance from the freezing pipes. Consequently, the internal strength of the frozen wall is non-uniform. The difference between the frozen surrounding rock of a tunnel and an artificial frozen wall is shown in Figure 2. If a frozen wall is treated as a homogeneous material to determine its mechanical properties, errors would be encountered in the calculation. [PERSON] [PERSON] (2014); [PERSON] (2015) derived a stress model for the frozen wall of single-row and double-row pipes using the M-C yield criterion. [PERSON] et al.