## 1 Introduction  
Artificial ground freezing [1; 2; 3; 4] is a special construction technology often used in the construction of underground structures in weak, water-bearing strata, such as shaft engineering, tunnel construction, and foundation pit excavation. Research on the stress and deformation of the frozen wall created during the artificial ground freezing process while performing excavations has always been a topic of interest in the field of underground freezing engineering; herein, the size of the plastic zone of the frozen wall is an important measure to evaluate the stability of the frozen wall. It should be noted that, in this paper, plastic zones refer to compression plastic zones. In the literature, in the case of both vertical or horizontal frozen walls, research on their plastic zone radius and stress expression for the case wherein the lateral pressure coefficient \(\lambda=1\)--i.e., on the elastic or elastic-plastic characteristics of frozen wall under uniform load--is most common [5; 6; 7; 8; 9; 10; 11]. However, numerous measured data indicate that the in situ stress includes a certain degree of inhomogeneity, i.e., the ratio of horizontal stress to vertical stress (which is referred to as the lateral pressure coefficient \(\lambda\)) is not equal to 1 [12; 13]. This inhomogeneous pressure plays an important role in the failure of underground structures. Considering this, worldwide, research has been conducted considering nonuniform formation pressure on the surrounding rock of roadways [14; 15; 16; 17; 18; 19]. Indeed, analytical solutions of models of excavation of circular tunnels in elastic homogeneous materials subject to nonuniform initial stresses (i.e., the solution known as Kirsch's solution) [20; 21] are discussed in textbooks of rock mechanics and excavation analysis due to their importance as basic teaching/learning tools. Nevertheless, the research on the elastoplastic mechanical properties of the frozen wall formed by the artificial ground freezing method under nonuniform load conditions are still lacking and not explicitly developed [22; 23]. In particular, under nonuniform load, the initial stress field of the frozen stratum is considerably different from that of the surrounding rock of roadways (as shown in Figures 1 and 2). In particular, as is clear from Figure 2, it is important to consider the interaction between the frozen wall and unfrozen soil in the frozen stratum while studying the mechanical characteristics of the frozen wall before the installation of a lining. However, as shown in Figure 1, before this installation, the surrounding rock of the roadway is just a single medium in the stratum; therefore, the latter case only involves an inhomogeneous stress analysis of a single medium. Therefore, the existing research results of the surrounding rock of roadways under nonuniform load are not suitable for analyzing the mechanical characteristics of horizontal frozen walls. Figure 1: Initial stress field of the surrounding rock of the roadway. Figure 2: Initial stress field of the frozen stratum. In the design theory of frozen walls, the vertical frozen wall theory has become relatively mature and has formed a relatively complete set of design theories [24]. However, the design of the horizontal frozen wall is not yet mature, mainly relying on the calculation formula for the vertical frozen wall of the neutral shaft in mining construction and combining the on-site experience method for design [25; 26]. In recent years, scholars have focused on the design and research of horizontal frozen walls, which are still in the fields of numerical simulation, model testing, and on-site measurement. They mostly focus on the analysis of temperature and displacement fields [27; 28; 29; 30]. Theoretical analysis only includes partial elastic design research, while elastic-plastic analysis has not yet seen relevant theoretical analysis research [6; 22].