## 4 Frozen Wall Elastic-Plastic Problem
### Influence of External Load on the Plastic Zone Contour  
Figure 12 shows the changes in the shape of the plastic zone contour of the frozen wall with an increase in \(p_{0}\) for different \(\lambda\). If the nonuniformity of the two external loads is strong (as shown in Figure 12a with \(\lambda=0.3\)), the tensile stress zone will clearly appear in the vertical direction at the inner edge of the frozen wall. In particular, when the value of \(p_{0}\) is low, the frozen wall only appears as a crescent-shaped plastic zone within a certain range of the inner edge in the horizontal direction (e.g., contours of 2 MPa and 3 MPa in Figure 12a), and the radius of plastic zone in the direction of 0\({}^{\circ}\) and 180\({}^{\circ}\) is the largest. As \(p_{0}\) increases, the plastic zone expands along the horizontal direction towards the outer edge of the freezing wall; in addition, it simultaneously expands along the inner edge of the freezing wall in the vertical direction. The contour of the plastic zone of the frozen wall gradually evolves from the crescent shape under low load to that of an ear-like shape (e.g., the contours of 5 MPa and 12 MPa in Figure 12a), and the plastic zone in the direction of 30-60\({}^{\circ}\) (the other three quadrants are similar in terms of the radius of the plastic zone) has the largest radius, which is the weakest position for plastic failure on the frozen wall. Furthermore, the horizontal inner edge of the freezing wall is in the plastic zone, while the outer edge is still in the elastic state; in contrast, for the vertical direction of the frozen wall, the inner edge lies in the tensile stress zone, whereas the outer edge lies in the elastic zone. If the nonuniformity of the two external loads is neither strong nor weak (as shown in Figure 12c with \(\lambda=0.5\)), there is no tensile stress zone in the entire frozen wall. In  
Figure 12: Relationship between the external load and plastic zone contour of the frozen wall. particular, when the value of \(p_{0}\) is low, the shape of the plastic zone of the frozen wall is the same as that of \(\lambda=0.3\). With an increase in \(p_{0}\), the contour of the plastic zone of the frozen wall gradually evolves from a crescent shape to ear-like shape (e.g., contours of 5 MPa and 12 MPa in Figure 12c), and the plastic zone in the direction of 30-60\({}^{\circ}\) (the other three quadrants are similar in terms of radius of the plastic zone) has the largest radius, which is the weakest position of plastic failure of the frozen wall, but its evolution process is slower than the case of \(\lambda=0.3\). Furthermore, the vertical direction of the frozen wall is entirely in the elastic zone, whereas inner edge and outer edge of the horizontal frozen wall lie in the plastic and elastic zones, respectively. If the nonuniformity of the two external loads is weak (as shown in Figure 12e with \(\lambda=0.8\)), because the stress condition is similar to the stress state under uniform load (\(\lambda=1\)), the contour of the plastic zone evolves from a crescent shape at the horizontal inner edge to an ellipse with an increase in the external load \(p_{0}\) on the frozen wall (e.g., contour of 5 MPa in Figure 12e). Thus, in this case, no ear-shaped plastic zone is observed during the entire evolution process, and the radius of plastic zone in the direction of 0\({}^{\circ}\) and 180\({}^{\circ}\) is always the largest. Based on our analysis, \(\lambda=0.485\) is the critical point for which the inner edge of the frozen wall just develops tensile stress under the abovementioned working conditions; this critical point for tensile stress is located on the vertical inner edge of the frozen wall. In contrast, when \(\lambda\)<0.485, the tensile stress zone is clearly observed on the inner edge of the vertical direction of the frozen wall, whereas, when \(\lambda>0.485\), there is no tensile stress zone on the frozen wall. Furthermore, \(\lambda=0.61\) is the critical point between the ear-shaped plastic zone and elliptical plastic zone of the frozen wall.