## 3 Frozen Wall Elastic Problem  
### Solution of the Elastic Unloading Mechanical Model for the Frozen Wall  
The elastic unloading mechanical model problem of the frozen wall can be obtained from the superposition of the equivalent initial stress field before excavation (Figure 4) and virtual excavation disturbance stress field (Figure 5) after the formation of the frozen wall. Figure 4: Equivalent initial stress field of the frozen wall before excavation. Figure 4 shows that the initial stress \(\sigma^{0}\) acts on both the inner edge of the frozen wall and outer edge of the unfrozen soil before excavation. In contrast, Figure 5 shows the virtual excavation disturbance stress field after excavation following the formation of the frozen wall, which is equivalent to the unloading force given by \(\sigma^{0}_{r}\) and \(\tau^{0}_{r\theta}\) at the inner edge of the frozen wall, where \(\sigma^{0}_{r}\) is the equivalent radial load and \(\tau^{0}_{r\theta}\) is the equivalent shear load. In the unloading model depicted in Figure 5, because the equivalent radial load and equivalent shear load of the inner edge of the frozen wall are trigonometrically related to the polar coordinate angle, it is assumed that the load and angle at the contact surface between the frozen wall and surrounding unfrozen soil are also related trigonometrically. Depending on the characteristics of the model boundary conditions and plane strain problem, when the semi-inverse solution method is used to solve the abovementioned problems, the plane problem using the stress method can be generalized as solving a stress function that satisfies a compatible equation. Let \(\Phi\) denote the stress function, then, the compatible equation in polar coordinates is given as follows:  
\[\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r }+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}\right)\left(\frac{ \partial^{2}\[PERSON]}{\partial r^{2}}+\frac{1}{r}\frac{\partial\[PERSON]}{\partial r }+\frac{1}{r^{2}}\frac{\partial^{2}\[PERSON]}{\partial\theta^{2}}\right)=0 \tag{6}\]  
If the stress function is known, the stress components can be obtained as follows:  
\[\left\{ \begin{array}{l}\sigma_{r}=\frac{1}{r}\frac{\partial\[PERSON]}{\partial r }+\frac{1}{r^{2}}\frac{\partial^{2}\[PERSON]}{\partial\theta^{2}}\\ \sigma_{\theta}=\frac{\partial\[PERSON]}{\partial r^{2}}\\ \tau_{r\theta}=-\frac{1}{r}\frac{\partial^{2}\[PERSON]}{\partial\pi\partial\theta} +\frac{1}{r^{2}}\frac{\partial\[PERSON]}{\partial\theta}=-\frac{\partial}{\partial r }\left(\frac{1}{r}\frac{\partial\[PERSON]}{\partial\theta}\right)\end{array} \right. \tag{7}\]  
The stress functions of the unfrozen soil and frozen wall \(\Phi_{i}(r,\theta)(i=1,2)\) are represented using separable forms (including \(f_{i}(r)\) and \(g_{i}(r)\)con\(2\theta\)); these stress functions are similar and can be expressed as follows [36; 37]:  
\[\Phi_{i}(r,\theta)=f_{i}(r)+g_{i}(r)\cos 2\theta \tag{8}\]  
Substituting the stress functions of Equation (8) into the compatible Equation (6) yields the following result:  
\[\frac{d^{4}f_{i}(r)}{dr^{4}}+\frac{2}{r}\frac{d^{3}f_{i}(r)}{dr^{3}}-\frac{1} {r^{2}}\frac{d^{2}f_{i}(r)}{dr^{2}}+\frac{1}{r^{3}}\frac{df_{i}(r)}{dr}+\left[ \frac{d^{4}g_{i}(r)}{dr^{4}}+\frac{2}{r}\frac{d^{3}g_{i}(r)}{dr^{3}}-\frac{9} {r^{2}}\frac{d^{2}g_{i}(r)}{dr^{2}}+\frac{9}{r^{3}}\frac{dg_{i}(r)}{dr}\right] \cos 2\theta=0 \tag{9}\]  
Equation (9) is applicable to any arbitrary \(\theta\). Hence,  
\[\left\{ \begin{array}{l}\frac{d^{4}f_{i}(r)}{dr^{4}}+\frac{2}{r}\frac{d^{3}f _{i}(r)}{dr^{3}}-\frac{1}{r^{2}}\frac{d^{2}f_{i}(r)}{dr^{2}}-\frac{1}{r^{2}} \frac{d^{2}f_{i}(r)}{dr^{2}}+\frac{1}{r^{3}}\frac{df_{i}(r)}{dr}=0\\ \frac{d^{4}f_{i}(r)}{dr^{4}}+\frac{2}{r}\frac{d^{3}g_{i}(r)}{dr^{3}}-\frac{9} {r^{2}}\frac{dg_{i}(r)}{dr}+\frac{9}{r^{3}}\frac{dg_{i}(r)}{dr}=0\end{array} \right. \tag{10}\]  
Figure 5: Virtual excavation disturbance stress field after the formation of the frozen wall.