## 3 Frozen Wall Elastic Problem
### Determination of Undetermined Parameters  
Based on the single value and continuous displacement conditions, we can say that \(b_{i}=0\), i.e.,  
\[b_{1}=b_{2}=0 \tag{15}\]  
According to the Saint-Venant principle, the original rock stress field at a sufficiently far distance will be correspondingly less affected by the excavation and unloading effect inside the frozen wall.  
In particular, when \(r\rightarrow\infty\), the stress of the surrounding unfrozen soil after excavation and unloading of the frozen soil at the inner edge of the frozen wall can still be expressed by Equation (2). Comparing Equation (2) with Equation (12), we obtain:  
\[\left\{\begin{array}{l}2d_{2}=p_{0}(1-\lambda)/2\\ 2a_{2}=p_{0}(1+\lambda)/2\\ f_{2}=0\end{array}\right. \tag{16}\]  
Furthermore, as previously mentioned, because the radial stress and shear stress at the inner edge of the frozen wall are equal to zero after excavation and unloading, the following relationship can be obtained:  
\[\left\{\begin{array}{l}(-2d_{1}-\frac{4e_{1}}{r_{0}^{2}}-\frac{6g_{1}}{r_{0}^ {4}})\cos 2\theta+2a_{1}+\frac{c_{1}}{r_{0}^{2}}=0\\ (6f_{1}r_{0}^{2}+2d_{1}-\frac{2c_{1}}{r_{0}^{2}}-\frac{6g_{1}}{r_{0}^{4}})\sin 2 \theta=0\end{array}\right. \tag{17}\]  
In a similar vein, because the radial stress, shear stress, radial displacement, and circumferential displacement at the contact surface between the frozen wall and surrounding unfrozen soil are equal, we obtain the following:  
\[\left\{\begin{array}{l}2d_{1}+\frac{4e_{1}}{r_{1}^{2}}+\frac{6g_{1}}{r_{1}^ {4}}=2d_{2}+\frac{4e_{2}}{r_{1}^{2}}+\frac{6g_{2}}{r_{1}^{4}}\\ 2d_{1}+\frac{1}{r_{1}^{4}}=2a_{2}+\frac{c_{1}^{4}}{r_{1}^{2}}\\ 6f_{1}r_{1}^{2}+2d_{1}-\frac{2e_{1}}{r_{1}^{2}}-\frac{6g_{1}}{r_{1}^{4}}=6f_{2 }r_{1}^{2}+2d_{2}-\frac{2e_{2}}{r_{1}^{2}}-\frac{6g_{2}}{r_{1}^{4}}\\ \frac{1+\mu_{1}}{E_{1}}\left\{-\left[2d_{1}r_{1}-\frac{1}{1+\mu_{1}}\frac{4e_{ 1}}{r_{1}}-\frac{2g_{1}}{r_{1}^{4}}-\frac{p_{0}(1-\lambda)r_{1}}{2}+\frac{\mu _{1}}{1+\mu_{1}}4f_{1}r_{1}^{3}\right]\right\}\\ =\frac{1+\mu_{2}}{E_{2}}\left\{-\left[2d_{2}r_{1}-\frac{1}{1+\mu_{2}}\frac{4e_ {2}}{r_{1}}-\frac{2g_{2}}{r_{1}^{4}}-\frac{p_{0}(1-\lambda)r_{1}}{2}+\frac{\mu _{2}}{1+\mu_{2}}4f_{2}r_{1}^{3}\right]\right\}\\ \frac{1-\mu_{1}}{E_{1}}2a_{1}r_{1}-\frac{1-\mu_{1}}{E_{1}}\frac{p_{0}(1+ \lambda)r_{1}}{2}-\frac{1+\mu_{1}}{E_{1}}\frac{1-\mu_{2}}{E_{2}}2a_{2}r_{1}- \frac{1-\mu_{2}}{E_{2}}\frac{p_{0}(1+\lambda)r_{1}}{2}-\frac{1+\mu_{2}}{E_{2}} \frac{c_{1}}{r_{1}}\\ \frac{1+\mu_{1}}{E_{1}}\left[2d_{1}r_{1}+\frac{2f_{1}r_{1}^{3}(\mu_{1}+3)}{1+ \mu_{1}}+\frac{2g_{1}}{r_{1}^{3}}-\frac{1-\mu_{1}}{1+\mu_{1}}\frac{2g_{1}}{r _{1}}-\frac{p_{0}(1-\lambda)r_{1}}{2}\right]\\ =\frac{1+\mu_{2}}{E_{2}}\left[2d_{2}r_{1}+\frac{2f_{2}r_{1}^{3}(\mu_{2}+3)}{1 +\mu_{2}}+\frac{2g_{2}}{r_{1}^{2}}-\frac{1-\mu_{2}}{1+\mu_{2}}\frac{2e_{2}}{r_ {1}}-\frac{p_{0}(1-\lambda)r_{1}}{2}\right]\end{array}\right. \tag{18}\]  
Equations (8)-(11) are simultaneous equations that can be solved using MAPLE, which is a numerical software, to obtain the values of \(a_{i}\), \(b_{i}\), \(c_{i}\), \(d_{i}\), \(e_{i}\), \(f_{i}\), and \(g_{i}\) (\(i=1,2\)); then, the expressions for stress and displacement in the elastic zone of a horizontal frozen wall can be obtained by substituting the parameter values into Equations (5) and (7). The expressions for stress and displacement of unfrozen soil can be obtained in a similar manner.