## 3 Frozen Wall Elastic Problem
### Solution of the Elastic Unloading Mechanical Model for the Frozen Wall
It is known that the constant term in the abovementioned stress function has no effect on the stresses and displacements of the unfrozen soil and frozen wall; therefore, we set the constant term in the stress function given by Equation (8) to 0. Hence, by solving the above differential equations, the appropriate stress function expression is obtained as follows:  
\[\Phi_{i}(r,\theta)=a_{i}r^{2}+b_{i}r^{2}\ln r+c_{i}\ln r+(e_{i}+f_{i}r^{4}+ \frac{g_{i}}{r^{2}}+d_{i}r^{2})\cos 2\theta \tag{11}\]  
where \(a_{i}\), \(b_{i}\), \(c_{i}\), \(d_{i}e_{i}\), \(f_{i}\), and \(g_{i}(i=1,2)\) are undetermined parameters. When \(i=1\), the undetermined parameters in the stress function satisfying the semi-inverse solution of the frozen wall are expressed; in contrast, when \(i=2\), the undetermined parameters in the stress function satisfying the semi-inverse solution of the surrounding unfrozen soil are expressed. Substituting Equation (11) into the stress expression of Equation (7), the distribution function of the stress field for the frozen wall and surrounding unfrozen soil can be obtained as follows:  
\[\left\{ \begin{array}{l}\sigma_{ri}=-(2d_{i}+\frac{4e_{i}}{r^{2}}+\frac{6g_{i }}{r^{4}})\cos 2\theta+2a_{i}+2b_{i}\ln r+b_{i}+\frac{c_{i}}{r^{2}}\\ \sigma_{\theta i}=(2d_{i}+12f_{i}r^{2}+\frac{6g_{i}}{r^{4}})\cos 2\theta+2a_{i} +2b_{i}\ln r+3b_{i}-\frac{c_{i}}{r^{2}}\\ \tau_{\theta i}=(6f_{i}r^{2}+2d_{i}-\frac{2e_{i}}{r^{2}}-\frac{6g_{i}}{r^{4}}) \sin 2\theta\end{array} \right. \tag{12}\]  
Because the stresses of the surrounding unfrozen soil and frozen wall are solved for in the same manner, we only describe the solution for the stress field of the frozen wall below. The excavation of frozen soil inside the frozen wall can be considered equivalent to a stress relief field acting on the inner edge of the frozen wall, which induces stress redistribution and deformation of the frozen wall. After the excavation of frozen soil inside the frozen wall, the relief stress inside the frozen wall can be expressed as follows:  
\[\left\{ \begin{array}{l}\Delta\sigma_{r1}=\sigma_{r1}-\sigma_{r1}^{0}=-\left[ 2d_{1}+\frac{4e_{1}}{r^{2}}+\frac{6g_{1}}{r^{4}}-\frac{p_{0}(1-\lambda)}{2} \right]\cos 2\theta+2a_{1}+2b_{1}\ln r+b_{1}+\frac{c_{1}}{r^{2}}-\frac{p_{0}(1+ \lambda)}{2}\\ \Delta\sigma_{\theta1}=\sigma_{\theta1}-\sigma_{\theta1}^{0}=\left[2d_{1}+12 f_{1}r^{2}+\frac{6g_{1}}{r^{4}}-\frac{p_{0}(1-\lambda)}{2}\right]\cos 2\theta+2a_{1} +2b_{1}\ln r+3b_{1}-\frac{c_{1}}{r^{2}}-\frac{p_{0}(1+\lambda)}{2}\\ \Delta\tau_{r\theta 1}=\tau_{r\theta 1}-\tau_{r\theta 1}^{0}=\left[6f_{1}r^{2}+2d_{1}- \frac{2e_{1}}{r^{2}}-\frac{6g_{1}}{r^{4}}-\frac{p_{0}(1-\lambda)}{2}\right]\sin 2 \theta\end{array} \right. \tag{13}\]  
Furthermore, substituting Equation (13) into the physical as well as geometric equations and simplifying them, the elastic displacement of the frozen wall caused by the relief stress can be obtained as follows:  
\[\left\{ \begin{array}{l}\Delta u_{1}=\frac{1+\mu_{1}}{E_{1}}\Big{\{}-\Big{[}2 d_{1}r-\frac{1}{1+\mu_{1}}\frac{4e_{1}}{r}-\frac{2g_{1}}{r^{3}}-\frac{p_{0}(1- \lambda)r}{2}+\frac{p_{1}}{1+\mu_{1}}4f_{1}r^{3}\Big{]}\cos 2\theta\\ +\frac{1-\mu_{1}}{1+\mu_{1}}2a_{1}r+\frac{1-\mu_{1}}{1+\mu_{1}}2b_{1}r\ln r- b_{1}r-\frac{c_{1}}{r}-\frac{1-\mu_{1}}{1+\mu_{1}}\frac{p_{0}(1+\lambda)r}{2} \Big{\}}\\ \Delta v_{1}=\frac{1+\mu_{1}}{E_{1}}\Big{[}2d_{1}r+\frac{2f_{1}r^{3}(\mu_{1}+ 3)}{1+\mu_{1}}+\frac{2g_{1}}{r^{3}}-\frac{1-\mu_{1}}{1+\mu_{1}}\frac{2e_{1}}{ r}-\frac{p_{0}(1-\lambda)r}{2}\Big{]}\sin 2\theta+\frac{4b_{1}r\theta}{E_{1}}\end{array} \right. \tag{14}\]