## 4 Analysis of Viscoelastic-Plastic Stability of Frozen Wall
### Viscoelastic Analysis of Frozen Wall  
It is assumed that the frozen soil is incompressible [34], i.e., \(\varepsilon_{r}+\varepsilon_{\theta}+\varepsilon_{z}=0\), and the in-plane strain \(\varepsilon_{z}=0\). In this case, the geometric equation can be obtained as follows:  
\[\frac{du}{dr}+\frac{u}{r}=0 \tag{22}\]  
The solution of the above equation is  
\[u=\frac{k}{r} \tag{23}\]  
where \(k\) is the integral constant. The creep of frozen soil can be expressed by a unified rheological equation as follows [35]:  
\[\varepsilon=A\sigma^{B}t^{C} \tag{24}\]  
In practice, the frozen wall is in a triaxial stress state. Therefore, Equation (24) can be extended to a three-dimensional (3D) state through equivalent stress and equivalent strain. For the 3D state, the constitutive equation of frozen soil creep is  
\[\varepsilon_{i}=A\sigma_{i}^{B}t^{C} \tag{25}\]  
where \(\varepsilon_{i}=\frac{\sqrt{2}}{3}\sqrt{\left(\varepsilon_{\theta}- \varepsilon_{r}\right)^{2}+\left(\varepsilon_{r}-\varepsilon_{z}\right)^{2}+ \left(\varepsilon_{z}-\varepsilon_{\theta}\right)^{2}}\), \(\sigma_{i}=\frac{\sqrt{2}}{2}\sqrt{\left(\sigma_{\theta}^{ne}-\sigma_{r}^{ne} \right)^{2}+\left(\sigma_{r}^{ne}-\sigma_{z}^{ne}\right)^{2}+\left(\sigma_{z} ^{ne}-\sigma_{\theta}^{ne}\right)^{2}}\). Using the geometric equation and \(\sigma_{z}=\frac{1}{2}\left(\sigma_{r}^{ne}+\sigma_{\theta}^{ne}\right)\)[35], it can be obtained that \(\varepsilon_{i}=\frac{2}{\sqrt{3}}\frac{k}{r^{2}}\); \(\sigma_{i}=\frac{\sqrt{2}}{2}\left(\sigma_{\theta}^{ne}-\sigma_{r}^{ne}\right)\). Substituting the above expressions of equivalent strain \(\varepsilon_{i}\) and equivalent stress \(\sigma_{i}\) into Equation (25) and simplifying it, we get  
\[\sigma_{\theta}^{ne}-\sigma_{r}^{ne}=\left(\frac{2}{\sqrt{3}}\right)^{\frac{B +1}{B}}\left(\frac{k}{A}\right)^{\frac{1}{B}}r^{-\frac{2}{B}}t^{-\frac{C}{B}} \tag{26}\]  
Substituting Equation (26) into Equation (12), we obtain  
\[\frac{d\sigma_{r}^{ne}}{dr}=\left(\frac{2}{\sqrt{3}}\right)^{\frac{B+1}{B}} \left(\frac{k}{A}\right)^{\frac{1}{B}}r^{-1-\frac{2}{B}}t^{-\frac{C}{B}} \tag{27}\]  
The general solution of the above equation can be obtained as follows:  
\[\sigma_{r}^{ne}=-\frac{B}{2}\left(\frac{2}{\sqrt{3}}\right)^{\frac{B+1}{B}} \left(\frac{k}{A}\right)^{\frac{1}{B}}r^{-\frac{2}{B}}t^{-\frac{C}{B}}+d \tag{28}\]  
where \(d\) is the integral constant, which can be obtained by using boundary conditions. In other words, when \(r=a\), \(\sigma_{r}^{ne}=0\), then  
\[d=\frac{B}{2}\left(\frac{2}{\sqrt{3}}\right)^{\frac{B+1}{B}}\left(\frac{k}{A} \right)^{\frac{1}{B}}a^{-\frac{2}{B}}t^{-\frac{C}{B}} \tag{29}\]Substituting Equation (29) into Equation (28), we obtain  
\[\sigma_{r}^{ne}=-\frac{B}{2}\bigg{(}\frac{2}{\sqrt{3}}\bigg{)}^{\frac{B+1}{B}} \bigg{(}\frac{k}{A}\bigg{)}^{\frac{1}{B}}t^{-\frac{C}{B}}\left(r^{-\frac{2}{B}}- a^{-\frac{2}{B}}\right) \tag{30}\]  
where the constant \(K\) can be used as the boundary condition: when \(r=b\), \(\sigma_{r}=P\). \[k=\frac{\sqrt{3}}{2}At^{C}\Bigg{[}\frac{-\sqrt{3}P}{B(b^{-\frac{2}{B}}-a^{- \frac{2}{B}})}\Bigg{]}^{B} \tag{31}\]  
The radial creep displacement of the frozen wall can be obtained using Equations (23) and (31) as follows:  
\[u=\frac{k}{r} \tag{23}\]0  
The creep test parameters can be determined according to the creep test of frozen soil under uniaxial compression: \(A=0.00195\), \(B=1.591\), and \(C=0.317\). Substituting \(r=a=4.053\) m, \(b=8.403\) m, \(P=2.81\) MPa, and the values of \(A\), \(B\), and \(C\) into Equation (30), the variation in the side-wall displacement with time can be obtained, as shown in \[u=\frac{k}{r} \tag{23}\]1 It is evident that when the shaft wall is exposed for 30 h, its displacement is 26.85 mm. According to the Coal Mine Technical specification for vertical shaft Sinking by Freezing Method: MT/T1124-2011, in the thick clay layer with buried depth greater than 200 m, the exposure time of the shaft wall is limited to 30 h, and the maximum radial displacement of the shaft wall is not more than 50 mm.