## 4 Analysis of Viscoelastic-Plastic Stability of Frozen Wall
### Elastic-Plastic Analysis of Frozen Wall
#### 4.2.2 Elastic-Plastic Stress Analysis of Frozen Wall  
When the external load of the frozen wall exceeds its ultimate elastic load \(P_{e}\), the frozen wall enters the elastic-plastic state. The frozen wall begins to yield from the inner wall. With the increase in the external load of the frozen wall, the plastic zone continues to expand toward the outer wall, and the plastic zone radius of the frozen wall is assumed to be \(r_{c}\). Further, it is assumed that the frozen wall satisfies the Mohr-Coulomb criterion after entering the plastic state, i.e.,  
\[\sigma_{1}=\frac{1+\sin\varphi}{1-\sin\varphi}\sigma_{3}+\frac{2c\cos\varphi} {1-\sin\varphi} \tag{10}\]  
where \(c\) is cohesive force of frozen soil, MPa; \(\varphi\) is the angle of internal friction in the frozen soil, \({}^{\circ}\). For plane strain state, \(\sigma_{z}=\frac{1}{2}(\sigma_{r}+\sigma_{\theta})\), and it is assumed that the compression is positive and tension is negative. Therefore, \(\sigma_{\theta}>\sigma_{z}>\sigma_{r}\), i.e.\(\sigma_{1}=\sigma_{\theta}\), \(\sigma_{2}=\sigma_{z}\), and \(\sigma_{3}=\sigma_{r}\). In the elastic-plastic state, \(\sigma_{1}=\sigma_{\theta}^{p}\) and \(\sigma_{3}=\sigma_{r}^{p}\). Substituting these expressions in Equation (10), we obtain  
\[\sigma_{\theta}^{p}-\sigma_{r}^{p}=\frac{2\sin\varphi}{1-\sin\varphi}\sigma_{ r}^{p}+\frac{2c\cos\varphi}{1-\sin\varphi} \tag{11}\]  
For axisymmetric plane strain structure, the equilibrium equation is expressed as follows:  
\[\frac{d\sigma_{r}^{p}}{dr}+\frac{\sigma_{r}^{p}-\sigma_{\theta}^{p}}{r}=0 \tag{12}\]  
Substituting Equation (11) into Equation (12), we get  
\[\frac{d\sigma_{r}^{p}}{dr}-\frac{1}{r}\frac{2\sin\varphi}{1-\sin\varphi}= \frac{1}{r}\frac{2c\cos\varphi}{1-\sin\varphi} \tag{13}\]  
Figure 11: Control of the wellbore temperature of \(-216\) m sandy clay during excavation. Using the boundary condition: \(\sigma_{r}^{p}|_{r=a}=0\), it can be obtained that  
\[\sigma_{r}^{p}=\frac{c\cos\varphi}{\sin\varphi}\left[a^{-\frac{2\sin\varphi}{1- \sin\varphi}}r^{\frac{2\sin\varphi}{1-\sin\varphi}}-1\right] \tag{14}\]  
Substituting Equation (14) into Equation (11), we get  
\[\left\{ \begin{aligned} \sigma_{r}^{p}&=\frac{c\cos \varphi}{\sin\varphi}\left[a^{-\frac{2\sin\varphi}{1-\sin\varphi}}r^{\frac{2 \sin\varphi}{1-\sin\varphi}}-1\right]\\ \sigma_{\theta}^{p}&=\frac{c(1+\sin\varphi)\cos \varphi}{\sin\varphi(1-\sin\varphi)}\left[a^{-\frac{2\sin\varphi}{1-\sin \varphi}}r^{\frac{2\sin\varphi}{1-\sin\varphi}}-1\right]+\frac{2c\cos\varphi }{1-\sin\varphi}\end{aligned} \right.\quad r\in[a,r_{c}] \tag{15}\]  
When \(r\in(r_{c},b]\), the frozen wall can be regarded as a thick-walled elastic cylinder, whose inner and outer walls are subjected to \(\sigma_{r_{c}}\) and \(P\), respectively, and the analytical expression of its stress is [24]  
\[\left\{ \begin{aligned} \sigma_{r}^{e}&=\frac{b^{2}P-r_{c }^{2}\sigma_{r_{c}}}{b^{2}-r_{c}^{2}}-\frac{b^{2}r_{c}^{2}(P-\sigma_{r_{c}})}{ r^{2}(b^{2}-r_{c}^{2})}\\ \sigma_{\theta}^{e}&=\frac{b^{2}P-r_{c}^{2}\sigma_{r _{c}}}{b^{2}-r_{c}^{2}}+\frac{b^{2}r_{c}^{2}(P-\sigma_{r_{c}})}{r^{2}(b^{2}-r_ {c}^{2})}\end{aligned} \right.\quad r\in(r_{c},b] \tag{16}\]  
At the interface of the elastic-plastic zone of the frozen wall, when \(r=r_{c}\),  
\[\left\{ \begin{aligned} \sigma_{\theta}^{e}&=\sigma_{r_{c}} \\ \sigma_{\theta}^{e}&=\frac{2b^{2}P-\left(b^{2}+ \frac{r}{2}\right)\sigma_{r_{c}}}{b^{2}-r_{c}^{2}}\end{aligned} \right. \tag{17}\]  
Equation (17) satisfies Equation (10), and \(\sigma_{1}=\sigma_{\theta}^{e}\), \(\sigma_{3}=\sigma_{r_{c}}^{e}\).