## 3 Plastic Zone Expansion Law
### Theoretical Calculation of the Plastic Zone
\[\sin\varphi=\frac{\sqrt{\left(\sigma_{\theta}\ ^{e}-\sigma_{r}\ ^{e}\right)^{2}+ \left(2\tau^{e}\ _{r\theta}\right)^{2}}}{\sigma_{\theta}\ ^{e}+\sigma_{r}\ ^{e}+2 ccot\varphi} \tag{2}\]  
The plasticity criterion Equation (2) is rearranged to derive the boundary line Equation (3) for the tunnel envelope's plastic zone under thermal coupling. \[\cos^{2}2\theta\Big{[}a^{2}\big{(}4\alpha^{4}\sin^{2}\varphi-\omega_{1}\ ^{2}+\omega_{2}\big{)}\Big{]}+a\cos 2 \theta\Bigg{[} \begin{array}{l}2(b-2p_{\rm i})\omega_{1}-\\ 4\sin^{2}\varphi(b+2 ccot\varphi)\end{array} \Bigg{]}a^{2}+\Bigg{[} \begin{array} []{l}(b+2 ccot\varphi)^{2}\sin^{2}\varphi-(b-2p_{\rm i})^{2}\alpha^{4}-\\ a^{2}\omega_{2}^{2}-8\pi\rmi^{2}\alpha^{4}+4a\omega_{2}\tau\rmi^{2}\sin 2 \theta\end{array} \Bigg{]}=0 \tag{3}\]  
\[\text{where }\left\{ \begin{array}{l}\omega_{1}=\big{(}1+3\alpha^{4}-2\alpha^{2} \big{)}\\ \omega_{2}=\big{(}1-3\alpha^{4}+2\alpha^{2}\big{)}\\ a=(\lambda-1)\sigma_{2}-\sigma^{T}\\ b=(1+\lambda)\sigma_{2}+\sigma^{T}\\ \sigma^{T}=\frac{6 EZ(T-70)}{H-h}\\ \alpha=\frac{r_{0}}{R_{0}}\end{array} \right.\.\]  
In a non-uniform stress field, provided that the stress and the properties of the surrounding rock are known (as per Table 2), the boundary of the plastic zone can be determined by employing Equation (3), we can derive the implicit Equation (4) involving the binary octave equation concerning (\(r_{0}/R_{0}\)) and (\(\theta\)). This equation allows us to examine the morphology of the plastic zone surrounding the circular tunnel in a non-uniform stress field and analyze its expansion characteristics. \[A\bigg{(}\frac{r_{0}}{R_{0}}\bigg{)}^{8}+B\bigg{(}\frac{r_{0}}{R_{0}}\bigg{)} ^{6}+C\bigg{(}\frac{r_{0}}{R_{0}}\bigg{)}^{4}+D\bigg{(}\frac{r_{0}}{R_{0}} \bigg{)}^{2}+E=0 \tag{4}\]  
Figure 2: Mechanical model of circular tunnel considering thermal stress of rock mass. \[\text{where }\left\{ \begin{array}{l}A=27\big{(}(\lambda-1)\sigma_{z}-\sigma^{T} \big{)}^{2}+9\big{(}(\lambda-1)\sigma_{z}-\sigma^{T}\big{)}^{2}\cos 4\theta\\ B=-36\big{(}(\lambda-1)\sigma_{z}-\sigma^{T}\big{)}^{2}-12\big{(}(\lambda-1) \sigma_{z}-\sigma^{T}\big{)}^{2}\cos 4\theta\\ C=2\big{(}(\lambda-1)\sigma_{z}-\sigma^{T}\big{)}^{2}+2\big{(}(1+\lambda)\sigma _{z}+\sigma^{T}\big{)}^{2}+\\ 6\big{(}(\lambda-1)\sigma_{z}-\sigma^{T}\big{)}^{2}\cos 4\theta+\\ 8\big{(}(\lambda-1)\sigma_{z}-\sigma^{T}\big{)}\big{(}(1+\lambda)\sigma_{z}+ \sigma^{T}\big{)}\cos 2\theta\\ D=6\big{(}(\lambda-1)\sigma_{z}-\sigma^{T}\big{)}^{2}-2\big{(}(\lambda-1) \sigma_{z}-\sigma^{T}\big{)}^{2}\cos 4\theta-\\ 2\big{(}(\lambda-1)\sigma_{z}-\sigma^{T}\big{)}\big{(}(1+\lambda)\sigma_{z}+ \sigma^{T}\big{)}\cos 2\theta+\\ 4\big{(}(\lambda-1)\sigma_{z}-\sigma^{T}\big{)}c\cos 2\theta\cot\varphi\\ E=2\big{(}(\lambda-1)\sigma_{z}-\sigma^{T}\big{)}^{2}-\big{(}(1+\lambda)\sigma _{z}+\sigma^{T}\big{)}^{2}-4c^{2}+\\ \Big{(}\big{(}(1+\lambda)\sigma_{z}+\sigma^{T}\big{)}^{2}-4c^{2}\Big{)}\cos 2 \varphi-\\ 4\big{(}(1+\lambda)\sigma_{z}+\sigma^{T}\big{)}c\sin 2\varphi\end{array} \right..\]  
Because of the intricate and lengthy nature of the implicit equation, directly solving for the radius of the plastic zone presents considerable challenges. Wolfram Mathematica 13.0 software is utilized for processing the solution effectively. By inputting relevant parameters such as cohesion, internal friction angle, tunnel radius, and burial depth a numerical analytical solution for the radius of the plastic zone surrounding the rock is derived using an iterative method. This approach guarantees calculation accuracy to within 1%. Distribution Pattern and Expansion Law of Plastic Zone in High Geothermal and High Geostress Tunnels  
Changes in geostress and geothermal temperatures are the principal factors driving the expansion of the plastic zone. Their impact on the plastic zone surrounding the rock is manifested in the variations of lateral pressure coefficients and thermal stress within the rock, influenced by the stress characteristics of the surrounding rock.