## 2 Theoretical Prediction of Plastic Zone Range in High-Temperature Tunnels  
### Analysis of Temperature Field in Gateway-like Structure Tunnel  
In general, under the impact of temperature, rock mass will generate a steady-state temperature field related to radial distance along the tunnel wall when ignoring the con -vective heat transfer. In this study, a circular tunnel model was established to analyze the temperature field, and the thermal conductivity differential equation is [21]:  
\[\lambda\frac{1}{r}\frac{d}{d_{r}}(r\frac{d_{T}}{d_{r}})=0 \tag{1}\]  
where \(\lambda\) is the thermal conductivity, \(T\) is the temperature, and \(r\) is the diameter length.  
By conducting continuous integration on Equation (1), the general solution is:  
\[T=c_{1}\ln r+c_{2} \tag{2}\]  
where \(c_{1}\) and \(c_{2}\) are the indeterminate constants.  
The calculation model is shown in Figure 1. Based on the influence range of rock mass temperature, it is assumed that the temperature at the inner boundary radius \(R_{1}\) is \(t_{1}\), and the rock mass temperature at the maximum influence radius \(R_{2}\) is \(t_{2}\), that is, the boundary conditions of the rock mass temperature field are determined.  
By substituting the boundary conditions into Equation (2), the indeterminate constants are obtained as:  
\[\left\{\begin{aligned} c_{1}=\frac{t_{2}-t_{1}}{\ln\frac{c_{2}}{R_{1 }}}\\ c_{2}=t_{1}-\ln r_{1}\frac{t_{2}-t_{1}}{\ln\frac{c_{2}}{R_{1}}} \end{aligned}\right. \tag{3}\]  
By substituting Equation (3) into Equation (2), the temperature field distribution equation of the circular tunnel rock mass is obtained as:  
\[T=t_{1}+\ln\frac{r}{R_{1}}\frac{t_{2}-t_{1}}{\ln\frac{R_{2}}{R_{1}}} \tag{4}\]  
Since the influence radius of tunnel excavation disturbance on the temperature field of the rock mass is twice about the excavation diameter [22], \(R_{2}=4R_{1}\) is taken. The area of the gateway-like structure tunnel is set to \(A_{S}\), which is equivalent to a circular tunnel of the same area.  
\[\left\{\begin{aligned} R_{1}=\sqrt{\frac{A_{S}}{\pi}}\\ R_{2}=4\sqrt{\frac{A_{S}}{\pi}}\end{aligned}\right. \tag{5}\]  
Then, the temperature field distribution equation of rock mass in the equivalent gateway-like structure tunnel can be obtained as:  
\[T=t_{1}+\ln\frac{r}{\sqrt{\frac{A_{S}}{\pi}}}\frac{t_{2}-t_{1}}{\ln 4} \tag{6}\]  
Figure 1: Calculation model.  
According to elastic-plastic mechanics, the temperature stress equation of rock mass in the circular tunnel is obtained as [23]:  
\[\left\{\begin{array}{l}\sigma_{\text{r1}}=-\frac{\alpha E\Delta t}{2(1-v)} \left(\frac{\ln\frac{R_{\text{r2}}}{2}-1}{\ln\frac{\delta_{\text{r2}}}{\delta_{ \text{r1}}}}-\frac{\left(\frac{R_{\text{r2}}}{\gamma}\right)^{2}-1}{\left(\frac {R_{\text{r2}}}{\delta_{\text{r1}}}\right)^{2}-1}\right)\\ \sigma_{\text{r1}}=-\frac{\alpha E\Delta t}{2(1-v)}\left(\frac{\ln\frac{R_{ \text{r2}}}{2}-1}{\ln\frac{\delta_{\text{r2}}}{\delta_{\text{r1}}}}+\frac{ \left(\frac{R_{\text{r2}}}{\gamma}\right)^{2}+1}{\left(\frac{R_{\text{r2}}}{ \delta_{\text{r1}}}\right)^{2}-1}\right)\end{array}\right. \tag{7}\]  
where \(\alpha\) is the thermal expansion coefficient, \(E\) is the elastic modulus, \(\Delta t\) is the temperature difference, \(v\) is the Poisson's ratio, and \(r\) is a specific point in the tunnel.  
Let \(-\frac{\alpha E\Delta t}{2(1-v)}=k\) and substitute Equation (5) into Equation (7), and then the temperature stress associated with the gateway-like structure tunnel can be obtained as:  
\[\left\{\begin{array}{l}\sigma_{\text{r2}}=k\left(\frac{1}{15}\left(1-\frac{ 16A_{\text{r}}}{\pi r^{2}}\right)+\frac{\ln\frac{4\sqrt{A_{\text{r}}}}{\sqrt{ \pi}}}{\ln 4}\right)\\ \sigma_{\text{r2}}=k\left(\frac{1}{15}\left(1+\frac{16A_{\text{r}}}{\pi r^{2}} \right)+\frac{\ln\frac{4\sqrt{A_{\text{r}}}}{\sqrt{\pi}}-1}{\ln 4}\right)\end{array}\right. \tag{8}\]  
Prediction of Plastic Zone Range of Gateway-like Structure Tunnel under Thermo-Mechanical Coupling Effect