# Numerical Studies on Rockbursts in Tunnels with High In Situ Stresses and Geothermally Rich Areas
## 3 Model of Tunnel Excavation  
### Verification of the Accuracy of the Numerical Method  
To verify the accuracy of the numerical simulation method, a model of the inner plate with a hole under the combined action of heat and external force is established for verification (see Figure 3).  
The theoretical solution to the model can be seen in Equations (12)-(16):  
\[T=T_{a}+(T_{b}-T_{a})\frac{\ln\left(\frac{r}{a}\right)}{\ln\left(\frac{b}{a} \right)} \tag{12}\]  
\[\sigma_{r}^{T}=\frac{\alpha E}{1-\mu}\frac{(T_{a}-T_{b})}{2\ln\left(\frac{b}{ a}\right)}\bigg{[}-\ln\frac{b}{r}+\frac{a^{2}}{b^{2}-a^{2}}\bigg{(}\frac{b^{2}}{ r^{2}}-1\bigg{)}\ln\frac{b}{a}\bigg{]} \tag{13}\]  
\[\sigma_{\theta}^{T}=\frac{\alpha E}{1-\mu}\frac{(T_{a}-T_{b})}{2\ln\left(\frac {b}{a}\right)}\bigg{[}1-\ln\frac{b}{r}-\frac{a^{2}}{b^{2}-a^{2}}\bigg{(}\frac{ b^{2}}{r^{2}}+1\bigg{)}\ln\frac{b}{a}\bigg{]} \tag{14}\]  
\[\sigma_{r}^{M}=\bigg{[}1-\frac{a^{2}}{r^{2}}\bigg{]}\bigg{[}\frac{\sigma_{h}+ \sigma_{\varphi}}{2}\bigg{]}+\bigg{[}1-\frac{4a^{2}}{r^{2}}+\frac{3a^{4}}{r^{ 4}}\bigg{]}\bigg{[}\frac{\sigma_{h}-\sigma_{\varphi}}{2}cos2\theta\bigg{]} \tag{15}\]  
Figure 3: Sketch of the thermal-mechanical model.  
Figure 2: (**a**) Shear failure under large deviating stress after cooling. (**b**) Tensile failure under small deviational stress after cooling. \(t_{0}\): the moment when high-temperature rock starts to cool; \(t_{1}\): the moment when high-temperature rock fails after cooling; \(\sigma_{t}\): rock tensile strength; \(\sigma_{1-t_{0}}\): the first principal stress at \(t=t_{0}\); \(\sigma_{1-t_{1}}\): the first principal stress at \(t=t_{1}\); \(\sigma_{3-t_{0}}\): the third principal stress at \(t=t_{0}\); and \(\sigma_{3-t_{1}}\): the third principal stress at \(t=t_{1}\).  
\[\sigma_{\theta}^{M}=\left[1+\frac{a^{2}}{r^{2}}\right]\left[\frac{\sigma_{h}+\sigma _{v}}{2}\right]-\left[1+\frac{3a^{4}}{r^{4}}\right]\left[\frac{\sigma_{h}- \sigma_{v}}{2}cos2\theta\right] \tag{16}\]  
\(T_{a}\) is the temperature around the hole; \(T_{b}\) is the outer boundary temperature; \(E\) is the elastic modulus; \(\mu\) is Poisson's ratio; \(a\) is the radius of the hole; \(b\) is half the length of the outer boundary; \(\alpha\) is the thermal expansion coefficient; \(\sigma_{h}\) is the horizontal stress; \(\sigma_{v}\) is the vertical stress; \(\theta\) is the angle; \(r\) is the length along the diameter; and the thickness of the plate is \(1\) mm.  
When considering the temperature-stress coupling process, the stress distribution near the hole is as follows:  
\[\sigma_{r}^{TM}=\sigma_{r}^{T}+\sigma_{r}^{M} \tag{17}\]  
\[\sigma_{\theta}^{TM}=\sigma_{\theta}^{T}+\sigma_{\theta}^{M} \tag{18}\]  
Here, the parameters are set as follows: \(a\) = 30 mm; \(b\) = 750 mm; \(T_{a}\) = 100 \({}^{\circ}\)C; \(T_{b}\) = 0 \({}^{\circ}\)C; \(E\) = 50 GPa; \(\mu\) = 0.24; \(\sigma_{h}\) = 10 MPa; \(\sigma_{v}\) = 15 MPa; and \(\alpha\) = 10\({}^{-5}\)\({}^{\circ}\)C\({}^{-1}\). The thermal conductivity of the rock is 10 W/(m\(\cdot\)K), and the initial temperature of the whole model is 0 \({}^{\circ}\)C. Figure 4a-c shows the curves of the \(T\), \(\sigma_{r}^{TM}\) and \(\sigma_{\theta}^{TM}\) results when \(\theta\) = 0.  
Figure 4 shows that the numerical simulation results can better conform to the theoretical results and can be used for engineering research.  
Figure 4: Comparison of the numerical simulation results with the theoretical results. (**a**) The curves of the \(T\) when \(\theta\) = 0. (**b**) The curves of the \(\sigma_{r}^{TM}\) when \(\theta\) = 0. (**c**) The curves of the \(\sigma_{\theta}^{TM}\) when \(\theta\) = 0.