## 3 Results and Analysis  
### Temperature and Stress Distribution of Tunnel under Ventilation without TIL  
A uniform numerical model (\(m\) = 1000) is adopted to analyze the evolution of the temperature and stress field in the ventilated tunnel. Figure 6a shows the temperature variation in the \(A\)-\(A^{\prime}\) section under different ventilation times. Affected by the cold airflow, a cold front appears in the tunnel rock, disturbing the temperature field near the tunnel wall and forming a thermal disturbance circle [42]. The rock temperature in this area displays a decrease along the radial direction from the tunnel wall to the interior. The cold front spreads into the rock interior as the ventilation time goes, enlarging the thermal disturbance circle in which the rock temperature decreases further. The rock temperature \(T_{r}\) of the tunnel wall drops from 200 \({}^{\circ}\)C to 38.2 \({}^{\circ}\)C as the ventilation time \(t\) goes to 100 h. The temperature drop in the thermal disturbance zone causes a temperature gradient, thereby generating corresponding thermal stress (\(\sigma_{T}\)) [29; 37] (negative represent tensile stress). Figure 6b presents the thermal stress of the \(A\)-\(A^{\prime}\) section under different ventilation times. Similar to the temperature evolution, the thermal stress descends from the tunnel wall to the interior but ascends with the increased ventilation time. The stress \(\sigma_{T}\) on the tunnel wall increases from \(-116.11\) MPa to \(-123.48\) MPa when the ventilation time \(t\) increases from 10 h to 100 h. It exceeds the tensile strength of most rock mass, which can possibly cause rock failure and crack initiation. The temperature and stress evolution illustrates that the increase in the thermal stress corresponds well to the decrease in the rock temperature, showing that a greater temperature drop causes more thermal stress increase. The surface temperature drops the fastest, resulting in the maximum temperature gradient and associated maximum thermal stress, which is consistent with [PERSON] et al. [22]. Thus, reducing the temperature decrease rate of the tunnel rock, such as paving TIL on the tunnel wall, can effectively weaken the induced thermal stress and alleviate the associated rock damage. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Values** & **Parameters** & **Values** \\ \hline Rock compressive strength \(\sigma_{sr}\) & 233.05 MPa & TIL compressive strength \(\sigma_{si}\) & 200 MPa \\ Rock tensile strength \(\sigma_{tr}\) & 11.61 MPa & TIL tensile strength \(\sigma_{ti}\) & 30 MPa \\ Rock thermal conductivity \(\lambda_{r}\) & 2.86 W/m/K & TIL thermal conductivity \(\lambda_{i}\) & 0.1 W/m/K \\ Rock specific heat capacity \(c_{r}\) & 0.85 \(\times\) 106 \(\mathrm{J/(m^{3}\cdot K)}\) & TIL specific heat capacity \(c_{i}\) & 1 \(\times\) 106 \(\mathrm{J/(m^{3}\cdot K)}\) \\ Rock Young’s modulus \(E_{r}\) & 37.46 GPA & TIL Young’s modulus \(E_{i}\) & 20 GPa \\ Rock Poisson’s ratio \(\
u_{r}\) & 0.25 & TIL Poisson’s ratio \(\
u_{i}\) & 0.25 \\ Rock thermal expansion & 8 \(\times\) 10\({}^{-5}\) 1/K & TIL thermal expansion & 1 \(\times\) 10\({}^{-5}\) 1/K \\ coefficient \(\alpha_{r}\) & 8 \(\times\) 10\({}^{-5}\) 1/K & Coefficient \(\alpha_{i}\) & 1 \(\times\) 10\({}^{-5}\) 1/K \\ Rock initial temperature \(T_{r}\) & 200 \({}^{\circ}\)C & TIL initial temperature \(T_{i}\) & 200 \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical and mechanical parameters in the numerical model. The resulting temperature difference reduces the thermal stress induced by the temperature variation. Thus, the thermal stress in the tunnel with TIL is significantly lower than that without TIL. As shown in Figure 7b, the stress \(\sigma_{T}\) on the tunnel wall reaches \(-123.48\) MPa in the tunnel without TIL as the ventilation time \(t\) grows to \(100\) h, which reduces to \(-24.66\) MPa after paving a TIL. The decrease in thermal stress can remarkably reduce the possibility of rock damage [29].