## 3 Solution and Analysis of Longitudinal Deformation Curve of Surrounding Rock
### Data Comparison and Equation Optimization Analysis
#### 3.2.2 Equation Optimization and Improvement  
The comparison between the monitoring data and theoretical calculation shows that the deformation of the surrounding rock during the actual construction of the tunnel is earlier than the theoretical derivation: it shows that the displacement release coefficient of the surrounding rock is not only related to Poisson's ratio, but also related to the elastic-plastic modulus \(E\), internal friction angle \(\varphi\), cohesion \(c\) and blasting parameters. (1) The relationship between the shear strength and Mohr-Circle (M-C) is shown in Figure 5 (\(\sigma\) is the compressive strength of the rock mass; \(\tau\) is the shear strength of the rock mass; \(c\) is cohesion; \(\varphi\) is the internal friction angle): under the same internal friction angle \(\varphi\), the greater the cohesion \(c\) and the wider the strength envelope range of the surrounding rock, the better the bearing capacity of the surrounding rock; similarly, under the same cohesion \(c\), the larger the internal friction angle \(\varphi\) is, the wider the strength envelope range of the surrounding rock is, the better the bearing capacity of the surrounding rock is. The results show that the displacement release coefficient is no linearly proportional to the internal friction angle \(\varphi\) and cohesion \(c\). (2) The shear strength test of the rock mass shows that the strength of the surrounding rock is related to the bulk modulus \(K\) and shear modulus \(G\). When Poisson's ratio is constant, the bulk modulus \(K\) and shear modulus \(G\) are directly proportional to the elastic \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fitting** & \multicolumn{2}{c}{**III**} & \multicolumn{2}{c}{**IV**} \\ \cline{2-5}
**Parameters** & **YK30+960** & **YK30+911** & **YK31+645** & **YK31+422** \\ \hline RMSE & 0.0678 & 0.0706 & 0.0708 & 0.0601 \\ \hline SSE & 0.0456 & 0.0647 & 0.0653 & 0.0470 \\ \hline R-square & 0.7920 & 0.7920 & 0.7794 & 0.8415 \\ \hline \end{tabular} Notes: RMSE is mean square deviation, SSE is sum variance, R-square is correlation coefficient.  
\end{table} 
Table 2: Fitting parameters of surrounding rock. Figure 4: Monitoring and theoretical fitting diagram of vault displacement release coefficient: (a) Class\(-\)III surrounding rock; (b) Class\(-\)IV surrounding rock. modulus \(E\). It shows that the displacement release coefficient is nonlinearly proportional to the elastic modulus \(E\). (3) Blasting causes great damage to the integrity of the rock mass and greatly reduces the elastic modulus \(E\) and cohesion \(c\) of the rock mass. The results show that the displacement release coefficient is nonlinear and inversely proportional to the blasting parameters. The following rules exist near the excavation face: the rock mass is damaged by construction blasting, resulting in cracks in the rock mass, and the bearing capacity of the surrounding rock is reduced, which is the main reason for the deviation between the monitoring data and the theoretical calculation. Therefore, for the theoretical Equation (8), it is proposed to add an \"expanding convergence function\" in the section \(x\geq 0\) to improve the practicability of the theoretical equation. Another exponential function (\(y~{}=~{}e^{-x}\)) is a convergent subtractive function with a maximum value of (\(y_{0}~{}=~{}0\)) when \(x\geq 0\), and its deformation law meets the requirements of an \"expanded convergent function\". Equation (9) of the \"expanded convergent function\" is obtained as follows in Equation (9):  
\[k(x)~{}=~{}\xi+\exp(-\eta x/R) \tag{9}\]  
where, \(\xi\) and \(\eta\) are the fitting parameters related to the elastic modulus \(E\), internal friction angle \(\varphi\), cohesion \(c\) and blasting parameters.