# Study on the Damage Process and Numerical Simulation of Tunnel Excavation in Water-Rich Soft Rock
## 2 Staged Softening Process of Water-Rich Soft Surrounding Rock Based on Tunneling
### Damage Deterioration Equation of Surrounding Rock Parameters Considering Dual Softening Coupling  
As the damage variables of tensile damage and compression shear damage are different, the deterioration of the surrounding rock parameters caused by them is also different. Generally, after the ultimate tensile strain is reached, the damage variable reaches 1, which means the loss of bearing capacity, so the elastic modulus, cohesion, and internal friction angle parameters will become 0. However, in compression shear damage, there is an obvious strain softening and residual stage of rock, and the mechanical parameters of the surrounding rock when reaching the residual stage is subject to a different degree of deterioration, thus, it is necessary to distinguish their deterioration process and magnitude. In this paper, after coupling the dual softening of unloading and water, the deterioration equations of the mechanical parameters of rock are derived as follows:  
1. Water-encountered damage under tension \[E=E_{0}\times(1-D_{t})=\left\{ \begin{array}{cc}E_{0}&\bar{\epsilon}\leq \varepsilon_{t}\\ \frac{\sigma_{tt}\sigma_{tt}\sigma_{t+2}}{E_{0}\bar{\epsilon}^{2}}&\varepsilon _{t}<\bar{\epsilon}<\varepsilon_{tt}\\ 0&\bar{\epsilon}\geq\varepsilon_{tt}\end{array} \right.\] (19) \[c=c_{0}\times(1-D_{t})=\left\{ \begin{array}{cc}c_{0}&\bar{ \epsilon}\leq\varepsilon_{t}\\ \frac{\sigma_{tt}\sigma_{tt}\sigma_{t+2}}{E_{0}\bar{\epsilon}^{2}}c_{0}& \varepsilon_{t}<\bar{\epsilon}<\varepsilon_{tt}\\ 0&\bar{\epsilon}\geq\varepsilon_{tt}\end{array} \right.\] (20) \[\varphi=\varphi_{0}\times(1-D_{t})=\left\{ \begin{array}{cc} \varphi_{0}&\bar{\epsilon}\leq\varepsilon_{t}\\ \frac{\sigma_{tt}\sigma_{t+2}}{E_{0}\bar{\epsilon}^{2}}\varphi_{0}&\varepsilon _{t}<\bar{\epsilon}<\varepsilon_{tt}\\ 0&\bar{\epsilon}\geq\varepsilon_{tt}\end{array} \right.\] (21) \[2.\quad\text{Water-encountered damage under compression shear}\] \[E=E_{0}\times(1-D_{E12})=\left\{ \begin{array}{cc}\left[1-(D_{em1}+D_{ em2})\frac{\bar{\epsilon}^{p}}{\bar{\epsilon}^{p}}+D_{em1}D_{em2}\frac{(\bar{ \epsilon}^{p})^{2}}{(\bar{\epsilon}^{p})^{2}}\right]E_{0}&0\leq\bar{\epsilon}^ {p}<\bar{\epsilon}^{p}\\ (1-D_{em1}-D_{em2}+D_{em1}D_{em2})E_{0}&\bar{\epsilon}^{p}\geq\bar{\epsilon}^ {p}\end{array} \right.\] (22) \[ \begin{array}{cc}D_{em1}=1-\frac{E_{1}}{E_{0}}&D_{em2}=1-\frac{E_ {2}}{E_{0}}&\\ \end{array} \] \[c=c_{0}\times(1-D_{c12})=\left\{ \begin{array}{cc}\left[1-(D_{em1}+D_ {em2})\frac{\bar{\epsilon}^{p}}{\bar{\epsilon}^{p}}+D_{em1}D_{em2}\frac{(\bar {\epsilon}^{p})^{2}}{(\bar{\epsilon}^{p})^{2}}\right]c_{0}&0\leq\bar{\epsilon}^ {p}<\bar{\epsilon}^{p}\\ (1-D_{em1}-D_{em2}+D_{em1}D_{em2})c_{0}&\bar{\epsilon}^{p}\geq\bar{\epsilon}^ {p}\\ D_{em1}=1-\frac{c_{1}}{c_{0}}&D_{em2}=1-\frac{c_{2}}{c_{0}}&\end{array} \right.\] (23) \[\varphi=\varphi_{0}\times(1-D_{\varphi 12})=\left\{ \begin{array}{cc} \left[1-\left(D_{\varphi m1}+D_{\varphi m2}\right)\frac{\bar{\epsilon}^{p}}{ \bar{\epsilon}^{p}}+D_{\varphi m1}D_{\varphi m2}\frac{(\bar{\epsilon}^{p})^{2} }{(\bar{\epsilon}^{p})^{2}}\right]\varphi_{0}&0\leq\bar{\epsilon}^{p}<\bar{ \epsilon}^{p}\\ \left(1-D_{\varphi m1}-D_{\varphi m2}+D_{\varphi m1}D_{\varphi m2}\right) \varphi_{0}&\bar{\epsilon}^{p}\geq\bar{\epsilon}^{p}\\ D_{\varphi m1}=1-\frac{\varphi_{1}}{\bar{\epsilon}_{0}}&D_{\varphi m2}=1- \frac{\varphi_{2}}{\bar{\epsilon}_{0}}\end{array}