# 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 Considering Time Effect and Dual Softening Coupling  
The loosening circle of surrounding rock caused by excavation unloading needs some time to converge and the differential water-rock action makes the areas with more concentrated development and better hydraulic conductivity in the fracture networks, preferentially expand into a hydraulic fracture. Therefore, the unloading softening and water-softening damage of surrounding rock have a time effect, so the deterioration process of the surrounding rock parameters also varies with time. After considering the time effect, the equations of damage variable evolution and rock parameters' deterioration are improved as follows:  
\[D(\varepsilon,t)=D(\varepsilon)\cdot\Phi(t) \tag{25}\]  
\[E(t)=E_{0}\times[1-D_{E}(\varepsilon,t)]=E_{0}\times[1-D_{E}(\varepsilon)\Phi_{ E}(t)] \tag{26}\]  
\[c(t)=c_{0}\times[1-D_{c}(\varepsilon,t)]=c_{0}\times[1-D_{c}(\varepsilon)\Phi_ {c}(t)] \tag{27}\]  
\[\varphi(t)=\varphi_{0}\times\big{[}1-D_{\varphi}(\varepsilon,t)\big{]}= \varphi_{0}\times\big{[}1-D_{\varphi}(\varepsilon)\Phi_{\varphi}(t)\big{]} \tag{28}\]  
where, \(D(\varepsilon,t)\) is the damage variable considering both softening parameters and time, and \(D(\varepsilon)\) is the damage variable only considering softening parameters, see Section 3.1. \(\Phi(t)\) is a time-dependent segmental function, \(\Phi(0)=0\), \(\Phi(\infty)=1\). \(\Phi_{E}(t)\), \(\Phi_{c}(t)\), \(\Phi_{\varphi}(t)\) are the time effect functions of deterioration of elastic modulus, cohesion, and internal friction angle, respectively.  
Surrounding rock unloading softening time is short. This paper assumes that the damage variables in this period are linear, while the water-softening time is longer. [PERSON] [53] studied the deformation and strength parameters of saturated rock samples with water immersion time and concluded that each physical and mechanical parameter of the rock is exponentially related to time. In this paper, assuming that the damage variables of the surrounding rock obey exponential variation with water-softening time, then:  
\[Y=\left\{\begin{array}{cc}-\frac{Y_{0}-Y_{1}}{t_{1}}t+Y_{0}&0\leq t<t_{1}\\ A\exp[-B(t-t_{1})]+C&t_{1}\leq t\leq t_{2}\\ Y_{2}&t_{2}\leq t\end{array}\right. \tag{29}\]  
where, \(Y\) denotes the parameters of surrounding rock, including elastic modulus, cohesion, and internal friction angle, etc. \(Y_{0}\) denotes the initial parameters of the surrounding rock, \(Y_{1}\) denotes the final parameters after unloading softening, and denotes the final parameters after water-softening. When \(t=t_{1}\), \(Y\)=\(Y_{1}\). When \(t=t_{\infty}\), \(Y\)=\(Y_{2}\). It is easy to obtain \(A=Y_{1}-Y_{2}\), \(C=Y_{2}\), so Equation (29) can be transformed into:  
\[Y=\left\{\begin{array}{cc}-\frac{Y_{0}-Y_{1}}{t_{1}}t+Y_{0}&0\leq t<t_{1}\\ (Y_{1}-Y_{2})\exp[-B(t-t_{1})]+Y_{2}&t_{1}\leq t\leq t_{2}\\ Y_{2}&t_{2}\leq t\end{array}\right. \tag{30}\]  
where, \(t=0\) indicates the start of unloading softening of surrounding rock. \(t=t_{1}\) indicates the end of unloading softening, and the start of water-softening. \(t=t_{2}\) indicates the end of water-softening. Usually, \(t_{1}\) is much smaller than \(t_{2}\).  
According to the damage variable \(D=1-\frac{Y}{Y_{0}}\), \(D_{1}=1-\frac{Y_{1}}{Y_{0}}\), \(D_{2}=1-\frac{Y_{2}}{Y_{1}}\), \(D_{12}=D_{1}+D_{2}-D_{1}D_{2}\), the segmental expression of the damage variable \(D(\varepsilon,t)\) considering time and dual softening can be obtained as follows:  
\[D(\varepsilon,t)=\left\{\begin{array}{cc}\frac{D_{1}}{D_{12}t_{1}}t&0\leq t <t_{1}\\ -\Big{(}1-\frac{D_{1}}{D_{12}}\Big{)}\exp[-B(t-t_{1})]+1&t_{1}\leq t<t_{2}\\ 1&t_{2}\leq t\end{array}\right. \tag{31}\]  
where, \(B\) is derived from the test, reflecting the rapidity of damage to surrounding rock under water-softening. The larger the \(B\) is, the more rapid the damage to surrounding rock under water-softening, and the faster the damage reaches stability.