# 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
### Evolution Equation of Damage Variables Considering Dual Softening Coupling  
In the excavation of the water-rich soft-rock tunnel, according to the analysis in Section 2, the surrounding rock environment is in order of the natural undisturbed state, excavation unloading damage soften state, unloading and water-softening state. Assuming that \(D_{1}\) is the unloading damage factor of the surrounding rock, \(D_{2}\) is the water-softening factor, \(D_{12}\) is the damage factor of two-damage coupling. Figure 7 shows the relationship between damage factors and surrounding rock states.  
The rock mechanical properties will change under the dual softening of excavation unloading and water and the change in elastic modulus reflects the degree of damage inside the material. Based on [PERSON]'s equivalent strain principle, the damage variable equation for the surrounding rock under unloading and water-softening effects is derived [59]:  
\[D_{1} =1-E_{1}/E_{0} \tag{1}\] \[D_{2} =1-E_{2}/E_{1}\] (2) \[D_{12} =1-E_{2}/E_{0} \tag{3}\]  
Figure 6: Stress-seepage–damage coupling interaction mode.  
Figure 7: Different damage factors of surrounding rock.  
\[\varepsilon_{01}=\frac{\sigma_{1}}{E_{0}}=\frac{\sigma_{0}}{E_{0}(1-\text{D}_{1})} \tag{4}\]  
\[\varepsilon_{12}=\frac{\sigma_{2}}{E_{1}}=\frac{\sigma_{1}}{E_{1}(1-\text{D}_{2})} \tag{5}\]  
\[\varepsilon_{02}=\frac{\sigma_{2}}{E_{0}}=\frac{\sigma_{0}}{E_{0}(1-\text{D}_{1 2})} \tag{6}\]  
where, \(\sigma_{0}\), \(E_{0}\) are the original effective stress and elastic modulus of surrounding rock; \(\sigma_{1}\), \(E_{1}\) are the effective stress and elastic modulus of surrounding rock after unloading softening; \(\sigma_{2}\), \(E_{2}\) are the effective stress and elastic modulus of surrounding rock after unloading and water-softening, \(\varepsilon_{01},\varepsilon_{12}\), \(\varepsilon_{02}\) are the strain generated by effective stress \(\sigma_{1}\) in the original state of rock, effective stress \(\sigma_{2}\) in the unloading softening state of rock, effective stress \(\sigma_{2}\) in the original state rock.  
From Equations (4)-(6), we obtain that:  
\[(1-D_{1})\sigma_{1}=\sigma_{0} \tag{7}\]  
\[(1-D_{2})\sigma_{2}=\sigma_{1} \tag{8}\]  
\[(1-D_{12})\sigma_{2}=\sigma_{0} \tag{9}\]  
From Equations (7) and (8), we have:  
\[(1-D_{1})(1-D_{2})\sigma_{2}=\sigma_{0} \tag{10}\]  
Combining Equations (9) and (10), it can be obtained that:  
\[D_{12}=D_{1}+D_{2}-D_{1}D_{2} \tag{11}\]  
[PERSON] et al. [60] and [PERSON] [61] also derived similar equations for water-softening and uniaxial loading coupling. Equation (11) is derived based on the elastic modulus. In this paper, unloading and water-softening coupled effects on cohesion and internal friction angle are also calculated according to Equation (11), although there is a certain error the engineering needs are met.  
At present, most of the damage studies of surrounding rock focus on compression shear yield damage, and the study of tensile yield damage is insufficient. Therefore, this paper considers both compression shear and tensile damage modes. Some scholars add the maximum tensile strength criterion to the Mohr-Coulomb strength criterion, trying to make material yield in tension by setting the tensile strength [62]. However, in the simulation, the damage caused by tensile strain is usually the result of compressive stress and excavation unloading, so the tensile strain should be used as an indicator to determine the tensile damage of the surrounding rock.