# Dynamically Triggered Damage Around Rock Tunnels: An Experimental and Theoretical Investigation
## 5 Discussion
### The Role of Geo-Stress and Dynamic Disturbances in Tunnel Failure
As per the strain energy theory, at the edge of the tunnel, the static strain energy density (SED) can be expressed in terms of the stress components and elastic parameters as [21]:  
\[SED_{s}=\frac{1}{2E}\Big{[}\sigma_{\theta}^{2}+\sigma_{z}^{2}-2\
u\sigma_{ \theta}\sigma_{z}\Big{]} \tag{25}\]  
where \(E\) and \(v\) are the elastic modulus and Poisson's ratio of the surrounding rock, respectively. Under coupled high stress and dynamic perturbation, the coupled strain energy density (SEDsd) in the surrounding rock can be simply written as:  
\[SED_{sd}=\frac{1}{2E}\bigg{\{}\Big{(}\sigma_{\theta}^{d}+\sigma_{\theta}^{s} \Big{)}^{2}+\sigma_{z}^{2}-2\
u\Big{(}\sigma_{\theta}^{d}+\sigma_{\theta}^{s} \Big{)}\sigma_{z}\bigg{\}} \tag{26}\]  
When [49]:  
\[SED_{\text{sd}}>SED_{\text{c}} \tag{27}\]  
the damage to the surrounding rock occurs. Here, SEDc is the ultimate energy storage capacity, which can be determined by indoor experiments. As shown in Figure 14b, the burial depth increases, resulting in an augmentation of the static geo-stress and strain energy in the tunnel surrounding rock. In conjunction with the additional stress and energy inputs caused by dynamic perturbations, this leads to a critical stress state and the accumulation of strain energy. Furthermore, the destructive process of the surrounding rock releases a substantial amount of energy [50], intensifying the violence and severity of the dynamic perturbation-induced damage in the deep rock mass [21; 42]. Numerical simulations offer a viable means of exploring the dynamic response of arched tunnels under varying initial stress conditions; as in our previous research, a finite element model has been constructed, incorporating diverse stress states and orientations. A vertical stress magnitude of 15 MPa has been prescribed, \(\sigma_{x}\) and \(\sigma_{y}\) represent the horizontal and vertical stresses, respectively, while the lateral stress coefficients, defined as k = \(\sigma_{x}\)/\(\sigma_{y}\), are designated as k = 0.5, 1.0, and 1.5 to represent distinct stress configurations. Tunnel inclination angles of \(0^{\circ}\), \(45^{\circ}\), and \(90^{\circ}\) have been selected to simulate the dynamic responses under horizontal, inclined, and  
Figure 15: Changes in the stress (**a**) and strain energy (**b**) states at the edge of an arched tunnel under dynamic disturbances. vertical impacts, respectively. Figure 16 illustrates the distribution of the plastic strain induced by dynamic disturbances. As shown in the figures, static stress concentrations were observed around the tunnel in the initial stress state, and higher SED occurs with higher initial static stresses according to Equation (25). Subsequently, under the impact of dynamic disturbances, additional dynamic stress concentrations arose due to the scattering of stress waves. Consequently, the surrounding rock experienced a combined dynamic and static stress state. As the dynamic load approached the tunnel perimeter, the stress within the surrounding rock was either intensified or alleviated. Specifically, when the incident direction aligned with the static maximum principal stress, the overlapping region of the static and dynamic stress concentrations was elevated, resulting in heightened tensile and compressive stresses around the tunnel. This phenomenon exacerbated the damage to the surrounding rock. As a comparison, Figure 17 shows two numerical simulation results from RFPA and LS-DYNA for dynamically triggered damage to the tunnel under different initial stress. The figures clearly visualize how, as the initial stress level increases, the damage becomes increasingly severe under the same incident stresses, which is in good agreement with the analysis provided above. Thus far, it is established that the combination of geo-stress and dynamic disturbances alters the stress state and energy storage of the surrounding rock.