# Dynamically Triggered Damage Around Rock Tunnels: An Experimental and Theoretical Investigation
## 5 Discussion
### The Role of Geo-Stress and Dynamic Disturbances in Tunnel Failure  
In engineering construction, there are preexisting initial stresses that must be taken into consideration. Therefore, the calculation of the stress concentration around the arch-shaped hole under different static stress conditions was conducted, revealing that a significant static stress concentration occurs around the chamber under different lateral pressure coefficients, as shown in Figure 14.  
Figure 13: Failure illustrations (**a**) and support systems (**b**) of an arched tunnel under dynamic disturbances. As shown in Figure 14, a static stress concentration always produces larger compressive stress concentrations in the direction perpendicular to the maximum principal stress. Under these conditions, when the chamber is subjected to external dynamic disturbances, the surrounding rock mass is in a state of superposition of dynamic and static stresses. When the dynamic load reaches the vicinity of the chamber, the circumferential stress in the surrounding rock is either enhanced or weakened. When the incident direction is parallel to the static maximum principal stress, the overlapping region of static and dynamic stress concentrations increases, leading to enhanced tensile and compressive stresses around the chamber, which results in more severe damage to the surrounding rock. Conversely, when the direction of the dynamic disturbance is neither parallel nor perpendicular to the static maximum principal stress, the overlapping region of the static stress concentration decreases and part of the dynamic stress concentration is mitigated. Under the same initial stress conditions, as the angle of dynamic incidence increases, the overlapping area of the stress concentrations decreases. Therefore, in the design of deep rock excavation, it is essential to consider the potential directions of dynamic disturbances and the direction of geo-stress. When the direction of dynamic disturbance is perpendicular to the static maximum principal stress, the stresses generated by dynamic loads and static initial stresses do not overlap at the same location around the chamber, and the dynamic stress concentration may even be offset by the static stress. By applying the Mohr-Coulomb criterion, the failure criterion at the edge of the tunnel can be expressed as follows:  
\[\tau=\sigma_{n}\tan\phi+c \tag{22}\]  
where \(\tau\) represents the shear stress on the element and \(\sigma_{n}\) represents the normal stress on the element, \(c\) and \(\phi\) is the rock cohesion and angle of internal friction, respectively. Taking into account both dynamic loading and geo-stress, in the \(\tau\)-\(\sigma\) plane, the change in the stress state at the edge of the tunnel is illustrated in Figure 14a, both in shallow and deep burial conditions. In the absence of dynamic disturbance, the radius of the Mohr stress circle can be determined by:  
\[R_{1}=\sqrt{\frac{\left(\sigma_{\theta}^{s}+\sigma_{z}\right)^{2}}{2}+\tau_{s }^{2}} \tag{23}\]  
where \(\sigma_{\theta}^{s}\) is the static circumferential stress at the tunnel edge. As a dynamic perturbation occurs, it generates additional tangential and circumferential stresses at the tunnel edge. At this point, the radius of the Mohr circle can be determined as follows:  
\[R_{2}=\sqrt{\frac{\left(\sigma_{\theta}^{s}+\sigma_{\theta}^{d}+\sigma_{z} \right)^{2}}{2}+\left(\tau_{s}+\tau_{d}\right)^{2}} \tag{24}\]  
where the superscripts \({}^{8}\) and \({}^{d}\) indicate the static and dynamic stress. As the depth of the tunnel increases, there is a corresponding increase in the geo-stress. This leads to an increase  
Figure 14: Static stress concentration around the arch chamber under different lateral pressure coefficients. in \(R_{1}\), and the dynamic disturbances cause additional stresses, leading to an increase in \(R_{2}\). Consequently, the deeply buried tunnel becomes more susceptible to damage under dynamic disturbances. Correspondingly, due to the increase in stress, the strain energy is also increased, as shown in Figure 15b.