# Dynamically Triggered Damage Around Rock Tunnels: An Experimental and Theoretical Investigation
## 4 Stress State Around the Arched Hole Under Dynamic Impact
### Dynamic Stress Distribution Around an Arched Hole  
The dynamic impact is a transient process and it is necessary to first obtain the steady-state dynamic stress distribution around the hole and then use Fourier transform technology to bridge the gap between the steady state and the transient state. The complex function method has long been employed to solve the scattering of stress waves and the stress distribution around complex boundary shapes in steady-state dynamics problems. Building upon [PERSON] et al. 's [37] research, this section focuses on resolving the transient dynamic stress distribution around an arched hole. A simple harmonic plane P-wave incident at circular frequency \(\omega\) and angle of incidence \(\beta\), omitting the time factor eit, can be expressed in the complex plane as [37]:  
\[\varphi^{i}=\varphi_{0}\sum_{n=-\infty}^{n=\infty}i^{n}J_{n}\left(k_{p}|w( \zeta)|\right)\left\{\begin{array}{l}\frac{w(\zeta)}{1\,w(\zeta)\,  
The dynamic stress concentration factor (DSCF) is defined as the ratio of the circumferential stress around the hole and the maximum stress generated by the incident P-wave, expressed as:  
\[\sigma_{\theta}^{*}=\frac{\sigma_{\theta}}{\sigma_{0}} \tag{16}\]  
where:  
\[\left\{ \begin{array}{l}\sigma_{0}=-\mu\beta^{2}\\ \sigma_{\theta}=-2\alpha^{2}(\lambda+\mu)\varphi\{\omega(\zeta),t\}\end{array} \right. \tag{17}\]  
From Equations (16) and (17), one obtains:  
\[\sigma_{\theta}^{*}(\omega,\theta)=\frac{2\big{(}\kappa^{2}-1\big{)}}{\kappa^ {2}}\text{Re}\Bigg{[}\left(\sum_{n=-\infty}^{\infty}A_{n}H_{n}^{(1)}(\alpha \Big{|}w(\zeta)\Big{|}\right)\bigg{\{}\frac{w(\zeta)}{\lceil w(\zeta)\rceil} \bigg{\}}^{n}+\varphi^{i}\bigg{)}\Bigg{]}e^{-i\omega t} \tag{18}\]  
where:  
\[\kappa^{2}=\frac{2(1-v)}{(1-2v)^{2}} \tag{19}\]  
For this study, the transient incident wave generated by the SHPB can be simplified as a half-sine excitation function:  
\[f(t)=\left\{ \begin{array}{l}\sin(\frac{\pi t}{t_{0}}),\quad 0\leq t<t_{0}\\ 0\quad\quad,\quad\quad t\geq t_{0}\end{array} \right. \tag{20}\]  
Here, \(t_{0}\) is the duration of the incident wave. Meanwhile, according to the Fourier transform techniques and Duhamel integral method, the transient DSCF can be obtained [9; 38]:  
\[\overline{\sigma}_{\theta\theta}^{*}=\left\{ \begin{array}{l}\int_{0}^{\infty }\frac{\sigma_{\theta}^{*}(\omega,\theta)}{\omega}\Bigg{[}\frac{\cos\omega t- \cos(\pi t/t_{0})}{\pi-\omega t_{0}}-\frac{\cos\omega t-\cos(\pi t/t_{0})}{ \pi+\omega t_{0}}\Bigg{]}\text{d}\omega,\quad\quad\quad\quad\quad\quad\quad 0\leq t<t_{0}\\ \int_{0}^{\infty}\frac{\sigma_{\theta}^{*}(\omega,\theta)}{\omega}\Bigg{[} \frac{\cos\omega t-\cos(\pi+\omega t-\omega t_{0})}{\pi-\omega t_{0}}-\frac{ \cos\omega t-\cos(\pi+\omega t_{0}-\omega t)}{\pi+\omega t_{0}}\Bigg{]}\text{ d}\omega\,,\quad\quad\quad t\geq t_{0}\end{array} \right. \tag{21}\]  
The stress distribution around the arched hole at different moments during impact is obtained by programming the calculations of Equations (18) and (21), as plotted in Figure 10. In a theoretical analysis, the stress distribution around a hole is solely proportional to the ratio between the wavelength and the size of the hole, irrespective of the actual dimensions of the hole. Therefore, the theoretical results are applicable to a hole or a tunnel. Figure 10: Theoretical results of the stress distribution around the arched holes (Shading indicates tunnel boundaries). When a dynamic load interacts with the arched hole surroundings, it becomes evident that the stresses undergo both amplification and attenuation. The location of these changes is closely linked to the direction of incidence. Irrespective of the angle of incidence, in the incidence direction, there is a noticeable weakening of stress or even a concentration of tensile stress, while in the perpendicular direction of incidence, compressive stress concentration occurs.