## 2 Theoretical Derivation of Analytical Solutions for Blasting Stress Fields in Adjacent Tunnels
### Blasting Stress Field of a Single-Hole Cylindrical Charge Based on [PERSON]'s Short-Cylinder Solution  
1.1 Stress Field of a Short-Cylinder Charge Incorporating Instantaneous Detonation Axial Pressure and Rock Mass Damage  
[PERSON] modeled the stress induced by the instantaneous detonation of a short cylindrical cavity charge as a transient pressure applied to the boundary of a short cylindrical cavity (Figure 1), from which displacement solutions were derived. The results show that detonation generates both P-waves and S-waves, with the latter further decomposing into SH-waves (parallel to the horizontal plane) and SV-waves (perpendicular to it). Figure 3 shows the particle vibration direction as the angle between the wave propagation direction and the short cylindrical cavity wall varies. The displacement solutions for these three wave types are given by [9]:  
\[\left[ \begin{matrix}\overset{\rightarrow}{u_{p}}\\ \overset{\leftarrow}{\omega_{P}}\end{matrix} \right]=\left(\frac{F_{1}(\phi) }{R}\frac{\mathrm{d}}{\mathrm{d}t}\bigg{\{}p(t-\frac{R}{C_{p}})\bigg{\}}+\frac {G_{1}(\phi)}{R}q(t-\frac{R}{C_{p}})\right)\left[ \begin{matrix}\sin\phi\\ -\cos\phi\end{matrix} \right] \tag{3}\]  
\[\left[ \begin{matrix}\overset{\rightarrow}{u_{SV}}\\ \overset{\rightarrow}{u_{SV}}\end{matrix} \right]=\left(\frac{F_{2}(\phi)}{R} \frac{\mathrm{d}}{\mathrm{d}t}\bigg{\{}p(t-\frac{R}{C_{s}})\bigg{\}}+\frac{G_ {2}(\phi)}{R}q(t-\frac{R}{C_{s}})\right)\left[ \begin{matrix}\cos\phi\\ \sin\phi\end{matrix} \right] \tag{4}\]  
\[\overset{\rightarrow}{v_{SH}}=\frac{K(\phi)}{R}\frac{\mathrm{d}}{\mathrm{d}t }\bigg{\{}s(t-\frac{R}{C_{s}})\bigg{\}} \tag{5}\]  
where \(\overset{\rightarrow}{u_{p}}\) and \(\overset{\rightarrow}{u_{SV}}\) are the horizontal displacements of the P-wave and SV-wave (m); \(\overset{\rightarrow}{\omega_{P}}\) and \(\overset{\rightarrow}{\omega_{SV}}\) are their vertical displacements (m); and \(\overset{\rightarrow}{v_{SH}}\) is the tangential displacement of the SH-wave (m). In tunnel face blasting, \(\overset{\rightarrow}{u}\), \(\overset{\rightarrow}{w}\) and \(\overset{\rightarrow}{v}\) represent the radial, axial and hoop displacements, respectively (m); \(C_{p}\) and \(C_{s}\) are the propagation velocities of the P-wave and S-wave (m/s); \(p(t)\), \(q(t)\), \(s(t)\), \(\phi\) and \(R\) are defined as in Figure 1; \(F_{1}(\phi)\) and \(G_{1}(\phi)\) are the source functions for the P-wave; \(F_{2}(\phi)\) and \(G_{2}(\phi)\) are for the SV-wave; and \(K(\phi)\) is for the SH-wave. Source function definitions [9]:  
\[\left\{ \begin{array}{l}F_{1}(\phi)=\frac{\Delta}{4\pi GC_{p}}\bigg{(}1- \frac{2C_{s}^{2}cos^{2}\phi}{C_{p}^{2}}\bigg{)}\\ G_{1}(\phi)=-\frac{\Delta}{4\pi GC_{s}}C_{s}^{2}cos\phi\\ F_{2}(\phi)=\frac{\Delta}{4\pi GC_{s}}\sin 2\phi\\ G_{2}(\phi)=\frac{\Delta}{4\pi GC_{s}}\sin\phi\\ K(\phi)=\frac{\Delta}{4\pi GC_{s}}\sin\phi\end{array} \right. \tag{6}\]  
Figure 3: The propagation direction of three types of waves and the vibration direction of particles. where \(\Delta\) is the volume of the short-cylinder cavity (m\({}^{3}\)), \(A\) is its lateral surface area (m\({}^{2}\)), and \(G\) is the shear modulus of the surrounding rock mass (GPa). The rock damage factor D is defined as follows [31]:  
\[D=1-\left(\frac{v_{p,post}}{v_{p,pre}}\right)^{2} \tag{7}\]  
where \(v_{p,pre}\) and \(v_{p,post}\) are the P-wave velocities of the rock before and after blasting, respectively (m/s), measured on-site using acoustic testing. The uncertainty in the damage factor D is \(\pm 0.01\). Since the P-wave velocity in rock mass after blasting is generally lower than that before blasting, the damage factor D ranges from 0 to 1, with higher values indicating more severe damage to the surrounding rock. Based on the existing formula [32], and incorporating the damage factor, \(p(t)\), \(q(t)\) and \(s(t)\) can be expressed as follows:  
\[\left\{