## 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
#### 2.1.2 Superposition of Blasting Stress Fields Considering Phase Delay Effects  
The detonation of an elongated cylindrical charge can be modeled as the superposition of instantaneous detonations from multiple short-cylinder segments that detonate sequentially. Accounting for time delays due to detonation wave propagation, the stress field for cylindrical charge blasting (e.g., with bottom initiation) can be derived by combining the superposition model in Figure 4 with Equations (17)-(19):  
Based on this criterion, elongated cylindrical charges can be divided and calculated. By substituting q(t) = \(\mu_{\mathrm{d}}\) (\(1-\lambda\))p(t), and s(t) = \(-\lambda\)p(t) (from Equation (8)), the stress field of the columnar charges in matrix form is as follows:  
\[\mathbf{\sigma}_{i}=\begin{pmatrix}\sigma_{ri}\\ \tau_{z_{i}}\\ \tau_{\theta_{i}}\end{pmatrix}=\rho\mathbf{C}\begin{pmatrix}\begin{pmatrix}\frac{ F_{2}(\phi_{i})}{R_{i}}\frac{d^{2}}{dt^{2}}+\mu_{d}(1-\lambda)\frac{C_{2}(\phi_{i})}{R_{i}} \frac{d}{dt}\bigg{\}}\bigg{\{}p\big{(}t-\frac{(i-1)L}{mD_{c}}-\frac{R_{i}}{C_{p }}\big{)}\\ \frac{F_{1}(\phi_{i})}{R_{i}}\frac{d^{2}}{dt^{2}}+\mu_{d}(1-\lambda)\frac{C_{1 }(\phi_{i})}{R_{i}}\frac{d}{dt}\bigg{\}}\bigg{\{}p\big{(}t-\frac{(i-1)L}{mD_{c} }-\frac{R_{i}}{C_{s}}\big{)}\bigg{\}}\\ \lambda\frac{K(\phi_{i})}{R_{i}}\frac{d^{2}}{dt^{2}}\bigg{\{}p\big{(}t-\frac{( i-1)L}{mD_{c}}-\frac{R_{i}}{C_{s}}\big{)}\bigg{\}}\end{pmatrix}\mathbf{R}_{\theta} \tag{20}\]  
\[\mathbf{C}=\begin{pmatrix}C_{p}&0&0\\ 0&-C_{s}&0\\ 0&0&-C_{s}\end{pmatrix},\mathbf{F}_{i}=\begin{pmatrix}\begin{pmatrix}\frac{F_{2}( \phi_{i})}{R_{i}}\frac{d^{2}}{dt^{2}}+\mu_{d}(1-\lambda)\frac{C_{2}(\phi_{i})}{ R_{i}}\frac{d}{dt}\bigg{\}}\bigg{\{}p\big{(}t-\frac{(i-1)L}{mD_{c}}-\frac{R_{i}}{C_{p }}\big{)}\\ \frac{F_{1}(\phi_{i})}{R_{i}}\frac{d^{2}}{dt^{2}}+\mu_{d}(1-\lambda)\frac{C_{1 }(\phi_{i})}{R_{i}}\frac{d}{dt}\bigg{\}}\bigg{\{}p\big{(}t-\frac{(i-1)L}{mD_{c} }-\frac{R_{i}}{C_{s}}\big{)}\bigg{\}}\\ \lambda\frac{K(\phi_{i})}{R_{i}}\frac{d^{2}}{dt^{2}}\bigg{\{}p\big{(}t-\frac{( i-1)L}{mD_{c}}-\frac{R_{i}}{C_{s}}\big{)}\bigg{\}}\end{pmatrix},\mathbf{R}_{\theta}= \begin{pmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1\end{pmatrix} \tag{21}\]  
\[\sigma_{i}=\rho\mathbf{C}\mathbf{F}_{i}\mathbf{R}_{\theta},\ \mathbf{\sigma}=\sum_{i=1}^{m}\mathbf{\sigma}_{i} \tag{22}\]  
In existing studies, the analytical solutions for derived blasting stress fields based on short cylindrical charges are formulated as follows [19]:  
\[\begin{pmatrix}\sigma_{ri}\\ \tau_{i}\end{pmatrix}=\rho\begin{pmatrix}C_{p}&0\\ 0&-C_{s}\end{pmatrix}\begin{pmatrix}\begin{pmatrix}sin\phi_{i}\\ cos\phi_{i}\end{pmatrix}\left(\frac{F_{1}(\phi_{i})}{R_{i}}\frac{d^{2}}{dt^{2}} \bigg{\{}p(t-\frac{(i-1)L}{mD_{c}}-\frac{R_{i}}{C_{p}})\bigg{\}}\right)+\rho \begin{pmatrix}C_{p}&0\\ 0&C_{s}\end{pmatrix}\begin{pmatrix}cos\phi_{i}\\ sin\phi_{i}\end{pmatrix}\left(\frac{F_{2}(\phi_{i})}{R_{i}}\frac{d^{2}}{dt^{2}} \bigg{\{}p(t-\frac{(i-1)L}{mD_{c}}-\frac{R_{i}}{C_{s}})\bigg{\}}\right) \tag{23}\]  
Figure 4: Calculation model for superposition of the stress field in columnar charge blasting: L is the total length of the cylindrical charge (m); and \(m\) is the number of segmented short-cylinder charges into which the elongated charge is divided. According to reference [17], explosive charges with a length-to-diameter ratio (L/D) of less than 10 are classified as short cylindrical charges.  
\[\begin{bmatrix}\sigma_{r}\\ \tau\end{bmatrix}=\sum_{i=1}^{m}\begin{bmatrix}\sigma_{ri}\\ \tau_{i}\end{bmatrix} \tag{24}\]  
Compared to the existing research Formulas (23) and (24), the Equations (20)-(22) derived in this study introduce two key advancements: the inclusion of instantaneous detonation axial pressure \(q(t)\), and the rock mass damage factor D.