## 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
\begin{array}{l}p(t)=P_{m}e^{-\frac{(at+b)}{(1-D)}}\\ q(t)=\mu_{d}(1-\lambda)p(t)\\ s(t)=-\lambda p(t)\\ P_{m}=\frac{1}{8}n\rho_{e}D_{e}^{2}\left(\frac{d_{e}}{d_{b}}\right)^{3}\end{array} \right. \tag{8}\]  
where \(P_{m}\) is the maximum pressure accounting for the combined effects of explosive stress waves and detonation product expansion (MPa); \(a\) and \(b\) are detonation decay coefficients of the explosion [33]; \(\mu_{d}\) is the dynamic Poisson's ratio of the rock (\(\mu_{d}=0.8\mu\)); \(\lambda\) is the lateral pressure coefficient, defined as \(\lambda=\mu_{d}/(1-\mu_{d})\); \(n\) is the pressure amplification coefficient of the detonation products; \(\rho_{e}\) is the explosive density (kg/m\({}^{3}\)); \(D_{e}\) is the explosive detonation velocity (m/s); \(d_{c}\) is the charge diameter (m); and \(d_{b}\) is the borehole diameter (m). During cylindrical charge detonation, the rock mass within the blast volume is subjected first to the shock front and then to explosive gas pressure. This process creates \"crushed\" and \"fractured\" zones, characterized by radial cracks, circumferential cracks, and non-radial fractures [34; 35]. While blasting can effectively break rock within the intended excavation zone, it may also damage surrounding rock that must be preserved, compromising its mechanical properties, bearing capacity, and stability [36]. To assess such effects, [PERSON] and [PERSON] developed the Blast Damage Index (BDI) for mining applications [37]. \[\text{BDI}=\frac{\text{IS}}{\text{DR}}=\frac{\text{VdC}}{\text{KT}} \tag{9}\]  
where IS is the induced stress; DR is the damage resistance; V is the vector sum of PPV (mm/s); d is the specific gravity of the rock mass (kg/m\({}^{3}\)); C is the compressional wave velocity of the rock mass (mm/s); K is the site quality constant; and T is the dynamic tensile strength of the rock mass (N/m\({}^{2}\)). This relationship accounts for both the rock mechanics and wave propagation effects, but it depends on numerous site-specific parameters. To simplify the representation of the rock mass, this paper introduces the damage factor D (0 \(<\) D \(<\) 1). By incorporating D into the exponent of the stress attenuation term (Equation (8)), the model quantitatively captures the influence of rock degradation on stress wave propagation. Physically, increasing D reflects progressively damaged material, such as the formation and growth of microcracks, voids, or other defects. These features act as energy dissipation mechanisms by (i) reflecting, refracting, and scattering incident stress waves, which reduces the measured wave amplitude; (ii) converting mechanical wave energy into heat and other dissipated forms through crack friction and plastic deformation; and (iii) reducing the rock mass's overall load bearing capacity. The expression \(\text{e}^{-(4+b)/(1-\text{D})}\) models this behavior. The term \(1/(1-\text{D})\) amplifies the decay rate: as D increases towards 1 (severely damaged rock), \((1-\text{D})\) decreases, making the exponent more negative and leading to faster stress amplitude decay of the stress amplitude \(P_{\text{m}}\). This matches physical observations--higher damage leads to stronger attenuation of stress waves. Thus, \(1/(1-\text{D})\) effectively quantifies how damage progressively accelerates wave energy dissipation and scattering. During tunnel face blasting, the radial, axial, and hoop displacements of the surrounding rock are expressed as follows:  
\[