Research on Analytical Solution of Stress Fields in Adjacent Tunnel Surrounding Rock Under Blasting and Verification Analysis

Abstract

In tunnel blasting, an analytical solution for dynamic stress in the surrounding rock of adjacent tunnels is critical for dynamic response analysis, mechanical evaluations, and crack propagation control. Previous studies on stress field analytical solutions primarily modeled rock as a linear elastic material, focusing mainly on the P-wave effects from instantaneous detonation. Based on Heelan’s short cylindrical cavity model, this paper derives an analytical solution for blast-induced dynamic stresses in adjacent tunnel rock, incorporating both induced SV-waves and a rock mass damage factor through rigorous theoretical analysis. Numerical case studies and field measurements were used to analyze stress propagation during tunnel blasting, and theoretical results were compared with measured data. The key findings were as follows: Radial stress > axial stress > hoop stress. All three stresses decay with increasing distance and damage factor, following an inversely proportional relationship with distance. Radial stress decays faster than axial and hoop stresses. Stress also decays exponentially over time, with the peak occurring after the transverse wave arrival. The theoretical results show approximately 10% deviation from the existing empirical formulas, while field measurements closely match the theoretical model, showing consistent stress trends and an average error of 7.02% (radial), 7.56% (axial) and 7.05% (hoop), confirming the reliability of the proposed analytical solution.

1. Introduction

Tunnel engineering, a key technology for traversing complex terrains and reducing transportation distances, is widely applied in expressways, railways, and urban transit systems. Among construction methods, drilling and blasting methods remain predominant due to their advantages in speed, cost-effectiveness, and adaptability to varying geological conditions [1]. However, the transient stress waves generated during tunnel blasting operations can significantly affect existing tunnels [2], with potential damage requiring careful evaluation. In closely spaced adjacent tunnels, dynamic loading from subsequent blasts can pose serious safety risks to neighboring tunnel structures, including lining failures. As the primary source of dynamic loading, blast-induced stress waves critically impact surrounding rock masses and support systems. Understanding their response and control has become a key technical challenge in safe tunnel construction. Therefore, obtaining analytical solutions for dynamic stresses in surrounding rocks under blasting loads is essential for understanding stress propagation, controlling blast vibrations, and ensuring the integrity of existing tunnel support structures.

Current research challenges in analytical solutions for blast-induced stress mainly fall into two areas: The first is the complexity of stress superposition from transient P- and S-wave interactions, compounded by stress propagation through heterogeneous and discontinuous materials [3,4]. The second is difficulties in validation due to limitations in in situ stress measurement embedding sensors in rock, which is technically challenging, and field data are often affected by interference from air shock waves and construction equipment during signal acquisition [5]. Analytical solutions for dynamic stresses in surrounding rocks of adjacent tunnels are crucial for understanding stress evolution, calculating dynamic responses, analyzing mechanical behavior, and optimizing blast control parameters.

As a form of transient energy excitation, blast-induced stress waves have long been a central topic in explosion theory and its applications. Based on charge geometry, blasting can be classified into spherical charges and cylindrical charges. Tunnel excavation blasting typically uses cylindrical charges, and extensive international research has focused on their associated stress field characteristics.

Theoretical developments have led to three mainstream models for calculating blast-induced stress fields: the Starfield superposition method, the ideal fluid medium model, and Heelan’s short-cylinder solution. Starfield [6] proposed decomposing cylindrical charges into multiple equivalent spherical charges. Experimental studies on stress wave propagation, showed that this superposition method yields acceptable agreement with measured data in the medium-to-far fields. Based on this framework, Liu [7] derived computational formulas for stress fields generated by cylindrical charges. Neiman et al. [8] treated rock media as incompressible ideal fluids under transient blast loading, where particle velocity fields satisfy Laplace’s equations. Using boundary mapping methods, they developed solutions for stress fields around cylindrical charges under various borehole boundary conditions. Heelan [9] established a classical analytical solution for cylindrical charge blasting using elastodynamic theory. This model assumes an infinite, isotropic rock mass and provides an expression for radial stress (${\sigma }_r$) distribution due to single hole blasting:

$${\sigma }_r=P_0{\left(\mathrm{r}/r_0\right)}^{\alpha }{e}^{-\beta (r-r_0)}$$

(1)

where

• $P_0$ is the initial blast pressure (MPa);
• $r_0$ is the charge radius (m);
• $\alpha$ and $\beta$ are rock mass attenuation coefficients.

Because the mechanical properties of the rock mass will change after blasting [10], this model accounts for rock mass damage or jointed fissures, leading to prediction errors exceeding 40% in weak surrounding rock. To address this, Henrych [11] introduced an attenuation factor and proposed a modified stress wave propagation equation for heterogeneous rock masses. The relationship between the attenuation coefficient $\beta$ and the rock mass integrity coefficient $K_v$ is expressed as follows:

$$\beta =0.85\times (1-K_v)\times \ln \left({\displaystyle \frac{r}{r_0}}\right)$$

(2)

This model has shown good accuracy in hard rock tunnels but lacks adaptability in fractured zones. To address this, Zhang et al. [12] introduced a damage factor to revise the stress wave equation, establishing a nonlinear attenuation model that accounts for fracture propagation. Their results showed that stress attenuation in jointed rock masses exceeds that in intact rock by over 50%. Ma et al. [13] discretized cylindrical charges into multiple linked spherical charges and derived the dynamic stress field using vector superposition methods. Lei et al. [14] developed analytical solutions for stress fields under different detonation transmission directions using an equivalent unit spherical charge superposition method, incorporating directional effects. Favreau [15], Graff [16], and Achenbach [17] analyzed stress wave fields induced by cylindrical charges based on spherical cavity expansion models. Blair [18] extended Heelan’s short-cylinder solution by integrating finite detonation velocity and charge length, advancing a superposition model for long charges. Gao et al. [19] further partitioned elongated cylindrical charges into a series of instantaneously detonated short charges, deriving vibration velocity formulas for arbitrary points using time-delayed superposition of detonation wave propagation.

In experimental and field-testing research, advanced techniques such as dynamic caustics and ultra-dynamic strain measurement have been widely adopted. Yang et al. [20] systematically investigated stress–strain fields in centrally initiated cylindrical charges and the evolution of local stress fields near blast-induced crack tips using dynamic caustics. Their study revealed significant principal stress differences across orientations and pronounced end effects. Ding [21] conducted scaled model experiments to analyze stress wave superposition between adjacent boreholes. Qiu [22] used physical modeling to simulate blast-induced disturbances in underground tunnels, mapping surface strain distributions at various blast locations. Yang [23] employed a high-speed camera-based digital image correlation system in dual-borehole blast tests, showing that peak stress at the midpoint between two charges reached 2.4–2.7 times that of single-hole blasting. Liu et al. [24] developed an embedded strain bar system incorporating MEMS sensors and wireless transmission, enabling real-time monitoring of dynamic stress fields during construction of the Qingdao Jiaozhou Bay Tunnel. Their signal processing algorithm combining wavelet denoising with Kalman filtering improved the signal-to-noise ratios from 15 dB to 28 dB. Zhang [25] optimized field blast monitoring protocols to capture dynamic strain on tunnel primary lining (TSL) surfaces. Luo et al. [26,27,28] performed in situ dynamic strain measurements on existing tunnel surfaces, quantitatively confirming strain amplitudes on blast-facing sides.

In a numerical simulation, Gao et al. [19] used 3D dynamic finite element software to model cylindrical charge blasting in infinite rock masses. By developing a quarter-symmetry finite element model, they systematically examined the effects of initiation positions on explosive energy transmission and stress field distribution. Liang et al. [29] applied numerical simulation to analyze rock fragmentation under stress wave superposition from multi-source charge detonations, confirming that stress wave interactions from spherical charges align with theoretical rock-breaking mechanics. Xiang et al. [30] proposed a numerical analysis model for strip charge stress fields by integrating the Starfield superposition method with dynamic finite element analysis, providing a comprehensive understanding of the formation and evolution mechanisms of blast-induced stress fields under diverse initiation modes.

In summary, despite substantial progress in studying stress fields from cylindrical charge blasting, three critical limitations remain:

• Methodological limitations: The widespread use of equivalent spherical charge models—which inherently omit S-wave generation—fails to reflect actual field conditions, where cylindrical charges produce both P- and S-waves.
• Dynamic modeling oversights: Current superposition approaches for elongated charges, based on short-cylinder solutions often overlook two key factors: (a) axial pressure effects from instantaneous detonation of short cylindrical cavities, and (b) the evolution of rock mass damage in media treated as continuous, homogeneous materials.
• Validation constraints: Most existing measurements focus on surface particles, lacking in situ dynamic stress data from within the rock mass, which limits the verification of theoretical formulations models.

To address these gaps, this paper builds on explosion stress wave propagation theory and Heelan’s short-cylinder solution [9], incorporating the axial pressure generated by the instantaneous detonation of short cylindrical cavities (Figure 1) and the effects of rock mass damage. Based on the phase delay effect and stress wave propagation intervals, a porous superposition method is used to derive an analytical solution for the stress in the surrounding rock of adjacent tunnels under blasting. Theoretical formulas are employed to calculate stress magnitude and propagation patterns of radial particles in adjacent tunnels. These results are then validated through field stress measurements and numerical simulations. The findings provide a theoretical basis for calculating dynamic stress in surrounding rock, understanding the propagation of internal cracks and fissures, and improving surrounding rock reinforcement and blast control.

2. Theoretical Derivation of Analytical Solutions for Blasting Stress Fields in Adjacent Tunnels

During tunnel face blasting, both surface waves (Rayleigh and Love) and body waves are generated and propagate through the surrounding rock mass. Body waves include P-waves and S-waves, with the latter divided into SV and SH components. P-waves travel faster than S-waves. The combined action of P-, SV-, and SH-waves induces three-dimensional dynamic stresses in the surrounding rock of adjacent tunnels: radial stress (${\sigma }_r$), axial stress (${\tau }_z$), and hoop stress (${\tau }_{\theta }$). The mechanical model for an adjacent tunnel’s surrounding rock under coupled wave actions is shown in Figure 2.

2.1. Blasting Stress Field of a Single-Hole Cylindrical Charge Based on Heelan’s Short-Cylinder Solution

2.1.1. Stress Field of a Short-Cylinder Charge Incorporating Instantaneous Detonation Axial Pressure and Rock Mass Damage

Heelan 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]:

$$\left[\begin{array}{c}\overrightarrow{u_P}\\ \overrightarrow{{\omega }_P}\end{array}\right]=\left({\displaystyle \frac{F_1(\varphi )}{R}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left\{p(t-{\displaystyle \frac{R}{C_p}})\right\}+{\displaystyle \frac{G_1(\varphi )}{R}}q(t-{\displaystyle \frac{R}{C_p}})\right)\left[\begin{array}{c}\sin \varphi \\ -\cos \varphi \end{array}\right]$$

(3)

$$\left[\begin{array}{c}\overrightarrow{u_{SV}}\\ \overrightarrow{{\omega }_{SV}}\end{array}\right]=\left({\displaystyle \frac{F_2(\varphi )}{R}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left\{p(t-{\displaystyle \frac{R}{C_s}})\right\}+{\displaystyle \frac{G_2(\varphi )}{R}}q(t-{\displaystyle \frac{R}{C_s}})\right)\left[\begin{array}{c}\cos \varphi \\ \sin \varphi \end{array}\right]$$

(4)

$$\overrightarrow{v_{SH}}={\displaystyle \frac{K(\varphi )}{R}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left\{s(t-{\displaystyle \frac{R}{C_{\mathrm{s}}}})\right\}$$

(5)

where $\overrightarrow{u_P}$ and $\overrightarrow{u_{SV}}$ are the horizontal displacements of the P-wave and SV-wave (m); $\overrightarrow{{\omega }_P}$ and $\overrightarrow{{\omega }_{SV}}$ are their vertical displacements (m); and $\overrightarrow{v_{SH}}$ is the tangential displacement of the SH-wave (m). In tunnel face blasting, $\overrightarrow{u}$, $\overrightarrow{w}$ and $\overrightarrow{v}$ 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); $p(t)$, $q(t)$, $s(t)$, $\varphi$ and $R$ are defined as in Figure 1; $F_1(\varphi )$ and $G_1(\varphi )$ are the source functions for the P-wave; $F_2(\varphi )$ and $G_2(\varphi )$ are for the SV-wave; and $K(\varphi )$ is for the SH-wave.

Source function definitions [9]:

$$\left\{\begin{array}{c}{F}_1(\varphi )={\displaystyle \frac{\Delta }{4\pi G{C}_p}}\left(1-{\displaystyle \frac{2C_s^{2}co{s}^2\varphi }{C_p^2}}\right)\hfill \\ G_1(\varphi )=-{\displaystyle \frac{A}{4\pi G{C}_p^2}}{C}_s^{2}cos\varphi \hfill \\ F_2(\varphi )={\displaystyle \frac{\Delta }{4\pi G{C}_s}}\sin 2\varphi \hfill \\ G_2(\varphi )={\displaystyle \frac{A}{4\pi G}}\sin \varphi \hfill \\ K(\varphi )={\displaystyle \frac{\Delta }{4\pi G{C}_s}}\sin \varphi \hfill \end{array}\right.$$

(6)

where $\Delta$ is the volume of the short-cylinder cavity (m³), A is its lateral surface area (m²), and G is the shear modulus of the surrounding rock mass (GPa). The rock damage factor D is defined as follows [31]:

$$D=1-({\displaystyle \frac{v_{p,post}}{v_{p,pre}}}{)}^2$$

(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 ±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:

$$\left\{\begin{array}{c}p(t)=P_m{e}^{-{\displaystyle \frac{(at+b)}{(1-D)}}}\hfill \\ q(t)={\mu }_d\left(1-\lambda \right)p(t)\hfill \\ s(t)=-\lambda p(t)\hfill \\ P_m={\displaystyle \frac{1}{8}}n{\rho }_e{D}_e^2{\left({\displaystyle \frac{d_c}{d_b}}\right)}^3\hfill \end{array}\right.$$

(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; ${\rho }_{\mathrm{e}}$ is the explosive density (kg/m³); $D_{\mathrm{e}}$ 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, Yu and Vongpaisal developed the Blast Damage Index (BDI) for mining applications [37].

$$\mathrm{BDI}={\displaystyle \frac{\mathrm{IS}}{\mathrm{DR}}}={\displaystyle \frac{\mathrm{VdC}}{\mathrm{KT}}}$$

(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/m³); 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/m²).

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 micro-cracks, 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 e^{−(at+b)/(1−D)} models this behavior. The term 1/(1 − D) amplifies the decay rate: as D increases towards 1 (severely damaged rock), (1 − D) decreases, making the exponent more negative and leading to faster stress amplitude decay of the stress amplitude $P_{\mathrm{m}}$. This matches physical observations—higher damage leads to stronger attenuation of stress waves. Thus, 1/(1 − 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:

$$\begin{array}{c}\overrightarrow{u}=\overrightarrow{u_P}+\overrightarrow{u_{SV}}=\left({\displaystyle \frac{F_1(\varphi )}{R}}{\displaystyle \frac{d}{dt}}\left\{p(t-{\displaystyle \frac{R}{C_p}})\right\}+{\displaystyle \frac{G_1(\varphi )}{R}}q(t-{\displaystyle \frac{R}{C_p}})\right)\sin \varphi \hfill \\ +\left({\displaystyle \frac{F_2(\varphi )}{R}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left\{p(t-{\displaystyle \frac{R}{C_s}})\right\}+\right.\left.{\displaystyle \frac{G_2(\varphi )}{R}}q(t-{\displaystyle \frac{R}{C_s}})\right)cos\varphi \hfill \end{array}$$

(10)

$$\begin{array}{c}\overrightarrow{\omega }=\overrightarrow{{\omega }_P}+\overrightarrow{{\omega }_{SV}}=-cos\varphi \left({\displaystyle \frac{F_1(\varphi )}{R}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left\{p(t-{\displaystyle \frac{R}{C_p}})\right\}+{\displaystyle \frac{G_1(\varphi )}{R}}q(t-{\displaystyle \frac{R}{C_p}})\right)\hfill \\ +\left({\displaystyle \frac{F_2(\varphi )}{R}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left\{p(t-{\displaystyle \frac{R}{C_s}})\right\}+{\displaystyle \frac{G_2(\varphi )}{R}}q(t-{\displaystyle \frac{R}{C_s}})\right)\sin \varphi \hfill \end{array}$$

(11)

$$\overrightarrow{v}=\overrightarrow{v_{\mathrm{SH}}}={\displaystyle \frac{K(\varphi )}{R}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left\{s(t-{\displaystyle \frac{R}{c_s}})\right\}$$

(12)

By differentiating the displacement field with respect to time, the vibration velocity field of the short-cylinder charge is derived as follows:

$$\begin{array}{c}{\overrightarrow{V}}_r={\displaystyle \frac{\partial u}{\partial t}}=\left({\displaystyle \frac{F_1(\varphi )}{R}}{\displaystyle \frac{d^2}{\mathrm{d}{t}^2}}\left\{p(t-{\displaystyle \frac{R}{c_p}})\right\}+{\displaystyle \frac{G_1(\varphi )}{R}}{\displaystyle \frac{d}{dt}}\left\{q(t-{\displaystyle \frac{R}{c_p}})\right\}\right)\sin \varphi \hfill \\ +\left({\displaystyle \frac{F_2(\varphi )}{R}}{\displaystyle \frac{d^2}{\mathrm{d}{t}^2}}\left\{p(t-{\displaystyle \frac{R}{c_s}})\right\}\right.\left.+{\displaystyle \frac{G_2(\varphi )}{R}}{\displaystyle \frac{d}{dt}}\left\{q(t-{\displaystyle \frac{R}{c_s}})\right\}\right)cos\varphi \hfill \end{array}$$

(13)

$$\begin{array}{c}\overrightarrow{V_{\mathrm{z}}}={\displaystyle \frac{\partial \omega }{\partial t}}=-cos\varphi \left({\displaystyle \frac{F_1(\varphi )}{R}}{\displaystyle \frac{d^2}{\mathrm{d}{t}^2}}\left\{p(t-{\displaystyle \frac{R}{C_{\mathrm{p}}}})\right\}+{\displaystyle \frac{G_1(\varphi )}{R}}{\displaystyle \frac{d}{dt}}\left\{q(t-{\displaystyle \frac{R}{C_p}})\right\}\right)\hfill \\ +\left({\displaystyle \frac{F_2(\varphi )}{R}}{\displaystyle \frac{d^2}{\mathrm{d}{t}^2}}\left\{p(t-{\displaystyle \frac{R}{C_s}})\right\}\right.\left.+{\displaystyle \frac{G_2(\varphi )}{R}}{\displaystyle \frac{d}{dt}}\left\{q(t-{\displaystyle \frac{R}{c_s}})\right\}\right)\sin \varphi \hfill \end{array}$$

(14)

$$\overrightarrow{V_{\theta }}={\displaystyle \frac{\partial v}{\partial t}}={\displaystyle \frac{K(\varphi )}{R}}{\displaystyle \frac{d^2}{\mathrm{d}{t}^2}}\left\{s(t-{\displaystyle \frac{R}{C_s}})\right\}$$

(15)

where $\overrightarrow{V_r}$, $\overrightarrow{V_z}$ and $\overrightarrow{V_{\theta }}$ represent the radial, axial, and hoop vibration velocities, respectively, in units of m/s.

During blasting operations, the relationships between the stresses (${\sigma }_r$, ${\tau }_z$, ${\tau }_{\theta }$) generated by P-waves, SV-waves, and SH-waves and the corresponding particle vibration velocities ($\overrightarrow{V_r}$, $\overrightarrow{V_z}$, $\overrightarrow{V_{\theta }}$) can be expressed as follows [38]:

$$\left\{\begin{array}{c}{\sigma }_r=\overrightarrow{V_r}\rho C_p\hfill \\ {\tau }_z=\overrightarrow{V_z}\rho C_s\hfill \\ {\tau }_{\theta }=\overrightarrow{V_{\theta }}\rho C_s\hfill \end{array}\right.$$

(16)

where $\rho$ is the rock density, kg/m³.

By synthesizing Equations (13)–(16), the stress field induced by a short-cylinder charge blasting can be derived as follows:

$$\begin{array}{c}{\sigma }_r=\left({\displaystyle \frac{F_1(\varphi )}{R}}{\displaystyle \frac{d^2}{\mathrm{d}{t}^2}}\left\{p(t-{\displaystyle \frac{R}{C_p}})\right\}+{\displaystyle \frac{G_1(\varphi )}{R}}{\displaystyle \frac{d}{dt}}\left\{q(t-{\displaystyle \frac{R}{C_p}})\right\}\right)\rho C_p\sin \varphi \hfill \\ +\left({\displaystyle \frac{F_2(\varphi )}{R}}{\displaystyle \frac{d^2}{\mathrm{d}{t}^2}}\left\{p(t-{\displaystyle \frac{R}{C_s}})\right\}\right.+\left.{\displaystyle \frac{G_2(\varphi )}{R}}{\displaystyle \frac{d}{dt}}\left\{q(t-{\displaystyle \frac{R}{C_s}})\right\}\right)\rho C_{p}cos\varphi \hfill \end{array}$$

(17)

$$\begin{array}{c}{\tau }_Z=-\left({\displaystyle \frac{F_1(\varphi )}{R}}{\displaystyle \frac{d^2}{\mathrm{d}{t}^2}}\left\{p(t-{\displaystyle \frac{R}{C_p}})\right\}+{\displaystyle \frac{G_1(\varphi )}{R}}{\displaystyle \frac{d}{dt}}\left\{q(t-{\displaystyle \frac{R}{C_p}})\right\}\right)\rho C_{s}cos\varphi \hfill \\ +\left({\displaystyle \frac{F_2(\varphi )}{R}}{\displaystyle \frac{d^2}{\mathrm{d}{t}^2}}\left\{p(t-{\displaystyle \frac{R}{C_s}})\right\}+\right.\left.{\displaystyle \frac{G_2(\varphi )}{R}}{\displaystyle \frac{d}{dt}}\left\{q(t-{\displaystyle \frac{R}{C_s}})\right\}\right)\rho C_s\sin \varphi \hfill \end{array}$$

(18)

$${\tau }_{\theta }={\displaystyle \frac{\rho C_{s}K(\varphi )}{R}}{\displaystyle \frac{d^2}{\mathrm{d}{t}^2}}\left\{s(t-{\displaystyle \frac{R}{C_s}})\right\}$$

(19)

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) = µ_d (1 − λ)p(t), and s(t) = −λp(t) (from Equation (8)), the stress field of the columnar charges in matrix form is as follows:

$${\mathit{\sigma }}_i=\left(\begin{array}{c}{\sigma }_{ri}\\ {\tau }_{z_i}\\ {\tau }_{{\theta }_i}\end{array}\right)=\rho \mathit{C}\left(\begin{array}{c}\left({\displaystyle \frac{F_2({\varphi }_i)}{R_i}}{\displaystyle \frac{d^2}{d{t}^2}}+{\mu }_d(1-\lambda ){\displaystyle \frac{G_2({\varphi }_i)}{R_i}}{\displaystyle \frac{d}{dt}}\right)\left\{p\left(t-{\displaystyle \frac{(i-1)L}{m{D}_e}}-{\displaystyle \frac{R_i}{C_p}}\right)\right\}\\ \left({\displaystyle \frac{F_1({\varphi }_i)}{R_i}}{\displaystyle \frac{d^2}{d{t}^2}}+{\mu }_d(1-\lambda ){\displaystyle \frac{G_1({\varphi }_i)}{R_i}}{\displaystyle \frac{d}{dt}}\right)\left\{p\left(t-{\displaystyle \frac{(i-1)L}{m{D}_e}}-{\displaystyle \frac{R_i}{C_s}}\right)\right\}\\ \lambda {\displaystyle \frac{K({\varphi }_i)}{R_i}}{\displaystyle \frac{d^2}{d{t}^2}}\left\{p\left(t-{\displaystyle \frac{(i-1)L}{m{D}_e}}-{\displaystyle \frac{R_i}{C_s}}\right)\right\}\end{array}\right){\mathit{R}}_{\theta }$$

(20)

$$\mathit{C}=\left(\begin{array}{ccc}{C}_p& 0& 0\\ 0& -C_s& 0\\ 0& 0& -C_s\end{array}\right),{\mathit{F}}_i=\left(\begin{array}{c}\left({\displaystyle \frac{F_2\left({\varphi }_i\right)}{R_i}}{\displaystyle \frac{d^2}{d{t}^2}}+{\mu }_d\left(1-\lambda \right){\displaystyle \frac{G_2\left({\varphi }_i\right)}{R_i}}{\displaystyle \frac{d}{dt}}\right)\left\{p\left(t-{\displaystyle \frac{\left(i-1\right)L}{m{D}_e}}-{\displaystyle \frac{R_i}{C_p}}\right)\right\}\\ \left({\displaystyle \frac{F_1\left({\varphi }_i\right)}{R_i}}{\displaystyle \frac{d^2}{d{t}^2}}+{\mu }_d\left(1-\lambda \right){\displaystyle \frac{G_1\left({\varphi }_i\right)}{R_i}}{\displaystyle \frac{d}{dt}}\right)\left\{p\left(t-{\displaystyle \frac{\left(i-1\right)L}{m{D}_e}}-{\displaystyle \frac{R_i}{C_s}}\right)\right\}\\ \lambda {\displaystyle \frac{K\left({\varphi }_i\right)}{R_i}}{\displaystyle \frac{d^2}{d{t}^2}}\left\{p\left(t-{\displaystyle \frac{\left(i-1\right)L}{m{D}_e}}-{\displaystyle \frac{R_i}{C_s}}\right)\right\}\end{array}\right),{\mathit{R}}_{\theta }=\left(\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right)$$

(21)

$${\sigma }_i=\rho \mathit{C}{\mathit{F}}_i{\mathit{R}}_{\theta },\hspace{1em}\mathit{\sigma }={\displaystyle \sum _{i=1}^m}{\mathit{\sigma }}_i$$

(22)

In existing studies, the analytical solutions for derived blasting stress fields based on short cylindrical charges are formulated as follows [19]:

$$\left(\begin{array}{c}{\sigma }_{ri}\\ {\tau }_i\end{array}\right)=\rho \left(\begin{array}{cc}{C}_p& 0\\ 0& -C_s\end{array}\right)\left(\begin{array}{c}sin{\varphi }_i\\ cos{\varphi }_i\end{array}\right)\left({\displaystyle \frac{F_1({\varphi }_i)}{R_i}}{\displaystyle \frac{d^2}{\mathrm{d}{t}^2}}\left\{p(t-{\displaystyle \frac{(i-1)L}{m{D}_e}}-{\displaystyle \frac{R_i}{C_p}})\right\}\right)+\rho \left(\begin{array}{cc}{C}_p& 0\\ 0& C_s\end{array}\right)\left(\begin{array}{c}cos{\varphi }_i\\ sin{\varphi }_i\end{array}\right)\left({\displaystyle \frac{F_2({\varphi }_i)}{R_i}}{\displaystyle \frac{d^2}{\mathrm{d}{t}^2}}\left\{p(t-{\displaystyle \frac{(i-1)L}{m{D}_e}}-{\displaystyle \frac{R_i}{C_s}})\right\}\right)$$

(23)

$$\left[\begin{array}{c}{\sigma }_r\\ \tau \end{array}\right]={\displaystyle {\sum }_{i=1}^m}\left[\begin{array}{c}{\sigma }_{ri}\\ {\tau }_i\end{array}\right]$$

(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.

2.2. Multi-Borehole Stress Field Superposition Based on Stress Wave Propagation Time Intervals

In multi-borehole blasting, stress field superposition must account for time interval effects caused by differences in stress wave propagation paths (see Figure 5). These delays result from variations in propagation distances and anisotropic wave velocities in the rock mass, necessitating precise phase corrections for accurate dynamic stress predictions.

Based on the single-hole stress field derived above, the total stress field at a given observation point under multi-borehole blasting is obtained by superposition:

$$\left[\begin{array}{c}{\sigma }_r\\ {\tau }_Z\\ {\tau }_{\theta }\end{array}\right]={\displaystyle \sum _{j=1}^n}\left[\begin{array}{c}{\sigma }_{rj}(t-t_j)\\ {\tau }_{Zj}(t-t_j)\\ {\tau }_{\theta j}(t-t_j)\end{array}\right]$$

(25)

where n is the number of boreholes; $t_j$ is the time required for stress waves generated from the j-th borehole to reach the observation point, given by $t_j=R_j/C_{p,s}$; $R_j$ is the distance from the j-th borehole to the observation point (m); and C is the corresponding P-wave or S-wave propagation velocity (m/s).

3. Application Examples and Comparative Analysis

3.1. Selection of Relevant Parameters and Calculation Points

A tunnel in Fujian Province is a double-track highway tunnel with a 10 m net spacing that was constructed using the upper- and lower-step method, with simultaneous construction of the left and right tunnels. The surrounding rock is weathered granite with well-developed joint fissures; its physical and mechanical properties are listed in

Table 1. Physical and mechanical parameters of tunnel surrounding rock.

Density (kg·m−3)	Compressive Strength (MPa)	Tensile Strength (MPa)	Elastic Modulus (GPa)	P-Wave Velocity (m/s)	S-Wave Velocity (m/s)	Poisson’s Ratio
2650	54.50	7.50	30.70	3036	1704	0.23

. Engineering practice indicates that slotted holes in a single section carry a high charge, producing the greatest stress and vibration. Therefore, as an example, eight secondary slotted holes were detonated simultaneously on the upper steps. Each borehole (dot) has a diameter of 40 mm, an explosive charge diameter of 32 mm, a hole depth of 3 m, a charge of 2.4 kg per hole, and bottom initiation. The layout is shown in Figure 6. The explosion has a detonation velocity of 3200–3800 m/s, a density of 1200 kg/m³ and a detonation product pressure coefficient (n) of 10 [31]. Due to the extensive jointing in the rock mass, the rock damage factor is taken as D = 0.2.

According to existing studies [39], the blast-facing sidewall of the adjacent tunnel is a critical risk area. To evaluate the stress magnitude and distribution in the rock mass at this location, six calculation points (Points A–F) are placed at 1-m intervals along a horizontal line within the sidewall (see Figure 7). The first point (Point A) is located 11.42 m from the center of the cut holes. According to reference [17], a single cylindrical charge with an aspect ratio of 10 was divided into 15 short cylindrical charge units (m = 15), each 16 cm long with a base radius of 1.6 cm. These parameters were then used in Equations (20)–(22) and (25) for calculation.

3.2. Calculation Results and Comparative Analysis

The peak stress points A~F are shown in

Table 2. Stress calculation results.

Calculation Point	Distance to the Cut Hole Center (m)	Peak Stress (MPa)
		Radial	Axial	Hoop
A	11.42	1.60	0.93	0.76
B	12.38	1.37	0.78	0.65
C	13.35	1.18	0.68	0.56
D	14.32	1.02	0.59	0.49
E	15.29	0.90	0.52	0.43
F	16.26	0.80	0.46	0.38

. The uncertainty in the peak stress is ±0.02. The results indicate that during blasting at the excavation face, the peak stresses in the rock mass follow the order: radial stress > axial stress > hoop stress.

To further examine stress evolution, Figure 8 shows the triaxial stress variations of radial points A–F along the tunnel axis, based on the data in Table 2. The results indicate that radial stress decays inversely with distance and attenuates more rapidly than axial stress. Hoop stress remains consistently lower than both radial and axial stresses. Based on Table 2, the time history curve of radial stress at Point A is plotted in Figure 9. This curve shows that radial stress decreases over time and exhibits a double peak. The peak occurs after the arrival of the S-wave, as the superposition of P- and S-waves causes a sudden increase in stress, indicating their combined effect on the surrounding rock particles. According to Table 2, radial, axial, and hoop stresses decay with distance following the power law σ = σ₀d^α, where σ₀ is the stress at d = 0, d is the distance from the cut hole center, and α = −2.004 ± 0.01063 is the common power coefficient, identical for all stresses (see Figure 7). The initial stresses are σ_r0 = 215.427 ± 5.89566 MPa, σ_a0 = 123.68921 ± 3.36189 MPa, and σ_h0 = 102.10915 ± 2.85727 MPa.

To compare the stress results, parameters from the engineering case were applied to Equations (20)–(22) and (25) from this study—which account for the instantaneous axial pressure from short-cylinder charges and the rock mass damage factor D—and to Equations (23) and (24) from previous studies. The computed stress values are shown in Figure 10 and Figure 11.

Comparative analysis of instantaneous detonation axial pressure ($q(t)$): As shown in Figure 10, radial stress is higher when instantaneous detonation axial pressure is considered. The maximum difference reaches 12.42%, with an average difference of 11.25% across all calculation points. Similarly, Figure 10 shows that axial stress is also higher with this factor included, with a maximum difference of 10.36%, and an average difference of 9.68%.

Comparative analysis of rock mass damage (D): As illustrated in Figure 11, incorporating the rock mass damage factor D reduces all the stress components. Stress attenuation follows a nonlinear trend as D increases. At D = 0.4, radial stress decreases by 44.58%, axial stress by 29.55%, and hoop stress by 27.18%, compared to the undamaged case (D = 0).

Based on the results in Table 2 and the comparative analysis in Figure 10 and Figure 11, the formulas derived in this study (Equations (20)–(22) and (25))—which account for instantaneous detonation axial pressure and rock mass damage—demonstrate greater accuracy and rationality than existing models.

4. Field Monitoring and Stress Analysis

4.1. Engineering Situations

In a twin-tunnel highway project in Fujian Province, the left and right tunnels were excavated simultaneously, with the left tunnel advancing more rapidly. The upper bench excavation area of the active tunnel is approximately 77 m². The blasting operations use digital electronic detonators, with 40 mm diameter boreholes and No. 2 rock emulsion explosive cartridges (32 mm diameter). The initiation sequence follows a ring-by-ring (row-by-row) pattern with a 50 ms inter-row delay, achieving a 3.0 m cyclic advance. Detailed blasting parameters are provided in

Table 3. Blasting parameters of upper bench.

Borehole Type	Number of Boreholes	Charge per Hole (kg)	Total Charge (kg)	Delay Segment
Cut Holes	6	1.5	9.0	Segment 1 (0 ms)
	8	2.4	19.2	Segment 2 (50 ms)
Auxiliary Holes	11	1.8	19.8	Segment 3 (100 ms)
	13		23.4	Segment 4 (150 ms)
	16		28.8	Segment 5 (200 ms)
	24		43.2	Segment 6 (250 ms)
Contour Holes	18	1.5	27.0	Segment 7 (300 ms)
	18		27.0	Segment 8 (350 ms)
Floor Holes	16	1.8	28.8	Segment 9 (400 ms)
Total	130		226.2

, and the blasting network is shown in Figure 12.

4.2. Monitoring Test Setup, Materials, and Methods

The purpose of dynamic strain monitoring is to measure stress in the surrounding rock of the adjacent tunnel during blasting, verify the accuracy of the previously derived analytical stress solutions, and analyze stress variation patterns induced by blasting. An ultra-dynamic strain measurement system (Figure 13) was used to monitor dynamic strain in the tunnel surrounding rock. The testing procedure, shown in Figure 14, operates as follows: voltage signals were collected via embedded sensors, converted into strain values using Equation (26), and dynamic stress peaks are then calculated based on Hooke’s law (Equation (27)) [40].

$$\epsilon =4\times U/\mathrm{K}\times U_0\times n\times G$$

(26)

$$\sigma =\mathrm{E}\epsilon$$

(27)

where $U$ is the peak output voltage (V), $\mathrm{K}$ is the sensitivity coefficient of the strain gauge, $U_0$ is the bridge voltage of the strain amplifier (V), $n$ is the number of effective bridge arms, $G$ is the gain of the strain amplifier, and $\mathrm{E}$ is the elastic modulus of the rock mass (GPa). The dynamic strain test uses a half-bridge circuit, with the instrument parameters listed in

Table 4. Key parameters of the instrument.

K	U0/V	n	G
2.0	4.0	1	100

.

The operating principle of the triggered voltage signal recording system is illustrated in Figure 15. When blast loads act on the strain bar, it undergoes minor deformation, causing the bonded strain gauges to change resistance. The resistance change produces a microvolt-level output via a Wheatstone bridge. The signal is then amplified by an ultra-dynamic strain amplifier, triggering a pulse that converts the microscopic deformation into a millivolt-level voltage signal recorded by the system.

The measurement point layout is shown in Figure 16. Horizontal boreholes are drilled into the sidewall of the adjacent existing tunnel, with five strain measurement points spaced at 1 m intervals (Point 1 is 1 m from the sidewall). Prefabricated strain bars, matched to the physical and mechanical properties of the tunnel surrounding rock, are used. Triaxial resistance strain gauges are mounted on the bars to measure strain via electrical resistance (Figure 17). The instrumented bars are inserted into the boreholes and fixed with grout. A schematic diagram of the strain bar inserted into the borehole is shown in Figure 18, where the specifications of the PVC pipe in the diagram are consistent with those of the on-site drilling. To determine the rock mass damage factor D at the same locations, a YL-LCT single-transmitter dual-receiver sonic probe (Figure 19) is used to measure P-wave velocities. Sonic testing boreholes are drilled 0.5 m from the strain measurement holes (Figure 20), ensuring spatial alignment for accurate comparative analysis.

During on-site dynamic strain testing, intense blasting vibrations caused collisions between data cables and surrounding structures, while mechanical equipment and welding operations introduced significant signal interference. To ensure measurement accuracy, the following measures were implemented before testing: data cables at borehole exits were wrapped with foam sleeves, and the remaining segments were enclosed in stainless steel junction boxes to prevent impact damage. Testing instruments were grounded using steel rebars to eliminate electromagnetic interference. All mechanical equipment was relocated, and welding operations were suspended during testing. Instrument connections were established and activated only after confirming the absence of interference sources. Sonic testing was completed prior to blasting. The ultra-dynamic strain amplifier was activated just before the blast and deactivated afterward. The field-testing configuration is shown in Figure 21. Each blasting round required approximately 15 min for complete monitoring, and the procedure was repeated over five consecutive rounds.

4.3. Monitoring Results

The wave velocity test results of the surrounding rock prior to the first blasting cycle are shown in Figure 22. The extracted results of measuring points 1–5 are shown in

Table 5. Sonic wave test results for measuring points 1–5.

	Point 1	Point 2	Point 3	Point 4	Point 5
hole depth (m)	1.0	2.0	3.0	4.0	5.0
original rock wave velocity (km/s)	3.036	3.036	3.036	3.036	3.036
wave velocity before the first blasting (km/s)	2.716	2.750	2.816	2.881	2.897
D	0.20	0.18	0.14	0.10	0.09

. Substituting the data into Equation (7), the calculated damage factors for monitoring Points 1 to 5 are 0.20, 0.18, 0.14, 0.10, and 0.09, respectively. Taking the radial strain test at monitoring Point 1 during the first blasting cycle as an example, the measured voltage signal is shown in Figure 23. The stress peak occurs at approximately 55 ms, closely aligning with the 50 ms delay associated with the highest charge, a single time-delay interval of the cut holes, confirming the reliability of the dynamic strain measurements. The wave velocities and dynamic strain data from the remaining four points were substituted into Equations (7), (26) and (27) to calculate the corresponding damage factors and dynamic stresses. The results are presented in

Table 6. Stress measurement results.

Blasting Times	Damage Factor D	Measure Point	Peak Voltage (mV)			Peak Stress (MPa)
			Radial	Axial	Hoop	Radial	Axial	Hoop
1	0.20	1	5.77	3.19	2.72	0.89	0.49	0.42
	0.18	2	6.27	3.57	2.94	0.96	0.55	0.46
	0.14	3	6.76	—	3.20	1.04	—	0.50
	0.10	4	7.17	4.20	3.41	1.10	0.65	0.52
	0.09	5	—	4.79	3.89	—	0.74	0.60
2	0.22	1	5.70	3.14	2.70	0.88	0.48	0.42
	0.20	2	6.19	3.55	—	0.95	0.54	—
	0.15	3	6.73	3.76	3.18	1.03	0.58	0.49
	0.10	4	6.99	4.10	3.43	1.07	0.63	0.51
	0.10	5	7.97	4.69	3.78	1.22	0.72	0.59
3	0.23	1	5.57	3.02	2.70	0.87	0.48	0.41
	0.21	2	6.08	3.49	—	0.93	0.54	—
	0.16	3	6.68	3.74	3.17	1.03	0.58	0.49
	0.14	4	—	4.04	3.32	—	0.62	0.50
	0.10	5	7.83	4.61	3.71	1.20	0.71	0.57
4	0.23	1	5.73	3.14	2.70	0.87	0.48	0.41
	0.21	2	5.85	3.34	2.81	0.90	0.51	0.43
	0.17	3	6.53	3.65	3.09	1.00	0.56	0.47
	0.15	4	6.66	3.90	3.16	1.02	0.60	0.49
	0.12	5	7.56	4.45	3.60	1.16	0.68	0.55
5	0.25	1	5.68	3.12	2.63	0.85	0.46	0.40
	0.23	2	5.85	—	2.78	0.90	—	0.43
	0.15	3	6.62	3.69	3.12	1.02	0.57	0.48
	0.12	4	6.89	4.03	3.25	1.06	0.62	0.50
	0.10	5	7.81	4.58	—	1.20	0.70	—

, where “—” indicates that dynamic strain data were not obtained at certain points due field complexity and signal interference. The uncertainty in the peak stress and peak voltage is ±0.02.

As shown in Table 6, at each monitoring point, the stresses hierarchy is as follows: radial stress > axial stress > hoop stress. Additionally, all three stresses decrease with the components of increasing rock mass damage and with greater distance from the blast source.

4.4. Stress Propagation Law and Comparative Analysis

Based on the data in Table 6, the average damage factors and stress values at monitoring points 1–5 over five blasting cycles are summarized

Table 7. The average value of measured damage factor and stress.

Measure Point	Damage Factor D	Average Stress (MPa)			Corresponding Theoretical Calculation Points	Theoretical Stress (MPa)
		Radial	Axial	Hoop		Radial	Axial	Hoop
1	0.23	0.88	0.48	0.41	F	0.80	0.46	0.38
2	0.21	0.93	0.54	0.44	E	0.90	0.52	0.43
3	0.15	1.03	0.58	0.49	D	1.02	0.59	0.49
4	0.12	1.06	0.62	0.50	C	1.18	0.68	0.56
5	0.10	1.20	0.71	0.57	B	1.37	0.78	0.65

. To further examine the stress propagation behavior in the surrounding rock of the adjacent tunnel with respect to distance from the blast center, Figure 24 was generated using the data in Table 7. As shown in Figure 24, radial, axial, and hoop stresses all decrease with increasing distance, with radial stress showing the fastest attenuation rate, followed by axial and hoop stresses.

The mean relative error (MRE) between measured and theoretical stress values is defined as follows:

$${\mathrm{MRE}}_{\left(i\right)}=\left|{\displaystyle \frac{{\overline{\sigma }}_{\mathrm{measured}}^{(i)}-{\sigma }_{\mathrm{theoretical}}^{(i)}}{{\sigma }_{\mathrm{theoretical}}^{(i)}}}\right|\times 100\%$$

(28)

$${\overline{\sigma }}_{\mathrm{measured}}^{(i)}={\displaystyle \frac{1}{n}}{\displaystyle {\displaystyle \sum _{j=1}^n}{\sigma }_{ij}}$$

(29)

where ${\overline{\sigma }}_{\mathrm{measured}}^{(i)}$ is the mean measured stress at the i-th monitoring point, ${\sigma }_{\mathrm{theoretical}}^{(i)}$ is the theoretical stress at the i-th monitoring point, n is the number of blast events, and ${\sigma }_{ij}$ is the measured stress at the i-th monitoring point during the j-th blast.

And the average error (AE) of all measuring points is defined as follows:

$$\mathrm{AE}={\displaystyle \frac{1}{k}}{\displaystyle {\displaystyle \sum _{i=1}^k}{\mathrm{MRE}}_{(i)}}$$

(30)

where k is the number of measuring points.

A comparison between measured and theoretical stress values is presented in Figure 25. The results show that the measured stresses are slightly lower than theoretical predictions, with average errors of 7.02% for radial stress, 7.56% for axial stress, and 7.05% for hoop stress.

Analysis of discrepancies: According to Reference [41], borehole stemming can improve explosive energy utilization by approximately 7.5%. In this test, however, the cut holes were unstemmed, leading to energy loss. Additionally, signal attenuation occurred due to excessive cable lengths at the signal receivers. According to reference [40], the voltage drop ($\Delta U$) along the cable is calculated as follows [42]:

$$\Delta U=\Delta U_0\%\times I\times L$$

(31)

where $\Delta U_0\%$ is the percentage voltage drop per kilometer of cable, I is the load current (A), and L is the cable length (km). Based on data from reference [42], $\Delta U_0\%$ was taken as 0.766%. With a measured load current of I = 0.52 A and cable length L = 0.1 km, the voltage drop calculated using Equation (31) is 0.3983 mV. This voltage drop reduced the peak voltage during measurements, resulting in values lower than theoretical predictions and contributing to the previously noted error, further lowering the reduced measured stress values. These factors collectively explain the observed discrepancies.

Despite these minor deviations, the measured and theoretical values align well, with errors within acceptable limits (≤8%), confirming the reliability of the analytical stress field solutions developed in this study. In tunnel engineering, the MRE between theoretical and measured stress values typically remains below 10%, a range acceptable for meeting project reliability requirements [43,44].

5. Conclusions

Based on Heelan’s short cylindrical cavity solution, this study derives an analytical model for dynamic stresses in the surrounding rock of adjacent tunnels by incorporating the axial pressure $q(t)$ generated by instantaneous detonation and the rock mass damage factor D. A case study was conducted to investigate stress magnitudes and propagation behavior, with validation through field measurements. The theoretical stress values derived from the model show consistent trends and minor numerical deviations when compared to measured data.

(1)

Stress magnitude and propagation laws: Both theoretical and measured results exhibit consistent triaxial stress trends in the following order: radial stress > axial stress > hoop stress. All three stresses decrease with increasing distance and damage factor, showing an inverse relationship with distance. Radial stress decays at the highest rates, followed by axial and hoop stresses. Stress also decays exponentially over time, with the peak occurring after the arrival of the transverse S-wave.

(2)

Advantages of proposed stress model: Incorporating instantaneous detonation pressure significantly improves accuracy compared to conventional models that neglect this factor, reducing average errors by 11.25% in radial stress and 9.68% in axial stress. Including the damage factor D reveals nonlinear stress attenuation with increasing damage. At D = 0.4, compared to the intact rock condition (D = 0), stress reductions reach 44.58% (radial), 29.55% (axial), and 27.18% (hoop), confirming the importance of accounting for detonation pressure and rock mass damage in stress prediction.

(3)

Validation through field measurements: Measured stresses show small deviations from theoretical values, with average errors of 7.02% (radial), 7.56% (axial), and 7.05% (hoop). This close agreement validates the reliability of the proposed analytical solution.

Note: The primary limitations of the stress calculation model in this study are its initial assumption of heterogeneous, anisotropic rock masses as homogeneous media, with stress computed through the introduction of a damage factor. This elasticity-based model simplifies reality and does not fully capture the complex behavior of anisotropic rock. In practice, stress wave propagation in such media is nonlinear, and influenced by transmission, refraction, and reflection at rock mass discontinuities, as well as strain-softening effects in tunnel rock [45]. During multi-borehole stress superposition, the rock between the cut holes and the central pillar is treated as a continuous medium. However, auxiliary (relief) and perimeter pilot holes cause additional stress attenuation due to wave scattering at these cavities. To address these limitations, future research will focus on the following: (1) Stress wave propagation in anisotropic and fractured rock masses using a true triaxial Hopkinson bar system to simulate in situ stress conditions, and ultrasonic transmission techniques to monitor P-wave velocity evolution under varying stress paths [46]. (2) Discrete element modeling (DEM) to simulate contact mechanics across rock discontinuities. (3) Numerical simulation tools to study stress wave behavior in rock masses containing pre-existing cavities. These approaches aim to improve theoretical accuracy and better reflect real-world conditions.