Study on the Three-Dimensional Behavior of Blasting Considering Non-Uniform In-Situ Stresses Distributed along the Blasthole Axis

Abstract

For roof-cutting by blasting in the gob-side entry under an overhanging hard roof, studies on the impacts of in-situ stresses on the propagation of blast-induced cracks have typically focused on uniform stresses but ignored the effects of non-uniform in-situ stresses (NIS) distributed along the blasthole axis. Therefore, the distribution patterns of hoop stress and rock damage caused by NIS distributed along the blasthole axis were investigated using numerical modeling and theoretical analysis. The results illustrate that with the rising NIS for the cross section along the blasthole axis, the peak values of hoop compressive stress at the same distance from the blasthole’s center gradually increase, resulting in a nonlinear attenuation trend in the damage range of the rock. Consequently, the spacing between blastholes should be determined based on the average length of the primary cracks under the maximum confining pressure. Additionally, for the cross section perpendicular to the blasthole axis, as the lateral pressure coefficient increases from 0.25 to 2, the damage range in the vertical direction significantly decreases. This results in varying extents of blast-induced cracks within the coal pillar, providing a reference for the design of shallow-borehole crack filling.

1. Introduction

Fragmentation of materials in mining practice is commonly achieved with blasting [1,2]. Researchers have increasingly recognized that the interaction between in-situ stress and blast loading results in a more intricate dynamic fracture behavior of rock [3,4]. Taking the gob-side entry under an overhanging hard roof as an example in Figure 1, the core approach for maintaining the stability of a roadway is roof-cutting with blasting, which cuts off the stress transfer path between the roadway roof and the gob roof [5,6]. Due to the excavation of the working face, the side abutment pressure in the gob is unevenly distributed along the blasthole axis, affecting the three-dimensional (3D) damage evolution of the rock. This further influences the spacing of blastholes and the extent of blast-induced cracks within the coal pillar. Therefore, it is essential to analyze the blasting mechanism of rock under the influence of non-uniform in-situ stresses (NIS) distributed along the blasthole axis to guide the arrangement of blastholes and the shallow-borehole crack filling for the coal pillar.

The effect of in-situ stress on crack expansion resulting from blasting has been extensively studied by researchers using theoretical calculations and experiments over the past few decades. As an illustration, Simha et al. analyzed the response of rock to blasting in situations of high pressure with a series of experiments; they recommended that two stages can be distinguished in the behavior of dynamic fracture: initiation and propagation. The initiation stage is primarily dominated by the stress transients generated by blasting, and the propagation stage is mainly controlled by high-stress conditions due to the rapid decay of these stress transients [7]. Numerous studies conducted using the dynamic caustics experimental method have shown that under initial confining pressure, stress concentrations of varying magnitudes are produced in the vicinity of the blasthole. Furthermore, with an increase in stress concentration, the propagation of blast-generated primary cracks is gradually deflected, aligning with the maximum principal stress direction at an increasing maximum deflection angle, and type II cracks become more prominent [8,9]. The development of an elastodynamic framework by Tao et al. was instrumental in improving the mechanical understanding of the dynamic behavior of rock caused by varying in-situ stresses by enabling the exploration of blast wave propagation in pre-stressed rock [10]. Yang et al. examined the stress concentration phenomenon around a blasthole under high static stress conditions using a theoretical analysis of stress distribution and then utilized a dynamic caustics experimental approach to explore how slit orientations and high static stress conditions influence the behavior of cracks caused by slit charge blasting [11]. Li et al. observed that the attenuation effect of jointed rock on stress waves increased as in-situ stress levels rose, corresponding to an enhanced attenuation by intact rock and a weakened attenuation by rock joints [12]. Zhang et al. used the digital image correlation (DIC) technique to study the evolution of the strain field of rock blasting under high stress [13]. To sum up, the aforementioned studies demonstrate that the in-situ stresses play a substantial role in both the creation and spread of cracks caused by blast loading.

Several interesting results have been obtained from numerical studies on rock responses considering in-situ stress and blast loading. Yi et al. calibrated the parameters of the Jones-Wilkins-Lee (JWL) equation of state using cylinder expansion tests to provide a numerical evaluation of the damage propagation under confining pressure. [1]. Through numerical simulations of blasting on a rock slope, Wang et al. discovered that the effect of repeated blast loading on rock damage was minimal in the minimum principal stress direction. Notably, by employing a plum-shaped blasthole arrangement and delayed detonation, it is possible to achieve better blasting results in pre-stress conditions [14]. Yilmaz et al. conducted 3D finite difference numerical simulations of the dynamic behavior of rock to discuss the impact of anisotropic in-situ stresses and loading rate considering explosive types and site conditions [15]. Under the in-situ stress environment, Li et al. suggested that the compressive stress induced by blast loading can be increased while the corresponding tensile stress can be reduced [16]. The impact of the lateral pressure coefficient and the isotropic stress field on crack behavior triggered by cut blasting was investigated by Xie et al. with the Riedel–Hiermaier–Thoma (RHT) model in LS-DYNA. The authors then proposed a design approach for cut blasting in high-stress environments [17]. Aliabadian et al. examined how fault orientation affects the fracturing of rock masses when subjected to high in-situ stresses. They observed that the blast-generated cracks were confined around the blasthole and near the fault due to the fault and high-stress field [18]. Han et al. employed a combined finite-discrete element method to simulate the development of the damage zone in a roadway with high in-situ stresses resulting from contour blasting, providing a potent instrument for investigating rock blasting [19]. Significantly, previous numerical studies have typically utilized a two-dimensional plane strain model to analyze the distribution of damage in rock resulting from blasting under high in-situ stress conditions.

Critically, previous studies about blast-induced crack propagation have not considered the NIS distributed along the blasthole axis, inevitably leading to inaccurate predictions for the damage evolution of rock. This omission will negatively impact the effectiveness of inter-borehole crack penetration and the determination of the grouting filling range for blast-induced cracks within the coal pillar. Therefore, this study first established a 3D mechanical model of the blasthole and obtained the hoop stress distribution of rock under the action of NIS distributed along the blasthole axis. A numerical model of rock blasting was then established to investigate the relationship between the 3D damage distribution around the blasthole and the NIS distributed along the blasthole axis by comparing two working conditions: no pressure and isotropic confining pressures. Following the presentation of these results, a comprehensive discussion of the relationship between lateral pressure coefficient and rock damage is undertaken.

2. Theoretical Analysis

2.1. Stress Distribution of Rock under NIS Distributed along the Blasthole Axis

As can be observed from Figure 1, to sever the stress transmission path between the gob and the roadway roof, a blasthole is typically installed in the roof for a coal pillar by drilling from the roadway into the gob at a fixed angle. The blasting with NIS distributed along the blasthole axis can be simplified to a 3D model, as shown in Figure 2. To realize a convenient calculation, the angle of the blasthole relative to the horizontal direction was simplified to 0°, and the orifice was positioned at the model boundary near the roadway. The boundary of the model near the gob was chosen as the location to establish the origin of a coordinate system. In this coordinate system, the x and y axes were positioned horizontally, while the z axis was positioned vertically. The y and z axes were perpendicular to the blasthole axis, while the x axis was aligned with it. The side abutment pressure in the gob ${\sigma }_v$ was imposed on the model’s upper and lower boundaries, and the model’s front and rear boundaries were subjected to the application of horizontal stress ${\sigma }_h$. Both stresses were non-uniformly distributed along the blasthole axis. Cross section I was perpendicular to the blasthole axis, while cross sections II and III were along the blasthole axis. The normal line of II was consistent with the z axis, and the normal line of III was consistent with the y axis.

For the sake of computational simplicity, the influence of gob-side entry on the side abutment pressure in gob was ignored. Then, according to limit equilibrium theory, the side abutment pressure in the gob and horizontal stress can be expressed as follows [20]:

$${\sigma }_v=\left\{\begin{array}{cc}\xi \left(P_i+Ccot\phi \right)e^{{\displaystyle \frac{2\xi xtan\phi }{m}}}-Ccot\phi \hfill & 0\le x\le x_0\hfill \\ \left(K-1\right)\gamma H{e}^{b\left(x_0-x\right)}+\gamma H\hfill & x>x_0\hfill \end{array}\right.$$

(1)

$${\sigma }_h=\lambda {\sigma }_v$$

(2)

$x_0$ can be defined as the distance between the maximum value of side abutment pressure in gob and the boundary of the model adjacent to gob [20]:

$$x_0={\displaystyle \frac{m}{2\xi tan\phi }}ln{\displaystyle \frac{K\gamma H+Ccot\phi }{\xi \left(P_i+Ccot\phi \right)}}$$

(3)

where $\lambda$ is the lateral pressure coefficient; $H$ is the burial depth; $\gamma$ is the formation density; $m$ is the height mining; $P_i$ is the support intensity; $K$ is the stress concentration factor; $C$ is the cohesion between the layers; $\phi$ is the internal friction angle; $\xi$ is the triaxial stress coefficient, $\xi =(1+sin\phi )/(1-sin\phi )$; and $b$ is the attenuation coefficient.

The blasting of a single hole caused by static stress was reduced to a plane strain problem for ease of analysis, as presented in Figure 3. Assuming a circular blasthole of radius R exists in an infinite isotropic rock, and the vertical and horizontal stresses for cross section I are imposed on the boundaries of the plane. By using Kirsch’s elastodynamic equations, the stress distribution surrounding the circular blasthole can be determined as shown below [21]:

$$\left\{\begin{array}{c}{\sigma }_{\rho }={\displaystyle \frac{{\sigma }_v+{\sigma }_h}{2}}(1-{\displaystyle \frac{R^2}{{\rho }^2}})+{\displaystyle \frac{{\sigma }_h-{\sigma }_v}{2}}(1-4{\displaystyle \frac{R^2}{{\rho }^2}}+3{\displaystyle \frac{R^4}{{\rho }^4}})cos2\theta \hfill \\ {\sigma }_{\theta }={\displaystyle \frac{{\sigma }_v+{\sigma }_h}{2}}(1+{\displaystyle \frac{R^2}{{\rho }^2}})-{\displaystyle \frac{{\sigma }_h-{\sigma }_v}{2}}(1+3{\displaystyle \frac{R^4}{{\rho }^4}})cos2\theta \hfill \\ {\tau }_{\rho \theta }={\displaystyle \frac{{\sigma }_h-{\sigma }_v}{2}}(1+2{\displaystyle \frac{R^2}{{\rho }^2}}-3{\displaystyle \frac{R^4}{{\rho }^4}})sin2\theta \hfill \end{array}\right.$$

(4)

where ${\sigma }_v$ is the vertical stress; ${\sigma }_h$ is the horizontal stress; $\rho$ is the polar radius; $\theta$ is the polar angle; ${\tau }_{\rho \theta }$ is the shear stress; ${\sigma }_{\theta }$ is the hoop stress; ${\sigma }_{\rho }$ is the radial stress; and $R$ is the circular borehole’s radial.

The hoop stress will affect the damage distribution around a blasthole; therefore, it was necessary to analyze the hoop stress distribution of rock when the action of NIS is distributed along the blasthole axis. Thus, it can be obtained with the following equations:

$${\sigma }_{\theta }=\left\{\begin{array}{cc}\left(\xi \left(P_i+Ccot\phi \right)e^{{\displaystyle \frac{2\xi tan\phi }{m}}x}-Ccot\phi \right)\left({\displaystyle \frac{1}{2}}\left(1+cos2\theta \left(1-\lambda \right)+\lambda \right)+{\displaystyle \frac{1}{2}}\left(1+\lambda \right){\displaystyle \frac{R^2}{{\rho }^2}}+{\displaystyle \frac{3}{2}}\left(1-\lambda \right)cos2\theta {\displaystyle \frac{R^4}{{\rho }^4}}\right)\hfill & 0\le x\le x_0\hfill \\ \left(\gamma H+\left(K-1\right)\gamma H{e}^{b\left(x_0-x\right)}\right)\left({\displaystyle \frac{1}{2}}\left(1+cos2\theta \left(1-\lambda \right)+\lambda \right)+{\displaystyle \frac{1}{2}}\left(1+\lambda \right){\displaystyle \frac{R^2}{{\rho }^2}}+{\displaystyle \frac{3}{2}}\left(1-\lambda \right)cos2\theta {\displaystyle \frac{R^4}{{\rho }^4}}\right)\hfill & x>x_0\hfill \end{array}\right.$$

(5)

2.2. Case Study

The selected mine had an average burial depth of 400 m, a mining height of 3 m, and a coal pillar width of 15 m. The cohesion between the layers was assumed to be 1.13 MPa, and the internal friction angle was 20.25° for the calculation. A formation density of 23.5 kN/m³, a stress concentration factor of 2.7, and a support intensity of 0 were used in the analysis. Furthermore, the attenuation coefficient was set to 0.3. Finally, a blasthole measuring 12 m in length and 0.048 m in diameter was drilled, with its bottom located at x = 3 m and its orifice located at x = 15 m. The hoop stresses around the blasthole were obtained for lateral pressure coefficients of 1 and 0.5 as follows:

$$\begin{array}{cc}{\sigma }_{\theta }=\left(15.98e^{0.3(3-x)}+9.4\right)\left(1+{\displaystyle \frac{5.76\times {10}^{-4}}{{\rho }^2}}\right)& x\ge 3,\lambda =1\end{array}$$

(6)

$$\begin{array}{cc}{\sigma }_{\theta }=\left(15.98e^{0.3\left(3-x\right)}+9.4\right)\left(0.75+{\displaystyle \frac{4.32\times {10}^{-4}}{{\rho }^2}}+\left({\displaystyle \frac{2.49\times {10}^{-7}}{{\rho }^4}}+0.25\right)cos2\theta \right)& x\ge 3,\lambda =0.5\end{array}$$

(7)

As seen in Figure 4, the hoop stress distribution around the blasthole was obtained based on Equations (6) and (7), where hoop compressive stress is positive. The distribution of hoop stress is depicted in Figure 4a at the blasthole wall for cross section I located at the orifice, where ${\sigma }_v$ = 10 MPa; $\lambda$ = 1, ${\sigma }_{\theta }$ is compressive on the blasthole wall, exhibiting a circular distribution with a magnitude of 2${\sigma }_v$ = 20 MPa; and $\lambda$ = 0.5, ${\sigma }_{\theta }$ is also compressive on the blasthole wall with a magnitude of 2.5${\sigma }_v$ = 25 MPa in the horizontal direction ($\theta$ = 0° and 180°) and 0.5${\sigma }_v$ = 5 MPa in the vertical direction ($\theta$ = 90° and 270°). Furthermore, as $\theta$ changes from 90° to 0°, the magnitude of ${\sigma }_{\theta }$ on the blasthole wall gradually increases. Hoop compressive stress inhibits crack propagation [11]. Therefore, these results illustrate that the crack has a tendency to propagate in the maximum principal stress direction.

Notably, since the magnitude of hoop stress on the blasthole wall is unrelated to the polar angle when the lateral pressure coefficient is 1, the hoop compressive stress distribution around the blasthole for cross section II is indicated in Figure 4b. The hoop compressive stress distribution on the surrounding area of blasthole for cross section II and III is presented in Figure 4c,d, respectively, for a lateral pressure coefficient of 0.5. Figure 4b,c shows that as $\rho$ increases, the hoop compressive stress in the horizontal direction declines sharply and then gradually approaches ${\sigma }_v$ for the same $x$, when $\lambda$ = 1 or 0.5. As can be observed from Figure 4d, at a given $x$, the hoop compressive stress that is aligned with the vertical axis initially increases slowly from 0.5 ${\sigma }_v$ to 0.7 ${\sigma }_v$, then decays back to 0.5 ${\sigma }_v$ before finally remaining stable with the rising of $\rho$, when $\lambda$ = 0.5. In addition, for a given $\rho$, the hoop compressive stress gradually decreases from the bottom of the blasthole to the orifice with the increase of $x$. Therefore, as the NIS distributed along the blasthole axis decrease, the extent of rock damage gradually goes up in the axial direction owing to the hoop compressive stress.

3. Material Behavior Models for Numerical Analysis

LS-DYNA R11.0 software is a versatile tool that can effectively solve complex nonlinear dynamics problems and be applied to address various challenges in the field of blasting engineering [22]. It primarily comprises three algorithms: Eulerian formulation, Lagrange formulation, and Arbitrary Lagrangian-Eulerian (ALE) formulation, among which the ALE can prevent excessive mesh distortion problems caused by large cell deformation. Thus, the ALE formulation is more suitable for simulating the behavior of rock resulting from blast loading [23].

3.1. Material Model for the Rock

An advanced damage plasticity model called the RHT model is commonly used in simulations of blasting response for materials that are prone to brittle failure, such as concrete or rock [24]. In contrast to other material models, the RHT model incorporates damage softening, strain hardening, confining pressure, and strain rate to assess the tensile-compressive damage of rock exposed to blast loading. The material damage of the model can be described as follows:

$$D={\displaystyle \sum \frac{\Delta {\epsilon }_P}{{\epsilon }_P^f}}$$

(8)

where $D$ is the damage; $\Delta {\epsilon }_P$ is used to quantify the amount of plastic strain that has accumulated; and ${\epsilon }_P^f$ is the plastic strain equivalent at the point of failure. Values for D can fall between 0 and 1, where $D$ = 0 indicates undamaged, and $D$ = 1 represents fully damaged.

The yield surface of this model is characterized by the Willam–Warnke function, a regularized yield function, and compressive strength [25]:

$${\sigma }_y(P_0^{*},{\dot{\epsilon }}_p,{\epsilon }_p^{*})=f_c{\sigma }_y^{*}(P_0^{*},F_r({\dot{\epsilon }}_p),{\epsilon }_p^{*})R_3({\theta }_1,P_0^{*})$$

(9)

where $F_r$ is the dynamic increase factor; $R_3$ is the Lode angle factor; ${\sigma }_y^{*}$ is a function that normalizes the yield; ${\epsilon }_p^{*}$ is the effective plastic strain; $P_0^{*}$ is the normalized pressure, $P_0^{*}=P_0/f_c$; ${\dot{\epsilon }}_p$ is the strain rate; ${\theta }_1$ is the Lode angle; $P_0$ is the hydrostatic pressure, $P_0=\left({\sigma }_1+2{\sigma }_3\right)/3$; ${\sigma }_1$ and ${\sigma }_3$ is the maximum and minimum principal stress, in that order; and $f_c$ is the uniaxial compressive strength.

The pressure in the RHT model is described using the Mie–Gruneisen equation of state, which includes a p-α compaction relation and a polynomial Hugoniot curve. The EOS is defined as follows [26]:

$$P_R={\displaystyle \frac{(B_{\Gamma 0}+B_{\Gamma }\mu ){\alpha }_0{\rho }_{0}e+A_1\mu +A_2{\mu }^2+A_3{\mu }^3}{\alpha }}$$

(10)

where ${\alpha }_0$ is the porous material’s initial porosity; ${\rho }_0$ is the porous material’s initial density; $P_R$ is the EOS pressure; $A_1$, $A_2$, $A_3$ are the polynomial coefficients; $\mu$ is volumetric strain; and $B_{\Gamma 0}$ and $B_{\Gamma }$ are material constants.

In this study, the Barre granite was chosen. As can be seen in

Table 1. Parameters that describe the properties of the rock material.

Parameter	Value	Parameter	Value
Erosion plastic strain EPSF	2	Relative shear strength $F{S}^{\ast }$	0.18
Parameter for polynomial EOS $B_0$	1.22	Hugoniot polynomial coefficient $A_3$ (GPa)	21.29
Mass density RO (kg/m3)	2660	Break tensile strain rate ET	3 × 1025
Elastic shear modulus SHEAR (GPa)	21.9	Reference tensile strain rate EOT	3 × 10−6
Parameter for polynomial EOS $B_1$	1.22	Reference compressive strain rate EOC	3 × 10−5
Parameter for polynomial EOS $T_1$ (GPa)	25.7	Volumetric plastic strain fraction in tension PTF	0.001
Compressive strength $FC$ (MPa)	167.8	Tensile strain rate dependence exponent BETAT	0.036
Minimum damaged residual strain EPM	0.015	Compressive strain rate dependence exponent BETAC	0.032
Gruneisen gamma GAMMA	0	Break compressive strain rate EC	3 × 1025
Lode angle dependence factor $B$	0.01	Damage parameter $D_1$	0.04
Residual surface parameter AN	0.62	Tensile yield surface parameter $G{T}^{\ast }$	0.7
Failure surface parameter $A$	2.44	Shear modulus reduction factor XI	0.5
Residual surface parameter AF	0.25	Crush pressure PEL (MPa)	125
Relative tensile strength $F{T}^{\ast }$	0.04	Damage parameter $D_2$	1
Hugoniot polynomial coefficient $A_1$ (GPa)	25.7	Compressive yield surface parameter $G{C}^{\ast }$	0.53
Parameter for polynomial EOS $T_2$	0	Initial porosity ALPHA	1
Hugoniot polynomial coefficient $A_2$ (GPa)	37.84	Compaction pressure PCO (GPa)	6
Failure surface parameter $N$	0.76	Porosity exponent NP	3
Lode angle dependence factor $Q_0$	0.68

, Xie’s paper provided some parameters for the RHT model, and others were determined using trial and error [17].

3.2. Material Model for the Explosive

A rapid chemical reaction takes place when an explosive is detonated during blasting operations. The explosive considered in this study was pentaerythritol tetranitrate (PETN). The choice of a material model for the simulation was MAT_HIGH_EXPLOSIVE_BURN, and the relationship between the volume and pressure of the explosive gas was defined through the use of the EOS_JWL equation of state, which can be expressed as follows [27]:

$$P=A(1-{\displaystyle \frac{\omega }{R_{1}V}})e^{-R_{1}V}+B(1-{\displaystyle \frac{\omega }{R_{2}V}})e^{-R_{2}V}+{\displaystyle \frac{\omega E}{V}}$$

(11)

where $P$ is the detonation pressure; $A$, $B$, $R_1$ and $R_2$ are the material constants selected in the experiment; $E$ is the amount of internal energy contained in each unit volume; and $V$ is the relative volume.

Table 2. Material and JWL EOS parameters for PETN.

Density/(kg/m3)	Velocity of Detonation/(m/s)	${\mathit{P}}_{\mathit{cj}}$/GPa	$\mathit{A}$/GPa	$\mathit{B}$/GPa	${\mathit{R}}_1$	${\mathit{R}}_2$	$\mathit{\omega }$	${\mathit{E}}_0$/GPa
1320	6690	16	575	22.4	5.73	1.81	0.276	7.42

displays the parameters obtained from Banadaki’s paper [28].

3.3. Calibration of Simulation Parameters

To assess the precision of the RHT model, a comparison was made between the outcomes of physical testing carried out by Banadaki [28] and the numerical simulation results. The physical test is shown in Figure 5a to comprise a cylindrical sample of Barre granite measuring 144 mm in diameter and 150 mm in height, with a centrally located blasthole of 6.45 mm in diameter. To prevent explosive gases from leaking, a copper tube (6.45 mm for outer diameter and 5.15 mm for inner diameter) was inserted into the blasthole. Air and polyethylene were employed as the coupling medium. As depicted in Figure 5b, an equivalent planar numerical model was constructed for validation with the test results. This model comprised five materials: the PETN explosive, polyethylene, air, copper, and Barre granite. The ALE algorithm was used to simulate the former three materials, while the copper and Barre granite were simulated using the Lagrangian descriptions. The copper tube was modeled with a plasticity kinematic material model, the parameters of which have been presented in Koneshwaran et al. [29], as can be seen in

Table 3. Material parameters for copper.

Density/(kg/m3)	Young’s Modulus/GPa	Poisson’s Ratio	Yield Stress/MPa	Tangent Modulus/MPa	$\mathit{P}$	$\mathit{C}$/s−1	$\mathit{\beta }$
8330	138	0.36	390	110	5.286	1.346 × 106	0

.

According to Li et al. [16], the damage threshold was set to 0.2. To ensure a more accurate comparison between the experimental and numerical results, the rock elements that were severely damaged beyond a certain threshold were excluded from the analysis. Figure 6 presents the crack pattern of the Barre granite after blasting. Figure 6a,b shows the final image obtained from pictures taken under high-intensity UV light and the experimental crack pattern based on Banadaki’s results [28], respectively. Figure 6c illustrates the spatial distribution of cracks at a damage level of D = 0.2, and Figure 6d depicts the simulated cracks pattern. These images illustrate that the high-intensity shock waves generated by the explosive detonation formed a compressive crushed zone near the blasthole. The shock waves dissipated into stress waves during propagation, with a strength exceeding that of the rock’s tensile strength, thereby forming a cracked zone. Reflection of the compressive stress waves at the free boundary of the specimen led to the generation of tensile stress waves and the consequent formation of a spalling damage zone. Overall, the simulation outcomes concurred with the experimental results, which served to authenticate the RHT model.

4. Numerical Analysis

A 3D numerical model was constructed to provide a visual assessment of the behavior of rock to blast loading under the action of NIS distributed along the blasthole axis. The research results can visually assess the impact of stress on the penetration of horizontal cracks and the extent of vertical cracks within the coal pillar, thereby providing guidance for determining the spacing of blastholes and the scheme for shallow-borehole crack filling in the coal pillar.

4.1. Numerical Model Set-Up

As illustrated in Figure 7, based on the mechanical model, an LS-DYNA model was created with a 4 m × 4 m × 12 m geometry. The x and y axes were positioned horizontally, while the z axis was positioned vertically, with the coordinate origin situated at the bottom of the blasthole. The y and z axes were perpendicular to the blasthole axis, while the x axis was aligned with it. The model comprised three materials: rock, explosive, and air. A 12 m long blasthole with a 0.048 m diameter was drilled in the center of the rock, and a 0.04 m diameter explosive was loaded into the hole for a charge length of 12 m. The average mesh size of the hexahedral meshes was set to 0.01 m. To eliminate the influences of reflected stress waves, non-reflecting boundaries were established on the model’s surfaces. The simulation of the explosive and air was carried out using the ALE algorithm, while the Lagrange algorithm was employed to model the rock material. Finally, both monitoring points A and B were placed 0.4 m from the blasthole’s center, with point A positioned horizontally and point B positioned vertically.

In this study, stress initialization was considered crucial to carry out the simulations accurately. The purpose of stress initialization is to set a pre-defined stress state in the numerical model, which represents the pre-existing in-situ conditions before blasting. In the first step, the NIS distributed along the blasthole axis were applied to the numerical model to simulate the in-situ stress conditions. The calculation results of this stress initialization were output into a dynain file using the keyword *INTERFACE_SPRINGBACK_LSDYNA. This file captures the stress distribution within the model, ensuring that the subsequent simulations start from a realistic stress state. In the second step, the dynain file was input into the model using the keyword *INCLUDE, and the dynamic analysis was performed. This analysis involved the inclusion of both explosive and air materials to accurately model the blast process. By initializing the stress state, the simulation can more accurately predict the behavior of the rock mass and the effectiveness of the blast, providing valuable insights into the stress redistribution and potential fracturing patterns post-blast.

4.2. Discussion on the Side Abutment Pressure

Two sets of working conditions were simulated using the numerical model developed in Section 4.1: in Case 1, without stress was applied to the model before blasting, i.e., ${\sigma }_h$ = ${\sigma }_v$ = 0; in Case 2, the curve of side abutment pressure in gob obtained by the theoretical calculation was simplified to Equation (12), in which $x$ is the distance from the bottom of the blasthole, ranging from 0 m to 12 m. Then, the NIS distributed along the blasthole axis were applied to the boundary of the model.

$${\sigma }_h={\sigma }_v=-1.25x+25$$

(12)

Following stress initialization, a local coordinate system was employed to obtain the distribution of hoop stress surrounding the blasthole in Case 2 prior to blasting. Four cross sections perpendicular to the blasthole axis were taken, and the distances between cross sections $S_1$, $S_2$, $S_3$, and $S_4$ and the bottom of the blasthole were 12, 8, 4, and 0 m, respectively, and the corresponding confining pressure was varied as 10, 15, 20, and 25 MPa. Figure 8 illustrates the hoop stress distribution around the blasthole for each cross section due to the influence of static stress, where a negative value presents compressive stress. Under isotropic in-situ stresses, the hoop stresses were all compressive with a circular distribution. Furthermore, the hoop compressive stress decreased sharply with increasing distance from the blasthole’s center, tending to ${\sigma }_v$. As the side abutment pressure in the gob increased, the hoop compressive stress gradually increased for locations equidistant from the blasthole’s center. The hoop compressive stresses on the blasthole wall for $S_1$, $S_2$, $S_3$, and $S_4$ were 20, 30, 40, and 50 MPa, respectively, all of which were equivalent to 2 ${\sigma }_v$; the hoop compressive stresses in the distant rock were 10, 15, 20, and 25 MPa, respectively, all of which were equivalent to ${\sigma }_v$. The hoop stress distribution supported the theoretical calculations presented in Section 2.2 and confirmed the feasibility of the stress initialization used.

The damage distribution of rock for the cross sections along the blasthole axis under the two considered working conditions is illustrated in Figure 9. In Case 1, the rock damage exhibited a uniform distribution within approximately 130 cm from the blasthole axis. In Case 2, the rock damage was unevenly distributed, and the extent of damage gradually declined with the increase in NIS distributed along the blasthole axis. The damage ranges at the orifice, and the bottom of the blasthole were 100 cm and 65 cm from the blasthole axis respectively, which was not conducive to the fragmentation of rock at the bottom of the blasthole. Furthermore, both damage ranges were smaller than those under the stress-free condition (Case 1). Therefore, it can be inferred that the NIS distributed along the blasthole axis inhibit the extension of rock damage, and the inhibiting effect rises with increasing stresses. For gob-side entry under the overhanging hard roof, the side abutment pressure affects the horizontal crack propagation of the blasthole. Therefore, to ensure optimal penetration, the spacing of blastholes should be estimated based on the extent of blast-induced cracks under the maximum confining pressure.

To analyze the effects of NIS distributed along the blasthole axis on the damage induced by blasting, a cross section $S_0$ perpendicular to the blasthole axis was taken at any location in Case 1, and four cross sections $S_1$, $S_2$, $S_3$, and $S_4$ perpendicular to the blasthole axis were taken in Case 2, consistent with the locations employed when calculating the hoop stress above. The damage distribution characteristics for cross sections are presented in Figure 10, which depicts similar damage patterns; i.e., after the explosion, the crushed zone was formed within a certain range around the blasthole, dense secondary cracks and radial primary cracks were distributed in the more distant rock, and the damage was arrayed in a circular shape with the blasthole at the center. The quantitative relationship between NIS distributed along the blasthole axis and the extent of damage around the blasthole was investigated next. The average lengths of the primary cracks for $S_0$, $S_1$, $S_2$, $S_3$, and $S_4$ were 132 cm, 105 cm, 91 cm, 67 cm, and 63 cm, which decreased by 20.5%, 13.3%, 26.4%, and 6.0% compared with the adjacent cross section in front, respectively. These results indicate that the extent of damage dropped sharply with an increase in confining pressure when it was less than 20 MPa and slowly decreased when it was greater than 20 MPa. Thus, with the increase in NIS distributed along the blasthole axis, the attenuation of the damage extent along the axis changed nonlinearly.

The hoop stresses at A for the four cross sections taken in Case 2 were extracted, as presented in Figure 11, to explore the evolution of hoop stress in rock resulting from the combination of NIS distributed along the blasthole axis and blast loading. The hoop stress evolution trends for the different cross sections were similar; i.e., before blasting, the hoop stresses for $S_1$, $S_2$, $S_3$, and $S_4$ exhibited stable curves with magnitudes of 10, 15, 20, and 25 MPa, respectively, which are equivalent to the hoop stress values induced by side abutment pressure in the gob. This indicated that the numerical model successfully applied the confining pressure. After blasting, the hoop stresses first declined to their minimum values, then rapidly increased to their maximum values before decreasing again and finally remaining stable. The peak values of hoop compressive stress at A for $S_1$, $S_2$, $S_3$, and $S_4$ exhibited a gradual upward trend with increasing NIS distributed along the blasthole axis, which were 29.5, 32.8, 36.4, and 40.3 MPa, respectively. This finding is supported by the theoretical calculation presented in Section 2.2 for a lateral pressure coefficient of 1.

4.3. Discussion on Lateral Pressure Coefficient

Section 4.2 analyzed the effects of NIS distributed along the blasthole axis on the 3D damage distribution of rock caused by a lateral pressure coefficient of 1. However, the actual geological environment is often complex, and anisotropic in-situ stresses have meaningful effects on the crushed zone characteristics and the crack patterns around blasthole. Therefore, the vertical in-situ stress was defined as the side abutment pressure in the gob at a burial depth of 400 m using the numerical model established in Section 4.1. Five different groups of lateral pressure coefficients were established by varying the magnitude of the horizontal in-situ stress, namely $\lambda$ = 0.25, $\lambda$ = 0.5, $\lambda$ = 1.0, $\lambda$ = 1.5, and $\lambda$ = 2.0.

Similar to the investigation discussed in Section 4.2, cross sections $S_1$, $S_2$, $S_3$, and $S_4$ were taken in each of the five models with varying lateral pressure coefficients. Figure 10b and Figure 12 illustrate the damage distribution around the blasthole for these cross sections. It can be deduced that when $\lambda$ ≠ 1, the primary crack tended to expand parallel to the maximum principal stress vector. To measure the degree of damage along the blasthole axis, the average lengths of primary cracks were determined in both horizontal and vertical directions for the four cross sections at different lateral pressure coefficients and used to generate variation curves (see Figure 13). When the action of all five lateral pressure coefficients, the gradual increase of damage ranges in both horizontal and vertical directions of each cross section was observed from the bottom of the blasthole to the orifice, which was attributed to the Influence of NIS distributed along the blasthole axis. The result corresponds to the investigation presented in Section 4.2. Therefore, the average length of the primary cracks should be considered according to the maximum confining pressure when determining the blasthole spacing. When the vertical loading remained constant and $\lambda$ increased from 0.25 to 2, the changes of the damage ranges in the horizontal direction stayed within 10 cm, while the damage ranges in the vertical direction of $S_1$, $S_2$, $S_3$, and $S_4$ decreased by 64, 58, 51 and 49 cm respectively. In other words, as the lateral pressure coefficient increases, the damage in the vertical direction of the blasthole shows a significant attenuation trend. However, for non-directional blasting, the expansion of vertical blast-induced cracks may affect the stability of the coal pillar (Figure 14). If necessary, shallow-borehole crack filling should be performed on the coal pillar, and the specific parameters depend on the distribution range of blast-induced cracks within the coal pillar. Therefore, it is essential to guide the shallow-borehole crack-filling scheme based on the influence of NIS distributed along the blasthole axis and the lateral pressure coefficient on vertical cracks.

Throughout the blasting simulations, changes in the hoop stress were monitored along both the vertical and horizontal directions of $S_1$, $S_2$, $S_3$, and $S_4$ because the lateral pressure coefficients varied. As indicated in Figure 15, the maximum hoop compressive stress was extracted at each monitoring point by the combined actions of blast loading and NIS distributed along the blasthole axis. Regardless of the lateral pressure coefficient, the maximum hoop compressive stresses at A and B gradually decreased from the bottom of the blasthole to the orifice, explaining the observed phenomenon in which the damage ranges in the vertical and horizontal directions of each cross section gradually increased as the NIS declined. When λ rose from 0.25 to 2, the maximum hoop compressive stresses at B for $S_1$, $S_2$, $S_3$, and $S_4$ increased by 19.1, 24.6, 31.9, and 33.4 MPa, respectively. In contrast, the maximum hoop compressive stresses at A for cross sections all decreased by less than 10 MPa as $\lambda$ increased from 0.25 to 2, indicating insignificant changes. For the cross section perpendicular to the blasthole axis, these laws illustrate the essence of the relationship between the damage extent around the blasthole and the lateral pressure coefficient. Comparison of the maximum hoop compressive stress at A and B of the same cross section under various lateral pressure coefficients reveals that lower hoop compressive stress is observed in the direction of maximum principal stress, which diminishes the resistance to crack propagation.

5. Discussion and Conclusions

The NIS distributed along the blasthole axis significantly affect the 3D dynamic response of rock subjected to blasting. Using a theoretical model, this study initially examined the distribution of hoop stress around the blasthole, considering NIS distributed along the blasthole axis. Then, an RHT model was calibrated using the results of blasting experiments provided in the literature. Building on this, the study investigated the impact of blast loading and NIS distributed along the blasthole axis on rock damage patterns under different lateral pressure coefficients. The findings suggest that when determining a reasonable blasthole spacing and the shallow-borehole crack-filling scheme within the coal pillar, the influences of NIS distributed along the blasthole axis and the lateral pressure coefficient should be considered. The following conclusions were obtained.

(1) When the action of isotropic and anisotropic confining pressures, for the cross section along the blasthole axis, the peak values of hoop compressive stress for locations equidistant from the blasthole’s center gradually increased with the rising of NIS. The damage extent exhibited a nonlinear decrease, which was smaller than that caused by the stress-free condition. Therefore, the extent of the cracks corresponding to the maximum confining pressure should be considered when determining the blasthole spacing due to the influences of NIS distributed along the blasthole axis.

(2) For the cross sections perpendicular to the blasthole axis, with a constant in-situ stress in the vertical direction, an increase in the lateral pressure coefficient from 0.25 to 2 caused a 19–33 MPa increase in the maximum hoop compressive stresses at the vertical monitoring points, and the corresponding damage ranges exhibited an obvious decrease of approximately 49–64 cm. However, the peak values of hoop compressive stress at the horizontal monitoring points indicated insignificant changes, about 4 MPa–9 MPa, and the variations of the corresponding damage ranges were within 10 cm. When necessary, the shallow-borehole crack-filling scheme within the coal pillar should be developed considering the combined influence of NIS distributed along the blasthole axis and the lateral pressure coefficient.

This paper represents an initial exploration of the 3D dynamic response of rock subjected to the coupling effect of NIS distributed along the blasthole axis and blast loading. Further research is required to expand on these findings. For instance, the effect of the initial blasting position on non-uniform damage along the blasthole axis can also be considered. Field blasting tests should be carried out to validate numerical simulation results.