Abstract
This work uses a combination of simulations performed via numerical models and field observations studied the attenuation of deep-hole blasting stress waves and the evolution mechanism of cracks in a jointed rock mass. First, we conclude that the larger the joint angle is, the larger is the transmission coefficient and smaller is the fractal dimension. Second, the time difference between the peak stress difference and the maximum principal stress on both sides of the blasting hole in the horizontal direction of the rock mass with joints is relatively large, but there is no significant difference in the vertical direction. Finally, an unjointed-rock-mass model and multiple parallel joint model are established to explore the attenuation of stress waves and damage effect of multiple joint rock mass, it is concluded that the larger the angle, the smaller is the particle peak velocity and amplitude attenuation, and as the number of stress waves passing through the joints increases, the amplitude gradually decreases and the high-frequency amplitude decreases more significantly than the low-frequency amplitude. The research conclusions of this paper further reveal the damage mechanism induced by a blasting stress wave on jointed rock masses and the law of stress wave propagation and attenuation.
1. Introduction
As mining depth increases, mining activities are exposed to increasing risks of mining-induced earthquakes and rock bursts caused by elevated in situ stresses. To reduce the possibility of rock bursts and ensure safe production at the work face, several methods have been proposed to relieve such highly elevated in situ stresses, among which blasting has been identified as the most effective and the most commonly used method [1].
Blasting inevitably induces stress waves. The propagation of these stress waves depends on the presence of rock discontinuities, such as cracks or joints, which are ubiquitous in natural rock masses and in situ stress conditions [2, 3]. For example, the characteristics of discontinuities, such as joint angle, joint number, joint stiffness, or joint spacing, impose anisotropy in the direction of stress wave propagation and attenuation [4–7]. The attenuation characteristics are usually studied by comparing the frequency-spectrum evolution before and after the joint formation [8, 9]. The in situ stress conditions, such as the length and velocity of the induced cracks and the dynamic stress intensity factor at the crack tip, affect the stress wave propagation [10–12].
Numerous investigations on blasting mechanisms are available in the literature [13–17], which focus mainly on the following five aspects: (1) the damage mechanism and evolution characteristics of cracks caused by blasting [18–20]; (2) the influence of joints and microscopic parameters on the blasting effect [21–23], although there are few studies on the quantitative analysis of the damage degree of the rock mass under different joint angles; (3) the blasting-stress-wave attenuation and determination of the law and safety threshold [24], although there is still little research on the attenuation law of stress waves at high and low frequencies in multiple joint rock masses with different joint angles; (4) also, many scholars who have studied research fracturing behavior of 3d printed artificial rocks containing single and double 3d internal flaws under static uniaxial compression [25–27]; and (5) the initiation, expansion, bifurcation, and crack arrest mechanism of cracks induced by blasting [28].
For example, Jayasinghe et al. [29] used the Riedel–Hiermaier–Thoma (RHT) material model to study the effect of joints and elevated in situ stresses on the evolution of blasting damage. Their results indicate that the effect of joints and elevated in situ stresses in rock bridges have a considerable influence on the evolution of blasting-induced damage. Tao et al. [30] found that the effective dimensionality and material brittleness are the main factors that control the size of rock fragments. Furthermore, they proved that brittleness is an inherent physical property that affects material crushing. Li et al. [31] revealed the initiation and propagation processes of rock cracks under elevated in situ stresses and discussed the mechanism of blasting-induced crack evolution under different hydrostatic and nonhydrostatic pressures. Yang et al. [32] studied rock damage under the action of in situ stresses and the changes in critical peak particle velocity (PPV) and stress release time. Additionally, the critical PPV of blasting-induced damage in prestressed rock masses was studied numerically. Zhu et al. [33] studied the propagation of tensile stress waves in jointed rock masses and evaluated their influence on the response, stability, and support of underground chambers. Liu et al. [34] analyzed the influence of the joint distance from the blasting hole, joint length, joint number, and joint angle on a fractured rock mass. Li et al. [35] conducted a quantitative analysis on the interaction between obliquely incident blasting-induced P- or S-waves and a linear elastic rock and deduced a wave propagation equation to evaluate the effect of the incident wave transmission and reflection on joint stiffness. Shao et al. [36] conducted a series of blasting tests on plexiglass samples to study the crack-initiation, -expansion, and -penetration characteristics of discontinuous jointed rock masses under explosive load. Yu et al. [37] proposed a new prediction formula for calculating the attenuation parameters of blasting vibration in jointed rock masses. A parameter sensitivity study was conducted considering multiple influencing factors, such as joint angle, joint spacing, and joint stiffness. Some scholars have studied the mechanism of initiation and expansion of presplitting explosion cracks [38, 39].
Previous scholars have mainly focused on the damage mechanism of blasting cracks and the influence of joint microscopic parameters on the blasting effect. However, there is still little research on the attenuation law of stress waves at high and low frequencies in multiple joint rock masses with different joint angles. In this paper, we investigated the effect of joint angles on damage mechanism and the stress wave attenuation of jointed rock masses via a numerical simulation model and at the same time analyzed the stress changes on both sides of the joint. In order to quantify the effect of joint angles on the damage mechanism, we used the fractal dimensions to quantify the induced fracture evolution, and the results were used to quantify the effect of jointed rock masses under a blasting load [40–42]. By studying the relationships among fractal dimension, joint angle, and transmission coefficient under a blasting load for a single-jointed rock mass, the damage mechanism of a blasting stress wave on jointed rock masses and the law of stress wave propagation and attenuation are further revealed. Additionally, field observations on the working face of a mine were reported, which correspond to the results of numerical simulation, verifying the rationality of the simulations.
2. Numerical Model Setup and Calibration
Owing to the complexity of geotechnical engineering, conventional numerical simulation methods, such as the finite element method (FEM) and finite difference method, cannot solve the problems of large-scale deformation and failure of geotechnical media satisfactorily. In recent years, particle flow code (PFC) has been extensively developed and applied to the traditional geotechnical theories effectively, revealing the damage mechanism of microscopic media under complex geotechnical engineering conditions [43, 44].
The PFC particle flow software can directly simulate the explosive process. The principle of blasting simulation by this software is as follows: when blasting occurs, the particle radius is gradually enlarged, and the pressure generated is transferred to the surrounding medium with the explosion point as the center, to achieve the blasting effect. Figure 1 shows a schematic of a spherical charge explosion. The explosion can be simplified as a pulsed stress wave, which is a sine wave with equal rise and fall times. The specific expression is shown in Figure 2. In the equation shown in Figure 2, is the peak stress in the blasting hole, which is 500 MPa; is the stress wave action time, which generally has the conventional blasting action time less than 10 ms, so in this case, we take 10 ms as the value; is the duration; is the blasting gas pressure. This explosion adopts decoupled charging method, and will quickly decay after the impact on the blasting hole wall because , which is calculated through the observation of particle vibration velocity changes at the boundary between the crushed and the fractured zones, as shown in Figure 2. The rationality of the stress wave propagation under this model is verified.
2.1. Model and Boundary Condition Setup
We used the commercial particle flow software PFC2D to simulate rock fracturing processes [45] and to investigate the attenuation characteristics of stress waves induced by blasting: the rock mass that encloses a blasting hole has a single joint with various joint inclination angles (Figure 3(a)). During the simulation, the stress and particle vibration velocity are monitored at eight different points that are uniformly distributed vertically (-1~-4) and horizontally (-1~-4).
(a) Single-jointed rock mass
(b) Multiple parallel joint rock mass
The size of this simulation model is . First, 79366 particles with radius of 5 to 7.5 mm are generated, and the particle radius is proportionally enlarged to achieve the target porosity. Additionally, a blasting hole with a radius of 15 mm is generated at the center of the model (0, 0). The joint angles are set as 0°, 25°, 50°, 75°, or 90°, and the joint length is 0.12 m. The vertical stress applied to the model is 10 MPa, and the horizontal stress is 5 MPa. The model adopts a stress loading wall to apply confining pressure. The particles are connected using a linear parallel bond model, whereas the joint adopts a smooth-joint contact model. Thus, the normal force, tangential force, and moment can be transferred between particles.
To simulate the infinite medium condition, artificial boundaries (generally, including the incident boundary and the transmission boundary) are established at the model boundary to absorb the incident wave energy. A nonreflecting boundary can be simulated if the contact forces of the boundary particles are specified at the model boundary. If there is an incident wave at the boundary and considering the transmission effect, the incident amplitude needs to be doubled to prevent the energy from being absorbed and causing the amplitude to be halved; then, the contact force between particles with input stress is as follows: . Concurrently, the dispersion effect of stress wave propagation cannot be ignored so it is necessary to consider a correction coefficient to achieve the desired result. The specific expression is as follows: , where and are the correction coefficients of the P- and S-wave dispersion effects, respectively, and both take the value of 0.35; and are the P- and S-wave velocities, respectively; and are the normal and tangential movement velocities of the particle, respectively.
2.2. Calibration of the Model Parameters
The parameters of the model were calibrated using the uniaxial compressive strength (UCS) and elastic modulus of rock. Both UCS in the laboratory were 58.11 MPa, which were approximately consistent with the value of 57.10 MPa obtained from the simulations. Similarly, the elastic modulus in the laboratory was 6.620 GPa, which agreed with the value of 6.170 GPa of rock from the simulations. Figure 4 shows the calibration results. Therefore, the parameters listed in Table 1 were determined to be reasonable. Since our research direction and geological conditions are similar to the reference [34], we have both studied the damage mechanism induced by blasting in jointed rock, but different joint parameter settings have different effects on the blasting damage, which needs to be further studied.
2.3. Fracture Growth Rate and Stress Changes in PFC2D
In the right-most section of Figure 2, the expansion of the main cracks in the rock mass under the action of the explosion stress wave can be seen. The evolution characteristics of the cracks in the crushed zone and fractured zone, as well as radial main crack around the blasting hole, are fully presented, and four cracks are formed. After the F-1 main crack expanded to 0.05 m, the monitoring circle was checked every 0.3 ms to monitor the fracture growth rate and horizontal stress change at the crack initiation point. As shown in Figure 5, the F-1 crack propagation has gone through two stages. In the early stage, the fracture growth rate decreased from high to low. In the later stage, the peak stress fluctuated under the action of explosive gas.
3. Simulation Results and Discussion
3.1. Damage Effect of Single-Jointed Rock Mass
3.1.1. Crack Development and Joint Force
Figure 6 shows contour maps of the evolution of single-jointed rock mass cracks under explosive loading. The simulation results show that in the early stage of explosive loading (0–1.5 ms), the joint has a significant attenuation effect on the stress wave. When the joint angle is small, stress concentration is likely to occur on the side of the joint away from the blasting hole. When the joint angle is large, the joint shows a significant attenuation effect. Under the action of ground stress, radial microcracks were first generated at the upper and lower ends of the hole wall and no local damage occurred at the joint ends. From 4.3 ms to 10 ms, the original cracks expanded to the joint and gradually penetrated the joint.
According to the form of fracture damage under the action of the blasting stress wave, the internal damage of rock mass is mainly caused by tensile cracks. The crushed zone is formed by the shear and tensile stresses, and the number of tensile cracks is slightly larger than the number of shear cracks. The cracks in the fractured zone are mainly caused by tensile failure, and the number of tensile cracks is much larger than shear cracks. Concurrently, the number of shear cracks on the jointed side of the blasting hole is greater than that on the unjointed side.
3.1.2. Fractal-Dimension Changes
Using the MATLAB fractal-dimension-calculation program, images of the cracks under the impact of the blasting load for the five different joint inclination angles in the test from Section 3.1.1 are processed and the calculation results are shown in Figure 7.
The box-counting dimension method is also called the covering method, in which the explosive cracked rock mass is divided into a grid, containing several sides of length . By adjusting the side length () to count the number of squares occupied by the cracks, the following fractal-dimension-calculation formula can be obtained:
A series of (, ) data obtained in the process of counting the grids are drawn as a double-logarithmic-coordinate-graph function, and the absolute value of the slope is the fractal dimension of the set. According to the calculation results, the correlation coefficient () of the curves is above 0.99, indicating that the fitting effect is good and showing that the internal crack evolution of different joint inclination angles under the action of the blasting load has good fractal characteristics [46]. Figure 8 shows the variation of crack fractal dimensions under different joint inclination angles.
Figure 8 shows that the crack evolution for different joint inclination angles under the action of the blasting load has obvious fractal characteristics, and the change of fractal dimension has the following behavior. Because the fractal dimension is positively correlated with crack complexity, when gradually increases, the fractal dimension shows a nonlinear trend that first increases and then decreases. This means that there is a joint inclination that maximizes the fractal dimension. According to Figure 8, when, the fractal dimension reaches the maximum value. This is mainly because the smaller the joint inclination is, the more concentrated is the stress at the crack tip under anisotropic confining pressure. Therefore, the first crack occurs at the joint tip and gradually penetrates. It has an obvious guiding effect on the expansion of cracks in other directions. Conversely, as the joint angle increases, the reflection coefficient gradually decreases. Thus, the smaller the joint angle is, the larger is the reflection coefficient.
3.1.3. Maximum Principal Stress Change around the Blasting Hole
Figures 9 and 10 show the variation trends of the maximum principal stress in the horizontal and vertical directions of the blasting hole. The joints have a significant influence on the stress distribution in the rock mass, particularly in the nonequivalence of the maximum principal stresses (on the side with joints in the horizontal direction and on the side with no joints) and the difference in the stress stabilization time, that is, the maximum principal stress on the side with joints in the horizontal direction is significantly smaller than that on the side with no joints ( and ). However, maximum principal stress attained during the stress stabilization time measured in the vertical direction for -1 as well as -3 and -2 as well as -4. When , the stress difference between -1 and -3 is smaller. This is mainly affected by crack propagation. Concurrently, Figure 9 shows that the peak stress difference () of the monitoring points on both sides of the blasting hole is inversely proportional to the time difference () of the maximum principal stress, which reaches the relative equilibrium. A small indicates that the blasting stress wave disturbs the joint for a longer time, and the stress state of the joint surface is more likely to change under the action of an external load. The reflection coefficient gradually decreases as the angle increases, so the combined action of the reflected stress wave and the ground stress can promote the further expansion of the main crack. The expansion of the main crack has an obvious influence on the change of the maximum principal stress, and this is reflected in the degree of curve change under different joint angles.
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3.2. Damage Effect of Multiple Joint Rock Mass
Figure 11 shows the crack propagation and changes in velocity contour maps of a multiple joint rock mass under the action of an explosive stress wave. Figure 11 shows that the stress wave attenuation and concentration show different trends for different joint inclination angles. (1) By observing the 1.5−3 s velocity contour maps, as the inclination of the joint increases, the maximum velocity gradually shifts from the middle of the joint to one end of the joint and the velocity attenuation is the greatest before and after the first joint. (2) The crack propagation evolution indicates that the concentration of joint stress can cause the explosion-induced crack to expand. The larger the joint angle, the more obvious is the blocking effect on crack propagation. When the joint inclination angles are 25°, 50°, 75°, and 90°, the explosion-induced cracks penetrated three, two, one, and one joints, respectively. Therefore, the number of joints that can be penetrated increases with decrease in the joint angles. (3) The joint force in multiple joint rock mass is obviously more complicated than in a single-jointed rock mass model because it is affected by the penetration and reflection of the joint surface. The closer the joints to the blasting hole, the greater is the joint force. Additionally, the force on the first joint is much larger than that on other joints (other joints have a small difference in force), indicating that the attenuation degree of the first joint is much greater than that of other joints.
3.3. Attenuation Characteristics of Blasting Stress Wave
3.3.1. Attenuation Characteristics of Blasting Stress Wave in Unjointed Rock Mass
Figure 12 shows the vibration velocity and frequency-spectrum change of particles under the action of the blasting stress wave in an unjointed rock mass. Figure 12 shows that the particle vibration velocity gradually decreases with the increase of the distance, and the attenuations are 0.24, 0.17, and 0.11 m/s, respectively. The time difference between the peak velocities is 0.06 ms, and the maximum amplitude attenuations are 0.00136, 0.0046, and 0.0028, respectively. As the distance increases, the particle vibration velocity and amplitude attenuation become very small. For this model, the distance is relative to P-1, P-2, P-3, and P-4 stress wave attenuations and has a very small effect.
(a) Particle vibration velocity
(b) Frequency-spectrum evolution
3.3.2. Attenuation Characteristics of Blasting Stress Wave in Multiple Parallel Joint Rock Mass
To simulate the attenuation characteristics of the blasting stress wave of a rock mass with multiple parallel joints, according to Figure 3(b), four joints were established on one side of the blasting hole and monitoring points (P-1, P-2, P-3, and P-4) were arranged on both sides of the joints to obtain the vibration velocity of the particles and the frequency spectrum on both sides of the joint.
Figure 13 shows the changes in vibration velocity of the particles on both sides of the joint under the action of the blasting stress wave. The figure shows that the attenuation degree of the vibration velocity of the particles in the rock mass is different under different joint angles. When the stress wave travels through the joints, multiple transmissions and reflections will occur between the joints, which will change the transmission coefficient to some extent. The stress wave attenuation is obviously complicated. An increase in the number of joints will cause the particle vibration velocity to decrease. The greater the joint angle, the smaller is the attenuation. According to the frequency-spectrum changes under different joint angles as shown in Figure 14, it can be seen that the number of stress waves passing through the joints increases and the amplitude gradually decreases. The high-frequency amplitude decreases more significantly than the low-frequency amplitude. The main reason for this is that the joints filter out the high frequencies, so there is high frequency attenuation is more.
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In summary, the joint energy transfer coefficient increases as the joint angle increases, the stress wave attenuation is the smallest when passing through vertical joints. Therefore, in the actual process, it is necessary to ensure that the center line of blasting hole is perpendicular to the main structural surface in the surrounding rock to maximize the efficiency of the blasting energy transfer [47, 48].
4. Field Observations of P-Wave Attenuation Characteristics
The contact surface between different rock strata can be simplified into a joint. When the blasting stress wave passes through the contact surface between different rock strata, the incident angles formed are different, and the transmission and reflection phenomena like the chapter 3 (numerical simulation results) will occur. To verify the results obtained from the numerical simulation, the attenuation characteristics of the blasting stress waves generated by the working face blasting were tested, and the attenuation law of the blasting stress waves between the joints was summarized. Therefore, the working face (Figure 15) is selected to study the changes of P-wave velocity and frequency in the joints in the rock mass under different incident wave angles. The coal seam mined at this working face is the #3 coal seam, and the mining level is −430 m. A total of five geophones are arranged around the working face. Their positions and three-dimensional coordinates are shown in Figure 15 and Table 2. The range of vibration energy that can be monitored is greater than 100 J, the frequency is between 0.1 Hz and 600 Hz, and the parameters of the roof above the coal seam and the blasting strata are shown in Table 3.
According to the three-dimensional position coordinates of the geophones in Table 2, there were two joints between the blasting source and the geophones because the five geophones were all arranged in the coal seam and the blasting position was 16.85 m above the coal seam. The distances between the #1 and #3 geophones and blasting source #1 were 310.73 and 308.16 m, respectively, and a difference in the distance between them was 2.57 m. Because the propagation velocity of the stress wave in the rock mass is about 4400 m/s, the distance difference can be ignored. Similarly, the distances between the #2 and #3 blasting sources and the #1 and #3 geophones can also be ignored. According to the positional relationships among the blasting sources and the geophones, the incident angle formed by the stress wave propagating to the #1 geophone was greater than the incident angle to the #3 geophone. Therefore, according to the stress wave signals received by the two geophones, the variations of P-wave velocity and frequency under different angles between the joint and the incident wave can be studied.
Figure 16 shows the waveforms of P-waves, and Figure 17 shows the spectrum amplitude changes as monitored by the #1 and #3 geophones. The selected #1, #2, and #3 blasting sources correspond to the times of 2019-10-16 03 : 48 : 13, 2019-10-20 05 : 44 : 32, and 2019-10-22 03 : 19 : 23, respectively. The waveforms monitored by the three geophones were all very clear, and there was an intact coal and rock mass between the blasting source and the geophone. Figure 17 shows that the first arrival times of the P-wave waveforms recorded by the two geophones are different. The peak velocities at the #1 geophone for these three times were , , and m/s. The peak velocities detected by the #3 geophone were obviously less than that of #1 and measured as , , and m/s. The main reason is that compared with the #3 geophone, the incident angle formed by the blasting source propagating to the #1 geophone is larger. The larger the incident angle, the greater is the transmission coefficient. Therefore, the peak velocity detected by the #1 geophone is greater than that detected by the #3 geophone. According to Figure 13, the greater the incident angle, the greater is the amplitude. Although the incident angle is different, the high-frequency amplitude decreases more significantly than the low-frequency amplitude. When the stress wave passes through the joints, it fully reflects how the different incident angles are significantly different about stress wave attenuation. The high-frequency attenuation is relatively large, which is consistent with the numerical simulation results. Thus, in the field, making the center line of the blast hole perpendicular to the main structural surface in the rock can maximize the efficiency of blasting energy transmission.
(a) 2019-10-16 03 : 48 : 13
(b) 2019-10-20 05 : 44 : 32
(c) 2019-10-22 03 : 19 : 23
5. Conclusions
(1)The growth rate of cracks shows a high–low–high trend, whereas the peak stress change is the opposite, and it is concluded that the larger the joint angle is, the larger is the transmission coefficient and smaller is the fractal dimension. The number of tensile cracks in the crushed zone is slightly greater than the number of shear cracks, whereas crack type changes in the opposite direction in the fractured zone. The time difference between the peak stress difference and the maximum principal stress on both sides of the blasting hole in the horizontal direction of the rock mass with joints is relatively large, but there is no significant difference in the vertical direction(2)The stress wave attenuation of a jointed rock mass presents an obvious complexity. The larger the angle, the smaller is the particle peak velocity and amplitude attenuation. As the number of stress waves passing through the joints increases, the high-frequency amplitude decreases more significantly than the low-frequency amplitude(3)According to the field analysis, the larger the joint angle, the smaller is the P-wave attenuation, which verified the results of the numerical simulation. That is, the smaller the angle between the joints and the horizontal plane (), the greater is the influence of the joint on the stress wave and the high-frequency amplitude reduction is obviously greater than that of the low-frequency. Therefore, in the actual process, it is necessary to ensure that the blasting hole center line should be perpendicular to the main structural surface in rock to maximize the efficiency of blasting energy transfer
Data Availability
The data in this manuscript are available from the authors.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
We gratefully wish to acknowledge the collaborative funding support from the National Natural Science Foundation of China (51574225).