Renshu Yang, Jinjing Zuo, Liwei Ma, Yong Zhao, Zhen Liu, and Quanmin Xie, Analysis of explosion wave interactions and rock breaking effects during dual initiation, Int. J. Miner. Metall. Mater., 31(2024), No. 8, pp.1788-1798. https://dx.doi.org/10.1007/s12613-024-2830-y
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Cylindrical charge blasting has a significant role in practical engineering, and the excitation stress field is the key to influence the blasting effect. The location of the detonated point influences the distribution of the explosion stress field and the final rock breaking effect. The frequently used detonated-point layouts are forward, reverse, and double-end detonations [1–4], of which double-end-detonated explosives completely explode in a shorter time than the other types. It is imperative to investigate the explosion wave process and superimposed stress field distribution of double-end-detonated explosives. The influence of the detonated-point location on the blasting effect can be explored by using small-sized model laboratory tests and numerical simulations. Zhang et al. [5–7] developed a multiprimer initiation blasting method based on the shock wave collision principle and used it in a field engineering scenario. They found that the problem of pillarless sublevel caving of a suspended roof can be effectively solved by multiprimer initiation. Leng et al. [8–9] analyzed the tension and compression shear failure zones in an exploded medium under different initiation modes and found that the spatial distribution of the explosion energy can be adjusted by changing the number of initiation points in practical blasting engineering. Onederra et al. [10] coupled their developed hybrid stress blasting model with experiments and observed that the damage range of the bottom of a blast hole is significantly smaller than that of its mouth when bottom initiation is adopted. Liu et al. [11] independently developed a set of tension and compression damage models with finite element software and then assessed the influences of various initiation point positions on bench blasting. Gao et al. [12–13] investigated the crushing effects of rock at different initiation points and measured that the blasting vibration of the bottom initiation was 61.3% smaller than that of the top initiation. Miao et al. [14–15] proposed a novel method named the symmetric bilinear initiation system based on the shock wave collision principle, which was found to increase the effect of rock fragmentation. Haeri [16] examined the effects of crack positions on the fracturing path in the bridge areas of double-cracked beam specimens.
To explore an explosion wave field, Zuo et al. [17–18] analyzed the propagation characteristics of the explosion shock waves and gases during the end and center initiations of a cylindrical charge and found that the damage range of the medium in each initiation mode was approximately five times the blast hole diameter and analyzed the explosion wave characteristics under an eccentric decoupled charge. Yang and Zuo [19] explored the explosion wave field propagation of a shaped charge explosion, determined the evolution characteristics of the explosion shock waves in the shaped and vertical directions, and found that the intensity of the explosion shock wave in the shaped direction was 2.3 times that in the vertical-shaped direction. Gerasimovet and Trepalov [20] employed the background-oriented schlieren technique to probe the movement law of an explosion shock wave, obtained a reference to observe the propagation of an explosion shock wave in space, estimated the explosion energy, and obtained the shock wave parameters.
Herein, we built a collision principle structure diagram of two oppositely spreading explosion shock waves based on the shock wave collision characteristics and Hugoniot curves. A high-speed schlieren experiment system was utilized, a shock wave collision experiment model was constructed, and the evolution of the explosion shock waves and gas over the entire time domain was examined. A mechanics model of the role of shock wave collision with the wall of a blast hole was designed, and force characteristic analysis of the collision superposition area was conducted. A numerical simulation analysis model was built, the development processes of the effective stress and damage of a specimen were examined, and the pressure–time history curve of the blast hole wall was constructed. Generally, the contribution of this work is to explore the explosion wave field and rock breaking effect of the two ends of the detonated explosive in the blast hole in terms of shock wave collision, stress change of the hole wall in the collision zone, and crack propagation in the collision zone.
Fig. 1 illustrates a schematic of the collision of two shock waves, where S1 and S2 propagate to the left and right, respectively. Based on the stress and particle velocity on the collision surface, the stress state in the collision is related to the intersection of the P–u Hugoniot curves.
The Hugoniot curve of a shock wave can be expressed as [1,21]:
| P=ρ0C0(u−u0)+ρ0s(u−u0) | (1) |
where P refers to the pressure, C0 refers to the sonic velocity of the medium, s refers to an empirical constant, u refers to the particle propagation velocity, u0 refers to the initial particle propagation velocity, and ρ0 refers to the initial density.
At initial time, u0 = 0, and for shock waves S1 and S2,
| {ρ0su2+ρ0C0u−P1=0ρ0su2+ρ0C0u−P2=0 | (2) |
By solving Eq. (2),
| {P1 = ρ0C0(2u1−u) + ρ0s(2u1−u)2P2 = ρ0C0(2u2−u) + ρ0s(2u2−u)2 | (3) |
When shock waves S1 and S2 pass through P = 0, Eq. (1) can be expressed as
| {P1 = ρ0C0(2u1−u) + ρ0s(2u1−u)2P2 = ρ0C0(2u2−u) + ρ0s(2u2−u)2 | (4) |
On the basis of the superposition curve, the particle propagation velocity, u3, at the collision intersection point is
| u3=u1+u2 | (5) |
Substituting Eq. (5) into Eq. (4) calculates the interface pressure P3:
| P3=ρ0C0(2u1−u3)+ρ0s(2u1−u3)2 | (6) |
| P3−(P1+P2)=2ρ0u2(C0+su1) | (7) |
Based on Eq. (3), u2<0 and C0+su1>0; thus, Eq. (7) becomes
| P3>(P1+P2) | (8) |
In particular, when P1=P2, u1=−u2, and u3=0, the following result can be obtained:
| P3>2P1 | (9) |
After the shock waves collide, the intensity of the shock wave produced on the collision surface becomes greater than the sum of the intensities of the two shock waves rather than their simple linear superposition.
Fig. 2 diagrams the shock wave convergence and stacking during the detonation of an explosive from both ends. The total energy (ET) of the explosive is the sum of the potential energy (EP) and the kinetic energy (EK) [22] in terms of energy conservation,
| ET=EP+EK=Pγ−1+12ρu2 | (10) |
where γ indicates the adiabatic factor, the first term is the potential energy (pressure energy), and the second term is the kinetic energy. Pressure energy is beneficial for improving rock breaking ability. When two shock waves propagate to each other, the particle velocity on the collision surface is u3=u1+u2=0, and the pressure increases to P3. According to the energy conservation equation, the kinetic energy is converted into potential energy with decreasing particle velocity and increasing rock breaking ability.
Fig. 3 displays the high-speed schlieren experiment system used here [23–25]. Its laser generates a group of stable lasers that pass through a beam-expanding mirror, resulting in astigmatism and pass through plane and concave mirrors, generating a group of parallel light fields. The parallel light fields pass through the plane mirror and reach the high-speed camera through a knife edge. A detonator detonates an explosive, and a synchronous controller controls the detonator in conjunction with a high-speed camera, which shoots at 100000 frames per second. The explosive was dinitrodiazophenol, which was detonated from both ends.
Fig. 4 presents the propagation of the explosion waves and gas produced by the double-end detonated explosive. At t = 10 μs, a high-voltage electric spark detonates the explosive. Then, two groups of explosion waves propagate in opposite directions, and each detonation wave and the charge axis form a certain angle α.
The explosion shock wave on each side extends radially and axially. At t = 30 μs, the explosion shock waves and the detonation gas separate, and the explosion energy starts to decay along the axial reverse direction. The axial direction has a higher decay rate than the radial direction. After the collision and superposition of the explosion waves, the superimposed shock wave is dumbbell shaped, which is kept during expansion. At t = 80 μs, the radial shock waves are separated from the explosion gas, demonstrating the start of the explosion energy decay.
Fig. 5 illustrates the explosion wave velocity. By taking images, the tip of an explosion shock wave can be obtained, and then, the propagation explosion shock wave velocity can be computed.
Fig. 6 is the evolution of the explosion shock wave velocity with time on both-end explosive detonation. At 0–30 μs, the explosion waves on both sides propagate along the charge direction with a propagation velocity of approximately 2500 m/s. During the experiments, the velocity of dinitrodiazophenol was less than the theoretical velocity because it was not compacted in the bulk state. After 30 μs, the explosion waves on both sides collide and superimpose, and the intensity of the superimposed shock wave in the collision region increases. The superimposed shock wave velocity in the collision region is greater than the radial shock wave velocity, which is greater than the axial shock wave velocity. The mean of the explosion wave transmission angles, α, is 60°.
The two groups of explosion waves transmitted in opposite directions produce a strong-impact shock wave in the collision superposition area, and the superimposed explosion shock wave is linear (Fig. 7). The above analysis indicates that the intensity of the superimposed shock wave on the collision surface is greater than the sum of the intensities of the two shock waves after their frontal collision. The strong superimposed shock wave on the collision surface acts on the blast hole wall, increasing its damage effect.
Fig. 8(a) and (b) displays the interaction between the explosion shock waves and the blast hole wall. The explosion shock waves are transmitted toward each other at an angle α, and the angle between each explosion shock wave front and the blast hole wall is β. The pressure angle of the blast hole wall under an explosion shock wave is δ.
| δ=90∘−β=90∘+12α | (11) |
The explosion waves after the both-end explosive detonation compress in the middle region, producing a 90° incidence angle to the blast hole wall in the superimposed region. The stress element of the blast hole wall in the superimposed area in Fig. 9 is used as an example.
Let {\boldsymbol{\sigma}} be the stress state at any point and {{\boldsymbol{\sigma}}_{{\theta }1}} (compressive stress) and {{\boldsymbol{\sigma}}_{t1}} (tensile stress) be the rightward explosion shock wave and blast hole wall stresses, respectively. Let {{\boldsymbol{\sigma}}_{{\theta }2}} (compressive stress) and {{\boldsymbol{\sigma}}_{t2}} (tensile stress) be the leftward explosion shock wave and blast hole wall stresses, respectively. Based on the stress superposition, the stress states at the blast hole wall in the final superposition area are {\boldsymbol{\sigma}}_{\textit{θ}'} (compressive stress) and {\boldsymbol{\sigma}}_{t'} (tensile stress). By vector calculation, the stress states in the superposition area are
| \left\{ {\begin{array}{*{20}{l}} {{{\boldsymbol{\sigma}} _{{\theta ^\prime }}} = \sqrt {{{\left| {{{\boldsymbol{\sigma}} _{\theta 1}}} \right|}^2} + {{\left| {{{\boldsymbol{\sigma}} _{\theta 2}}} \right|}^2} + 2\left| {{{\boldsymbol{\sigma}} _{\theta 1}}} \right|\left| {{{\boldsymbol{\sigma}} _{\theta 2}}} \right|\cos \left( {{{\boldsymbol{\sigma}} _{\theta 1}},{{\boldsymbol{\sigma}} _{\theta 2}}} \right)} }\\ {{{\boldsymbol{\sigma}} _{{t^\prime }}} = \sqrt {{{\left| {{{\boldsymbol{\sigma}} _{t1}}} \right|}^2} + {{\left| {{{\boldsymbol{\sigma}} _{t2}}} \right|}^2} + 2\left| {{{\boldsymbol{\sigma}} _{t1}}} \right|\left| {{{\boldsymbol{\sigma}} _{t2}}} \right|\cos \left( {{{\boldsymbol{\sigma}} _{t1}},{{\boldsymbol{\sigma}} _{t2}}} \right)} } \end{array}} \right. | (12) |
Due to a single and continuous explosion, theoretically, the forces of the explosion waves on both sides are equal, i.e., {{\boldsymbol{{\sigma }}}_{\theta 1}} = {{\boldsymbol {\sigma }}_{\theta 2}} = {{\boldsymbol {\sigma }}_\theta } and {{\boldsymbol {\sigma }}_{{t}1}} = {{\boldsymbol {\sigma }}_{t2}} = {{\boldsymbol {\sigma }}_t} ; thus,
| \left\{ {\begin{array}{*{20}{l}} {{{\boldsymbol{\sigma}} _{{\theta ^\prime }}} = \sqrt {{{\left| {{{\boldsymbol{\sigma}} _{\theta 1}}} \right|}^2} + {{\left| {{{\boldsymbol{\sigma}} _{\theta 2}}} \right|}^2} + 2\left| {{{\boldsymbol{\sigma}} _{\theta 1}}} \right|\left| {{{\boldsymbol{\sigma}} _{\theta 2}}} \right|\cos \left( {{{\boldsymbol{\sigma}} _{\theta 1}},{{\boldsymbol{\sigma}} _{\theta 2}}} \right)} }\\ {{{\boldsymbol{\sigma}} _{{t' }}} = \sqrt {{{\left| {{{\boldsymbol{\sigma}} _{t1}}} \right|}^2} + {{\left| {{{\boldsymbol{\sigma}} _{t2}}} \right|}^2} + 2\left| {{{\boldsymbol{\sigma}} _{t1}}} \right|\left| {{{\boldsymbol{\sigma}} _{t2}}} \right|\cos \left( {{{\boldsymbol{\sigma}} _{t1}},{{\boldsymbol{\sigma}} _{t2}}} \right)} } \end{array}} \right. | (13) |
We take the explosion shock wave propagating to the left in Fig. 10 as an example. This explosion shock wave is incident on the blast hole wall, producing an included angle δ with the blast hole wall. Similarly, the included angle of the explosion shock wave propagating to the right with the blast hole wall is δ.
Furthermore, it can be concluded that
| \cos ({{\boldsymbol{\sigma}}_{\theta 1}},{{\boldsymbol{\sigma}}_{\theta 2}}) = \cos (\text{π} - 2\delta ) = \cos \alpha | (14) |
Substituting Eq. (14) into Eq. (13) obtains the relationship between the superimposed and incident stresses:
| \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} {{\boldsymbol{\sigma }}_{{\theta ^\prime }}} = \sqrt {2{{\left| {{{\boldsymbol{\sigma }}_\theta }} \right|}^2} + 2{{\left| {{{\boldsymbol{\sigma }}_\theta }} \right|}^2}\cos \left( {{{\boldsymbol{\sigma }}_\theta },{{\boldsymbol{\sigma }}_\theta }} \right)} = \\ \sqrt {2{{\left| {{{\boldsymbol{\sigma }}_\theta }} \right|}^2} + 2{{\left| {{{\boldsymbol{\sigma }}_\theta }} \right|}^2}\cos \alpha } = \sqrt {2 + 2\cos \alpha } \times {{\boldsymbol{\sigma }}_\theta } \end{array}\\ \begin{array}{l} {{\boldsymbol{\sigma }}_{t'}} = \sqrt {2{{\left| {{{\boldsymbol{\sigma }}_t}} \right|}^2} + 2{{\left| {{{\boldsymbol{\sigma }}_t}} \right|}^2}\cos \left( {{{\boldsymbol{\sigma }}_t},{{\boldsymbol{\sigma }}_t}} \right)} = \\ \sqrt {2{{\left| {{{\boldsymbol{\sigma }}_t}} \right|}^2} + 2{{\left| {{{\boldsymbol{\sigma }}_t}} \right|}^2}\cos \alpha } = \sqrt {2 + 2\cos \alpha } \times {{\boldsymbol{\sigma }}_t} \end{array} \end{array}} \right. | (15) |
where α indicates the included angle of the explosion wave head, i.e., the superimposed and initial stress intensities have a certain coefficient relationship that is associated with the included angle of the explosion wave transmission. In the schlieren experiments, α = 60°, {\boldsymbol{\sigma }}_{\theta '}=\sqrt{\text{3}}{\boldsymbol{\sigma }}_{\theta } , and {\boldsymbol{\sigma }}_{t'}=\sqrt 3{\boldsymbol{\sigma }}_{t} .
It has been revealed that poly(methyl methacrylate) (PMMA) has similar dynamic fracture characteristics to rock materials under dynamic loading [26–28]. PMMA was employed as an experimental material in the model tests. The model had a length of 400 mm and a width of 300 mm. The cylindrical blast hole had a length of 100 mm and a width of 4 mm. The explosive used was dinitrodiazophenol, and two initiation points were placed at both ends of the charge, as shown in Fig. 11(a).
Fig. 11(b) and (c) presents the experimental results and binarization diagram of an explosion crack. Large-scale radial explosive cracks were generated on the blast hole wall in the superimposed area during both-end explosive detonation, and the crack widths in the radial fracture expansion area were different. The explosive crack width in the strong-impact region was approximately 3 times that in the weak-impact region, approximately 11 times the blast hole diameter, and approximately 5% of the blast hole length. From Fig. 12, this difference is because the explosion products superimpose and form a wide fracture zone in the fracture region of the strong-impact loading region. With the reduction of the superimposed stress and under the action of the explosion gas wedge, the explosion crack width is reduced.
LS-DYNA, a finite element numerical simulation software program, was employed to develop a test model in a 1:1 ratio. Because of the large deformation and displacement in the simulation of an explosion problem, to avoid calculation failure resulting from grid distortion, we employ the arbitrary Lagrange–Euler algorithm. The Euler algorithm was employed for the explosive, which was simulated using the *MAT_HIGH_EXPLOSIVE_BURN keyword. The Jones–Wilkins–Lee equation of state was applied to simulate the pressure–volume relationship in the explosion process [29]:
| P = A\left( {1 - \frac{\varpi }{{{R_1}}}} \right){{\mathrm{e}}^{ - {R_1}V}} + B\left( {1 - \frac{\varpi }{{{R_2}}}} \right){{\mathrm{e}}^{ - {R_2}V}} + \frac{{\varpi {E_0}}}{V} | (16) |
where P indicates the pressure, V indicates the relative volume, R1, R2, A, B, and \varpi indicate the constants, i.e., R1 = 4.8, R2 = 1.2, \varpi=0.3 , A = 405 GPa, and B = 0.43 GPa, and E0 is the initial internal energy density.
Herein, PMMA and organic glass were simulated using the *MAT_JOHNSON_HOLMQUIST_CERAMICS keyword and the Lagrange algorithm, respectively. In the *MAT_JOHNSON_HOLMQUIST_CERAMICS material model, D is the cumulative damage and primarily the plastic strain accumulation [30–31].
| D = \sum {\frac{{\Delta \mathop \varepsilon \nolimits^{\text{p}} }}{{\Delta \mathop \varepsilon \nolimits_{\text{f}}^{\text{p}} }}} | (17) |
where \Delta \mathop \varepsilon \nolimits^{\text{p}} refers to the accumulated plastic strain in the loading process and \Delta \mathop \varepsilon \nolimits_{\text{f}}^{\text{p}} refers to the plastic strain during fracture under pressure P. Plexiglass plate has parameters of density \rho = 2.53 g·cm−3, shear modulus G = 0.304 GPa, complete strength parameter A = 0.93, crack strength parameter B = 0.088, fracture strength parameter M = 0.35, strength index N = 0.77, elastic limit HEL = 0.0595 GPa, strain rate C = 0.03, and failure criterion Fs = 0.8.
HYPERMESH preprocessing software was employed for model establishment, mesh division, algorithm definition, and boundary condition setting, as presented in Fig. 13. Quadrilateral (quads) and triangular (trias) elements were employed to divide the grid into front units. Different detonation points were set based on the coordinate position using the *INITIAL_DETONATION keyword.
Fig. 14 illustrates the Mises equivalent stress propagation process during both-end explosive detonation. In the stress cloud diagram, the stress field intensities shown in red, green, and blue successively decrease. The Mises stress waveform is triangular before the superimposed collision and extends primarily along the charge section and radial directions. At t = 10 μs, the effective stresses superpose in the middle of the blast hole, and the stress intensity on superposition reaches a maximum. At t = 15 μs, an effective stress with a higher intensity is observed in the superposition area, which presents a rhomboid expansion mode and finally produces an oval distribution.
Fig. 15 presents the medium damage around the blast hole. The red and green areas indicate the crushing and crack areas, respectively. The numerical simulation results align with the model experimental results and demonstrate that large-scale radial explosive cracks are produced in the central superimposed area. Before 10 μs, with the detonation transfer, an outstanding crushing zone is found around the blast hole under the strong explosive dynamic load, and explosive cracks start to emerge at the end of the blast hole. At t = 10 μs, explosive cracks start to emerge in the superposition region. Due to the strong dynamic loading resulting from the superposition, the crack size is large at the start of the superposition, which aligns with the model test results. After 24 μs, the crack propagation width in the superimposed region decreases because of stress attenuation in the superimposed region.
On the basis of the model test and numerical simulation results, large-scale radial cracks are generated in the superimposed area, as depicted in Fig. 16. With the initiation of superimposed cracks on the blast hole wall, crack propagation shows a change from wide to narrow cracks. This is caused by the strong dynamic loading on superposition, which forms a strong-impact region in a certain area of the blast hole wall, resulting in the instantaneous destruction of the medium in that area and the formation of a large-width crack. Then, due to the attenuation of the explosion, an excessive crack area is produced, and the crack propagation width is reduced.
Fig. 17(a) presents the arrangement of the measuring points on the blast hole wall along the axial direction. Considering the explosion distribution symmetry during both-end explosive detonation, the measuring points are set from one end of the blast hole at 5-mm intervals, with the first point being measuring point 1 and the collision position in the middle being measuring point 11. Fig. 17(b) displays the pressure distribution of the blast hole wall along the axial direction. The peak pressure value of each measurement point was extracted. After the detonation at the initiation points, each peak pressure value at the blast hole wall quickly increases and then gradually stabilizes with the propagation of the explosion stress waves. At each detonation point, the blast hole wall pressure is the minimum, and the peak pressure is 190 MPa. In contrast, the blast hole wall pressure is the maximum, and the peak pressure is 540 MPa at the charge center. The pressure in the central superposition area is approximately 2.84 times greater than that at the initiation ends.
To more accurately examine the pressure of the blast hole wall in the superposition area, four measuring points were arranged between measuring points 10 (corresponding to 45 mm of charge) and 11 (corresponding to 50 mm of charge) (Fig. 18(a)). The extracted pressure curve of the blast hole wall in this section is displayed in Fig. 18(b). The blast hole wall pressure gradually increases with a decrease in distance from the measuring point to the center. The superposition area of the blast hole wall is significantly improved in the center 5-mm range, accounting for 5% of the total charge length.
The points around the blast hole at which the peak pressure is extracted are depicted in Fig. 19(a), and the fitted peak pressure–blast hole length relationship is presented in Fig. 19(b). The distribution of the pressure shows a valley pattern, and the distribution characteristics of the peak pressure of the foundation are divided into three areas: nonsuperimposed, weakly superimposed, and strongly superimposed areas, which account for 80%, 15%, and 5% of charge length, respectively. The average peak pressures are 274, 341, and 504 MPa, respectively. The mean peak pressures in the strongly superimposed area are 1.48 and 1.84 times greater than those in the weakly superimposed and nonsuperimposed areas, respectively.
This work aims to explore the explosion wave field and rock breaking effect of the two ends of the detonated explosive in a blast hole in terms of shock wave collision, stress change of the hole wall in the collision zone, and crack propagation in the collision zone.
(1) The intensity of the superimposed shock wave on the collision surface was greater than the sum of the intensities of the colliding shock waves. At the collision site, the kinetic energy was transformed into potential energy with a decrease in particle velocity on the wave front, and the wave front pressure increased.
(2) The expansion form of the superimposed shock wave was dumbbell shaped, and when the two shock waves collided and superimposed, the intensity in the collision area rose. The shock wave velocity in the collision area was greater than the radial shock wave velocity, and the average propagation angle of the explosion shock waves was approximately 60°.
(3) A stress model of the blast hole wall in the superimposed area was established. A fitted relationship between blast hole wall stress and explosion wave propagation angle in the superimposed area was plotted. Under the experimental conditions, the superimposed stress of the blast hole wall was approximately 1.73 times the single-explosion wave incident stress.
(4) The results of the model test and numerical simulations revealed that large-scale radial fracture cracks were produced on the blast hole wall in the superimposed area. The width of the large-scale radial fracture cracks resulting from the strong impact was approximately 5% of the blast hole length. The mean peak pressures in the strongly superimposed area were approximately 1.48 and 1.84 times greater than those in the weakly superimposed and nonsuperimposed areas, respectively.
This research was financially supported by the National Natural Science Foundation of China (Nos. 52208384 and 51934001), the National Key Research and Development Program of China (No. 2021YFB3401501), the State Key Laboratory of Precision Blasting and Hubei Key Laboratory of Blasting Engineering, Jianghan University (No. PBSKL2022C05).
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