# Rock Dynamic Fracture Characteristics by Mini-Linear Shaped Charge Jet Penetration: A Case Study

Authors: Zhenyang Xu; Xin Liu; Hongyi Mo; Lianjun Guo; Jun Yang
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

An insignificant number of studies have focused on employing penetration using mini-linear shaped charge jets for directionally controlled splitting of massive rocks. In this study, we adopt a numerical calculation method to simulate the penetration and formation of the main splitting surface of a concrete specimen, considering a wedge angle of 45° in both processes. The surface on the principal axial plane is found to split first due to linear jet penetration. In the case of single primary-plane splitting, cracks appear at both ends of the long axis of the penetration crack and the splitting surface extends diagonally from the center of the penetration. A transverse crack separates the splitting surface and the radial-fracture surface, and the degree of fracture decreases along the direction of the height of the specimen. Finally, a realistic physical model demonstrating penetration using a mini-linearshaped charge jet is established. It is a rapid and safe blasting technology to handle hazardous massive rocks during emergency rescuing or mining.


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## Body

## 1. Introduction

Massive rocks displaced by landslides, collapses, mudslides, and earthquakes typically cause roadblocks and obstruct traffic. Such displacement severely affects construction progress and threatens the safety of personnel. To ensure the security of people, it is important to effectively handle hazardous massive rocks. To this end, an efficient and safe technique should be developed to directionally split and blast these rocks for their rapid removal. Directional fracture blasting employs the explosionlike charge effect of a slotted cartridge [1, 2] and the mechanism of in-hole directional breaking [3, 4] to achieve splitting of these rocks. However, few studies have focused on directionally controlled splitting by linear jet penetration on the exterior of massive rocks, which employs a rapid and safe blasting technology and has the potential to replace traditional drilling and blasting methods; the risks associated with blasting can also be controlled.The shaped charge effect was proposed by Birkhoff et al. [5] based on the hydrodynamic theory. The use of a shaped charge jet significantly enhances the local effect of the explosion in one direction. The most observable effect of the shaped charge jet can be achieved only when the physical and mechanical properties of the metal match. The effectiveness of the shaped charge jet is indicated by its penetration depth. Chanteret [6] studied the relationship between the charge jet density and the penetration depth and concluded that increasing the density of a fractured jet does not result in an increase in its penetration depth. Markelov [7] observed that the penetration depth of a shaped charge jet could increase upon heating the jet to a certain temperature before detonation. Haugstad [8] proposed that the compressibility of a target could reduce jet penetration depth, and Flis and Chou [9] pointed out that the porosity of a target has significant effect on penetration depth. To determine the diameter of jet penetration, Held [10] analyzed the transverse velocity of a jet during penetration. Cornish et al. [11] explored various potential degradation mechanisms to identify those that have the most significant effects on the penetration process. Rokni et al. [12] studied the process of penetration of a fractured jet and determined the factors responsible for the narrower penetration diameter of the fracture jet compared to that of a continuous jet.A linear shaped charge (LSC) can be used to construct a knifelike jet, which can penetrate a target using a wedge-shaped metallic cartridge cover. LSC is extensively applied to aerospace and military fields. In civilian blasting engineering, LSC is used in bridge demolition [13], destruction of waste ammunition [14], and mining [15]. A stress gradient is formed in the rock during linear jet penetration, exhibiting dynamic crack propagation. Wu et al. [16] observed that the primary and random secondary cracks were formed in different directions. Wu et al. [17] demonstrated that the propagation length of the crack formed in the energy-accumulation direction of the shaped charge energy is greater than that in the nonenergy-accumulation direction. Due to the concentration of the stress field, the splitting crack preferentially initiates and expands in the plane of the linear jet penetration axis. The effect of the tensile stress field is experienced at the center for a short period even after the penetration stops, and the propagation direction of the splitting crack remains constant. Furthermore, the preexisting fracture network in the rock affects crack propagation. Zhang et al. [18] used the preexisting fracture network (CCPF) to explore the effects of closed cemented natural fractures on crack propagation.Directionally controlled splitting is a promising blasting technology with notable economic and social benefits, which are attributable to its safety of operation and reliability of performance. However, the mechanism of splitting cannot be comprehensively explained based only on experiments and research. Numerical simulations using modern computer technology can facilitate the improvement and practical application of the theory. Cong et al. [19] established a fracture-propagation model that involved temperature-pressure phase coupling. Wan et al. [20] studied the fracture toughness of mode I cracks under blast load. The accuracy of the finite element method plays a significant role in the experimental-numerical analysis method. Wang et al. [21] studied the dynamic fracture propagation in rocks under impact loads. Smirnov et al. [22] focused on the instabilities in crack propagation velocity. In one study, the following observations were made: (1) at optimal blasting height, a rock specimen can be split into pieces when the wedge angle is 90°; (2) a uniform radial fracture surface can be produced when the wedge apex angle is 60; and (3) the primary plane of the specimen can be symmetrically split into two parts without generating any flying fragments when the wedge angle is 45° [23]. Numerical methods can be used to simulate splitting and analyze the dynamic fracture characteristics in rocks resulting from the penetration of mini-LSC jets. In this study, we induced a mini-LSC by top-center detonation in order to study jet penetration depth, fracture formation, and stress field in concrete specimens.


## 2. Physical Model of Mini-LSC Jet Penetration Induced by Top-Center Detonation

The volume of a penetration cavity can indicate the kinetic energy of the jet, which can help determine the basic parameters of an LSC structure. The jet kinetic energy accounts for only 10–20% of the explosive chemical energy, and the conversion of jet kinetic energy to explosive energy is different in different charge structures, as shown in Figure1. The shaped charge model shown in Figure 1(f) combines the advantages of the ones shown in Figures 1(d) and 1(e). Therefore, the model of the mini-LSC in Figure 1(f) was selected for this study. A wood shell with 5 mm thickness was used to process the LSC outer shell, because it can enhance the performance of the jet by reducing the interference of the sparse wave in detonation. In the selected structure, the top angle of the wedge-shaped cover was equal to the top angle of the wood shell. Xu et al. [23] found that the jet stretch is the longest and the jet head velocity is the highest when the wedge angle is 45°.Figure 1
Structure of the shaped charge. (a) At the highest jet energy utilization rate. (b) At the highest jet head velocity. (c) To reduce the interference of the sparse wave, the abrupt change points of the shell shape are selected at a slightly inclined position and at the apex of the shell. (d–g) Shape of the tip can cause the detonation pressure to increase uniformly and produce a localized high-pressure effect; cutting depth can be maintained within a large blasting height range [24, 25].
(a)(b)(c)(d)(e)(f)(g)To avoid significant errors due to the small size of the model specimen, all specimens were designed to have the same diameter and height. According to the concrete mechanical test standard, the concrete specimens were cured for 30 days to achieve the standard strength grade C40, with concrete strength exceeding 95%, as shown in Figure2(a); the ratio of cement, sand, stone, and water was 1 : 2:2.15 : 0.40 [26].Figure 2
Physical model of mini-linear shaped charge jet penetration induced by top-center detonation.
(a)(b)According to the frictional movement model, the resistant pressure that arises from the shaped charge jet penetration in the sample is the shearing yield stressσs. The expression of the kinematics of the convergent jet is [27, 28](1)mjsdvdt=2Sσs,where mjs is the mass of the jet. Assuming the penetration depth is the same, the contact area S of the jet with the sample is equal to the product of the penetration depth h and the penetration length l (Figure 2(b)).(2)dvdt=vdvdh.The evolution of equation (1) can be written as follows:(3)mjsvdvdh=2Sσs=2σslh.Integration of equation (3) yields(4)Ekjs=12mjsv02−12mjsv2=σslh2,where Ekjs is kinematic consumptions.


## 3. Simulation Analysis

### 3.1. Simulation Model

The MAT_HIGH_EXPLOSIVE_BURN [29] model and the JWL equation are adopted to describe the explosive:(5)P=A1−ωR1Ve−R1V+B1−ωR2Ve−R2V+ωE0V,where P is the pressure of the detonation product; A, B, R1, R2, and ω are the test fitting parameters; e is the specific energy; E0 is the initial internal energy of a unit volume explosive; ρe is the density of the explosive (HMX); VOD is the detonation velocity; PCJ is the Chapman–Jouguet pressure; and V is the relative volume, and the explosive material parameters are shown in Table 1.Table 1
Properties of the explosive (HMX) and the JWL state equation.
ρe (g/cm3)VOD (m/s)PCJ (GPa)A (GPa)B (GPa)R1R2ωE0 (J/m3)1.1991042778.37.14.21.00.310.5The JOHNSON_COOK [30] model and the Grüneisen [31] equation of state are adopted to describe the shaped charge cover. Also, the JOHNSON_COOK model is suitable for most metal materials under large deformation, high strain rate, and high temperature, the explosive formation of metal is its typical application, and the parameters of copper chosen in the numerical simulation are shown in Tables 2 and 3.Table 2
Material parameters of copper in the model.
ρm (g/cm3)G (GPa)A (MPa)B (MPa)nCmEPS08.96469002900.310.0251.090.1 × 10−5PCSPALLITD1D2D3D4D5−93030000Note:ρm is the density; A, B, and C are the constants of the material and can be determined by torsion tests under different strain rates.Table 3
State equation parameters of copper.
CS1S2S3GAMAOAV00.3941.489002.020.471.00Note:C is the intercept of the shear-compression wave velocity curve; S1, S2, and S3 are the slope coefficients of the above curve.The Johnson–Holmquist concrete (HJC) [32] is adopted to simulate concrete specimens. The large strain, high strain rate, and high-pressure effect are considered in the HJC model comprehensively:(6)σ∗=A1−D+BP∗N1−clnε•∗,where, σ∗=σ/fc, which demonstrates the ratio of actual equivalent stress to the static yield; D is the damage parameter; p∗=p/fc, which demonstrates dimensionless pressure; ε∗•=ε•/ε•0, which demonstrates dimensionless strain rate; A is the normal viscosity coefficient; B is the normal pressure hardening coefficient; and C is the strain rate coefficient. The keyword ∗MAT_ADD_EROSION was used to compute as tensile failure criterion for concrete specimens, the tensile failure strength is 4 MPa, and the parameters of the specimen model are shown in Table 4.Table 4
Mechanical parameter of the concrete.
ρ (g/cm3)G (GPa)ABCNFC (MPa)EPS02.4414.860.791.60.0070.61481 × 10−6T (MPa)EFMINSFMAXPC (MPa)UCPL (MPa)UL40.017.0160.0018000.1D1D2K1 (GPa)K2 (GPa)K3 (GPa)0.041.085171208As is shown in Figures3(a) and 3(b), a 1/2 model was established, wherein the formation of the jet was controlled by an explosive pressure wave and the air domain mesh was compacted along with the jet formation and penetration path. With charge maintained at a constant value, the formation of the main splitting surface was simulated at a wedge angle of 45° [23] in the two cases.Figure 3
The 1/2 model of numerical simulation. (a) Case 1:H = 30 mm, l = 40 mm. (b) Case 2: H = 40 mm, l = 30 mm.
(a)(b)


### 3.2. Fracture Formation

An increase in the area of the crushing zone resulted in stress wave dissipation at 80μs and a decrease in crack-propagation velocity, as shown in Figure 4(a). A linear notch is clearly visible at the site of penetration, which promotes the growth of a circumferential crack advantage at this point. A central longitudinal crack tip propagates rapidly; the propagation of the splitting surface cracks also tends to accelerate. Consequently, the crack tip moves to the circumferential crack region. Gradually, the central longitudinal crack penetrates the specimen at 130 μs; however, the profile surface crack does not reach the side of the specimen. After penetrating the specimen, the central longitudinal crack continues to propagate from the bottom of the specimen to both sides of the main fracture surface. The spalling is generated symmetrically when the splitting crack does not extend to the profile of the specimen. Elliptical crushing occurs in a small area in the middle of the bottom surface of the specimen. The propagation direction of the splitting crack tip is slightly disturbed, after which the crack continues to grow steadily and the displacement of the crack opening increases. Crack P0, which did not propagate from the surface splitting cracks, originates in the middle of the profile of the specimen at 150 μs. The tensile stress field generated due to cavity expansion leads to an increase in the propagation velocity of splitting cracks inside the specimen. The profile splitting crack reaches the boundary of the specimen and intersects with the bottom splitting crack, concluding the splitting process of the specimen.Figure 4
The process of fracture formation. (a) Case 1. (b) Case 2.
(a)(b)The fragmentation degree in Case 2 (Figure4(b)) is significantly higher than that in Case 1 (Figure 4(a)). The penetration depth increases to 17.1 cm, reducing the range of the crushing zone; the plot representing the penetration is linear. During free motion at 44 μs, the central part of the jet is unsteady due to torsion; meanwhile, particle dispersion is intensified, and the radial penetration effect of the jet is significantly enhanced in Case 1. The jet remains stable while moving, and the tail particles scatter less in Case 2. The tensile strength is also higher than that in Case 1. The maximum velocities of the jet head in Cases 1 and 2 are 3169.8 m/s and 3203.4 m/s, respectively [23]. An increase in the height of the charge position leads to an increase in the velocity and total kinetic energy of the jet head. Meanwhile, the impulse dynamic pressure and the fluid characteristics at the penetration site are enhanced, and the penetration gap narrows with the increase in penetration depth.In Case 2, the distribution of the stress field is uniform at the initial stage of penetration and the crack tip propagates in a straight line along the direction of crack initiation. The uniform distribution of the stress field is disrupted when the stress wave propagates to the boundary and undergoes reflection and superposition. The crack propagation occurs along anS-shaped path and presents a curved plot. Due to tensile stress concentration on the main fracture surface, the crack rapidly begins to propagate in the original direction after the first curved crack. A second curved crack occurs with an offset time of crack propagation longer than that of the first one. Therefore, the second curved crack is longer than the first one. The propagation path is found to be S-shaped after the crack is connected. Thus, the crack tip propagates both on the surface and the interior at the same time, and the stability of the crack tip propagation determines whether the direction of crack propagation will be offset.


### 3.3. Stress Nephogram

As shown in Figure5, the stress field is evenly distributed in the penetration stage at 30 μs, an elliptical tensile stress zone is formed at the penetration site, and the tensile stress is concentrated in the axial direction on both sides symmetrically. The stress field gradually elongates on both sides, and the radial penetration effect of the jet is enhanced. The tensile stress is concentrated to the maximum value at the end of the long axis of the jet. Due to the propagation of the surface splitting crack, the area of concentration of tensile stress forms a downward-protruding spherical shape, with the maximum value of tensile stress at the splitting crack tip. With an increase in the length of the penetration crack, the splitting crack inside the specimen propagates several times faster than the surface crack. The stress wave is reflected to the free surface at 65 μs. Subsequently, a circumferential crack originates, which prevents the propagation of the surface crack to some extent. Nevertheless, the propagation of the main splitting surface is not disrupted because of the high tensile-stress concentration. At 100 μs, the tensile-stress concentration has a stronger effect at the bottom of the penetration pit, which makes the internal splitting crack continue to propagate at a high velocity. The superposition of the reflected tensile stress causes the intensity of the surface tensile-stress field to decrease; the resistance to crack propagation increases, resulting in a significant decrease in the crack propagation velocity. The jet stops penetrating at 275 μs, and the intensity of the stress field decreases significantly. Meanwhile, the main splitting surface is penetrated, and the tensile stress, which is distributed over the main splitting area and the area adjacent to it, has a certain strength on the main splitting surface.Figure 5
Stress nephogram: the marked point indicates the maximum value of tensile stress, which always appears close to the main splitting surface.Since the fracture in the specimen is mostly a tensile-shear failure, it is considered a combination of mode I and mode II fractures, as is shown in Figure6. According to our analysis results, the crack tip experiences a compound stress field of modes I and II when the shock wavefront is incident on the crack tip. The stress field of mode I has a more significant effect than that of mode II at the early stage of splitting crack propagation. The splitting crack, which is a compound crack, exhibits behavior that is macroscopically similar to that of an open brittle fracture. The splitting crack tip can steadily propagate in a particular direction for a relatively long time after crack initiation, with a high propagation velocity. With an increase in crack propagation time, the stress field intensity of mode I starts to decrease in an intermediate stage of crack propagation, which directly results in a mode II fracture. The time point of velocity mutation of the crack tip of propagation is the moment at which the crack tip deflects from its original direction of propagation. As the splitting crack continues to propagate, the internal pressure and shear stress of the specimen act in combination to cause a stepwise decrease in the crack propagation rate. The stress field distribution within the specimen considerably changes at the final stage of splitting, and the primary effect on the crack tip is the generation of a redistributed stress field in the lower part of the specimen. Stress attenuation has a negligible effect on the distribution gradient of the tensile-shear stress field. Due to splitting in the specimen, the mode II fracture causes a deflection crack.Figure 6
The crack tip is in the compound stress field.
(a)(b)


## 3.1. Simulation Model

The MAT_HIGH_EXPLOSIVE_BURN [29] model and the JWL equation are adopted to describe the explosive:(5)P=A1−ωR1Ve−R1V+B1−ωR2Ve−R2V+ωE0V,where P is the pressure of the detonation product; A, B, R1, R2, and ω are the test fitting parameters; e is the specific energy; E0 is the initial internal energy of a unit volume explosive; ρe is the density of the explosive (HMX); VOD is the detonation velocity; PCJ is the Chapman–Jouguet pressure; and V is the relative volume, and the explosive material parameters are shown in Table 1.Table 1
Properties of the explosive (HMX) and the JWL state equation.
ρe (g/cm3)VOD (m/s)PCJ (GPa)A (GPa)B (GPa)R1R2ωE0 (J/m3)1.1991042778.37.14.21.00.310.5The JOHNSON_COOK [30] model and the Grüneisen [31] equation of state are adopted to describe the shaped charge cover. Also, the JOHNSON_COOK model is suitable for most metal materials under large deformation, high strain rate, and high temperature, the explosive formation of metal is its typical application, and the parameters of copper chosen in the numerical simulation are shown in Tables 2 and 3.Table 2
Material parameters of copper in the model.
ρm (g/cm3)G (GPa)A (MPa)B (MPa)nCmEPS08.96469002900.310.0251.090.1 × 10−5PCSPALLITD1D2D3D4D5−93030000Note:ρm is the density; A, B, and C are the constants of the material and can be determined by torsion tests under different strain rates.Table 3
State equation parameters of copper.
CS1S2S3GAMAOAV00.3941.489002.020.471.00Note:C is the intercept of the shear-compression wave velocity curve; S1, S2, and S3 are the slope coefficients of the above curve.The Johnson–Holmquist concrete (HJC) [32] is adopted to simulate concrete specimens. The large strain, high strain rate, and high-pressure effect are considered in the HJC model comprehensively:(6)σ∗=A1−D+BP∗N1−clnε•∗,where, σ∗=σ/fc, which demonstrates the ratio of actual equivalent stress to the static yield; D is the damage parameter; p∗=p/fc, which demonstrates dimensionless pressure; ε∗•=ε•/ε•0, which demonstrates dimensionless strain rate; A is the normal viscosity coefficient; B is the normal pressure hardening coefficient; and C is the strain rate coefficient. The keyword ∗MAT_ADD_EROSION was used to compute as tensile failure criterion for concrete specimens, the tensile failure strength is 4 MPa, and the parameters of the specimen model are shown in Table 4.Table 4
Mechanical parameter of the concrete.
ρ (g/cm3)G (GPa)ABCNFC (MPa)EPS02.4414.860.791.60.0070.61481 × 10−6T (MPa)EFMINSFMAXPC (MPa)UCPL (MPa)UL40.017.0160.0018000.1D1D2K1 (GPa)K2 (GPa)K3 (GPa)0.041.085171208As is shown in Figures3(a) and 3(b), a 1/2 model was established, wherein the formation of the jet was controlled by an explosive pressure wave and the air domain mesh was compacted along with the jet formation and penetration path. With charge maintained at a constant value, the formation of the main splitting surface was simulated at a wedge angle of 45° [23] in the two cases.Figure 3
The 1/2 model of numerical simulation. (a) Case 1:H = 30 mm, l = 40 mm. (b) Case 2: H = 40 mm, l = 30 mm.
(a)(b)


## 3.2. Fracture Formation

An increase in the area of the crushing zone resulted in stress wave dissipation at 80μs and a decrease in crack-propagation velocity, as shown in Figure 4(a). A linear notch is clearly visible at the site of penetration, which promotes the growth of a circumferential crack advantage at this point. A central longitudinal crack tip propagates rapidly; the propagation of the splitting surface cracks also tends to accelerate. Consequently, the crack tip moves to the circumferential crack region. Gradually, the central longitudinal crack penetrates the specimen at 130 μs; however, the profile surface crack does not reach the side of the specimen. After penetrating the specimen, the central longitudinal crack continues to propagate from the bottom of the specimen to both sides of the main fracture surface. The spalling is generated symmetrically when the splitting crack does not extend to the profile of the specimen. Elliptical crushing occurs in a small area in the middle of the bottom surface of the specimen. The propagation direction of the splitting crack tip is slightly disturbed, after which the crack continues to grow steadily and the displacement of the crack opening increases. Crack P0, which did not propagate from the surface splitting cracks, originates in the middle of the profile of the specimen at 150 μs. The tensile stress field generated due to cavity expansion leads to an increase in the propagation velocity of splitting cracks inside the specimen. The profile splitting crack reaches the boundary of the specimen and intersects with the bottom splitting crack, concluding the splitting process of the specimen.Figure 4
The process of fracture formation. (a) Case 1. (b) Case 2.
(a)(b)The fragmentation degree in Case 2 (Figure4(b)) is significantly higher than that in Case 1 (Figure 4(a)). The penetration depth increases to 17.1 cm, reducing the range of the crushing zone; the plot representing the penetration is linear. During free motion at 44 μs, the central part of the jet is unsteady due to torsion; meanwhile, particle dispersion is intensified, and the radial penetration effect of the jet is significantly enhanced in Case 1. The jet remains stable while moving, and the tail particles scatter less in Case 2. The tensile strength is also higher than that in Case 1. The maximum velocities of the jet head in Cases 1 and 2 are 3169.8 m/s and 3203.4 m/s, respectively [23]. An increase in the height of the charge position leads to an increase in the velocity and total kinetic energy of the jet head. Meanwhile, the impulse dynamic pressure and the fluid characteristics at the penetration site are enhanced, and the penetration gap narrows with the increase in penetration depth.In Case 2, the distribution of the stress field is uniform at the initial stage of penetration and the crack tip propagates in a straight line along the direction of crack initiation. The uniform distribution of the stress field is disrupted when the stress wave propagates to the boundary and undergoes reflection and superposition. The crack propagation occurs along anS-shaped path and presents a curved plot. Due to tensile stress concentration on the main fracture surface, the crack rapidly begins to propagate in the original direction after the first curved crack. A second curved crack occurs with an offset time of crack propagation longer than that of the first one. Therefore, the second curved crack is longer than the first one. The propagation path is found to be S-shaped after the crack is connected. Thus, the crack tip propagates both on the surface and the interior at the same time, and the stability of the crack tip propagation determines whether the direction of crack propagation will be offset.


## 3.3. Stress Nephogram

As shown in Figure5, the stress field is evenly distributed in the penetration stage at 30 μs, an elliptical tensile stress zone is formed at the penetration site, and the tensile stress is concentrated in the axial direction on both sides symmetrically. The stress field gradually elongates on both sides, and the radial penetration effect of the jet is enhanced. The tensile stress is concentrated to the maximum value at the end of the long axis of the jet. Due to the propagation of the surface splitting crack, the area of concentration of tensile stress forms a downward-protruding spherical shape, with the maximum value of tensile stress at the splitting crack tip. With an increase in the length of the penetration crack, the splitting crack inside the specimen propagates several times faster than the surface crack. The stress wave is reflected to the free surface at 65 μs. Subsequently, a circumferential crack originates, which prevents the propagation of the surface crack to some extent. Nevertheless, the propagation of the main splitting surface is not disrupted because of the high tensile-stress concentration. At 100 μs, the tensile-stress concentration has a stronger effect at the bottom of the penetration pit, which makes the internal splitting crack continue to propagate at a high velocity. The superposition of the reflected tensile stress causes the intensity of the surface tensile-stress field to decrease; the resistance to crack propagation increases, resulting in a significant decrease in the crack propagation velocity. The jet stops penetrating at 275 μs, and the intensity of the stress field decreases significantly. Meanwhile, the main splitting surface is penetrated, and the tensile stress, which is distributed over the main splitting area and the area adjacent to it, has a certain strength on the main splitting surface.Figure 5
Stress nephogram: the marked point indicates the maximum value of tensile stress, which always appears close to the main splitting surface.Since the fracture in the specimen is mostly a tensile-shear failure, it is considered a combination of mode I and mode II fractures, as is shown in Figure6. According to our analysis results, the crack tip experiences a compound stress field of modes I and II when the shock wavefront is incident on the crack tip. The stress field of mode I has a more significant effect than that of mode II at the early stage of splitting crack propagation. The splitting crack, which is a compound crack, exhibits behavior that is macroscopically similar to that of an open brittle fracture. The splitting crack tip can steadily propagate in a particular direction for a relatively long time after crack initiation, with a high propagation velocity. With an increase in crack propagation time, the stress field intensity of mode I starts to decrease in an intermediate stage of crack propagation, which directly results in a mode II fracture. The time point of velocity mutation of the crack tip of propagation is the moment at which the crack tip deflects from its original direction of propagation. As the splitting crack continues to propagate, the internal pressure and shear stress of the specimen act in combination to cause a stepwise decrease in the crack propagation rate. The stress field distribution within the specimen considerably changes at the final stage of splitting, and the primary effect on the crack tip is the generation of a redistributed stress field in the lower part of the specimen. Stress attenuation has a negligible effect on the distribution gradient of the tensile-shear stress field. Due to splitting in the specimen, the mode II fracture causes a deflection crack.Figure 6
The crack tip is in the compound stress field.
(a)(b)


## 4. Discussion

### 4.1. Shaped Charge with Wedge Angles 60° (Case 3) and 90° (Case 4)

Different splitting effects can be obtained by varying the wedge angle of the shaped charge, as shown in Figure7. Results indicate that while the main splitting surface is formed, the propagation of several radial cracks can cause radial fracture in the specimen when the wedge angle is 60° and the specimen typically breaks when the wedge angle is 90°. However, the change in the penetration depth is negligible. Furthermore, a funnel-shaped cavity is formed. With an increase in the radial penetration depth, the stress concentration distribution in the specimen becomes sparse. Because a guiding stress concentration does not form on the main splitting surface, fractures occur at multiple locations, developing a large number of horizontal cracks in the specimen and thus making it fragmented.Figure 7
The effect of the shaped charge wedge angle. (a) Case 3. (b) Case 4.
(a)(b)


### 4.2. Physical Model of Penetration Depth

The head of the linear jet is incident on the rock surface at a considerably high velocity, acting as a massive compressive load and immediately imparting a fluidic property to a specific zone in the rock; the zone is called the hydrodynamic or the fluid zone, which can be approximated as an ellipse due to the limited length of the linear concentrator charge in this study. A shock wave resembling an ellipse originates in this zone and propagates within the rock, being subject to sparse waves. The energy density in the fluid zone decreases rapidly, resulting in a transient phenomenon in the open pit. At this stage, the penetration depth is limited. In a study on penetration channel profiles, Chuan et al. [33] conducted penetration experiments on rocks of different strengths, which resulted in the formation of funnel-shaped pits of different diameters. Joo and Choi [34] investigated the penetration performance of shaped charge-adopted inhibitors. Zhu et al. [35] proposed a semiempirical model for the calculation of the penetration depths and cavity diameters of shaped charge jet into high- and ultrahigh-strength RPC targets. In fact, due to greater rock extrusion at the ends of the linear length of the jet and the superposition of the individual unit jets involved in penetration, the central penetration is stronger than that at the lengthwise ends of the charge. By superimposing the cavities penetrated by each unit jet, we can approximate that both the penetration depth and pit show a linearly decreasing trend from the center to both ends, which is consistent with practical results. Therefore, a realistic physical model of penetration depth is shown as follows:(7)S=lnhn=Δlhn,(8)ΔEkjsi=σslnhn2=σsΔlhn2,(9)Ekjs′=ΔEkjs1+ΔEkjs2…+ΔEkjsn−1+ΔEkjsn=σsΔlh12+σsΔlh22…+σsΔlhn−12+σsΔlhn2=σsΔl∑1nhn2=A⋅Ekjs,where A is the constant.


### 4.3. Critical Length of the Crack

Maximum splitting size depends on critical fracture stress and crack length. When the length of the LSC increases, penetration depth also increases. The corresponding critical fracture stress decreases, making splitting easier. For large cylindrical rocks, when the ratio of the LSC length to the rock bottom diameter exceeds a critical value, the jet energy is wasted. Considering energy losses during actual splitting, the critical value can be slightly increased for practical applications.The critical stress required for crack propagation decreases with increasing crack length [36]; however, it does not decrease to a value lower than the theoretical tensile strength of the material. The critical length determines if the tensile stress at the crack tip allows continued propagation of the crack. The critical length cc of the crack is calculated using equation (9) and is the critical value of the linear shaped charge length.(10)cc=2Eγπσθ2,where d=20/ρCKIC/ε•2/3d=20/ρCKIC/ε•2/3; ε• is the constant; KIC is the elastic modulus of the specimen; C is surface energy per unit area; and KPa is circumferential stress. The calculated critical length of the crack can be used as the critical LSC length.


### 4.4. Path of Split Development

For the crack tip to propagate further after crack initiation, the stress wave should be maintained at a certain intensity. Stress wave propagation gradually decays with increasing distance, and the propagation distance of the crack tip depends on the stress wave energy within the specimen as well as the size of the specimen. The manner in which the splitting crack extends to any location within the rock is consistent with that described in the theory of gap [36]. A similar analysis considering different cross sections is sufficient. The theory of gap also proposes a critical value for the height of the rock beyond which the stress wave cannot cause the splitting crack to continue expanding. In Case 1, the jet is twisted and incurs accumulation in the head; this leads to an increase in the degree of internal crushing in the specimen. Particle scattering at the tail causes the surface crushing zone to extend further. The above phenomena can be summarized as follows: (1) the cohesiveness of the jet determines the degree of specimen crushing and (2) the linear length of the jet determines the stability of the directional propagation of the splitting crack, as is shown in Figure 8.Figure 8
The path of split development.The narrower the intrusion crack, the smaller the volume of the hydrodynamic zone and the smaller the crushing zone. Furthermore, the wider the crack, the larger the crushing zone. Finally, the longer the intrusion crack, the more stable the directional splitting crack propagation.


### 4.5. Control of Rock Fragmentation Size

In the kinetic energy model, it is assumed that a significant number of defects must exist within the bulk rock. Due to the defects, defective cracks are likely to expand before the stress wave attains the theoretical tensile strength. A fracture pattern originates in the tension between defective cracks and cleavage cracks. The fragmentation of the specimen also shows a symmetrical pattern in the primary plane of linear jet penetration in the axial direction. Meanwhile, stress concentration still plays a controlling role in the fragmentation process, and the defects dominate the local fragmentation pattern. Moreover, compared to the defects, energy is a more significant factor controlling the dynamic fracture process [37].In summary, the final fragmentation size of a rock is related to the jet kinetic energy, the stress concentration zone formed due to penetration, and the number of defects activated within the stress concentration zone. In engineering applications, the size of the fragments can be controlled by varying the structure of the LSC and using linear jets with different kinetic energies and morphologies to penetrate large blocks of rock. The penetration effect of the jets can be employed to control the density and distribution pattern of stress wave energy in rocks.


## 4.1. Shaped Charge with Wedge Angles 60° (Case 3) and 90° (Case 4)

Different splitting effects can be obtained by varying the wedge angle of the shaped charge, as shown in Figure7. Results indicate that while the main splitting surface is formed, the propagation of several radial cracks can cause radial fracture in the specimen when the wedge angle is 60° and the specimen typically breaks when the wedge angle is 90°. However, the change in the penetration depth is negligible. Furthermore, a funnel-shaped cavity is formed. With an increase in the radial penetration depth, the stress concentration distribution in the specimen becomes sparse. Because a guiding stress concentration does not form on the main splitting surface, fractures occur at multiple locations, developing a large number of horizontal cracks in the specimen and thus making it fragmented.Figure 7
The effect of the shaped charge wedge angle. (a) Case 3. (b) Case 4.
(a)(b)


## 4.2. Physical Model of Penetration Depth

The head of the linear jet is incident on the rock surface at a considerably high velocity, acting as a massive compressive load and immediately imparting a fluidic property to a specific zone in the rock; the zone is called the hydrodynamic or the fluid zone, which can be approximated as an ellipse due to the limited length of the linear concentrator charge in this study. A shock wave resembling an ellipse originates in this zone and propagates within the rock, being subject to sparse waves. The energy density in the fluid zone decreases rapidly, resulting in a transient phenomenon in the open pit. At this stage, the penetration depth is limited. In a study on penetration channel profiles, Chuan et al. [33] conducted penetration experiments on rocks of different strengths, which resulted in the formation of funnel-shaped pits of different diameters. Joo and Choi [34] investigated the penetration performance of shaped charge-adopted inhibitors. Zhu et al. [35] proposed a semiempirical model for the calculation of the penetration depths and cavity diameters of shaped charge jet into high- and ultrahigh-strength RPC targets. In fact, due to greater rock extrusion at the ends of the linear length of the jet and the superposition of the individual unit jets involved in penetration, the central penetration is stronger than that at the lengthwise ends of the charge. By superimposing the cavities penetrated by each unit jet, we can approximate that both the penetration depth and pit show a linearly decreasing trend from the center to both ends, which is consistent with practical results. Therefore, a realistic physical model of penetration depth is shown as follows:(7)S=lnhn=Δlhn,(8)ΔEkjsi=σslnhn2=σsΔlhn2,(9)Ekjs′=ΔEkjs1+ΔEkjs2…+ΔEkjsn−1+ΔEkjsn=σsΔlh12+σsΔlh22…+σsΔlhn−12+σsΔlhn2=σsΔl∑1nhn2=A⋅Ekjs,where A is the constant.


## 4.3. Critical Length of the Crack

Maximum splitting size depends on critical fracture stress and crack length. When the length of the LSC increases, penetration depth also increases. The corresponding critical fracture stress decreases, making splitting easier. For large cylindrical rocks, when the ratio of the LSC length to the rock bottom diameter exceeds a critical value, the jet energy is wasted. Considering energy losses during actual splitting, the critical value can be slightly increased for practical applications.The critical stress required for crack propagation decreases with increasing crack length [36]; however, it does not decrease to a value lower than the theoretical tensile strength of the material. The critical length determines if the tensile stress at the crack tip allows continued propagation of the crack. The critical length cc of the crack is calculated using equation (9) and is the critical value of the linear shaped charge length.(10)cc=2Eγπσθ2,where d=20/ρCKIC/ε•2/3d=20/ρCKIC/ε•2/3; ε• is the constant; KIC is the elastic modulus of the specimen; C is surface energy per unit area; and KPa is circumferential stress. The calculated critical length of the crack can be used as the critical LSC length.


## 4.4. Path of Split Development

For the crack tip to propagate further after crack initiation, the stress wave should be maintained at a certain intensity. Stress wave propagation gradually decays with increasing distance, and the propagation distance of the crack tip depends on the stress wave energy within the specimen as well as the size of the specimen. The manner in which the splitting crack extends to any location within the rock is consistent with that described in the theory of gap [36]. A similar analysis considering different cross sections is sufficient. The theory of gap also proposes a critical value for the height of the rock beyond which the stress wave cannot cause the splitting crack to continue expanding. In Case 1, the jet is twisted and incurs accumulation in the head; this leads to an increase in the degree of internal crushing in the specimen. Particle scattering at the tail causes the surface crushing zone to extend further. The above phenomena can be summarized as follows: (1) the cohesiveness of the jet determines the degree of specimen crushing and (2) the linear length of the jet determines the stability of the directional propagation of the splitting crack, as is shown in Figure 8.Figure 8
The path of split development.The narrower the intrusion crack, the smaller the volume of the hydrodynamic zone and the smaller the crushing zone. Furthermore, the wider the crack, the larger the crushing zone. Finally, the longer the intrusion crack, the more stable the directional splitting crack propagation.


## 4.5. Control of Rock Fragmentation Size

In the kinetic energy model, it is assumed that a significant number of defects must exist within the bulk rock. Due to the defects, defective cracks are likely to expand before the stress wave attains the theoretical tensile strength. A fracture pattern originates in the tension between defective cracks and cleavage cracks. The fragmentation of the specimen also shows a symmetrical pattern in the primary plane of linear jet penetration in the axial direction. Meanwhile, stress concentration still plays a controlling role in the fragmentation process, and the defects dominate the local fragmentation pattern. Moreover, compared to the defects, energy is a more significant factor controlling the dynamic fracture process [37].In summary, the final fragmentation size of a rock is related to the jet kinetic energy, the stress concentration zone formed due to penetration, and the number of defects activated within the stress concentration zone. In engineering applications, the size of the fragments can be controlled by varying the structure of the LSC and using linear jets with different kinetic energies and morphologies to penetrate large blocks of rock. The penetration effect of the jets can be employed to control the density and distribution pattern of stress wave energy in rocks.


## 5. Conclusion

In this study, different forms of splitting in a specimen under the effect of a mini-LSC jet were numerically simulated. The conclusions of the study are as follows:(1)
The main splitting strength of the surface tensile stress concentration on the distribution of the stress field of relative superiority degree determines the direction of splitting ability and the final form of the split.(2)
Under the action of the LSC jet, the main plane of penetration along the axial direction of the jet cracked first; the crack propagated along the diagonal from the penetration center, resulting in an open-fracture splitting pattern.(3)
Increasing the linear charge length can improve the control of the splitting direction. The cylindrical concrete specimen, with a bottom diameter of 30 cm, was split directionally when the linear charge length was 3-4 cm.(4)
A realistic physical model of the penetration depth is obtained. It is a rapid and safe blasting technology to handle hazardous, massive rocks during emergency rescuing or mining.



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*Source: 2039225-2022-03-30.xml*