#### Abstract

For the purpose of uncovering the influence law on the explosion stress wave transmission and blast-induced crack propagation by inclined natural fracture and water jet slot, the influential mechanism for the transmission law of explosion stress wave due to inclined natural fracture was analyzed based on stress wave theory in this article. Next, the evolution law of explosion stress wave and blast-induced crack propagation affected by inclined natural fracture and water jet slot were simulated by ANSYS/LS-DYNA, and an experiment was conducted to understand the crack propagation law under explosion shock wave loading. The results indicate that the existence of inclined natural fracture will affect the “guiding effect” of water jet slot on blast-induced crack propagation by changing the propagation path and mechanical properties of the explosion stress wave. When the distance between inclined natural fracture and blast hole is too small, the water jet slot cannot play a guiding role in propagation process of blast-induced main crack. With the increase of the distance, the propagation direction of blast-induced crack is affected by both inclined natural fracture and water jet slot. When the distance increases further and exceeds a certain value, inclined natural fracture cannot affect the guiding effect of water jet slot in propagation process of blast-induced main crack.

#### 1. Introduction

Borehole-blasting method is the main method of roadway excavation in underground mines in Chinasince its advantages of high efficiency and economy [1, 2]. In the actual construction process, there are two objectives that should be realized in roadway tunneling operation: one is to make the rock mass in the blasting area broken as far as possible under blasting action, and the other is to reduce the blasting damage of the retaining rock mass as much as possible. Based on this, roadway contour-controlled blasting technology came into being [3–5], and the directional controlled blasting technology with water jet assistance is one of them, its technical principle is guiding blast-induced crack propagate along the predetermined direction by the water jet slot, and the blasting damage of retaining rock mass is reduced, and finally directional controlled blasting is realized [6–9].

Generally speaking, the more uniform the rock is, the more favorable it is to control the propagation of blast-induced crack. However, rock mass contains a large number of joints and fissures, which has a great influence on the propagation of blast-induced crack [10–12]. Therefore, it is of great significance to analyze the effect of natural fracture on the blast-induced crack propagation. Aiming at this problem, many scholars have carried out research and made a series of original scientific achievements. Larson, Larson and Pueleise [13] analyzed the influence mechanism of relative position between blast hole and internal structural plane of rock on the blast effect, and the results show that when the direction of blasting hole is consistent with the internal structural plane of rock, the blast effect is better, but when the direction of blasting hole and rock internal structural plane is not consistent, the blast effect is relatively poor. Based on displacement discontinuity theory, Myer et al. [14–17] analyzed the deformation and failure of joint surface with filling material under explosive shock load by using Kelvin and Max models, and the attenuation of explosion stress wave is studied by acoustic emission detection technology, and the results show that the propagation velocity of explosion stress wave and the displacement amplitude of particle will be greatly reduced by the primary fracture in the rock mass. Li et al. [18, 19] analyzed the boundary conditions on the weak structural plane of the rock mass wheel and the boundary conditions of the weak structural plane of the rock mass, obtained the transmission reflection relationship of the explosion stress wave passing through the weak structural plane under the condition of friction and sliding, and established the transmission reflection relationship model of the stress wave obliquely incident on the wave potential. Lu and Tao [20] proposed the failure criterion of rock mass under the action of explosion stress wave and the propagation law of stress wave in the layered rock mass by analyzing the quasi-static force of detonation gas during crack propagation, and the relationship between the transmission wave and the reflected wave amplitude is obtained; and the results show that the rock mass structure plane has a filtering effect on the high-frequency part of the stress wave. Rossmanith et al. [21, 22] studied the influence mechanism of rock fracture on stress wave, and the failure model of rock mass internal crack interface under stress wave was established by using the continuum failure rule based on the constitutive equation of viscoelastic interface material. Zhao et al. [23–25] systematically studied the reflection and transmission of stress wave propagating through linear joints and also studied the attenuation law of stress wave propagation in inclined joint fissure and rock mass with multiple sets of joints. The results show that the crack spacing and stress wave length will have an important impact on the transmission coefficient of stress wave propagation at joint fissures. Wang et al. [26, 27] established a nonlinear displacement discontinuity model of joints based on the elastic displacement of waves and the assumption of equivalent displacement. On this basis, the influence of incident wave frequency and joint parameters on joint transmission and reflection coefficient is studied, and the results are compared with the existing numerical calculation results, which are in good agreement. Li et al. [28, 29] established the fractal damage constitutive relation based on fractal damage theory and gave the theoretical expression between the transmission and reflection coefficient and the fractal dimension. Li et al. [30–32] respectively studied the propagation characteristics of stress waves with different types and parameters when passing through different joints and fissures and established the energy evolution model of stress waves after passing through different joints and fissures. Zhu et al. [33, 34] studied the influence of the incident angle of explosion stress wave on the initiation angle and propagation angle of detonation crack with the aid of PMMA experimental test materials using a single electric detonator as the explosion shock wave source. At the same time, numerical simulation software is used to analyze the law of crack propagation caused by explosion in different positions. The results show that as the incident angle is 0°, crack initiation angle is 0°, the cracks propagate along the crack surface plane. As the incident angle is greater than 15° and less than 75°, wing cracks occur along the preexisting crack tips. The wing-crack property is related to the incident angle of stress waves. As the incident angle is greater than 75°, new developed cracks are very small. Yang et al. [35, 36] studied the influence of artificial cracks with different positions, different orientations, and different filling materials (water, air, clay) on the propagation of single ordinary blast hole explosion-induced crack, and the results show that blasting cracks almost cannot surmount prefabricated fissures, and total cracks and left and wing cracks of models with filling air almost are larger than the models with filling clay and water. By using a dynamic caustic line test system, Yue et al. [37–40] studied the propagation process of stress wave, crack propagation mechanism, and multiple joint fissure penetration mechanism in defective medium under explosion load. The results show that the initiation and propagation of secondary cracks depend on the concentration effect of the diffraction formation stress at the prefabricated through crack ends, and the secondary crack initiates along the direction of the maximum energy release rate and is a mode I type fracture.

Through the analysis of the above research results, it can be seen that the influence of natural fracture on the blast-induced crack propagation law of ordinary circular hole can be mainly summary into two aspects: On one hand, the existence of natural fracture can intensify the reflection of explosion stress wave, which let compressive stress wave and reflected tensile wave are superimposed on each other, thus affect the propagation law of blast-induced main crack and promote the generation of secondary cracks. On the other hand, wing cracks occur along the tips of natural fracture due to the concentration effect of diffraction formation stress at the natural fracture ends, and its property is related to the incident angle of stress waves and the filling material of natural fracture.

The above research focuses on the influence of natural fracture in rock mass on the crack growth of ordinary round hole blasting. However, for the water jet slotted, guided, controlled blasting technology, the existence of jet slots will affect the propagation law of blast-induced cracks [6, 7], so it is necessary to study the combined effect mechanism of natural fracture and water jet slot on the propagation of blast-induced crack. Based on this, an attempt was made in this article to study the propagation law for blast-induced crack affected by inclined natural fracture and water jet slot [11]. The stress wave theory was applied to characterize the explosion stress wave transmission law. Next, a numerical simulation for transmission and reflection of explosion stress wave and propagation law for blast-induced crack affected by inclined natural fracture and vertical natural fracture and water jet slot was performed using the FEM software ANSYS/LS-DYNA. Finally, an experiment was performed to investigate the crack propagation law affected by water jet slot and natural fracture under explosion shock wave loading.

#### 2. Analysis on the Propagation Law of Explosive Stress Wave Affected by Inclined Natural Fracture

According to blasting theory [41], reflection and transmission will occur on the interface when the explosion stress wave incident on natural fracture, as shown in Figure 1. The influence on transmission of blasting stress wave by natural fracture mainly includes two aspects: On one hand, the existence of natural fracture will change the propagation law of explosion stress wave, and the reflected wave and transmission wave occurred at the interface, then the property of explosion stress wave will be changed. On the other hand, the energy of explosion stress wave will be reduced to a great extent due to the interaction between explosion stress wave and natural fracture.

The reflection and transmission of explosion stress wave on the interface conform to Snell’s law, and the reflection and transmission of explosion stress wave meet two boundary conditions: On one hand, the stress state of explosion stress wave on both sides of boundary surface is equal. On the other hand, the velocities of particles perpendicular to the direction of the boundary surface must be equal. Therefore, the relationship between incident wave, reflected wave, and transmitted wave can be expressed as follows:where *σ _{i}* is the stress value of incident stress wave,

*σ*is the stress value of reflected wave,

_{r}*σ*is the stress value of transmitted wave,

_{t}*ρ*

_{1}is the density of transmission medium #1,

*c*

_{1}is the propagation velocity of stress wave in propagating medium #1,

*ρ*

_{2}is the density of transmission medium #2, and

*c*

_{2}is the propagation velocity of stress wave in propagating medium #2.

And the incident angle of P-wave satisfies following relation:

Therefore, the relationship between the incident angle and the reflection angle of reflected P-wave is as follows:

In addition, the relationship between parameters of each stress wave can be obtained according to the theory of stress wave, which is given as follows:where *σ*_{ip} is the stress value of incident P-wave; *σ*_{rp} is the stress value of reflected P-wave; *σ*_{tp} is the stress value of transmitted S-wave; *τ*_{rs} is the stress value of reflected S-wave; *τ*_{ts} is the stress value of transmitted S-wave; *ρ*_{r} is the density of rock; *c*_{p} is the propagation velocity of P-wave; *c*_{s} is the propagation velocity of S-wave; , and are the velocity of medium particles caused by the propagation of incident P-wave, reflected P-wave, and transmitted P-wave, respectively; and and are the velocity of medium particles is caused by the propagation of reflected S-wave and transmitted S-wave.

Therefore, assuming that the normal stress value and tangential stress value of the interaction between stress wave and natural fracture are *σ*_{nn} and *τ*_{nn}, respectively, the following relationship can be obtained:

By simplifying and merging above formulas, the relationship between them can be described as follows:where tan*φ* is the friction coefficient of natural fracture surface.

#### 3. Numerical Simulation of Explosion Stress Wave Evolution Law under Blast Loading

ANSYS/LS-DYNA is a dynamic software which can solve the dynamic problems of highly nonlinear structures, such as collision and explosion. Therefore, it was used to simulate the evolution law of explosion stress wave affected by natural fracture and water jet slot under blast loading.

##### 3.1. Numerical Simulation Model

In the actual blasting operation, the cylindrical charge structure was adopted. During the blasting, the force generated by the explosion of the explosive uniformly acts on the hole wall in the axial direction of the hole. Therefore, according to the characteristics of blast hole wall stress, it can be simplified as a plane strain problem in the analysis of the initiation and propagation law of blast-induced crack based on the theoretical analysis, and a “quasi-two-dimensional” analysis model was established, as shown in Figure 2. In which, *l*_{n} is the length of the natural fracture, *D*_{n} is the distance between natural fracture and blast hole, *α*_{n} is an angle between natural fracture plane and the horizontal direction of the borehole, *r*_{b} is the radius of blast hole, *l*_{s} is the length of water jet slot, and is the width of water jet slot.

Based on this, quasi-two-dimensional numerical simulation models were established. The parameters of each simulation case are shown in Table 1, and simulation model for case CN_{I}-3 is shown in Figures 3 and 4.

##### 3.2. Numerical Calculation Algorithm

In this article, fluid-solid coupling algorithm was adopted for the analysis of explosive detonation, of which, Arbitrary Eulerian–Lagrange (ALE) algorithm will be used for explosive and air which filled in the water jet slot and Lagrange algorithm will be used for rock. The material derivative equation and the governing equations of the ALE algorithm can be expressed as formulas (7)–(10), [27, 29]where *f* is the physical quantity, *X*, *ω*_{i} is the velocity of the computational grid, *x*_{i} is the Lagrangian coordinate system, *y*_{i} is the Eulerian coordinate system, *ν*_{i} is the velocity of particle, *ρ* is the density, is the particle velocity, *e* is the internal energy of unit mass, *σ*_{ij} is the Cauchy stress tensor, *f*_{i} is the body force, *q*_{i} is the heat flux, and subscripts *i* and *j* stand for the direction of coordinate.

In addition, meshes of explosive and the air were joined with common nodes. Then the fluid-solid coupling was defined between the meshes of the explosive, air, and rock by the keyword CONSTRAINED_LAGRANGE_IN_SOLID. According to the characteristics blasting process [41], the time step scale factor of the simulation is 0.67, and the calculation time is set to 0.0025 s.

##### 3.3. Simulation Material Model

###### 3.3.1. Explosive

Consider the characteristics of high temperature and high pressure of explosive explosion, MAT_HIGH_EXPLOSIVE_BURN was chosen as the explosive model, and the Jones–Wilkens–Lee (JWL) equation of state (EOS) was used to model the pressure released by the chemical energy in the engineering calculations [42], it can be written in formula (11). The following parameters were used: explosive density *ρ*_{e} of 1.93 × 10^{3} kg/m^{3}, detonation velocity of 9.93 × 10^{3} m/s, output pressure *p*_{cut} of 3.37 × 10^{4} MPa, and the input parameters were *A* of 3.7 × 10^{5} MPa, *B* of 7.43 × 10^{3} MPa, *R*_{1} of 4.15, *R*_{2} of 0.95, and *ω* of 0.30.where *p*_{e} is the pressure produced by the detonation products from the high explosive; *ω*, *A*, *B*, *R*_{1}, and *R*_{2} are user-defined input parameters; *V* is the relative volume; and *E*_{e} is the internal energy per initial volume.

###### 3.3.2. Rock

MAT_PLASTIC_KINEMATIC was adopted as the element material of rock modeling, and the following parameters of the rock modeling were used in the numerical calculation: density *ρ*_{r} of 2.55 × 10^{3} kg/m^{3}, elastic modulus *E*_{r} of 2.25 × 10^{4} MPa, Poisson’s ratio *P*_{R} of 0.22, yield stress SIGN of 3.24 MPa, tangent modulus ETAN of 4.25 × 10^{4} MPa, strain rate parameter SRC is 0.0, strain rate parameter SRP is 0.0, and failure strain *F*_{S} of 0.06.

###### 3.3.3. Air

Air was modeled by the material model of MAT_NULL with the Gruneisen equation, and the pressure *P*_{a} can be calculated by the following formula:where *ρ*_{a} is the density of air, which is 1.25 kg/m^{3}; *V*_{a} is the relative volume of air, which is 1.0; *E*_{a} is the internal energy of air, which is 2.5 × 10^{−6}; and *C*_{0}, *C*_{1}, *C*_{2}, *C*_{3}, *C*_{4}, *C*_{5}, and *C*_{6} are user-defined constants, of which *C*_{0} is −1×10^{−6}, *C*_{1} = *C*_{2} = *C*_{3} = *C*_{6} = 0, *C*_{4} = *C*_{5} = 0.4.

##### 3.4. Simulation Result Analysis

###### 3.4.1. Propagation of Blast-Induced Crack Affected by Natural Fracture and Water Jet Slot

Postprocessing software LS-Prepost was used to draw the pressure diagram of explosive stress waves for each case, as shown in Figures 5–14.

Figures 5–14 show the propagation law of explosion stress wave for case CN_{O}-1∼CN_{O}-8, from which we can see that reflection and diffraction occurred when the explosion stress wave propagates to natural fracture, and smaller the *D*_{n} is, the more intense the phenomenon of reflection and diffraction of stress wave becomes, which is consistent with theoretical analysis. If *D*_{n} is too small, the natural fracture can strongly affect the propagation law of blast-induced crack, which induce main crack and secondary crack propagates along the direction of line of hitch of least resistance, as shown in Figures 5 and 8. At the same time, the derived wing cracks generated by stress wave diffraction at the tip of natural fracture can also be induced by the main crack, thus forming large area of crushing and destruction. With the increase of *D*_{n}, it is difficult to form the crushing failure zone between natural fracture and blast hole, but the induction effect of natural fracture on main crack and secondary crack still exists. That is to say, the influence of natural fracture on the crack propagation law of water jet slotted, guided blasting is limited by the distance between natural fracture and water jet slotted blast hole.

The propagation law of explosion stress wave for case CH_{I}-3∼CH_{I}-5 are shown in Figures 5–8. It is can be seen that when *D*_{n} is small, the blast-induced crashing zone is formed before the growth of the wing crack which produced at the tip of natural fracture due to the blast stress concentration, and the formation of blast-induced crashing zone greatly reduce the explosive energy. With the increase of *D*_{n}, it is impossible to form the blast-induced crushing zone, and the wine crack generate at the tip of the natural fracture will interact with main crack and form blasted fragmentation. In addition, if the blast-induced crushing zone between blast hole and inclined natural fracture can be formed, the explosive energy is mainly consumed in the fragmentation of rock in the crushing zone; otherwise, if the blast-induced crushing zone cannot be formed, the explosive energy is mainly concentrated at the tip of the blast-induced cracks and consumed in the growth of cracks. Meanwhile, by comparative analysis, it can be seen that the smaller the distance from the tip of natural fracture, greater the concentration degree of explosion energy is, and the reverse is smaller.

By comparing Figures 5 and 9, it can be found that the blast-induced crack deflects towards the tip of natural fracture during the process of crack propagation, and the propagation length of the blast-induced crack which is located on the line of natural fracture and blast hole is relatively short. That is because the reflected tensile stress wave which produced when the explosion stress wave propagates to the natural fracture plays a different role in the propagation of the blast-induced crack. If the propagation direction of the tensile stress wave is opposite to the propagation direction of the blast-induced crack, the reflected tensile stress wave hinders the propagation of blast-induced crack. If the propagation direction of the tensile stress wave is the same as the propagation direction of blast-induced crack, the tensile stress wave will promote the propagation of blast-induced crack.

###### 3.4.2. Evolution of Explosion Stress Wave Affected by Natural Fracture and Water Jet Slot

Monitoring points were arranged at the front and back of natural fracture to further analyze the propagation law of explosion stress wave. The arrangement of monitoring point is shown in Figure 15, and the P-T curve for case CN_{V}-1∼CN_{V}-4 is shown in Figures 16–19.

**(a)**

**(b)**

By analyzing the P-T curves of each measurement point, it can be found that the waveform of each case which without natural fracture are very similar, and the maximum positive peak pressure and maximum negative peak pressure are 146 MPa and −140 MPa, respectively. With the increase of the distance from blast hole, both the maximum positive peak pressure and the maximum negative peak pressure decrease gradually, but the descender of peak pressure shows a huge difference. By comparing data from each case, it can be found when the distance increases from 10 cm to 40 cm, the maximum positive peak pressure decreases from 146 MPa to 79.8 MPa with a decay rate of 45.34%, and the maximum negative peak pressure decreases from −140 MPa to −3.48 MPa with a decay rate of 97.51%. In addition, when vertical natural fracture exists, there is no obvious change for the explosion stress waveform by comparing to normal controlled blasting with water jet slotting assistance. However, the maximum positive peak pressure and maximum negative peak pressure are 66.5 MPa and −9.71 MPa, respectively. Compared with normal controlled blasting with water jet slotting assistance, the maximum positive peak pressure and negative peak pressure are reduced by 54.45% and 93.06%, respectively. By comparing data from each group, it can be found when the distance increases from 10 cm to 40 cm, the maximum positive peak pressure decreases from 66.5 MPa to 12.9 MPa with a decay rate of 80.61%. That is to say, the explosion stress wave energy is greatly weakened when it propagates to natural fracture, and the propagation of stress wave can be almost completely blocked by natural fracture when a certain distance is exceeded. With the increase of distance from blast hole, the peak value of explosion stress wave decreases gradually, and the weakening effect of natural fracture on the explosion stress wave energy will be less.

#### 4. Experiment on Propagation Law of Blast-Induced Crack under Explosion Shock Wave Loading

##### 4.1. Introduction of Experiment

To study the distribution law and propagation law of blast-induced crack under different conditions, PMMA was chosen as specimen material by comprehensively analyzing the characteristics of various experimental materials, and the physical mechanical properties of PMMA were the density of 1.18 × 10^{3} kg/m^{3}, elastic modulus of 22.26 × 10^{3} MPa, compressive strength of 142.39 MPa, tensile strength of 62.48 MPa, shear strength of 125.42 MPa, Poisson’s ratio of 0.29, longitudinal wave velocity of 2319 m/s, and shear wave velocity of 1252 m/s. The schematic diagram of the blasting experiment is shown in Figure 20, and the introduction of each experiment test case is shown in Table 2.

No. 8 electronic detonator was chosen as a source of the explosion shock wave, it had the following parameters: charge weight of 9×10^{−4 } kg, explosive density of 1.82 × 10^{3} kg/m^{3}, explosion heat of 5.392 J/kg, detonation velocity of 8.75 × 10^{3} m/s, and delay time of 7.5×10^{−8}s.

##### 4.2. Experiment Result Analysis

###### 4.2.1. Blasting Effect Affected by Natural Fracture and Water Jet Slot

The blasting effect for each case is shown in Figures 21–26. By comparing Figures 21, 23, and 25, we can see that the blast-induced crushing zone formed near the natural fracture when the distance from the blast hole is too small. That is because the strength of tensile stress wave generated by the explosion stress wave propagating to the natural fracture exceeds the tensile strength of rock mass. In addition, the blast-induced crack on the side of natural fracture cannot propagate along the direction of water jet slot.

Derived wing cracks are generated at the tip of natural fracture due to the stress concentration effect of explosion stress wave, and the length of derived wing cracks for case SN_{V}-1 is slightly larger than the length of derived wing cracks for case SN_{I}-1 and case SN_{I}-3. That is because the existence of water jet slot causes explosion stress wave concentrate in this direction of water jet slot [11]. If the distance between blast hole and natural fracture, the derived wing cracks will no longer occur.

By comparing Figures 21, 22, and 26, we can see that the blast-induced crack deflects towards the tip of natural fracture in the process of crack propagation, that is it indicates that the stress concentration effect at tip of natural fracture has a guiding effect on the propagation of blast-induced crack. However, by comparing Figure 22, 24, and 26, it can be found that the blast-induced crack located in the direction of center line between natural fracture and blast hole will be hindered by tensile stress wave reflected from explosion stress wave propagating to natural fracture, and its propagation direction will be changed, which is similar to those of numerical simulation.

###### 4.2.2. Evolution of Blast Strain Wave Affected by Natural Fracture and Water Jet Slot

Figures 27 and 28 are explosion strain time history curves of monitoring point #1 and monitoring point #1 for case SN_{V}-1 and SN_{V}-2. The analysis shows that the radial strain values for case SN_{V}-1 is between 4891.73*με* and 4721.01*με*, and the radial strain values for case SN_{V}-1 is between 5188.08*με* and 4796.21*με*. By comparing to normal controlled blasting with water jet slotting assistance, there is no significant difference between the radial strain values of monitoring point at the same position. However, the tangential strain values of monitoring point at the same position were increased by 47.6% and 17.11%, respectively. That is because the reflected stress wave interacts with the blast-induced crack in non-slotted direction, which further promote the propagation of blast-induced crack in non-slotted direction, and the tangential strain of monitoring point in vertical direction changes obviously.

**(a)**

**(b)**

**(a)**

**(b)**

In the direction of water jet slot, the strain value of monitoring point #2 decreased sharply due to the barrier effect of vertical natural fracture. The radial strain values of monitoring point #2 for case SN_{V}-1 is between 2605.26*με* and −2773.57*με*, and the tangential strain value of monitoring point #2 for case SN_{V}-1 is between 3113.29*με* and −1833.1089*με*. By comparing with the strain values of monitoring point at the same position of normal controlled blasting with water jet slotting assistance, the radial strain values are reduced by 55.03% and 59.27%, respectively, and the tangential strain value are reduced by 43.74% and 50.97%, respectively. In addition, the radial strain values of monitoring point #2 for case SN_{V}-2 is between 4141.54*με* and −2680.70*με*, and the tangential strain value of monitoring point #2 for case SN_{V}-2 is between 3722.62 *με* and −1935.72*με*. By comparing with the strain values of monitoring point at the same position of normal controlled blasting with water jet slotting assistance, the radial strain values are reduced by 28.51% and 60.65%, respectively, and the tangential strain value are reduced by 32.73% and 48.25%, respectively. In contrast, the closer the vertical natural crack is to the borehole, the stronger its energy weakening effect will be.

###### 4.2.3. Uncertainty Quantification of Experiment Result

Blasting experiment test results often have certain randomness, which is due to some uncertainty and variability factors affecting the experimental results [43–45]. There will inevitably be differences between actual size and design size in the manufacturing process of specimens, and the length of water jet slot and radius of blast hole are two very important parameters, which determine the size of the stress intensity factor at the tip of water jet slot, and then affect the initiation and propagation of blast-induced crack, and ultimately affect the blasting experimental results. Based on fracture mechanics and Irwin fracture mechanics theory, the fracture will propagate when the stress intensity factor *K*_{I} reaches critical value *K*_{IC} for model I crack, otherwise it will stop [8, 46]. The *K*_{I} is calculated as follows:where *K*_{I} is the stress intensity factor of Mode I, and *K*_{IC} is the critical stress intensity factor (fracture toughness), *P*_{0} is the pressure produced by explosion, *F* is the correction factor of stress intensity factor, which is a function of the blast hole radius and length of crack, *r*_{b} is the radius of blast hole, and *l*_{s} is the length of water jet slot.

In this article, these parameters were assumed to obey normal distribution with user-defined mean values and standard deviations, and the final distributions defined for *l*_{s} and *r*_{b} are shown in Figure 29, and the related parameters are listed in Table 3.

**(a)**

**(b)**

In this article, the difference between the standard value and the actual value of stress intensity factor of tip of water jet slot was studied. Figure 30 shows the probability description of the occurrence of value under different conditions, from which we can see that value obeys a semi-normal distribution, and the probability of occurrence of value increases first and then decreases with its increase. When value is 15, the probability of its occurrence reaches the maximum of 17.75%.

#### 5. Conclusions

The existence of inclined natural fracture will change the propagation path of explosion stress wave and its mechanical properties and then change the propagation law of blast-induced crack, which finally affect the “guiding effect” of water jet slot on blast-induced crack propagation. The influence of inclined natural fracture on propagation of blast-induced crack depends on the distance between inclined natural fracture and blast hole. When the distance is too small, explosion stress wave reflection occurred and the tensile stress wave will generate when it propagates to inclined natural fracture, and crushing zone will be formed between inclined natural fracture and blast hole due to Hopkinson effect. Under this condition, the water jet slot cannot play a guiding role in the blast-induced main crack propagation process. With the increase of the distance, the propagation direction of blast-induced crack is affected by both inclined natural fracture and water jet slot. On one hand, the stress concentration effect formed by explosion stress wave at the tip of inclined natural fracture will induce blast-induced crack to propagate in this direction. The blast-induced crack located in the direction of center line between inclined natural fracture and blast hole will be hindered by tensile stress wave reflected from explosion stress wave propagating to inclined natural fracture, and its propagation direction will be changed. On the other hand, if the induced effect of inclined natural fracture plays a role in the propagation process of secondary crack on both sides of blast-induced crack, the propagation direction of secondary crack is determined by inclined natural fracture and the propagation direction of blast-induced crack is determined by water jet slot. If the induced effect of inclined natural fracture directly affects blast-induced crack, the propagation direction of blast-induced main crack is determined by water jet slot in initial stage, and the propagation direction of blast-induced main crack is determined by inclined natural fracture in subsequent stage. When the distance increases further and exceeds a certain value, inclined natural fracture cannot affect the guiding effect of water jet slot in propagation process of blast-induced main crack. In addition, it should be noted that the effect of strain rate on the propagation law of blast-induced crack is not considered in this article, which will be studied in the follow-up work.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

Financial support for this work was provided by the Natural Science Foundation of Southwest University of Science and Technology (18zx7124).