#### Abstract

The open-pit mining slopes continue to become higher and steeper with the continuous exploitation of near-surface mineral resources. The blasting excavation exerts a significance influence on the slope stability. In fact, intact rock slopes do not exist and slope stability is controlled by the geological defects. In this paper, the stability of a rock slope imbedded with a fault is considered. The P-wave component of blasting seismic waves is focused on and the fault is simplified as a semi-infinite crack. In background of Daye iron mine, the peak particle velocity (PPV) threshold is determined based on the linear elastic fracture dynamics. The influence of frequency, Young's modulus, and Poisson's ratio is studied to modify the PPV threshold. Results show that (1) the PPV threshold decreases with the increasing Young's modulus and Poisson's ratio, but increases with the increasing frequency; (2) the initiation angle is immune to Young's modulus and the frequency, and only depends on the Poisson's ratio; (3) the PPV criterion is finally determined as 1.47 cm/s when the frequency *f* ≤ 10 Hz, 1.47 cm/s–3.30 cm/s when 10 Hz < *f* ≤ 50 Hz and 3.37 cm/s–6.59 cm/s when *f* > 50 Hz, which are far less than that of intact rock slopes; (4) The north slope is quite safe if the proposed PPV threshold is not violated due to the variation range of the initiation angle *θ*_{0}.

#### 1. Introduction

With the continuous exploitation of near-surface mineral resources in China, the open-pit mining slopes continue to become higher and steeper, whose stability is declining [1]. Due to depletion, underground mining will be adopted in more and more mines. As the necessary means in mining, rock excavation, and other constructions, blasting brings huge benefits; on the other hand, blasting-induced geological disasters are nothing new across the world [2].

Dynamic stability has been extensively studied around the world. Ausilioe et al. studied the dynamic stability of reinforced slopes by considering different failure modes [3, 4]. Li et al. studied the dynamic stability of a rock slope based the Hoek-Brown failure criterion [5]. Chen et al. proposed the equivalent acceleration and Sarma method to analyze the dynamic stability of a rock slope in Jinping I hydropower station [6, 7]. Numerical simulation is a powerful method to analyze and evaluate the slope stability. Deb et al. studied the dynamic stability of the rock slope in Pasir coal mine based the finite element method and in-situ vibration tests [8]. Morales-Esteban et al. studied the stability of the San Pedro cliff at the Alhambra in Granada, Spain, based on the finite element method [9]. Kumar et al. [10] analyzed the stability of rock slopes under the impact of underground blasting by the finite difference method. Zheng et al. [11] adopted the discrete element method and the strength reduction method to analyze the stability of the slope in an iron mine. Liu et al. [12] adopted the discontinuous deformation analysis method to evaluate the slope stability under bench blasting. Cui et al. [13, 14] adopted the particle flow to simulate the failure process of pit slopes.

In fact, intact rock slopes do not exist. All the slopes contain geological defects, such as fault, potential sliding surface. Consequently, the slope stability is controlled by these geological defects. Some but insufficient studies on slope stability have been carried out in the recent years. Ouyang et al. [15] constructed a similar laboratory slope model and discussed the influence of weak layers on slope stability subjected to a single borehole basting. Huang and Wu [16] derived the formula of slope stability factor under the incidence of blasting waves and discussed the influence of duration and dominant frequency.

In blasting engineering, the peak particle velocity (PPV) is always selected as the index for the dynamic stability of rock slopes [17–24]. In general, blasting seismic waves consist of body waves (including P- and S-waves) and surface waves. Both body and surface waves exert a significant influence on the fault. Because the tip of fault is always deeply buried, we can focus on the interaction between body waves and the fault tip and neglect the influence of surface waves. Consequently, the fault can be simplified as semi-infinite crack and the blasting P-wave is selected to study the interaction based on the dynamic fracture mechanics in this paper. The formula of PPV threshold is derived to evaluate slope stability. In background on Daye iron mine, the PPV threshold is proposed.

#### 2. Interaction of P-Wave and Fault

##### 2.1. The Simplified Model

A right-handed Cartesian coordinate system *Oxy* is established, as shown in Figure 1. We suppose that a semi-infinite crack occupies the negative *x-*axis and the crack tip is at the original point. A time-harmonic plane P-wave impinges on the crack. The incident angle is denoted as *γ*_{1}. In order to analyze the stress field near the crack tip, a polar coordinate system (*r*, *θ*) is also set up. The potential function of the time-harmonic plane P-wave can be expressed in terms of the displacement potential aswhere *φ*_{0} is the amplitude of the incident P-wave, *α*_{1} is the wave number of P-waves and , *ω* is the circular frequency of incident wave, *c*_{1} is the wave speed of P-waves and , and are Lame’s constants, is the density of rock mass, and is the unit of complex.

In general, when the plane P-wave arrives at the crack, two diffracted waves will be generated. They are the diffracted P-wave *φ*^{(d)} and the diffracted SV-wave *ψ*^{(d)}. Let *φ* denote the linear sum of the potentials of the incident P-wave and the diffracted P-wave and *ψ* denote the linear sum of the potential of the diffracted SV-wave, thenwhere the diffracted field must satisfy the condition *φ*^{(d)}, *ψ*^{(d)} ⟶ 0 as (*x*^{2} *+* *y*^{2})^{1/2} ⟶ ∞.where *c*_{2} is the wave speed of SV-waves and .

Due to that the incident P-wave is time-harmonic, the diffracted waves should be time-harmonic. Then the potentials of diffracted waves can be expressed aswhere and are governed by the following Helmholtz equations:

The boundary condition on the crack can be written as:

Since and are known, the boundary conditions are only a problem for diffracted waves. In order to facilitate the analysis, the problem can be divided into a symmetric part about the *x-*axis and the antisymmetric part. As a consequence, the diffracted filed is divided into two parts: *φ*^{(s)}, *ψ*^{(s)} is symmetric about the *x-*axis, and *φ*^{(a)}, *ψ*^{(a)} is antisymmetric. Then the corresponding boundary conditions arefor the symmetric part andfor the antisymmetric part. Then the solutions can be expressed aswhere , the integral path is shown in Figure 2.

Let and , the boundary conditions can be derived as the two following dual integral equations:where , , , .

Equations (11) and (12) can be solved by the Wiener-Hopf technique [25, 26] and the solutions can be expressed aswhere , *α*_{R} is the wave number of Rayleigh wave and given by the minimum positive real root of the equation .

Because the incident wave will not cause the stress concentration near the crack tip, the stress field immediate near the crack tip is only determined by the diffracted waves. With the boundary conditions taken into consideration, the mode I stress intensity factor (SIF) is given byand the mode II SIF is given bywhere and .

The PPV normal to the wavefront of the incident wave can be expressed as

Substituting equation (16) into (14) and (15), we can obtain the SIFs in terms of as follows:

##### 2.2. The Safety Criterion Model

The stress field near the crack tip in the polar coordinate system can be expressed as [27]

In fracture mechanics, the maximum circumferential tensile stress criterion is always adopted to analyze fracture cracking. As a consequence, the safety criterion can be expresses aswhere and *K*_{IC} is the measured fracture toughness.

#### 3. Numerical Results and Discussions

Daye iron mine is located in Hubei Province, China. At its east open pit, the stability of the north slope is controlled by the faults F9 and F9′, as shown in Figure 3. The blasting excavation area is between the north and south slopes. According to geological data, the incident angle of blasting waves varies from 0 to 8^{o} at the tip of F9′ and from 48^{o} to 60^{o} at the tip of F9.

It is obvious that the PPV threshold depends on the mechanical parameters of rock mass and the property of blasting waves. The mode I fracture toughness *K*_{IC} is measured as 1 MPa∙m^{1/2}. The parameters are selected as *ρ* *=* 2710 kg/m^{3}, *ν* = 0.25. The PPV threshold vs. incident angle *γ*_{1} curves are plotted in Figure 4.

It can be seen from Figure 4 that decreases with the increasing incident angle *γ*_{1} for the same *E* and also decreases with the increasing *E* for the same *γ*_{1}. The least occurs when *γ*_{1} = 60^{o} and it increases from 2.24 cm/s to 10.02 cm/s when the frequency *f* increases from 10 Hz to 200 Hz. Due to the sensitivity, the PPV criterion should be strictly controlled and given in accordance with the variation range of frequency. With the *Blasting Safety Regulation* (GB6722-2014) in China taken into consideration, the PPV criterion is determined as 2.24 cm/s when *f* ≤ 10 Hz, 2.24 cm/s–5.01 cm/s when 10 Hz < *f* ≤ 50 Hz, and 5.01 cm/s–10.02 cm/s when *f* > 50 Hz.

The variation of the initiation angle *θ*_{0} with *γ*_{1} for different frequencies is shown in Figure 5. It can be seen from Figure 5 that the curves totally overlap for different frequencies, indicating that *θ*_{0} is immune to *f.* The initiation angle varies from −48.3° to 0° and we can claim that the north slope is quite safe if the proposed PPV threshold is not violated.

In order to get a more general and more applicable threshold, the influence of Young’s modulus *E* and Poisson’s ratio will be discussed.

##### 3.1. The Influence of Young’s Modulus

As mentioned before, decreases monotonously with increasing *f.* The parameters are selected as *ρ* *=* 2710 kg/m^{3}, *ν* = 0.25. The variation of with *E* for different *f* when *γ*_{1} = 60^{o} is shown in Figure 6.

It can be seen from Figure 6 that decreases monotonously and quickly with the increasing *E.* decreases from 5.88 cm/s to 1.53 cm/s when *E* increases from 10 GPa to 30 GPa and *f* *=* 10 Hz, decreases from 13.14 cm/s to 3.43 cm/s when *f* *=* 50 Hz, and decreases from 26.29 cm/s to 6.86 cm/s when *f* *=* 200 Hz. As a consequence, the PPV criterion should be modified as 1.53 cm/s when *f* ≤ 10 Hz, 1.53 cm/s–3.43 cm/s when 10 Hz < *f* ≤ 50 Hz, and 3.43 cm/s–6.86 cm/s when *f* > 50 Hz.

The variation of the initiation angle *θ*_{0} with *γ*_{1} for different *E* when *f* *=* 10 Hz is shown in Figure 7. It can be seen from Figure 7 that *θ*_{0} is immune to *E.* The initiation angle still varies from −48.3° to 0°, and we can claim that the north slope is quite safe if the proposed PPV threshold is not violated.

##### 3.2. The Influence of Poisson’s Ratio

As mentioned before, decreases monotonously with the increasing *E.* For the safety, we choose the parameters as *E* = 30 GPa, *f* = 10 Hz, 50 Hz and 200 Hz. The variation of with Poisson’s ratio for different frequencies is shown in Figure 8.

It can be seen from Figure 8 that decreases monotonously and slowly with the increasing decreases from 1.59 cm/s to 1.47 cm/s when increases from 0.10 to 0.30 and *f* *=* 10 Hz, decreases from 3.57 cm/s to 3.30 cm/s when *f* *=* 50 Hz, and decreases from 7.15 cm/s to 6.59 cm/s when *f* *=* 200 Hz. As a consequence, the PPV criterion should be modified as 1.47 cm/s when *f* ≤ 10 Hz, 1.47 cm/s–3.30 cm/s when 10 Hz < *f* ≤ 50 Hz, and 3.37 cm/s–6.59 cm/s when *f* > 50 Hz.

The variation of the initiation angle *θ*_{0} with *γ*_{1} for different *E* when *f* *=* 10 Hz is shown in Figure 9. It can be seen from Figure 9 that *θ*_{0} changes obviously with the increasing . The maximum *θ*_{0} varies from −54.7° to −39.3°. Although the variation range of *θ*_{0} narrows, no positive *θ*_{0} occurs and we can claim that the north slope is quite safe if the proposed PPV threshold is not violated.

#### 4. Conclusions

The following conclusions can be drawn:(1)The PPV threshold decreases with the increasing Young’s modulus *E* and the Poisson’s ratio , but increases with the increasing frequency *f.*(2)The initiation angle *θ*_{0} is immune to Young's modulus *E* and the frequency *f* and only depends on Poisson’s ratio .(3)The PPV criterion is finally determined as 1.47 cm/s when *f* ≤ 10 Hz, 1.47 cm/s–3.30 cm/s when 10 Hz < *f* ≤ 50 Hz, and 3.37 cm/s–6.59 cm/s when *f* > 50 Hz, which are far less than that of intact rock slopes.(4)The north slope is quite safe if the proposed PPV threshold is not violated due to the variation range of the initiation angle *θ*_{0}.

#### Data Availability

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

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

#### Acknowledgments

This work was supported by the Hubei Natural Science Foundation Project (Grant no. 2019CFB224), the scientific research project of Hubei Provincial Department of Education (Grant no. Q20191308), and the Jingzhou Science and Technology Project (Grant no. 2019Z18001).These grants are gratefully acknowledged.