PPV Criterion of a Rock Slope Imbedded with a Fault subjected to Blasting P-Waves

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 θ0.


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][18][19][20][21][22][23][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. e formula of PPV threshold is derived to evaluate slope stability. In background on Daye iron mine, the PPV threshold is proposed.

e 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 xaxis and the crack tip is at the original point. A time-harmonic plane P-wave impinges on the crack. e incident angle is denoted as c 1 . In order to analyze the stress field near the crack tip, a polar coordinate system (r, θ) is also set up. e potential function of the time-harmonic plane P-wave can be expressed in terms of the displacement potential as where φ 0 is the amplitude of the incident P-wave, α 1 is the wave number of P-waves and α 1 � (w/c 1 ), ω is the circular frequency of incident wave, c 1 is the wave speed of P-waves and c 1 � �������� � (λ + 2μ)/ρ, λ and μ are Lame's constants, ρ is the density of rock mass, and i is the unit of complex.
In general, when the plane P-wave arrives at the crack, two diffracted waves will be generated. ey 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, then where the diffracted field must satisfy the condition φ (d) , where c 2 is the wave speed of SV-waves and c 2 � ���� � (μ/ρ).
Due to that the incident P-wave is time-harmonic, the diffracted waves should be time-harmonic. en the potentials of diffracted waves can be expressed as where φ (d) (x, y) and ψ (d) (x, y) are governed by the following Helmholtz equations: e boundary condition on the crack can be written as: Since σ (i) yy and σ (i) xy 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.
en the corresponding boundary conditions are for the symmetric part and for the antisymmetric part. en the solutions can be expressed as path is shown in Figure 2.

e Safety Criterion
Model. e 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 as where θ 0 � 2 arctan(1 − ������������ � 1 + 8(K II /K I ) 2 /((4K II /K I ))) and K IC is the measured fracture toughness.

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. e 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 v P depends on the mechanical parameters of rock mass and the property of blasting waves. e mode I fracture toughness K IC is measured as 1 MPa•m 1/2 . e parameters are selected as ρ � 2710 kg/m 3 , ] � 0.25. e PPV threshold v P vs. incident angle c 1 curves are plotted in Figure 4.
It can be seen from Figure 4 that v P decreases with the increasing incident angle c 1 for the same E and also de-   e variation of the initiation angle θ 0 with c 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. e 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 v will be discussed.

e Influence of Young's Modulus.
As mentioned before, v P decreases monotonously with increasing f. e parameters are selected as ρ � 2710 kg/m 3 , ] � 0.25. e variation of v P with E for different f when c 1 � 60 o is shown in Figure 6.
It can be seen from Figure 6 that v P decreases monotonously and quickly with the increasing E. v P decreases from 5.88 cm/s to 1.53 cm/s when E increases from 10 GPa to 30 GPa and f � 10 Hz, v P decreases from 13.14 cm/s to 3.43 cm/s when f � 50 Hz, and v P 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. e variation of the initiation angle θ 0 with c 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. e 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.

e Influence of Poisson's Ratio.
As mentioned before, v P 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. e variation of v P with Poisson's ratio v for different frequencies is shown in Figure 8.
It can be seen from Figure 8 that v P decreases monotonously and slowly with the increasing v.v P decreases from 1.59 cm/s to 1.47 cm/s when v increases from 0.10 to 0.30 and f � 10 Hz, v P decreases from 3.57 cm/s to 3.30 cm/s when f � 50 Hz, and v P 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    Shock and Vibration 5 when 10 Hz < f ≤ 50 Hz, and 3.37 cm/s-6.59 cm/s when f > 50 Hz. e variation of the initiation angle θ 0 with c 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 v. e 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.

Conclusions
e following conclusions can be drawn: (1) e PPV threshold decreases with the increasing Young's modulus E and the Poisson's ratio v, but increases with the increasing frequency f. (2) e initiation angle θ 0 is immune to Young's modulus E and the frequency f and only depends on Poisson's ratio v. (3) e 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) e 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
e data used to support the findings of this study are included within the article.   Shock and Vibration