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

Authors: Shiwei Lu; Chuanbo Zhou; Zhen Zhang; Ling Ji; Nan Jiang
Journal: Shock and Vibration
 (2020)

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## 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 frequencyf ≤ 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.


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## Body

## 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 systemOxy 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 γ1. 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 as(1)φix,y,t=φ0e−iα1xcosγ1+ysinγ1+ωt,where φ0 is the amplitude of the incident P-wave, α1 is the wave number of P-waves and α1=w/c1, ω is the circular frequency of incident wave, c1 is the wave speed of P-waves and c1=λ+2μ/ρ, λ and μ are Lame’s constants, ρ is the density of rock mass, and i is the unit of complex.Figure 1
Interaction between P-wave and fault.In general, when the plane P-wave arrives at the crack, two diffracted waves will be generated. They are the diffracted P-waveφ(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(2)φx,y,t=φix,y,t+φdx,y,t,ψx,y,t=ψdx,y,t,where the diffracted field must satisfy the condition φ(d), ψ(d) ⟶ 0 as (x2 + y2)1/2 ⟶ ∞.(3)∇2φ=1c12∂2φ∂t2,∇2ψ=1c22∂2ψ∂t2,where c2 is the wave speed of SV-waves and c2=μ/ρ.(4)ux=∂φ∂x+∂ψ∂y,uy=∂φ∂y−∂ψ∂x,σxx=λ∇2φ+2μ∂2φ∂x2+∂2ψ∂x∂y,σyy=λ∇2φ+2μ∂2φ∂y2−∂2ψ∂x∂y,σxy=μ2∂2φ∂x∂y−∂2ψ∂x2+∂2ψ∂y2.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 as(5)φdx,y,t=φ^dx,ye−iωt,ψdx,y,t=ψ^dx,ye−iωt,where φ^dx,y and ψ^dx,y are governed by the following Helmholtz equations:(6)∇2+α12φ^dx,y=0,∇2+α22ψ^dx,y=0.The boundary condition on the crack can be written as:(7)σyyix,0,t+σyydx,0,t=0,σxyix,0,t+σyydx,0,t=0.x<0.Sinceσyyi and σxyi 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 are(8)σxysx,0,t=0,uysx,0,t=0,x>0,σyyix,0,t+σyysx,0,t=0,σxysx,0,t=0,x<0for the symmetric part and(9)σyyax,0,t=0,uxax,0,t=0,x>0,σxyix,0,t+σxyax,0,t=0,σyyax,0,t=0,x<0,for the antisymmetric part. Then the solutions can be expressed as(10)φsx,y,t=12π∫−∞+∞A11se−β1y−isx−iωtds,ψsx,y,t=12π∫−∞+∞A12se−β2y−isx−iωtds,φsx,y,t=12π∫−∞+∞A21se−β1y−isx−iωtds,ψsx,y,t=12π∫−∞+∞A22se−β2y−isx−iωtds,where βj=s2−αj2s≥αj−iαj2−s2s<αj,j=1,2, the integral path is shown in Figure 2.Figure 2
Integral path ins-plane.LetA11sA12s=2/α22A1ss2−1/2α22−isβ1 and A21sA22s=2/α22A2sisβ2s2−1/2α22, the boundary conditions can be derived as the two following dual integral equations:(11)12π∫−∞+∞β1A1se−isxds=0,x>0,12π∫−∞+∞fsA1se−isxds=P1e−iα1xcosγ1,x<0,(12)12π∫−∞+∞β2A2se−isxds=0,x>0,12π∫−∞+∞fsA2se−isxds=Q1e−iα1xcosγ1,x<0,where P1=α24φ01−2κ2cos2γ1, Q1=κ2α24φ0sin2γ1, κ=α1/α2, fs=2s2−α222−4s2β1β2.Equations (11) and (12) can be solved by the Wiener-Hopf technique [25, 26] and the solutions can be expressed as(13)A1s=P12i⋅α1+α1cosγ1α12−α22αR+α1cosγ1F+α1cosγ1⋅1s−αRs−α1cosγ1s+α1F−s,A2s=Q12i⋅α2+α1cosγ1α12−α22αR+α1cosγ1F+α1cosγ1⋅1s−αRs−α1cosγ1s+α2F−s,where F±s=exp−1/π∫α1α2arctan4z2z2−α12α22−z2/2z2−α222dz/z±s, αR is the wave number of Rayleigh wave and given by the minimum positive real root of the equation 2−α22/αR22−41−α12/αR21−α22/αR2=0.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 by(14)KI=limx⟶02πxσyyx,0,t=μφ0α22πλ1KI1e−iωt+3π/4,and the mode II SIF is given by(15)KII=limx⟶02πxσxyx,0,t=μφ0α22πλ1KII1e−iωt+3π/4,where KI1=1−2κ2cos2γ11+cosγ1/παR/α1+cosγ1F+α1cosγ1 and KII1=κ2sin2γ1κ−1+cosγj/παR/α1+cosγ1F+α1cosγ1.The PPV normal to the wavefront of the incident wave can be expressed as(16)vP=κωα2φ0.Substituting equation (16) into (14) and (15), we can obtain the SIFs in terms of vP as follows:(17)KI=μvPα2ωκπλ1KI1e−iωt+3π/4,KII=μvPα2ωκπλ1KII1e−iωt+3π/4.


### 2.2. The Safety Criterion Model

The stress field near the crack tip in the polar coordinate system can be expressed as [27](18)σrr=122πrKI3−cosθcosθ2+KII3cosθ−1sinθ2,σθθ=122πrcosθ2KI1+cosθ−3KIIsinθ,σrθ=122πrcosθ2KIsinθ+KII3cosθ−1.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(19)vP=2KICωκμα2πλ1cosθ0/2KI11+cosθ0−3sinθ0KII1,where θ0=2arctan1−1+8KII/KI2/4KII/KI and KIC is the measured fracture toughness.


## 2.1. The Simplified Model

A right-handed Cartesian coordinate systemOxy 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 γ1. 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 as(1)φix,y,t=φ0e−iα1xcosγ1+ysinγ1+ωt,where φ0 is the amplitude of the incident P-wave, α1 is the wave number of P-waves and α1=w/c1, ω is the circular frequency of incident wave, c1 is the wave speed of P-waves and c1=λ+2μ/ρ, λ and μ are Lame’s constants, ρ is the density of rock mass, and i is the unit of complex.Figure 1
Interaction between P-wave and fault.In general, when the plane P-wave arrives at the crack, two diffracted waves will be generated. They are the diffracted P-waveφ(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(2)φx,y,t=φix,y,t+φdx,y,t,ψx,y,t=ψdx,y,t,where the diffracted field must satisfy the condition φ(d), ψ(d) ⟶ 0 as (x2 + y2)1/2 ⟶ ∞.(3)∇2φ=1c12∂2φ∂t2,∇2ψ=1c22∂2ψ∂t2,where c2 is the wave speed of SV-waves and c2=μ/ρ.(4)ux=∂φ∂x+∂ψ∂y,uy=∂φ∂y−∂ψ∂x,σxx=λ∇2φ+2μ∂2φ∂x2+∂2ψ∂x∂y,σyy=λ∇2φ+2μ∂2φ∂y2−∂2ψ∂x∂y,σxy=μ2∂2φ∂x∂y−∂2ψ∂x2+∂2ψ∂y2.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 as(5)φdx,y,t=φ^dx,ye−iωt,ψdx,y,t=ψ^dx,ye−iωt,where φ^dx,y and ψ^dx,y are governed by the following Helmholtz equations:(6)∇2+α12φ^dx,y=0,∇2+α22ψ^dx,y=0.The boundary condition on the crack can be written as:(7)σyyix,0,t+σyydx,0,t=0,σxyix,0,t+σyydx,0,t=0.x<0.Sinceσyyi and σxyi 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 are(8)σxysx,0,t=0,uysx,0,t=0,x>0,σyyix,0,t+σyysx,0,t=0,σxysx,0,t=0,x<0for the symmetric part and(9)σyyax,0,t=0,uxax,0,t=0,x>0,σxyix,0,t+σxyax,0,t=0,σyyax,0,t=0,x<0,for the antisymmetric part. Then the solutions can be expressed as(10)φsx,y,t=12π∫−∞+∞A11se−β1y−isx−iωtds,ψsx,y,t=12π∫−∞+∞A12se−β2y−isx−iωtds,φsx,y,t=12π∫−∞+∞A21se−β1y−isx−iωtds,ψsx,y,t=12π∫−∞+∞A22se−β2y−isx−iωtds,where βj=s2−αj2s≥αj−iαj2−s2s<αj,j=1,2, the integral path is shown in Figure 2.Figure 2
Integral path ins-plane.LetA11sA12s=2/α22A1ss2−1/2α22−isβ1 and A21sA22s=2/α22A2sisβ2s2−1/2α22, the boundary conditions can be derived as the two following dual integral equations:(11)12π∫−∞+∞β1A1se−isxds=0,x>0,12π∫−∞+∞fsA1se−isxds=P1e−iα1xcosγ1,x<0,(12)12π∫−∞+∞β2A2se−isxds=0,x>0,12π∫−∞+∞fsA2se−isxds=Q1e−iα1xcosγ1,x<0,where P1=α24φ01−2κ2cos2γ1, Q1=κ2α24φ0sin2γ1, κ=α1/α2, fs=2s2−α222−4s2β1β2.Equations (11) and (12) can be solved by the Wiener-Hopf technique [25, 26] and the solutions can be expressed as(13)A1s=P12i⋅α1+α1cosγ1α12−α22αR+α1cosγ1F+α1cosγ1⋅1s−αRs−α1cosγ1s+α1F−s,A2s=Q12i⋅α2+α1cosγ1α12−α22αR+α1cosγ1F+α1cosγ1⋅1s−αRs−α1cosγ1s+α2F−s,where F±s=exp−1/π∫α1α2arctan4z2z2−α12α22−z2/2z2−α222dz/z±s, αR is the wave number of Rayleigh wave and given by the minimum positive real root of the equation 2−α22/αR22−41−α12/αR21−α22/αR2=0.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 by(14)KI=limx⟶02πxσyyx,0,t=μφ0α22πλ1KI1e−iωt+3π/4,and the mode II SIF is given by(15)KII=limx⟶02πxσxyx,0,t=μφ0α22πλ1KII1e−iωt+3π/4,where KI1=1−2κ2cos2γ11+cosγ1/παR/α1+cosγ1F+α1cosγ1 and KII1=κ2sin2γ1κ−1+cosγj/παR/α1+cosγ1F+α1cosγ1.The PPV normal to the wavefront of the incident wave can be expressed as(16)vP=κωα2φ0.Substituting equation (16) into (14) and (15), we can obtain the SIFs in terms of vP as follows:(17)KI=μvPα2ωκπλ1KI1e−iωt+3π/4,KII=μvPα2ωκπλ1KII1e−iωt+3π/4.


## 2.2. The Safety Criterion Model

The stress field near the crack tip in the polar coordinate system can be expressed as [27](18)σrr=122πrKI3−cosθcosθ2+KII3cosθ−1sinθ2,σθθ=122πrcosθ2KI1+cosθ−3KIIsinθ,σrθ=122πrcosθ2KIsinθ+KII3cosθ−1.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(19)vP=2KICωκμα2πλ1cosθ0/2KI11+cosθ0−3sinθ0KII1,where θ0=2arctan1−1+8KII/KI2/4KII/KI and KIC 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 Figure3. 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 8o at the tip of F9′ and from 48o to 60o at the tip of F9.Figure 3
Profit of engineering geological conditions of Daye iron mine.It is obvious that the PPV thresholdvP depends on the mechanical parameters of rock mass and the property of blasting waves. The mode I fracture toughness KIC is measured as 1 MPa∙m1/2. The parameters are selected as ρ = 2710 kg/m3, ν = 0.25. The PPV threshold vP vs. incident angle γ1 curves are plotted in Figure 4.Figure 4
Variation ofvP with γ1 for different f.It can be seen from Figure4 that vP decreases with the increasing incident angle γ1 for the same E and also decreases with the increasing E for the same γ1. The least vP occurs when γ1 = 60o 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 angleθ0 with γ1 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 θ0 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.Figure 5
Variation ofθ0 with γ1 for different f.In order to get a more general and more applicable threshold, the influence of Young’s modulusE and Poisson’s ratio v will be discussed.


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

As mentioned before,vP decreases monotonously with increasing f. The parameters are selected as ρ = 2710 kg/m3, ν = 0.25. The variation of vP with E for different f when γ1 = 60o is shown in Figure 6.Figure 6
Variation ofvP with E for different f when γ1 = 60o and v = 0.25.It can be seen from Figure6 that vP decreases monotonously and quickly with the increasing E.vP decreases from 5.88 cm/s to 1.53 cm/s when E increases from 10 GPa to 30 GPa and f = 10 Hz, vP decreases from 13.14 cm/s to 3.43 cm/s when f = 50 Hz, and vP 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 angleθ0 with γ1 for different E when f = 10 Hz is shown in Figure 7. It can be seen from Figure 7 that θ0 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.Figure 7
Variation ofθ0 with γ1 for different E.


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

As mentioned before,vP 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 vP with Poisson’s ratio v for different frequencies is shown in Figure 8.Figure 8
Variation ofvP with v for different f when γ1 = 60o and E = 30 GPa.It can be seen from Figure8 that vP decreases monotonously and slowly with the increasing v.vP decreases from 1.59 cm/s to 1.47 cm/s when v increases from 0.10 to 0.30 and f = 10 Hz, vP decreases from 3.57 cm/s to 3.30 cm/s when f = 50 Hz, and vP 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 angleθ0 with γ1 for different E when f = 10 Hz is shown in Figure 9. It can be seen from Figure 9 that θ0 changes obviously with the increasing v. The maximum θ0 varies from −54.7° to −39.3°. Although the variation range of θ0 narrows, no positive θ0 occurs and we can claim that the north slope is quite safe if the proposed PPV threshold is not violated.Figure 9
Variation ofθ0 with γ1 for different v.


## 3.1. The Influence of Young’s Modulus

As mentioned before,vP decreases monotonously with increasing f. The parameters are selected as ρ = 2710 kg/m3, ν = 0.25. The variation of vP with E for different f when γ1 = 60o is shown in Figure 6.Figure 6
Variation ofvP with E for different f when γ1 = 60o and v = 0.25.It can be seen from Figure6 that vP decreases monotonously and quickly with the increasing E.vP decreases from 5.88 cm/s to 1.53 cm/s when E increases from 10 GPa to 30 GPa and f = 10 Hz, vP decreases from 13.14 cm/s to 3.43 cm/s when f = 50 Hz, and vP 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 angleθ0 with γ1 for different E when f = 10 Hz is shown in Figure 7. It can be seen from Figure 7 that θ0 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.Figure 7
Variation ofθ0 with γ1 for different E.


## 3.2. The Influence of Poisson’s Ratio

As mentioned before,vP 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 vP with Poisson’s ratio v for different frequencies is shown in Figure 8.Figure 8
Variation ofvP with v for different f when γ1 = 60o and E = 30 GPa.It can be seen from Figure8 that vP decreases monotonously and slowly with the increasing v.vP decreases from 1.59 cm/s to 1.47 cm/s when v increases from 0.10 to 0.30 and f = 10 Hz, vP decreases from 3.57 cm/s to 3.30 cm/s when f = 50 Hz, and vP 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 angleθ0 with γ1 for different E when f = 10 Hz is shown in Figure 9. It can be seen from Figure 9 that θ0 changes obviously with the increasing v. The maximum θ0 varies from −54.7° to −39.3°. Although the variation range of θ0 narrows, no positive θ0 occurs and we can claim that the north slope is quite safe if the proposed PPV threshold is not violated.Figure 9
Variation ofθ0 with γ1 for different v.


## 4. Conclusions

The following conclusions can be drawn:(1)
The PPV threshold decreases with the increasing Young’s modulusE and the Poisson’s ratio v, but increases with the increasing frequency f.(2)
The initiation angleθ0 is immune to Young's modulus E and the frequency f and only depends on Poisson’s ratio v.(3)
The PPV criterion is finally determined as 1.47 cm/s whenf ≤ 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 angleθ0.



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