Application of FDEM in Blasting Fragmentation of Jointed Rock Mass

Chenyu Xu, Yingguo Hu, Peng Li, Meishan Liu, Genquan Li
Key Laboratory of Geotechnical Mechanics and Engineering of the Ministry of Water Resources, Changjiang River Scientific Research Institute, Wuhan, Hubei 430010, China

Abstract

The finite element and discrete element coupling method (FDEM) is an advanced numerical calculation method that is highly suitable for simulating the entire rock blasting process. Considering that the rock mass contains many joints, the present study introduces a rock joint constitutive model to restore the transmission and reflection phenomena of blasting stress waves when they reach the joint. At the same time, based on the original FDEM code, an optimized blasting calculation model is proposed. This model considers the effect of detonation gas and accurately describes the physical relationship between the detonation gas pressure and the change in the blasting chamber area caused by crack propagation. To overcome the limitation of previous blasting models that could only apply the pressure of the detonation gas to the borehole wall, the present study optimized the determination conditions for crack penetration and the calculation method for the blasting chamber area, and further considered the influence of the embedding effect of the detonation gas on the crack propagation. Finally, through two examples, the transmission and reflection laws of stress waves at joints and the entire process of rock mass throw blasting are simulated. The results illustrate that this model can capture the propagation of stress waves, the gas wedge effect of detonation gas, and the entire process of crack initiation, propagation, and penetration in the rock mass during the explosion, demonstrating the unlimited potential of FDEM in blasting simulation.

Keywords: FDEM; jointed rock mass; rock blasting; detonation gas

Introduction

Currently, simulation methods for rock blasting fragmentation can be divided into three major categories [1,2]: continuous simulation methods based on continuum mechanics (FEM, BEM, etc.), discontinuous methods based on Newtonian mechanics (DDA, DEM, etc.), and coupled continuous-discontinuous methods (SPH-FEM, FDEM, etc.). Continuous simulation methods offer shorter computation times and good stability for far-field blasting simulation, yielding satisfactory fragmentation results. However, the blasting process involves numerous discontinuous phenomena, and these methods lack quantitative description of crack initiation and propagation. Discontinuous simulation methods provide better phenomenological representation of blasting but are computationally expensive and exhibit insufficient stability and accuracy in mechanical indicators such as stress, velocity, and acceleration time histories. While they demonstrate clear advantages in simulating blasting throwing and crack propagation, their accuracy heavily depends on element/block size and contact parameter settings.

Both continuous and discontinuous numerical methods have inherent limitations when simulating blasting processes, making it significant to develop coupled continuous-discontinuous simulation methods that combine their advantages. Two main strategies exist for such methods. The first involves coupling established continuous and discontinuous methods: Saiang [3] coupled PFC2D and FLAC2D to study the spatial distribution characteristics of blast-induced damage and parameter degradation mechanisms; Wang Zhiliang et al. [4] and Huang Yonghui et al. [5] implemented SPH-FEM coupling for rock blasting simulation, using SPH in the near-field and FEM in the far-field. The second strategy establishes new calculation methods: Feng Chun et al. [6] simulated three-dimensional bench blasting in open-pit mines using CDEM; Yan Chengzeng et al. [7] proposed a rock fracture simulation method in FDEM that considers gas-driven crack propagation; Ren Huimin et al. [8] simulated rock foundation pit blasting using CDEM and conducted comparative analysis of average fragmentation size, limiting fragmentation size, boulder yield, and system fragmentation degree. Such methods, combining the advantages of continuous and discontinuous approaches, are becoming important tools for rock blasting fragmentation simulation.

This study improves the FDEM source code by introducing a real joint constitutive model and a blasting calculation model that considers the dynamic effect of detonation gas on rock mass. Through examples, the improved FDEM demonstrates accurate reproduction of the entire rock blasting process and shows sufficient application potential.

2.1 Representation of Structural Planes in FDEM

Currently, the representation of structural planes (faults, joints, bedding, weak layers) in FDEM is shown in Figure 1 [FIGURE:1]. Thick solid elements comprising triangular and joint elements are primarily used for simulating weak underlying layers, while zero-thickness joint elements can simulate real joints (bedding) through parameter redefinition and modification. Based on the cementation and filling conditions between real joints, these can be further classified into rigid structural planes and weak structural planes. Rigid structural planes are considered as already damaged joint elements, whereas weak structural planes have intact joint elements with parameters determined according to actual joint conditions. For comprehensive analysis, this study selects the weak structural plane approach to represent real joints.

Utilizing established and validated real joint constitutive models (linear elastic model [9], BB model [10], etc.), this study introduces a stress-based failure criterion and incorporates a linear joint constitutive relationship into FDEM, expressed as follows:

$$
\begin{cases}
\sigma = k_n d \
\tau = k_s s \quad \text{when } \tau \le \tau_s
\end{cases}
$$

where $d$ and $s$ are the normal and shear displacements of the joint, respectively; $\sigma$ and $\tau$ are the normal and shear stresses; $k_n$ and $k_s$ are the normal and shear stiffness of the joint; and $\tau_s$ is the maximum shear stress.

3 Blasting Process Simulation Considering Detonation Gas

Numerous studies [11,12] have demonstrated that rock failure results from the combined action of stress waves and detonation gas. However, almost all current commercial software [13] cannot accurately characterize both the action process of detonation gas on rock fracture and the entire crack evolution cycle simultaneously. As established, FDEM is an advanced numerical method highly suitable for simulating the complete blasting process, yet the original FDEM requires optimization to achieve refined simulation of the entire blasting process. This study improves upon the single-hole blasting simulation method of Yan Chengzeng et al. [7], simplifies the blasting chamber area calculation process to facilitate consideration of multi-hole blasting scenarios, and further extends this method to additional application contexts.

Figure 2 [FIGURE:2] presents a cross-sectional schematic of borehole chamber expansion during rock blasting in FDEM. $R_0$ is the initial borehole radius, and $R$ is the diffusion radius of detonation gas in fractures at time $t$, where $R = V_p \cdot t$ and $V_p$ is the gas diffusion velocity. The detonation gas is assumed to be an ideal gas diffusing uniformly at constant velocity. The chamber area calculation assumes that triangular elements are relatively rigid, ignoring area changes from mutual embedding, and that triangular elements on the gas diffusion line contribute half their area. The final chamber area is given by Equation (2), approximated as the area after gas diffusion minus the initial area and triangular element area.

After determining the total area occupied by detonation gas, the current gas pressure is determined based on the ratio of the current area to the initial gas area. The instantaneous chamber pressure is calculated using the exponential pressure equation shown in Equation (3) [11], and gas pressure is then applied to all fracture boundaries connected to the chamber. For isolated blocks surrounded by detonation gas, gas pressure is applied uniformly to their outer boundaries.

$$
P = P_0 \left(\frac{V_0}{V}\right)^\gamma
$$

where $P$ and $P_0$ are the current and initial detonation gas pressures, respectively; $V$ and $V_0$ are the current and initial chamber areas; and $\gamma$ is a constant related to explosive and rock properties.

4.1 P-Wave Propagation at a Single Linear Deformable Joint

Schoenberg et al. [9] and Pyrak-Nolte et al. [14] derived the transmission and reflection coefficients for a normally incident harmonic P-wave crossing a single linear deformable joint using the displacement discontinuity model:

$$
T_{\text{lin}} = \frac{2k_n}{2k_n + i\omega z_p}
$$

$$
R_{\text{lin}} = \frac{-i\omega z_p}{2k_n + i\omega z_p}
$$

where $T_{\text{lin}}$ and $R_{\text{lin}}$ are the transmission and reflection coefficients for a P-wave crossing a single linear joint; $k_n$ is the joint normal stiffness; $\omega$ is the angular frequency of the harmonic wave; and $z_p$ is the P-wave impedance, defined as the product of P-wave velocity and rock density. The magnitudes $|T_{\text{lin}}|$ and $|R_{\text{lin}}|$ depend on the normalized joint normal stiffness $K_n = k_n/(\omega z_p)$. $|T_{\text{lin}}|$ increases with increasing $K_n$, approaching 1 when $K_n$ is large, and satisfies $|T_{\text{lin}}|^2 + |R_{\text{lin}}|^2 = 1$.

The numerical analysis model for stress wave propagation across a joint is shown in Figure 3 [FIGURE:3], with the coordinate origin at the lower left corner. The model measures 300 m in length and 1 m in width, with the joint located at the center. A half-cycle sinusoidal wave with a frequency of 20 Hz and velocity amplitude of 1 m/s is vertically incident from the left boundary and propagates along the x-direction. Viscous boundary conditions are applied at both ends of the model to avoid wave reflections from artificial boundaries. The entire model has constrained displacement in the y-direction and free displacement in the x-direction. Monitoring points A (75, 0.5) and B (225, 0.5) record the time histories of incident, reflected, and transmitted waves. During numerical simulation, rock material is assumed to be perfectly elastic because wave attenuation in rock mass is primarily caused by geological structures such as joints [15,16], with material attenuation having minimal effect. Since this study focuses on joint effects on wave propagation, rock material attenuation is neglected. The variation characteristics of transmission (reflection) coefficients (ratio of transmitted/reflected wave amplitude to incident wave amplitude) are investigated by varying joint normal stiffness from 5×10⁷ to 5×10⁹ Pa/m (corresponding to normalized joint normal stiffness of 0.049–4.989). Specific parameters are listed in Table 1 [TABLE:1].

Table 1 Parameters for numerical model of P-wave crossing a single joint

Figure 4 [FIGURE:4] presents the velocity time histories at monitoring points A and B for P-waves crossing single joints with different normal stiffness values. Figure 5 [FIGURE:5] compares the theoretical solutions and FDEM numerical solutions for transmission coefficients at different normalized joint normal stiffness values. The results indicate that when the incident wave reaches the joint, transmitted waves (monitored at point A) and reflected waves (monitored at point B) are generated, both with peaks smaller than the incident wave. As k_n increases, the transmitted wave peak increases while the reflected wave peak decreases, meaning the transmission coefficient increases and the reflection coefficient decreases. The variation of transmission coefficients with normalized joint normal stiffness obtained by FDEM is consistent with theoretical solutions.

4.2 Throwing Blasting

A rock blasting model was established (Figure 6 [FIGURE:6]) with dimensions of 16 m in length and 7 m in height. The borehole radius is R₀ = 0.05 m, the borehole center coordinate is O(0,0), and the minimum burden distance is 2 m. The model comprises 8,950 triangular elements and 13,542 zero-thickness joint elements. The joint elements employ the real joint constitutive model introduced above, with a joint normal stiffness of 6×10⁹ Pa/m. The initial peak pressure of detonation gas is P₀ = 0.2 GPa, the gas diffusion velocity is v_p = 200 m/s, and the constant related to explosive and rock properties is γ = 1.4.

Two scenarios were simulated: without and with detonation gas. The entire blasting process is illustrated in Figure 7 [FIGURE:7]. The crack evolution process is shown in Figure 8 [FIGURE:8]. Rock mass around the borehole first develops radial initial shear cracks under the action of blasting stress waves, while tensile cracks dominate in the far field. The detonation gas diffuses slower than the stress wave and subsequently creates numerous tensile cracks, playing a primary role in rock fragmentation. Figure 8(a) shows the time histories of crack numbers for both scenarios: 1,901 cracks in the case without gas and 3,367 cracks with gas, indicating that detonation gas can significantly increase the number of blasting-induced cracks. Figure 8(b) presents the velocity time history at monitoring point A, demonstrating that detonation gas accelerates rock throwing. Figure 8(c) shows the time histories of chamber area and detonation gas pressure. Due to the approximate calculation of chamber volume, the gas pressure curve exhibits significant fluctuations but shows an overall exponential decay trend consistent with Equation (3). Overall, the improved FDEM proposed in this study can capture the entire process of stress wave propagation, crack initiation and propagation under the combined action of stress waves and detonation gas, as well as subsequent block separation and throwing, demonstrating the method's potential for blasting simulation. The limitation of this example is that it does not comparatively analyze the influence of joint presence on blasting effects, which will be investigated in future studies.

Conclusions

(1) This study determined the representation method for real joints in FDEM and established a real joint constitutive model, thereby enabling simulation of the complete transmission and reflection process of blasting stress waves at joint planes. The simulation results demonstrate that FDEM can accurately reproduce the propagation patterns and attenuation characteristics of blasting stress waves in joints, providing a necessary computational foundation for subsequent research on the complete blasting process in jointed rock mass.

(2) Building upon previous work, this study further optimized the FDEM blasting calculation model. The model can consider the effect of detonation gas, with improved determination conditions for crack penetration and calculation methods for blasting chamber area. This enables accurate capture of chamber volume and detonation gas pressure, thereby reproducing the promoting effect of detonation gas on crack development and demonstrating the method's potential for blasting simulation.

(3) The study discussed the rock failure mechanism in rock blasting, concluding that after stress waves create initial radial fractures, detonation gas expansion and embedding into fractures play a primary role in rock fragmentation and throwing. Future research will focus on three-dimensional FDEM and GPU acceleration for large-scale computations to further advance FDEM application in practical blasting engineering.

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