Hybrid Finite-Discrete Element Modelling of Dynamic Fracture of Rock and Resultant Fragment Arching by Rock Blast H.M. An a, b and H.Y. Liu a * a School of Engineering and ICT, the University of Tasmania, Hobart, Australia b School of Civil & Environmental Engineering, University of Science and Technology Beijing, China * Corresponding author's Email: Hong.Liu@utas.edu.au Abstract A hybrid finite-discrete element method is introduced to simulate dynamic fracture of rock and resultant fragment arching caused by rock blast. Three typical examples, i.e. blast in a rock mass with a single borehole and a free surface, simultaneous blast, and consecutive blast with various delay times, are modelled and the obtained stress wave propagation, fracture process and resultant fragment movement are compared with those well documented in literatures. It is found that the hybrid finite-discrete element method reproduces the dynamic fracture of rock in rock blast from the stress wave propagation to the formations of crushed zone and cracked zone around the boreholes, the propagation of long radial cracks resulting in rock fracture, the casting of resultant fragments into air, and the fragment arching. It is concluded that the hybrid finite-discrete element method is a valuable numerical tool for the study on rock blast and a more advanced numerical method compared with the finite element method or discrete element method in terms of modelling dynamic fracture of rock. Keywords: Hybrid FEM/DEM, Rock blast, Dynamic fracture, Stress wave, Fragment arching 1. Introduction Blasting is a popular method in modern civil and mining industries, which is frequently employed in rock fragmentation, hard rock tunnelling and structure demolition. However, rock blasting is an extremely complex process and generally, involves explosive detonation, gas expansion, stress wave propagation, rock fracturing and resultant rock fragment flowing and muck-piling. Lack of understanding the complex process of the rock blasting has limited engineers to optimize rock blast design. Thus, it is imperative to study the rock blasting process. Many researchers have focused their study on the rock blasting process in order to understand the rock fragmentation mechanism and then improve rock blasting efficiency. In early stages, empirical model was put forward by some researchers for the daily design such as Kuz-Ram model in 1980. Because the empirical model requires only a few of input parameters from the engineering applications, it can be easily applied in routine blast design layout spreadsheets (Cunningham, 1983; 1987). However, limited input parameters may lead to inaccurate prediction. Moreover, the empirical model may not be able to satisfy the requirements of the modern rock blasting engineering since rock fracture and fragmentation progresses are totally ignored in the empirical model. Thanks to the fast development of the computer technology, it is nowadays possible to complete large-scale numerical calculations in a short time. Correspondingly, many numerical methods have been developed to simulate the rock blasting processes. In general, the numerical methods available in literatures can be classified according to their hypothesis that whether the rock is modelled as a continuous or discontinuous material. The representative finite element software for modelling rock blasting includes ANSYS and ABAQUS. These simulations can reflect in some degree the stress wave propagation and fracture initiation and propagation. However, rock fragmentation process and resultant fragment movement can't be captured using the finite element methods due to their continual hypothesis. The typical discrete element method is DEM and DDA. The discrete element method is suitable to model the movement of blasting resultant rock fragment after rock fracture but has the limitation in modelling the transition from continua to discontinua through fracture and fragmentation (Ning et al., 2011). The emerging hybrid finite element and discrete element method may be one of the best methods for modelling the rock blasting process since it combines the advantages of both FEM and DEM but overcomes their shortcomings (Munjiza, 2004). This paper aims to model a series
8 th Asian Rock Mechanics Symposium ARMS8 of rock blasting using the hybrid finite and discrete element method implemented by Liu et al. (2013) in order to understand rock blasting mechanisms and then improve rock blasting efficiency. 2. Hybrid finite-discrete element method Hybrid finite-discrete element method usually consists of the following components: contact detection, contact interaction between individual bodies, deformability and transition from continuum to discontinuum, temporal integration scheme, and computational fluid dynamics (Mohammadi, 2003; Munjiza, 2004; Morin and Ficarazzo, 2006). A hybrid finite-discrete element method including all of components mentioned above has been developed by Liu et al., (2013) on the basis of their previous enriched finite element codes RT 2D (Liu et al., 2004) and TunGeo 3D (Liu, 2010) and the open-source combined finite-discrete element libraries Y 2D and Y 3D originally developed by Munjiza (2004) and Xiang et al., (2009), respectively. Here only the key component related to rock blasting, i.e. the simplified fluid pressure expansion model implemented into the hybrid finite-discrete element method, is introduced, as shown in Fig. 1. At step A, the blast has just occurred and fluid pressure is uniformly applied to the edge of the blasthole. Then these pressurized fluids push the surrounding rock outward and cracks form when the stresses satisfy the strength of the surrounding rock. Once the cracks are formed, their geometry is added to the blasthole and filled with fluids for the current step calculations. Accordingly, the borehole expands and obtains a new geometry with gas pressure acting on it as shown in Step B. The geometry of the blasthole changes significantly at each step since they are exposed to high gas pressures. As the borehole expands, the gas pressure decreases according to a user-defined equation of state. Because of the expansion of the gas pressure, cracks are initiated in rock surrounding the borehole and driven to propagate to form radial cracks as shown in Step C. Then the gas enters the formed radial cracks and further drives the radial cracks to propagate to form long radial cracks. a) Step A b) Step B c) Step C Fig. 1. Simplified fluid pressure expansion model of the hybrid finite-discretee element method 3. Hybrid finite-discrete element modelling of rock blasting In this section, the stress wave propagation and fracture process induced by blasting in a rock mass with one free surface is first modelled using the hybrid finite-discrete element method. The obtained results will be compared with those well documented in literatures to calibrate the hybrid finite-discrete element method. After calibration, the hybrid method will then be applied to model simultaneous detonations and consecutive detonations with various delay strategies. a) Geometrical model b) Finite-discrete element mesh around borehole Fig. 2. Geometrical and numerical models for rock blast in a rock mass with a single borehole and one free surface.
a) Stress wave propagation b) Fracture progressive process Fig. 3. Stress wave propagation and failure process by blast with one free surface 3.1 Stress wave propagation and fracture process by blast in a rock mass with one free surface One of the basic principles of designing rock blasting is the presence of a free face parallel or sub-parallel to the blastholes as detonation occurs. Thus a rock mass with one free surface and a borehole parallel to the free surface is first modelled using the hybrid finite-discrete element method. Because the denotation velocity of the explosive along the length of the borehole is very fast, it can be supposed that the explosive along the length of the borehole is detonated at the same time and the rock fracture mechanisms and the resultant fragmentation are similar in each section along the length of the borehole. Thus, the problem can be simplified as a two-dimensional plane strain problem, as shown in Fig. 2a. The numerical model comprises a borehole with a radius of 50 mm, which is 2 m far
8 th Asian Rock Mechanics Symposium ARMS8 from the free surface. The boundaries of the rock mass are put far away from the borehole in order to eliminate any influence caused by the reflected waves from the boundary except the free surface. As shown in Fig. 2b, triangular elements are used to discretize the model, and dense elements are employed in the interested area, i.e. the vicinity of the borehole. Fig. 3 shows the temporal distribution of the stress wave propagation and failure development induced by the rock blasting. It should be noted that, following the solid mechanics regulations, the compressive stresses are taken as negative, while the tensile stresses are regarded as negative. Moreover, in Fig. 3a, the colour represents the size of minor principal stress while, in Fig. 3b, the tensile and compressive failures are marked using the blue and red colours, respectively. As well recognized, the failures of rock mass are mainly caused by radial cracks induced by the tensile stresses. While the explosive is detonated, a stress wave is produced around the wall of the borehole and propagates radially out of the borehole. The highest pressure at the front of the stress wave may cause the rock mass around the borehole to crack. As shown in Fig. 3, the borehole wall began to crack at 0.15 ms and the borehole continuously expands due to the high gas pressure from 0.15 ~0.45 ms. After that, the stress wave propagates along the radical direction of the borehole in the form of circles, as shown in Fig. 3a at 0.55 ms. It is well known that the rock mass can sustain the compressive stresses several times larger than the tensile stress. Thus the tensile stress is easier to cause the rock crack compared with the compressive stress, which is why the crack propagates along the radial direction of the borehole. Meanwhile the high pressure gas may penetrate into the crack and accelerate the crack propagation. As the compressive stress wave propagates, it reaches the free surface at around 0.55 ms. The compressive stress wave is reflected in the free surface to become the tensile stress wave propagating toward the borehole, as shown in Fig. 3a at 0.555 ms. The rock mass around the free surface starts to crack by the tensile stress due to the reflection of the compressive stress wave at the free surface. Meanwhile, the long radial cracks initiated from the cracked zone around the bore hole propagate to coalesce with the tensile cracks initiated at the free surface. Finally, fragments are formed around the free surface by coalescence of the long radial cracks and the tensile cracks, which are be casted into air by the combination of the gas pressure and the stress wave. Fig. 4. Simulated crushed zone and cracked zone around the borehole and stress wave reflection around the free surface and effect of initial detonation pressure on them The hybrid finite-discrete element simulation reproduces the rock fragmentation process by rock blast in the rock mass with one free surface. First of all, a crushed area is produced by the stress wave propagation and the high pressure gas expansion, as shown in Fig. 4. Then the stress wave propagates into the rock mass inducing long radial cracks together with the gas pressure penetrating into radial cracks. After that, the compressive stress wave propagates to reach the free surface and reflect back to form a tensile stress wave, which cause the rock mass around the free surface fails due to the tensile stress. Finally, the radial cracks initiated from the cracking zone coalesce with the tensile cracks near the free surface to form rock fragments, which are casted into air due to the combination of the gas pressure and the stress wave. It is noted from Fig. 4 that the initial detonation pressure has effects on the final fracture pattern. 3.2 Stress wave propagation and fracture process by simultaneous blast In this section, a model with two borehole and one free surface is considered. In order to eliminate the influence caused by the reflected waves from the boundaries other than the free surface, the boundaries except the free surface are put far away from the borehole vicinity. Triangular elements are also used to discretize the model and the dense elements are employed in the vicinity of the borehole. Fig. 5 depicts the modelled stress wave propagation and fracture process caused by simultaneous blast. It can be seen from Fig. 5 that as soon as the boreholes are detonated, two independent stress fields and fracture zones are formed around the two boreholes. At 0.25 ms, the
fronts of the stress wave initiated from the two boreholes reach each other. The stress wave and fractures initiated from the two boreholes then interact with each other. After that, due to the close positions of the two boreholes, it seems that stress wave and fractures are initiated from a single borehole, as shown in Fig. 5. The final fracture pattern is also quite similar to that caused by blasting in a single borehole. Fig. 5 Stress wave propagation and failure process caused by simultaneous blast Fig. 6 highlights the interaction of the stress waves and fractures initiated from the two boreholes. It can be seen from Fig. 6 a), the stress waves first superimpose in the area between the two boreholes, labelled as the area No. 2 in Fig. 6 a). After that, the stress waves superimposed in the areas No. 1 and No. 3 at the same time. As soon as the stress waves superimpose, the fractures initiated from the two
8 th Asian Rock Mechanics Symposium ARMS8 borehole also interact with each other. At 0.75 ms, the rock mass between the two boreholes is cracked, which makes the high pressure gases in two separate boreholes perforatee with each other and forms a relative big fracture zone, as shown in Fig. 6 b). a) Interaction of stress waves b) Interaction of fractures Fig. 6 Superimposed stresses and fractures caused by simultaneouss blasts 3.3 Stress wave propagation and fracture process by consecutive blast with various delay times Fig. 7 summarizes the modelled stress wave propagation and fracture process caused by consecutive blast with a delay time of 0.4 ms. It can be seen from Fig. 7 that the stress wave caused by the left borehole reaches the free surface at 0.4 ms while there is nothing happed in the second borehole since there is a delay time of 0.4 ms. After that, the right borehole is denoted and the stress waves and fractures induced by the two boreholes interact with each other resulting in final fracture pattern shown in Fig. 7. However, due to short delay time, the final fracture pattern is quite similar to those by the blast in a single borehole or the simultaneous blast although the sizess of the crushed zone, the cracked zone, and the crater are different and the height of fragment casting into area varies. Fig. 7 Stress wave propagation and failure process caused by consecutive blast with 0.4 ms delay
Fig. 8 Stress wave propagation and failure process caused by consecutive blast with 1.0 ms delay Fig. 8 depicts the stress wave propagation and failure process caused by consecutive blast with a delay time of 1.0 ms. The comparison of Figs. 7 and 8 reveals that consecutive blast with an optimum delay time will result in better fragmentation. Moreover, it is found that with a short delay time, the fracture pattern caused by consecutive blast is close to that by simultaneous blast while, with a longer delay time, the fracture pattern around the second borehole is greatly influenced by that caused by the first borehole. 4. Conclusions A hybrid finite-discrete element method with a simplified gas pressure expansion model is used to model dynamic fracture of rock and resultant fragment arching in rock blast. The numerical simulation reproduces the dynamic fracture of rock in rock blast from the stress wave propagation to the formations of crushed zone and cracked zone around the boreholes, the propagation of long radial cracks resulting in rock fracture, the casting of resultant fragments into air, and the fragment arching. It is concluded that the hybrid finite-discrete element method is a valuable numerical tool since it naturally deal with the transition of rock mass from continua to discontinuum in rock blast compared with the traditional finite element method and discrete element method. Acknowledgements The first author's study at the University of Tasmania is partly supported by a visiting PhD scholarship provided by China Scholarship Council, which is greatly appreciated. References Cunningham, C., 1983, The Kuz-Ram model for prediction of fragmentation from blasting, Proceedings of the first international symposium on rock fragmentation by blasting, Lulea, Sweden. Cunningham, C., 1987, Fragmentation estimations and the Kuz-Ram model-four years on, Proc. 2nd Int. Symp. on Rock Fragmentation by Blasting. Liu, H.Y., Kou, S.Q., Lindqvist, P.A., and Tang, C.A., 2004, Numerical studies on the failure process and associated microseismicity in rock under triaxial compression, Tectonophysics 384: 149-174 Liu, H.Y., 2010, A numerical model for failure and collapse analysis of geostructures, Australian Geomechanics, 45(3): 11-19 Liu, H.Y., Kang, Y.M., and Lin, P., 2013, Hybrid finite-discrete element modeling of geomaterials fracture and fragment muck-piling, International Journal of Geotechnical Engineering, pp. 1-17. Mohammadi, S., 2003, Discontinuum mechanics: Using finite and discrete elements, WIT press Southampton. Morin, M.A., and Ficarazzo, F., 2006, Monte Carlo simulation as a tool to predict blasting fragmentation based on the Kuz Ram model, Computers & Geosciences 32(3): 352-359. Munjiza, A., 2004, The Combined Finite-Discrete Element Method, Wiley Online Library. Munjiza, A., Andrews, K.R.F., and White, J.K., 1999, Combined single and smeared crack model in combined finite-discrete element analysis, International Journal for Numerical Methods in Engineering, 44(1): 41-57. Ning, Y., Yang, J., Ma, G., and Chen P., 2011, Modelling rock blasting considering explosion gas penetration using discontinuous deformation analysis, Rock mechanics and Rock Engineering 44(4): 483-490. Xiang, J., Munjiza, A., and Latham, J.P., 2009, Finite strain, finite rotation quadratic tetrahedral element for the combined finite-discrete element method, International Journal for Numerical Methods in Engineering, 79, 946-978