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DEM method with Voronoi grains
Engineering Analysis with Boundary Elements 108 (2019) 472–483
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Micro-mechanism study on rock breaking behavior under water jet impact using coupled SPH-FEM/DEM method with Voronoi grains Zhijun Wu a, Fangzheng Yu a, Penglin Zhang a, Xuewei Liu b,∗ a b
School of Civil Engineering, Wuhan University, Wuhan 430072, China Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China
a r t i c l e
i n f o
Keywords: SPH-FEM/DEM Water jet impact Rock breaking behavior Micro-structure Micro-mechanical properties
a b s t r a c t In this study, a two-dimensional SPH-FEM/DEM coupled model is developed to investigate rock breaking behavior under water jet impact based on the LS-DYNA software, where the SPH method is adopted to model the water jet and the FEM/DEM method is adopted to simulate rock breaking response, respectively. To better approximate the micro-structure of rock, a Voronoi tessellation technique is adopted to generate the random polygonal grains. A zero-thickness cohesive element is inserted along the boundaries of the Voronoi grains to model the mechanical interaction between grains as well as the breaking process of rock. Numerical water jet impact tests on sandstone are firstly conducted to verify the proposed method as well as calibrate the corresponding micro-parameters. Then, using the verified method, the effects of micro-structure and micro-mechanical properties on the rock breaking performance under water jet impact are systematically investigated. The numerical results show that the rock breaking performance is greatly affected by grain size and irregularity, ductility, microscopic strength and the heterogeneity of micro-parameters, whereas the contact stiffness ratio has little effect on the rock breaking performance under water jet impact.
1. Introduction Benefiting from the significant advantages, such as low cutting force, selective removal capability, high efficiency, dust-free, heat-free and vibration-free performance [1,2], water jet technology has been applied widely in many industry fields, especially in rock breaking and mining engineering. Rock breaking under water jet impact is a complex fluidsolid coupling process, involving the deformation and fracture of rocks, crack initiation and propagation and fluid flowing. At the microscopic scale, rock is not homogeneous but composed by mineral grains and micro-structures, such as grain interfaces and micro defects, which has a great influence on the macro-mechanical property of rock [3–7]. Therefore, in order to better understand the rock breaking behavior under the impact of water jet, it is necessary to consider the effects of rock microproperties. However, due to the complexity of rock micro-structures and the difficulty to obtain the micro-mechanical properties, revealing the micro-mechanism of rock breaking behavior under water jet impact is still very difficult. In the past decades, many methods have been proposed to investigate the rock breaking behavior under water jet impact. Among these methods, experimental research shows the most visible and reliable information. Lu and Huang [8] carried out impact tests on three kinds of sandstones at different velocities and concluded three failure patterns of ∗
sandstone: center broken pit caused by shear stress at low velocity, internal fractures caused by tensile stress at moderate velocity and macro cracks on the side surface at high velocity. In order to investigate the effect of pulsed length and pulsation frequency, Dehkhoda and Hood [9] conducted pulsed water jet impact on granite and marble and found that pulsation frequency influences fracture initiation and pulse length is associated with the crack opening energy. To investigate the effects of rock properties and operating parameters on the cutting depth, Engin [10] performed abrasive water jet (AWJ) cutting on 42 different types of natural rocks and the results indicated that the Shore hardness, Bohme surface abrasion resistance and density are the most significant rock properties while the working pump pressure, traverse velocity are the most significant operating parameters affecting the cutting depth. Using mass removal measurements, microscopical observations and the mercury penetration technique, Momber and Kovacevic [11] investigated the influence of grain inclusions on the fracture behavior of a multiphase brittle material exposed to water jet impact and found that the increase of inclusions leads to a reduction of the threshold destruction energy and finally controls the fracture evolution and materials removal performance. These experimental studies do have promoted the understanding of rock breaking mechanism under water jet impact. However, due to the opaque characteristic and complex micro-structure of rock, nowadays it is still challenging to experimentally establish the detailed
Corresponding author. E-mail addresses: [email protected] (Z. Wu), [email protected] (F. Yu), [email protected] (P. Zhang), [email protected] (X. Liu).
https://doi.org/10.1016/j.enganabound.2019.08.026 Received 15 May 2019; Received in revised form 15 July 2019; Accepted 24 August 2019 0955-7997/© 2019 Elsevier Ltd. All rights reserved.
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Engineering Analysis with Boundary Elements 108 (2019) 472–483
relationship between the rock micro-characteristics and the macroscopic breaking behavior under water jet impact. Recently, with the development of computer technology, more and more researchers are devoted to investigating the rock breaking behavior under water jet impact by means of numerical simulation. However, due to the huge mechanical differences of rock and water during rock breaking process under water jet impact, it may be better to adopt different methods to simulate their behavior [12,13]. For modeling water, due to the large deformation, the traditional Lagrangian grid-based finite element method (FEM) will be faced with the problems of precision decrease and even calculation termination induced by the distortion of the grids [14]. By allowing grid points separating and moving independently from material points, the Arbitrary Lagrange Euler (ALE) method can easily solve the large deformation problems and has been successfully applied to the water jet model [15–18]. However, when dealing with the coupling problem with rocks, the results of ALE method are very sensitive to the coupling parameters, which are difficult to be determined [19]. Alternatively, due to no meshing and simple coupling process, a meshless method, smooth particle hydrodynamics (SPH), shows good adaptation for dynamic large distortion problems such as the liquid-solid impact and has been widely applied into the water jet model [20–24]. Therefore, in this study, the SPH method is adopted to simulate the water jet process. However, modeling the rock breaking behavior by water jet also requires to accurately capture the rock failure process. Although the SPH method can also simulate the rock breaking process [25,26], it suffers from high computational expense and difficulty in enforcing the boundary conditions [27]. Therefore, in recent years, some SPH-coupled methods which take the advantages of SPH in modeling the water behavior and other methods in simulating the rock breaking process have been developed. Among them, the coupled FEM-SPH method is the most commonly used one [21,22]. However, the FEM model treats rocks into continuous and uniform grids, which deviates from the micro-structure of rocks, and even worse the continuous assumption makes FEM fail to realistically simulate the evolution of cracks during rock breaking process [28–30]. In addition to continuum-based methods represented by FEM, discontinuum-based methods and other meshless methods, such as discrete element method (DEM), discontinuous deformation analysis (DDA), and element bending group (EBG), can be also coupled with SPH to investigate the fluid-solid interactions [31–35] due to its powerful capability in representing the rock micro-structure. However, the discontinuum-based methods still have some limitations, such as difficulties in determining microscopic parameters and capturing the deformation and breaking process itself. Actually, rock is neither a complete continuum nor discontinuum but a combination of these two [36,37]. Therefore, the rock breaking process can be better simulated by the coupled continuous-discontinuous method [38–41]. As a such coupled method, the FEM/DEM model regards the material as a number of discrete interactive grains, so the key feature of rock failure process modeling, which involves the transition from continuum to discontinuum through deformation, fracture and fragmentation, can be easily achieved. With a contact algorithm, the grain separation and interaction can be correctly considered. Based on these features, the FEM/DEM model has been widely applied to simulate the rock breaking process [42–44]. Therefore, in this study, to study the micro-mechanism of rock breaking behavior under water jet impact, the coupled SPH-FEM/DEM model is implemented in LS-DYNA software, where the FEM/DEM method is adopted to simulate rock breaking response and the SPH method is adopted to model the water jet. To implement the FEM/DEM, two FORTRAN subroutines are self-developed and embedded into LS-DYNA. Firstly, the random polygonal Voronoi tessellation program is adopted to generate the micro-grains of rock for better representing the rock micro-structure and modeling the mechanical behavior of rock grains. Then, zerothickness cohesive elements are inserted along the boundaries of the Voronoi grains by the inserting program to model the mechanical in-
teraction between grains as well as the breaking process of rock. To verify the proposed model, a series of water jet impact simulations are conducted and the predicted rock breaking depth are compared with those given by previous studies. Finally, based on the proposed coupled SPH-FEM/DEM model, the effects of the rock micro-structure and micro-mechanical properties on rock breaking performance under water jet impact are analyzed. 2. Numerical model 2.1. Basic theory of SPH In SPH method, a kernel approximation is used based on randomly distributed interpolation points. The properties of each particle are evaluated via the integrals or the sums over the values of its neighboring particles. Considering a problem domain Ω discretized by a group of particles, and assuming the kernel function W has a compact supporting domain with a radius of h. The approximation of a function f(xi ) and its differential form ∇f(xi ) at point i can be respectively expressed by the discretized particles as follows [45–47]: 𝑓 (𝑥𝑖 ) ≈
𝑁 ∑ 𝑚𝑗 𝑗=1
∇𝑓 (𝑥𝑖 ) ≈ −
𝜌𝑗
( ) 𝑓 (𝑥𝑗 )𝑊 𝑥𝑖 − 𝑥𝑗 , ℎ
𝑁 ∑ 𝑚𝑗 𝑗=1
𝜌𝑗
𝑓 (𝑥𝑗 )∇𝑊 (𝑥𝑖 − 𝑥𝑗 , ℎ)
(1)
(2)
where W is the kernel function; h is the smooth length which defines the range of supporting domain; N is the total number of the particles within the smoothing length that affects particle i; j represents those influenced particles nearby the particle i; mj is the mass of particle j; 𝜌j is the density of particle j. By discretizing the Navier–Stokes equation with Eqs. (1) and (2), the SPH formulation of the governing equation for fluid can be derived: Conservation of mass: 𝑁
𝜕 𝑊𝑖𝑗 𝑑 𝜌𝑖 ∑ = 𝑚𝑗 𝑣𝛽𝑖𝑗 𝑑𝑡 𝜕𝑥𝛽 𝑗=1
(3)
𝑖
Conservation of momentum: 𝑑𝑣𝛼𝑖 𝑑𝑡
=
⎛ 𝜎 𝛼𝛽 𝜎𝑗𝛼𝛽 ⎞ 𝜕 𝑊𝑖𝑗 ⎟ 𝑚𝑗 ⎜ 𝑖 + ⎜ 𝜌2 𝜌2𝑗 ⎟ 𝜕𝑥𝛽𝑖 𝑗=1 ⎝ 𝑖 ⎠
𝑁 ∑
Conservation of energy: ( ) 𝑁 𝑝𝑗 𝜕 𝑊𝑖𝑗 𝑑 𝑒𝑖 𝑝𝑖 𝜇 1∑ = 𝑚 + 𝑣𝛽𝑖𝑗 + 𝑖 𝜀𝛼𝛽 𝜀𝛼𝛽 𝛽 𝑑𝑡 2 𝑗=1 𝑗 𝜌2 𝜌2 2 𝜌𝑖 𝑖 𝑗 𝜕𝑥𝑖 𝑖 𝑗
(4)
(5)
Where 𝜌i , 𝑣𝛼𝑖 and ei are the density, momentum and internal energy of the particle i respectively, p is the fluid pressure, 𝑥𝛽𝑖 is the coordinate of particle i in direction of 𝛽, 𝜎𝑖𝛼𝛽 and 𝜀𝛼𝛽 𝑖 are the stress and strain tensor of particle i respectively, 𝜇 is the viscosity coefficient of fluid, 𝑣𝛽𝑖𝑗 represents the relative velocity between two particles in direction of 𝛽. 2.2. Voronoi tessellation technique In order to represent the micro-structure of the rock grain more realistically [48,49], in this study, a Voronoi tessellation technique is added to LS-DYNA by a FORTRAN program, which can be divided into the following steps: Firstly, a series of randomly distributed control points are generated in the problem domain, as shown in Fig. 1(a). Then, the adjacent control points are triangulated based on the Delaunay triangulation. When the circumference of the newly formed triangle contains other control points in addition to the vertices, the triangle will be discarded, and the corresponding vertex will be re-formed a new triangle with other control points until the above criterion is met, as shown in 473
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Fig. 1. Generation procedure of random Voronoi polygons (a) generation of control points (b) triangulation of problem domain (c) connection of circumcenters (d) generation of random Voronoi grains.
Fig. 2. Schematic diagram of the numerical modeling process (a) generation of Voronoi grains (b) meshing of Voronoi grains (c) insertion of cohesive elements (d) establishment of FEM-DEM model.
Fig. 1(b). Furthermore, the circumcenters of adjacent triangles are connected each other, as shown in Fig. 1(c). Finally, the triangles and control points are removed, and the Voronoi grains are generated, as shown in Fig. 1(d). In order to generate granular rock model with different grain size and regularity, two control parameters d0 and Ri are introduced in the process of generating random control points. The parameter d0 is the average distance between adjacent control points, which controls the average size of Voronoi grains. The average grain size will increase with the increase of d0 . The parameter Ri is the fluctuation amplitude of the distance between control points. By adjusting the fluctuation range above and below the given average distance (d0 ), the regularity of Voronoi grain can be changed. With a fixed average distance (d0 ), a larger value of Ri will lead to a rock model with more irregular grains.
2.3.2. Constitutive model of the cohesive element model In this study, a bilinear traction-separation law, which describes the loss of the load bearing capacity of the material as a function of separation, is adopted in the cohesive model, as shown in Fig. 3. The failure pattern can be divided into two types: tensile behavior (as shown in Fig. 3(a)) and shear behavior (as shown in Fig. 3(b)). For case when the cohesive element is under tension, the tensile stress-displacement law can be expressed as [50]: ⎧ eff ⎪𝑘𝑛 𝛿𝑛 ⎪ 𝑡𝑛 = ⎨(1 − 𝐷𝑛 )𝑘𝑛 𝛿𝑛𝑐 ⎪ ⎪0 ⎩
eff
𝛿𝑛 ≤ 𝛿𝑛𝑐 𝛿𝑛𝑐 ≤ 𝛿𝑛 ≤ 𝛿𝑛𝑓 eff
𝛿𝑛𝑓
≤
(6)
eff 𝛿𝑛
eff
where 𝛿𝑛 is the effective relative tensile displacement, 𝛿𝑛𝑐 is the critical tensile displacement beyond which the softening happens, 𝛿𝑛𝑓 is the ultimate tensile displacement at which the cohesive element entirely loses its strength, kn is the normal initial stiffness coefficient. eff In softening stage (𝛿𝑛𝑐 ≤ 𝛿𝑛 ≤ 𝛿𝑛𝑓 ), the contact stress of the cohesive element is controlled by a damage variable Dn which is defined as:
2.3. Rock failure model 2.3.1. Numerical modeling In this study, to simulate the dynamic breaking behavior of rock under water jet impact, the zero-thickness cohesive element is inserted along the boundaries of the Voronoi grains. By simulating the failure of the cohesive element, the rock breaking process can be realized, such as the crack initiation and propagation. In order to realize this process, the numerical model of FEM-DEM is established, as shown in Fig. 2. Firstly, random Voronoi grains are generated in the certain region based on the Voronoi tessellation technique, as shown in Fig. 2(a). Subsequently, the generated Voronoi grains are meshed with triangular grids, which forms the FEM model, as shown in Fig. 2(b). By a FORTRAN insertion program, the FEM model can be dispersed and the zero-thickness cohesive element is generated between the grain boundaries, as shown in Fig. 2(c). Finally, the FEM/DEM model, which contains both solid grains and zero-thickness cohesive elements, is established as shown in Fig. 2(d).
eff
𝐷𝑛 =
𝛿𝑛 − 𝛿𝑛𝑐
(7)
𝛿𝑛𝑓 − 𝛿𝑛𝑐
Similarly, for case when the cohesive element is sheared, its shear stress ts can be obtained as: ⎧ eff ⎪𝑘𝑠 𝛿𝑠 ⎪ 𝑡𝑠 = ⎨(1 − 𝐷𝑠 )𝑘𝑠 𝛿𝑠𝑐 ⎪ ⎪0 ⎩ eff
eff
𝛿𝑠 ≤ 𝛿𝑠𝑐 𝛿𝑠𝑐 ≤ 𝛿𝑠 ≤ 𝛿𝑠𝑓 eff
𝛿𝑠𝑓
≤
(8)
eff 𝛿𝑠
where 𝛿𝑠 is the effective relative shear displacement, 𝛿𝑠𝑐 is the critical shear displacement, 𝛿𝑠𝑓 is the ultimate shear displacement, ks is the 474
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Table 1 State equation parameters of water. 𝜌(kg m−3 )
C(m s−1 )
S1
S2
S3
𝛾0
𝛼
1000
1480
2.56
−1.986
0.2286
0.49
1.397
The deformation characteristic before the rock failure is mainly controlled by solid elements, while the mechanical response after the rock failure is mainly achieved by the failure of cohesive elements. Therefore, the linear elastic material of MAT_ELASTIC is applied to rock elements and the material of MAT_COHESIVE_GENERAL is used for the cohesive failure model in LS-DYNA. In the cohesive material model, a dimensionless effective separation parameter 𝜆 is used, which grasps for the interaction between the relative displacements in the normal and tangential directions: √ √( )2 ( )2 √ √ 𝛿 𝛿2 𝜆 = √ 𝑓1 + (14) 𝛿𝑛 𝛿𝑠𝑓 where 𝛿 1 is the normal displacement and 𝛿 2 is the tangential displacement. The cohesive element will reach failure when 𝜆= 1. 2.4. State equation of water jet The material of MAT_NULL in LS-DYNA is used for the water jet, and the GRUNEISEN equation is used to describe the relationship between pressure and density, which is shown as follow [51]: [ ( ) ] 𝛾 𝜌𝐶 2 𝜇 1 + 1 − 20 𝜇 − 𝛼2 𝜇 2 𝑃 = [ (15) ]2 + (𝛾0 + 𝛼𝜇)𝐸 3 𝜇2 − 𝑆3 𝜇 2 1 − (𝑆1 − 1)𝜇 − 𝑆2 𝜇+1 (𝜇+1)
where P is the pressure; 𝜌 is the density; E is internal energy per unit volume; C is the intercept of the curve 𝜇 s -𝜇 p ; S1 , S2 and S3 are the slopes of the curve 𝜇 s -𝜇 p ; 𝛾 0 is the GRUNEISEN coefficient, 𝛼 is the correction coefficient for the relationship between the 𝛾 0 and volume. The parameters of the state equation are shown in Table 1 [23,24].
Fig. 3. Constitutive model of the cohesive element (a) tensile behavior (b) shear behavior.
2.5. Coupling algorithm
tangential initial stiffness coefficient. And the damage variable Ds in eff softening stage (𝛿𝑠𝑐 ≤ 𝛿𝑠 ≤ 𝛿𝑠𝑓 ) is defined as:
Based on the LS-DYNA platform. a node to surface contact algorithm is adopted to implement the coupling of SPH and FEM/DEM, which is based on the penalty method. Assuming a virtual spring existed between SPH particles and the face of finite elements and the penetration between them will be checked in every time step. The contact force is imposed when there occurs a penetration, whereas nothing will be done. In addition, the FEM/DEM model based on Voronoi grains will be exerted a single surface contact algorithm, which allows automatic searching for penetrations of all grains after cohesive elements failed.
eff
𝐷𝑠 =
𝛿𝑠 − 𝛿𝑠𝑐 𝛿𝑠𝑓 − 𝛿𝑠𝑐
(9)
With the stress-displacement laws, the area under the stressdisplacement curve is the fracture energy which can be calculated in terms of the material mechanical properties through the following relationship: 𝐺𝐼𝐶 = 𝐺𝐼 𝐼 𝐶 =
2 𝐾𝐼𝐶
𝐸
𝐾𝐼2𝐼 𝐶 𝐸
3. Numerical validation of SPH-FEM/DEM (10) In this section, the capability of the SPH-FEM/DEM method in modeling the dynamic response and breaking behavior of rock under water jet impact is verified by reproducing the laboratory water jet impact test on sandstone. The test was conducted by Lu et al. [8] with the high-precision computerized numerical control (CNC) water jet cutting (WJC) machine manufactured by OMAX Corporation. The rock samples are short cylindrical sandstone cores of 50 mm in diameter and 50 mm in length, and the mechanical properties are as follows: Young’s modulus E = 57.62 GPa, Shear modulus G = 22.1 GPa, Poisson’s ratio 𝜐 = 0.23, density 𝜌 = 2370 kg/m3 , uniaxial compressive strength 𝜎 c = 68 MPa and uniaxial tensile strength 𝜎 t = 2.17 MPa. The diameter of water jet is 2 mm, and the standoff distance is 3 mm. In addition, seven different impact velocities were selected in the test, i.e., 157 m/s, 316 m/s, 447 m/s, 547 m/s, 632 m/s, 707 m/s and 774 m/s, respectively.
(11)
where GIC and GIIC are the fracture energies for mode I and mode II, respectively. As a result, the peak tractions and ultimate displacements are as follows: 𝛿𝑛𝑓 =
2𝐺𝐼𝐶 𝜎max
(12)
𝛿𝑠𝑓 =
2𝐺𝐼 𝐼 𝐶 𝜏max
(13)
where 𝜎 max and 𝜏 max are the normal peak traction and tangential peak traction, respectively. 475
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Fig. 5. Comparison of predicted rock broken depths with test results.
Fig. 4. Numerical model for simulation of impact test on sandstone.
Considering computational efficiency, a simplified 2-dimentional numerical model is established to reproduce the laboratory test, as shown in Fig. 4. The numerical model of rock is 50 mm in length and 50 mm in high, which is generated by Voronoi grains with the size parameter d0 of 1 mm and the regularity parameter Ri of 1.5. The water jet is 2 mm in diameter and modeled by SPH particles. The standoff distance and the impact velocity of water jet are consistent with the laboratory test. Besides, numerical simulation of FEM/DEM requires proper selection of the micro-mechanical parameters. In this study, by assuming the appropriate micro-mechanical parameters, such as normal and shear initial stiffness (kn and ks ), peak tensile and shear traction (𝜎 max and 𝜏 max ) and fracture energy (GIC , GIIC ), the numerical reproduction of the laboratory test is realized, and the rock broken depths in sandstone samples of the two conditions are compared and analyzed. After a series of trial-and-error tests, the micro-mechanical properties of cohesive elements are determined, based on which the numerical predicted rock broken depths can approximate the results of laboratory test at different impact velocity, as shown in Fig. 5. As illustrated in the figure, the rock broken depths of test and simulation both increase linearly with the increase of velocity, where the test data increases from 6.8 mm to 43.9 mm and the simulation data increases from 6.24 mm to 41.2 mm (with the average deviation of 5.77%). In addition, Fig. 6 demonstrates the qualitative results of numerical simulation and laboratory test with impact velocity of 447 m/s. As illustrated in the Fig. 6(a), a numerical predicted crushing pit is formed in the rock, which is surrounded by two inner cracks. In addition, a penetrating crack initiates and extends to the side surface of rock specimen. Fig. 6(b) shows the processing image of test qualitative result with impact velocity of 447 m/s. It can be observed from the figure that the rock is broken obviously and a penetrating crack extends to the side surface as well. However, because the test image is not from inner section but outer contour, the internal failure state is difficult to observe. The results indicate that there
Fig. 6. Qualitative results of numerical simulation and laboratory test (447 m/s) (a) numerical result (b) test result.
is a good agreement between numerical simulation and laboratory test, which verifies the capability of the developed method in modeling the dynamic breaking response of sandstone under water jet impact. The calibrated micro-mechanical parameters are as follows: kn = 433 GPa/mm, ks = 131 GPa/mm, 𝜎 max = 2.6 MPa, 𝜏 max = 10.2 MPa, GIC = 54 J/m2 , and GIIC = 70 J/m2 . 4. Numerical results In this part, to investigate the micro-mechanism of rock breaking behavior under water jet impact, the macro and micro-breaking process of rock is firstly analyzed. Then, by a series of numerical simulations with different microscopic conditions, the effects of the micro-structure (such as grain size, regularity) and the micro-mechanical properties (such as ductility, microscopic strength, contact stiffness ratio, and the heterogeneity of micro-parameters) on rock breaking performance under water jet impact are investigated. To conduct the investigation, numerical models which are all 60 mm in length and 30 mm in high, respectively, are built by random Voronoi grains with different micro-structure and micro-properties (as shown in Fig. 7). By changing the control parameters d0 and Ri , the size and regularity of rock micro-grains can be adjusted easily. Then, the generated Voronoi grains are further divided into triangular elements with a mesh size of 0.25 mm. In addition, the water jet in all models is simplified as a rectangle with size of 1 mm × 50 mm, which is meshed by 504 SPH particles under a uniform distribution. For all cases, the distance between water jet and rock target is 1 mm and the velocity of water jet is set 476
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4.2. Effects of the micro-structure on rock breaking performance 4.2.1. The effect of grain size In this section, the effect of the grain size on the rock breaking performance under water jet impact is analyzed. With the regularity parameter Ri = 1.5, five different d0 , i.e., 0.5 mm, 0.75 mm, 1.0 mm, 1.25 mm, 1.5 mm, are used in the numerical model, respectively. Besides, two parameters, namely broken depth (D) and broken area (A) are defined to evaluate the rock breaking performance quantitatively, as shown in Fig. 9. Fig. 10 presents the relationship between grain size and broken depth and area. As shown in the figure, the rock broken depth and area increase approximately linearly with the growth of grain size. When the d0 is 0.5 mm, the rock broken depth is 4.8 mm and broken area is 67.6 mm2 . As the d0 increases to 1.5 mm, the broken depth and area increase to 7.93 mm and 204.68 mm2 respectively, which indicates the grain size has a great effect on the development of crushing pit. In addition, the initiation and propagation of macro cracks also play an important role in rock breaking process under water jet impact. Therefore, the initiation amount (N) and maximum propagation length (L) of macro cracks (as defined in Fig. 9) outside the crushing pit with different grain size are statistically analyzed, as shown in Fig. 11. When d0 is less than 1.0 mm, no obvious macro cracks initiate in the rock. Then, macro cracks develop rapidly with the increase of d0 . As d0 increases to 1.5 mm, three macro cracks initiate in the rock, and the maximum propagation length of the cracks reaches 22.27 mm. The numerical results indicate that the grain size has a great effect on rock breaking performance under water jet impact. As the grain size increases, more severe rock breakage will be induced.
Fig. 7. Numerical model (d0 = 1 mm Ri = 2.5).
4.2.2. The effect of regularity The regularity is also an important parameter affecting mechanical properties of rock. In order to investigate the effect of regularity on rock breaking performance under water jet impact, in this section, five different values of regularity parameter Ri , i.e., 1.0, 1.5, 2.0, 2.5, 3.0 are used in the numerical model respectively. In addition, the grain size parameter d0 is set at 1.0 mm uniformly. The statistics of the rock broken depth and area with different Ri are shown in Fig. 12. As illustrated in the figure, the rock broken depth and area increase with the increase of regularity parameter. When Ri increases from 1.0 to 3.0, the rock broken depth changes from 5.99 to 8.15 mm, and broken area changes from 108.4 mm2 to 200.8 mm2 . Fig. 13 demonstrates the effect of regularity on the initiation amount and maximum propagation length of macro cracks. It can be found that as the grains change from regular to irregular, both the initiation amount and maximum propagation length of macro cracks increase. There are no obvious macro cracks initiated in the rock when Ri = 1.0 and 1.5. However, macro cracks develop rapidly after Ri reaches 2.0. When Ri increases to 3.0, four macro cracks initiate outside the crushing pit, and the maximum propagation length of the cracks reaches 23.61 mm. The numerical results indicate that the grain regularity has a significant effect on rock breaking performance. As the grains become more irregular, the range of crushing pit as well as the initiation and propagation of macro cracks all increase.
as 300 m/s. Besides, to avoid the boundary reflection effect, no-reflect boundary condition is set on both sides and bottom boundary for all numerical models. 4.1. Rock breaking process under water jet impact In this section, to investigate the detailed macro and micro breaking process of rock under water jet impact, the grain size and regularity parameter are adopted to be d0 = 1.0 mm and Ri = 2.5, respectively, as show in Fig. 7. The numerical predicted rock breaking process is shown in Fig. 8. As illustrated in the figure, at 1𝜇s, the water jet firstly impacted on the top surface of the rock, which induces compression-shear failure of the rock, resulting in initial crushing pit in the vicinity of the impact point due to the instantaneous high impact pressure. Meanwhile, the stress wave transmits into the rock, causing some micro cracks around the crushing pit, as shown in Fig. 8(a). As the water jet continues to advance, intense lateral flow is induced due to the severe compression of the injected liquid resulting from liquid and rock contact interaction, causing further tensile-shear failure of the rock. At this time, due to the horizontal free surface on the top, the crushing pit is developed rapidly horizontally, while the development in vertical direction is not obvious due to the constraint at the bottom, as shown in Fig. 8(b). Then, water flow invades along the expanded cracks, leading to further extension of cracks and expansion of crushing zone. The broken grains are washed away by the water flow, which forms the macro crushing pit. Besides, due to the intense compression of the water jet, the cracks at the bottom of the crushing pit expands obviously and form the macro crack (exceeding 5 mm), as shown in Fig. 8(c). With the continuous impact of water jet, the crushing pit and macro cracks at the bottom of the crushing pit are further developed. Finally, a crushing pit with the broken depth of 7.59 mm and the broken area of 152 mm2 is formed, which are surrounded by two obvious macro cracks with the length of 8.48 mm and 17.32 mm respectively, as shown in Fig. 8(d).
4.3. Effects of the micro-mechanical properties on rock breaking performance In this section, the effects of the micro-mechanical properties on rock breaking performance under water jet impact, such as ductility, microscopic strength, contact stiffness ratio and the heterogeneity of microparameters, are studied by the proposed numerical model. Uniformly, the grain size parameter d0 and regularity parameter Ri are set as 1.0 mm and 1.5 during the numerical simulation. 477
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Fig. 8. Macro and micro breaking process of rock at different time (a) initiation of micro cracks (b) widening of crushing pit (c) formation of macro cracks (d) further development of crushing pit and macro cracks.
4.3.1. The effect of ductility In order to investigate the effect of ductility on rock breaking performance under water jet impact, in this study, different levels of ductility are obtained by scaling the fracture energy GIC (GIIC ) with fixed peak tensile (shear) traction and initial normal (tangential) stiffness. The scaling ratio can be expressed by Sa . In this section, a series of numerical simulations of water jet impact are conducted with seven different levels of Sa , i.e., 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75.
Fig. 14 shows the statistical results of rock broken depth and area with different levels of ductility. As illustrated in the figure, as Sa increases from 0.25 to 1.75, the rock broken depth reduces from 12.88 mm to 4.63 mm, and the broken area reduces from 319.68 mm2 to 71.48 mm2 , but the reducing rates both decrease gradually. Fig. 15 presents the effect of ductility on the initiation amount and maximum propagation length of macro cracks. When Sa is equal to 0.25, eight obvious macro cracks initiate in the rock and the maximum propagation 478
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Fig. 9. Schematic diagram of indicators for evaluating rock breaking performance.
Fig. 10. The effect of grain size on broken depth and area.
Fig. 12. The effect of regularity on broken depth and area.
Fig. 11. The effect of grain size on the initiation amount and maximum propagation length of macro cracks.
Fig. 13. The effect of regularity on the initiation amount and maximum propagation length of macro cracks.
length of the cracks reaches 26.01 mm, which suggests that the rock broken degree is relatively severe. As Sa increases, both the initiation amount and maximum propagation length of macro cracks rapidly decrease. After Sa reaches 1.00, the ductility of the rock exceeds the
original value, which causes the impact energy of water jet not enough to expand micro cracks into macro cracks. At this time, the formation and expansion of the crushing pit are the main failure mode. The results show that when the rock becomes more ductile, the rock broken degree 479
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Fig. 14. The effect of material ductility on broken depth and area.
Fig. 16. The effect of microscopic strength on broken depth and area.
Fig. 15. The effect of material ductility on the initiation amount and maximum propagation length of macro cracks.
Fig. 17. The effect of microscopic strength on the initiation amount and maximum propagation length of macro cracks.
4.3.3. The effect of contact stiffness ratio ks /kn In this section, the effect of contact stiffness ratio ks /kn on rock breaking performance is investigated. The ductility and microscopic strength are fixed, and seven different values of ks /kn , i.e., 0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10.0, are adopted respectively in the study. Fig. 18 shows the relationship of rock broken depth, area and the value of ks /kn . As illustrated in the figure, there is no significant correlation between rock broken depth, area and the contact stiffness ratio. As the value of ks /kn changes from 0.1 to 10.0, the rock broken depth and area fluctuate between 6–7 mm and 130–150 mm2 , respectively. Meanwhile, in addition to a tiny macro crack initiated in the model of ks /kn = 2.0, there are no macro cracks generated in other models. It can be concluded that the contact stiffness ratio has no obvious effect on the rock breaking performance under water jet impact.
will reduce significantly, which indicates the material ductility has a great effect on the rock breaking performance under water jet impact.
4.3.2. The effect of microscopic strength In this section, the effect of microscopic strength on the rock breaking performance under water jet impact is investigated. With fixed ductility and initial stiffness, different level of microscopic strength can be obtained by scaling the peak traction 𝜎 max (𝜏 max ). The scaling ratio can be expressed by Sb and seven different levels of Sb , i.e., 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, are used in the numerical model respectively. Fig. 16 shows the effect of different level of microscopic strength on rock broken depth and area. As illustrated in the figure, both the rock broken depth and area reduce obviously as Sb increases, but the reducing rate decrease gradually. Fig. 17 presents the statistical curve of the initiation amount and maximum propagation length of macro cracks with different level of microscopic strength, which shows a similar result with that of the ductility. When the level of Sb is low, the development of macro cracks is relatively high. With the increase of Sb , both the initiation amount and maximum propagation length of macro cracks rapidly decrease. After Sb reaches 1.00, in addition to the crushing pit, no obvious macro cracks initiate in the rock. The results indicate that the increased microscopic strength will cause a decreasing rock breaking performance under water jet impact obviously.
4.3.4. The effect of the heterogeneity of micro-parameters Rock material is inherently inhomogeneous, which contains initial micro-defects, such as micro cracks, cleavages and grain boundaries. Mechanical behavior and breaking process of rock, as grained composite material, is deeply affected by such micro-defects [52]. To account for the micro-defects induced heterogeneity, the micro-mechanical parameters, including the peak traction 𝜎 max (𝜏 max ), initial stiffness kn (ks ), and fracture energy GIC (GIIC ) of the cohesive element, are assumed to conform to the Weibull distribution, as defined by the following 480
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Fig. 18. The effect of ks /kn on broken depth and area.
Fig. 20. The effect of the heterogeneity of micro-parameters on the initiation amount and maximum propagation length of macro cracks.
initiation amount reaching 4 and the maximum propagation length reaching 16.84 mm. As m increases, both the initiation amount and maximum propagation length of macro cracks gradually decrease. When m reaches to 10, no obvious macro cracks initiate in the rock. The numerical results indicate that as the heterogeneity index m increases, the rock broken degree under water jet impact will nonlinearly decrease first and then become stable, which means that the heterogeneity of micro-parameters has an obvious effect on rock breaking performance. 4.4. Discussion According to the above numerical results, both the micro-structure and micro-mechanical properties of the rock have significant effects on the rock breaking performance under water jet impact. For the microstructure effects, the rock broken degree increases with the increase of grain size, which may result from less linkage energy required for coarse-grained rocks. This result is consistent with the conclusion given by Dehkhoda and Hood [9]. Besides, as the grains become more irregular, the rock broken degree increases significantly. This may be due to the irregular grain shape result in more severe stress concentration at grain corners, which makes the micro cracks easily initiate. As for typical micro-mechanical properties, the ductility and microscopic strength have similar effects on the rock breaking performance under water jet impact. With the increase of ductility and microscopic strength, the energy and stress required for rock failure gradually increase, making the rock breaking performance gradually decreases under same water jet impact velocity. Besides, as the heterogeneity of microscopic parameters increases, the rock broken degree increases nonlinearly. This is because when the heterogeneity is very high, many micro cracks will be randomly initiated in the rock. As a result, when the water jet impacts rock, the micro cracks will propagate and coalesce with each other, which induce severe breakage of rock. Furthermore, the contact stiffness ratio has no obvious effect on the rock breaking performance, which indicates that the tensile failure plays a dominant role in the rock breaking process under water jet impact.
Fig. 19. The effect of the heterogeneity of micro-parameters on broken depth and area.
probability density function [53]: ( )𝑚−1 [ ( )𝑚 ] 𝑚 𝑥 𝑥 𝑊 (𝑥) = exp − 𝑢0 𝑢0 𝑢0
(16)
where x is the cohesive element micro-mechanical strength (such as peak traction, initial stiffness and fracture energy); u0 is the scale parameter related to the average micro-mechanical strength of cohesive element; m defines the shape of the distribution function, which controls the heterogeneity degree and is called the homogeneity index. An increase of homogeneity index leads to a more homogeneous numerical model. In order to investigate the effect of the heterogeneity of microparameters on rock breaking performance under water jet impact, six different heterogeneity indexes m, i.e., 1, 2, 3, 5, 10, 20 respectively, are adopted in this analysis. The statistics of the rock broken depth and area with different heterogeneity indexes m are shown in Fig. 19. As illustrated in the figure, when m is equal to 1.0, the rock broken depth is 9.18 mm and area is 177.92 mm2 . With the increase of m, both the rock broken depth and area reduce significantly, but the reducing rate gradually decrease. When m reaches 10, the rock broken depth and area become stable and stay around 6.9 mm and 130.2 mm2 , respectively, which approximates to that of homogeneous rock. The effect of the heterogeneity of micro-parameters on the initiation amount and maximum propagation length of macro cracks shows a similar rule. As illustrated in Fig. 20, macro cracks are most developed when m is equal to 1.0, with the
5. Conclusion In this study, the coupled SPH-FEM/FDM model has been developed in LS-DYNA software to simulate the dynamic behavior of rock under water jet impact. The SPH method was adopted to simulate the water jet behavior and FEM/DEM method was used to model the rock breaking process. In order to better represent the rock micro-structure and model the mechanical behavior of rock grains, the Voronoi tessellation 481
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technique was adopted to generate the micro-grains of rock. Zerothickness cohesive elements were inserted along the boundaries of the Voronoi grains to model the mechanical interaction between grains as well as the breaking process of rock. The proposed numerical scheme was verified by successfully reproducing the laboratory water jet impact test on sandstone. With the proposed coupled SPH-FEM/DEM model, the effects of micro-structure and micro-mechanical properties on the rock breaking performance under water jet impact have been systematically investigated. Based on the numerical results, the following conclusions can be drawn:
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(1) The grain size has a great effect on the rock breaking performance under water jet impact. As the grain size increases, more severe rock breakage will be induced. (2) The irregular shape of rock micro-grain makes the breaking more severe. When the irregularity parameter Ri increases from 1.0 to 3.0, the rock broken depth and area as well as the initiation amount and maximum propagation length of macro cracks developed all increase. (3) The ductility and microscopic strength of the rock have similar effects on rock breaking performance under water jet impact. When the ductility or microscopic strength of rock is low, relatively severe breakage will be induced. As the ductility or microscopic strength increases, the rock broken degree reduces significantly with the reducing rate gradually decreases. (4) The contact stiffness ratio ks /kn has little effect on the rock breaking performance under water jet impact. With the change of ks /kn , both the rock broken depth and area fluctuate within a certain range. (5) The heterogeneity of rock micro-parameters significantly affects its breaking performance under water jet impact. With the increase of the heterogeneity index m (the heterogeneity decreases), the rock broken degree reduces obviously first. When the heterogeneity index m reaches to 10, the rock breaking performance becomes stable, which approximates to that of homogeneous rock. Acknowledgment The research work is supported by the National Natural Science Foundation of China (Grant Nos. 41502283, 41772309, 41602324), National Key Research and Development Program of China (2017YFC1501302), the Major Program of Technological Innovation of Hubei Province, China (No. 2017ACA102) and the National Basic Research Program of China (973 Program) (Grant Nos. 2014CB046900, 2014CB046904). References [1] Momber AW. Fluid jet erosion as a non-linear fracture process: a discussion. Wear 2001;250(1):100–6. [2] Momber AW. An SEM-study of high-speed hydrodynamic erosion of cementitious composites. Compos Part B Eng 2003;34(2):135–42. [3] Lan H, Martin CD, Bo H. Effect of heterogeneity of brittle rock on micromechanical extensile behavior during compression loading. J Geophys Res Solid Earth 2010;115(B1):B01202. doi:10.1029/2009JB006496. [4] Peng J, Wong LNY, Teh CI. Influence of grain size heterogeneity on strength and micro-cracking behavior of crystalline rocks. J Geophys Res Solid Earth 2017;122(2):1054–73. [5] Sabri M, Ghazvinian A, Nejati HR. Effect of particle size heterogeneity on fracture toughness and failure mechanism of rocks. Int J Rock Mech Min Sci 2016;81:79–85. [6] Wasantha PLP, Ranjith PG, Zhao J, Shao SS, Permata G. Strain rate effect on the mechanical behaviour of sandstones with different grain sizes. Rock Mech Rock Eng 2015;48(5):1883–95. [7] Miao S, Pan PZ, Wu Z, Zhao S. Fracture analysis of sandstone with a single filled flaw under uniaxial compression. Eng Fract Mech 2018;204:319–43. [8] Lu Y, Huang F, Liu X, Ao X. On the failure pattern of sandstone impacted by high-velocity water jet. Int J Impact Eng 2015;76:67–74. [9] Dehkhoda S, Hood M. An experimental study of surface and sub-surface damage in pulsed water-jet breakage of rocks. Int J Rock Mech Min Sci 2013;63:138–47. [10] Engin IC. A correlation for predicting the abrasive water jet cutting depth for natural stones. S Afr J Sci 2012;108(9–10):69–79. 482
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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound
Micro-mechanism study on rock breaking behavior under water jet impact using coupled SPH-FEM/DEM method with Voronoi grains Zhijun Wu a, Fangzheng Yu a, Penglin Zhang a, Xuewei Liu b,∗ a b
School of Civil Engineering, Wuhan University, Wuhan 430072, China Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China
a r t i c l e
i n f o
Keywords: SPH-FEM/DEM Water jet impact Rock breaking behavior Micro-structure Micro-mechanical properties
a b s t r a c t In this study, a two-dimensional SPH-FEM/DEM coupled model is developed to investigate rock breaking behavior under water jet impact based on the LS-DYNA software, where the SPH method is adopted to model the water jet and the FEM/DEM method is adopted to simulate rock breaking response, respectively. To better approximate the micro-structure of rock, a Voronoi tessellation technique is adopted to generate the random polygonal grains. A zero-thickness cohesive element is inserted along the boundaries of the Voronoi grains to model the mechanical interaction between grains as well as the breaking process of rock. Numerical water jet impact tests on sandstone are firstly conducted to verify the proposed method as well as calibrate the corresponding micro-parameters. Then, using the verified method, the effects of micro-structure and micro-mechanical properties on the rock breaking performance under water jet impact are systematically investigated. The numerical results show that the rock breaking performance is greatly affected by grain size and irregularity, ductility, microscopic strength and the heterogeneity of micro-parameters, whereas the contact stiffness ratio has little effect on the rock breaking performance under water jet impact.
1. Introduction Benefiting from the significant advantages, such as low cutting force, selective removal capability, high efficiency, dust-free, heat-free and vibration-free performance [1,2], water jet technology has been applied widely in many industry fields, especially in rock breaking and mining engineering. Rock breaking under water jet impact is a complex fluidsolid coupling process, involving the deformation and fracture of rocks, crack initiation and propagation and fluid flowing. At the microscopic scale, rock is not homogeneous but composed by mineral grains and micro-structures, such as grain interfaces and micro defects, which has a great influence on the macro-mechanical property of rock [3–7]. Therefore, in order to better understand the rock breaking behavior under the impact of water jet, it is necessary to consider the effects of rock microproperties. However, due to the complexity of rock micro-structures and the difficulty to obtain the micro-mechanical properties, revealing the micro-mechanism of rock breaking behavior under water jet impact is still very difficult. In the past decades, many methods have been proposed to investigate the rock breaking behavior under water jet impact. Among these methods, experimental research shows the most visible and reliable information. Lu and Huang [8] carried out impact tests on three kinds of sandstones at different velocities and concluded three failure patterns of ∗
sandstone: center broken pit caused by shear stress at low velocity, internal fractures caused by tensile stress at moderate velocity and macro cracks on the side surface at high velocity. In order to investigate the effect of pulsed length and pulsation frequency, Dehkhoda and Hood [9] conducted pulsed water jet impact on granite and marble and found that pulsation frequency influences fracture initiation and pulse length is associated with the crack opening energy. To investigate the effects of rock properties and operating parameters on the cutting depth, Engin [10] performed abrasive water jet (AWJ) cutting on 42 different types of natural rocks and the results indicated that the Shore hardness, Bohme surface abrasion resistance and density are the most significant rock properties while the working pump pressure, traverse velocity are the most significant operating parameters affecting the cutting depth. Using mass removal measurements, microscopical observations and the mercury penetration technique, Momber and Kovacevic [11] investigated the influence of grain inclusions on the fracture behavior of a multiphase brittle material exposed to water jet impact and found that the increase of inclusions leads to a reduction of the threshold destruction energy and finally controls the fracture evolution and materials removal performance. These experimental studies do have promoted the understanding of rock breaking mechanism under water jet impact. However, due to the opaque characteristic and complex micro-structure of rock, nowadays it is still challenging to experimentally establish the detailed
Corresponding author. E-mail addresses: [email protected] (Z. Wu), [email protected] (F. Yu), [email protected] (P. Zhang), [email protected] (X. Liu).
https://doi.org/10.1016/j.enganabound.2019.08.026 Received 15 May 2019; Received in revised form 15 July 2019; Accepted 24 August 2019 0955-7997/© 2019 Elsevier Ltd. All rights reserved.
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Engineering Analysis with Boundary Elements 108 (2019) 472–483
relationship between the rock micro-characteristics and the macroscopic breaking behavior under water jet impact. Recently, with the development of computer technology, more and more researchers are devoted to investigating the rock breaking behavior under water jet impact by means of numerical simulation. However, due to the huge mechanical differences of rock and water during rock breaking process under water jet impact, it may be better to adopt different methods to simulate their behavior [12,13]. For modeling water, due to the large deformation, the traditional Lagrangian grid-based finite element method (FEM) will be faced with the problems of precision decrease and even calculation termination induced by the distortion of the grids [14]. By allowing grid points separating and moving independently from material points, the Arbitrary Lagrange Euler (ALE) method can easily solve the large deformation problems and has been successfully applied to the water jet model [15–18]. However, when dealing with the coupling problem with rocks, the results of ALE method are very sensitive to the coupling parameters, which are difficult to be determined [19]. Alternatively, due to no meshing and simple coupling process, a meshless method, smooth particle hydrodynamics (SPH), shows good adaptation for dynamic large distortion problems such as the liquid-solid impact and has been widely applied into the water jet model [20–24]. Therefore, in this study, the SPH method is adopted to simulate the water jet process. However, modeling the rock breaking behavior by water jet also requires to accurately capture the rock failure process. Although the SPH method can also simulate the rock breaking process [25,26], it suffers from high computational expense and difficulty in enforcing the boundary conditions [27]. Therefore, in recent years, some SPH-coupled methods which take the advantages of SPH in modeling the water behavior and other methods in simulating the rock breaking process have been developed. Among them, the coupled FEM-SPH method is the most commonly used one [21,22]. However, the FEM model treats rocks into continuous and uniform grids, which deviates from the micro-structure of rocks, and even worse the continuous assumption makes FEM fail to realistically simulate the evolution of cracks during rock breaking process [28–30]. In addition to continuum-based methods represented by FEM, discontinuum-based methods and other meshless methods, such as discrete element method (DEM), discontinuous deformation analysis (DDA), and element bending group (EBG), can be also coupled with SPH to investigate the fluid-solid interactions [31–35] due to its powerful capability in representing the rock micro-structure. However, the discontinuum-based methods still have some limitations, such as difficulties in determining microscopic parameters and capturing the deformation and breaking process itself. Actually, rock is neither a complete continuum nor discontinuum but a combination of these two [36,37]. Therefore, the rock breaking process can be better simulated by the coupled continuous-discontinuous method [38–41]. As a such coupled method, the FEM/DEM model regards the material as a number of discrete interactive grains, so the key feature of rock failure process modeling, which involves the transition from continuum to discontinuum through deformation, fracture and fragmentation, can be easily achieved. With a contact algorithm, the grain separation and interaction can be correctly considered. Based on these features, the FEM/DEM model has been widely applied to simulate the rock breaking process [42–44]. Therefore, in this study, to study the micro-mechanism of rock breaking behavior under water jet impact, the coupled SPH-FEM/DEM model is implemented in LS-DYNA software, where the FEM/DEM method is adopted to simulate rock breaking response and the SPH method is adopted to model the water jet. To implement the FEM/DEM, two FORTRAN subroutines are self-developed and embedded into LS-DYNA. Firstly, the random polygonal Voronoi tessellation program is adopted to generate the micro-grains of rock for better representing the rock micro-structure and modeling the mechanical behavior of rock grains. Then, zerothickness cohesive elements are inserted along the boundaries of the Voronoi grains by the inserting program to model the mechanical in-
teraction between grains as well as the breaking process of rock. To verify the proposed model, a series of water jet impact simulations are conducted and the predicted rock breaking depth are compared with those given by previous studies. Finally, based on the proposed coupled SPH-FEM/DEM model, the effects of the rock micro-structure and micro-mechanical properties on rock breaking performance under water jet impact are analyzed. 2. Numerical model 2.1. Basic theory of SPH In SPH method, a kernel approximation is used based on randomly distributed interpolation points. The properties of each particle are evaluated via the integrals or the sums over the values of its neighboring particles. Considering a problem domain Ω discretized by a group of particles, and assuming the kernel function W has a compact supporting domain with a radius of h. The approximation of a function f(xi ) and its differential form ∇f(xi ) at point i can be respectively expressed by the discretized particles as follows [45–47]: 𝑓 (𝑥𝑖 ) ≈
𝑁 ∑ 𝑚𝑗 𝑗=1
∇𝑓 (𝑥𝑖 ) ≈ −
𝜌𝑗
( ) 𝑓 (𝑥𝑗 )𝑊 𝑥𝑖 − 𝑥𝑗 , ℎ
𝑁 ∑ 𝑚𝑗 𝑗=1
𝜌𝑗
𝑓 (𝑥𝑗 )∇𝑊 (𝑥𝑖 − 𝑥𝑗 , ℎ)
(1)
(2)
where W is the kernel function; h is the smooth length which defines the range of supporting domain; N is the total number of the particles within the smoothing length that affects particle i; j represents those influenced particles nearby the particle i; mj is the mass of particle j; 𝜌j is the density of particle j. By discretizing the Navier–Stokes equation with Eqs. (1) and (2), the SPH formulation of the governing equation for fluid can be derived: Conservation of mass: 𝑁
𝜕 𝑊𝑖𝑗 𝑑 𝜌𝑖 ∑ = 𝑚𝑗 𝑣𝛽𝑖𝑗 𝑑𝑡 𝜕𝑥𝛽 𝑗=1
(3)
𝑖
Conservation of momentum: 𝑑𝑣𝛼𝑖 𝑑𝑡
=
⎛ 𝜎 𝛼𝛽 𝜎𝑗𝛼𝛽 ⎞ 𝜕 𝑊𝑖𝑗 ⎟ 𝑚𝑗 ⎜ 𝑖 + ⎜ 𝜌2 𝜌2𝑗 ⎟ 𝜕𝑥𝛽𝑖 𝑗=1 ⎝ 𝑖 ⎠
𝑁 ∑
Conservation of energy: ( ) 𝑁 𝑝𝑗 𝜕 𝑊𝑖𝑗 𝑑 𝑒𝑖 𝑝𝑖 𝜇 1∑ = 𝑚 + 𝑣𝛽𝑖𝑗 + 𝑖 𝜀𝛼𝛽 𝜀𝛼𝛽 𝛽 𝑑𝑡 2 𝑗=1 𝑗 𝜌2 𝜌2 2 𝜌𝑖 𝑖 𝑗 𝜕𝑥𝑖 𝑖 𝑗
(4)
(5)
Where 𝜌i , 𝑣𝛼𝑖 and ei are the density, momentum and internal energy of the particle i respectively, p is the fluid pressure, 𝑥𝛽𝑖 is the coordinate of particle i in direction of 𝛽, 𝜎𝑖𝛼𝛽 and 𝜀𝛼𝛽 𝑖 are the stress and strain tensor of particle i respectively, 𝜇 is the viscosity coefficient of fluid, 𝑣𝛽𝑖𝑗 represents the relative velocity between two particles in direction of 𝛽. 2.2. Voronoi tessellation technique In order to represent the micro-structure of the rock grain more realistically [48,49], in this study, a Voronoi tessellation technique is added to LS-DYNA by a FORTRAN program, which can be divided into the following steps: Firstly, a series of randomly distributed control points are generated in the problem domain, as shown in Fig. 1(a). Then, the adjacent control points are triangulated based on the Delaunay triangulation. When the circumference of the newly formed triangle contains other control points in addition to the vertices, the triangle will be discarded, and the corresponding vertex will be re-formed a new triangle with other control points until the above criterion is met, as shown in 473
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Fig. 1. Generation procedure of random Voronoi polygons (a) generation of control points (b) triangulation of problem domain (c) connection of circumcenters (d) generation of random Voronoi grains.
Fig. 2. Schematic diagram of the numerical modeling process (a) generation of Voronoi grains (b) meshing of Voronoi grains (c) insertion of cohesive elements (d) establishment of FEM-DEM model.
Fig. 1(b). Furthermore, the circumcenters of adjacent triangles are connected each other, as shown in Fig. 1(c). Finally, the triangles and control points are removed, and the Voronoi grains are generated, as shown in Fig. 1(d). In order to generate granular rock model with different grain size and regularity, two control parameters d0 and Ri are introduced in the process of generating random control points. The parameter d0 is the average distance between adjacent control points, which controls the average size of Voronoi grains. The average grain size will increase with the increase of d0 . The parameter Ri is the fluctuation amplitude of the distance between control points. By adjusting the fluctuation range above and below the given average distance (d0 ), the regularity of Voronoi grain can be changed. With a fixed average distance (d0 ), a larger value of Ri will lead to a rock model with more irregular grains.
2.3.2. Constitutive model of the cohesive element model In this study, a bilinear traction-separation law, which describes the loss of the load bearing capacity of the material as a function of separation, is adopted in the cohesive model, as shown in Fig. 3. The failure pattern can be divided into two types: tensile behavior (as shown in Fig. 3(a)) and shear behavior (as shown in Fig. 3(b)). For case when the cohesive element is under tension, the tensile stress-displacement law can be expressed as [50]: ⎧ eff ⎪𝑘𝑛 𝛿𝑛 ⎪ 𝑡𝑛 = ⎨(1 − 𝐷𝑛 )𝑘𝑛 𝛿𝑛𝑐 ⎪ ⎪0 ⎩
eff
𝛿𝑛 ≤ 𝛿𝑛𝑐 𝛿𝑛𝑐 ≤ 𝛿𝑛 ≤ 𝛿𝑛𝑓 eff
𝛿𝑛𝑓
≤
(6)
eff 𝛿𝑛
eff
where 𝛿𝑛 is the effective relative tensile displacement, 𝛿𝑛𝑐 is the critical tensile displacement beyond which the softening happens, 𝛿𝑛𝑓 is the ultimate tensile displacement at which the cohesive element entirely loses its strength, kn is the normal initial stiffness coefficient. eff In softening stage (𝛿𝑛𝑐 ≤ 𝛿𝑛 ≤ 𝛿𝑛𝑓 ), the contact stress of the cohesive element is controlled by a damage variable Dn which is defined as:
2.3. Rock failure model 2.3.1. Numerical modeling In this study, to simulate the dynamic breaking behavior of rock under water jet impact, the zero-thickness cohesive element is inserted along the boundaries of the Voronoi grains. By simulating the failure of the cohesive element, the rock breaking process can be realized, such as the crack initiation and propagation. In order to realize this process, the numerical model of FEM-DEM is established, as shown in Fig. 2. Firstly, random Voronoi grains are generated in the certain region based on the Voronoi tessellation technique, as shown in Fig. 2(a). Subsequently, the generated Voronoi grains are meshed with triangular grids, which forms the FEM model, as shown in Fig. 2(b). By a FORTRAN insertion program, the FEM model can be dispersed and the zero-thickness cohesive element is generated between the grain boundaries, as shown in Fig. 2(c). Finally, the FEM/DEM model, which contains both solid grains and zero-thickness cohesive elements, is established as shown in Fig. 2(d).
eff
𝐷𝑛 =
𝛿𝑛 − 𝛿𝑛𝑐
(7)
𝛿𝑛𝑓 − 𝛿𝑛𝑐
Similarly, for case when the cohesive element is sheared, its shear stress ts can be obtained as: ⎧ eff ⎪𝑘𝑠 𝛿𝑠 ⎪ 𝑡𝑠 = ⎨(1 − 𝐷𝑠 )𝑘𝑠 𝛿𝑠𝑐 ⎪ ⎪0 ⎩ eff
eff
𝛿𝑠 ≤ 𝛿𝑠𝑐 𝛿𝑠𝑐 ≤ 𝛿𝑠 ≤ 𝛿𝑠𝑓 eff
𝛿𝑠𝑓
≤
(8)
eff 𝛿𝑠
where 𝛿𝑠 is the effective relative shear displacement, 𝛿𝑠𝑐 is the critical shear displacement, 𝛿𝑠𝑓 is the ultimate shear displacement, ks is the 474
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Table 1 State equation parameters of water. 𝜌(kg m−3 )
C(m s−1 )
S1
S2
S3
𝛾0
𝛼
1000
1480
2.56
−1.986
0.2286
0.49
1.397
The deformation characteristic before the rock failure is mainly controlled by solid elements, while the mechanical response after the rock failure is mainly achieved by the failure of cohesive elements. Therefore, the linear elastic material of MAT_ELASTIC is applied to rock elements and the material of MAT_COHESIVE_GENERAL is used for the cohesive failure model in LS-DYNA. In the cohesive material model, a dimensionless effective separation parameter 𝜆 is used, which grasps for the interaction between the relative displacements in the normal and tangential directions: √ √( )2 ( )2 √ √ 𝛿 𝛿2 𝜆 = √ 𝑓1 + (14) 𝛿𝑛 𝛿𝑠𝑓 where 𝛿 1 is the normal displacement and 𝛿 2 is the tangential displacement. The cohesive element will reach failure when 𝜆= 1. 2.4. State equation of water jet The material of MAT_NULL in LS-DYNA is used for the water jet, and the GRUNEISEN equation is used to describe the relationship between pressure and density, which is shown as follow [51]: [ ( ) ] 𝛾 𝜌𝐶 2 𝜇 1 + 1 − 20 𝜇 − 𝛼2 𝜇 2 𝑃 = [ (15) ]2 + (𝛾0 + 𝛼𝜇)𝐸 3 𝜇2 − 𝑆3 𝜇 2 1 − (𝑆1 − 1)𝜇 − 𝑆2 𝜇+1 (𝜇+1)
where P is the pressure; 𝜌 is the density; E is internal energy per unit volume; C is the intercept of the curve 𝜇 s -𝜇 p ; S1 , S2 and S3 are the slopes of the curve 𝜇 s -𝜇 p ; 𝛾 0 is the GRUNEISEN coefficient, 𝛼 is the correction coefficient for the relationship between the 𝛾 0 and volume. The parameters of the state equation are shown in Table 1 [23,24].
Fig. 3. Constitutive model of the cohesive element (a) tensile behavior (b) shear behavior.
2.5. Coupling algorithm
tangential initial stiffness coefficient. And the damage variable Ds in eff softening stage (𝛿𝑠𝑐 ≤ 𝛿𝑠 ≤ 𝛿𝑠𝑓 ) is defined as:
Based on the LS-DYNA platform. a node to surface contact algorithm is adopted to implement the coupling of SPH and FEM/DEM, which is based on the penalty method. Assuming a virtual spring existed between SPH particles and the face of finite elements and the penetration between them will be checked in every time step. The contact force is imposed when there occurs a penetration, whereas nothing will be done. In addition, the FEM/DEM model based on Voronoi grains will be exerted a single surface contact algorithm, which allows automatic searching for penetrations of all grains after cohesive elements failed.
eff
𝐷𝑠 =
𝛿𝑠 − 𝛿𝑠𝑐 𝛿𝑠𝑓 − 𝛿𝑠𝑐
(9)
With the stress-displacement laws, the area under the stressdisplacement curve is the fracture energy which can be calculated in terms of the material mechanical properties through the following relationship: 𝐺𝐼𝐶 = 𝐺𝐼 𝐼 𝐶 =
2 𝐾𝐼𝐶
𝐸
𝐾𝐼2𝐼 𝐶 𝐸
3. Numerical validation of SPH-FEM/DEM (10) In this section, the capability of the SPH-FEM/DEM method in modeling the dynamic response and breaking behavior of rock under water jet impact is verified by reproducing the laboratory water jet impact test on sandstone. The test was conducted by Lu et al. [8] with the high-precision computerized numerical control (CNC) water jet cutting (WJC) machine manufactured by OMAX Corporation. The rock samples are short cylindrical sandstone cores of 50 mm in diameter and 50 mm in length, and the mechanical properties are as follows: Young’s modulus E = 57.62 GPa, Shear modulus G = 22.1 GPa, Poisson’s ratio 𝜐 = 0.23, density 𝜌 = 2370 kg/m3 , uniaxial compressive strength 𝜎 c = 68 MPa and uniaxial tensile strength 𝜎 t = 2.17 MPa. The diameter of water jet is 2 mm, and the standoff distance is 3 mm. In addition, seven different impact velocities were selected in the test, i.e., 157 m/s, 316 m/s, 447 m/s, 547 m/s, 632 m/s, 707 m/s and 774 m/s, respectively.
(11)
where GIC and GIIC are the fracture energies for mode I and mode II, respectively. As a result, the peak tractions and ultimate displacements are as follows: 𝛿𝑛𝑓 =
2𝐺𝐼𝐶 𝜎max
(12)
𝛿𝑠𝑓 =
2𝐺𝐼 𝐼 𝐶 𝜏max
(13)
where 𝜎 max and 𝜏 max are the normal peak traction and tangential peak traction, respectively. 475
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Fig. 5. Comparison of predicted rock broken depths with test results.
Fig. 4. Numerical model for simulation of impact test on sandstone.
Considering computational efficiency, a simplified 2-dimentional numerical model is established to reproduce the laboratory test, as shown in Fig. 4. The numerical model of rock is 50 mm in length and 50 mm in high, which is generated by Voronoi grains with the size parameter d0 of 1 mm and the regularity parameter Ri of 1.5. The water jet is 2 mm in diameter and modeled by SPH particles. The standoff distance and the impact velocity of water jet are consistent with the laboratory test. Besides, numerical simulation of FEM/DEM requires proper selection of the micro-mechanical parameters. In this study, by assuming the appropriate micro-mechanical parameters, such as normal and shear initial stiffness (kn and ks ), peak tensile and shear traction (𝜎 max and 𝜏 max ) and fracture energy (GIC , GIIC ), the numerical reproduction of the laboratory test is realized, and the rock broken depths in sandstone samples of the two conditions are compared and analyzed. After a series of trial-and-error tests, the micro-mechanical properties of cohesive elements are determined, based on which the numerical predicted rock broken depths can approximate the results of laboratory test at different impact velocity, as shown in Fig. 5. As illustrated in the figure, the rock broken depths of test and simulation both increase linearly with the increase of velocity, where the test data increases from 6.8 mm to 43.9 mm and the simulation data increases from 6.24 mm to 41.2 mm (with the average deviation of 5.77%). In addition, Fig. 6 demonstrates the qualitative results of numerical simulation and laboratory test with impact velocity of 447 m/s. As illustrated in the Fig. 6(a), a numerical predicted crushing pit is formed in the rock, which is surrounded by two inner cracks. In addition, a penetrating crack initiates and extends to the side surface of rock specimen. Fig. 6(b) shows the processing image of test qualitative result with impact velocity of 447 m/s. It can be observed from the figure that the rock is broken obviously and a penetrating crack extends to the side surface as well. However, because the test image is not from inner section but outer contour, the internal failure state is difficult to observe. The results indicate that there
Fig. 6. Qualitative results of numerical simulation and laboratory test (447 m/s) (a) numerical result (b) test result.
is a good agreement between numerical simulation and laboratory test, which verifies the capability of the developed method in modeling the dynamic breaking response of sandstone under water jet impact. The calibrated micro-mechanical parameters are as follows: kn = 433 GPa/mm, ks = 131 GPa/mm, 𝜎 max = 2.6 MPa, 𝜏 max = 10.2 MPa, GIC = 54 J/m2 , and GIIC = 70 J/m2 . 4. Numerical results In this part, to investigate the micro-mechanism of rock breaking behavior under water jet impact, the macro and micro-breaking process of rock is firstly analyzed. Then, by a series of numerical simulations with different microscopic conditions, the effects of the micro-structure (such as grain size, regularity) and the micro-mechanical properties (such as ductility, microscopic strength, contact stiffness ratio, and the heterogeneity of micro-parameters) on rock breaking performance under water jet impact are investigated. To conduct the investigation, numerical models which are all 60 mm in length and 30 mm in high, respectively, are built by random Voronoi grains with different micro-structure and micro-properties (as shown in Fig. 7). By changing the control parameters d0 and Ri , the size and regularity of rock micro-grains can be adjusted easily. Then, the generated Voronoi grains are further divided into triangular elements with a mesh size of 0.25 mm. In addition, the water jet in all models is simplified as a rectangle with size of 1 mm × 50 mm, which is meshed by 504 SPH particles under a uniform distribution. For all cases, the distance between water jet and rock target is 1 mm and the velocity of water jet is set 476
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4.2. Effects of the micro-structure on rock breaking performance 4.2.1. The effect of grain size In this section, the effect of the grain size on the rock breaking performance under water jet impact is analyzed. With the regularity parameter Ri = 1.5, five different d0 , i.e., 0.5 mm, 0.75 mm, 1.0 mm, 1.25 mm, 1.5 mm, are used in the numerical model, respectively. Besides, two parameters, namely broken depth (D) and broken area (A) are defined to evaluate the rock breaking performance quantitatively, as shown in Fig. 9. Fig. 10 presents the relationship between grain size and broken depth and area. As shown in the figure, the rock broken depth and area increase approximately linearly with the growth of grain size. When the d0 is 0.5 mm, the rock broken depth is 4.8 mm and broken area is 67.6 mm2 . As the d0 increases to 1.5 mm, the broken depth and area increase to 7.93 mm and 204.68 mm2 respectively, which indicates the grain size has a great effect on the development of crushing pit. In addition, the initiation and propagation of macro cracks also play an important role in rock breaking process under water jet impact. Therefore, the initiation amount (N) and maximum propagation length (L) of macro cracks (as defined in Fig. 9) outside the crushing pit with different grain size are statistically analyzed, as shown in Fig. 11. When d0 is less than 1.0 mm, no obvious macro cracks initiate in the rock. Then, macro cracks develop rapidly with the increase of d0 . As d0 increases to 1.5 mm, three macro cracks initiate in the rock, and the maximum propagation length of the cracks reaches 22.27 mm. The numerical results indicate that the grain size has a great effect on rock breaking performance under water jet impact. As the grain size increases, more severe rock breakage will be induced.
Fig. 7. Numerical model (d0 = 1 mm Ri = 2.5).
4.2.2. The effect of regularity The regularity is also an important parameter affecting mechanical properties of rock. In order to investigate the effect of regularity on rock breaking performance under water jet impact, in this section, five different values of regularity parameter Ri , i.e., 1.0, 1.5, 2.0, 2.5, 3.0 are used in the numerical model respectively. In addition, the grain size parameter d0 is set at 1.0 mm uniformly. The statistics of the rock broken depth and area with different Ri are shown in Fig. 12. As illustrated in the figure, the rock broken depth and area increase with the increase of regularity parameter. When Ri increases from 1.0 to 3.0, the rock broken depth changes from 5.99 to 8.15 mm, and broken area changes from 108.4 mm2 to 200.8 mm2 . Fig. 13 demonstrates the effect of regularity on the initiation amount and maximum propagation length of macro cracks. It can be found that as the grains change from regular to irregular, both the initiation amount and maximum propagation length of macro cracks increase. There are no obvious macro cracks initiated in the rock when Ri = 1.0 and 1.5. However, macro cracks develop rapidly after Ri reaches 2.0. When Ri increases to 3.0, four macro cracks initiate outside the crushing pit, and the maximum propagation length of the cracks reaches 23.61 mm. The numerical results indicate that the grain regularity has a significant effect on rock breaking performance. As the grains become more irregular, the range of crushing pit as well as the initiation and propagation of macro cracks all increase.
as 300 m/s. Besides, to avoid the boundary reflection effect, no-reflect boundary condition is set on both sides and bottom boundary for all numerical models. 4.1. Rock breaking process under water jet impact In this section, to investigate the detailed macro and micro breaking process of rock under water jet impact, the grain size and regularity parameter are adopted to be d0 = 1.0 mm and Ri = 2.5, respectively, as show in Fig. 7. The numerical predicted rock breaking process is shown in Fig. 8. As illustrated in the figure, at 1𝜇s, the water jet firstly impacted on the top surface of the rock, which induces compression-shear failure of the rock, resulting in initial crushing pit in the vicinity of the impact point due to the instantaneous high impact pressure. Meanwhile, the stress wave transmits into the rock, causing some micro cracks around the crushing pit, as shown in Fig. 8(a). As the water jet continues to advance, intense lateral flow is induced due to the severe compression of the injected liquid resulting from liquid and rock contact interaction, causing further tensile-shear failure of the rock. At this time, due to the horizontal free surface on the top, the crushing pit is developed rapidly horizontally, while the development in vertical direction is not obvious due to the constraint at the bottom, as shown in Fig. 8(b). Then, water flow invades along the expanded cracks, leading to further extension of cracks and expansion of crushing zone. The broken grains are washed away by the water flow, which forms the macro crushing pit. Besides, due to the intense compression of the water jet, the cracks at the bottom of the crushing pit expands obviously and form the macro crack (exceeding 5 mm), as shown in Fig. 8(c). With the continuous impact of water jet, the crushing pit and macro cracks at the bottom of the crushing pit are further developed. Finally, a crushing pit with the broken depth of 7.59 mm and the broken area of 152 mm2 is formed, which are surrounded by two obvious macro cracks with the length of 8.48 mm and 17.32 mm respectively, as shown in Fig. 8(d).
4.3. Effects of the micro-mechanical properties on rock breaking performance In this section, the effects of the micro-mechanical properties on rock breaking performance under water jet impact, such as ductility, microscopic strength, contact stiffness ratio and the heterogeneity of microparameters, are studied by the proposed numerical model. Uniformly, the grain size parameter d0 and regularity parameter Ri are set as 1.0 mm and 1.5 during the numerical simulation. 477
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Fig. 8. Macro and micro breaking process of rock at different time (a) initiation of micro cracks (b) widening of crushing pit (c) formation of macro cracks (d) further development of crushing pit and macro cracks.
4.3.1. The effect of ductility In order to investigate the effect of ductility on rock breaking performance under water jet impact, in this study, different levels of ductility are obtained by scaling the fracture energy GIC (GIIC ) with fixed peak tensile (shear) traction and initial normal (tangential) stiffness. The scaling ratio can be expressed by Sa . In this section, a series of numerical simulations of water jet impact are conducted with seven different levels of Sa , i.e., 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75.
Fig. 14 shows the statistical results of rock broken depth and area with different levels of ductility. As illustrated in the figure, as Sa increases from 0.25 to 1.75, the rock broken depth reduces from 12.88 mm to 4.63 mm, and the broken area reduces from 319.68 mm2 to 71.48 mm2 , but the reducing rates both decrease gradually. Fig. 15 presents the effect of ductility on the initiation amount and maximum propagation length of macro cracks. When Sa is equal to 0.25, eight obvious macro cracks initiate in the rock and the maximum propagation 478
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Fig. 9. Schematic diagram of indicators for evaluating rock breaking performance.
Fig. 10. The effect of grain size on broken depth and area.
Fig. 12. The effect of regularity on broken depth and area.
Fig. 11. The effect of grain size on the initiation amount and maximum propagation length of macro cracks.
Fig. 13. The effect of regularity on the initiation amount and maximum propagation length of macro cracks.
length of the cracks reaches 26.01 mm, which suggests that the rock broken degree is relatively severe. As Sa increases, both the initiation amount and maximum propagation length of macro cracks rapidly decrease. After Sa reaches 1.00, the ductility of the rock exceeds the
original value, which causes the impact energy of water jet not enough to expand micro cracks into macro cracks. At this time, the formation and expansion of the crushing pit are the main failure mode. The results show that when the rock becomes more ductile, the rock broken degree 479
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Fig. 14. The effect of material ductility on broken depth and area.
Fig. 16. The effect of microscopic strength on broken depth and area.
Fig. 15. The effect of material ductility on the initiation amount and maximum propagation length of macro cracks.
Fig. 17. The effect of microscopic strength on the initiation amount and maximum propagation length of macro cracks.
4.3.3. The effect of contact stiffness ratio ks /kn In this section, the effect of contact stiffness ratio ks /kn on rock breaking performance is investigated. The ductility and microscopic strength are fixed, and seven different values of ks /kn , i.e., 0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10.0, are adopted respectively in the study. Fig. 18 shows the relationship of rock broken depth, area and the value of ks /kn . As illustrated in the figure, there is no significant correlation between rock broken depth, area and the contact stiffness ratio. As the value of ks /kn changes from 0.1 to 10.0, the rock broken depth and area fluctuate between 6–7 mm and 130–150 mm2 , respectively. Meanwhile, in addition to a tiny macro crack initiated in the model of ks /kn = 2.0, there are no macro cracks generated in other models. It can be concluded that the contact stiffness ratio has no obvious effect on the rock breaking performance under water jet impact.
will reduce significantly, which indicates the material ductility has a great effect on the rock breaking performance under water jet impact.
4.3.2. The effect of microscopic strength In this section, the effect of microscopic strength on the rock breaking performance under water jet impact is investigated. With fixed ductility and initial stiffness, different level of microscopic strength can be obtained by scaling the peak traction 𝜎 max (𝜏 max ). The scaling ratio can be expressed by Sb and seven different levels of Sb , i.e., 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, are used in the numerical model respectively. Fig. 16 shows the effect of different level of microscopic strength on rock broken depth and area. As illustrated in the figure, both the rock broken depth and area reduce obviously as Sb increases, but the reducing rate decrease gradually. Fig. 17 presents the statistical curve of the initiation amount and maximum propagation length of macro cracks with different level of microscopic strength, which shows a similar result with that of the ductility. When the level of Sb is low, the development of macro cracks is relatively high. With the increase of Sb , both the initiation amount and maximum propagation length of macro cracks rapidly decrease. After Sb reaches 1.00, in addition to the crushing pit, no obvious macro cracks initiate in the rock. The results indicate that the increased microscopic strength will cause a decreasing rock breaking performance under water jet impact obviously.
4.3.4. The effect of the heterogeneity of micro-parameters Rock material is inherently inhomogeneous, which contains initial micro-defects, such as micro cracks, cleavages and grain boundaries. Mechanical behavior and breaking process of rock, as grained composite material, is deeply affected by such micro-defects [52]. To account for the micro-defects induced heterogeneity, the micro-mechanical parameters, including the peak traction 𝜎 max (𝜏 max ), initial stiffness kn (ks ), and fracture energy GIC (GIIC ) of the cohesive element, are assumed to conform to the Weibull distribution, as defined by the following 480
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Fig. 18. The effect of ks /kn on broken depth and area.
Fig. 20. The effect of the heterogeneity of micro-parameters on the initiation amount and maximum propagation length of macro cracks.
initiation amount reaching 4 and the maximum propagation length reaching 16.84 mm. As m increases, both the initiation amount and maximum propagation length of macro cracks gradually decrease. When m reaches to 10, no obvious macro cracks initiate in the rock. The numerical results indicate that as the heterogeneity index m increases, the rock broken degree under water jet impact will nonlinearly decrease first and then become stable, which means that the heterogeneity of micro-parameters has an obvious effect on rock breaking performance. 4.4. Discussion According to the above numerical results, both the micro-structure and micro-mechanical properties of the rock have significant effects on the rock breaking performance under water jet impact. For the microstructure effects, the rock broken degree increases with the increase of grain size, which may result from less linkage energy required for coarse-grained rocks. This result is consistent with the conclusion given by Dehkhoda and Hood [9]. Besides, as the grains become more irregular, the rock broken degree increases significantly. This may be due to the irregular grain shape result in more severe stress concentration at grain corners, which makes the micro cracks easily initiate. As for typical micro-mechanical properties, the ductility and microscopic strength have similar effects on the rock breaking performance under water jet impact. With the increase of ductility and microscopic strength, the energy and stress required for rock failure gradually increase, making the rock breaking performance gradually decreases under same water jet impact velocity. Besides, as the heterogeneity of microscopic parameters increases, the rock broken degree increases nonlinearly. This is because when the heterogeneity is very high, many micro cracks will be randomly initiated in the rock. As a result, when the water jet impacts rock, the micro cracks will propagate and coalesce with each other, which induce severe breakage of rock. Furthermore, the contact stiffness ratio has no obvious effect on the rock breaking performance, which indicates that the tensile failure plays a dominant role in the rock breaking process under water jet impact.
Fig. 19. The effect of the heterogeneity of micro-parameters on broken depth and area.
probability density function [53]: ( )𝑚−1 [ ( )𝑚 ] 𝑚 𝑥 𝑥 𝑊 (𝑥) = exp − 𝑢0 𝑢0 𝑢0
(16)
where x is the cohesive element micro-mechanical strength (such as peak traction, initial stiffness and fracture energy); u0 is the scale parameter related to the average micro-mechanical strength of cohesive element; m defines the shape of the distribution function, which controls the heterogeneity degree and is called the homogeneity index. An increase of homogeneity index leads to a more homogeneous numerical model. In order to investigate the effect of the heterogeneity of microparameters on rock breaking performance under water jet impact, six different heterogeneity indexes m, i.e., 1, 2, 3, 5, 10, 20 respectively, are adopted in this analysis. The statistics of the rock broken depth and area with different heterogeneity indexes m are shown in Fig. 19. As illustrated in the figure, when m is equal to 1.0, the rock broken depth is 9.18 mm and area is 177.92 mm2 . With the increase of m, both the rock broken depth and area reduce significantly, but the reducing rate gradually decrease. When m reaches 10, the rock broken depth and area become stable and stay around 6.9 mm and 130.2 mm2 , respectively, which approximates to that of homogeneous rock. The effect of the heterogeneity of micro-parameters on the initiation amount and maximum propagation length of macro cracks shows a similar rule. As illustrated in Fig. 20, macro cracks are most developed when m is equal to 1.0, with the
5. Conclusion In this study, the coupled SPH-FEM/FDM model has been developed in LS-DYNA software to simulate the dynamic behavior of rock under water jet impact. The SPH method was adopted to simulate the water jet behavior and FEM/DEM method was used to model the rock breaking process. In order to better represent the rock micro-structure and model the mechanical behavior of rock grains, the Voronoi tessellation 481
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technique was adopted to generate the micro-grains of rock. Zerothickness cohesive elements were inserted along the boundaries of the Voronoi grains to model the mechanical interaction between grains as well as the breaking process of rock. The proposed numerical scheme was verified by successfully reproducing the laboratory water jet impact test on sandstone. With the proposed coupled SPH-FEM/DEM model, the effects of micro-structure and micro-mechanical properties on the rock breaking performance under water jet impact have been systematically investigated. Based on the numerical results, the following conclusions can be drawn:
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(1) The grain size has a great effect on the rock breaking performance under water jet impact. As the grain size increases, more severe rock breakage will be induced. (2) The irregular shape of rock micro-grain makes the breaking more severe. When the irregularity parameter Ri increases from 1.0 to 3.0, the rock broken depth and area as well as the initiation amount and maximum propagation length of macro cracks developed all increase. (3) The ductility and microscopic strength of the rock have similar effects on rock breaking performance under water jet impact. When the ductility or microscopic strength of rock is low, relatively severe breakage will be induced. As the ductility or microscopic strength increases, the rock broken degree reduces significantly with the reducing rate gradually decreases. (4) The contact stiffness ratio ks /kn has little effect on the rock breaking performance under water jet impact. With the change of ks /kn , both the rock broken depth and area fluctuate within a certain range. (5) The heterogeneity of rock micro-parameters significantly affects its breaking performance under water jet impact. With the increase of the heterogeneity index m (the heterogeneity decreases), the rock broken degree reduces obviously first. When the heterogeneity index m reaches to 10, the rock breaking performance becomes stable, which approximates to that of homogeneous rock. Acknowledgment The research work is supported by the National Natural Science Foundation of China (Grant Nos. 41502283, 41772309, 41602324), National Key Research and Development Program of China (2017YFC1501302), the Major Program of Technological Innovation of Hubei Province, China (No. 2017ACA102) and the National Basic Research Program of China (973 Program) (Grant Nos. 2014CB046900, 2014CB046904). References [1] Momber AW. Fluid jet erosion as a non-linear fracture process: a discussion. Wear 2001;250(1):100–6. [2] Momber AW. An SEM-study of high-speed hydrodynamic erosion of cementitious composites. Compos Part B Eng 2003;34(2):135–42. [3] Lan H, Martin CD, Bo H. Effect of heterogeneity of brittle rock on micromechanical extensile behavior during compression loading. J Geophys Res Solid Earth 2010;115(B1):B01202. doi:10.1029/2009JB006496. [4] Peng J, Wong LNY, Teh CI. Influence of grain size heterogeneity on strength and micro-cracking behavior of crystalline rocks. J Geophys Res Solid Earth 2017;122(2):1054–73. [5] Sabri M, Ghazvinian A, Nejati HR. Effect of particle size heterogeneity on fracture toughness and failure mechanism of rocks. Int J Rock Mech Min Sci 2016;81:79–85. [6] Wasantha PLP, Ranjith PG, Zhao J, Shao SS, Permata G. Strain rate effect on the mechanical behaviour of sandstones with different grain sizes. Rock Mech Rock Eng 2015;48(5):1883–95. [7] Miao S, Pan PZ, Wu Z, Zhao S. Fracture analysis of sandstone with a single filled flaw under uniaxial compression. Eng Fract Mech 2018;204:319–43. [8] Lu Y, Huang F, Liu X, Ao X. On the failure pattern of sandstone impacted by high-velocity water jet. Int J Impact Eng 2015;76:67–74. [9] Dehkhoda S, Hood M. An experimental study of surface and sub-surface damage in pulsed water-jet breakage of rocks. Int J Rock Mech Min Sci 2013;63:138–47. [10] Engin IC. A correlation for predicting the abrasive water jet cutting depth for natural stones. S Afr J Sci 2012;108(9–10):69–79. 482
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