To investigate the rock fragmentation and its influence factors under the impact load of water jet, a numerical method which coupled finite element method (FEM) with smoothed particle hydrodynamics (SPH) was adopted to simulate the rock fragmentation process by water jet. Linear and shock equations of state were applied to describe the dynamic characteristics of rock and water, respectively, while the maximum principal stress criterion was used for the rock failure detection. The dynamic stresses at the selected element containing points in rock are computed as a function of time under the impact load of water jet. The influences of the factors of boundary condition, impact velocity, confining pressure, and structure plane on rock dynamic fragmentation are discussed.

Rock fragmentation by the water jet technology has been widely used in mining, petroleum drilling, civil construction, gas drainage, and cleaning operations [

The present paper is aiming at revealing the rock fragmentation mechanism and explaining the reasons for crushing zone formation, crack initiation, and propagation under the impact load of water jet. To achieve this goal, a numerical method coupled FEM with SPH was adopted and the following topics were investigated: (a) the fracture process of rock sample under water jet impact; (b) the mechanism of crushing zone formation, crack initiation, and propagation; (c) the effects of boundary condition, impact velocity, confining pressure, and structure plane on the rock failure and crack initiation and propagation.

As we all know, the large deformation of water exists in the rock fragmentation process by water jet impact. The calculation would easily terminate due to the mesh distortion while adopting the conventional Lagrangian FEM. The coupled Lagrangian/Eulerian method can effectively avoid the mesh distortion but would increase the computational cost and reduce the computational efficiency.

SPH is one of the mesh-free particle methods in Lagrangian frame, which has been widely used in various fields since it is first invented to solve the astrophysical problems. In the SPH method, a series of particles with some physical quantity, such as mass and velocity, are used to express the continuous material. The mesh distortion can be well avoided due to no structural mesh among these particles. Therefore, it is a powerful method for solving the multiphysics flow and large deformation problems. Meanwhile, the FEM is suitable for the simulation of mechanical characteristics of solid materials. Nowadays, the coupled SPH/FEM method has been verified to solve the fluid-solid interaction problem commendably, and it could overcome the disadvantages associated with the Lagrangian/Eulerian method [

Coupled SPH/FEM method.

Compared with the finite element method, a kernel approximation is used in SPH based on randomly distributed interpolation points with no assumptions about which points are neighbors to calculate spatial derivatives. Considering a problem domain

The equations of conservation governing the evolution of mechanical variables can be expressed as follows:

conservation of mass:

conservation of momentum:

conservation of energy:

As mentioned above, there is a large deformation of water in the rock fragmentation process. Therefore, the shock equation of state is adopted to describe the mechanical characteristics of water
^{3},

The pressure or deformation of the rock is relative small under the dynamic load by water jet impact, and their variations have less influence on the thermodynamic entropy. Therefore, the pressure variation could be deemed to be only related to the density and volume variations of the rock element. Consequently, the linear equation of state is adopted to describe the rock mechanical properties, which is very suitable for solving the dynamic problem with small deformation and pressure. The equation can be expressed as [

In order to investigate the mechanism of rock fragmentation under the dynamic load by water jet impact, the maximum principal stress criterion is adopted to describe the rock failure behavior. Once the maximum tensile or shear principal stresses exceed the rock dynamic tensile or shear strength, the rock element fails. The maximum principal stress criterion can be expressed as
^{3}, elasticity modulus = 58.9 GPa, dynamic tensile strength = 57 MPa, dynamic shear strength = 192 MPa, and poisson ratio = 0.22.

The geometric model of rock fragmentation under dynamic load by water jet impact is shown in Figure

Geometric model.

Figure

Rock status as a function of time by water jet impact.

1.2

3.1

6.2

20

50

Experimental result

Pressure versus time at the element containing point 1.

To investigate the formation mechanism of the crushing zone by water jet impact, the stresses of the element containing point 2 (0 mm, 27 mm) in the crushing zone as a function of time are recorded as shown in Figure

Stresses change of the element containing point 2.

In Figure

Stresses change of the element containing point 3.

The rock bottom is set as the free boundary in the simulation, which results in the appearance of spall crack at rock bottom. The element containing point 4 (6 mm, 7 mm) is selected to investigate the mechanism of spall crack and its stresses as a function of time are recorded as shown in Figure

Stresses change of the element containing point 4.

As shown in Figure

Stresses change of the element containing point 5.

Pressure versus time of the elements in different positions.

The element stresses in different positions (shown in Figure

Many factors influence the rock fragmentation process by water jet impact, such as boundary conditions, water jet velocity and diameter, rock confining pressure, and structure plane. Based the above-mentioned numerical model, the influence of these factors on rock fragmentation will be investigated in the following sections.

The rock bottom is set as the free boundary and no-reflection boundary, respectively; then the rock fragmentation with the two different boundary conditions is investigated based on the developed numerical model. Figure

Rock fragmentation status with free and no-reflection boundaries.

Free boundary

No-reflection boundary

The velocity of water jet determinates the impact energy; therefore, it has a direct effect on rock fragmentation. Three different velocities of 300 m/s, 500 m/s, and 800 m/s were selected in the simulation, and rock failure behaviors at 20

Rock fragmentation status at different impact velocities.

300 m/s

500 m/s

In the deep geotechnical engineering, the confining pressure greatly influences the rock fragmentation and the crack initiation and propagation. In order to avoid the effect of the reflected stress wave on the numerical simulation results, the rock bottom is set as the no-reflection boundary, and both two rock sides were applied with the confining pressure. Four confining pressures of 10 MPa, 20 MPa, 40 MPa, and 60 MPa were separately applied to both two rock sides, and the rock fragmentation statuses at 20

Rock fragmentation status with different confining pressures.

10 MPa

20 MPa

40 MPa

60 MPa

The relative position between the impact point and the structure plane also influences the rock fragmentation and crack initiation and propagation. The sandstone with a width of 0.5 mm was used to simulate the structure plane, and the distances of 10 mm, 15 mm, and 20 mm as well as the angles of 30° and 60° were set in the following simulations. The rock bottom and sides were all set as the no-reflection boundary to avoid the effect of the reflected stress wave on the simulation results. The rock fragmentation statuses at 20

Rock fragmentation with different structure planes.

Structural plane

10 mm

15 mm

20 mm

30°

60°

In the present study, the coupled SPH/FEM method is implemented to simulate the rock fragmentation under the impact load of water jet. The influence factors on the rock fragmentation and crack initiation and propagation are extensively investigated. From the numerical simulation results, we come to the following conclusions.

The rock nearby the impact point is crushed severely due to the extremely high local pressure at the initial impact stage. Then, the pressure attenuates sharply when it propagates in the rock medium due to the energy dissipation. Therefore, the rock fragmentation by water jet impact can also be regarded as a load/unload process.

The rock fragmentation modes at the upper and lower parts from the numerical simulation are consistent with that in the experiment. The rock fragmentation by water jet impact is due to the combined action of shear and tensile failure. The crushing zone nearby the impact point is mainly caused by the shear failure as a result of the high compressive stress, while the radial crack initiation and propagation is aroused by the tensile failure. However, spall crack nearby the rock bottom is caused by the reflected stress wave. At the same time, the micromechanism of the cracks initiation and propagation is investigated by analyzing the element stresses as a function of time in different positions (Figures

The effect of the free boundary on the rock fragmentation is very significant, and the spall crack can improve the fragmentation extent. The surface erosion of rock is primary at low impact velocity and the actual failure (such as radial and spall cracks) will occur only when impact velocity reaches up to a certain value. The confining pressure can restrain the stress wave and cracks propagation, but it will conduce to a severe rock fragmentation as a result of the stress concentration closed to the impact point. The effect of structure plane on rock fragmentation is similar to that of the free boundary. An optimal distance between the impact point and the rock structural plane can be obtained once the other conditions are determined. Meanwhile, the structure plane can lead the spall crack to propagate along the plane direction.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to acknowledge the Foundation of National 863 Plan of China (2012AA062104), the National Natural Science Foundation of China (51375478), the project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (SZBF2011-6-B35), and the Graduate Education Innovation Project of Jiangsu Province (CXLX12_0948).