The discrete element method (DEM) was first proposed by Cundall and Strack to model the behaviour of soil particles subject to dynamic loading conditions . Mishra and Rajamani pioneered the application of DEM to grinding mills and demonstrated that despite the fact that the DEM simulations are based on two dimensions (2D), the technique is able to predict the power draw of mills with reasonable accuracy over a wide range of mill diameters . More than 10 years since then, the DEM technique has been successfully applied to ball mills , SAG mills and centrifugal mills.
DEM has also been applied to study impact-induced particle breakage. Using DEM simulation of impact breakage of agglomerates and aggregates that are hardened by cement, different parameters that influence the impact fracture have been analysed. In some studies, the finite element method (FEM) is usually adopted to determine stress patterns, and DEM has been used to show crack distributions in rocks under loading. Also, using DEM modeling of the compressive strength and drop weight test, the relationship between strain rate, impact energy and the degree of fragmentation has been determined.
Nine rocks with different mechanical properties were modelled as granular assemblies in the shape of a sphere and/or a cube. Each rock as a single particle was modelled while it was crushed in a laboratory jaw crusher , and its fracture behaviour in the processing zone was studied using PFC3D software.
To verify the estimated results, the fracture mechanisms of the rocks in the jaw crusher were studied, and the results were discussed with respect to the fracture behaviour modelled by PFC3D software. Then, the determined work done by the jaws of the crusher were compared with those of the Bond crushing energies (estimated from the Bond crushability index) of the rocks. Also, the effect of the Bond crushability index on the estimated crushing energy of the rocks was studied.
Materials and methods
In this work, nine different rock specimens were obtained from some mines in Iran. The mechanical properties (elasticity modulus and uniaxial compressive strength) and the Bond crushability index of these rocks were experimentally evaluated. For this purpose a standard compaction test was carried out on five cylindrical specimens of 54 mm diameter and 135 mm height using an Instron servo hydraulic machine.
To measure the modulus of elasticity, five strain gauges were used. The Bond index was estimated using a standard impact crushability test that was applied to 15 cubic specimens of height 50 mm. The dry density of the rocks studied was estimated by the saturation and caliper technique, which was defined by the International Society for Rock Mechanics (ISRM).
Generally, there are three Bond indices: ball, rod and crushability. The Bond ball and rod indices are determined using standard ball and rod mill, respectively, and are suitable for grinding. The Bond crushability index, which is estimated using a standard impact crushability test, is suitable for crushing. Because the jaw crusher was modelled in this study, the Bond crushability index of the rocks studied was determined.
To study fracture mechanisms of the rocks in the processing zone of a crusher, a cubic and a spherical specimen of limestone were prepared, and their fracture process in a laboratory jaw crusher were studied using a high-speed camera.
PFC3D model of rock and jaw crusher
In order to examine the fracture behaviour of the rocks and to determine the applied energy in a jaw crusher, a PFC3D model of a jaw crusher was developed. This model is a single macro-particle simulation in a crusher, and multi-body interactions are ignored. The crushing process for a single cubic and/or a single spherical rock in the jaw crusher is simulated using the PFC3D model.
It is possible to model the behaviour of particles that may be enclosed within a finite volume by non-deformable walls. The code keeps a record of individual particles and updates any contact with other particles or walls. Each calculation step includes the solution of equations of motion to a particle, using a force-displacement law for each contact . The rock is modelled as an assembly of stiff particles (balls) that are bonded together. The properties of the bonds between balls can be defined in the PFC3D software . For the purpose of modelling, a cubic specimen 5 cm high and a spherical specimen 6 cm in diameter were prepared from nine various rocks that were studied.
A laboratory jaw crusher was then modelled using the PFC3D code. A jaw crusher has two plates (jaws), one of which is fixed and the other, swinging open and closed, trapping and crushing material between the two surfaces. The feeding entrance is 10 cm wide, and its maximum discharge aperture is 2.5 cm (open status). The minimum open space between the jaws during the crushing cycle is 17 mm (closed status). The rotational speed of the moving jaw is almost 300 rpm.
Various positions in which a cubic and a spherical specimen may be located in the jaw crusher are shown in Figure 3. For the cubic specimen, two types of contact between a rock and the vertical jaw are possible. These are described as planar and linear.
In order to computing the breakage energy in the DEM model, the concept of wall energy has been utilized. The wall energy is equal to the total accumulated work done by all walls (here the jaws of the crusher) on the granular model of rock specimens that is calculated by the PFC3D software . It is assumed the fracture energy of rocks is equal to the wall energy per mass of rocks. In this case dynamic and multiple impacts behaviour of the crusher has been neglected.