Discrete Element modeling and analysis of shielding effects during the crushing of a grain

The potential for a particle to crush under one-dimensional compression is critically dependent on the coordination number of that particle. Neighboring particles decrease deviatoric forces at contacts, which reduces tensile stress and subsequent fracture propagation in the crushable particle. This phenomenon is called “shielding effect”. In this paper, we model a sand particle as a spherical cluster of bonded, hexagonally packed, equally sized, non-breakable spheres with the Discrete Element Method (DEM). We use rigid walls to apply forces at the contact with neighboring particles. First, we calibrate the cluster mechanical parameters against published experimental results obtained during unconfined uniaxial compression tests. Then we propose a procedure employed in DEM to generate symmetric and random distributions of walls. We use two loading walls only: the remainder of the walls is used for passive shielding. Force-displacement curves obtained during the crushing simulations clearly show that the peak force reached when the cluster first splits increases with the number of shielding walls, which demonstrates shielding effects. The total resulting compression force applied by the walls increases linearly the coordination number. We expect that our computational method will allow the optimization of crushing in powder technology, and the prevention of crushing in geotechnical engineering. roughness[10], which have a strong influence on the initial rotation and chipping of asperities before the catastrophic crushing of the particle [6]. By idealizing particles as a hexahedron composed of two triangular pyramids, Cavarretta proposed a simplified model to calculate the vertical and horizontal displacements in the initial rotation stage. In DEM simulations (performed with PFC3D software), Cheng [2] also noticed rotation and slippage before breakage during uniaxial compression tests. Effect of the coordination number. A high coordination number is known to prevent crushing. The mechanism can be explained by the redistribution of concentrated compression forces at particle contacts into a distributed pressure that is close to hydrostatic conditions. Hence the induced tensile stress developed inside the grain is reduced. Tsoungui [5] presented a method to combine all the contact forces applied on a particle in order to calculate the principal stresses: the loading conditions with multiple contacts is simplified into a configuration with four contacts in two principal directions, which was used to calculate the tensile stress in the particle. The tensile stress in the particle was compared with the particle strength to predict fracture propagation in the particle and crushing. Unfortunately the author did not give the solution for 3D problems. Besides, in this method the particle will never break if the two principal stresses are equal, which is obviously not practical for sand or rock particles. Lim and McDowell [3] also mentioned the importance of the coordination number in a paper that presents a model of agglomerates that they used to simulate crushable particles. But they discussed the influence of the number of particles that make the cluster, and not on the number of contacts between agglomerates. The influence of the coordination number on particle crushing is still not fully understood. In this paper, we simulate the process of crushing of a single particle with different number of coordination numbers. We use PFC3D, a Distinct Element Method (DEM) software[11]. We model a sand particle as a spherical cluster of bonded, hexagonally packed, equally sized, non-breakable spheres. The compression forces applied by neighboring particles are accounted for by applying velocity boundary conditions to walls in contact with the cluster. The aim of our simulation is to testify the existence of shielding effect, which occurs when small particles act as a coating agent and prevent the crushing of larger particles. This paper is organized in this sequence: In the first part, we calibrate the cluster mechanical parameters against published experimental results obtained during unconfined uniaxial compression tests. In the second part, we simulate the shielding effects for a symmetric distribution of 2 to 135 walls and for a random distribution of 60 to 135 walls. Conclusions are drawn in the last part of the paper. 2. MODEL CALIBRATION The crushable particle was modeled by a spherical cluster 1.6mm in diameter, which consisted of around 11,000 rigid (uncrushable) spheres. The normal and shear strength of the bonds between these spheres obeyed a parallel bond model, which is already implemented in PFC3D and can be seen as a cement-like substance acting as a glue between the spheres. We arranged the rigid spheres into a Hexagonal Close Packing (HCP) because it gives highest density and proved to be effective in the simulation crushable particles in PFC3D [12-14]. We note that the HCP is not symmetric, therefore the packing is denser in some directions of space, which induces anisotropic cluster mechanical properties. Therefore, we compared a vertical and a horizontal HCP (shown in Fig. 1). Fig. 1. Vertical and horizontal HCP of spheres used to model the crushable clusters In order to apply the uniaxial compression force at the top and bottom of the cluster, we used disk walls (Fig. 1), which are more representative of the contact surface that develops between round particles than square walls. Disk walls had a diameter of 0.8mm. We checked that a diameter of 0.8mm size yields the same peak force at grain failure with infinite walls. This allowed us to use experimental test results obtained during the compression of individual grains between flat platens for model calibration. Wall stiffness was set to be much larger than that of spheres and bonds. After creating the cluster and the walls, we subjected the cluster to gravity forces. Then we imposed a controlled velocity to the walls to apply the uniaxial compression force. We simulated crushing at a very low speed, in order to remain in quasi-static conditions. The intensity of forces at the walls and the number of broken bonds were monitored during the crushing simulation. We model a sand particle similar to the one that Cil used in his experiment [13]. Experimental results used in our calibration are shown in Fig. 2. The peak compression force at failure is 146N. The size of the sand particle is between US sieves #20 (0.599 mm) and #30 (0.853 mm). For simplicity, we used the average value, i.e. 0.729mm. Fig. 2. Load-displacement curve obtained during the experiment of a uniaxial compression test performed on a crushable sand grain [13] In order to calculate the peak force of different sizes with Eq.(2), we had to determine the size effect parameter b . Lee’s compression tests of individual particles of Leighton Buzzard sand, oolitic limestone and carboniferous limestone revealed that the size effect parameters in Eq.(2) were -0.357, -0.343 and -0.420, respectively [4, 9]. To our best knowledge, more precise experimental data does not exist in the literature. Therefore, we assumed that b was between -0.343 and 0.420. According to Eq.(2), by computing the ratio of forces and cancelling K, the peak compression force for the same sand particle with a diameter of 1.6mm should between 506N and 531N. The calibration process is done by continuously adjusting parameters, i.e. normal and shear parallel bond strength and stiffness, to best fit the peak force and the shape of the displacement-force. A summary of final parameters used in the simulation is reported in Table 1, and the corresponding calibration results are shown in Fig. 4. Table 1. Parameters used in DEM simulation Input parameter Value Diameter of cluster: mm 1.6 Diameter of sphere: mm 3.2×10 Density of sphere: kg/m 3581 Normal and shear stiffness of each sphere: N/m 1×10 Normal and shear bond strength: MPa 170 Normal stiffness of parallel bond: N/m 3×10 Shear stiffness of parallel bond: N/m 1×10 Frictional coefficient of sphere 0.5 The pattern of Load-Displacement curve is similar to that in the experiment. The model captures and the breakage of the cluster into several fragments (Fig. 3) and the peak force when this catastrophic collapse occurs (Fig. 4). It is interesting to find that an initial peak is obtained with the horizontal HCP (point A in Fig. 4), and not with the vertical HCP. We also noted that the horizontal HCP cluster was rotating during the simulations. Similar initial peak forces and rotations were obtained and analyzed by other searchers [2, 15]. Rotation is interpreted as the minimization of the potential energy of the cluster, which gets closer to a stable equilibrium position. Despite this difference of curve shape between the two packing tested, the peak forces only exhibited a 4% difference: we obtained peak forces of 509N (for the vertical HCP) and 489N (for the horizontal HCP), which falls close to the range of 506N and 531N expected. We also note that the final number of broken bonds is the same for both packings.