Analysis of the forces in ordered FCC packings with different orientations

Abstract

Ordered packing of hard spheres, namely {111}- and {100}-oriented FCC, is numerically generated by means of Discrete Element Method (DEM). The probability density distribution P(F) and transmission of interparticle forces in such a packing is analyzed in detail. It is shown that P(F) can be described by a symmetric distribution irrespective of the orientation, consistent with the literature findings. However, different orientations lead to different force transmissions because of the effect of gravity and stability requirement. In the {111}-oriented FCC, forces transmit both horizontally (in-plane) and vertically (inter-plane), while only vertical inter-plane forces are observed in the {100}-oriented FCC. The analysis on the two resulting sandpiles shows that a force peak exists at the position right under the sandpile apex.

Introduction

Packing of particles is important to scientific research and industrial applications [1], [2], [3], [4], [5], [6]. Its study has a long history dating back to the Kepler conjecture proposed about four hundred years ago. Since the pioneer work of Bernal [1], particle packing has attracted enormous interests from various fields, such as physics, chemistry, materials and biology [4], [5], [6]. Some modern studies have recently been reviewed and discussed by Torquato and Stillinger [7].

One of the challenges in this area is to understand the highly heterogeneous force network in a packing with disordered structure. With an external force (e.g. gravity), the force inside the packing propagates along the so-called “force chains” which define the path of interparticle forces. Such a feature has been observed in physical experiments [8], [9], [10], [11], [12], [13], [14], [15], [16], [17] and numerical simulations [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]. The contact forces can be statistically quantified in terms of probability density functions (PDF). It is a general consensus that the distribution has exponential decay at large forces. Distribution at small forces is still unclear because of the limitation of experiments in measuring small forces. Some results showed that the distribution has a small peak near the mean value and then decreases toward zero, while others showed a plateau around the mean value and a slight increase with decreasing force.

Depending on structure, particle packing can be divided into two categories, random or disordered packing (e.g. the so called random loose packing and random close packing) and ordered packing (e.g. face-centered cubic (FCC) packing and hexagonal-close-packed (HCP) packing). Previous studies on force network were mainly on disordered packing and few are on ordered packing. Ideally, the forces in an ordered packing formed by rigid particles may have certain discrete values which can be analytically obtained. However, particles with finite hardness deform under external forces and the deformation is far less than particle size. Thus in reality, an ordered packing may not be perfect and its forces will have a distribution. To date, the force distribution and propagation in a slightly perturbed ordered packing is still an open issue [33], [34].

There are a few studies on the measurement of the force distribution in a packing which may be subjected to external stresses [35], [36], [37], [38]. For example, under uniaxial compression Blair et al. [35] studied the probability distribution of normal forces between particles in the HCP packing. The physical measurements by Chan and Ngan [36] on the probability distribution of contact forces in the HCP packing of polystyrene spheres identified that the force distribution exhibited a smooth peak. Mueggenburg et al. [37] measured the local contact forces at both the top and bottom boundaries of 3D FCC and HCP packing in response to an external force applied to a small area at the top surface. On the other hand, Spannuth et al. [38] studied the stress transmission through 3D FCC packing with stacking faults. However, in most of these studies, only the forces at the boundaries can be measured, i.e. only the forces between particles and container walls are determined. In general, internal particle–particle forces and external particle–wall forces are different [39], so it is not always correct to use the external forces to infer the internal ones. It is important to find a method to determine the internal forces among particles in a packing. Numerical simulations by means of discrete element method (DEM) can produce such information readily [40].

In this paper, we numerically investigate the force structure in slightly perturbed ordered packings. Two FCC packings with different orientations, namely {100} and {111}, are generated from the DEM simulation [41], [42]. Here their force distributions are analyzed in detail. The results are useful in constructing a comprehensive picture about the forces and their network in an ordered packing of particles.