Page 29 - The Combined Finite-Discrete Element Method

12 INTRODUCTION

with rapid density decrease towards the top of the pack. The final density is almost 10%
smaller than the theoretical density for the same problem with periodic boundaries.
A further demonstration of the influence of sphere size on packing density is obtained
through deposition of the same volume of solid, but comprised of larger spheres. Numer-
ical experiments included:
• 41472 spheres of d = 7.497 mm (Figure 1.15)
• 5184 spheres of d = 14.994 mm (Figure 1.15)
• 648 spheres of d = 29.988 mm (Figure 1.16)
• 192 spheres of d = 44.982 9 mm (Figure 1.16)
• 81 spheres of d = 59.976 mm (Figure 1.17)
• 50 spheres of d = 70.439 mm (Figure 1.17).
The final density profiles for each of the packs are given in Figure 1.18. The density
of packs comprising very large spheres is very far from the theoretical density. This is
due to the influence of the boundary conditions. For large spheres, the box is simply too
small, and theoretical packing cannot be achieved.
As the spheres get smaller, the influence of the box diminishes and the density gets
closer to the theoretical density, as shown in Figure 1.19, where packing density as a
function of normalised sphere diameter (sphere diameter divided by the size of the edge
of the box base–250 mm) is plotted. For large spheres the density can go either up or down
with a reduction in sphere size, i.e. discontinuum behaviour is strongly pronounced. As the
spheres get smaller, more uniform convergence toward the theoretical result is achieved.
Thus, at zero sphere diameter, theoretical density is achieved.

Figure 1.15 Final pack of spheres: left d = 7.497 mm, right d = 14.994 mm.