Shape influences on the packing density of frustums

The shape of a frustum is a bivariate function of aspect ratio (w) and base radii ratio (c). To investigate the shape influences on the packing density of frustums, random packings of cones (c=0), truncated cones and cylinders (c=1) with various heights and diameters are studied through numerical simulations. An improved relaxation algorithm with assembly sphere models for non-spherical particles is applied in the numerical simulations, and the randomness of the packings of the frustums considered is verified. Base on the simulation results, the relationship between the packing density and shape parameters is illustrated, and an empirical formula is proposed to reflect the correlation. It shows that, for the particles having the same w, truncated cones can be packed denser than cones, but looser than cylinders. The packing density of truncated cones first increases and then decreases with the growth of w, while the packing density increases monotonically with the increase of c. No obvious peak is found on the curve of packing density versus c for the random packing of truncated cones, which is different from the reported results for ordered packing where a peak was identified. Furthermore, we observe that the packing density of truncated cones is a linear superposition of the c effect on w effect, and the effects of c and w on the packing density can be treated independently. The optimal aspect ratios of truncated cones, which give the highest packing densities, are all around 0.8. The highest packing densities of cylinders and cones are also the upper and lower bound of the random packing density of truncated cones.