Determine the minimum possible voids ratio for a uniformly graded sand of perfectly spherical grains arranged in a rhombohedral array (densest packing).

Minimum Voids Ratio for Uniformly Graded Sand

Problem Statement

Determine the minimum possible voids ratio for a uniformly graded sand of perfectly spherical grains arranged in a rhombohedral array (densest packing).

Solution

1. Initial Cubical Packing (Maximum Voids)

In the original unit cube (loosest packing, \( \alpha = 90^\circ \)):

Number of particles \( = \frac{1}{d} \times \frac{1}{d} \times \frac{1}{d} = \frac{1}{d^3} \)

Volume of solids \( V_s = \frac{\pi}{6}d^3 \times \frac{1}{d^3} = \frac{\pi}{6} \approx 0.5236 \)

2. Rhombohedral Packing (Densest State)

Rearranged into a rhombohedral array (\( \alpha = 60^\circ \)):

Total volume \( V = 1 + 2\cos\alpha(1 – \cos\alpha) \)

For \( \alpha = 60^\circ \):

\( V = 1 + 2\cos60^\circ(1 – \cos60^\circ) = 0.7071 \)

3. Voids Ratio and Porosity

Volume of voids:

\( V_v = V – V_s = 0.7071 – 0.5236 = 0.1835 \)

Voids ratio:

\( e = \frac{V_v}{V_s} = \frac{0.1835}{0.5236} \approx 0.35 \)

Porosity:

\( n = \frac{e}{1 + e} = \frac{0.35}{1.35} \approx 25.95\% \)

Results:

Minimum voids ratio: \( e_{\text{min}} \approx 0.35 \)
Porosity: \( n \approx 25.95\% \)

Explanation

The minimum voids ratio occurs when spherical grains are packed in a rhombohedral array (\( \alpha = 60^\circ \)). This dense arrangement reduces the total volume (\( V \)) while keeping the solids’ volume (\( V_s \)) constant. The voids ratio (\( e \)) and porosity (\( n \)) are derived from the relationship \( V_v = V – V_s \).

Physical Meaning

The minimum voids ratio represents the densest achievable packing for spherical grains. In geotechnical engineering, this is critical for:

Designing stable foundations with minimal settlement.
Optimizing soil compaction for infrastructure projects.
Understanding theoretical limits of soil density and permeability.