An imaginary soil mass is contained in a container measuring 10 cm X 10 cm X 10 cm. The soil consists of spherical grains of size 1 cm in diameter. Determine the maximum possible voids ratio, porosity and percent solids.

Problem Statement

An imaginary soil mass is contained in a container measuring 10 cm × 10 cm × 10 cm. The soil consists of spherical grains of size 1 cm in diameter. Determine:

Maximum possible void ratio (\( e_{max} \))
Maximum porosity (\( n_{max} \))
Percent solids

Solution

Calculate Total Volume (\( V_t \))

\( V_t = 10 \times 10 \times 10 = 1000 \, \text{cm}^3 \)

Calculate Volume of Solids

Volume of one sphere = \( \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (0.5)^3 = 0.524 \, \text{cm}^3 \)
For simple cubic packing:
Number of spheres = \( \frac{1000}{0.524 \times 1.91} = 524 \) spheres
Total volume of solids = \( 524 \times 0.524 = 524 \, \text{cm}^3 \)

Calculate Void Volume (\( V_v \))

\( V_v = V_t - V_s = 1000 - 524 = 476 \, \text{cm}^3 \)

Calculate Void Ratio (\( e \))

\( e = \frac{V_v}{V_s} = \frac{476}{524} = 0.91 \)

Calculate Porosity (\( n \))

\( n = \frac{V_v}{V_t} \times 100 = \frac{476}{1000} \times 100 = 47.6\% \)

Calculate Percent Solids

Percent solids = \( \frac{V_s}{V_t} \times 100 = \frac{524}{1000} \times 100 = 52.4\% \)

Final Results:

Void ratio (\( e \)) = 0.91
Porosity (\( n \)) = 47.6%
Percent solids = 52.4%

Explanation

1. Packing Arrangement:
The solution considers the most efficient arrangement of spherical particles while maintaining realistic spacing. This arrangement allows for optimal filling of the container volume.

2. Volume Relationships:
The calculations show that approximately 52.4% of the total volume is occupied by solid particles, with the remaining 47.6% being void space. This distribution is typical for spherical particles of uniform size.

3. Void Ratio:
The void ratio of 0.91 indicates that the volume of voids is slightly less than the volume of solids, which is consistent with the geometric constraints of spherical particle packing.

Physical Meaning

1. Significance of Results:
The calculated values represent a realistic packing arrangement for uniform spheres:

The void ratio (0.91) indicates an efficient packing structure
The porosity (47.6%) shows balanced distribution between solids and voids
The percent solids (52.4%) indicates good space utilization

2. Practical Applications:
Understanding these parameters helps in:

Predicting soil behavior under loading
Estimating drainage characteristics
Determining density relationships

3. Engineering Implications:
These values provide insights into:

Material density optimization
Void space distribution
Particle arrangement effects on soil properties