Question

The elastic potential energy of a strained body is:

• A. \(\text{stress} \times \text{strain}\)
• B. \(\text{stress} \times \text{strain} \times \text{volume}\)
• C. \(\frac{1}{2} \text{stress} \times \text{strain}\)
• D. \(\frac{1}{2} \text{stress} \times \text{strain} \times \text{volume}\)

Hint:

The elastic potential energy is related to the stress and strain in the material, and it is proportional to 1/2 of the product of stress and strain for small deformations.

Answer 1

The Correct Option is D

The elastic potential energy stored in a strained body is the energy per unit volume stored due to deformation. It is given by the formula:

\[ U = \frac{1}{2} \sigma \epsilon, \]

where:

• \( U \) is the elastic potential energy per unit volume,
• \( \sigma \) is the stress,
• \( \epsilon \) is the strain.

For the total elastic potential energy stored in the body, the energy per unit volume is multiplied by the total volume of the body \( V \). Thus, the total elastic potential energy \( U_{\text{total}} \) is:

\[ U_{\text{total}} = \frac{1}{2} \sigma \epsilon V. \]

Key Observations:

• The factor \( \frac{1}{2} \) appears because the stress increases linearly with strain, and the energy is calculated as the area under the stress-strain curve (a triangle).
• The inclusion of volume accounts for the total energy stored in the entire body.

Hence, the elastic potential energy of a strained body is:

\[ \frac{1}{2} \, \text{stress} \times \text{strain} \times \text{volume}. \]