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

Elastic potential energy, the energy stored per unit volume within a deformed body, is defined by:

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

where:

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

The total elastic potential energy for the entire body is found by multiplying the energy per unit volume by the body's total volume \( V \). Therefore, the total elastic potential energy \( U_{\text{total}} \) is:

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

Key Points:

• The \( \frac{1}{2} \) factor arises because stress and strain have a linear relationship, and the energy is the area under the stress-strain curve (a triangle).
• The inclusion of volume calculates the total energy storage within the entire body.

In conclusion, the elastic potential energy of a strained body can be calculated as:

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