The elastic energy per unit volume is terms of longitudinal strain `sigma` and Young's modulus `Y` is

Options

A. `Ysigma^(2)//2`
B. `Ysigma//2`
C. `2Ysigma^(2)//2`
D. `Y^(2)sigma//2`

Explanation

To find the elastic energy per unit volume in terms of longitudinal strain \( \sigma \) and Young's modulus \( Y \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Elastic Potential Energy**:
The elastic potential energy per unit volume (denoted as \( E \)) is given by the formula:
\[ E = \frac{1}{2} \times \text{Stress} \times \text{Strain} \]

2. **Define Stress in Terms of Young's Modulus**:
Stress (\( \sigma \)) is related to Young's modulus (\( Y \)) and strain (\( \epsilon \)) by the equation:
\[ \text{Stress} = Y \times \text{Strain} \]
Here, we can denote strain as \( \epsilon \).

3. **Substituting Stress into the Energy Formula**:
Substitute the expression for stress into the energy formula:
\[ E = \frac{1}{2} \times (Y \times \text{Strain}) \times \text{Strain} \]
This simplifies to:
\[ E = \frac{1}{2} \times Y \times \text{Strain}^2 \]

4. **Replace Strain with Longitudinal Strain \( \sigma \)**:
Since the question specifies that the longitudinal strain is \( \sigma \), we substitute \( \sigma \) for strain:
\[ E = \frac{1}{2} \times Y \times \sigma^2 \]

5. **Final Expression**:
Therefore, the elastic energy per unit volume in terms of longitudinal strain \( \sigma \) and Young's modulus \( Y \) is:
\[ E = \frac{Y \sigma^2}{2} \]

### Conclusion:
The final answer is:
\[ E = \frac{Y \sigma^2}{2} \]