Strain energy per unit volume is given by

Options

A. `(1)/(2)xx(("stress")^(2))/(y)`
B. `(1)/(2)xx("stress")^(2)y`
C. `(1)/(2)xx("strain")/("stress")`
D. `(1)/(2)F l`

Answer

To solve the question regarding the strain energy per unit volume, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding Strain Energy Density**:
- Strain energy per unit volume is also known as energy density. It represents the amount of energy stored in a material due to deformation.

2. **Formula for Strain Energy Density**:
- The general formula for strain energy density (U) in terms of stress (σ) and strain (ε) is given by:
\[ U = \frac{1}{2} \times \text{stress} \times \text{strain} \]

3. **Relating Strain to Stress and Young's Modulus**:
- According to Hooke's Law, stress is proportional to strain, and this relationship can be expressed as:
\[ \text{stress} = \text{Young's Modulus} \times \text{strain} \quad (\sigma = E \cdot \epsilon) \]
- Rearranging this gives us:
\[ \text{strain} = \frac{\text{stress}}{\text{Young's Modulus}} \quad (\epsilon = \frac{\sigma}{E}) \]

4. **Substituting Strain in the Energy Density Formula**:
- We can substitute the expression for strain into the strain energy density formula:
\[ U = \frac{1}{2} \times \text{stress} \times \left(\frac{\text{stress}}{\text{Young's Modulus}}\right) \]
- This simplifies to:
\[ U = \frac{1}{2} \times \frac{\text{stress}^2}{\text{Young's Modulus}} \]

5. **Final Expression**:
- Therefore, the strain energy per unit volume (energy density) can be expressed as:
\[ U = \frac{1}{2} \times \frac{\sigma^2}{E} \]
- This matches with option A from the provided options.

### Conclusion:
The strain energy per unit volume is given by:
\[ U = \frac{1}{2} \times \frac{\sigma^2}{E} \]