If $$S$$ is stress and $$Y$$ is Young's modulus of material of a wire, the energy stored in the wire per unit volume is

The elastic potential energy stored per unit volume (also known as energy density, $$u$$) in a stretched wire is given by the standard formula:

$$u = \frac{1}{2} \times \text{Stress} \times \text{Strain}$$

By definition, Young's modulus ($$Y$$) is the ratio of longitudinal stress to longitudinal strain:

$$Y = \frac{\text{Stress}}{\text{Strain}}$$

Rearranging this formula to solve for Strain gives:

$$\text{Strain} = \frac{\text{Stress}}{Y}$$

We are given that the stress is denoted by $$S$$. Substituting $$S$$ into the strain equation yields:

$$\text{Strain} = \frac{S}{Y}$$

Now, substitute the expressions for Stress ($$S$$) and Strain ($$\frac{S}{Y}$$) back into the initial energy density formula:

$$u = \frac{1}{2} \times S \times \left( \frac{S}{Y} \right)$$

Simplifying the expression:

$$u = \frac{S^2}{2Y}$$