Define strain energy and derive the equation for the same. (or) Explain the concept of elastic potential energy in a stretched wire and hence obtain the expression for it?

Strain Energy :

The energy stored in a body due to its deformation is called strain energy. Expression for strain energy (or) work done in stretched wire:

Explanation : Let us consider a wire of length ‘L’ if force acts on the wire gradually increases,
‘e’ be the extension of wire. 
The average force on the wire $=\frac{\mathrm{O}+\mathrm{F}}{2}=\frac{\mathrm{F}}{2}$
Work done W=Average force x extension.

$\begin{array}{l}\mathrm{W}=\frac{\mathrm{F}}{2}\times \mathrm{e}\\ \mathrm{W}=\frac{1}{2}\left(\frac{\mathrm{F}}{\mathrm{A}}\right)\times \left(\frac{\mathrm{e}}{\mathrm{L}}\right)\times \mathrm{AL}\\ \mathrm{W}=\frac{1}{2}\times \text{ stress }\times \text{ strain }\times \text{ volume }\end{array}$

Strain energy in the wire per unit volume.
$\mathrm{W}=\frac{1}{2} \mathrm{stress} \times \mathrm{strain} \left[\therefore \mathrm{Y}=\frac{ \mathrm{stress} }{ \mathrm{strain} }\right]$

$\begin{array}{l}\mathrm{W}=\frac{1}{2}\times \mathrm{y}\text{ strain }\times \text{ strain }[\text{ stress }=\mathrm{y}\text{ strain] }\\ \mathrm{W}=\frac{1}{2}\text{ y }{\mathrm{strain}}^2\end{array}$