If a force F is applied along the length L of wire for stretching by an amount x, then Young’s modulus is given by
$Y=\frac{stress}{strain}=\frac{FL}{Ax}$
where, A = area of cross-sectional $\Rightarrow F=\frac{YA}{L}x$
The work done is stretching the wire is given by
$W=\underset{0}{\overset{x}{\int }}F.dx={\underset{0}{\overset{x}{\int }}\frac{YA}{L}x=\frac{YA}{L}\left[\frac{x^2}{2}\right]}_0=\frac{YA{x}^2}{2L}$