if S is the stress and Y is youngs modulus of the material of wire ,what is the energy stored in the wire per unit volume in terms of S and Y?PLEASE EXPLAIN IN DETAIL WITH DIAGRAM.

When a wire of original length L and area  of cross-section A is subjected to a deforming force F, the wire get elongated by $∆L$.
According to the Hook's law Y = stess/strain where stress =F/A and strain =​$∆L$/L. therfore 
Y =F/A /​(​$∆L$/L)
$F=YA\times \frac{∆L}{L}$
Now the work done 
dW=F.dL =​$F=YA\times \frac{∆L}{L}$dL now integrate the equation we get
W=$$\begin{gathered} \int dW=\int YA\frac{∆LdL}{L} \\ W=YA/L\int ∆LdLW=\frac{YA∆L^2}{2L}=\frac{1\times Y\times (∆L/L{)}^2\times AL}{2}=\frac{1}{2}\times Y\times (strain)2\times volume of wire=\frac{1\times stress\times strain\times volume of wire}{2} \end{gathered}$$$$\begin{gathered} \int dW=\int YA\frac{∆LdL}{L} \\ W=YA/L\int ∆LdLW=\frac{YA∆L^2}{2L}=\frac{1\times Y\times (∆L/L{)}^2\times AL}{2}=\frac{1}{2}\times Y\times (strain)2\times volume of wire=\frac{1\times stress\times strain\times volume of wire}{2} \end{gathered}$$
Elastic energy per unit volume =1/2 *stress * strain