Derive an expression for the elastic energy stored per unit volume of a wire.

Explanation

When a body is stretched, work is done against the restoring force (internal force). This work done is stored in the body in the form of elastic energy. Let us consider a wire whose un-stetch length is L and area of cross section is A. Let a force produce an extension l and it is assumed that the elastic limit of the wire has not been exceeded and there is no loss in energy. Then, the work done by the force F is equal to the energy gained by the wire. The work done in stretching the wire by dl,
dW = F dl
The total work done in stretching the wire from 0 to l is
`W=overset(l)underset(0)(int)Fdl" "...(1)`
From Young's modulus of elasticity,
`Y=(F)/(A)xx(L)/(l)rArrF=(YAl)/(L)" "...(2)`
Substituting equation (2) in equation (1), we get
`W=overset(l)underset(0)(int)(YAl)/(L)dl`
Since, l is the dummy variable in the integration, we can change l to l' (not in limits), therefore
`W=overset(l)underset(0)(int)(YAl')/(L)dl'=(YA)/(L)((l'^2)/(2))_(0)^(l)`
`=(YA)/(L)(l^2)/(2)=(1)/(2)((YAl)/(L))l`
`=(1)/(2)Fl`
`W=(1)/(2)Fl=` Elastic potential energy
Energy per unit volume is called energy density, `u=("Elastic potential energy")/("Volume")`
`=((1)/(2)Fl)/(AL)`
`(1)/(2)(F)/(A)(l)/(L)=(1)/(2)("Stress"xx"Strain")`