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. Consider a wire whose unstretch length is L and area of cross section is A. Let a force produce an extension l and further assume 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 = int_(0)^(l)F dl` ....(1)
From Young's modulus of elasticity,
`Y = (F)/(A) xx (L)/(l) dl = (YAl^(2))/(L.2) = (1)/(2).Fl`
`W = int_(0)^(l)(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")`