Question

Derive the expression for strain energy hence show that
strain energy per unit rolume is directly proportional
to square of stress.​

Answer 1

We know, Young's modulus is the ratio of stress to strain. e.g., Y = stress/strain.
if we assume a wire of length L, area of cross section A is fixed at one end and stretched by suspended a load M from the other end due to this extension of wire is x.

so, Young's modulus of wire , Y = FL/Ax
or, F = YAx/L
we know, workdone , W = F.dx
so, W = (YAx/L)dx
or, W = 1/2 × YAx²/L
W = 1/2 × {YAx/L} × x
W = 1/2 × F × x

hence, Work done = 1/2 × load × extension

now, potential energy = workdone by external applied force
= 1/2 × load × extension.
so, strain energy = 1/2 × load × extension
now, strain energy per unit volume = 1/2 × load × extension/volume
= 1/2 × load × extension/(cross section area × length)
= 1/2 × (load/cross section area) × (extension/length)
= 1/2 × stress × strain

hence, energy per unit volume = 1/2 × stress × strain

Hope this helps u
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