Strain Energy

When a wire is stretched by a gradually increasing force, work is done on it, and this work is stored in the wire as elastic potential energy, called strain energy — the same energy that is released again when the stretching force is removed and the wire (if it is behaving elastically) snaps back to its original length.

Consider a wire of original length $L$ and cross-sectional area $A$, stretched by a force $F$ acting along its length, producing a total elongation $l$. Because stress and strain increase together, proportionately, at every intermediate stage of the stretching, Young's modulus can be written as before,

so that, rearranged, the force needed for a given extension $x$ at any intermediate stage of the stretching is $f = \dfrac{YAx}{L}$.

To find the total work done, this expression is used for a small further extension $dx$ at the stage where the force is $f$: the incremental work is $dW = f\,dx = \dfrac{YAx}{L}\,dx$. Integrating this from $x = 0$ (unstretched) to $x = l$ (the final elongation) gives the total work done in stretching the wire:

This can be rewritten, using $Y = \dfrac{FL}{Al}$ from above, as

i.e. the compact result

Since this work done by the stretching force is exactly equal to the energy gained by the wire, it is also the strain energy stored in the wire:

It is often more useful to express this as strain energy per unit volume of the wire, so that the result no longer depends on the wire's particular dimensions. Dividing the work done by the wire's volume $A L$:

so that

$\text{Strain energy per unit volume} = \dfrac{1}{2}(\text{stress})(\text{strain})$ …