Elastic energy

When a wire is stretched, work is done against the internal restoring force, and this work is stored in the wire as ELASTIC POTENTIAL ENERGY (also simply called elastic energy). Consider a wire of un-stretched length L and cross-sectional area A, stretched by a force that produces an extension l, with the elastic limit not exceeded and no energy lost. The small work done in stretching the wire by a further $dl$ is $dW=F\,dl$, so the total work done in stretching it from 0 to l is $W=\int_0^l F\,dl$. Using Young's modulus, $Y=\dfrac{F/A}{l/L}$, so $F=\dfrac{YAl}{L}$; substituting and integrating gives $W=\int_0^l \dfrac{YAl'}{L}\,dl'=\dfrac{YAl^2}{2L}=\dfrac{1}{2}\left(\dfrac{YAl}{L}\right)l=\dfrac{1}{2}Fl$. So the elastic potential energy stored in the stretched wire is $W=\tfrac12 Fl$. The ENERGY DENSITY (energy stored per unit volume) is found by dividing by the wire's volume $AL$: $u=\dfrac{W}{AL}=\dfrac{\tfrac12 Fl}{AL}=\dfrac12\left(\dfrac{F}{A}\right)\left(\dfrac{l}{L}\right)=\dfrac12\times\text{stress}\times\text{strain}$. Since stress $=Y\ …