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# Elastic potential energy for Hookean materials

For a Hookean material, the elastic potential energy (in $\text{J}$) is given by: $$E_{\text{p}}=\frac{1}{2}kx^2$$

$k$ is the spring constant and $x$ is the magnitude of displacement of the elastic material in $\text{m}$. The potential energy is proportional to the square of the extension.

Recall that the restoring force of an elastic material is related to the displacement by $\vecphy{F}=-k\vecphy{x}$. The elastic potential energy therefore can be expressed in terms of force and the magnitude of displacement: $$E_{\text{p}}=\frac{1}{2}Fx$$

In general, the elastic potential energy is given by the area under the force-extension graph (force on $y$-axis, extension on $x$-axis). This reflects the definition of work as the force times the distance over which the force is applied.

The elastic potential energy is given by the area under the force-extension graph. For a Hookean material, this area is a right-angled triangle.