# 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.