Many elastic materials obey Hooke’s Law only to a certain extent. Empirically, we have found the force-extension graphs of materials such as metal wires and rubber bands to look something like the one below.
There are a few important landmarks on the graph:
Proportional limit: The material obeys Hooke’s Law only up till this point.
Elastic limit: Beyond this point, the material suffers permanent deformation.
Fracture point: At this point, the material breaks.
Below is a sketch of what may happen if the material is stretched beyond the elastic limit, and then allowed to un-stretch (the stretching force is gradually removed).
Notice that because the material has been stretched beyond its elastic limit, it does not un-stretch along the original path (in blue), but along the green line instead. In fact, it does not even return to its original length! It has been deformed permanently.
The sum of the red and the yellow areas represents the work done by the external force to stretch the material. The yellow area represents the fraction of the original work done that is recovered when the material un-stretches. The red area represents the energy that has been “lost” as heat.