I know that work is done on a rubber band to extend it, and then the rubber band does work to contract. However, then what is the net work done?
If it returns to its original length, is the area between loading and unloading curves the net work done / the energy dissipated as heat?
Also if for example, a material extends past its elastic limit and then returns with permanent deformation, what is the net work done? and what happens to that energy not recovered?
When you stretch a rubber band there is considerable deformation to the polymer molecules in the rubber. As a result some of the work you do on the rubber band goes into exciting molecular vibrations i.e. heat. Some of the work you do is stored as elastic energy and some is dissipated as heat.
As the band is allowed to relax the elastic energy stored within it does work on you. However, as before some of this energy goes into molecular vibrations and some into heat.
In both cases the work is the integral of force with distance, i.e. the area under a force-distance graph, however this graph will show some hysteresis due to the energy that is dissipated as heat:
Because of the hysteresis you don't get as much work out as you put in and the net work is not zero.
In the case of permanent deformation the physics is basically similar, except that the difference between the extension and relaxtion force-distance curves is greater so more work is lost as heat.
When you extend the rubber band, you store potential energy in it. Now when you release the rubber band this potential energy is converted into kinetic energy (the ends of the rubber band start moving). In an ideal situation the energy of the rubber band would then stay constant, (potential would convert to kinetic energy and then kinetic would convert to potential and so forth). But since there is friction and air resistance in real life the energy gets dissipated (non conservative force field) the rubber band will eventually come to a stop. And yes the energy released as heat would be equal to the energy you stored initially in it by pulling it apart.
As far as deformation is concerned, it is the potential energy that is stored in the deformed part. That is the only reason, why in inelastic collisions the kinetic energy of the system is not conserved. This is because the two bodies that are colliding stick to each other and get deformed at the point of contact.
