Thermodynamics: how does the deformation of partially elastic materials produce heat? When a perfectly elastic material is deformed, the energy associated with the strain remains stored in the body as elastic potential energy, called strain energy. Upon the removal of the deforming forces, the body completely regains its original shape, size and configuration with no further loss of heat. However, nothing is perfectly elastic! When a partially elastic material (e.g., a rubber band) is deformed, there is always a remnant deformation even upon the withdrawal of the deforming forces. It exhibits hysteresis and part of the energy converted to heat. Here is a quick reference. 
How does this remnant deformation responsible for the heating up of the material?
Edit According to one of the $TdS$ equations, $$dT=\frac{T}{C_V}\Big[dS-\Big(\frac{\partial P}{\partial T}\Big)_V dV\Big],$$ we note that a change in temperature is caused by either a change in volume $V$ or a change in entropy $S$ or both. Can we use this to understand what is going on? 
My guess is that for the elastic deformation of a perfectly elastic material, during the process of loading $dT$ is negative because $dV>0$ and during unloading $dT$ is positive because $dV<0$. In the whole cycle, when the system comes to its original state, it does not heat up. But for a partially elastic material, the residual permanent deformation may be responsible for a nonzero $dV$, a nonzero $dS$ or both?
 A: A partially elastic material (i.e., viscoelastic material) exhibits a combination of elastic and viscous behavior, and it is the viscous part that is responsible for the increase in internal energy (adiabatic case) or the emission of heat (isothermal case).  
Imagine a spring and a damper (dashpot) in series.  When the combination is stretched, both the spring and damper extend, but the extension of the damper is responsible for dissipation of mechanical energy to either internal energy of heat.  So, when the force extending the combination is released, the spring can return to its original length, but the new length of the damper remains locked in.  So the overall combination retains some residual extension, and some of the mechanical energy has been dissipated by the damper.
A: Simple: inelastic deformation is an irreversible process.
Rubber is perhaps a bad example, due to its hyperelasticity (rubber bands actually get colder when you un-stretch them!)
Instead, consider a rod of some metal. Pretend for a moment that it behaves perfectly elasticity. What would happen if you were to stretch it, and then release it? It would simply spring back and forth, as a simple harmonic oscillator indefinitely. This is clearly not what would happen in real life. The energy of stretching would be damped, and it the rod would simply go back to the unstretched state. The energy has to however go somewhere. It of course goes to heat. But how? Disregard for the moment dislocations, or the configurational changes of amorphous materials. The lattice of this material is subject to simple electric fields, and electric fields are conservative. Where then can the energy go? Why, magnetic fields. Moving charges generate magnetic fields, and magnetic fields are non-conservative! Thus, the energy can be transformed from bulk lattice vibrations to microscopic acoustic phonon vibrations, or what the thermodynamicists call temperature.
