Can model this as a ball colliding into a stationary spring compressing it. The spring has a latch mechanism which engages and holds the spring in its state of max compression.
If the collision and spring and latch are non-dissipative, all the (kinetic) energy of the ball is converted into the potential energy locked in the spring.
If a fraction of the energy is dissipated into heat and sound, that much less will end up locked into the spring in its final configuration.
The spring latch may never unlock -analogous to the metal of the car folding into a fixed shape.
The mass of the car will increase according to m = e/c^2.
Kinetics of stored and dissipated energies associated with
cyclic loadings of dry polyamide 6.6 specimens studies where the energy goes for a particular polymer. From the abstract:
From a thermodynamic viewpoint, it was shown that the dissipated
energy per cycle was always less than the mechanical energy that could
be associated with the area of the hysteresis loop. This energy
difference reflects the significant contribution of the stored energy
associated, cycle by cycle, with the microstructural changes.
The area under the stress strain curve is the energy absorbed during deformation. They made infrared observations of the heat dissipated, and could see how much less this was than the total energy absorbed during deformation (and unloading as they tracked around an entire Hysteresis loop).
A heuristic model here: the canonical metastable potential, where the compression is initially non-dissipative / reversable / elastic as one roles the ball up the hill, then becomes plastic / non-reversable as the ball roles into the higher little stable valley. The difference in energy between the initial and final potentials of the ball correspond to increased internal energy in the crystal lattice who's structure has changed(compressed). Some of the balls energy may also go into heat.