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If you want to store energy using a solid, you don't compress it, you bend/twist it as in a spring. You can then get a large force times distance (=work = energy).
When one stores energy in a spring by bending one is in effect partly compressing one side and stretching the other,
Effectively, you have a lot of internal friction as the bulk material moves and this will result in energy loss as the solid 'flows'. For instance, rubber is often used in vehicle suspensions because of its damping (energy absorbing) properties.
Does that mean that your car's rubber dampers will heat up if you wizz round a lot of S-bends?