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I vaguely remember reading that the elastic potential energy of a spring, $\frac{1}{2} k x^2$ comes from mass which is turned into energy according to the law $E=mc^2$. I also remember hearing that the elastic energy is stored as potential energy in the bonds. Which of these, if any, are right?

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Both are correct. The elastic energy of solid is stored in the chemical bound. On the other hand, it also contributes, although by very very tinny amount, to the total mass of the system due to $E=mc^2$. So combining both facts, we arrived at the conclusion that the chemical bound has mass, even though the mass of the chemical bound is about only $10^{-9}$ of the mass of the atoms.

Although it is legitimate (and maybe cool) to consider the mass equivalence of the elastic energy (as well as the energy of the chemical bounds), but this way of thinking usually does not gain us any deeper understanding only to cause confusions. Most mass of the matter are stored in the nucleus as the energy of the strong interaction between quarks, which is of the order GeV. While the energy of the chemical bound originated from a weaker type of interaction, the electromagnetic interaction, which is typically of the order eV. The elastic energy is just the deformation energy of the chemical bounds which is even weaker. So these mechanical and chemical energies will never contribute prominently to the mass of the matter. Therefore talking about their mass equivalences does not make a point to me.

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