People often talk about different forms of energy like thermal energy, chemical energy, nuclear energy, etc. as belonging to the internal energy of a physical system. However, I can't really remember having encountered these in rigorous treatments of thermodynamics. I know that the amount of internal energy of a system is much less important than its changes and that we can neglect things like rest energy if they are irrelevant for the situation at hand. But it makes me really suspicous that the energy forms which I cited above don't seem to play a role at all in the equations.
So my question is: Can the internal energy $U$ of a physical system actually be partioned into well-defined components like thermal, chemical, nuclear, etc.? Maybe it can be done in some situations but not in all? Or maybe they lose their distinction as we become more rigorous?