Towards a Fundamental Structure of Continuum Mechanics

For a class of systems obeying Euler's equation of motion the existence of a quantity to be named "proper mechanical energy" (PME) is shown which, together with internal energy, results in a quantity to be named "proper energy" (PE), which is conserved under conditions of time-dependent potentials. The appertaining formal structure for the continuum mechanics of such systems is the counterpart to Gibbs ' fundamental equation of thermodynamics and the relations deriving therefrom. Euler's equation of motion, in particular, corresponds to the Gibbs-Duhem equation of thermodynamics. The transport properties of P M E and PE are different from those of the corresponding conventional energies. The results point to a general structure of this kind for continuum mechanics.