= Lagrangian internal energy balance
{title2=$\rho_0\dot U=P_{Ii}\dot F_{iI}+\rho_0r-q_{I,I}$}
For the material-first <nominal stress tensor>, the displayed local <conservation of energy> law balances mechanical work, heat supply and outgoing <heat flux>. Integrating over a fixed reference material volume and using the <divergence theorem> gives its integral form. The full total-energy balance includes <kinetic energy> and external mechanical work; <momentum conservation> reduces it to this internal-energy balance.
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