Lagrangian internal energy balance (${\rho }_0\dot{U}=P_{Ii}{\dot{F}}_{iI}+{\rho }_{0}r-q_{I,I}$)

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.