# Explain $\rho_{0}\dot{e} - \bf{P}^{T} : \bf{\dot{F}}+\nabla_{0} \cdot \bf{q} -\rho_{0}S = 0$

I am trying to understand the balance of energy -law from continuum mechanics, fourth law here. Could someone break this a bit to help me understand it? From chemistry, I can recall $$dU = \partial Q + \partial W$$ where $U$ is the internal energy, $Q$ is heat and $W$ is the work. How is the fourth law of conservation in CM:

$$\rho_{0}\dot{e} - \bf{P}^{T} : \bf{\dot{F}}+\nabla_{0} \cdot \bf{q} -\rho_{0}S = 0$$

related to that?

Terms

• $e(\bar{x}, t) = \text{internal energy per mass}$
• $q(\bar{x}, t) = \text{heat flux vector}$
• $\rho(\bar{x}, t) = \text{mass density}$

Operations

• $: \text{ -operation} = \text{Frobenius inner product?}$ (related)
• $\dot{\text{v}} = \text{derivative of vector } v$
• $\dot{\text{M}} = \text{transpose of matrix } M$

1. change of internal energy $e$, per unit mass (so multiplied by density)
2. change of elastic energy (elastic potential, stored elastic energy, deformation energy); The $:$ means tensor contraction, $\mathbf{P}$ is Piola-Kirchhof stress, $\mathbf{\dot F}$ is rate of deformation gradient.
Going back to your chemistry equation, the second term is (mechanical) work, third is heat flux, and the first and last terms are internal energy (which is broken down in two sources, though can be written as $\rho_0(\dot e-\mathbf{S})$.