# 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)
• $\dot{\text{v}} = \text{derivative of vector } v$
• $\dot{\text{M}} = \text{transpose of matrix } M$
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Add bounty to your question if you want to get answers faster. – Masi Oct 9 '11 at 18:00
It would improve your question significantly if you can be a bit more specific about what you're looking for than just saying "how are they related?" – David Z Oct 9 '11 at 18:52

## 1 Answer

You could add reference to the wikipedia article where you got this equation from (the Lagrangian description is simpler to understand -- I think; the terms have the same meaning, but are in the current configuration). So going one after another:

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.
3. divergence of heat, i.e. change of heat in the volume element
4. material energy change (think e.g. of chemical reaction going on at the point, which produces energy, not taken in account by other terms)

All those terms have to balance each other, i.e. sum to (scalar) zero.

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 $\rho_0(\dot e-\mathbf{S})$.

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