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I know two different equations that provides the joule heating effect. The first one only depends on the potential ($\varphi$) and the electrical conductivity ($\gamma$) and can de modeled based on the Ohm's law as follows:

$$ P = \gamma||\nabla \varphi ||^{2} $$

while the second one desribes the joule heating based on the plastic deformation and the vacancy flux:

$$ P = \rho Q $$

where $\rho$ is the mass density and $Q$ is the heat generated within the body, which is decomposed into two componenets: $$Q = Q_P + Q_V$$ Where $Q_P$ is the heat generated by plastic deformation at step $n+1$ described as $$Q_P = \sigma_{n+1}:\dot\epsilon_{n+_1}^{vp}$$

where $\dot\epsilon_{n+_1}^{vp}$ is the plastic strain rate. $Q_V$ is the heat generated due to vacancy flux, which can be expressed by $$Q_V = q:F_k$$ where $q$ and $F_k$ are the vacancy flux and the effective driving force as follows:

$$q =-D_v C_{v0}[\nabla c +\frac{c}{kT^2}Q^{*}\nabla T + \frac{cf\Omega}{kT}\nabla{\sigma^{sp}}]\\ F_k = -(f\Omega \nabla \sigma^{sp}+\frac{Q^*}{kT}\nabla T +\frac{kT}{c}\nabla c) $$

where $D_v, f, \Omega, k, \sigma^{sp}$ and $Q^*$ are the effective vacancy diffusivity, the vacancy relaxation ratio, ratio of atomic volume to the volume of a vacancy, the atomic volume, Boltzmann's constant, the absolute temperature, the spherical part of stress and the heat of transport respectively.

I am wondering which equation (or maybe both or neither) describes the joule heating effect more accurately?

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