Fundamentals of Continuum Mechanics

9.6.3.1 The deformation-temperature-electric field-magnetic field formulation

In this formulation, F, Θ, e*, and h* are the independent variables. We define the energy potential E^FΘeh as the Legendre transformation of internal energy $\epsilon =\breve{\epsilon }\left(\mathbf{F},\eta ,{\mathbf{p}}^{*}/\rho ,{\mathbf{m}}^{*}/\rho \right)$ with respect to the thermal, electrical, and magnetic variables, from η to Θ, p*/ρ to e*, and m*/ρ to h*, i.e.,

$E^{F\text{Θ}\mathit{eh}}=\epsilon -\text{Θ}\eta -{\mathbf{e}}^{*}·\frac{{\mathbf{p}}^{*}}{\rho }-{\mu }_{\text{o}}{\mathbf{h}}^{*}·\frac{{\mathbf{m}}^{*}}{\rho }.$

  (9.71)

Refer to Table 9.4. Taking the rate of (9.71) gives

${\stackrel{.}{E}}^{F\text{Θ}\mathit{eh}}=\stackrel{.}{\epsilon }-\text{Θ}\stackrel{.}{\eta }-\eta \stackrel{.}{\text{Θ}}-{\mathbf{e}}^{*}·\stackrel{·}{\overline{\left(\frac{{\mathbf{p}}^{*}}{\rho }\right)}}-\frac{{\mathbf{p}}^{*}}{\rho }·{\stackrel{.}{\mathbf{e}}}^{*}-{\mu }_{\text{o}}{\mathbf{h}}^{*}·\stackrel{·}{\overline{\left(\frac{{\mathbf{m}}^{*}}{\rho }\right)}}-{\mu }_{\text{o}}\frac{{\mathbf{m}}^{*}}{\rho }·{\stackrel{.}{\mathbf{h}}}^{*},$

and use of this result in (9.61 ...