It is important to note that the electromagnetic equations (9.46f)–(9.46j) are not independent. For instance, it can be shown that conservation of charge (9.46f) is implicitly contained in Maxwell's equations (9.46g)–(9.46j); refer to Problem 9.6.

9.2.2 Lagrangian fundamental laws

Recall from Section 9.1 the Lagrangian integral forms of the first principles:

Conservation of mass

${\displaystyle \underset{\mathcal{P}}{\int }\rho \text{d}v}={\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }{\rho }_{\text{R}}\text{d}V},$

Balance of linear momentum

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\mathbf{v}{\rho }_{\text{R}}\text{d}V}={\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\left({\mathbf{f}}^{\text{m}}+{\mathbf{f}}^{\text{em}}\right){\rho }_{\text{R}}\text{d}V}+{\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }{\mathbf{t}}_{\text{R}}\text{d}A},$

Balance of angular momentum

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\mathbf{x}\times \mathbf{v}{\rho }_{\text{R}}\text{d}V}={\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\mathbf{x}\times \left({\mathbf{f}}^{\text{m}}+{\mathbf{f}}^{\text{em}}\right){\rho }_{\text{R}}\text{d}V}+{\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }\mathbf{x}\times {\mathbf{t}}_{\text{R}}\text{d}A}+{\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }{\mathbf{c}}^{\text{em}}{\rho }_{\text{R}}\text{d}V},$

First law of thermodynamics

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\frac{1}{2}\mathbf{v}·\mathbf{v}{\rho }_{\text{R}}\text{d}V}+\frac{\text{d}}{\text{d}t}{\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\epsilon {\rho }_{\text{R}}\text{d}V}={\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\left({\mathbf{f}}^{\text{m}}+{\mathbf{f}}^{\text{em}}\right)·\mathbf{v}{\rho }_{\text{R}}\text{d}V}+{\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }{\mathbf{t}}_{\text{R}}·\mathbf{v}\text{d}A}+{\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\left(r^{\text{t}}+r^{\text{em}}\right){\rho }_{\text{R}}\text{d}V}-{\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }{h}_{\text{R}}\text{d}A},$

Second law of thermodynamics

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\eta {\rho }_{\text{R}}\text{d}V}\ge {\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\frac{r^t}{\text{Θ}}{\rho }_{\text{R}}\text{d}V}-{\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }\frac{h_{\text{R}}}{\text{Θ}}\text{d}A},$

Conservation of electric charge

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }{\sigma }_{\text{R}}\text{d}V}=-{\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }{\mathbf{j}}_{\text{R}}·\mathbf{N}\text{d}A},$

Gauss's law for magnetism

${\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }{\mathbf{b}}_{\text{R}}·\mathbf{N}\text{d}A}=0,$

Faraday's law

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{{\mathcal{Q}}_{\text{R}}}{\int }{\mathbf{b}}_{\text{R}}·\mathbf{N}\text{d}A}=-{\displaystyle \underset{\partial {\mathcal{Q}}_{\text{R}}}{\int }{\mathbf{e}}_{\text{R}}·{\mathbf{l}}_{\text{R}}\text{d}L},$

Gauss's law for electricity

${\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }{\mathbf{d}}_{\text{R}}·\mathbf{N}\text{d}A}=-{\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }{\sigma }_{\text{R}}\text{d}V},$

Ampère-Maxwell law

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{{\mathcal{Q}}_{\text{R}}}{\int }{\mathbf{d}}_{\text{R}}·\mathbf{N}\text{d}A}+{\displaystyle \underset{{\mathcal{Q}}_{\text{R}}}{\int }{\mathbf{j}}_{\text{R}}·\mathbf{N}\text{d}A}={\displaystyle \underset{\partial {\mathcal{Q}}_{\text{R}}}{\int }{\mathbf{h}}_{\text{R}}·{\mathbf{l}}_{\text{R}}\text{d}L}.$

An important feature of the Lagrangian integral conservation laws (9.48a)–(9.48j) is that the regions ${\mathcal{P}}_{\text{R}}$ and ${\mathcal{Q}}_{\text{R}}$ occupied by subsets ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ in the reference configuration are fixed. Hence, the regions of integration ${\mathcal{P}}_{\text{R}}$ and ${\mathcal{Q}}_{\text{R}}$ do not change with time, so time derivatives of Lagrangian surface and volume integrals can be passed directly inside the integrals. Conversely, with the Eulerian integral conservation laws (9.42a)–(9.42j), the regions of integration $\mathcal{P}$ and $\mathcal{Q}$ change with time. Thus, to take time derivatives of these time-varying integrals, the transport theorem for surface integrals (9.43) and the transport theorem for volume integrals (4.20) must be employed. The transport theorems are analogous to Leibniz's rule in multivariable calculus.

The continuous, bounded nature of the integrands in (9.48a)–(9.48j) enables the divergence theorem and Stokes's theorem, and the requirement that (9.48a)–(9.48j) be continuous and global, i.e., true for the entire body and all subsets, enables the localization theorem. With the traction t_R and heat flux h_R dependent on surface geometry only through the outward unit normal N, so t_R = PN and h_R = q_R · N, it can be shown that application of the divergence theorem, Stokes's theorem, and the localization theorem to the Lagrangian integral equations (9.48a)–(9.48j) leads to the Lagrangian pointwise equations

$\rho J={\rho }_{\text{R}},$

${\rho }_{\text{R}}\stackrel{.}{\mathbf{v}}={\rho }_{\text{R}}\left({\mathbf{f}}^{\mathbf{m}}+{\mathbf{f}}^{\text{em}}\right)+\text{Div}\mathbf{P},$

${\rho }_{\text{R}}{\text{Γ}}^{\text{em}}+\mathbf{P}{\mathbf{F}}^{\text{T}}-\mathbf{F}{\mathbf{P}}^{\mathbf{T}}=\mathbf{0},$

${\rho }_{\text{R}}\stackrel{.}{\epsilon }=\mathbf{P}·\text{Grad}\mathbf{v}+{\rho }_{\text{R}}\left(r^{\text{t}}+r^{\text{em}}\right)-\text{Div}{\mathbf{q}}_{\text{R}},$

${\rho }_{\text{R}}\stackrel{.}{\eta }\ge {\rho }_{\text{R}}\frac{r^{\text{t}}}{\text{Θ}}-\text{Div}\left(\frac{{\mathbf{q}}_{\text{R}}}{\text{Θ}}\right),$

${\stackrel{.}{\sigma }}_{\text{R}}+\text{Div}{\mathbf{j}}_{\text{R}}=0,$

$\text{Div}{\mathbf{b}}_{\text{R}}=0,$

$\text{Curl}{\mathbf{e}}_{\text{R}}=-{\stackrel{.}{\mathbf{b}}}_{\text{R}},$

$\text{Div}{\mathbf{d}}_{\text{R}}={\sigma }_{\text{R}},$

$\text{Curl}{\mathbf{h}}_{\text{R}}={\stackrel{.}{\mathbf{d}}}_{\text{R}}+{\mathbf{j}}_{\text{R}}.$

Note that it can be shown that conservation of charge (9.49f) is not independent of Maxwell's equations(9.49g)–(9.49j), but rather is implicitly contained in them. Recall that P is the first Piola-Kirchhoff stress, q_R is the referential heat flux vector, “Div” is the Lagrangian divergence (i.e., the divergence calculated with respect to the reference configuration), “Curl” is the Lagrangian curl, and

$\stackrel{.}{\mathbf{a}}=\frac{\partial }{\partial t}\widehat{\mathbf{a}}\left(\mathbf{X},t\right)$

is the material derivative of an arbitrary vector $\mathbf{a}=\widehat{\mathbf{a}}\left(\mathbf{X},t\right)$, i.e., the partial derivative of the referential description of a with respect to time t. Also,

${\text{Γ}}^{\text{em}}\mathbf{a}={\mathbf{c}}^{\text{em}}\times \mathbf{a}$

for any vector a, i.e., Γ^em is a skew tensor whose corresponding axial vector is c^em. Note that in deriving the pointwise version of the first law of thermodynamics (9.49d) from its integral counterpart (9.48d), we have made use of the Lagrangian form of the energy theorem for continuum electrodynamics:

${\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\left({\mathbf{f}}^{\text{m}}+{\mathbf{f}}^{\text{em}}\right)·\mathbf{v}{\rho }_{\text{R}}\text{d}V}+{\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }{\mathbf{t}}_{\text{R}}·\mathbf{v}\text{d}A}-\frac{\text{d}}{\text{d}t}{\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\frac{1}{2}\mathbf{v}·\mathbf{v}{\rho }_{\text{R}}\text{d}V}={\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\mathbf{P}·\text{Grad}\mathbf{v}\text{d}V}.$

Compare (9.50) with (4.54), the Lagrangian form of the energy theorem for mechanics.

9.3.1 Minkowski model

The Minkowski model [32, 33, 38] is motivated by Einstein's special theory of relativity. In this approximation, the effective fields are related to the standard fields through semirelativistic inverse Lorentz transformations:

$\begin{array}{l}\begin{array}{llll}e*=e_{\text{M}}+v\times b_{\text{M}},\hfill & h*=h_{\text{M}}-v\times d_{\text{M}},\hfill & d*=d_{\text{M}},\hfill & b*=b_{\text{M}}\hfill \end{array},\hfill \\ \begin{array}{llll}p*=p_{\text{M}},\hfill & m*=m_{\text{M}}+v\times p_{\text{M}},\hfill & j=j_{\text{M}}-{\sigma }_{\text{M}}v,\hfill & \sigma *={\sigma }_{\text{M}}\hfill \end{array},\hfill \end{array}$

where (·)_M represents a standard electromagnetic field corresponding to the Minkowski model.

9.3.2 Lorentz model

The Lorentz model is motivated by Lorentz's theory of electrons [39], which postulates that a body consists of an infinitely large number of rapidly moving charged particles. The motion of these charged particles, in turn, generates rapidly fluctuating microscopic electromagnetic fields. The microscopic fields averaged over a small time interval and an infinitesimal volume are defined as the corresponding macroscopic fields. The aforementioned postulates lead to the following relationships:

$\begin{array}{l}\begin{array}{llll}\text{e}*={\text{e}}_{\text{L}}+\text{v}\times {\text{b}}_{\text{L}},\hfill & \text{h}*={\text{h}}_{\text{L}}-{\in }_{\text{o}}\text{v}\times {\text{e}}_{\text{L}},\hfill & \text{d}*={\text{d}}_{\text{L}},\hfill & \text{b}*={\text{b}}_{\text{L}},\hfill \end{array}\hfill \\ \begin{array}{llll}\text{p}*={\text{p}}_{\text{L}},\hfill & \text{m}*={\text{m}}_{\text{L}},\hfill & \text{j}*={\text{j}}_{\text{L}}-{\sigma }_{\text{L}}\text{v},\hfill & \sigma *\hfill \end{array}={\sigma }_{\text{L}},\hfill \end{array}$

where (·)_L represents a standard electromagnetic field corresponding to the Lorentz model.

9.3.3 Statistical model

The statistical model [40] is a modification of Lorentz's theory, wherein the charged particles are grouped into stable structures (e.g., atoms, molecules, or ions). The field effects of the charged particles within each stable structure are represented by microscopic electric and magnetic multipole moments (e.g., dipole, quadrupole, and octupole moments), and the macroscopic polarization and magnetization fields are defined as statistical averages of these multipole moments over a large number of stable structures. The transformations presented in this model are identical to Minkowski's, i.e.,

$\begin{array}{l}\begin{array}{llll}\text{e}*={\text{e}}_{\text{S}}+\text{v}\times {\text{b}}_{\text{S}},\hfill & \text{h}*={\text{h}}_{\text{S}}-\text{v}\times {\text{d}}_{\text{S}},\hfill & \text{d}*={\text{d}}_{\text{S}},\hfill & \text{b}*={\text{b}}_{\text{S}},\hfill \end{array}\hfill \\ \begin{array}{llll}\text{p}*={\text{p}}_{\text{S}},\hfill & \text{m}*={\text{m}}_{\text{S}}+\text{v}\times {\text{p}}_{\text{S}},\hfill & \text{j}*={\text{j}}_{\text{S}}-{\sigma }_{\text{S}}\text{v},\hfill & \sigma *\hfill \end{array}={\sigma }_{\text{S}},\hfill \end{array}$

where ( )_s represents a standard electromagnetic field corresponding to the statistical model.

9.3.4 CHU model

The Chu model [41] is based on the postulate that deforming bodies contribute to electromagnetic phenomena by acting, in a macroscopic sense, as electric and magnetic dipole sources for the electromagnetic fields. The transformations for the Chu model are

$\begin{array}{l}\begin{array}{llll}\text{e}*={\text{e}}_{\text{C}}+{\mu }_{\text{O}}\times {\text{h}}_{\text{C}},\hfill & \text{h}*={\text{h}}_{\text{C}}-{\in }_{\text{o}}\text{v}\times {\text{e}}_{\text{C}},\hfill & \text{d}*={\text{d}}_{\text{C}},\hfill & \text{b}*={\text{b}}_{\text{C}},\hfill \end{array}\hfill \\ \begin{array}{llll}\text{p}*={\text{p}}_{\text{C}}+{\text{m}}_{\text{C}}\times \frac{\text{v}}{c^2},\hfill & \text{m}*={\text{m}}_{\text{C}},\hfill & \text{j}*={\text{j}}_{\text{C}}-{\sigma }_{\text{C}}\text{v},\hfill & \sigma *\hfill \end{array}={\sigma }_{\text{C}},\hfill \end{array}$

where ( )_c represents a standard electromagnetic field corresponding to the Chu model.

9.3.5 A comparison of the four models

Table 9.2 catalogs Maxwell's equations for each of the four models, deduced by substituting each set of transformations (i.e., (9.51)–(9.54)) into (9.46g)–(9.46j). Recall that since each of the four models is based on a different set of postulates, the standard electromagnetic fields generally have different physical connotations from model to model. We can deduce relationships between the standard fields by comparing the equations in Table 9.2. For instance, the Minkowski and statistical models, although developed from different perspectives, are mathematically equivalent (i.e., m_M = m_s, p_M = p_s, etc.).

9.4.1 An alternative approach

Electromagnetic effects in the thermomechanical balance laws can be modeled as an electromagnetically induced body force, body couple, and energy supply (as discussed previously; see also [32, 33, 44]) or, alternatively, incorporated into the constitutive response (see, for instance, [34–36]). The latter is accomplished by defining a symmetrictotal stress tensor

$\tau =\mathbf{T}+{\mathbf{T}}^{\text{em}}$

that consists of contributions from the cauchy stress tensor T and the electromagnetic Maxwell stress tensor T^em. The Maxwell stress T^em is defined so that twice its skew part is ρΓ^em and its divergence is ρf^em; refer to Problem 9.8. For the Minkowski formulation (9.55a)–(9.55c), it can be shown that

${\mathbf{T}}^{\text{em}}={\mathbf{e}}^{*}\otimes {\mathbf{d}}^{*}+{\mathbf{h}}^{*}\otimes {\mathbf{b}}^{*}-\frac{1}{2}\left({\epsilon }_{\text{o}}{\mathbf{e}}^{*}·{\mathbf{e}}^{*}+{\mu }_{\text{o}}{\mathbf{h}}^{*}·{\mathbf{h}}^{*}\right)\mathbf{I}.$

By formulating the first principles and the constitutive equations in terms of the total stress τ instead of the Cauchy stress T, we eliminate explicit coupling between the electromagnetic fields and the thermomechanical fields in the first principles (9.57a)–(9.57c). This coupling is instead accounted for implicitly through the constitutive equation for τ.

9.6.1 The reduced clausius-duhem inequality, work conjugates

We algebraically combine the Eulerian forms of the Clausius-Duhem inequality (9.46e) and the first law of thermodynamics (9.57c) to produce the reduced Clausius-Duhem inequality

$-\stackrel{.}{\epsilon }+\frac{1}{{\rho }_{\text{R}}}\mathbf{P}·\stackrel{.}{\mathbf{F}}+\text{Θ}\stackrel{.}{\eta }+{\mathbf{e}}^{*}·\stackrel{·}{\overline{\left(\frac{\mathbf{p}*}{\rho }\right)}}+{\mu }_{\text{o}}{\mathbf{h}}^{*}·\stackrel{·}{\overline{\left(\frac{\mathbf{m}*}{\rho }\right)}}+\frac{1}{\rho }{\mathbf{j}}^{*}·{\mathbf{e}}^{*}-\frac{1}{\rho \text{Θ}}\mathbf{q}·\text{grad}\text{Θ}\ge 0.$

Analogously to classical thermodynamics [61], this fundamental statement of the second law consists of contributions from conjugate pairs of thermal, electrical, magnetic, and mechanical quantities. Each of these conjugate pairs (or work conjugates) is the product of the rate of an extensive quantity in rate form ($\stackrel{.}{\mathbf{F}}$, $\stackrel{.}{\eta }$, $\left({\stackrel{.}{\mathbf{p}}}^{*}/\rho \right)$, and $\left({\stackrel{.}{\mathbf{m}}}^{*}/\rho \right)$) and an intensive quantity (P, Θ, e*, and h*).¹⁰ The utility of the second law, and our identification of the extensive and intensive quantities, will become apparent in the following section.

9.6.2 The all-extensive formulation

Recall that we specialized to reversible TEME material response, which implies that the deformation, although it may be large, is elastic and fully recoverable, and the material only undergoes nondissipative TEMM processes. It follows, then, that these TEMM processes can be completely described through the principles of classical equilibrium thermodynamics¹¹ [61].

Analogously to classical equilibrium thermodynamics, our fundamental energy potential is the specific internal energy ε, which employs extensive quantities as its independent variables [61]. Hence, for the fundamental formulation, the natural independent variables are the extensive quantities F, η, p*/ρ, and m*/ρ appearing as rates in (9.61), and the natural dependent variables are the conjugate intensive quantities P, Θ, e*, and h*. Thus, for thermodynamic consistency, and to respect the reversible elastic nature of the process, we adjust the division (9.60) so that the dependence of the response on the present position x, polarization p*, and magnetization m* is through the deformation gradient F = Grad x, p*/ρ, and m*/ρ, respectively. Additionally, we demand that the response depends on these independent variables only through their values at the present time t, not their histories, rates, or gradients, i.e.,

$\mathbf{P}=\breve{\mathbf{P}}\left(\mathbf{F},\eta ,\frac{{\mathbf{p}}^{*}}{\rho },\frac{{\mathbf{m}}^{*}}{\rho }\right),\text{Θ}=\breve{\text{Θ}}\left(\mathbf{F},\eta ,\frac{{\mathbf{p}}^{*}}{\rho },\frac{{\mathbf{m}}^{*}}{\rho }\right),{\mathbf{e}}^{*}=\breve{\mathbf{e}}*\left(\mathbf{F},\eta ,\frac{{\mathbf{p}}^{*}}{\rho },\frac{{\mathbf{m}}^{*}}{\rho }\right),{\mathbf{h}}^{*}=\breve{\mathbf{h}}*\left(\mathbf{F},\eta ,\frac{{\mathbf{p}}^{*}}{\rho },\frac{{\mathbf{m}}^{*}}{\rho }\right),\epsilon =\breve{\epsilon }\left(\mathbf{F},\eta ,\frac{{\mathbf{p}}^{*}}{\rho },\frac{{\mathbf{m}}^{*}}{\rho }\right).$

Notation (9.62) indicates that P, Θ, e*, h*, and ε are explicit functions of F, η, p*/p, and m*/ρ evaluated at present position x and time t, and implicit functions of x and t. Note that a superscript breve is used to distinguish a function from its value (or output). As with all materials in the TEMM theory, the response functions (9.62) for a TEME material must satisfy the second law of thermodynamics, invariance requirements, conservation of angular momentum, and material symmetry conditions.

In what follows, we demonstrate how restrictions imposed by the second law of thermodynamics (9.61) yield a set of constitutive equations that provide the dependent variables P, Θ, e*, and h* (the intensive quantities) as partial derivatives of the specific internal energy ε (the fundamental thermodynamic energy potential) with respect to the independent variables F, η, p*/ρ, and m*/ρ (the extensive quantities), respectively. Use of the chain rule on $\epsilon =\breve{\epsilon }\left(\mathbf{F},\eta ,{\mathbf{p}}^{*}/\rho ,{\mathbf{m}}^{*}/\rho \right)$ gives

$\stackrel{.}{\epsilon }=\frac{\partial \breve{\epsilon }}{\partial \mathbf{F}}·\stackrel{.}{\mathbf{F}}+\frac{\partial \breve{\epsilon }}{\partial \eta }\stackrel{.}{\eta }+\frac{\partial \breve{\epsilon }}{\partial \left(\frac{\mathbf{p}*}{\rho }\right)}·\stackrel{·}{\overline{\left(\frac{\mathbf{p}*}{\rho }\right)}}+\frac{\partial \breve{\epsilon }}{\partial \left(\frac{\mathbf{m}*}{\rho }\right)}·\stackrel{·}{\overline{\left(\frac{\mathbf{m}*}{\rho }\right)}},$

and substitution of this result into the second law (9.61) leads to

$\left(\frac{1}{{\rho }_{\text{R}}}\mathbf{P}-\frac{\partial \breve{\epsilon }}{\partial \mathbf{F}}\right)·\stackrel{.}{\mathbf{F}}+\left(\text{Θ}-\frac{\partial \breve{\epsilon }}{\partial \eta }\right)\stackrel{.}{\eta }+\left({\mathbf{e}}^{*}-\frac{\partial \breve{\epsilon }}{\partial \left(\frac{\mathbf{p}*}{\rho }\right)}\right)·\stackrel{·}{\overline{\left(\frac{\mathbf{p}*}{\rho }\right)}}+\left({\mu }_{\text{o}}{\mathbf{h}}^{*}-\frac{\partial \breve{\epsilon }}{\partial \left(\frac{\mathbf{m}*}{\rho }\right)}\right)·\stackrel{·}{\overline{\left(\frac{\mathbf{m}*}{\rho }\right)}}+\frac{1}{\rho }{\mathbf{j}}^{*}·{\mathbf{e}}^{*}-\frac{1}{\rho \text{Θ}}\mathbf{q}·\text{grad}\text{Θ}\ge 0.$

Inequality (9.63) must hold for all processes. Since the coefficients of the rates ($\stackrel{.}{\mathbf{F}}$, $\stackrel{.}{\eta }$, etc.) in (9.63) are independent of the rates themselves, and the rates may be varied independently and are arbitrary, it follows that the coefficients vanish, i.e.,

$\mathbf{P}={\rho }_{\text{R}}\frac{\partial \breve{\epsilon }}{\partial \mathbf{F}},\text{Θ}=\frac{\partial \breve{\epsilon }}{\partial \eta },{\mathbf{e}}^{*}=\frac{\partial \breve{\epsilon }}{\partial \left(\frac{\mathbf{p}*}{\rho }\right)},{\mathbf{h}}^{*}=\frac{1}{{\mu }_{\text{o}}}\frac{\partial \breve{\epsilon }}{\partial \left(\frac{\mathbf{m}*}{\rho }\right)}.$

What remains of inequality (9.63), i.e.,

${\mathbf{j}}^{*}·{\mathbf{e}}^{*}-\frac{1}{\text{Θ}}\mathbf{q}·\text{grad}\text{Θ}\ge 0,$

is called the residual dissipation inequality. The residual dissipation inequality quantifies irreversibilities in a thermodynamic process. In this case, the first term in (9.65) represents Joule heating due to current flow, and the second term represents heat conduction; both are transport processes. Accordingly, unlike the other dependent variables (see (9.64)), the conductive current j* and heat flux q are not derivable from a thermodynamic energy potential. Instead, their constitutive equations are specified empirically (e.g., Ohm's law for j* and Fourier's law for q), with the caveat that they must satisfy inequality (9.65).

We collectively coin the set of extensive independent variables {F, η, p*/ρ, m*/ρ}, the thermodynamic energy potential $\left\{\mathbf{F},\eta ,{\mathbf{p}}^{*}/\rho ,{\mathbf{m}}^{*}/\rho \right\}$, and the constitutive equations (9.64) the all-extensive formulation. The all-extensive formulation, along with the other formulations that we present in this section, correlate with a particular experiment, the independent variables being controlled and the dependent variables being the measured responses.