9.1.1 Notation and nomenclature

We now present the notation necessary to describe the geometry of the deformable TEMM body. We label the body $\mathbb{B}$ and two arbitrary subsets ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$.² Subset ${\mathcal{S}}_1$ is bounded by a closed material surface, while subset ${\mathcal{S}}_2$ is bounded by a closed material curve; see Figure 9.1. As will soon be evident, both closed material surfaces and closed material curves are required to formulate integral statements of the fundamental laws of continuum electrodynamics.

In the reference configuration, body $\mathbb{B}$ occupies open volume ${\mathcal{R}}_{\text{R}}$ of e $\partial {\mathcal{R}}_{\text{R}}$. Subset ${\mathcal{S}}_1$ occupies open volume ${\mathcal{P}}_{\text{R}}\subset {\mathcal{R}}_{\text{R}}$, bounded by closed surface $\partial {\mathcal{P}}_{\text{R}}$, and subset ${\mathcal{S}}_2$ occupies open surface ${\mathcal{Q}}_{\text{R}}\subset {\mathcal{R}}_{\text{R}}$, bounded by closed curve $\partial {\mathcal{Q}}_{\text{R}}$.

In the present configuration at time t, body $\mathbb{B}$ occupies open material volume $\mathcal{R}$, bounded by closed material surface $\partial \mathcal{R}$. Subset ${\mathcal{S}}_1$ occupies open material volume $\mathcal{P}\subset \mathcal{R}$, bounded by closed material surface $\partial \mathcal{P}$, and subset ${\mathcal{S}}_2$ occupies open material surface $\mathcal{Q}\subset \mathcal{R}$, bounded by closed material curve $\partial \mathcal{Q}$.³

9.1.2 Conservation of mass

Primitively, conservation of mass postulates that the mass $\mathcal{M}$ of every subset of the body $\mathbb{B}$ is constant throughout its motion, or, equivalently, the time rate of change of the mass of every subset is zero. Applying this primitive statement to arbitrary subset ${\mathcal{S}}_1$ allows us to express conservation of mass mathematically in material form:

$\frac{\text{d}}{\text{d}t}\mathcal{M}\left({\mathcal{S}}_1,t\right)=0\text{or}\mathcal{M}\left({\mathcal{S}}_1\right)\equiv \text{independent of}t.$

Specializing to a continuum, the approach taken heretofore in this book, allows us to express the mass $\mathcal{M}$ of subset ${\mathcal{S}}_1$ in Eulerian and Lagrangian integral forms, i.e.,

$\mathcal{M}\left({\mathcal{S}}_1\right)={\displaystyle \underset{{\mathcal{S}}_1}{\int }\text{d}m}=\{\begin{array}{l}{\displaystyle \underset{\mathcal{P}}{\int }\rho \text{d}v,}\hfill \\ {\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }{\rho }_{\text{R}}\text{d}V.}\hfill \end{array}$

Thus, specializing to a continuum is tantamount to a smoothness assumption on the mass $\mathcal{M}$; refer to Section 4.2. The Eulerian integral representation (top of (9.2)) corresponds to subset ${\mathcal{S}}_1$ as seen in its present configuration, while the Lagrangian integral representation (bottom of (9.2)) corresponds to ${\mathcal{S}}_1$ as seen in its reference configuration.⁴ In (9.2), dv and dV are volume elements in the present and reference configurations (refer to Section 3.6), and ρ and ρ_R are the mass densities in the present and reference configurations (refer to Section 4.1). Note that ρ has units of mass per present volume, while ρ_R has units of mass per reference volume (see Table 9.1). Both ρ and ρ_R are bounded, continuous functions of space and time.

Table 9.1

Units for Thermal, Electrical, Magnetic, and Mechanical Quantities

Type	Quantity	Representation	Symbol	Fundamental Units	Derived Units	SI Units
Electrical	Vacuum permittivity		εo	$\frac{{\text{C}}^2{\text{T}}^2}{{\text{ML}}_{\text{P}}^3}$	$\frac{\text{Capacitance}}{\text{Present length}}$	$\frac{\text{Farad}}{\text{Meter}}$
	Electric field	Referential	eR	$\frac{{\text{ML}}_{\text{P}}^2}{{\text{L}}_{\text{R}}{\text{T}}^2\text{C}}$	$\frac{\text{Force}}{\text{Charge}}·\frac{\text{Present length}}{\text{Reference length}}$	$\frac{\text{Newton}}{\text{Coulomb}}\equiv \frac{\text{Volt}}{\text{Meter}}$
		Spatial	e*	$\frac{{\text{ML}}_{\text{P}}}{{\text{CT}}^2}$	$\frac{\text{Force}}{\text{Charge}}$	$\frac{\text{Newton}}{\text{Coulomb}}\equiv \frac{\text{Volt}}{\text{Meter}}$
	Electric polarization	Referential	pR	$\frac{\text{C}}{{\text{L}}_{\text{R}}^2}$	$\frac{\text{Charge}}{\text{Reference area}}$	$\frac{\text{Coulomb}}{{\text{Meter}}^2}$
		Spatial	p*	$\frac{\text{C}}{{\text{L}}_{\text{P}}^2}$	$\frac{\text{Charge}}{\text{Present area}}$	$\frac{\text{Coulomb}}{{\text{Meter}}^2}$
	Electric displacement	Referential	dR	$\frac{\text{C}}{{\text{L}}_{\text{R}}^2}$	$\frac{\text{Charge}}{\text{Reference area}}$	$\frac{\text{Coulomb}}{{\text{Meter}}^2}$
		Spatial	d*	$\frac{\text{C}}{{\text{L}}_{\text{P}}^2}$	$\frac{\text{Charge}}{\text{Present area}}$	$\frac{\text{Coulomb}}{{\text{Meter}}^2}$
	Free charge density	Referential	σR	$\frac{\text{C}}{{\text{L}}_{\text{R}}^3}$	$\frac{\text{Charge}}{\text{Reference volume}}$	$\frac{\text{Coulomb}}{{\text{Meter}}^3}$
		Spatial	σ*	$\frac{\text{C}}{{\text{L}}_{\text{P}}^3}$	$\frac{\text{Charge}}{\text{Present volume}}$	$\frac{\text{Coulomb}}{{\text{Meter}}^3}$
	Conductive current density	Referential	jR	$\frac{\text{C}}{{\text{L}}_{\text{R}}^2\text{T}}$	$\frac{\text{Current}}{\text{Reference area}}$	$\frac{\text{Ampere}}{{\text{Meter}}^2}$
		Spatial	j*	$\frac{\text{C}}{{\text{L}}_{\text{P}}^2\text{T}}$	$\frac{\text{Current}}{\text{Present area}}$	$\frac{\text{Ampere}}{{\text{Meter}}^2}$
Magnetic	Vacuum permeability		μo	$\frac{{\text{ML}}_{\text{P}}}{{\text{C}}^2}$	$\frac{\text{Inductance}}{\text{Present length}}$	$\frac{\text{Henry}}{\text{Meter}}$
	Magnetic field	Referential	hR	$\frac{\text{C}}{{\text{L}}_{\text{R}}\text{T}}$	$\frac{\text{Current}}{\text{Reference length}}$	$\frac{\text{Ampere}}{\text{Meter}}$
		Spatial	h*	$\frac{\text{C}}{{\text{L}}_{\text{P}}\text{T}}$	$\frac{\text{Current}}{\text{Present length}}$	$\frac{\text{Ampere}}{\text{Meter}}$
	Magnetization	Referential	mR	$\frac{\text{C}}{{\text{L}}_{\text{R}}\text{T}}$	$\frac{\text{Current}}{\text{Reference length}}$	$\frac{\text{Ampere}}{\text{Meter}}$
		Spatial	m*	$\frac{\text{C}}{{\text{L}}_{\text{P}}\text{T}}$	$\frac{\text{Current}}{\text{Present length}}$	$\frac{\text{Ampere}}{\text{Meter}}$
	Magnetic flux density	Referential	bR	$\frac{{\text{ML}}_{\text{P}}^2}{{\text{L}}_{\text{R}}^2\text{TC}}$	$\frac{\text{Magnetic flux}}{\text{Reference area}}$	$\frac{\text{Weber}}{{\text{Meter}}^2}\equiv \text{Tesla}$
		Spatial	b*	$\frac{\text{M}}{\text{TC}}$	$\frac{\text{Magnetic flux}}{\text{Present area}}$	$\frac{\text{Weber}}{{\text{Meter}}^2}\equiv \text{Tesla}$
Mechanical	Mass density	Referential	ρR	$\frac{\text{M}}{{\text{L}}_{\text{R}}^3}$	$\frac{\text{Mass}}{\text{Reference volume}}$	$\frac{\text{Kilogram}}{{\text{Meter}}^3}$
		Spatial	ρ	$\frac{\text{M}}{{\text{L}}_{\text{P}}^3}$	$\frac{\text{Mass}}{\text{Present volume}}$	$\frac{\text{Kilogram}}{{\text{Meter}}^3}$
	Traction	Referential	tR	$\frac{{\text{ML}}_{\text{P}}}{{\text{L}}_{\text{R}}^2{\text{T}}^2}$	$\frac{\text{Force}}{\text{Reference area}}$	$\frac{\text{Newton}}{{\text{Meter}}^2}\equiv \text{Pascal}$
		Spatial	t	$\frac{\text{M}}{{\text{L}}_{\text{P}}{\text{T}}^2}$	$\frac{\text{Force}}{\text{Present area}}$	$\frac{\text{Newton}}{{\text{Meter}}^2}\equiv \text{Pascal}$
	Stress	Referential	P	$\frac{{\text{ML}}_{\text{P}}}{{\text{L}}_{\text{R}}^2{\text{T}}^2}$	$\frac{\text{Force}}{\text{Reference area}}$	$\frac{\text{Newton}}{{\text{Meter}}^2}\equiv \text{Pascal}$
		Spatial	T	$\frac{\text{M}}{{\text{L}}_{\text{P}}{\text{T}}^2}$	$\frac{\text{Force}}{\text{Present area}}$	$\frac{\text{Newton}}{{\text{Meter}}^2}\equiv \text{Pascal}$
	Velocity		v	$\frac{{\text{L}}_{\text{P}}}{\text{T}}$		$\frac{\text{Meter}}{\text{Second}}$
	Deformation gradient		F	$\frac{{\text{L}}_{\text{P}}}{{\text{L}}_{\text{R}}}$
	Green's deformation		C	$\frac{{\text{L}}_{\text{P}}^2}{{\text{L}}_{\text{R}}^2}$
	Volumetric deformation		J	$\frac{{\text{L}}_{\text{P}}^3}{{\text{L}}_{\text{R}}^3}$
Thermal	Heat flux rate	Referential	hR	$\frac{{\text{ML}}_{\text{P}}^2}{{\text{L}}_{\text{R}}^2{\text{T}}^3}$	$\frac{\text{Energy}}{\text{Reference area}·\text{time}}$	$\frac{\text{Watt}}{{\text{Meter}}^2}$
		Spatial	h	$\frac{\text{M}}{{\text{T}}^3}$	$\frac{\text{Energy}}{\text{Present area}·\text{time}}$	$\frac{\text{Watt}}{{\text{Meter}}^2}$
	Heat flux vector	Referential	qR	$\frac{{\text{ML}}_{\text{P}}^2}{{\text{L}}_{\text{R}}^2{\text{T}}^3}$	$\frac{\text{Energy}}{\text{Reference area}·\text{time}}$	$\frac{\text{Watt}}{{\text{Meter}}^2}$
		Spatial	q	$\frac{\text{M}}{{\text{T}}^3}$	$\frac{\text{Energy}}{\text{Present area}·\text{time}}$	$\frac{\text{Watt}}{{\text{Meter}}^2}$
	Temperature		Θ	θ		Kelvin
	Specific internal energy		ε	$\frac{{\text{L}}_{\text{P}}^2}{{\text{T}}^2}$	$\frac{\text{Energy}}{\text{Mass}}$	$\frac{\text{Joule}}{\text{Kilogram}}$
	Specific entropy		η	$\frac{{\text{L}}_{\text{P}}^2}{{\text{T}}^2\theta }$	$\frac{\text{Energy}}{\text{Mass}·\text{temperature}}$	$\frac{\text{Joule}}{\text{Kilogram}·\text{kelvin}}$

Note that M is mass, L_Pis present length, L_Ris reference length, T is time, θ is temperature, and C is charge.

To perform the integrations in (9.2), it is natural to consider ρ in its Eulerian description, i.e., as a function of x and t, and to consider ρ_R in its Lagrangian description, i.e., as a function of X and t, although both ρ and ρ_R can be expressed using either an Eulerian or a Lagrangian description. Recall that X and x are the reference and present positions of a continuum particle, related through the motion x = χ(X, t) (refer to Section 3.1).

Use of (9.2) in (9.1) leads to Eulerian and Lagrangian integral representations of conservation of mass:

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{P}}{\int }\rho \text{d}v}=0,$

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

These integral statements are valid for any open volume $\mathcal{P}$ in the present configuration and corresponding open volume ${\mathcal{P}}_{\text{R}}$ in the reference configuration. Note that (9.3a) and (9.3b) are identical to their counterparts in the thermomechanical theory (refer to Chapter 4).

9.1.3 Balance of linear momentum

Balance of linear momentum postulates that the time rate of change of the linear momentum $\mathcal{L}$ of any subset of the body is equal to the resultant external force f acting on that subset. Applying this primitive statement to subset ${\mathcal{S}}_1$ gives the material form:

$\frac{\text{d}}{\text{d}t}\mathcal{L}\left({\mathcal{S}}_1,t\right)=\mathbf{f}\left({\mathcal{S}}_1,t\right).$

Assuming smoothness of $\mathcal{L}$, we write its Eulerian and Lagrangian integral representations:

$\mathcal{L}\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{P}}{\int }\mathbf{v}\rho \text{d}v,}\hfill \\ {\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\mathbf{v}{\rho }_{\text{R}}\text{d}V,}\hfill \end{array}$

where v is the velocity of a continuum particle at the present time t, and the integrands are continuous, bounded functions of space and time. As was done in Section 4.2, it is assumed that the resultant external force f can be additively decomposed into a body force and a contact force. The effects of electromagnetism are modeled through an electromagnetic contribution to the body force [32, 33] so that

$\mathbf{f}\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{P}}{\int }\left({\mathbf{f}}^{\text{m}}+{\mathbf{f}}^{\text{em}}\right)\rho \text{d}v}+{\displaystyle \underset{\partial \mathcal{P}}{\int }\mathbf{t}\text{d}a},\hfill \\ {\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\left({\mathbf{f}}^{\text{m}}+{\mathbf{f}}^{\text{em}}\right){\rho }_{\text{R}}\text{d}V}+{\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }{\mathbf{t}}_{\text{R}}\text{d}A},\hfill \end{array}$

where f^m is the mechanically induced body force per unit mass, f^em is the electro-magnetically induced body force per unit mass, t and t_R are the spatial and referential tractions (refer to Section 4.9), and da and dA are area elements in the present and reference configurations (refer to Section 3.6).⁵ Elaborating, t is the traction acting on surface $\partial \mathcal{P}$ in the present configuration measured per unit area of $\partial \mathcal{P}$, whereas t_R is the traction acting on surface $\partial \mathcal{P}$ in the present configuration but measured per unit area of the corresponding surface $\partial {\mathcal{P}}_{\text{R}}$ in the reference configuration. Thus, t has units of force per present area, while t_R has units of force per reference area (see Table 9.1).

Use of (9.5) and (9.6) in (9.4) gives Eulerian and Lagrangian integral representations of balance of linear momentum:

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

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

Comparing (9.7a) and (9.7b) with their counterparts in the thermomechanical theory (refer to Chapter 4), we see that (9.7a) and (9.7b) contain an additional body force f^em.

9.1.4 Balance of angular momentum

Balance of angular momentum postulates that the time rate of change of the angular momentum H₀ of any subset of the body about the origin 0 is equal to the resultant external moment M₀ acting on that subset about the origin 0. In material form:

$\frac{\text{d}}{\text{d}t}{\mathbf{H}}_{\mathbf{0}}\left({\mathcal{S}}_1,t,\mathbf{0}\right)={\mathbf{M}}_{\mathbf{0}}\left({\mathcal{S}}_1,t,\mathbf{0}\right).$

Assuming that H₀ is smooth leads to Eulerian and Lagrangian integral representations of the angular momentum about 0, i.e.,

${\mathbf{H}}_0\left({\mathcal{S}}_1,t,\mathbf{0}\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{P}}{\int }\mathbf{x}\times \mathbf{v}\rho \text{d}v},\hfill \\ {\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\mathbf{x}\times \mathbf{v}{\rho }_{\text{R}}\text{d}V}.\hfill \end{array}$

The integrands in (9.9) are continuous, bounded functions of space and time. Similarly, smoothness of M₀ implies that

${\mathbf{M}}_0\left({\mathcal{S}}_1,t,\mathbf{0}\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{P}}{\int }\mathbf{x}\times \left({\mathbf{f}}^{\text{m}}+{\mathbf{f}}^{\text{em}}\right)\rho \text{d}v}+{\displaystyle \underset{\partial \mathcal{P}}{\int }\mathbf{x}\times \mathbf{t}\text{d}a}+{\displaystyle \underset{\mathcal{P}}{\int }{\mathbf{c}}^{\text{em}}\rho \text{d}v},\hfill \\ {\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\mathbf{x}\times \left({\mathbf{f}}^{\text{m}}+{\mathbf{f}}^{\text{em}}\right){\rho }_{\text{R}}\text{d}V}+{\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }\mathbf{x}\times {\mathbf{t}}_{\text{R}}\text{d}A}+{\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }{\mathbf{c}}^{\text{em}}{\rho }_{\text{R}}\text{d}V}.\hfill \end{array}$

Following [32, 33], an electromagnetically induced body couple per unit mass c^em is included in (9.10) to model the effects of electromagnetism. Note that the first two terms in (9.10)₁ and (9.10)₂ represent the moment about 0 due to the resultant external force f, the first term being a contribution from the body force and the second term being a contribution from the contact force.

Use of (9.9) and (9.10) in (9.8) gives Eulerian and Lagrangian integral representations of balance of angular momentum:

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

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

Comparing (9.11a) and (9.11b) with their counterparts in the thermomechanical theory (refer to Chapter 4), we see that (9.11a) and (9.11b) contain additional moments due to (1) the electromagnetic body force f^em and (2) the electromagnetic body couple c^em.

9.1.5 First law of thermodynamics

The first law of thermodynamics (or conservation of energy) postulates that the time rate of change of the total energy (i.e., kinetic energy K plus internal energy E) of any subset of the body is equal to the rate of work R generated by the resultant external force acting on that subset plus the rate of all other energies $\mathcal{A}$ (e.g., heat, electromagnetic, chemical) entering or exiting that subset. In material form,

$\frac{\text{d}}{\text{d}t}\left(K\left({\mathcal{S}}_1,t\right)+E\left({\mathcal{S}}_1,t\right)\right)=R\left({\mathcal{S}}_1,t\right)+\mathcal{A}\left({\mathcal{S}}_1,t\right).$

Assuming that the kinetic and internal energies of part ${\mathcal{S}}_1$ are smooth implies that

$K\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{P}}{\int }\frac{1}{2}\mathbf{v}·\mathbf{v}\rho \text{d}v},\hfill \\ {\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\frac{1}{2}\mathbf{v}·\mathbf{v}{\rho }_{\text{R}}\text{d}V},\hfill \end{array}E\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{P}}{\int }\epsilon \rho \text{d}v},\hfill \\ {\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\epsilon {\rho }_{\text{R}}\text{d}V},\hfill \end{array}$

where ε is the specific internal energy (or internal energy per unit mass), and the integrands are bounded, continuous functions of space and time. The rate of work R generated by the resultant external force f can be additively decomposed into contributions from the body force and the contact force, i.e.,

$R\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{P}}{\int }\left({\mathbf{f}}^{\text{m}}+{\mathbf{f}}^{\text{em}}\right)·\mathbf{v}\rho \text{d}v}+{\displaystyle \underset{\partial \mathcal{P}}{\int }\mathbf{t}·\mathbf{v}\text{d}a},\hfill \\ {\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\left({\mathbf{f}}^{\text{m}}+{\mathbf{f}}^{\text{em}}\right)·\mathbf{v}{\rho }_{\text{R}}\text{d}V}+{\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }{\mathbf{t}}_{\text{R}}·\mathbf{v}\text{d}A}.\hfill \end{array}$

Following [32, 33], the auxiliary energy rate $\mathcal{A}$ is additively decomposed into three contributions, two from radiation and one from conduction: the rate of heat absorption throughout the volume, the rate of electromagnetic energy absorption throughout the volume, and the rate of heat entering through the boundary, i.e.,

$\mathcal{A}\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{P}}{\int }{r}^{\text{t}}\rho \text{d}v}+{\displaystyle \underset{\mathcal{P}}{\int }{r}^{\text{em}}\rho \text{d}v}-{\displaystyle \underset{\partial \mathcal{P}}{\int }h\text{d}a},\hfill \\ {\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }{r}^{\text{t}}{\rho }_{\text{R}}\text{d}V}+{\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }{r}^{\text{em}}{\rho }_{\text{R}}\text{d}V}-{\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }{h}_{\text{R}}\text{d}A},\hfill \end{array}$

where r^t is the specific heat supply rate, r^em is the specific electromagnetic energy supply rate, and h_R and h are the referential and spatial heat flux rates (refer to Section 4.9.3).⁶ Elaborating, h is the rate of heat flow out of the present boundary $\partial \mathcal{P}$ measured per unit area of the present boundary $\partial \mathcal{P}$. Conversely, h_R is the rate of heat flow out of the present boundary $\partial \mathcal{P}$, but measured per unit area of the corresponding boundary $\partial {\mathcal{P}}_{\text{R}}$ in the reference configuration. Thus, h has units of energy per time per present area, while h_R has units of energy per time per reference area (see Table 9.1).

Use of (9.13)–(9.15) in (9.12) yields Eulerian and Lagrangian integral representations of the first law of thermodynamics:

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

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

Comparing (9.16a) and (9.16b) with their counterparts in the thermomechanical theory (refer to Chapter 4), we see that (9.16a) and (9.16b) contain additional energy contributions from (1) the work due to the electromagnetic body force f^em and (2) the electromagnetic energy supply r^em.

9.1.6 Second law of thermodynamics

In this chapter—as was done in the thermomechanical theory of Chapter 4—we adopt the Clausius-Duhem inequality as our particular statement of the second law of thermodynamics. The Clausius-Duhem inequality postulates that the rate of change of the entropy $\mathcal{N}$ of any subset of the body is greater than or equal to the rate of entropy generation $\mathcal{R}$ due to the radiative heat supply minus the rate of entropy loss $\mathcal{H}$ due to the outward heat flux. Applying this primitive statement of the second law to subset ${\mathcal{S}}_1$ yields the material form:

$\frac{\text{d}}{\text{d}t}\mathcal{N}\left({\mathcal{S}}_1,t\right)\ge \mathcal{R}\left({\mathcal{S}}_1,t\right)-\mathcal{H}\left({\mathcal{S}}_1,t\right).$

Specializing to a continuum and assuming smoothness of $\mathcal{N}\left({\mathcal{S}}_1,t\right)$, $\mathcal{R}\left({\mathcal{S}}_1,t\right)$, and $\mathcal{H}\left({\mathcal{S}}_1,t\right)$, we can write

$\mathcal{N}\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{P}}{\int }\eta \rho \text{d}v},\hfill \\ {\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\eta {\rho }_{\text{R}}\text{d}V},\hfill \end{array}\mathcal{R}\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{P}}{\int }\frac{r^{\text{t}}}{\text{Θ}}\rho \text{d}v},\hfill \\ {\displaystyle \underset{{\mathcal{P}}_R}{\int }\frac{r^{\text{t}}}{\text{Θ}}{\rho }_{\text{R}}\text{d}V},\hfill \end{array}$

and

$\mathcal{H}\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}{\displaystyle \underset{\partial \mathcal{P}}{\int }\frac{h}{\text{Θ}}\text{d}a},\hfill \\ {\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }\frac{h_{\text{R}}}{\text{Θ}}\text{d}A},\hfill \end{array}$

where η is the specific entropy (or entropy per unit mass) and Θ is the absolute temperature. The Eulerian integral representations of $\mathcal{N}\left({\mathcal{S}}_1,t\right)$, $\mathcal{R}\left({\mathcal{S}}_1,t\right)$, and $\mathcal{H}\left({\mathcal{S}}_1,t\right)$ (top of (9.18) and (9.19)) correspond to subset ${\mathcal{S}}_1$ as seen in its present configuration, while the Lagrangian integral representations of these quantities (bottom of (9.18) and (9.19)) correspond to subset ${\mathcal{S}}_1$ as seen in its reference configuration.

Use of (9.18) and (9.19) in (9.17) leads to Eulerian and Lagrangian integral representations of the Clausius-Duhem inequality:

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{P}}{\int }\eta \rho \text{d}v}\ge {\displaystyle \underset{\mathcal{P}}{\int }\frac{r^{\text{t}}}{\text{Θ}}\rho \text{d}v}-{\displaystyle \underset{\partial \mathcal{P}}{\int }\frac{h}{\text{Θ}}\text{d}a},$

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

Note that these integral statements are valid for any open volume $\mathcal{P}$ bounded by closed surface $\partial \mathcal{P}$ in the present configuration, or corresponding open volume ${\mathcal{P}}_{\text{R}}$ bounded by closed surface $\partial {\mathcal{P}}_{\text{R}}$ in the reference configuration. Also note that (9.20a) and (9.20b) are identical to their counterparts in the thermomechanical theory (refer to Chapter 4).

9.1.7 Conservation of electric charge

Conservation of charge postulates that the time rate of change of the total electric charge (i.e., free charge Σ plus bound charge Σ_b) within any closed material surface is equal to the sum of the free (or conductive) current $\mathcal{J}$ and the polarization current ${\mathcal{J}}_{\text{p}}$ entering that surface. Applying this primitive statement of the law to subset ${\mathcal{S}}_1$ (an open material volume bounded by a closed material surface) allows us to express conservation of charge mathematically in material form:

$\frac{\text{d}}{\text{d}t}\left(\text{Σ}\left({\mathcal{S}}_1,t\right)+{\text{Σ}}_{\text{b}}\left({\mathcal{S}}_1,t\right)\right)=\mathcal{J}\left({\mathcal{S}}_1,t\right)+{\mathcal{J}}_{\text{p}}\left({\mathcal{S}}_1,t\right).$

Loosely, free charges are unpaired and “free” to move; this motion gives rise to the conductive current. Conversely, bound charges are paired, and are thus “bound” to a particular atom. When a material experiences a spatially varying polarization, the bound charges realign; if this polarization is also time varying, a polarization current arises.

Physically, the conductive current $\mathcal{J}\left({\mathcal{S}}_1,t\right)$ always enters and exits subset ${\mathcal{S}}_1$ at time t through its present surface $\partial \mathcal{P}$, but we are free to label this surface by its reference location $\partial {\mathcal{P}}_{\text{R}}$ instead. Similarly, the free charge $\sum \left({\mathcal{S}}_1,t\right)$ always resides in present volume $\mathcal{P}$, but we are free to label this volume by its reference location ${\mathcal{P}}_{\text{R}}$ instead. Exploiting this freedom in how the geometry of ${\mathcal{S}}_1$ is labeled allows us to write Eulerian and Lagrangian integral representations of the free charge, bound charge, conductive current, and polarization current, i.e.,

$\text{Σ}\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{P}}{\int }{\sigma }^{*}\text{d}v},\hfill \\ {\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }{\sigma }_{\text{R}}\text{d}V},\hfill \end{array}{\text{Σ}}_{\text{b}}\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}-{\displaystyle \underset{\mathcal{P}}{\int }\text{div}{\mathbf{p}}^{*}\text{d}v},\hfill \\ -{\displaystyle \underset{{\mathcal{P}}_{\text{R}}}{\int }\text{Div}{\mathbf{p}}_{\text{R}}\text{d}V},\hfill \end{array}$

$\mathcal{J}\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}-{\displaystyle \underset{\partial \mathcal{P}}{\int }{\mathbf{j}}^{*}·\mathbf{n}\text{d}a},\hfill \\ -{\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }{\mathbf{j}}_{\text{R}}·\mathbf{N}\text{d}A},\hfill \end{array}{\mathcal{J}}_{\text{p}}\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}-\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\partial \mathcal{P}}{\int }{\mathbf{p}}^{*}·\mathbf{n}\text{d}a},\hfill \\ -\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }{\mathbf{p}}_{\text{R}}·\mathbf{N}\text{d}A},\hfill \end{array}$

where n and N are outward unit normals in the present and reference configurations (refer to Section 3.6), “div” denotes the Eulerian divergence (i.e., the divergence calculated with respect to the present configuration), and “Div” denotes the Lagrangian divergence (i.e., the divergence calculated with respect to the reference configuration). Note that the minus signs in (9.23) are required to maintain consistency in our sign convention: positive $\mathcal{J}$ denotes current flowing into the boundary (see (9.21)), whereas positive j* · n implies that current is flowing out of the boundary (n is an outward unit normal).

In (9.22) and (9.23), σ* and σ_R are denoted the spatial free charge density and referential free charge density, j* and j_R the spatial conductive current density and referential conductive current density, and p* and p_R the spatial electric polarization and referential electric polarization. All are bounded, continuous functions of space and time. Recall that the spatial and referential representations of a particular quantity are different since they are associated with different labels for the geometry of the subset: σ* has units of charge per present volume, while σ_R has units of charge per reference volume; j* has units of current per present area, while j_R has units of current per reference area; and p* has units of charge per present area, while p_R has units of charge per reference area (see Table 9.1).

σ*, j*, and p* are often called effective electromagnetic fields in the literature [33] to signify that they are measured with respect to a co-moving frame, or rest frame, i.e., one affixed to but not deforming with the continuum. In this book, an effective electromagnetic field is denoted by a superscript asterisk. In Section 9.3, we present transformations that relate the effective electromagnetic fields to the standard electromagnetic fields, the latter being measured with respect to a stationary frame, or laboratory frame, rather than a co-moving frame.

Use of (9.22) and (9.23) in (9.21) leads to Eulerian and Lagrangian integral representations of conservation of charge:

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{P}}{\int }{\sigma }^{*}\text{d}v}=-{\displaystyle \underset{\partial \mathcal{P}}{\int }{\mathbf{j}}^{*}·\mathbf{n}\text{d}a},$

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

These integral statements are valid for any open volume $\mathcal{P}$ bounded by closed surface $\partial \mathcal{P}$ in the present configuration, or corresponding open volume ${\mathcal{P}}_{\text{R}}$ bounded by closed surface $\partial {\mathcal{P}}_{\text{R}}$ in the reference configuration. Note that (9.24a) and (9.24b) involve only free charge and free current. Also note that to perform the integrations in (9.24a) and (9.24b), it is more natural to consider σ* and j* in their Eulerian descriptions, i.e., as functions of x and t, and σ_R and j_R in their Lagrangian descriptions, i.e., as functions of X and t.

9.1.8 Faraday's law

Faraday's law postulates that the time rate of change of the magnetic flux $\mathcal{B}$ through any open material surface is equal to and opposite the electromotive force $\mathcal{E}$ induced in the closed material curve bounding that surface. Mathematically, in material form for subset ${\mathcal{S}}_2$ (an open material surface bounded by a closed material curve), this amounts to

$\frac{\text{d}}{\text{d}t}\mathcal{B}\left({\mathcal{S}}_2,t\right)=-\mathcal{E}\left({\mathcal{S}}_2,t\right).$

Smoothness of $\mathcal{B}$ and of $\mathcal{E}$ imply that

$\mathcal{B}\left({\mathcal{S}}_2,t\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{Q}}{\int }{\mathbf{b}}^{*}·\mathbf{n}\text{d}a},\hfill \\ {\displaystyle \underset{{\mathcal{Q}}_{\text{R}}}{\int }{\mathbf{b}}_{\text{R}}·\mathbf{N}\text{d}A},\hfill \end{array}\mathcal{E}\left({\mathcal{S}}_2,t\right)=\{\begin{array}{l}{\displaystyle \underset{\partial \mathcal{Q}}{\int }{\mathbf{e}}^{*}·\mathbf{l}\text{d}l},\hfill \\ {\displaystyle \underset{\partial {\mathcal{Q}}_{\text{R}}}{\int }{\mathbf{e}}_{\text{R}}·{\mathbf{l}}_{\text{R}}\text{d}L},\hfill \end{array}$

where dl and dL are line elements in the present and reference configurations, I and I_R are unit tangents in the present and reference configurations, b* and b_R are the spatial magnetic flux density and referential magnetic flux density (or magnetic induction), and e* and e_R are the spatial electric field and referential electric field. Refer to Table 9.1 for their respective units. Use of (9.26) in (9.25) leads to Eulerian and Lagrangian representations of Faraday's law:

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{Q}}{\int }{\mathbf{b}}^{*}·\mathbf{n}\text{d}a}=-{\displaystyle \underset{\partial \mathcal{Q}}{\int }{\mathbf{e}}^{*}·\mathbf{l}\text{d}l},$

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

Note that these integral statements are valid for any open surface $\mathcal{Q}$ bounded by a closed curve $\partial \mathcal{Q}$ in the present configuration, or corresponding open surface ${\mathcal{Q}}_{\text{R}}$ bounded by a closed curve $\partial {\mathcal{Q}}_{\text{R}}$ in the reference configuration.

9.1.9 Gauss's law for magnetism

Gauss's law for magnetism (a statement of conservation of magnetic flux, or, alternatively, the absence of magnetic monopoles) postulates that the magnetic flux $\mathcal{B}$ through any closed material surface is zero, i.e.,

$\mathcal{B}\left({\mathcal{S}}_1,t\right)=0.$

Assuming that the magnetic flux $\mathcal{B}\left({\mathcal{S}}_1,t\right)$ through the surface of ${\mathcal{S}}_1$ (recall that ${\mathcal{S}}_1$ consists of an open volume bounded by a closed surface) at time t is smooth, we can write

$\mathcal{B}\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}{\displaystyle \underset{\partial \mathcal{P}}{\int }{\mathbf{b}}^{*}·\mathbf{n}\text{d}a},\hfill \\ {\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }{\mathbf{b}}_{\text{R}}·\mathbf{N}\text{d}A}.\hfill \end{array}$

Use of (9.29) in (9.28) gives Eulerian and Lagrangian representations of Gauss's law for magnetism:

${\displaystyle \underset{\partial \mathcal{P}}{\int }{\mathbf{b}}^{*}·\mathbf{n}\text{d}a}=0,$

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

9.1.10 Gauss's law for electricity

Gauss's law for electricity postulates that the electric flux $\mathcal{F}$ through any closed material surface is proportional to the total electric charge (i.e., free charge Σ plus bound charge Σ_b) enclosed within that surface, i.e.,

$\mathcal{F}\left({\mathcal{S}}_1,t\right)=\frac{\text{Σ}\left({\mathcal{S}}_1,t\right)+{\text{Σ}}_{\text{b}}\left(S_1,t\right)}{{\epsilon }_{\text{o}}},$

where ε_o is the electric permittivity in vacuo. Assuming that the electric flux $\mathcal{F}$ is smooth implies that

$\mathcal{F}\left({\mathcal{S}}_1,t\right)=\{\begin{array}{l}{\displaystyle \underset{\partial \mathcal{P}}{\int }{\mathbf{e}}^{*}·\mathbf{n}\text{d}a},\hfill \\ {\displaystyle \underset{\partial {\mathcal{P}}_{\text{R}}}{\int }J{\mathbf{C}}^{-1}{\mathbf{e}}_{\text{R}}·\mathbf{N}\text{d}A},\hfill \end{array}$

where J is the determinant of the deformation gradient F, and C⁻¹ is the inverse of the right Cauchy-Green deformation tensor C = F^TF. Recall that the free charge Σ and the bound charge Σ_b associated with subset ${\mathcal{S}}_1$ at time t are given in (9.22). Subsequent use of (9.22) and (9.32) in (9.31) leads to Eulerian and Lagrangian representations of Gauss's law for electricity:

${\displaystyle \underset{\partial \mathcal{P}}{\int }{\mathbf{d}}^{*}·\mathbf{n}\text{d}a}={\displaystyle \underset{\mathcal{P}}{\int }{\sigma }^{*}\mathit{dv}},$

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

where the spatial electric displacement d* and referential electric displacement d_R are introduced through the algebraic relationships (see, for instance, [37])

${\mathbf{d}}^{*}={\mathbf{p}}^{*}+{\epsilon }_{\text{o}}{\mathbf{e}}^{*},{\mathbf{d}}_{\text{R}}={\mathbf{p}}_{\text{R}}+{\epsilon }_{\text{o}}J{\mathbf{C}}^{-1}{\mathbf{e}}_{\text{R}}.$

9.1.11 Ampère-maxwell law

The Ampère-Maxwell law postulates that the time rate of change of the electric flux $\mathcal{F}$ through any open material surface plus the conductive current $\mathcal{J}$, polarization current ${\mathcal{J}}_{\text{p}}$, and magnetization current ${\mathcal{J}}_{\text{m}}$ passing through that surface is proportional to the magnetic field $\mathcal{T}$ around the closed material curve bounding that surface. Application of this primitive statement of the law to subset ${\mathcal{S}}_2$ (an open material surface bounded by a closed material curve) allows us to express the Ampère-Maxwell law mathematically in material form:

${\mu }_{\text{o}}\left(\left({\epsilon }_{\text{o}}\right)\frac{\text{d}}{\text{d}t}\mathcal{F}\left({\mathcal{S}}_2,t\right)+\mathcal{J}\left({\mathcal{S}}_2,t\right)+{\mathcal{J}}_{\text{p}}\left({\mathcal{S}}_2,t\right)+{\mathcal{J}}_{\text{m}}\left({\mathcal{S}}_2,t\right)\right)=\mathcal{T}\left({\mathcal{S}}_2,t\right),$

where μ_o is the magnetic permeability in vacuo. Smoothness allows us to write

$\mathcal{F}\left({\mathcal{S}}_2,t\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{Q}}{\int }{\mathbf{e}}^{*}·\mathbf{n}\text{d}a},\hfill \\ {\displaystyle \underset{{\mathcal{Q}}_{\text{R}}}{\int }J{\mathbf{C}}^{-1}{\mathbf{e}}_{\text{R}}·\mathbf{N}\text{d}A},\hfill \end{array}\mathcal{J}\left({\mathcal{S}}_2,t\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{Q}}{\int }{\mathbf{j}}^{*}·\mathbf{n}\text{d}a},\hfill \\ {\displaystyle \underset{{\mathcal{Q}}_{\text{R}}}{\int }{\mathbf{j}}_{\text{R}}·\mathbf{N}\text{d}A},\hfill \end{array}$

${\mathcal{J}}_{\text{p}}\left({\mathcal{S}}_2,t\right)=\{\begin{array}{l}\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{Q}}{\int }{\mathbf{p}}^{*}·\mathbf{n}\text{d}a},\hfill \\ \frac{\text{d}}{\text{d}t}{\displaystyle \underset{{\mathcal{Q}}_{\text{R}}}{\int }{\mathbf{p}}_{\text{R}}·\mathbf{N}\text{d}A},\hfill \end{array}{\mathcal{J}}_{\text{m}}\left({\mathcal{S}}_2,t\right)=\{\begin{array}{l}{\displaystyle \underset{\mathcal{Q}}{\int }\left(\text{curl}{\mathbf{m}}^{*}\right)·\mathbf{n}\text{d}a},\hfill \\ {\displaystyle \underset{{\mathcal{Q}}_{\text{R}}}{\int }\left(\text{Curl}{\mathbf{m}}_{\text{R}}\right)·\mathbf{N}\text{d}A},\hfill \end{array}$

$\mathcal{T}\left({\mathcal{S}}_2,t\right)=\{\begin{array}{l}{\displaystyle \underset{\partial \mathcal{Q}}{\int }{\mathbf{b}}^{*}·\mathbf{l}\text{d}l},\hfill \\ {\displaystyle \underset{\partial {\mathcal{Q}}_{\text{R}}}{\int }\frac{1}{J}\mathbf{C}{\mathbf{b}}_{\text{R}}·{\mathbf{l}}_{\text{R}}\text{d}L},\hfill \end{array}$

where m* and m_R are the spatial magnetization and referential magnetization (or magnetic polarization), “curl” is the Eulerian curl (i.e., the curl calculated with respect to the present configuration), and “Curl” is the Lagrangian curl (i.e., the curl calculated with respect to the reference configuration). Use of (9.36)–(9.38) in (9.35) leads to

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{\mathcal{Q}}{\int }{\mathbf{d}}^{*}·\mathbf{n}\text{d}a}+{\displaystyle \underset{\mathcal{Q}}{\int }{\mathbf{j}}^{*}·\mathbf{n}\text{d}a}={\displaystyle \underset{\partial \mathcal{Q}}{\int }{\mathbf{h}}^{*}·\mathbf{l}\text{d}l},$

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

where we have used (9.34) to introduce d* and d_R, and the algebraic relationships (see, for instance, [36])

${\mathbf{h}}^{*}=\frac{1}{{\mu }_{\text{o}}}{\mathbf{b}}^{*}-{\mathbf{m}}^{*},{\mathbf{h}}_{\text{R}}=\frac{1}{{\mu }_{\text{o}}J}\mathbf{C}{\mathbf{b}}_{\text{R}}-{\mathbf{m}}_{\text{R}}$

to introduce the spatial magnetic field h* and referential magnetic field h_R.

9.1.12 Transformations between spatial and referential TEMM quantities

The spatial (Eulerian) and referential (Lagrangian) TEMM quantities appearing in Sections 9.1.2–9.1.11, along with their corresponding units, are listed in Table 9.1. (In Table 9.1, we employ the following notation for the fundamental units: M is mass, L_P is present length, L_R is reference length, T is time, θ is temperature, and C is charge.)⁷ It can be shown that these spatial and referential quantities are related through the following linear algebraic transformations:

$\begin{array}{llll}{e}_{\text{R}}=F^{\text{T}}e*,\hfill & p_{\text{R}}=J{F}^{-1}p*,\hfill & d_{\text{R}}=J{F}^{-1}d*,\hfill & {\sigma }_{\text{R}}=J\sigma *,\hfill \\ h_{\text{R}}=F^{\text{T}}h*,\hfill & m_{\text{R}}=F^{\text{T}}m*,\hfill & b_{\text{R}}=J{F}^{-1}b*\hfill & {\rho }_{\text{R}}=J\rho ,\hfill \\ j_{\text{R}}=J{F}^{-1}j*,\hfill & P=JT{F}^{-\text{T}},\hfill & q_{\text{R}}=J{F}^{-1}q.\hfill \end{array}$

Recall that J is the determinant of the deformation gradient F, and is a measure of dilatation or volume change (refer to Section 3.6). Several of these results—namely, (9.41)₈, (9.41)₁₀, and (9.41)₁₁—were obtained in Sections 4.10 and 4.12. The other transformations can be obtained in a similar manner; refer to Problems 9.1–9.3.