Thermo-electrodynamics of conductive media based on the nonlinear viscoelastic Cosserat continuum of a special type

Abstract

We develop a general nonlinear theory of thermo-electrodynamics. We show that two theories constructed in our previous works, namely the linear theory of thermo-electrodynamics and the nonlinear theory of electromagnetism, can be obtained from the general nonlinear theory proposed in the present paper. We also make some assumptions about how our model can be used to describe the fields corresponding to strong and weak interactions. Our approach is based on using of the Cosserat continuum of a special type as a mechanical model and some analogues between mechanical and physical quantities.

Appendices

A Transformation of the energy balance equation

Let us rewrite Eq. (17) as

$$\begin{aligned} \nabla {\textbf{v}} = \frac{\delta {\textbf{g}}^{-1}}{\delta t} \cdot {\textbf{g}}, \quad \nabla \varvec{\omega } = \frac{\delta \varvec{\varTheta }}{\delta t} + \varvec{\varTheta } \times \varvec{\omega } + \frac{\delta {\textbf{g}}^{-1}}{\delta t} \cdot {\textbf{g}} \cdot \varvec{\varTheta }. \end{aligned}$$

(178)

In view of Eq. (178), the energy balance equation (11) takes the form

$$\begin{aligned} \rho \frac{\delta U}{\delta t} = \varvec{\tau }^T \cdot \cdot \left( \frac{\delta {\textbf{g}}^{-1}}{\delta t} \cdot {\textbf{g}} + {\textbf{E}} \times \varvec{\omega } \right) + {\textbf{T}}^T \cdot \cdot \left( \frac{\delta \varvec{\varTheta }}{\delta t} + \varvec{\varTheta } \times \varvec{\omega } + \frac{\delta {\textbf{g}}^{-1}}{\delta t} \cdot {\textbf{g}} \cdot \varvec{\varTheta }\right) . \end{aligned}$$

(179)

Next, we transform Eq. (179) as follows

$$\begin{aligned} \rho \frac{\delta U}{\delta t} = \varvec{\tau }^T \cdot \cdot \left( \frac{\delta {\textbf{g}}^{-1}}{\delta t} \cdot {\textbf{g}} + {\textbf{E}} \times \varvec{\omega } \right) + {\textbf{T}}^T \cdot \cdot \left( \frac{\delta \varvec{\varTheta }}{\delta t} + \varvec{\varTheta } \times \varvec{\omega } - \varvec{\omega } \times \varvec{\varTheta } + \left[ \frac{\delta {\textbf{g}}^{-1}}{\delta t} \cdot {\textbf{g}} + \varvec{\omega } \times {\textbf{E}} \right] \cdot \varvec{\varTheta }\right) .\nonumber \\ \end{aligned}$$

(180)

In view of identity ${\textbf{E}} \times \varvec{\omega } = \varvec{\omega } \times {\textbf{E}}$ and the properties of the double dot product, Eq. (180) can be reduced to the form

$$\begin{aligned} \rho \frac{\delta U}{\delta t} = \left( {\textbf{g}}^T \cdot \left[ \varvec{\tau } + {\textbf{T}} \cdot \varvec{\varTheta }^T\right] \right) ^T \cdot \cdot \left( \frac{\delta {\textbf{g}}^{-1}}{\delta t} + {\textbf{g}}^{-1} \times \varvec{\omega } \right) + {\textbf{T}}^T \cdot \cdot \left( \frac{\delta \varvec{\varTheta }}{\delta t} + \varvec{\varTheta } \times \varvec{\omega } - \varvec{\omega } \times \varvec{\varTheta }\right) . \end{aligned}$$

(181)

Performing some transformation of Eq. (181), we arrive at

$$\begin{aligned} \rho \frac{\delta U}{\delta t} = \left( {\textbf{g}}^T \cdot \left[ \varvec{\tau } + {\textbf{T}} \cdot \varvec{\varTheta }^T\right] \cdot {\textbf{P}} \right) ^T \cdot \cdot \, \frac{\delta \left( {\textbf{g}}^{-1} \cdot {\textbf{P}}\right) }{\delta t} + \left( {\textbf{P}}^T \cdot {\textbf{T}} \cdot {\textbf{P}}\right) ^T \cdot \cdot \, \frac{\delta \left( {\textbf{P}}^T \cdot \varvec{\varTheta }\cdot {\textbf{P}}\right) }{\delta t}. \end{aligned}$$

(182)

B Transformation of the strain balance equations

We start with the equations for the stretch tensor ${\textbf{g}}$. It is easy to see that the first equation in (17) can be rewritten as

$$\begin{aligned} \displaystyle \frac{d {\textbf{g}}}{d t} = - {\textbf{v}} \cdot \nabla {\textbf{g}} - (\nabla {\textbf{v}}) \cdot {\textbf{g}}\Rightarrow & {} \frac{d {\textbf{g}}}{d t} = - \nabla \left( {\textbf{v}} \cdot {\textbf{g}}\right) + \left( \nabla {\textbf{g}}^T\right) \cdot {\textbf{v}} - {\textbf{v}} \cdot \nabla {\textbf{g}} \quad \nonumber \\\Rightarrow & {} \displaystyle \frac{d {\textbf{g}}}{d t} = - \nabla \left( {\textbf{v}} \cdot {\textbf{g}}\right) + {\textbf{v}} \times (\nabla \times {\textbf{g}}). \end{aligned}$$

(183)

Taking into account the first equation in (12) we can show that $\nabla \times {\textbf{g}} = 0$. In this case, the last equation in (183) turns to the first equation in (29).

Analogously, the first equation in (35) can be transformed as

$$\begin{aligned} \displaystyle \frac{d {\textbf{g}}}{d t} = - (\nabla {\textbf{v}}) \cdot {\textbf{g}} - \left( \nabla {\textbf{g}}^T\right) \cdot {\textbf{v}} + \varvec{\varUpsilon }_g\Rightarrow & {} \frac{d {\textbf{g}}}{d t} = - (\nabla {\textbf{v}}) \cdot {\textbf{g}} - {\textbf{v}} \times (\nabla \times {\textbf{g}}) - {\textbf{v}} \cdot \nabla {\textbf{g}} + \varvec{\varUpsilon }_g \nonumber \\ {}\Rightarrow & {} \displaystyle \frac{\delta {\textbf{g}}}{\delta t} = - (\nabla {\textbf{v}}) \cdot {\textbf{g}} - {\textbf{v}} \times (\nabla \times {\textbf{g}}) + \varvec{\varUpsilon }_g. \end{aligned}$$

(184)

The last equation in (184) is, in fact, the first equation in (36).

Now, we turn to the equations for the wryness tensor $\varvec{\varTheta }$. The second equation in (17) can be rewritten as

$$\begin{aligned} \displaystyle \frac{d \varvec{\varTheta }}{d t} = \nabla \varvec{\omega } - \varvec{\varTheta } \times \varvec{\omega } - {\textbf{v}} \cdot \nabla \varvec{\varTheta } - (\nabla {\textbf{v}}) \cdot \varvec{\varTheta }\Rightarrow & {} \frac{d \varvec{\varTheta }}{d t} = - \nabla \left( {\textbf{v}} \cdot \varvec{\varTheta } - \varvec{\omega }\right) - \varvec{\varTheta } \times \varvec{\omega } + \left( \nabla \varvec{\varTheta }^T\right) \cdot {\textbf{v}} - {\textbf{v}} \cdot \nabla \varvec{\varTheta } \nonumber \\ \displaystyle\Rightarrow & {} \frac{d \varvec{\varTheta }}{d t} = - \nabla \left( {\textbf{v}} \cdot \varvec{\varTheta } - \varvec{\omega }\right) - \varvec{\varTheta } \times \varvec{\omega } + {\textbf{v}} \times (\nabla \times \varvec{\varTheta }). \end{aligned}$$

(185)

Taking into account the second equation in (12) we can show that ${\displaystyle \nabla \times \varvec{\varTheta } = \frac{1}{2}\,\varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta }}$. In this case, the last equation in (185) takes the form

$$\begin{aligned} \frac{d \varvec{\varTheta }}{d t} = - \nabla \left( {\textbf{v}} \cdot \varvec{\varTheta } - \varvec{\omega }\right) - \varvec{\varTheta } \times \varvec{\omega } + {\textbf{v}} \times \frac{1}{2}\left( \varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta }\right) \Rightarrow \frac{d \varvec{\varTheta }}{d t} = - \nabla \left( {\textbf{v}} \cdot \varvec{\varTheta } - \varvec{\omega }\right) + \varvec{\varTheta } \times \left( {\textbf{v}} \cdot \varvec{\varTheta } - \varvec{\omega }\right) .\nonumber \\ \end{aligned}$$

(186)

It is easy to see that the last equation in (186) coincides with the second equation in (29).

Next, we transform the second equation in (35) as follows

$$\begin{aligned} \displaystyle \frac{d \varvec{\varTheta }}{d t}= & {} - \nabla (\varvec{{\textbf{v}} \cdot \varvec{\varTheta } - \omega }) + \varvec{\varTheta } \times ({\textbf{v}} \cdot \varvec{\varTheta } - \varvec{\omega }) + \varvec{\varUpsilon }_\varTheta \nonumber \\ {}\Rightarrow & {} \displaystyle \frac{d \varvec{\varTheta }}{d t} = - \nabla \left( {\textbf{v}} \cdot \varvec{\varTheta } - \varvec{\omega }\right) - \varvec{\varTheta } \times \varvec{\omega } + {\textbf{v}} \times \frac{1}{2}\left( \varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta }\right) + \varvec{\varUpsilon }_\varTheta \nonumber \\ {}\Rightarrow & {} \displaystyle \frac{d \varvec{\varTheta }}{d t} = - \nabla \left( {\textbf{v}} \cdot \varvec{\varTheta } - \varvec{\omega }\right) - \varvec{\varTheta } \times \varvec{\omega } + {\textbf{v}} \times \left( \nabla \times \varvec{\varTheta }\right) - {\textbf{v}} \times \left( \nabla \times \varvec{\varTheta } - \frac{1}{2}\, \varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta }\right) + \varvec{\varUpsilon }_\varTheta \nonumber \\ {}\Rightarrow & {} \displaystyle \frac{d \varvec{\varTheta }}{d t} = \nabla \varvec{\omega } - \varvec{\varTheta } \times \varvec{\omega } - {\textbf{v}} \cdot \nabla \varvec{\varTheta } - (\nabla {\textbf{v}}) \cdot \varvec{\varTheta } - {\textbf{v}} \times \left( \nabla \times \varvec{\varTheta } - \frac{1}{2}\, \varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta }\right) + \varvec{\varUpsilon }_\varTheta \nonumber \\ {}\Rightarrow & {} \displaystyle \frac{\delta \varvec{\varTheta }}{\delta t} = \nabla \varvec{\omega } - \varvec{\varTheta } \times \varvec{\omega } - (\nabla {\textbf{v}}) \cdot \varvec{\varTheta } - {\textbf{v}} \times \left( \nabla \times \varvec{\varTheta } - \frac{1}{2}\, \varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta }\right) + \varvec{\varUpsilon }_\varTheta . \end{aligned}$$

(187)

The last equation in (187) is the second equation in (36).

C The energy balance equation in the case of the modified strain tensors

Let us rewrite Eq. (37) as

$$\begin{aligned} \nabla {\textbf{v}} = \frac{\delta {\textbf{g}}^{-1}}{\delta t} \cdot {\textbf{g}} + \varvec{\varUpsilon }_g^* \cdot {\textbf{g}}^{-1}, \quad \nabla \varvec{\omega } = \frac{\delta \varvec{\varTheta }}{\delta t} + \varvec{\varTheta } \times \varvec{\omega } + \left( \frac{\delta {\textbf{g}}^{-1}}{\delta t} \cdot {\textbf{g}} + \varvec{\varUpsilon }_g^* \cdot {\textbf{g}}^{-1}\right) \cdot \varvec{\varTheta } - \varvec{\varUpsilon }_\varTheta ^*. \end{aligned}$$

(188)

Inserting Eq. (188) in the energy balance equation (11), we obtain

$$\begin{aligned} \rho \frac{\delta U}{\delta t}= & {} \varvec{\tau }^T \cdot \cdot \left( \frac{\delta {\textbf{g}}^{-1}}{\delta t} \cdot {\textbf{g}}+ \varvec{\varUpsilon }_g^* \cdot {\textbf{g}}^{-1} + {\textbf{E}} \times \varvec{\omega } \right) \nonumber \\{} & {} + {\textbf{T}}^T \cdot \cdot \left( \frac{\delta \varvec{\varTheta }}{\delta t} + \varvec{\varTheta } \times \varvec{\omega } + \left[ \frac{\delta {\textbf{g}}^{-1}}{\delta t} \cdot {\textbf{g}} + \varvec{\varUpsilon }_g^* \cdot {\textbf{g}}^{-1}\right] \cdot \varvec{\varTheta } - \varvec{\varUpsilon }_\varTheta ^*\right) . \end{aligned}$$

(189)

After simple transformations Eq. (189) can be rewritten as

$$\begin{aligned} \begin{array}{c} {\displaystyle \rho \frac{\delta U}{\delta t} = \varvec{\tau }^T \cdot \cdot \left( \frac{\delta {\textbf{g}}^{-1}}{\delta t} \cdot {\textbf{g}} + {\textbf{E}} \times \varvec{\omega } \right) + {\textbf{T}}^T \cdot \cdot \left( \frac{\delta \varvec{\varTheta }}{\delta t} + \varvec{\varTheta } \times \varvec{\omega } + \frac{\delta {\textbf{g}}^{-1}}{\delta t} \cdot {\textbf{g}} \cdot \varvec{\varTheta } \right) } \\ {\displaystyle +\, \left[ \,{\textbf{g}}^{-1} \cdot \left( \varvec{\tau }^T + \varvec{\varTheta } \cdot {\textbf{T}}^T\right) \right] \cdot \cdot \,\varvec{\varUpsilon }_g^* - {\textbf{T}}^T \cdot \cdot \,\varvec{\varUpsilon }_\varTheta ^*.} \end{array} \end{aligned}$$

(190)

Next, performing the same transformations as in [53], we obtain Eq. (39). Performing the same transformations as in “Appendix A,” we reduce Eq. (190) to the form

$$\begin{aligned} \begin{array}{c} {\displaystyle \rho \frac{\delta U}{\delta t} = \left( {\textbf{g}}^T \cdot \left[ \varvec{\tau } + {\textbf{T}} \cdot \varvec{\varTheta }^T\right] \cdot {\textbf{P}} \right) ^T \cdot \cdot \, \frac{\delta \left( {\textbf{g}}^{-1} \cdot {\textbf{P}}\right) }{\delta t} + \left( {\textbf{P}}^T \cdot {\textbf{T}} \cdot {\textbf{P}}\right) ^T \cdot \cdot \, \frac{\delta \left( {\textbf{P}}^T \cdot \varvec{\varTheta }\cdot {\textbf{P}}\right) }{\delta t} } \\ {\displaystyle +\, \left[ \,{\textbf{g}}^{-1} \cdot \left( \varvec{\tau }^T + \varvec{\varTheta } \cdot {\textbf{T}}^T\right) \right] \cdot \cdot \,\varvec{\varUpsilon }_g^* - {\textbf{T}}^T \cdot \cdot \,\varvec{\varUpsilon }_\varTheta ^*.} \end{array} \end{aligned}$$

(191)

This equation coincides with Eq. (40).

D Transformation of the angular momentum balance equation

If we use restriction (49), we should consider the model based on the energy moment stress tensor. In this case the angular momentum balance equation (43) takes the form

$$\begin{aligned} \nabla \cdot {\textbf{T}} + \rho {\textbf{L}} = \rho J\, \frac{d \varvec{\omega }}{d t}. \end{aligned}$$

(192)

In virtue the relations ${\textbf{T}} = {\textbf{T}}_e \cdot {\textbf{P}}^T$ and $\varvec{\omega } = \varvec{\varOmega } \cdot {\textbf{P}}^T$, we can rewrite Eq. (192) as

$$\begin{aligned} \nabla \cdot ( - {\textbf{M}}_e \times {\textbf{P}}^T) + \rho {\textbf{L}} = \rho J\, \frac{d (\varvec{\varOmega } \cdot {\textbf{P}}^T)}{d t}. \end{aligned}$$

(193)

Let us take the dot product of Eq. (193) on tensor ${\textbf{P}}$ and perform some simple transformations in view of the kinematic relation (42). As a result, we have

$$\begin{aligned} - \nabla \times {\textbf{M}}_e + {\textbf{M}}_e \cdot \left( \nabla \times {\textbf{P}}^T\right) \cdot {\textbf{P}} + \rho {\textbf{L}}_e = \frac{d \varvec{{\mathcal {K}}}_e}{d t}, \end{aligned}$$

(194)

where we use the notation $\varvec{{\mathcal {K}}}_e = \rho J \varvec{\varOmega }$ and $\rho {\textbf{L}}_e = \rho {\textbf{L}} \cdot {\textbf{P}}$. We can specify the constitutive equation for the external moment $\rho {\textbf{L}}$, e.g., as the linear function of $\varvec{\omega }$. In this case, the constitutive equation for $\rho {\textbf{L}}_e$ will be the linear function of $\varvec{\varOmega }$. Thus, choosing ${\textbf{M}}_e$, $\varvec{{\mathcal {K}}}_e$, $\varvec{\varOmega }$ and $\varvec{\varTheta }_e$ as the basic variables, we have transformed the angular momentum balance equation to the form, where only one term contains the rotation tensor. Now, we show that we can rewrite the expression $\left( \nabla \times {\textbf{P}}^T\right) \cdot {\textbf{P}}$ in terms of tensor $\varvec{\varTheta }_e$ only if we use the equation $\nabla {\textbf{P}} = \varvec{\varTheta } \times {\textbf{P}}$. Let us rewrite this equation as

$$\begin{aligned} \frac{\partial {\textbf{P}}}{\partial q^i} = \varvec{\varTheta }_i \times {\textbf{P}}, \quad \varvec{\varTheta } = {\textbf{r}}^i \varvec{\varTheta }_i. \end{aligned}$$

(195)

Then, we can perform the following transformations:

$$\begin{aligned} \displaystyle \left( \nabla \times {\textbf{P}}^T\right) \cdot {\textbf{P}}= & {} {\textbf{r}}^i \times \left( \frac{\partial {\textbf{P}}}{\partial q^i}\right) ^T \cdot {\textbf{P}} = {\textbf{r}}^i \times \left( \varvec{\varTheta }_i \times {\textbf{P}}\right) ^T \cdot {\textbf{P}} = - {\textbf{r}}^i \times \left( {\textbf{P}}^T \times \varvec{\varTheta }_i\right) \cdot {\textbf{P}} \\ \displaystyle= & {} - {\textbf{r}}^i \times \left( ({\textbf{P}}^T \cdot {\textbf{P}}) \times (\varvec{\varTheta }_i \cdot {\textbf{P}})\right) = - {\textbf{r}}^i \times {\textbf{E}} \times (\varvec{\varTheta }_i \cdot {\textbf{P}}) = - (\varvec{\varTheta }_i \cdot {\textbf{P}}) {\textbf{r}}^i + (\varvec{\varTheta }_i \cdot {\textbf{P}}) \cdot {\textbf{r}}^i\, {\textbf{E}} \\ \displaystyle= & {} - \varvec{\varTheta } \cdot {\textbf{P}} + \textrm{tr}(\varvec{\varTheta } \cdot {\textbf{P}})\,{\textbf{E}} = - \left( \varvec{\varTheta }_e - \textrm{tr}\,\varvec{\varTheta }_e {\textbf{E}}\right) ^T. \end{aligned}$$

We emphasize that the last transformations are valid only in virtue of equation $\nabla {\textbf{P}} = \varvec{\varTheta } \times {\textbf{P}}$:

$$\begin{aligned} \nabla {\textbf{P}} = \varvec{\varTheta } \times {\textbf{P}} \quad \Rightarrow \quad \left( \nabla \times {\textbf{P}}^T\right) \cdot {\textbf{P}} = - \left( \varvec{\varTheta }_e - \textrm{tr}\,\varvec{\varTheta }_e {\textbf{E}}\right) ^T. \end{aligned}$$

(196)

If we reject the first equation in (196) we cannot use the second equation in (196), and hence we cannot eliminate tensor ${\textbf{P}}$ from Eq. (194).

E Derivation of the constitutive equations

Let us start with the energy balance equation (27), where $\varvec{\tau }_r$, ${\textbf{T}}_r$ are given by Eq. (25), and ${\textbf{g}}_r$, $\varvec{\varTheta }_r$ are given by Eq. (26). We assume that the structure of tensor ${\textbf{T}}$ is determined by Eq. (51). In this case tensors $\varvec{\tau }_r$, ${\textbf{T}}_r$ take the form

$$\begin{aligned} \varvec{\tau }_r = {\textbf{g}}^T \cdot \left( \varvec{\tau } + T \varvec{\varTheta }^T - {\textbf{M}} \times \varvec{\varTheta }^T \right) \cdot {\textbf{P}}, \quad {\textbf{T}}_r = T {\textbf{E}} - {\textbf{M}}_r \times {\textbf{E}}, \quad {\textbf{M}}_r = {\textbf{P}}^T \cdot {\textbf{M}}. \end{aligned}$$

(197)

In view of the second equation in (197), we can rewrite Eq. (27) as

$$\begin{aligned} \rho \,\frac{\delta U}{\delta t} = \varvec{\tau }_r^T \cdot \cdot \,\frac{\delta {\textbf{g}}_r}{\delta t} + (T {\textbf{E}} - {\textbf{M}}_r \times {\textbf{E}})^T \cdot \cdot \,\frac{\delta \varvec{\varTheta }_r}{\delta t}. \end{aligned}$$

(198)

After simple transformations Eq. (198) takes the form

$$\begin{aligned} \rho \,\frac{\delta U}{\delta t} = \varvec{\tau }_r^T \cdot \cdot \,\frac{\delta {\textbf{g}}_r}{\delta t} + T \frac{\delta \varTheta }{\delta t} + {\textbf{M}}_r \cdot \,\frac{\delta \varvec{\varPsi }_r}{\delta t}, \end{aligned}$$

(199)

where

$$\begin{aligned} \varTheta = \textrm{tr}\,\varvec{\varTheta }, \quad \varvec{\varPsi }_r = {\textbf{P}}^T \cdot \varvec{\varPsi }, \quad \varvec{\varPsi } = \varvec{\varTheta }_\times . \end{aligned}$$

(200)

We assume that the continuum is elastic. In this case, from Eq. (27) it follows that $U = U\bigl ({\textbf{g}}_r,\,\varTheta ,\,\varvec{\varPsi }_r\bigr )$. In addition, we assume that the internal energy does no depend on ${\textbf{g}}_r$. Then, by standard reasoning, we arrive at the Cauchy–Green relations

$$\begin{aligned} \varvec{\tau }_r = 0, \quad T = \frac{\partial \rho U\bigl (\varTheta ,\,\varvec{\varPsi }_r\bigr )}{\partial \varTheta } \quad {\textbf{M}}_r = \frac{\partial \rho U\bigl (\varTheta ,\,\varvec{\varPsi }_r\bigr )}{\partial \varvec{\varPsi }_r}. \end{aligned}$$

(201)

Next, we assume that function $\rho U\bigl (\varTheta ,\,\varvec{\varPsi }_r\bigr )$ is specified by Eq. (54). Then, from Eq. (201) it follows

$$\begin{aligned} \varvec{\tau }_r = 0, \quad T = T_* + C_\varTheta \,(\varTheta - \varTheta _*), \quad {\textbf{M}}_r = C_\varPsi \, \varvec{\varPsi }_r. \end{aligned}$$

(202)

Taking into account Eqs. (197), (200), we can transform Eq. (202) to the form of Eq. (55).

F Derivation of equations for the wryness tensor

Multiplying Eq. (62) by the unit tensor and calculating the difference of Eq. (59) and the obtained equation, we arrive at the equation

$$\begin{aligned} \frac{d (\varvec{\varTheta } - \textrm{tr}\,{\varvec{\varTheta }}\, {\textbf{E}})}{d t} = \nabla \cdot ({\textbf{E}}\,\varvec{\omega } - \varvec{\omega }\, {\textbf{E}}) - \left( \varvec{\varTheta } \times {\textbf{E}} - {\textbf{E}}\,\varvec{\varTheta }_\times \right) \cdot \varvec{\omega } - \frac{2}{3}\,\varUpsilon _\varTheta {\textbf{E}} - \frac{1}{2}\, \varvec{\varUpsilon }_\varPsi \times {\textbf{E}}. \end{aligned}$$

(203)

Next, we perform the following transformation:

$$\begin{aligned} \left( \varvec{\varTheta } \times {\textbf{E}} - {\textbf{E}}\,\varvec{\varTheta }_\times \right) \cdot \varvec{\omega }= & {} \varvec{\varTheta } \times \varvec{\omega } - \varvec{\varTheta }_\times \cdot \varvec{\omega }\, {\textbf{E}} = - \left( \varvec{\omega } \times \varvec{\varTheta }^T + \varvec{\varTheta }_\times \!\cdot \varvec{\omega }\, {\textbf{E}}\right) ^T \\= & {} - \bigl (\varvec{\omega } \times \varvec{\varTheta } + \varvec{\omega } \times {\textbf{E}} \times \varvec{\varTheta }_\times + \varvec{\varTheta }_\times \!\cdot \varvec{\omega }\, {\textbf{E}} \bigr )^T = - \bigl (\varvec{\omega } \times \varvec{\varTheta } + \varvec{\varTheta }_\times \varvec{\omega }\bigr )^T. \end{aligned}$$

In view of the last transformation, Eq. (203) takes the form of Eq. (63).

Let us take the double cross product of tensor $\varvec{\varTheta }^T$ and Eq. (59), then take the double cross product of transposed Eq. (44) and tensor $\varvec{\varTheta }$. Adding up the obtained equations, we obtain

$$\begin{aligned} \displaystyle \varvec{\varTheta }^T \!\times \times \, \frac{d \varvec{\varTheta }}{d t} + \frac{d \varvec{\varTheta }^T}{d t} \!\times \times \, \varvec{\varTheta }= & {} \varvec{\varTheta }^T \!\times \times \, (\nabla \varvec{\omega }) + (\nabla \varvec{\omega })^T \!\times \times \, \varvec{\varTheta } - \varvec{\varTheta }^T \!\times \times \, (\varvec{\varTheta } \times \varvec{\omega }) - (\varvec{\varTheta } \times \varvec{\omega })^T \nonumber \\{} & {} \displaystyle \!\times \times \, \varvec{\varTheta } + \varvec{\varTheta }^T \!\times \times \, \varvec{\varUpsilon }_\varTheta + \varvec{\varUpsilon }_\varTheta ^T \!\times \times \, \varvec{\varTheta }. \end{aligned}$$

(204)

Taking into account the identity ${\textbf{A}}^T \!\times \times \, {\textbf{B}} = {\textbf{B}}^T \!\times \times \, {\textbf{A}}$, which is valid for arbitrary tensors ${\textbf{A}}$ and ${\textbf{B}}$, we rewrite Eq. (204) as

$$\begin{aligned} \frac{d }{d t} \left( \frac{1}{2}\,\varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta } \right) = \varvec{\varTheta }^T \!\times \times \, (\nabla \varvec{\omega }) - \varvec{\varTheta }^T \!\times \times \, (\varvec{\varTheta } \times \varvec{\omega }) + \varvec{\varTheta }^T \!\times \times \, \varvec{\varUpsilon }_\varTheta . \end{aligned}$$

(205)

The first term on the right-hand side of Eq. (205) can be transformed as follows:

$$\begin{aligned} \varvec{\varTheta }^T \!\times \times \, (\nabla \varvec{\omega }) = - \nabla \times (\varvec{\varTheta } \times \varvec{\omega }) + (\nabla \times \varvec{\varTheta }) \times \varvec{\omega }. \end{aligned}$$

The second term on the right-hand side of Eq. (205) can be rewritten as

$$\begin{aligned} \varvec{\varTheta }^T \!\times \times \, (\varvec{\varTheta } \times \varvec{\omega }) = \frac{1}{2} \left( \varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta }\right) \times \varvec{\omega }. \end{aligned}$$

In view of these identities Eq. (205) takes the form

$$\begin{aligned} \frac{d }{d t} \left( \frac{1}{2}\,\varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta } \right) = - \nabla \times (\varvec{\varTheta } \times \varvec{\omega }) + \left( \nabla \times \varvec{\varTheta } - \frac{1}{2}\, \varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta } \right) \times \varvec{\omega } + \varvec{\varTheta }^T \!\times \times \, \varvec{\varUpsilon }_\varTheta . \end{aligned}$$

(206)

If the source term $\varvec{\varUpsilon }_\varTheta$ has the form of Eq. (52), then

$$\begin{aligned} \varvec{\varTheta }^T \!\times \times \, \varvec{\varUpsilon }_\varTheta = - \frac{1}{3}\, \varUpsilon _\varTheta \left( \varvec{\varTheta } - \textrm{tr}\, \varvec{\varTheta }\, {\textbf{E}}\right) ^T - \frac{1}{2}\,\varvec{\varUpsilon }_\varPsi \cdot \bigl (\varvec{\varTheta } \times {\textbf{E}} - {\textbf{E}} \varvec{\varTheta }_\times \bigr ). \end{aligned}$$

(207)

Let us perform the following transformation:

$$\begin{aligned} \begin{array}{c} \varvec{\varUpsilon }_\varPsi \cdot \bigl (\varvec{\varTheta } \times {\textbf{E}} - {\textbf{E}}\, \varvec{\varTheta }_\times \bigr ) = {\textbf{E}} \times \left( \varvec{\varTheta }^T \cdot \varvec{\varUpsilon }_\varPsi \right) - \varvec{\varUpsilon }_\varPsi \varvec{\varTheta }_\times = {\textbf{E}} \times (\varvec{\varTheta } \cdot \varvec{\varUpsilon }_\varPsi ) + {\textbf{E}} \times (\varvec{\varTheta }_\times \times \varvec{\varUpsilon }_\varPsi ) - \varvec{\varUpsilon }_\varPsi \varvec{\varTheta }_\times \\ = {\textbf{E}} \times \varvec{\varTheta } \cdot \varvec{\varUpsilon }_\varPsi + \varvec{\varUpsilon }_\varPsi \varvec{\varTheta }_\times - \varvec{\varTheta }_\times \varvec{\varUpsilon }_\varPsi - \varvec{\varUpsilon }_\varPsi \varvec{\varTheta }_\times = \left( {\textbf{E}} \times \varvec{\varTheta } - \varvec{\varTheta }_\times {\textbf{E}} \right) \cdot \varvec{\varUpsilon }_\varPsi . \end{array} \end{aligned}$$

Then Eq. (207) takes the form

$$\begin{aligned} \varvec{\varTheta }^T \!\times \times \, \varvec{\varUpsilon }_\varTheta = - \frac{1}{3}\, \varUpsilon _\varTheta \left( \varvec{\varTheta } - \textrm{tr}\, \varvec{\varTheta }\, {\textbf{E}}\right) ^T - \frac{1}{2}\left( {\textbf{E}} \times \varvec{\varTheta } - \varvec{\varTheta }_\times {\textbf{E}} \right) \cdot \varvec{\varUpsilon }_\varPsi . \end{aligned}$$

(208)

Inserting Eq. (208) in Eq. (206), we arrive at Eq. (65).

Now, we take the curl of Eq. (59). As a result, we have

$$\begin{aligned} \frac{d }{d t} (\nabla \times \varvec{\varTheta }) = - \nabla \times (\varvec{\varTheta } \times \varvec{\omega }) + \nabla \times \varvec{\varUpsilon }_\varTheta . \end{aligned}$$

(209)

Calculating the difference of Eq. (209) and Eq. (206) yields

$$\begin{aligned} \frac{d }{d t} \left[ \nabla \times \varvec{\varTheta } - \frac{1}{2}\, \varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta } \right] + \left[ \nabla \times \varvec{\varTheta } - \frac{1}{2}\, \varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta } \right] \times \varvec{\omega } = \nabla \times \varvec{\varUpsilon }_\varTheta - \varvec{\varTheta }^T \!\times \times \, \varvec{\varUpsilon }_\varTheta .\qquad \qquad \qquad \end{aligned}$$

(210)

If the source term $\varvec{\varUpsilon }_\varTheta$ is given by Eq. (52), then

$$\begin{aligned} \nabla \times \varvec{\varUpsilon }_\varTheta = \frac{1}{3}\, (\nabla \varUpsilon _\varTheta ) \times {\textbf{E}} + \frac{1}{2}\, \bigl ((\nabla \cdot \varvec{\varUpsilon }_\varPsi ) {\textbf{E}} - \nabla \varvec{\varUpsilon }_\varPsi \bigr )^T. \end{aligned}$$

(211)

Inserting Eqs. (208), (211) in Eq. (210), we arrive at Eq. (68).

Now, we turn to equations that were used in our previous models, see [60, 63], and we show how these equations can be obtained from the above equations. If we suppose that $\varUpsilon _\varTheta = 0$ and $\varvec{\varUpsilon }_\varPsi = 0$, then Eq. (68) takes the form

$$\begin{aligned} \frac{d }{d t} \left[ \nabla \times \varvec{\varTheta } - \frac{1}{2}\, \varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta } \right] + \left[ \nabla \times \varvec{\varTheta } - \frac{1}{2}\, \varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta } \right] \times \varvec{\omega } = 0. \end{aligned}$$

(212)

Using the notation

$$\begin{aligned} {\textbf{X}} = \nabla \times \varvec{\varTheta } - \frac{1}{2}\, \varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta }, \end{aligned}$$

(213)

we can rewrite Eq. (212) as

$$\begin{aligned} \frac{d {\textbf{X}}}{d t} + {\textbf{X}} \times \varvec{\omega } = 0. \end{aligned}$$

(214)

Let us represent ${\textbf{X}}$ as ${\textbf{X}} = {\textbf{Y}} \cdot {\textbf{P}}^T$, where ${\textbf{P}}$ is the rotation tensor corresponding to the angular velocity vector $\varvec{\omega }$, i.e., ${\displaystyle \frac{d {\textbf{P}}}{d t} = \varvec{\omega } \times {\textbf{P}}}$. Then Eq. (214) takes the form

$$\begin{aligned} \frac{d {\textbf{Y}}}{d t} \cdot {\textbf{P}}^T = 0. \end{aligned}$$

(215)

If $\left. {\textbf{X}}\right| _{t=0} = 0$, and hence, $\left. {\textbf{Y}}\right| _{t=0} = 0$, from Eq. (215) follows that ${\textbf{Y}} = 0$, and hence, ${\textbf{X}} = 0$. Thus, solving Eq. (212) with the initial condition ${\displaystyle \left. \left( \nabla \times \varvec{\varTheta } - \frac{1}{2}\, \varvec{\varTheta }^T \!\times \times \, \varvec{\varTheta }\right) \right| _{t=0} = 0}$, we obtain Eq. (14).

G Algebraic relations between physical quantities

Relations between the charge densities ${\mathscr {Q}}$, $\varvec{{\mathscr {Q}}}_m$ and $\varvec{{\mathscr {Q}}}_g$, between the inductions $\varvec{{\mathscr {D}}}$, $\varvec{\mathscr {D}}_m$, $\varvec{{\mathscr {D}}}_g$ and the entropy per unit volume $\varTheta _a$ have the form

$$\begin{aligned} {\mathscr {Q}} = \textrm{tr}\,\varvec{{\mathscr {Q}}}_g, \quad \varvec{{\mathscr {Q}}}_m = \left( \varvec{\mathscr {Q}}_g\right) _\times , \quad \varTheta _a = - \frac{a}{\chi }\,\frac{\textrm{tr}\,\varvec{{\mathscr {D}}}_m}{2}, \quad \varvec{{\mathscr {D}}} = \left( \varvec{{\mathscr {D}}}_m\right) _\times , \quad \varvec{{\mathscr {D}}} = \varvec{{\mathscr {D}}}_g \!\cdot \cdot \, {\textbf{E}}, \quad \varvec{{\mathscr {D}}}_m^T = \varvec{\mathscr {D}}_g \cdot \times \, {\textbf{E}}.\nonumber \\ \end{aligned}$$

(216)

Expressions for the electric charge density ${\mathscr {Q}}$ and the magnetic charge density vector $\varvec{{\mathscr {Q}}}_m$ in terms of the entropy and electromagnetic induction tensor $\varvec{{\mathscr {D}}}_m$, the electric induction vector $\varvec{{\mathscr {D}}}$ and the entropy per unit volume $\varTheta _a$ are

$$\begin{aligned} {\mathscr {Q}} = - \frac{1}{4 \chi }\,\varvec{{\mathscr {D}}}_m^T \!\cdot \cdot \, \varvec{{\mathscr {D}}}_m + \frac{\chi }{2 a^2}\,\varTheta _a^2, \quad \varvec{{\mathscr {Q}}}_m = \frac{1}{2 \chi }\,\varvec{\mathscr {D}}_m \cdot \varvec{{\mathscr {D}}} + \frac{1}{2 a}\, \varTheta _a \varvec{{\mathscr {D}}}. \end{aligned}$$

(217)

Two expressions for the generalized charge density tensor $\varvec{{\mathscr {Q}}}_g$ in terms of the entropy and electromagnetic induction tensor $\varvec{{\mathscr {D}}}_m$ and the entropy per unit volume $\varTheta _a$ have the form

$$\begin{aligned} \begin{array}{c} {\displaystyle \varvec{{\mathscr {Q}}}_g = \frac{1}{2 \chi }\,\varvec{{\mathscr {D}}}_m^T \times \times \, \varvec{{\mathscr {D}}}_m - \frac{1}{a}\, \varTheta _a \varvec{{\mathscr {D}}}_m^T - \frac{\chi }{a^2}\,\varTheta _a^2\, {\textbf{E}},} \\ {\displaystyle \varvec{{\mathscr {Q}}}_g = \frac{1}{2 \chi }\,\left( \varvec{{\mathscr {D}}}_m^T \cdot \varvec{\mathscr {D}}_m^T - \frac{1}{2} (\varvec{{\mathscr {D}}}_m \!\cdot \cdot \, \varvec{{\mathscr {D}}}_m) {\textbf{E}}\right) + \frac{1}{2 a}\, \varTheta _a \varvec{{\mathscr {D}}}_m^T + \frac{\chi }{2 a^2}\,\varTheta _a^2\, {\textbf{E}}.} \end{array} \end{aligned}$$

(218)

Relations between the current densities have the form

$$\begin{aligned} \varvec{{\mathscr {J}}}_I = \left( \varvec{{\mathscr {J}}}_m\right) _\times , \quad \varvec{{\mathscr {J}}}_I = \varvec{{\mathscr {J}}}_g \!\cdot \cdot \, {\textbf{E}}, \quad \varvec{{\mathscr {J}}}_m^T = \varvec{{\mathscr {J}}}_g \cdot \times \, {\textbf{E}}. \end{aligned}$$

(219)

Two expressions for the electromagnetic current density tensor $\varvec{{\mathscr {J}}}_m$ are

$$\begin{aligned} \varvec{{\mathscr {J}}}_m = - \frac{1}{\chi } \left[ \varvec{{\mathscr {H}}} \times \left( \varvec{{\mathscr {D}}}_m + \frac{\chi }{a}\,\varTheta _a {\textbf{E}}\right) + \varvec{\mathscr {D}} \varvec{{\mathscr {H}}}\, \right] ^T, \quad \varvec{{\mathscr {J}}}_m = \frac{1}{\chi } \left[ \left( \varvec{{\mathscr {D}}}_m + \frac{\chi }{a}\,\varTheta _a {\textbf{E}}\right) \times \varvec{{\mathscr {H}}} - (\varvec{{\mathscr {H}}} \cdot \varvec{{\mathscr {D}}}) {\textbf{E}}\, \right] .\nonumber \\ \end{aligned}$$

(220)

Two additional relations containing the electromagnetic current density tensor are written as

$$\begin{aligned} \varvec{{\mathscr {J}}}_m = - \frac{1}{\chi }\left( \varvec{{\mathscr {H}}} \cdot \varvec{{\mathscr {D}}}_g + \varvec{{\mathscr {D}}} \varvec{{\mathscr {H}}}\right) ^T, \quad \textrm{tr}\,\varvec{{\mathscr {J}}}_m = - \frac{2}{\chi }\, \varvec{{\mathscr {D}}} \cdot \varvec{{\mathscr {H}}}. \end{aligned}$$

(221)

Expressions for the internal current density vector $\varvec{{\mathscr {J}}}_I$ and the internal voltage density vector $\varvec{{\mathscr {V}}}_I$ have the form

$$\begin{aligned} \varvec{{\mathscr {J}}}_I = \frac{1}{\chi }\,\varvec{{\mathscr {H}}} \cdot \varvec{\mathscr {D}}_m, \quad \varvec{{\mathscr {V}}}_I = - \frac{1}{\chi }\left( \varvec{{\mathscr {E}}} \cdot \varvec{{\mathscr {D}}}_m + \frac{a}{\chi }\, T_a \varvec{{\mathscr {D}}}\right) . \end{aligned}$$

(222)

Three additional relations containing the magnetic field vector are

$$\begin{aligned} \varvec{{\mathscr {H}}}_m = \varvec{{\mathscr {H}}} {\textbf{E}} - {\textbf{E}} \varvec{{\mathscr {H}}}, \quad \varvec{{\mathscr {J}}}_g = \frac{1}{\chi }\,\varvec{\mathscr {D}}_g \times \varvec{{\mathscr {H}}}, \quad {\textbf{h}}_\varTheta = - \frac{a}{\chi }\,\varvec{{\mathscr {H}}}. \end{aligned}$$

(223)

We remind the reader that $\varvec{{\mathscr {H}}}_m$ is the magnetic flux tensor, $\varvec{{\mathscr {D}}}_g$ is the generalized induction tensor, $\varvec{{\mathscr {J}}}_g$ is the generalized current density tensor, and ${\textbf{h}}_\varTheta$ is the entropy flux vector.