Modeling of thermal and electrical conductivities by means of a viscoelastic Cosserat continuum

Modeling of physical fields by means of the Cosserat continuum

“…In order to explain the processes occurring in the electromagnet within the framework of our theory, it is necessary to add the conduction current $$\bm{\mathcal J}_c$$ to the second equation in Equation (). Now, we do so by referring to Ivanova [45], where we introduce the conduction current and obtain the differential equations that describe electromagnetic processes in conductors. Thus, we replace Equation () by the equations $$$$\begin{equation} \nabla \cdot \bm{\mathcal B} = 0, \qquad \nabla \times \bm{\mathcal H} = \bm{\mathcal J} + \bm{\mathcal J}_c, \qquad \bm{\mathcal J} = \frac{1}{\chi }\, \bm{\mathcal H} \cdot {\bm{\mathcal D}}_m, \qquad \nabla \cdot {\bm{\mathcal D}}_m^T = {\bm{\mathcal Q}}_m, \qquad \bm{\mathcal B} = \mu _0 \Bigl [\mathbf {E} + \mathbf {f}({\bm{\mathcal D}}_m) \cdot \mathbf {f}^T({\bm{\mathcal D}}_m)\Bigr ] \cdot \bm{\mathcal H}.$$…”

“…In order to explain the processes occurring in the electromagnet within the framework of our theory, it is necessary to add the conduction current 𝓙 𝑐 to the second equation in Equation ( 87). Now, we do so by referring to Ivanova [45], where we introduce the conduction current and obtain the differential equations that describe electromagnetic processes in conductors. Thus, we replace Equation (87) by the equations…”

“…We emphasize that vectors $$\bm{\mathcal J}$$ and $$\bm{\mathcal V}$$ determined by Equations () and () should not be confused with the conduction current density proportional to the electric field vector and the conduction voltage density proportional to the magnetic induction vector. The aforementioned conduction current density and conduction voltage density are introduced in Ivanova [45], where we consider a model taking into account dissipative processes. In the present paper, we deal with the conservative model, where the conduction processes are ignored.…”

“…We are also convinced that differential equations relating these quantities to each other describe real physical processes and phenomena. Inserting Equation (45) into Equations ( 18), (21), and ( 22) or inserting Equation ( 46) into Equations ( 30), (33), and (34), we obtain…”

“…where 𝐽 0 = 𝜒 2 𝜇 0 ∕𝜌 and 𝐟 * is a tensor function. In view of Equations ( 42), (45), and ( 46), from Equation (77), it follows…”