Modeling of electrodynamic processes by means of mechanical analogies

Abstract

This study continues the line of earlier research in mechanical models of electrodynamic processes suggested in previous works. The basic steps we take to construct these models are: to formulate equations of a special type Cosserat continuum, and then to suggest analogies between quantities characterizing the stress–strain state of the continuum and quantities characterizing electrodynamic processes. In addition to the previously introduced mechanical analogies of the electric field vector and the magnetic induction vector, in this paper we provide mechanical analogies of the magnetic field vector, the electric induction vector, the electric current density and the electric charge density. In the framework of the suggested model, we obtain a set of differential equations that coincide with the first Maxwell's equation (the one with the displacement current term), the Gauss law for electric field and the charge conservation law. As a debatable question, we discuss the possibility of modifying the Maxwell–Faraday equation and the Gauss law for magnetic field.

Modeling of physical fields by means of the Cosserat continuum

“…These consequences are important for us because below we use them for physical interpretation of the theory. As shown in Ivanova [19], taking the vector invariant of Equation () yields $$$$\begin{equation} \nabla \times \bm{\omega } = \bm{\omega } \cdot \bigl (\bm{\Theta } - {\rm tr}\,\bm{\Theta }\, \mathbf {E} \bigr ) + \frac{d \bm{\Theta }_{\times }}{d t}. \end{equation}$$$$If we rewrite term $$\nabla \times \bm{\omega }$$ as $$\nabla \cdot (\mathbf {E} \times \bm{\omega })$$ and interpret tensor $$\mathbf {E} \times \bm{\omega }$$ as a flux tensor, then we can treat Equation () as a balance equation for vector $$\bm{\Theta }_{\times }$$.…”

“…The only difference is the rank of the tensor quantities. As shown in Ivanova [19], from Equations () and (), it follows that $$$$\begin{equation} \frac{d }{d t} {\left(\frac{1}{2}\,\bm{\Theta }^T \!\times \times \, \bm{\Theta }\right)} = - \nabla \times (\bm{\Theta } \times \bm{\omega }). \end{equation}$$$$Taking the trace of Equation (), we arrive at the conservation law for the second principal scalar invariant of tensor Θ : $$$$\begin{equation} \frac{d}{d t} {\left[\frac{1}{2} {\left(\bigl ({\rm tr}\,\bm{\Theta }\bigr )^2 - \bm{\Theta } \cdot \cdot \, \bm{\Theta } \right)}\right]} = - \nabla \cdot \Bigl [\,\bm{\omega } \cdot \bigl (\bm{\Theta } - {\rm tr}\,\bm{\Theta }\, \mathbf {E} \bigr ) \Bigr ].$$…”

“…Now, we consider some consequences of Equations () and (). As shown in Ivanova [19], taking the vector invariant of Equation (), we obtain $$$$\begin{equation} \nabla \times \bm{\Omega } = - \bm{\Omega } \cdot \bigl ({\bm{\Theta }}_e - {\rm tr}\,{\bm{\Theta }}_e\, \mathbf {E} \bigr ) + \frac{d \bigl ({\bm{\Theta }}_e\bigr )_{\times }}{d t}. \end{equation}$$$$It is easy to see that Equation () is analogous to Equation () in the first theory.…”

“…In fact, the only difference is the minus signs on the right‐hand sides of Equations () and (). As shown in Ivanova [19], from Equations () and (), it follows that $$$$\begin{equation} \frac{d }{d t} {\left(\frac{1}{2}\,\bm{\Theta }_e^T \!\times \times \, \bm{\Theta }_e\right)} = - \nabla \times {\left(\bm{\Theta }_e \times \bm{\Omega }\right)}. \end{equation}$$$$This equation has exactly the same form as Equation () in the first theory.…”

“…Then second‐rank tensor $$\bm{\Theta } \times \bm{\omega }$$ can be treated as a source term. We also note that the strain tensor Θ satisfies the strain compatibility equation, see Ivanova [19]: $$$$\begin{equation} \nabla \times \bm{\Theta } = \frac{1}{2}\,\bm{\Theta }^T \!\times \times \, \bm{\Theta }. \end{equation}$$$$Next, we turn to some consequences of Equations () and ().…”