Modeling of orientational polarization within the framework of extended micropolar theory

Abstract

In this paper the process of polarization of transversally polarizable matter is investigated based on concepts from micropolar theory. The process is modeled as a structural change of a dielectric material. On the microscale it is assumed that it consists of rigid dipoles subjected to an external electric field, which leads to a certain degree of ordering. The ordering is limited, because it is counteracted by thermal motion, which favors stochastic orientation of the dipoles. An extended balance equation for the microinertia tensor is used to model these effects. This balance contains a production term. The constitutive equations for this term are split into two parts, one , which accounts for the orienting effect of the applied external electric field, and another one, which is used to represent chaotic thermal motion. Two relaxation times are used to characterize the impact of each term on the temporal development. In addition homogenization techniques are applied in order to determine the final state of polarization. The traditional homogenization is based on calculating the average effective length of polarized dipoles. In a non-traditional approach the inertia tensor of the rigid rods is homogenized. Both methods lead to similar results. The final states of polarization are then compared with the transient simulation. By doing so it becomes possible to link the relaxation times to the finally observed state of order, which in terms of the finally obtained polarization is a measurable quantity.

Modeling of physical fields by means of the Cosserat continuum

“…Therefore, in the spatial description, it makes no sense to calculate the inertia tensor of the elementary volume as the inertia tensor of a rigid body. A new approach to the introduction of inertia characteristics of the elementary volume is developed in [46–51]. According to this approach, tensor $$\mathbf {J}$$ is defined as $$$$\begin{equation} \mathbf {J} = \frac{1}{m N} \sum _{i=1}^{N} \hat{\mathbf {J}}_i, \qquad \hat{\mathbf {J}}_i = \mathbf {P}_i \cdot \hat{\mathbf {J}}_i^0 \cdot \mathbf {P}_i^T.$$…”

“…Therefore, in the spatial description, it makes no sense to calculate the inertia tensor of the elementary volume as the inertia tensor of a rigid body. A new approach to the introduction of inertia characteristics of the elementary volume is developed in [46][47][48][49][50][51].…”

“…In the present study, following ideas of Refs. [46][47][48][49][50][51][52], we suppose that the inertia tensor 𝐉 depends on the strain tensor 𝚯, and the inertia tensor 𝐉 𝑒 depends on the strain tensor 𝚯 𝑒 . We suggest that the following expressions are appropriate:…”

“…In the present study, following ideas of Refs. [46–52], we suppose that the inertia tensor $$\mathbf {J}$$ depends on the strain tensor Θ , and the inertia tensor $$\mathbf {J}_e$$ depends on the strain tensor $${\bm{\Theta }}_e$$. We suggest that the following expressions are appropriate: $$$$\begin{equation} \mathbf {J} = J_0 {\left[ \mathbf {E} + \mathbf {f}_*(\bm{\Theta } - {\rm tr}\, \bm{\Theta }\, \mathbf {E}) \cdot \mathbf {f}_*^T(\bm{\Theta } - {\rm tr}\, \bm{\Theta }\, \mathbf {E}) \right]}, \qquad \mathbf {J}_e = J_0 {\left[ \mathbf {E} + \mathbf {f}_*{\left({\bm{\Theta }}_e - {\rm tr}\,{\bm{\Theta }}_e\, \mathbf {E}\right)} \cdot \mathbf {f}_*^T{\left({\bm{\Theta }}_e - {\rm tr}\,{\bm{\Theta }}_e\, \mathbf {E}\right)} \right]}, \end{equation}$$$$where $$J_0 = \chi ^2 \mu _0 / \rho$$ and $$\mathbf {f}_*$$ is a tensor function.…”

Linear theory of micropolar media with internal nonlocal potential interactions

“…This significantly reduces the breadth of practical application of the relevant effects. Therefore, studies were carried out, for example, by [10] and are currently continued to eliminate this drawback (see, e.g., [11][12][13][14][15][16][17][18][19][20][21][22]).…”