Communications in Mathematical Sciences

Volume 20 (2022)

Number 6

On variational principles for polarization responses in electromechanical systems

Pages: 1541 – 1550

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n6.a3

Authors

Bob Eisenberg (Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Il., U.S.A.; and Department of Physiology and Biophysics, Rush University, Chicago Illinois, U.S.A.)

Chun Liu (Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Il., U.S.A.)

Yiwei Wang (Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Il., U.S.A.)

Abstract

Electromechanical systems can be found in many physical and biological applications, such as ion transport in membranes, batteries, and dielectric elastomers. Classical electrodynamics uses a dielectric constant to describe the polarization response of an electromechanical system to charges in an electric field. We generalize that description to include a wide variety of responses to charges in the electric field by a unified, thermodynamically consistent, variational framework. This framework is motivated and developed using the classical energetic variational approach (EnVarA). The coupling between the electrical part and the chemo-mechanical parts of the system is described either by Lagrange multipliers or various energy relaxations. The classical polarization and its dielectrics and dielectric constants appear as outputs of this analysis. The Maxwell equations then become universal conservation laws of charge and current, conjoined to an electromechanical description of the polarization of materials.

Keywords

electromechanical coupling, electrodynamics, energetic variation approach

2010 Mathematics Subject Classification

70H30, 78A02, 78A30

Received 17 September 2021

Received revised 9 January 2022

Accepted 10 January 2022

Published 14 September 2022