On equilibrium equations and their perturbations using three different variational formulations of nonlinear electroelastostatics

TL;DR

This paper derives equilibrium equations for nonlinear electroelastostatics using three variational formulations involving electric field, displacement, or polarization, highlighting their differences and implications for stability analysis.

Contribution

It introduces three variational formulations for nonlinear electroelastostatics and analyzes their equivalence, differences, and suitability for stability and bifurcation studies.

Findings

Maxwell stress naturally emerges from variational principles.

Formulations based on electric field and displacement are simpler for stability analysis.

Certain bifurcation terms are difficult to obtain via ordinary perturbation methods.

Abstract

We derive the equations of nonlinear electroelastostatics using three different variational formulations involving the deformation function and an independent field variable representing the electric character - considering either one of the electric field $\mathbb{E}$, electric displacement $\mathbb{D}$, or electric polarization $\mathbb{P}$. The first variation of the energy functional results in the set of Euler-Lagrange partial differential equations which are the equilibrium equations, boundary conditions, { and certain constitutive equations} for the electroelastic system. The partial differential equations for obtaining the bifurcation point have been also found using the second variation based bilinear functional. We show that the well-known Maxwell stress in vacuum is a natural outcome of the derivation of equations from the variational principles and does not depend on the formulation used. As a result of careful analysis it is found that there are certain terms in the bifurcation equation which appear difficult to obtain by an ordinary perturbation based analysis of the Euler-Lagrange equation. From a practical viewpoint, the formulations based on $\mathbb{E}$ and $\mathbb{D}$ result in simpler equations and are anticipated to be more suitable for analysing problems of stability as well as post-buckling behaviour.