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

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 the electric field $\mathbb{E}$ , the electric displacement $\mathbb{D}$ or the 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 also been found using the bilinear functional based on the second variation. We show that the well-known Maxwell stress in a 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 that appear to be difficult to obtain using 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 expected to be more suitable for analysing problems of stability as well as post-buckling behaviour.