On equilibrium equations and their perturbations using three different variational formulations of nonlinear electroelastostatics
arXiv:1911.07090 · doi:10.1177/1081286520911073
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 , electric displacement , or electric polarization . 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 and result in simpler equations and are anticipated to be more suitable for analysing problems of stability as well as post-buckling behaviour.
Accepted for publication in Mathematics and Mechanics of Solids