Analysis of axisymmetric necking of a circular dielectric membrane based on a one-dimensional model

TL;DR

This paper develops a one-dimensional model for axisymmetric necking in dielectric membranes under electric and mechanical loads, simplifying analysis while maintaining accuracy across different regimes, aiding in predicting breakdown and designing resilient actuators.

Contribution

A variational asymptotic 1d model for axisymmetric necking in dielectric membranes derived from 3d theory, validated against finite-element simulations, and applicable to soft materials.

Findings

The 1d model matches 3d theory in linear and weakly nonlinear regimes.

It provides accurate predictions in the fully nonlinear regime.

The model enables precise calculation of minimum membrane thickness during necking.

Abstract

To facilitate the understanding of the mechanisms underlying the electric breakdown of dielectric elastomers, we derive a one-dimensional (1d) model for axisymmetric necking in a dielectric membrane subjected to equibiaxial stretching and an electric field, starting from the three-dimensional (3d) nonlinear electroelasticity theory. Our reduction is built on the variational asymptotic method, so that the resulting 1d model is asymptotically self-consistent. The 1d model offers an easier and more efficient way to analyze axisymmetric necking in a dielectric membrane in the linear, weakly nonlinear and fully nonlinear regimes. It delivers results identical to the 3d theory in the linear and weakly nonlinear regimes, and near-identical results in the fully nonlinear regime due to its asymptotic self-consistency. We demonstrate the straightforward implementation of this 1d model by solving it using the Rayleigh--Ritz method and validate it by comparison with finite-element simulations. The 1d model enables a precise calculation of the minimum thickness that a dielectric membrane can reach when necking instability occurs and quantitative assessment of the effects of imperfections so that the integrity of a dielectric elastomer actuator can be evaluated with respect to electric breakdown. The developed methodology is not problem-specific and can also be applied to analyze similar phenomena in other soft materials subjected to any fields (e.g., the axisymmetric necking of stretched plastic membranes).