# Stable wrinkling in voltage and charge controlled dielectric membranes

## TL;DR

This paper analyzes the stability of dielectric elastomers under voltage and charge control, revealing fundamental energetic differences and conditions for stable wrinkling, with implications for tuning membrane instabilities.

## Contribution

It provides an energetic framework for understanding dielectric elastomer stability, highlighting differences between voltage and charge control and illustrating how boundary conditions influence instabilities.

## Key findings

- Wrinkling stability depends on tension-extension inequality in voltage control.
- Wrinkling is always stable under charge control.
- Boundary conditions can tune the type and hierarchy of instabilities.

## Abstract

Thin dielectric elastomers with compliant electrodes exhibit various types of instability under the action of electromechanical loading. Guided by the thermodynamically-based formulation of Fosdick and Tang (J. Elasticity 88, 255-297, 2007), here we provide an energetic perspective on the stability of dielectric elastomers and we highlight the fundamental energetic divide between voltage control and charge control. By using the concept of energy relaxation, we describe wrinkling for neo-Hookean ideal elastomers, and we show that in voltage control wrinkling is stable as long as the tension-extension inequality holds, whereas wrinkling is always stable in charge control. We finally illustrate some examples involving both homogeneous and inhomogeneous deformations, showing that the type and hierarchy of instabilities taking place in dielectric membranes can be tuned by suitable choices of the boundary conditions.