Effective behavior of nematic elastomer membranes

TL;DR

This paper derives the effective energy density of nematic elastomer membranes, accounting for fine-scale features like wrinkling and nematic oscillations, and identifies four distinct deformation regimes.

Contribution

It provides an explicit characterization of the effective energy and stress states, including the regimes with microstructure and wrinkling effects, based on Gamma-convergence analysis.

Findings

Four deformation regimes identified with distinct microstructure and stress behaviors.

Explicit formulas for effective energy density and stress as functions of deformation.

Discovery of a shear strain regime with no shear stress and in-plane features.

Abstract

We derive the effective energy density of thin membranes of liquid crystal elastomers as the Gamma-limit of a widely used bulk model. These membranes can display fine-scale features both due to wrinkling that one expects in thin elastic membranes and due to oscillations in the nematic director that one expects in liquid crystal elastomers. We provide an explicit characterization of the effective energy density of membranes and the effective state of stress as a function of the planar deformation gradient. We also provide a characterization of the fine-scale features. We show the existence of four regimes: one where wrinkling and microstructure reduces the effective membrane energy and stress to zero, a second where wrinkling leads to uniaxial tension, a third where nematic oscillations lead to equi-biaxial tension and a fourth with no fine scale features and biaxial tension. Importantly, we find a region where one has shear strain but no shear stress and all the fine-scale features are in-plane with no wrinkling.