Effective Behavior of Nematic Elastomer Membranes

Abstract

We derive the effective energy density of thin membranes of liquid crystal elastomers as the -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.

Variational modelling of nematic elastomer foundations

“…In order to render the construction admissible, as already done e.g. in [16] and [9], we introduce the standard stereographic projection with pole z π z : S 2 → R 2 .…”

“…Mathematically, this phenomenon has been rigorously studied in [18] where the relaxation of the elastic energy of [27] has been obtained explicitly, thus explaining the occurrence of nematic microstructure in terms of minimising sequences of energy functionals lacking lower-semicontinuity with the tools of the calculus of the variations. Building upon this result, a substantial body of literature has appeared on the analytical modelling of NLCEs in both non-linear and linearised elasticity [18,12,17,8,6,5,7,1,9,25].…”

Microstructure-enabled control of wrinkling in nematic elastomer sheets

“…In this work, we systematically develop a two dimensional Koiter theory for nematic elastomer sheets by both (i) starting from an appropriate three dimensional description of these elastomers [1,11,29,30], and (ii) building off of the development of the effective or relaxed membrane theory [4]. Thus, in the course of this development, we find it natural to introduce several variants of strain energy densities modeling nematic elastomers, for which their is a well-characterized hierarchy.…”

“…To develop this theory, we build on the work of Cesana et al [4] who started from an appropriate three dimensional formulation and derived from it the relaxed membrane energy. They also provided the explicit characterization of the instabilities or oscillations (Young measures) that underly the relaxation.…”

An electro‐elastic phase‐field model for nematic liquid crystal elastomers based on Landau‐de‐Gennes theory

“…Fundamental theoretical studies on the effective elastic response of polydomain nematics with different cross-linking history have been documented in, for example, UCHIDA [59] and BIGGINS, WARNER & BHATTACHARYA [3,4]. The effective behavior of nematic elastomer membranes was recently analyzed by CESANA, PLUCINSKY & BHATTACHARYA [9].…”