Variational modelling of nematic elastomer foundations

TL;DR

This paper derives a simplified two-dimensional model for thin nematic elastomer membranes on elastic foundations, capturing complex microstructure and pre-strain effects through variational analysis.

Contribution

It provides the first rigorous derivation of a membrane model for nematic elastomers with active foundations using $ ext{Gamma}$-convergence, incorporating both Frank and De Gennes theories.

Findings

The limit energy functional is a linear membrane model with nematic active foundation.

Microstructure formation can suppress shear energy in the system.

The model captures pre-strain effects transmitted from the foundation to the membrane.

Abstract

We compute the $\Gamma$-limit of energy functionals describing mechanical systems composed of a thin nematic liquid crystal elastomer sustaining a homogeneous and isotropic elastic membrane. We work in the regime of infinitesimal displacements and model the orientation of the liquid crystal according to the order tensor theories of both Frank and De Gennes. We describe the asymptotic regime by analysing a family of functionals parametrised by the thickness of the membranes and the relative ratio of the elastic constants, establishing that, in the limit, the system is represented by a two-dimensional integral functional interpreted as a linear membrane on top of a nematic active foundation involving an effective De Gennes optic tensor which allows for low order states. The latter can suppress shear energy by formation of microstructure as well as act as a pre-strain transmitted by the foundation to the overlying film.