A nonlinear bending theory for nematic LCE plates

TL;DR

This paper develops a nonlinear bending theory for nematic liquid crystal elastomer plates, deriving a 2D model from 3D elasticity, and demonstrates its application through numerical simulations of complex deformations.

Contribution

It introduces a novel nonlinear bending model for nematic LCE plates, coupling elastic and director energies, derived rigorously via Gamma limits, and provides a new numerical algorithm for analysis.

Findings

The model captures non-flat equilibrium deformations influenced by material and boundary conditions.

Numerical simulations reveal complex mechanical behaviors and rich deformation patterns.

The derived model effectively predicts the influence of nematic orientation on plate bending.

Abstract

In this paper, we study an elastic bilayer plate composed of a nematic liquid crystal elastomer in the top layer and a nonlinearly elastic material in the bottom layer. While the bottom layer is assumed to be stress-free in the flat reference configuration, the top layer features an eigenstrain that depends on the local liquid crystal orientation. As a consequence, the plate shows non-flat deformations in equilibrium with a geometry that non-trivially depends on the relative thickness and shape of the plate, material parameters, boundary conditions for the deformation, and anchorings of the liquid crystal orientation. We focus on thin plates in the bending regime and derive a two-dimensional bending model that combines a nonlinear bending energy for the deformation, with a surface Oseen-Frank energy for the director field that describes the local orientation of the liquid crystal elastomer. Both energies are nonlinearly coupled by means of a spontaneous curvature term that effectively describes the nematic-elastic coupling. We rigorously derive this model as a {\Gamma}-limit from three-dimensional, nonlinear elasticity. We also devise a new numerical algorithm to compute stationary points of the two-dimensional model. We conduct numerical experiments and present simulation results that illustrate the practical properties of the proposed scheme as well as the rich mechanical behavior of the system.