Numerical Study of Liquid Crystal Elastomer Using Mixed Finite Element Method

TL;DR

This paper develops a mixed finite element method to simulate liquid crystal elastomers, capturing semi-soft elasticity and analyzing the mathematical properties of the numerical approach, though stripe domains were not observed.

Contribution

It introduces a finite element simulation framework for liquid crystal elastomers based on an energy functional, with analysis of existence, uniqueness, and convergence of the method.

Findings

Semi-soft elasticity observed and linked to director rotation

Mathematical properties of the numerical method established

Stripe domain phenomena not observed, possibly due to mesh coarseness

Abstract

We aimed to use finite element method to simulate the unique behaviors of liquid crystal elastomer, such as semi-soft elasticity, stripe domain instabilities etc. We started from an energy functional with the 2D Bladon-Warner-Terentjev stored energy of elastomer, the Oseen-Frank energy of liquid crystals, plus the penalty terms for the incompressibility constraint on the displacement, and the unity constraint on the director. Then we applied variational principles to get the differential equations. Next we used mixed finite element method to do the numerical simulation. The existence, uniqueness, well-posedness and convergence of the numerical methods were investigated. The semi-soft elasticity was observed, and can be related to the rotation of the directors. The stripe domain phenomenon, however, wasn't observed. This might due to the relative coarse mesh we have used.