Reduced Membrane Model for Liquid Crystal Polymer Networks: Asymptotics and Computation

TL;DR

This paper derives a simplified membrane model for liquid crystal polymer networks using asymptotics, and develops computational methods to solve the resulting non-convex minimization problem, demonstrating its effectiveness through numerical simulations.

Contribution

It provides a formal derivation of a 2D membrane model from 3D elasticity and introduces a finite element method with regularization for solving the non-convex problem.

Findings

Successful approximation of solutions with point defects

Effective numerical simulations capturing physical phenomena

Demonstration of the model's ability to describe complex behaviors

Abstract

We examine a reduced membrane model of liquid crystal polymer networks (LCNs) via asymptotics and computation. This model requires solving a minimization problem for a non-convex stretching energy. We show a formal asymptotic derivation of the 2D membrane model from 3D rubber elasticity. We construct approximate solutions with point defects. We design a finite element method with regularization, and propose a nonlinear gradient flow with Newton inner iteration to solve the non-convex discrete minimization problem. We present numerical simulations of practical interests to illustrate the ability of the model and our method to capture rich physical phenomena.