Mechanics of nematic membranes: Euler-Lagrange equations, Noether charges, stress, torque and boundary conditions of the surface Frank nematic field

TL;DR

This paper develops a comprehensive variational framework for understanding the mechanics of nematic membranes, deriving key equations, stress tensors, and boundary conditions, and exploring implications of nematic order on membrane behavior.

Contribution

It introduces a generalized elasticity theory for nematic membranes, deriving Euler-Lagrange equations, Noether charges, and boundary conditions from first principles.

Findings

Derived explicit forms of nematic stress and torque tensors.

Established boundary conditions for free edges of nematic membranes.

Visualized effects of nematic order on revolution surfaces with axial symmetry.

Abstract

The mechanics of a flexible membrane decorated with a nematic liquid-crystal texture is considered in a variational framework. The variations on the splay, twist and the bend energy of the nematics are obtained from the local deformations leading to changes in the shape membrane. The Euler-Lagrange derivatives and the Noether charges are identified from the variational equations. The nematic stress tensor is obtained as a consequence of translational invariance. Likewise, the rotational invariance implies the torque nematic tensor. The corresponding boundary conditions are obtained for free edges in the open-membrane configuration. These results constitute the basis of a generalized theory of elasticity for anisotropic nematic membranes. Some relevant consequences of the presence of nematic ordering are visualized at revolution surfaces with axial symmetry.