# Membrane stress and torque induced by Frank's nematic textures: A geometric perspective using surface-based constraints

## TL;DR

This paper develops a geometric framework to analyze how nematic textures in elastic membranes induce anisotropic stresses and torques, extending classical membrane elasticity models with insights from liquid crystal theory.

## Contribution

It introduces analytical expressions linking nematic director deformations to membrane stresses and torques using differential geometry, highlighting the impact of internal nematic ordering on membrane mechanics.

## Key findings

- Analytical formulas for stress and torque from nematic textures
- Visualization of forces on spherical and cylindrical membranes
- Demonstration of anisotropic stress contributions beyond classical elasticity

## Abstract

An elastic membrane with embedded nematic molecules is considered as a model of anisotropic fluid membrane with internal ordering. By considering the geometric coupling between director field and membrane curvature, the nematic texture is shown to induce anisotropic stresses additional to Canham-Helfrich elasticity. Building upon differential geometry, analytical expressions are found for the membrane stress and torque induced by splaying, twisting and bending of the nematic director as described by the Frank energy of liquid crystals. The forces induced by prototypical nematic textures are visualized on the sphere and on cylindrical surfaces.