On the stress and torque tensors in fluid membranes

TL;DR

This paper derives explicit formulas for stress and torque tensors in fluid membranes using the Helfrich model, extending previous results and enabling analysis of membrane forces and torques through balance equations.

Contribution

It provides simple, explicit expressions for membrane stress and torque tensors, including Gaussian rigidity and surface potential effects, facilitating practical force and torque analysis.

Findings

Derived explicit stress and torque tensors in local and fixed frames.

Extended previous covariant geometric results to include surface potential effects.

Demonstrated applications to membrane force and torque scenarios.

Abstract

We derive the membrane elastic stress and torque tensors using the standard Helfrich model and a direct variational method in which the edges of a membrane are infinitesimally translated and rotated. We give simple expressions of the stress and torque tensors both in the local tangent frame and in projection onto a fixed frame. We recover and extend the results of Capovilla and Guven [J. Phys. A, 2002, \textbf{35}, 6233], which were obtained using covariant geometry and Noether's theorem: we show that the Gaussian rigidity contributes to the torque tensor and we include the effect of a surface potential in the stress tensor. Many interesting situations may be investigated directly using force and torque balances instead of full energy minimization. As examples, we consider the force exerted at the end of a membrane tubule, membrane adhesion and domain contact conditions.