Shear stresses in fluid and solid membranes with bending elasticity

TL;DR

This paper compares models of fluid and solid membranes to understand how bending energies induce shear stresses, highlighting differences in stress responses and the implications for membrane mechanics.

Contribution

It demonstrates how bending energies in membranes generate shear stresses and clarifies the differences between fluid-like and solid-like membrane models.

Findings

Fluid-like Helfrich energy contributes shear stresses.

Solid-like energy produces purely isotropic tangential stress tensor.

Distinction between resistance to tangential flows and tangential stress support.

Abstract

Comparison of a few simple models of fluid and solid membranes illustrates how shear stresses can arise from a bending energy through a coupling between curvature and surface stresses, a feature incidental to the fluid or solid nature of the material. In particular, it is shown how a fluid-like Helfrich bending energy contributes shear stresses, while a related solid-like energy, the correct continuum limit of a Seung-Nelson discrete bending term, produces a stress tensor with purely isotropic tangential part. A distinction is noted between the resistance to tangential flows on the surface and the ability to support tangential stresses. The tangential projection of the divergence of the stress, or the pseudomomentum balance, is viewed in the light of some statements and observations in the literature. Different types of constraints, invariance properties, and shape equations are briefly discussed. The state of a patch of unidirectionally-bent material is connected with prior analyses of the pulling of membrane tethers.