Shear stresses in fluid and solid membranes with bending elasticity

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.

significant expansions and changes

References in corpus (16)

• Fluid-membrane tethers: minimal surfaces and elastic boundary layers
• Dynamics of filaments and membranes in a viscous fluid
• Stresses in lipid membranes
• Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics
• A stabilized finite element formulation for liquid shells and its application to lipid bilayers
• Arbitrary Lagrangian--Eulerian finite element method for curved and deforming surfaces. I. General theory and application to fluid interfaces
• How paper folds: bending with local constraints
• Descriptions of membrane mechanics from microscopic and effective two-dimensional perspectives
• Contrasting bending energies from bulk elastic theories
• A constrained Cosserat shell model up to order $O(h^5)$: Modelling, existence of minimizers, relations to classical shell models and scaling invariance of the bending tensor
• Strain tensor selection and the elastic theory of incompatible thin sheets
• Dilation-invariant bending of elastic plates, and broken symmetry in shells
• Force dipoles and stable local defects on fluid vesicles
• Direct Computation of Two-Phase Icosahedral Equilibria of Lipid Bilayer Vesicles
• Quadratic-stretch elasticity
• Energies for elastic plates and shells from quadratic-stretch elasticity