Shear stresses in fluid and solid membranes with bending elasticity

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 fluidlike Helfrich bending energy contributes shear stresses, while a related solidlike energy, the correct continuum limit of a Seung-Nelson discrete bending term, produces a stress tensor with a 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.