I present a development of the modern theories of elastic shells, regarded as mathematical surfaces endowed with kinematical and constitutive structures deemed sufficient to represent many of the features of the response of thin shell-like bodies. The emphasis is on Cosserat theory, specialized to obtain a model of the Kirchhoff-Love type through the introduction of appropriate constraints. Noll's concept of material symmetry, adapted to surface theory by Cohen and Murdoch, is used to derive new constitutive equations for elastic surfaces having hemitropic, isotropic and unimodular symmetries. The last of these furnishes a model for fluid films with local bending resistance, which may be used to describe the response of certain fluid microstructures and biological cell membranes.

Introduction

I use the nonlinear Kirchhoff-Love theory of shells to describe the mechanics of a number of phenomena including elastic surface-substrate interactions and the equilibria of fluid-film microstructures. The Kirchhoff-Love shell may be interpreted as a one-director Cosserat surface (Naghdi 1972) with the director field constrained to coincide with the local orientation field.

The phenomenology of surfactant fluid-film microstructures interspersed in bulk fluids poses significant challenges to continuum theory. By using simple models of elastic surfaces, chemical physicists have been partially successful in describing the qualitative features of the large variety of equilibrium structures observed (Kellay et al. 1994, Gelbart et al. 1994). The basic constituent of such a surface is a polar molecule composed of hydrophilic head groups attached to hydrophobic tail groups.