In this chapter we give a simple account of the theory of isotropic nonlinear elastic membranes. Firstly we look at both two-dimensional and three-dimensional theories and highlight some of the differences. A number of examples are then used to illustrate the application of various aspects of the theory. These include basic finite deformations, bifurcation problems, wrinkling, cavitation and existence problems.

Introduction

The aim of this chapter is to give a simple basic account of the theory of isotropic hyperelastic membranes and to illustrate the application of the theory through a number of examples. We do not aim to supply an exhaustive list of all relevant references, but, conversely, we give only a few selected references which should nevertheless provide a suitable starting point for a literature search.

The basic equations of motion can be formulated in two distinct ways; either by starting from the three-dimensional theory as outlined in Chapter 1 of this volume and then making assumptions and approximations appropriate to a very thin sheet; or from first principles by forming a theory of twodimensional sheets. The former approach leads to what might be called the three-dimensional theory and can be found in Green and Adkins (1970), for example. A clear derivation of the two-dimensional theory can be found in the paper of Steigmann (1990). Since there are two different theories attempting to model the same physical entities it is natural to compare and contrast these two theories.