A membrane, the two-dimensional analog of a string, is a thin-walled body that can only carry internal forces tangent to its deformed shape. Thus, a membrane is either a shell so thin that it is, essentially, incapable of supporting stress couples or else it is a shell with sensible bending stiffness but which is subject to a combination of external loads and boundary/initial conditions which produce negligible bending. If we wish to distinguish between these two models, we shall refer to the former as a true membrane and to the latter as a (solutiondependent) shell-membrane. True membranes model certain biological tissues, inflatables, soap bubbles, shells subject to very high pressure and very large strains, and the like; shell-membranes model regions of shells that cannot undergo (nearly) inextensional bending and which are not too close to geometric or material discontinuities such as crowns of toroidal shells, cracks, or shell boundaries. Moreover, true membranes cannot support compressive stresses and thus may wrinkle whereas shell-membranes can resist this type of instability.

In various parts of Section V.S we looked at the membrane limit of the simplified, nonlinear, small-strain, axisymmetric Reissner equations. In Section V.T, we developed the general theory of aximembranes undergoing large strains and gave many examples and references. In this chapter, we develop the general theory of membranes, including the subtheory of wrinkling, a topic both mathematically challenging and of growing applicability to natural and manufactured objects.