Slumping instabilities of elastic membranes holding liquids and gases

Abstract

This paper treats the plane-strain free-boundary problem describing the equilibrium of a closed cylindrical non-linearly elastic membrane that contains a heavy liquid and a weightless compressible gas and that sits upon a rigid horizontal plane. The material points of the membrane in contact with the rigid plane, the liquid, and the gas are unknowns of the problem. We carefully formulate the geometrically exact equations, and then show that solutions must have attractive regularity, symmetry, and similarity properties, which are not intuitively obvious. There is an interesting interplay between the material behavior and the live pressure loadings generated by gases and heavy liquids. The properties of the equilibrium states depend crucially on the material behaviour of the membrane. In particular, for certain materials the number of equilibrium states changes abruptly as certain parameters describing the liquid and gas change. Thus the system can suffer instabilities of snapping type (with their attendant hysteresis effects), which we term slumping instabilities.