On the analysis of non-linear membranes

Abstract

This paper presents a non-linear analysis of hinged flat circular and spherical membranes undergoing arbitrarily large axisymmetric deformations when subjected to internal pressure. No assumptions as to the magnitude of strains and displacement gradients are made and the non-linear constitutive relation for a highly elastic incompressible solid is used. The non-linear membrane equilibrium equations are formulated in terms of the displacement components and the known geometry of the undeformed surface for the general nonorthogonal midsurface coordinate case and are simplified for the axisymmetric shell of revolution problem, a particular case of which is the flat circular membrane. Because of the type of constitutive relation used in the formulation, the inflation pressure is a highly nonlinear function of the displacements. Using finite differences and a Newton Raphson iterative scheme, a technique is developed whereby all three equilibrium configurations, associated with a particular value of the inflation pressure, are established. Results are presented graphically in a nondimensionalized form and where possible are compared with solutions obtained by others.