Linearly elastic annular and circular membranes under radial, transverse, and torsional loading. Part I: large unwrinkled axisymmetric deformations

Abstract

Three theories for determination of the equilibrium states of initially flat, linearly elastic, rotationally symmetric, taut membranes are considered: Foppl-von Karman theory, Reissner’s theory, and a new generalization of Reissner’s theory that does not restrict the strains to be small. Attention is focused on annular membranes, but circular membranes are also treated. Large deformations are allowed, and the equilibrium equations are written in terms of transverse, radial, and circumferential displacements. Problems considered include radial stretching, transverse displacement of the inner edge, an adhesive punch pull-off test on a circular blister, transverse pressure, ponding of annular and circular membranes, a vertical distributed load with a vertically sliding outer membrane edge, pull-in (snap-down, jump-to-contact) instability of a MEMS device, torsion of the inner or outer edge of a stretched membrane, and a combination of radial stretching, vertical displacement, and torsion. Results for the three theories are compared. Closed-form solutions are available in a few cases, but usually a shooting method is utilized to obtain numerical solutions for displacements, strains, and stresses. Conditions for the onset of wrinkling are determined. In the second part of this two-part study, small vibrations about equilibrium configurations are analyzed.