On Axisymmetrical Deformations of Nonlinear Membranes

Abstract

The mechanics problem concerning large axisymmetric deformations of nonlinear membranes is reformulated in terms of a system of three first-order ordinary differential equations with explicit derivatives. With a set of proper boundary conditions, arrangements are made to change the boundary-value problem into the form of an initial value problem such that the solution can be obtained by standard numerical methods for integrating ordinary differential equations. The system of equations derived applies to the class of all axisymmetric deformations of membranes with a general elastic stress-strain relation. Three examples are given on inflating of a flat membrane, longitudinal stretching of a tube, and flattening of a semispherical cap. In the examples, the Mooney model are assumed to describe the material behavior of the membranes. The solution on the flat membrane serves to compare with an existing one in literature. The solutions on the tube and the cap are new.