The Application of the Minimum Potential Energy Principle to Nonlinear Axisymmetric Membrane Problems

Abstract

The deformed configuration of nonlinear axisymmetric membranes is obtained by potential energy minimization via the Ritz method. The solution technique presented in this paper departs from the usual application of the potential energy principle in that the deformed configuration is minimizing instead of the displacement field. This approach is advantageous for problems in which the final configuration is easy to represent by a series of coordinate functions while the displacement field is complex and difficult to approximate. The potential energy functional minimized in the example problems is based on the Mooney strain-energy density for an incompressible material. In that no approximation is made in the large deformation extension ratios written in terms of final position, the solution may justifiably be compared to a numerical integration of the corresponding equilibrium equations. The energy solution is in excellent agreement with previously published solutions for the inflation of a circular membrane sheet. Solutions to two new example problems are presented.