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.

Deflections of a rubber membrane

“…Further results were obtained by Feng and Yang (1973), Yang and Lu (1973), Feng et al (1974), and Feng and Huang (1975) for inflation and inflation-induced contact problems related to both circular and rectangular membranes. Problems related to the axisymmetric inflation of a circular membrane containing a rigid disc inclusion and the inflation of an ellipsoidal membrane were examined by Tielking and Feng (1974), using a minimum potential energy approach. Feng et al (1974) extended these studies to develop a solution to the problem of an edge-constrained square membrane subjected to uniform pressure and included a situation where a contact constraint is induced by a rigid immovable smooth obstacle interacting with the deforming membrane.…”

Review of the Energy Limiters Approach to Modeling Failure of Rubber

“…51,[54][55][56][57][58][59][60][61][62][63] The topic culminated in the recent book by Muller and Strehlow, 64 which is completely devoted to rubber balloons.…”

Thermoformability of Thermoplastic Sheets: Towards a Standard Method of Assessment

“…A variety of authors have analyzed the inflation of axisymmetric membranes [28][29][30][31][32][33]. Studies extended to include non-axisymmetric geometries or constrained contact include those by Williams [25], Wu [14], among others [34][35][36][37].…”