Pressurized axisymmetric membrane deforming into a prescribed shape

Abstract

This paper presents a new approach to the optimal design of an axisymmetric membrane with variable thickness, which has potential applications in the development of active optical elements (liquid lenses). The governing equations are based on the Saint Venant-Kirchhoff material law, which postulates a linear relation between the Green-Lagrange strains and the second Piola-Kirchoff stresses, combined with the exact description of geometric nonlinearity, without any simplifying assumptions. It is shown that the membrane thickness can be designed such that the prestressed membrane subjected to a given uniform liquid pressure deforms into a prescribed rotationally symmetric shape, e.g., a spherical or parabolic cap. For the special but important case of a spherical cap, a closed-form solution is derived. A numerical procedure is developed for the general case, and its high accuracy and efficiency is demonstrated by examples. The sensitivity of the optimal design to material parameters and prestressing displacement is assessed.