Pressurized axisymmetric membrane deforming into a prescribed shape. (1st August 2020)

• Record Type: Journal Article
• Authors: Jirásek, Milan; Šmejkal, Filip; Horák, Martin
• Is Part Of: International journal of solids and structures. Volume 198(2020)
• Page Start: 1
• Page End: 16
• Publication Date: 2020-08-01
• Subjects: Axisymmetric membrane -- Variable thickness -- Prescribed shape -- Finite differences -- Saint Venant-Kirchhoff model -- Green-Lagrange strain -- Second Piola-Kirchoff stress; Mechanics, Applied -- Periodicals; Structural analysis (Engineering) -- Periodicals; Elastic solids -- Periodicals; Mécanique appliquée -- Périodiques; Constructions, Théorie des -- Périodiques; Solides élastiques -- Périodiques; Elastic solids; Mechanics, Applied; Structural analysis (Engineering); Periodicals; 624.18
• DOI: 10.1016/j.ijsolstr.2020.04.021
• Languages: English
• ISSNs: 0020-7683

Abstract

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.