Inflation mechanics of a membrane reflector supported by an inflated toroidal rim Soham Roychowdhury * , Anirvan DasGupta Department of Mechanical Engineering, and Center for Theoretical Studies, Indian Institute of Technology Kharagpur, 721302, India article info Article history: Received 30 January 2017 Received in revised form 22 August 2017 Accepted 23 August 2017 Available online 30 August 2017 Keywords: Inflatable structure Tunable structure Inflatable reflector Parabolic reflector Wrinkling Pull-in instability abstract A stack of two identical flat circular membranes, bonded along the periphery can be inflated to form a reflector. To maintain the desired shape of the reflector surface, a supporting outer rim is required. An inflatable membrane reflector supported by an outer inflated toroidal rim structure is considered. Both the reflector and the toroidal rim are considered to be geometrically flat in the uninflated state. Initially, the circular membranes are pre-stretched and joined with the inner equator of the inflated torus, causing an inward radial force on the toroidal rim. The inflated shapes of the circular membranes (reflectors) under uniform pressure are obtained by an iterative solution scheme. The inflation problem is solved to determine the inflated shapes and the possible occurrence of wrinkling instability of the reflector or pull- in instability of the toroidal rim. The shape of the reflector is found to be close to a paraboloid whose focal length depends on the level of inflation and pre-stretch. Lower inflation pressure of the inflatable reflector is found to result in a better parabolic approximation of the reflecting surface. © 2017 Elsevier Masson SAS. All rights reserved. 1. Introduction Inflatable structures have gained a lot of attention for applica- tions in tunable devices (Carpi et al., 2011; Blum et al., 2011; Xiao et al., 2013) and space structures (Jenkins, 2001, 2006). Two cir- cular membranes bonded at the periphery and inflated can be used to construct inflatable antenna/reflectors, inflatable solar concen- trators and tunable lenses. For such inflatable devices, a supporting rim structure is required to maintain the shape of the inflatable reflector/antenna surface. An inflated toroidal membrane structure can be used as the outer rim of the inflatable reflector. In the pre- sent study, the mechanics of such complex inflatable structures is analyzed, where the interaction between the reflector and the supporting toroidal rim is studied. The proposed inflatable struc- ture is flat (both intrinsically and extrinsically) in its uninflated configuration, which is advantageous from the manufacturing point of view. The pressurization limits are determined to avoid the wrinkling of the reflecting surface or pull-in of the toroidal rim, and the effective zones within which the reflecting surfaces can be approximated as paraboloids are determined. Membranes are preferable in space and terrestrial applications as they are pliable, light in weight and cost competitive. Inflatable space structures have high reliability of deployment and can be packed and stored compactly. The use of inflatable structures in space applications are discussed by Freeland et al. (1998), Cassapakis and Thomas (1995) and Belvin (2004). Remotely deployable precision antennas (Hedgepeth, 1989) and technology for highly accurate inflatable reflectors (Thomas and Veal, 1984) have been proposed. L'Garde has developed a new concept for power antenna, which utilizes an inflatable reflector to concur- rently concentrate solar energy for space electrical power genera- tion, while acting as a large aperture high gain antenna (Lichodziejewski and Cassapakis, 1999). Later, further in- vestigations were carried out to implement the concept of large, lightweight, deployable space reflectors using mesh reflector technology, inflatable membrane reflector technology and Shape Memory Polymer reflector technology (Im et al., 2007). Nonlinear structural analysis of an inflatable parabolic antenna was carried out both experimentally and computationally (using Finite Element Method) by Sreekantamurthy and Gaspar (Sreekantamurthy et al., 2007; Gaspar et al., 2006). For all proposed inflatable reflector/antenna technologies, a supporting structure that provides stability and stiffness to the reflector/antenna is essential. We propose to use an inflated toroidal rim as supporting structure, which must be capable of carrying the inward pulling force caused by the pre-stretching and * Corresponding author. E-mail address: soham28feb@yahoo.in (S. Roychowdhury). Contents lists available at ScienceDirect European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol http://dx.doi.org/10.1016/j.euromechsol.2017.08.012 0997-7538/© 2017 Elsevier Masson SAS. All rights reserved. European Journal of Mechanics A/Solids 67 (2018) 34e44