Analysis of Nonlinear Electrostatic Membranes John Edmiston and David Steigmann Department of Mechanical Engineering University of California at Berkeley, USA E-Mail: steigman@newton.berkeley.edu Abstract. We give a concise treatment of the interaction of a nonlinear elastic membrane with an electrostatic field. The focus is on the generation of reduced dimension model equations for contin- uum electrodynamics based on the methodology used by Steigmann (2007) in the purely mechanical context. We formulate the back- ground theory for reduced dimension models pertaining to contin- uum electrostatics and implement the resulting equations numeri- cally for an example problem of practical interest. 1 Introduction In this chapter we describe a systematic approach to the generation of re- duced dimension model equations which govern equilibrium for a nonlinear elastic membrane acted on by an electrostatic field. The primary purpose of this work is two-fold – to contribute to the strong foundation laid by Kovetz (2000) for work in continuum electrodynamics by adopting conventions used there, and to apply the technique of Steigmann (2007) for generating re- duced dimension equations pertaining to equilibrium electro-elastostatics problems. To implement the theory we present, the system we study in some detail is that of an elastomeric dielectric membrane with deformable electrodes fixed to opposing lateral surfaces. The similarly treated magnetostatic case has been previously considered by Steigmann (2004). For the academic, the combination of finite deformation and electric field interaction presents a challenging system to analyze, one for which many fundamental ideas from continuummechanics are required. From a practical standpoint, the combination of an elastomeric dielectric material along with a coexisting system of charged electrodes is an active area of research and development, R. W. Ogden et al. (eds.), Mechanics and Electrodynamics of Magneto- and Electro-elastic Materials, © CISM, Udine 2011