Nonlinear-electrostatic analysis of micro-actuated beams based on couple stress and surface elasticity theories

Abstract

In this paper, a size-dependent electrostatic model for cantilever micro-actuated beams is investigated considering the microstructure and surface energy effects. The modified couple stress theory is used to capture the microstructure effects while surface effects are incorporated into the model based on the Gurtin and Murdoch surface elasticity model. The electrostatic energy considering the fringing field effect forces the beam to a self-excited nonlinear beam. The governing nonlinear ordinary differential equation (ODE) of micro-actuated beams is derived, in which additional stiffnesses are added to incorporate surface energy and microstructure effects. Two solutions are proposed for the governing equation: linear exact solution and nonlinear numerical solution. At first, a linearization scheme is suggested to simplify the ODE to obtain an exact analytical solution. Then, a numerical technique, based on a finite difference method, is proposed to solve the general nonlinear ODE. The exact analytical solution of the linear ODE is used as an initial guess to numerically solve the extracted set of equations using Newton's method. The present model is verified by comparing its estimations with the available numerical, analytical and experimental results. Finally, a parametric study is provided to show effects of the couple stress and surface energy on the electrostatic behavior of micro-beams. Moreover, a comparison between the two proposed solution schemes is provided which allows defining the limit of applicability of each one of the proposed solutions.