Static Pull-in Analysis of Electrostatically Actuated Functionally Graded Micro-Beams Based on the Modified Strain Gradient Theory

Abstract

In this paper, the static pull-in behavior of electrostatically actuated functionally graded (FG) micro-beams resting on an elastic medium is studied using the modified strain gradient (MSG) theory. To this end, the equilibrium equation along with classical and non-classical boundary conditions is obtained by considering the fringing field and elastic foundations effects within the principle of minimum total potential energy. Also, the elastic medium is composed of a shear layer (Pasternak foundation) and a linear normal layer (Winkler foundation). The governing differential equation is solved for cantilever and doubly fixed FG beams using an iterative numerical method. This method is a combination of the reduced-order technique, the fourth-order Runge–Kutta and shooting methods. The pull-in voltages of silicon beams obtained from the present model are compared with the experimental and theoretical results reported in the literature. It is seen that the MSG theory is able to reduce significantly the difference between pull-in voltages predicted by theoretical approaches based on the classical continuum theory and the experimental observations. Finally, a parametric study is carried out to analyze in detail the influences of power index, length scale parameters, coefficients of elastic foundations and boundary conditions on the pull-in behavior of FG micro-beams. Findings indicate that the size effect on the pull-in instability of FG micro-beams with both boundary conditions is significant; however, it is seen that the variation with the normalized length scale parameter of static pull-in voltages for doubly fixed beams is larger than cantilever beams. Also, it is shown that the increase of ceramic volume fraction can improve the pull-in resistance of beams. However, this influence becomes smaller with rise of power index for both cases.