Nonlinear Instability Modeling of a Nonlocal Strain Gradient Functionally Graded Capacitive Nano-Bridge in Thermal Environment

Abstract

This paper develops a theoretical model directed towards investigation of the static pull-in instability of a functionally graded (FG) electrostatically actuated nano-bridge via nonlocal strain gradient theory (NLSGT) of elasticity and Euler–Bernoulli beam theory in thermal environment. The nano-beam is under the influence of electrostatic and van der Waals (vdW) forces. In addition to the nonlinear nature of the electrostatic force, the other type of nonlinearity namely geometric nonlinearity resulting from the mid-plane stretching is considered. Material properties of FG nano-beam are assumed to vary gradually along the thickness direction according to simple power-law form. With the purpose of eliminating the coupling between the stretching and bending due to the asymmetrical material variation along the thickness, a new surface reference is introduced. The nonlinear integro-differential governing equation is derived utilizing minimum total potential energy principle, linearized by means of the step-by-step linearization method (SSLM) and solved by Galerkin-based weighted residual method. The numerical investigations are performed while the emphasis is placed on studying the effect of various parameters including: nonlocal parameter, material characteristic length scale, material gradient index, thermal effect and intermolecular force on the static pull-in instability of FG nano-beam. To establish the validity of the present formulation, a comparison is conducted with experimental and numerical results reported in previous studies.