Softening effect of nonlocality against the hardening effect of stretching in a capacitive micro-beam

This paper investigates the nonlinear resonant behavior of a capacitive microbeam based on the nonlocal theory of elasticity. The micro-beam is de ected by a DC voltage, where it acts as a micro-resonator by superimposing an AC voltage. Taking into account stretching eects, the Galerkin projection method is used to discretize the partial dierential equations into a set of nonlinear, ordinary dierential equations. The multiple-scales method is used to obtain an approximate analytical solution to construct the nonlinear resonant curves of the transverse vibration amplitude. Taking into account the classical and nonlocal elasticity theories, the frequency response curves are plotted for dierent values of DC voltage. Eects of mid-plane stretching on the resonant curves are also examined. The hardening behavior of the system is shown to decrease due to the presence of the nonlocality, as well as the DC voltage. However, mid-plane stretching increases the hardening eects. The results show that, despite the existence of nonlinearity in the system, this con ict eect can result in a linear frequency response curve for some values of the nonlocal parameter.