Effect of surface stresses on pull-in instability of a nanocantilever under electrostatic and intermolecular forces

The problem of pull-in instability of an electrostatically actuated nanocantilever is investigated here by considering the effect of the residual surface stress and surface attractions. A novel approach is developed by replacing the original differential equation with an equivalent integral equation for the deflection, obtained by using the Green’s function of the nanocantilever. Moreover, the resultant lateral force is approximated by a power function of the axial coordinate containing two unknown parameters, namely the power-law exponent and the tip deflection. These two unknowns can be found from a matching procedure by requiring that the approximated distribution of the lateral force calculated at the midspan and at the free tip must coincide with the actual load distribution calculated from the deflection predicted by the governing integral equation when the approximated load distribution is considered. In this way, a system of two nonlinear algebraic equations for the two unknown parameters as functions of the applied voltage is derived. The maximum attained by the electrostatic voltage then provides the approximated values of the pull-in voltage and the pull-in deflection. The plotted results show the effects of positive and negative residual surface stress and surface attractions on the pull-in parameters. A practical application is also considered for a nanocantilever made of Silicon with crystallographic direction [100] on faces. It is observed that for a very thin Si[100] nanocantilever there exists a critical length at which the nanobeam buckles without any applied electrostatic voltage and for any gap distance between movable and fixed electrodes.