Effects of local surface residual stresses on the near resonance vibrations of nano-beams

The effects of surface residual stresses on nano-beams including mid-plane stretching under near resonance vibrations, are studied. The nonlinear vibration equations are separated into two complementary parts: static, which includes the surface residual moments and yields a residual deflection, and dynamic, for the beam vibrations associated with the residual deflection via geometrical nonlinearity. The dynamic response is expressed by a shape mode expansion with time dependent coefficients, governed by a set of coupled ordinary differential equations. An approximated solution is extracted analytically by a combination of the asymptotic straight-forward expansion and multiple scale method. Results exhibit fine correspondence with finite element simulations. It is found that the non-uniformities in the surface residual stresses change the resonance frequency of the beam, shifts its amplitude-response curve and reduce its phase. Applications for nano-sensors are demonstrated and optimization possibilities are discussed.