The size-dependent analysis of multilayer micro-cantilever plate with piezoelectric layer incorporated voltage effect based on a modified couple stress theory

In this paper, the modified couple stress theory has been employed to analyze the size-dependent behavior of a multilayer micro-plate with piezoelectric layers in the presence of electrical voltage as a model for the Atomic Force Microscope cantilever. By using the Hamilton's principle, the equations of motion considering the material length scale parameter and the effects of voltage for the piezoelectric micro-plate have been derived based on the Kirchhoff's plate model. The equations have been solved by applying the GDQR as a semi-analytical method for different boundary conditions. The static and free vibration behaviors of the micro-cantilever are investigated to signify the size effect, piezoelectric properties and external voltage. The results indicate when the material length scale parameter is comparable with plate thickness; the size effect will be significant. By separating the effects of the induced electrical field to the voltage effects and the structural effects of the piezoelectric material, the influence of the applied voltage on vibration frequency as well as the size effects at different boundary conditions have been investigated. Also it is shown that in the piezoelectric micro-cantilever the voltage effect on the size-dependent natural frequency differs significantly for the positive and negative applied voltage.