Size-dependent free vibration analysis of electrostatically pre-deformed rectangular micro-plates based on the modified couple stress theory

Abstract

The modified couple stress theory (MCST) is a non-classical continuum theory which is capable of capturing size-dependent behavior of small structures occurring in micron and sub-micron scales. The objective of the present paper is to investigate the size-dependent free vibration characteristics of rectangular micro-plates pre-deformed by an electric field based on the MCST. To this end, a size-dependent Kirchhoff׳s plate model is considered and the equation of motion which accounts for the effect of residual and couple stress components as well as the inherent non-linearity of distributed electrostatic excitation is derived using Hamilton׳s principle. The eigenvalue equation corresponding to the free vibration of electrostatically pre-deformed rectangular micro-plates is also extracted from the equation of motion. This equation is solved numerically using the finite element method (FEM). The results are compared and validated by available analytical and semi-analytical findings for flat micro-plates as well as empirical pull-in observations in the literature and a very good agreement between them is observed. A parametric study is also conducted to investigate the effects of couple stress components as well as electrostatic attraction on both natural frequencies and mode-shapes of fully clamped and simply supported micro-plates. It is found that the size effect on natural frequencies is quite negligible for flat and electrostatically pre-deformed micro-plates when the ratio of plate thickness to the material length scale parameter is larger than 10 and 20, respectively. Furthermore, it is found that couple stress components and electrostatic attraction do not have a sizeable effect on micro-plate mode-shapes.