The size-dependent static deflection, pull-in instability and resonant frequency of a circular microplate under capillary force have been studied using modified couple stress elasticity theory. Size-dependency is a phenomenon in which the normalized quantities that classical elasticity theory predicts to be independent of the structure size, such as normalized deflection or normalized frequency, vary significantly as the structure size changes. This phenomenon has been observed in micro-scale structures such as micro-electro-mechanical-systems (MEMS). Since classical elasticity theory is unable to predict the size-dependency, non-classical elasticity theories such as modified couple stress theory have been developed recently.
In this paper, modified couple stress theory is used for the first time to develop the governing equation and boundary conditions of circular microplates when subjected to capillary force. Consideration of capillary force is important since it is has a significant role in the mechanical behavior and stability of micro-scale structures in the presence of a liquid bridge. We investigated the static deflection and pull-in instability of microplates using the Galerkin method to assess the effect of size-dependency for static deflection. We observed that, as the ratio of the microplate thickness to length scale parameter (an additional material property suggested in modified couple stress theory to capture the size-dependency) decreases, the normalized deflection of the microplate also decreases. We further observed that the difference between the normalized deflection predicted by classical elasticity theory and the one evaluated using modified couple stress theory is significant when thickness of the microplate is small, but diminishes as thickness increases. Furthermore, we defined a dimensionless number called the dimensionless capillary tension (DCT) as a function of the mechanical, geometrical and size-dependent properties of the microplate as well as the characteristics of the liquid bridge such as the contact angle and the interfacial tension. We showed that for DCT values greater than a threshold evaluated in this paper, pull-in instability happens and the microplate collapses to the substrate. Moreover, we evaluated the size-dependent resonant frequency of the microplate under capillary force as a function of the DCT and obtained the result that the frequency decreases as DCT increases. In addition, our investigation of size-dependency revealed that as the ratio of the microplate thickness to length scale parameter increases, the frequency decreases in a way that for large values of the ratio, it asymptotically approaches the value predicted by classical elasticity theory.
Boundary Layers on Rough Compressor Blades
