Bending and vibration of nanoplates: Symmetry lowering and surface elastic effects

Abstract

In this study, we investigate the bending and vibration of nanoplates under the coupled effects of surface elasticity, residual surface stress, symmetry lowering, and additional elastic constants. A significant size dependence is observed for the maximum deflections and natural frequencies at nanoscale. Increasing nanoplate thickness reduces surface effects and restores symmetry, causing deflections and frequencies to converge to the conventional Kirchhoff plate limit. Si 〈100〉 oriented nanoplates with various surface reconstructions were adopted as examples. In all cases except the (2 × 1)-type reconstruction on both surfaces, the maximum deflection not only decreases in magnitude but also reverses direction. The resultant positive residual stresses, resulting from specific combinations of surface reconstruction on the upper and lower surfaces, can increase the nanoplate's deflection and vibration frequency, while negative values have the opposite effect. The resultant negative residual stresses, arising from surface reconstruction on the upper and lower surfaces, are a more critical factor influencing the bending characteristics than surface elasticity and symmetry lowering, whereas the reverse holds for resultant positive residual stresses. A decrease in the maximum deflection is associated with both a reduction in resultant residual stresses and an increase in bending stiffness. Due to the reduced influence of surface effects and symmetry lowering on higher-order vibration modes, the normalized natural frequency of the nanoplates tends to a stable value in higher-order modes.