On the mechanics of Kirchhoff and Mindlin plates incorporating surface energy

In this paper, size-dependent Kirchhoff and Mindlin plate models are developed to investigate the coupling effects of nonlocal stress, strain gradient and surface energy on the dynamic response of nanoplate. The nonlocal stress and strain gradient effects are captured by nonlocal strain gradient theory, while the surface energy effects are incorporated by surface elasticity theory. The governing equations of motion and related boundary conditions are derived from Hamilton's principle. Analytical solutions for the natural frequencies of simply supported nanoplate are obtained through the Navier approach. A good agreement between the results of the present models and those available in literatures are observed. Selected examples are presented to show the influences of nonlocal parameter, material length scale parameter, length-to-thickness ratio, aspect ratio, surface energy and shear deformation on the vibration behavior of nanoplate. It is found that for nanoplate with lower length-to-thickness ratio, the nonlocal stress, strain gradient and surface energy have remarkable influences on the vibration characteristic simultaneously. While surface effects play a dominant role in the vibration behavior for nanoplate with higher length-to-thickness ratio.