Axisymmetric bending and vibration of circular nanoplates with surface stresses

It is recognized that nanoscale structures exhibit strong surface effects. This article studies axisymmetric bending and free vibration of circular nanoplates with consideration of surface stresses. Using the Gurtin- Murdoch surface elasticity theory and the well-known first-order shear deformation plate theory, a single fourth-order partial differential equation governing axisymmetric bending and free vibration of circular nanoplates is derived. The effect of the surface material properties on the deflection and natural frequencies is analyzed. For a circular nanoplate subjected to a centrally-loaded concentrated force and uniformly distributed loading, explicit expressions for the transverse deflection and its maximum are determined for different boundary conditions. The frequency equations are obtained for the axisymmetric free vibration of free, simply supported, and clamped circular nanoplates. The natural frequencies and mode shapes are presented. The classical results of Mindlin plates are recovered from the present only if removing the surface effects. The effects of surface properties on the static deflection and natural frequencies are presented graphically.