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. •Axisymmetric static and dynamic problems of circular nanoplates with surface stresses.•A single governing equation for moderately thick nanoplates with surface stresses.•Exact solution of bending of circular plates under concentrated and uniformly distributed forces.•Frequency equations for free vibration of free, simply-supported and clamped circular Mindlin nanoplates.•Effect of surface elasticity on the natural frequencies of axisymmetric free vibration.