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.

人们认识到纳米级结构表现出强烈的表面效应。本文研究了考虑表面应力的圆形纳米板的轴对称弯曲和自由振动。使用 Gurtin-Murdoch 表面弹性理论和众所周知的一阶剪切变形板理论，推导出控制圆形纳米板轴对称弯曲和自由振动的单个四阶偏微分方程。分析了表面材料特性对挠度和固有频率的影响。对于承受中心载荷集中力和均匀分布载荷的圆形纳米板，确定了不同边界条件下横向挠度及其最大值的显式表达式。获得了自由轴对称自由振动的频率方程，简单支撑和夹紧的圆形纳米板。给出了固有频率和模态振型。只有去除表面效应，Mindlin 板的经典结果才能从现在恢复。表面特性对静态挠度和自然频率的影响以图形方式呈现。