Finite-Element Modeling of Circular Nanoplates

Abstract

A finite-element method for static and free vibration analysis of axisymmetric circular nanoscale plates is presented based on the Gurtin-Murdoch continuum theory accounting for surface energy. Explicit solutions for stiffness and mass matrices and the load vector of a two-node plate finite element are derived by using a weighted residual formulation. The accuracy of the present finite-element model is verified by comparing with a set of analytical solutions for nanoplates. Simply supported plates show more influence of the surface energy effect compared with clamped plates. The first natural frequency is also affected, but higher natural frequencies as well as all mode shapes show minor influence of surface energy effects. The surface residual stress is found to be the key parameter that influences the deflections and natural frequencies. The current solutions clearly show the size-dependent response of nanoplates.