Three-Dimensional Static Analysis of Nanoplates and Graphene Sheets by Using Eringen's Nonlocal Elasticity Theory and the Perturbation Method

A three-dimensional (3D) asymptotic theory is reformulated for the static analysis of simply-supported, isotropic and orthotropic single-layered nanoplates and graphene sheets (GSs), in which Eringen's nonlocal elasticity theory is used to capture the small length scale effect on the static behaviors of these. The perturbation method is used to expand the 3D nonlocal elasticity problems as a series of two-dimensional (2D) nonlocal plate problems, the governing equations of which for various order problems retain the same differential operators as those of the nonlocal classical plate theory (CST), although with different nonhomogeneous terms. Expanding the primary field variables of each order as the double Fourier series functions in the in-plane directions, we can obtain the Navier solutions of the leading-order problem, and the higher-order modifications can then be determined in a hierarchic and consistent manner. Some benchmark solutions for the static analysis of isotropic and orthotropic nanoplates and GSs subjected to sinusoidally and uniformly distributed loads are given to demonstrate the performance of the 3D nonlocal asymptotic theory.