Bending analysis of embedded nanoplates based on the integral formulation of Eringen's nonlocal theory using the finite element method

Highlights: • The bending analysis of embedded nanoplates is presented based on the integral formulation of Eringen's nonlocal theory. • The governing equations are presented for both integral and differential forms of Eringen's nonlocal theory. • The formulation is presented in a general form and arbitrary kernel functions can be considered. • The finite element method is applied to solve the integral formulation of nonlocal model. - Abstract: Due to the capability of Eringen's nonlocal elasticity theory to capture the small length scale effect, it is widely used to study the mechanical behaviors of nanostructures. Previous studies have indicated that in some cases, the differential form of this theory cannot correctly predict the behavior of structure, and the integral form should be employed to avoid obtaining inconsistent results. The present study deals with the bending analysis of nanoplates resting on elastic foundation based on the integral formulation of Eringen's nonlocal theory. Since the formulation is presented in a general form, arbitrary kernel functions can be used. The first order shear deformation plate theory is considered to model the nanoplates, and the governing equations for both integral and differential forms are presented. Finally, the finite element method is applied to solve the problem. Selected results are given to investigate the effects of elastic foundation and to compare the predictions of integral nonlocal model with those of its differential nonlocal and local counterparts. It is found that by the use of proposed integral formulation of Eringen's nonlocal model, the paradox observed for the cantilever nanoplate is resolved.