A Nonlinear Shear Deformable Nanoplate Model Including Surface Effects for Large Amplitude Vibrations of Rectangular Nanoplates with Various Boundary Conditions

Abstract

In this paper, a geometrically nonlinear first-order shear deformable nanoplate model is developed to investigate the size-dependent geometrically nonlinear free vibrations of rectangular nanoplates considering surface stress effects. For this purpose, according to the Gurtin–Murdoch elasticity theory and Hamilton's principle, the governing equations of motion and associated boundary conditions of nanoplates are derived first. Afterwards, the set of obtained nonlinear equations is discretized using the generalized differential quadrature (GDQ) method and then solved by a numerical Galerkin scheme and pseudo arc-length continuation method. Finally, the effects of important model parameters including surface elastic modulus, residual surface stress, surface density, thickness and boundary conditions on the vibration characteristics of rectangular nanoplates are thoroughly investigated. It is found that with the increase of the thickness, nanoplates can experience different vibrational behavior depending on the type of boundary conditions.