Size dependency in vibration analysis of nano plates; one problem, different answers

In addition to the classical (local) case, several non-classical constitutive equations consisting first order strain gradient (five versions), second order strain gradient (four versions), differential form of non-local and implicit gradient (IG) were employed to model the vibrational behavior of plates. Employing these models, vibration governing differential equations were developed. In addition to the classical material characteristics, a non-classical property is interfered into the equations. To solve the PDEs under two different boundary conditions, Navier approach was employed to obtain parametric answers for the plate natural frequencies. Moreover, Galerkin approach was employed to solve the PDEs numerically. The later approach was also applied to obtain approximated parametric answers. The results obtained via the introduced theories in the current research and the relevant ones reported in the literature were compared. According to the obtained parametric answers, the roles of different parameters on the natural frequencies were explained. In the part of numerical studies, influence of different parameters consisting classical and non-classical properties, boundary conditions and the plate size on the natural frequencies were studied. (C) 2016 Elsevier Masson SAS. All rights reserved.