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

• In addition to classical, several non-classical constitutive equations were employed to model mechanical behavior of nano plates.
• The role of constitutive equations in the vibration governing equations of the nanoplates was investigated.
• Parametric answers are obtained for the governing equations.
• Moreover, Galerkin's approach is used to solve the governing equations.
• According to the numerical results, differential non-local approach predicts softening whereas the strain gradient, couple stress and implicit gradient approaches predict stiffening behavior while non-classical constant increases.

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.