Geometrically nonlinear vibration analysis of sandwich nanoplates based on higher-order nonlocal strain gradient theory

In this paper, a new model for studying the effects of small-scale parameters simultaneously, on large amplitude vibrations of sandwich plates is developed using the higher-order nonlocal strain gradient theory. Considering the higher-order theories for capturing the size effects of nanostructures results in a set of nonlinear partial differential (PD) equations, including bi-nonlocal terms. By employing Hamilton's principle, the equations of motion for symmetric and anti-symmetric sandwich plates are derived based on the classical plate theory. The partial nonlinear differential equations of motion are reduced to an ordinary differential equation for transverse vibrations of nanoplates using the Galerkin's method. An analytical solution procedure is employed to obtain the closed-form frequency equation as a function of the vibration amplitude, small-scale parameters and sandwich layers elasticity, density and thickness coefficients. Numerical results are presented in order to investigate the sandwich layers coefficients on nonlinear vibrational behavior of nanoplates as same as small-scale parameters and the amplitude of vibrations. It is found that the vibration amplitude plays the main role in nonlinear vibrational behavior of nanoplates in which, nonlinear frequency and its ratio to linear frequency will be increased by increasing it. Moreover, there are non-uniform behaviors by increasing the sandwich layers coefficients and small-scale parameters. In addition, in the case of large amplitude vibrations, effects of sandwich layers' coefficients and small-scale parameters on the nonlinear frequency and its ratio to linear frequency will become more noticeable. In order to validate the present solution procedure, the results are compared with those obtained from molecular dynamics simulations, the higher-order nonlocal strain gradient theory and the higher-order shear deformation plate theory.