Free vibration analysis of FG nanoplates embedded in elastic medium based on second-order shear deformation plate theory and nonlocal elasticity

In this paper, the second-order shear deformation plate theory is developed for a study of free vibration analysis of the functionally graded (FG) nanoplates embedded in an elastic medium based on nonlocal elasticity of Eringen. The material distributions of FG nanoplates are considered as power law, sigmoid and exponential models. The complete governing equations of FG nanoplates are derived by using the Hamilton's principle. The analytical solution for natural frequencies and corresponding mode shapes of simply supported FG nanoplates are established. Furthermore, the expressions of the elements of stiffness and mass matrices are completely manifested. The influences of the power index, exponential index, nonlocal parameter, the stiffness of an elastic medium and the environmental temperature on free vibration responses are investigated. The obtained analytical results show an excellent agreement with other available solutions. The formulation and these analytical results of the proposed method could serve as a benchmark for an evaluation of future research.