Bending and Free Vibration Analysis of Functionally Graded Nano-plate Using Trigonometric Higher-Order Plate Theory

In this paper, bending and free vibration analyses of functionally graded nano-plates are investigated using a new trigonometric higher-order plate theory. The governing equations are developed by employing Hamilton’s principle and then a Navier-type analytical solution for bending and free vibration of simply supported rectangular FG nano-plates is obtained. Furthermore, the nonlocal theory of elasticity is used to take into account the small scale effects. The mechanical properties of the functionally graded nano-plates are assumed to vary by a power law function through the thickness. In order to confirm the accuracy of the present theory, the obtained results from the present solution are compared with the existed results, and a very good agreement is achieved. Moreover, the effects of length-to-thickness ratio, aspect ratio and nonlocal parameter on the bending and free vibration solutions are investigated. The results demonstrate that the inclusion of nonlocal parameter in governing equations or increasing the power index, leads to reduction of the natural frequency and increasing of the deflection and in another word the nano-plate stiffness is reduced. Also, the impact of the nonlocal parameter and size effects is reduced by increasing the length of the nano-plate or aspect ratio. Furthermore, the present theory not only provides exact solution for the bending and free vibration of thick and moderately thick functionally graded nano-plates with minimum computational cost, but also exhibits the parabolic distribution of the shear stress through the thickness.