H. Salehipour, H. Nahvi, and A. R. Shahidi, Exact analytical solution for free vibration of functionally graded micro/nano plates via three-dimensional nonlocal elasticity, Accepted for publication in Physica E, (2014).
Using three-dimensional (3-D) nonlocal elasticity theory of Eringen, this paper presents closed-form solutions for in-plane and out-of-plane free vibration of simply supported functionally graded (FG) rectangular micro/nano plates. Elasticity modulus and mass density of FG material are assumed to vary exponentially through the thickness of micro/nano plate, whereas Poisson’s ratio is considered to be constant. By employing appropriate displacement fields for the in-plane and out-of-plane modes that satisfy boundary conditions of the plate, ordinary differential equations of free vibration are obtained. Boundary conditions on the lateral surfaces are imposed on the analytical solutions of the equations to yield the natural frequencies of FG micro/nano plate. The natural frequencies of FG micro/nano plate are obtained for different values of nonlocal parameter and gradient index of material properties. The results of this investigation can be used as a benchmark for the future numerical, semi-analytical and analytical studies on the free vibration of FG micro/nano plates.