Decoupling the nonlocal elasticity equations for three dimensional vibration analysis of nano-plates

In this paper, a three dimensional vibration analysis of nano-plates is studied by decoupling the field equations of Eringen theory. Considering the small scale effect, the three dimensional equations of nonlocal elasticity are obtained. At first, three decoupled equations in terms of displacement components and three decoupled equations in terms of rotation components are obtained. In order to find the solution for a nano-plate based on the presented formulation, one of the three equations in terms of displacement components and corresponding rotation equation should be solved independently. Using some relations, the other two displacement components can be obtained in terms of the mentioned displacement and rotation component. A Navier-type method for finding the exact three dimensional solution of a nano-plate is presented using the Fourier series technique. Exact natural frequencies of nano-plates are presented and compared with the results of nonlocal first order and third order shear deformation theories.