Free vibration analysis of the macro-micro-nano plates and shells made of a material with functionally graded porosity: A closed-form solution

This article deals with introducing a closed-form solution for considering the vibration of macro-micro-nano plates and shells made of unit material with functionally graded porosity (FGP). Classic continuum theory beside modified nonlocal strain gradient theory are used at macro and micro/nano scales, respectively. The governing equations are obtained in the frame work of third order plate and shell theories. In the solution procedure, a differential operator is defined and by using it, five equations of motion are converted to the five algebraic equations. Then, by satisfying the equations of boundary conditions and achieving the nonzero solution, the frequencies are extracted.