Nonlinear Dynamic Behavior of Porous and Imperfect Bernoulli-Euler Functionally Graded Nanobeams Resting on Winkler Elastic Foundation

Nonlinear free vibrations of functionally graded porous Bernoulli–Euler nano-beams resting on an elastic foundation through a stress-driven nonlocal elasticity model are studied taking into account von Kármán type nonlinearity and initial geometric imperfection. By using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural flexural frequency is then established using the Hamiltonian approach to nonlinear oscillators. Several comparisons with existing models in the literature are performed to validate the accuracy and reliability of the proposed approach. Finally, a numerical investigation is developed in order to analyze the effects of the gradient index coefficient, porosity volume fraction, initial geometric imperfection, and the Winkler elastic foundation coefficient, on the nonlinear flexural vibrations of metal–ceramic FG porous Bernoulli–Euler nano-beams.