Nonlinear Free Vibration of Buckled Size-Dependent Functionally Graded Nanobeams Using Homotopy Perturbation Method

The present study aims at investigating nonlinear free vibration of thermally buckled functionally graded nanobeam. The nonlocal nonlinear Euler-Bernoulli beam theory as well as linear eigenmodes of a functionally graded nanobeam vibrating around the first buckling configuration are employed to derive a system of ordinary differential equations via the Galerkin method. Semi-analytical solutions are obtained based on both the homotopy perturbation method and the variational iteration method. Results show that the difference between nonlinear and linear frequencies increases with a rise in the maximum lateral initial deflection, small scale parameter value, and index of the power law. Investigating the effect of the ratio of length to thickness on the variance between the nonlinear and linear frequencies shows that the aspect ratio makes no difference on the classical ratio of nonlinear to linear frequencies although the difference between the nonlocal nonlinear and linear frequencies decreases with a rise in the aspect ratio. In contrast to the ratio of the first nonlinear frequency to the first linear one which will decrease if compressive axial load increases, the values of the compressive axial load which are beyond the load bearing capacity of the functionally graded nanobeam do not affect the ratio of the second nonlinear to linear frequencies.