Free Vibration of Bi-Directional Functionally Graded Nanobeams Resting on Winkler–Pasternak Foundations

This study investigates the free vibration characteristics of a simply supported bi-directional functionally graded nanobeam using the Euler-Bernoulli beam theory. The aim is to understand how various factors influence the vibration behavior of these types of nanobeam.

The material properties of the nanobeam are assumed to vary exponentially in the axial direction and follow a power law in the thickness direction. To incorporate small-scale effects, the nonlocal strain gradient theory is utilized. The nanobeam is positioned on a Winkler-Pasternak elastic foundation. Hamilton’s principle is employed to derive the governing equations, considering the physical neutral surface, and the Rayleigh-Ritz method is used to obtain the frequency parameters.

The accuracy of the obtained results is confirmed through convergence studies and comparison with existing literature in specific cases. The study also analyzes the influence of various factors such as material in-homogeneity constant, material parameter, nonlocal parameter, material length scale parameter, Winkler elastic constant, and Pasternak elastic constant on the frequency parameters. The findings can contribute to the design and optimization of nanoscale devices and structures that utilize bi-directional functionally graded materials.