Analytical solutions for vibration of Bi-directional functionally graded nonlocal nanobeams

Abstract

This article provides analytical solutions for the vibration of bidirectional functionally graded (BDFG) nanobeams. The material characteristics of BDFG nanobeams vary along the axial and the thickness directions. The nonlocal elasticity theory of Eringen is followed to model the small-scale effects. The nanobeam's vibrational behavior is formulated by Euler-Bernoulli and Timoshenko beam theories. Hamilton's principle is used to derive the governing equations of motion. Using the Laplace transform, analytical solutions to the governing equations are established, and sets of closed-form frequency equations for four boundary conditions of nonlocal nanobeams are presented. The effects of material constants in both the axial and the thickness directions are investigated. The accuracy of the presented frequency equations is demonstrated by comparisons with a few numerical results available in the literature. It is discovered that when the material parameter along the axis increases, clamped nanobeams have a higher natural frequency, but simply supported, cantilevered, and propped cantilevered nanobeams have a lower. Nonetheless, a rising material parameter along the thickness direction exhibited identical decreasing patterns in natural frequency for all boundary conditions.