Analytical buckling analysis of bidirectional functionally graded nonlocal nanobeams

Abstract

This research paper aims to present closed-form analytical equations that predict the buckling behavior of bidirectional functionally graded (BDFG) nanobeams characterized by varying the material properties along axial and thickness directions. The small-scale effects inherent in nanobeams are captured by Eringen’s nonlocal elasticity theory, and the displacement field of the nanobeam is assumed by the Euler–Bernoulli beam theory. The governing equations of motion are derived by applying Hamilton’s principle and variational formulation. Analytical solutions to the governing equations are obtained by using the Mellin transform. Analytical expressions are derived as stability criteria for four distinct boundary conditions of BDFG nonlocal nanobeams. A comprehensive investigation is conducted into the effects of material constants in axial and thickness directions. To validate the proposed analytical method, critical buckling loads obtained from the presented stability criteria equations are compared with pertinent results in existing literature computed through various numerical schemes. The study reveals that variations in the material’s thickness parameter have a consistent stiffening impact on the critical buckling load across all boundary conditions and buckling modes, with an optimal magnitude closer to unity for optimal buckling stiffness. Additionally, changes in the material’s axial parameter enhance the nanobeam’s buckling stiffness, exhibiting varying increase rates across boundary conditions and buckling modes.