Surface Elasticity Based Nonlocal Vibration Analysis of Bidirectional Functionally Graded Tapered Nanobeam

Abstract

The present paper proposes the impact of the mutual interplay of nonuniform geometry with surface and nonlocal stresses on the vibration characteristics of bi-directional functionally graded tapered nanobeam with surface layers. The material composition of nanobeam is assumed to follow a power-law distribution along the thickness and exponential along the length. The nonuniformity in the geometry of nanobeam arises due to the linear variation of thickness along its length. The considered nanobeam is modeled as a Timoshenko nanobeam with surface layers of negligible thickness. The nonlocal and surface effects are incorporated using Eringen's nonlocal theory with Gurtin-Murdoch's surface elasticity theory. Hamilton's energy principle is employed to derive the nonlocal equations of motion with boundary conditions. The differential quadrature method is exploited to obtain the natural frequencies and the convergence of the method is demonstrated. A parametric study is introduced to investigate the influence of critical parameters such as taper parameter, surface parameter and nonlocal parameter on the vibration characteristics of bi-directionally graded nanobeam. This work explains that the nonuniformity in the geometry of nanobeam significantly influences the frequency range of tapered nanobeam with surface layers. These results will serve as a benchmark for future work on nonuniform nanostructures.