Applicability and efficacy of Galerkin-based approximation for solving the buckling and dynamics of nanobeams with higher-order boundary conditions

Abstract

In this research effort, the general nonlocal elasticity theory is applied to an axially-loaded Euler-Bernoulli nanobeam with varying boundary conditions. In applying the general nonlocal elasticity theory, a modified version of Eringen's nonlocal elasticity theory, a sixth-order partial differential equation is obtained for the transverse governing equation of motion. Obtaining the sixth-order equation created the need for additional non-classical beam boundary conditions. These boundary conditions are determined using the weighted residual approach and are physically interpreted as higher-order force and moment terms. This study investigates the buckling and dynamic responses of nanobeams using a Galerkin-based approximation with different trial mode shapes, as it relates to the exact solution. As such, comparisons are made between using three distinct types guess functions in the approximation, namely, the exact unloaded classical mode shapes, exact unloaded Eringen nonlocal mode shapes, and exact unloaded general nonlocal mode shapes. Unloaded mode shapes refer to the mode shapes in which the applied axial load is neglected or set to zero. Additional studies are conducted focusing on direct comparison between Eringen nonlocal and general nonlocal beams for both equal and unequal nonlocal parameters. It is shown that selection of an appropriate guess function that satisfies all boundary conditions is necessary for the approximate solution. Additionally, the response of Eringen and general nonlocal nanobeams with equal nonlocal parameters and hinged-hinged boundary conditions is identical using both the exact and approximate solutions. Finally, it is demonstrated that the general nonlocal dynamic response follows a different trajectory than the Eringen nonlocal dynamic response as a function of the applied axial load. The findings of this investigation may be used by other researchers to simplify analysis procedures and to better understand the fundamental relationships existing in the size-dependent governing equations and boundary conditions for nanobeam-based systems.