Analytical Solutions for Bending of Nanoscaled Bars Based on Eringen’s Nonlocal Differential Law

Abstract

The one-dimensional nanoscaled bar is commonly seen in current nanodevices, and it plays a significant role in nanoelectromechanical system (NEMS) and related nanotechnology. Motivated by this, the paper is concerned with the bending and stability of a nanoscaled bar especially a flexible bar subjected to vertical concentrated, vertical linearly distributed, and horizontal concentrated forces simultaneously. The theoretical model is developed by employing Eringen’s nonlocal differential law. Hence, the nonlocal differential constitutions in terms of stress and bending moment at a nanoscale are applied to the classical equilibrium formulation in order to construct the nonlocal constitutive model and then determine the analytical solutions for bending of such a nanoscale bar. Subsequently, the effect of a nonlocal scale on the midpoint deflections and internal forces is shown numerically, and the internal and external characteristic length scales are also examined. It is revealed that the capacity of resisting compression decreases with increasing the nonlocal effect since the internal characteristic scale cannot be neglected by comparing with the external characteristic scale. There exists a threshold value for bending stiffness. Both the midpoint deflection and bending moment vary monotonously when the material bending stiffness exceeds the threshold value, while the mechanical quantities fluctuate when the bending stiffness is less than its threshold value, namely, for a flexible nanoscaled bar. Accordingly, two different kinds of nonlocal predictions and the related disputes are resolved in the context of Eringen’s nonlocal differential law. The work is expected to be useful for the design, application, and optimization of nanoscaled bars sustaining bending with linear vertical and horizontal forces.