Observations on the general nonlocal theory applied to axially loaded nanobeams

Abstract

Static and dynamic analyses for the pre- and post-buckling behavior of an axially loaded nanobeam modeled with a general form of Eringen’s nonlocal theory are conducted and discussed. The general nonlocal theory is a modified form of Eringen’s nonlocal elasticity theory that, contrary to Eringen’s, considers separate attenuation functions to account for the long-range interactions for the two different material moduli and thus introduces an additional nonlocal parameter. The nanobeam is modeled considering Euler–Bernoulli beam theory and the von Karman geometric nonlinearity for midplane stretching in end-constrained beams. Three sets of boundary conditions are studied, namely clamped–clamped, clamped-hinged, and hinged–hinged. The governing equations are derived by virtue of the Hamilton’s principle. Introducing the general nonlocal theory into a beam system increases the order of the resultant transverse equation of motion from fourth to sixth, thus requiring two additional higher-order boundary conditions. These higher-order boundary conditions are derived using a weighted residual approach and are thus variationally consistent. Through this effort, a focus is placed on the effects that these higher-order boundary conditions have on the static and dynamic responses of the system. Specifically, the nonlocal parameters, Poisson ratio, boundary conditions, and axial load are varied. The static critical buckling loads, bifurcation diagrams, and static post-buckling configurations are examined and presented. Additionally, through a linear eigenvalue analysis, the natural frequencies and mode shapes in the pre- and post-buckling regimes are examined. It is shown that the system is very sensitive to the selected nonlocal parameters and higher-order boundary conditions for both the static and dynamic responses. It is the hope that this expanded model can be used by other researchers to accurately model nanobeams composed of materials outside the limits of applicability of Eringen’s nonlocal theory.