Surface Elasticity Effects Can Apparently Be Explained Via Their Nonconservativeness

Abstract

Recently, considerable attention has been given to investigating the surface effects on nanoscale materials. These effects can be predominant for small-scale structures, such as nanobeams, nanoplates, and nanoshells. In this paper, surface elasticity effects are considered for small scale beam structures based on the Laplace–Young equation, which results in an equivalent distributed loading term in the beam equation. We show that these effects are explained by their nonconservative nature that can be essentially modeled as a follower tensile loading for inextensible beams. The buckling and vibrations of small scale beams in the presence of surface elasticity effects is studied for various boundary conditions. It is shown that the surface elasticity effects may significantly affect the buckling and vibrations behavior of small scale beams. For clamped-free boundary conditions, we show that the buckling load is reduced compared to the one without this surface effect. This result is consistent with some recent numerical results based on surface Cauchy–Born model and with experimental results available in the literature. It appears that this result cannot be obtained if surface elasticity effects are modeled as a conservative-type loading. For other boundary conditions such as hinge–hinge and clamped–clamped boundary conditions, the results are identical to the ones already published. We explain in this paper the surprising results observed in the literature that surface elasticity effects may soften a nanostructure for some specific boundary conditions (due to the nonconservative nature of its loading application). The same conclusions are obtained for the vibrations of small scale beams with surface elasticity effects, where the natural frequency tends to decrease with surface elasticity effects for clamped-free conditions.