Surface stress effects may induce softening: Euler–Bernoulli and Timoshenko buckling solutions

Abstract

Surface elasticity effects may be significant for small scale structures. In this paper, the effect of surface elasticity effects is investigated for the buckling of Euler–Bernoulli and Timoshenko beams. Engesser and Haringx theory of Timoshenko shear beams are studied. The 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 non-conservative nature that can be essentially modelled as a follower tensile loading for inextensible beams. As a consequence the usual paradigm that smaller is stiffer is not necessarily found for these structural problems. The buckling small scale shear beams in presence of surface elasticity effects is studied for various boundary conditions. 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. 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 non-conservative nature of its loading application). Furthermore, self-instability is theoretically noticed for small shear beams.