On boundary conditions for buckling and vibration of nonlocal beams

In this paper, we argue that the boundary conditions for Eringen's nonlocal beam theory have to take on the discrete forms similar to those for the first order central finite difference beam model for beam buckling and vibration problems. The latter finite difference beam model has been found to be analogous to the physical Hencky bar-chain and therefore these two models can be regarded as one discrete beam model. Based on the phenomenological similarities between this discrete beam model and the Eringen's nonlocal beam theory, one may calibrate the Eringen's small length scale coefficient e0 which is expected to be a constant appropriate to each material. When using the classical continuous nonlocal boundary conditions, we find that e0 is not a constant but it depends on the cases of boundary conditions (such as pinned-pinned, clamped-clamped, clamped-pinned ends). However, this conundrum may be resolved if we use the discrete boundary conditions instead. By adopting the discrete boundary conditions, analytical solutions for e0 may be obtained and the e0 values finally converge to 0.289 for buckling problems and 0.408 for free vibration problems regardless of the cases of boundary conditions.