Asymptotic derivation of nonlocal beam models from two-dimensional nonlocal elasticity

The paper is focused on the possible justification of nonlocal beam models (at the macroscopic scale) from an asymptotic derivation based on nonlocal two-dimensional elasticity (at the material scale). The governing partial differential equations are expanded in Taylor series, through the dimensionless depth ratio of the beam. It is shown that nonlocal Bernoulli-Euler beam models can be asymptotically obtained from nonlocal two-dimensional elasticity, with a nonlocal length scale at the beam scale (macroscopic length scale) that may differ from the nonlocal length scale at the material scale. Only when the nonlocality is restricted to the axial direction are the two length scales coincident. In this specific nonlocal case, the nonlocal Bernoulli-Euler model emerged at the zeroth order of the asymptotic expansion, and the nonlocal truncated Bresse-Timoshenko model at the second order. However, in the general case, some new asymptotically-based nonlocal beam models are built which may differ from existing references nonlocal structural models. The natural frequencies for simply supported nonlocal beams are determined for each nonlocal model. The comparison shows that the models provide close results for low orders of frequencies and the difference increases with the order.