Comparison of nonlocal continualization schemes for lattice beams and plates

International audienceThis paper investigates both stability and vibration of nonlocal beams or plates in the presence of compressive forces. Various nonlocal structural models have been proposed to capture the inherent scale effects of lattice-based beams or plates. These nonlocal models are either based on continualization of the difference equations of the original lattice problem (labeled as continualized nonlocal models), or developed from phenomenological nonlocal approaches such as Eringen's type nonlocality. Considered herein are several continualization schemes that lead to either fourth or sixth order spatial governing differential or partial differential equation. Even if the new continualized nonlocal plate models differ in their mathematical description, they appear to furnish very close macroscopic results as shown from asymptotic expansion arguments. The continualized nonlocal beam and plate models and the phenomenological approaches are also introduced from variational arguments. The key role of boundary conditions is shown especially for Eringen's nonlocal model that is not necessarily variationally based. For each of them, the buckling load and the natural frequencies are determined for simply supported beams and plates and compared to their counterparts obtained from the lattice model. The small length scale coefficient of the nonlocal beam or plate models is intrinsically constant and problem independent for the continualized approaches, whereas it is calibrated for the phenomenological models based on the equivalence with the reference microstructured model and consequently, depends on the load, the buckling or vibration mode or the aspect ratio. It is found that the nonlocal continualized approaches are more efficient than the nonlocal phenomenological ones. For beam problems, continualized nonlocal and phenomenological approaches such as Eringen's nonlocal theory can become the same. However, for plate problems, phenomenological approaches may differ significantly from continualized nonlocal ones; thereby offering one the opportunity to have a new class of two-dimensional nonlocal models