Characterization of the Bernoulli-Navier model for a rectangular section beam as the limit of the Kirchhoff-Love model for a plate

Abstract

In this paper we compare the Kirchhoff-Love model for a linearly elastic rectangular plate Ωtε = (0, L) × (−t, t) × (−ε, ε) of thickness 2ε with the Bernoulli-Navier model for the same solid considered as a linearly elastic beam of length L and cross-section ω tε 1 = (−t, t) × (−ε, ε). We assume that the solid is clamped on both ends {0, L} × [−t, t] × [−ε, ε]. We show that the scaled version of the displacements field ζ t in the middle plane, solution of the Kirchhoff-Love model, converges strongly to the unique solution of a one-dimensional problem when the plate width parameter t tends to zero. Moreover, after re-scaling this limit, we show that, as a matter of fact, it is the solution of the Bernoulli-Navier model for the beam. This means that, under appropriate assumptions on the order of magnitude of the data, the Bernoulli-Navier displacement field is the natural approximation of the Kirchhoff-Love displacement field when the cross-section of the plate is rectangular and its width is sufficiently small and homothetic to thickness.