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

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.This research was partially supported by the Research Centre of Mathematics of the University of Minho through the FEDER Funds “Programa Operacional Factores de Competitividade COMPETE”, and by the Portuguese Funds through FCT - “Funda¸c˜ao para a Ciˆencia e a Tecnologia”, within the Project PEst-OE/MAT/UI0013/2014, and by the project “Modelizaci´on y simulaci´on num´erica de s´olidos y fluidos en dominios con peque˜nas dimensiones. Aplicaciones en estructuras, biomec´anica y aguas someras”, MTM2012-36452-C02-01 financed by the Spanish Ministry of Econom´ıa y Competitividad with the participation of FEDER