3 research outputs found
Asymptotic analysis of an elastic rod with rounded ends
We derive a one-dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one-dimensional image of the rod depends on the roundness exponent m is an element of(0,1). Error estimates are obtained in the case m is an element of(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m is an element of(0,1/4) but play a crucial role in the formulation of the limit problem for m >= 1/4.Peer reviewe
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