Reinforcement of a thin plate by a thin layer

Abstract

International audienceWe study the bending of a thin plate, stiffened with a thin elastic layer, of thickness delta. We describe the complete construction of an asymptotic expansion with respect to delta of the solution of the Kirchhoff-Love model and give optimal estimates for the remainder. We identify approximate boundary conditions, which take into account the effect of the stiffener at various orders. Thanks to the tools of multi-scale analysis, we give optimal estimates for the error between the approximate problems and the original one. We deal with a layer of constant stiffness, as well as with a stiffness in delta(-1

Modeling of the nonlinear vibrations of a stiffened moderately thick plate

“…On cherche alors à remplacer la couche mince par des conditions aux limites qui rendent compte de son effet. Cette idée a fait l'objet de plusieurs études (voir [1,2,[4][5][6][7]). Dans cette note, on se propose de modéliser le comportement d'une plaque renforcée par un raidisseur (couche mince très rigide) sur une partie de son bord.…”

“…Dans cette note, on se propose de modéliser le comportement d'une plaque renforcée par un raidisseur (couche mince très rigide) sur une partie de son bord. Dans [5], [6] et [7], des conditions aux limites approchées ont été justifiées pour des modèles bidimensionnels provenant de la théorie classique des plaques minces. Dans ce qui suit, on considère un modèle non linéaire qui relève de la théorie des plaques modérément épaisses et pour laquelle les effets de rotation et du cisaillement transverse ne peuvent être négligés.…”

Multi-scale asymptotic expansion for a singular problem of a free plate with thin stiffener

“…In a previous work (see [30]), we have addressed the problem of a plate surrounded by a thin stiffener and derived approximate boundary conditions. The technique we have used is based on a construction of an asymptotic expansion of the solution of the problem in powers of the thickness δ of the layer.…”

“…We first try to start with the algorithm we have already used for the case of a plate surrounded by a thin layer (see [30]) and seek an expansion of the form w δ + = n 0 δ n w n + and W δ − = n 0 δ n W n − . The algorithm of construction of the functions w n + and W n − consists in inserting the above ansatz into the boundary value problem (6) and identifying the terms of same power of δ.…”

Modelling of the Effect of a Thin Stiffener on the Boundary of a Nonlinear Thermoelastic Plate

“…[1,4,5]). It has been also investigated by several authors problems of reinforcement by thin layers in structural mechanics, (see [9,10,11,13]). …”