In this paper, we derive and analyze a Reissner-Mindlin-like model for isotropic heterogeneous linearly elastic plates. The modeling procedure is based on a Hellinger-Reissner principle, which we modify to derive consistent models. Due to the material heterogeneity, the classical polynomial profiles for the plate shear stress are replaced by more sophisticated choices, that are asymptotically correct. In the homogeneous case we recover a Reissner-Mindlin model with as shear correction factor. Asymptotic expansions are used to estimate the modeling error. We remark that our derivation is not based on asymptotic arguments only. Thus, the model obtained is more sophisticated (and accurate) than simply taking the asymptotic limit of the three dimensional problem. Moreover, we do not assume periodicity of the heterogeneities.
@article{M2AN_2004__38_5_877_0,
author = {Auricchio, Ferdinando and Lovadina, Carlo and Madureira, Alexandre L.},
title = {An asymptotically optimal model for isotropic heterogeneous linearly elastic plates},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {38},
year = {2004},
pages = {877-897},
doi = {10.1051/m2an:2004042},
mrnumber = {2104433},
zbl = {1130.74403},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2004__38_5_877_0}
}
Auricchio, Ferdinando; Lovadina, Carlo; Madureira, Alexandre L. An asymptotically optimal model for isotropic heterogeneous linearly elastic plates. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) pp. 877-897. doi : 10.1051/m2an:2004042. http://gdmltest.u-ga.fr/item/M2AN_2004__38_5_877_0/
[1] ,, and, Derivation and Justification of Plate Models by Variational Methods. Centre de Recherches Mathematiques, CRM Proceedings and Lecture Notes (1999). | MR 1696513 | Zbl 0958.74033
[2] and, Asymptotic analysis of the boundary layer for the Reissner Mindlin plate model. SIAM J. Math. Anal. 27 (1996) 486-514. | Zbl 0846.73027
[3] , and, On the range of applicability of the Reissner-Mindlin and Kirchhoff-Love plate bending models. J. Elasticity 67 (2002) 171-185. | Zbl 1089.74595
[4] and, Partial-mixed formulation and refined models for the analysis of composite laminates within FSDT. Composite Structures 46 (1999) 103-113.
[5] , Thin elastic and periodic plates. Math. Meth. Appl. Sci. 6 (1984) 159-191. | Zbl 0543.73073
[6] , Mathematical Elasticity, volume II: Theory of Plates, North-Holland Publishing Co., Amsterdam. Stud. Math. Appl. 27 (1997). | MR 1477663 | Zbl 0953.74004
[7] and, A justification of the two dimensional linear plate model. J. Mécanique 18 (1979) 315-344. | Zbl 0415.73072
[8] and, Asymptotics of arbitrary order for a thin elastic clamped plate, I. Optimal error estimates. Asymptotic Anal. 13 (1996) 167-197. | Zbl 0856.73029
[9] , Sur une justification des modèles de plaques et de coques par les méthodes asymptotiques. Ph.D. thesis, Université Pierre et Marie Curie - Paris, France (1980).
[10] , and, A high-order theory of plate deformation. J. Appl. Mech. 46 (1977) 663-676. | Zbl 0369.73053
[11] and, Justification of the Kirchhoff hypotheses and error estimation for two-dimensional models of anisotropic and inhomogeneous plates, including laminated plates. IMA J. Appl. Math. 65 (2000) 1-28. | Zbl 0985.74037
[12] and, Asymptotic consistency of the polynomial approximation in the linearized plate theory application to the Reissner-Mindlin model. ESAIM: Proc. 2 (1997) 203-213. | MR 1486082 | Zbl 0897.73033
[13] and, Introduction aux méthodes asymptotiques et à l'homogénéisation, Masson, Paris (1992).