ESAIM: Mathematical Modelling and Numerical Analysis

Table of Contents - September 2004 - Volume 38 - Issue05

Research Article

An asymptotically optimal model for isotropic heterogeneous linearly elastic plates

Auricchio, Ferdinandoa1, Lovadina, Carloa2 and Madureira, Alexandre L.a3

a1 Dipartimento di Meccanica Strutturale, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy. auricchio@unipv.it.

a2 Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy. lovadina@dimat.unipv.it.

a3 Departamento de Matemática Aplicada e Computacional, Laboratório Nacional de Computação Científica, Av. Getúlio Vargas 333, Petrópolis - RJ, Brazil. alm@lncc.br.

Abstract

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 5/6 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.

(Received May 29 2003)

(Revised August 31 2004)

(Online publication October 15 2004)

Key Words:

  • Reissner;
  • Mindlin;
  • plate;
  • heterogeneous plates;
  • asymptotic analysis.

Mathematics Subject Classification:

  • 35B40;
  • 74K20
  • [1] S.M. Alessandrini, D.N. Arnold, R.S. Falk and A.L. Madureira, Derivation and Justification of Plate Models by Variational Methods. Centre de Recherches Mathematiques, CRM Proceedings and Lecture Notes (1999).
  • [2] D.N. Arnold and R.S. Falk , Asymptotic analysis of the boundary layer for the Reissner Mindlin plate model. SIAM J. Math. Anal. 27 (1996) 486–514. [OpenURL Query Data]  [CrossRef]  [MathSciNet]  [Google Scholar]
  • [3] D.N. Arnold , A.L. Madureira and S. Zhang , On the range of applicability of the Reissner-Mindlin and Kirchhoff-Love plate bending models. J. Elasticity 67 (2002) 171–185. [OpenURL Query Data]  [CrossRef]  [MathSciNet]  [Google Scholar]
  • [4] F. Auricchio and E. Sacco , Partial-mixed formulation and refined models for the analysis of composite laminates within FSDT. Composite Structures 46 (1999) 103–113. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [5] D. Caillerie , Thin elastic and periodic plates. Math. Meth. Appl. Sci. 6 (1984) 159–191. [OpenURL Query Data]  [CrossRef]  [MathSciNet]  [Google Scholar]
  • [6] P.G. Ciarlet, Mathematical Elasticity, volume II: Theory of Plates, North-Holland Publishing Co., Amsterdam. Stud. Math. Appl. 27 (1997).
  • [7] P.G. Ciarlet and Ph. Destuynder , A justification of the two dimensional linear plate model. J. Mécanique 18 (1979) 315–344. [OpenURL Query Data]  [MathSciNet]  [Google Scholar]
  • [8] M. Dauge and I. Gruais , Asymptotics of arbitrary order for a thin elastic clamped plate, I. Optimal error estimates. Asymptotic Anal. 13 (1996) 167–197. [OpenURL Query Data]  [Google Scholar]
  • [9] P. Destuynder, 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] K.H. Lo , R.M. Christensen and E.M. Wu , A high-order theory of plate deformation. J. Appl. Mech. 46 (1977) 663–676. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [11] O.V. Motygin and S.A. Nazarov , 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. [OpenURL Query Data]  [CrossRef]  [MathSciNet]  [Google Scholar]
  • [12] J.C. Paumier and A. Raoult , Asymptotic consistency of the polynomial approximation in the linearized plate theory application to the Reissner-Mindlin model. ESAIM: Proc. 2 (1997) 203-213. [OpenURL Query Data]  [CrossRef]  [Google Scholar]
  • [13] J. Sanchez Hubert and E. Sanchez Palencia, Introduction aux méthodes asymptotiques et à l'homogénéisation, Masson, Paris (1992).