An asymptotically optimal model for isotropic heterogeneous linearly elastic plates

Ferdinando Auricchio; Carlo Lovadina; Alexandre L. Madureira

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 38, Issue: 5, page 877-897
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Auricchio, Ferdinando, Lovadina, Carlo, and Madureira, Alexandre L.. "An asymptotically optimal model for isotropic heterogeneous linearly elastic plates." ESAIM: Mathematical Modelling and Numerical Analysis 38.5 (2010): 877-897. <http://eudml.org/doc/194244>.

@article{Auricchio2010,
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. },
author = {Auricchio, Ferdinando, Lovadina, Carlo, Madureira, Alexandre L.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Reissner; Mindlin; plate; heterogeneous plates; asymptotic analysis.},
language = {eng},
month = {3},
number = {5},
pages = {877-897},
publisher = {EDP Sciences},
title = {An asymptotically optimal model for isotropic heterogeneous linearly elastic plates},
url = {http://eudml.org/doc/194244},
volume = {38},
year = {2010},
}

TY - JOUR
AU - Auricchio, Ferdinando
AU - Lovadina, Carlo
AU - Madureira, Alexandre L.
TI - An asymptotically optimal model for isotropic heterogeneous linearly elastic plates
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 38
IS - 5
SP - 877
EP - 897
AB - 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.
LA - eng
KW - Reissner; Mindlin; plate; heterogeneous plates; asymptotic analysis.
UR - http://eudml.org/doc/194244
ER -

References

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  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).  Zbl0958.74033
  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.  Zbl0846.73027
  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. Elasticity67 (2002) 171–185.  Zbl1089.74595
  4. F. Auricchio and E. Sacco, Partial-mixed formulation and refined models for the analysis of composite laminates within FSDT. Composite Structures46 (1999) 103–113.  
  5. D. Caillerie, Thin elastic and periodic plates. Math. Meth. Appl. Sci.6 (1984) 159–191.  Zbl0543.73073
  6. P.G. Ciarlet, Mathematical Elasticity, volume II: Theory of Plates, North-Holland Publishing Co., Amsterdam. Stud. Math. Appl.27 (1997).  Zbl0888.73001
  7. P.G. Ciarlet and Ph. Destuynder, A justification of the two dimensional linear plate model. J. Mécanique18 (1979) 315–344.  
  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.  Zbl0856.73029
  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.  Zbl0369.73052
  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.  Zbl0985.74037
  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.  Zbl0897.73033
  13. J. Sanchez Hubert and E. Sanchez Palencia, Introduction aux méthodes asymptotiques et à l'homogénéisation, Masson, Paris (1992).  

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