Skip to main content
SpringerLink
Log in

Mathematical Justifications of Plate Models

  • Living reference work entry
  • First Online:
Encyclopedia of Continuum Mechanics

  • 251 Accesses

Synonyms

Formal asymptotic expansions; Lower-dimensional models

Definition

Justifying plate models consists in deriving limit bidimensional models (membrane or bending models) starting from the complete three-dimensional models considering the thickness as a small parameter supposed to vanish. The mathematical approach avoids any mechanical or geometrical simplification.

Introduction

The interest of lower dimensional models for elastic structures relies on simpler mathematical formulation and on gain of space and time in numerical computations. In order to get bidimensional models, some mechanical or geometrical assumptions are made; for example, by imposing vanishing in-plane stresses, or preventing any rotation of the normal to the cross section, one can obtain bending models for plates. In this chapter the formal asymptotic expansion method is investigated to justify rigorously (i.e., without any mechanical assumptions) the equilibrium equations of the Kirchhoff-Love elastic plate...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  • Anzellotti G, Baldo S, Percivale D (1994) Dimension reduction in variational problems, asymptotic developments in Γ-convergence and thin structures in elasticity. Asymptot Anal 9:61–100

    MathSciNet  MATH  Google Scholar 

  • Bourquin F, Ciarlet PG, Geymonat G, Raoult A (1992) Γ-convergence et analyse asymptotique des plaques minces, Comptes rendus de l’Acadžmie des sciences. Sžrie 1, Mathžmatique 315(9):1017–1024

    MATH  Google Scholar 

  • Ciarlet PG, Destuynder P (1979a) A justification of the two-dimensional linear plate model. J Mécanique 18:315–344

    MathSciNet  MATH  Google Scholar 

  • Ciarlet PG, Destuynder P (1979b) A justification of a nonlinear model in plate theory. Comput Methods Appl Mech Eng 17/18:227–258

    Article  MATH  Google Scholar 

  • Figueiredo IN, Trabucho L (1992) A Galerkin approximation for linear elastic shallow shells. Comput Mech 10(2):107–119

    Article  MathSciNet  MATH  Google Scholar 

  • Fox DD, Raoult A, Simo JC (1993) A justification of nonlinear properly invariant plate theories. Arch Ration Mech Anal 124:157–199

    Article  MathSciNet  MATH  Google Scholar 

  • Lods V, Miara B (1998) Nonlinearly elastic shell models: a formal asymptotic approach II. The flexural model. Arch Ration Mech Anal 142:355–374

    Article  MathSciNet  MATH  Google Scholar 

  • Miara B (1989) Optimal spectral approximation in linearized plate theory. Appl Anal 87:291–307

    Article  MathSciNet  MATH  Google Scholar 

  • Miara B (1994) Justification of the asymptotic analysis of elastic plates, II. The non-linear case. Asymptot Anal 9:119–134

    MathSciNet  MATH  Google Scholar 

  • Miara B, Trabucho L (1992) A Galerkin spectral approximation in linearized beam theory. ESAIM Math Model Numer Anal 26(3):425–446

    Article  MathSciNet  MATH  Google Scholar 

  • Mielke A (1995) On the justification of plate theories in linear elasticity theory using exponential decay estimates. J Elast 38(2):165–208

    Article  MathSciNet  MATH  Google Scholar 

  • Paroni R, Tomassetti G, Podio-Guidugli P (2007) A justification of the Reissner-Mindlin plate theory through variational convergence. Anal Appl 5(2):165–182

    Article  MathSciNet  MATH  Google Scholar 

  • Paumier J-C (1991) Existence and convergence of the expansion in the asymptotic theory of elastic thin plates. Math Model Numer Anal 25(3):371–391

    Article  MathSciNet  MATH  Google Scholar 

  • Schwab C (1993) The Fourier approach to the asymptotic analysis of plate models. Mathematics Research Report, pp 93–04. University of Maryland in Baltimore Country

    Google Scholar 

  • Trabucho L, Viano JM (1996) Mathematical modelling of rods. In: Handbook of numerical analysis, vol 4. North-Holland, Amsterdam. MR 1422507 — Zbl 0873.7304

    Google Scholar 

  • Veiga MF (1994) A Galerkin approximation for homogeneous anisotropic elastic beams. Appl Anal 53(1): 67–84

    Article  MathSciNet  MATH  Google Scholar 

  • Vogelius M, Babuska I (1981) On a dimensional reduction method. The optimal selection of basis functions. Math Comput 37(155):31–46

    MathSciNet  MATH  Google Scholar 

  • Xiao LM (1998) Asymptotic analysis of dynamic problems for linearly elastic shell justification of equation for dynamic membrane shell. Asymptot Anal 17:121–134

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université Paris-Est, Champs-sur-Marne, France

    Bernadette Miara

Authors
  1. Bernadette Miara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bernadette Miara .

Editor information

Editors and Affiliations

  1. Institut für Mechanik, Otto-von-Guericke-Universität Institut für Mechanik, Magdeburg, Germany

    Holm Altenbach

  2. Griffith University (Gold Coast Campus), Griffith School of Engineering Griffith University (Gold Coast Campus), Southport, Queensland, Australia

    Andreas Öchsner

Section Editor information

  1. Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTAR, Université Paris-Est, 77455, Marne-la-Vallée, France

    Karam Sab

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer-Verlag GmbH Germany

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Miara, B. (2018). Mathematical Justifications of Plate Models. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_138-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-53605-6_138-1

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53605-6

  • Online ISBN: 978-3-662-53605-6

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation