Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

3D Derivations of Static Plate Theories

  • A. LebéeEmail author
  • S. Brisard
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_132-1



  • Thin plate model: A model where the only kinematic d.o.f. is the transverse deflection. It neglects the shear energy.

  • Thick plate model: A model including also two in-plane rotation d.o.f. and including shear deflection.


Plates are three-dimensional structures with a small dimension compared to the other two dimensions. Numerous approaches were suggested in order to replace the three-dimensional problem by a two-dimensional problem while guaranteeing the accuracy of the reconstructed three-dimensional fields. Turning the 3D problem into a 2D plate model is known as dimensional reduction.

The approaches for deriving a plate model from 3D elasticity may be separated in two main categories: axiomatic and asymptotic approaches. Axiomatic approaches start with ad hoc assumptions on the 3D field representation of the...

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  1. Alessandrini SM, Arnold DN, Falk RS, Madureira AL (1999) Derivation and justification of plate models by variational methods. In: Fortin M (ed) Plates and shells, vol 21. American Mathematical Society, Providence, pp 1–21CrossRefGoogle Scholar
  2. Babuška I, Li L (1992) The-p-version of the finite-element method in the plate modelling problem. Commun Appl Numer Methods 8(1):17–26MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bollé L (1947) Contribution au problème linéaire de flexion d’une plaque élastique. Bulletin technique de la Suisse romande 73(21):281–285Google Scholar
  4. Braess D, Sauter S, Schwab C (2010) On the justification of plate models. J Elast 103(1):53–71MathSciNetCrossRefzbMATHGoogle Scholar
  5. Ciarlet PG (1997) Mathematical elasticity – volume II: theory of plates. Elsevier Science Bv, AmsterdamzbMATHGoogle Scholar
  6. Ciarlet PG, Destuynder P (1979) Justification Of the 2-dimensional linear plate model. Journal de Mecanique 18(2):315–344MathSciNetzbMATHGoogle Scholar
  7. Hencky H (1947) Über die Berücksichtigung der Schubverzerrung in ebenen Platten. Ingenieur- Archiv 16(1):72–76CrossRefzbMATHGoogle Scholar
  8. Kirchhoff G (1850) Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal für die reine und angewandte Mathematik (Crelles Journal) 1850(40):51–88CrossRefGoogle Scholar
  9. Lebée A, Sab K (2017a) On the generalization of Reissner plate theory to laminated plates, part I: theory. J Elast 126(1):39–66MathSciNetCrossRefzbMATHGoogle Scholar
  10. Lebée A, Sab K (2017b) On the generalization of Reissner plate theory to laminated plates, part II: comparison with the bending-gradient theory. J Elast 126(1):67–94MathSciNetCrossRefzbMATHGoogle Scholar
  11. Love AEH (1888) The small free vibrations and deformation of a thin elastic shell. Philos Trans R Soc Lond A 179:491–546CrossRefzbMATHGoogle Scholar
  12. Mindlin R (1951) Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 18:31–38zbMATHGoogle Scholar
  13. Paumier JC, Raoult A (1997) Asymptotic consistency of the polynomial approximation in the linearized plate theory. Application to the Reissner-Mindlin model. Elast Viscoelast Optim Control 2:203–214MathSciNetzbMATHGoogle Scholar
  14. Reissner E (1944) On the theory of bending of elastic plates. J Math Phys 23:184–191MathSciNetCrossRefzbMATHGoogle Scholar
  15. Yang P, Norris CH, Stavsky Y (1966) Elastic wave propagation in heterogeneous plates. Int J Solids Struct 2(4):665–684CrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  1. 1.Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTARUniversité Paris-EstMarne-la-ValléeFrance

Section editors and affiliations

  • Karam Sab
    • 1
  1. 1.Laboratoire Navier (UMR 8205), CNRS, ENPC, IFSTTARUniversité Paris-EstMarne-la-ValléeFrance