Journal of Elasticity

, Volume 103, Issue 1, pp 53–71 | Cite as

On the Justification of Plate Models

  • Dietrich Braess
  • Stefan Sauter
  • Christoph SchwabEmail author


In this paper, we will consider the modelling of problems in linear elasticity on thin plates by the models of Kirchhoff–Love and Reissner–Mindlin. A fundamental investigation for the Kirchhoff plate goes back to Morgenstern (Arch. Ration. Mech. Anal. 4:145–152, 1959) and is based on the two-energies principle of Prager and Synge. This was half a century ago.

We will derive the Kirchhoff–Love model based on Morgenstern’s ideas in a rigorous way (including the proper treatment of boundary conditions). Our derivation provides insights (a) into the relation of the (1,1,0)-model with the (1,1,2)-model which differs by a quadratic term in the ansatz for the third component of the displacement field and (b) into the rôle of the shear correction factor. A further advantage of the approach by the two-energies principle is that the extension to the Reissner–Mindlin plate model becomes very transparent and easy. Our study includes plates with reentrant corners with any interior opening angle <2π.


Plate models Justification Hypercircle inequality Prager–Synge Shear correction factor Reissner–Mindlin model 

Mathematics Subject Classification (2000)

74K20 35Q74 74B05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alessandrini, A.L., Arnold, D.N., Falk, R.S., Madureira, A.L.: Derivation and justification of plate models by variational methods. In: Fortin, M. (ed.) Plates and Shells, Quebec 1996. CRM Proceeding and Lecture Notes, vol. 21, pp. 1–20. American Mathematical Society, Providence (1999) Google Scholar
  2. 2.
    Arnold, D.N., Falk, R.S.: The boundary layer for the Reissner–Mindlin plate model. SIAM J. Math. Anal. 21, 281–312 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Arnold, D.N., Madureira, A.L., Zhang, S.: On the range of applicability of the Reissner–Mindlin and Kirchhoff–Love plate bending models. J. Elast. 67, 171–185 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Babuška, I., d’Harcourt, J.M., Schwab, Ch.: Optimal shear correction factors in hierarchical plate modelling. Math. Model. Sci. Comput. 1, 1–30 (1993) zbMATHGoogle Scholar
  5. 5.
    Babuška, I., Pitkäranta, J.: The plate paradox for hard and soft simple support. SIAM J. Math. Anal. 21, 551–576 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Braess, D.: Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, Cambridge (2007) zbMATHCrossRefGoogle Scholar
  7. 7.
    Braess, D., Ming, P., Shi, Z.: An error estimate for a plane elasticity problem and the enhanced strain method. SIAM J. Numer. Anal. 47, 4473–4491 (2010) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Braess, D., Pillwein, V., Schöberl, J.: Equilibrated residual error estimates are p-robust. Comput. Methods Appl. Mech. Eng. 198, 1189–1197 (2009) zbMATHCrossRefADSGoogle Scholar
  9. 9.
    Ciarlet, P.G.: Mathematical Elasticity, Vol. II: Theory of Plates. North-Holland, Amsterdam (1997) Google Scholar
  10. 10.
    Ciarlet, P.G., Destuynder, P.: A justification of the two-dimensional linear plate model. J. Méc. 18, 315–344 (1979) zbMATHMathSciNetGoogle Scholar
  11. 11.
    Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dolejsi, V., Feistauer, M., Schwab, Ch.: A finite volume discontinuous Galerkin scheme for nonlinear convection diffusion problems. Calcolo 39, 1–40 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Grisvard, P.: Singularities in Boundary Value Problems. Masson, Paris (1992) zbMATHGoogle Scholar
  14. 14.
    Jerison, D., Kenig, C.E.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kirchhoff, G.: Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine angew. Math. 40, 51–58 (1850) zbMATHCrossRefGoogle Scholar
  16. 16.
    Melenk, M., Schwab, Ch.: hp-FEM for reaction-diffusion equations, robust exponential convergence. SIAM J. Numer. Anal. 35, 1520–1557 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn. Dover, New York (1927) zbMATHGoogle Scholar
  18. 18.
    Morgenstern, D.: Herleitung der Plattentheorie aus der dreidimensionalen Elastizitätstheorie. Arch. Ration. Mech. Anal. 4, 145–152 (1959) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Prager, W., Synge, J.L.: Approximations in elasticity based on the concept of function spaces. Q. Appl. Math. 5, 241–269 (1947) zbMATHMathSciNetGoogle Scholar
  20. 20.
    Reissner, E.: On bending of elastic plates. Q. Appl. Math. 5, 55–68 (1947) zbMATHMathSciNetGoogle Scholar
  21. 21.
    Repin, S., Sauter, S.: Computable estimates of the modeling error related to the Kirchhoff–Love plate model. Preprint 11-2009, Institut für Mathematik, University of Zurich (to appear in Anal. Appl. 2010) Google Scholar
  22. 22.
    Rössle, A., Bischoff, M., Wendland, W., Ramm, E.: On the mathematical foundation of the (1,1,2)-plate model. Int. J. Solids Struct. 36, 2143–2168 (1999) zbMATHCrossRefGoogle Scholar
  23. 23.
    Schwab, Ch.: Hierarchical models of plates—Fourier analysis and a-posteriori error estimation. Habilitation Thesis, Stuttgart, Germany (1995) Google Scholar
  24. 24.
    Stein, E.: Private communication Google Scholar
  25. 25.
    Triebel, H.: Interpolation, Function Spaces and Differential Operators, 2nd edn. Joh. A. Barth Publ., Leipzig (1995) Google Scholar
  26. 26.
    Zhang, S.: On the accuracy of Reissner–Mindlin plate model for stress boundary conditions. M2AN 40, 269–294 (2006) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Dietrich Braess
    • 1
  • Stefan Sauter
    • 2
  • Christoph Schwab
    • 3
    Email author
  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  2. 2.Institut für MathematikUniversität ZürichZürichSwitzerland
  3. 3.ETH ZürichZurichSwitzerland

Personalised recommendations