On the Justification of Plate Models


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.

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  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)

    MATH  Article  MathSciNet  Google 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)

    MATH  Article  MathSciNet  Google 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)

    MATH  Google Scholar 

  5. 5.

    Babuška, I., Pitkäranta, J.: The plate paradox for hard and soft simple support. SIAM J. Math. Anal. 21, 551–576 (1990)

    MATH  Article  MathSciNet  Google Scholar 

  6. 6.

    Braess, D.: Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, Cambridge (2007)

    Google 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)

    MATH  Article  MathSciNet  Google 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)

    MATH  Article  ADS  Google 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)

    MATH  MathSciNet  Google Scholar 

  11. 11.

    Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)

    MATH  Article  MathSciNet  Google 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)

    MATH  Article  MathSciNet  Google Scholar 

  13. 13.

    Grisvard, P.: Singularities in Boundary Value Problems. Masson, Paris (1992)

    Google Scholar 

  14. 14.

    Jerison, D., Kenig, C.E.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  15. 15.

    Kirchhoff, G.: Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine angew. Math. 40, 51–58 (1850)

    MATH  Article  Google Scholar 

  16. 16.

    Melenk, M., Schwab, Ch.: hp-FEM for reaction-diffusion equations, robust exponential convergence. SIAM J. Numer. Anal. 35, 1520–1557 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  17. 17.

    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn. Dover, New York (1927)

    Google Scholar 

  18. 18.

    Morgenstern, D.: Herleitung der Plattentheorie aus der dreidimensionalen Elastizitätstheorie. Arch. Ration. Mech. Anal. 4, 145–152 (1959)

    MATH  Article  MathSciNet  Google 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)

    MATH  MathSciNet  Google Scholar 

  20. 20.

    Reissner, E.: On bending of elastic plates. Q. Appl. Math. 5, 55–68 (1947)

    MATH  MathSciNet  Google 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)

  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)

    MATH  Article  Google Scholar 

  23. 23.

    Schwab, Ch.: Hierarchical models of plates—Fourier analysis and a-posteriori error estimation. Habilitation Thesis, Stuttgart, Germany (1995)

  24. 24.

    Stein, E.: Private communication

  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)

    MATH  Article  Google Scholar 

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Correspondence to Christoph Schwab.

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Braess, D., Sauter, S. & Schwab, C. On the Justification of Plate Models. J Elast 103, 53–71 (2011). https://doi.org/10.1007/s10659-010-9271-8

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

Mathematics Subject Classification (2000)

  • 74K20
  • 35Q74
  • 74B05