Homogenization of the nonlinear bending theory for plates

  • Stefan NeukammEmail author
  • Heiner Olbermann


We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of \(\Gamma \)-convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a “cylinder” or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions of class \(W^{2,2}\), using two-scale convergence. This is a non-trivial task, since one has to treat two-scale convergence in connection with a nonlinear differential constraint.

Mathematics Subject Classification

35B27 49J45 74E30 74Q05 



The authors would like to thank an anonymous referee for pointing out a mistake in an earlier version of this manuscript, cf. Remark 2. This work was initiated while the first author was employed at the Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany. The second author gratefully acknowledges the hospitality of the Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany.


  1. 1.
    Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55(11), 1461–1506 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Hornung, P.: Approximation of flat \(W^{2,2}\) isometric immersions by smooth ones. Arch. Ration. Mech. Anal. 199(3), 1015–1067 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Hornung, P.: Fine level set structure of flat isometric immersions. Arch. Ration. Mech. Anal. 199(3), 943–1014 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Fonseca, I., Krömer, S.: Multiple integrals under differential constraints: two-scale convergence and homogenization. Indiana Univ. Math. J. 59(2), 427–457 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Anza Hafsa, O., Mandallena, J.-P.: Homogenization of nonconvex integrals with convex growth. Journal de Mathématiques Pures et Appliquées 96(2), 167–189 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Hornung, P., Neukamm, S., Velčić, I.: Derivation of a homogenized nonlinear plate theory from 3d elasticity. Calc Var Partial Differ Equ 1–23 (2013)Google Scholar
  7. 7.
    Hornung P., Velcic I.: Derivation of a homogenized von-Karman shell theory from 3d elasticity. arXiv preprint arXiv:1211.0045 (2012)
  8. 8.
    Neukamm, S.: Homogenization, linearization and dimension reduction in elasticity with variational methods. PhD thesis, Technische Universität München (2010)Google Scholar
  9. 9.
    Neukamm, S.: Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from three-dimensional elasticity. Arch. Ration. Mech. Anal. 206(2), 645–706 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Neukamm, S., Velčić, I.: Derivation of a homogenized von-kármán plate theory from 3d nonlinear elasticity. Math. Models Methods Appl. Sci. 23(14), 2701–2748 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Velcic I.: A note on the derivation of homogenized bending plate model. arXiv preprint arXiv:1212.2594 (2012)
  12. 12.
    Kirchheim B.: Geometry and rigidity of microstructures. Habilitation Thesis, Universität Leipzig (2001)Google Scholar
  13. 13.
    Pakzad, M.R.: On the Sobolev space of isometric immersions. J. Differ. Geom. 66(1), 47–69 (2004)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Müller, S., Pakzad, M.R.: Regularity properties of isometric immersions. Math. Z. 251(2), 313–331 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hartman, P., Nirenberg, L.: On spherical image maps whose Jacobians do not change sign. Am. J. Math. 81, 901–920 (1959)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Friesecke, G., James, R.D., Müller, S.: A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. 180(2), 183–236 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3), 608–623 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Visintin, A.: Towards a two-scale calculus. ESAIM Control Optim. Calc. Var. 12(3), 371–397 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Visintin, A.: Two-scale convergence of some integral functionals. Calc. Var. Partial Differ. Equ. 29(2), 239–265 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Attouch, H.: Variational convergence for functions and operators. Pitman (Advanced Publishing Program), Boston, MA, Applicable Mathematics Series (1984)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany
  2. 2.Department of MathematicsTechnische Universität DresdenDresdenGermany
  3. 3.Hausdorff Center for Mathematics and Institute for Applied MathematicsUniversity of BonnBonnGermany

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