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Derivation of a homogenized nonlinear plate theory from 3d elasticity

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Abstract

We derive, via simultaneous homogenization and dimension reduction, the \(\Gamma \)-limit for thin elastic plates whose energy density oscillates on a scale that is either comparable to, or much smaller than, the film thickness. We consider the energy scaling that corresponds to Kirchhoff’s nonlinear bending theory of plates.

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References

  1. Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Attouch, H.: Variational Convergence for Functions and Operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA (1984)

    Google Scholar 

  3. Babadjian, J.-F., Baía, M.: 3D–2D analysis of a thin film with periodic microstructure. Proc. R. Soc. Edinb. Sect. A 136(2), 223–243 (2006)

    Article  MATH  Google Scholar 

  4. Braides, A., Fonseca, I., Francfort, G.: 3D–2D asymptotic analysis for inhomogeneous thin films. Indiana Univ. Math. J. 49(4), 1367–1404 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hornung, P., Lewicka, M., Reza Pakzad, M.: Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells. J. Elast. 111(1), 1–19 (2013)

    Article  MATH  Google Scholar 

  8. Hornung, P.: Approximation of flat \(W^{2,2}\) isometric immersions by smooth ones. Arch. Ration. Mech. Anal. 199(3), 1015–1067 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hornung, P.: Fine level set structure of flat isometric immersions. Arch. Ration. Mech. Anal. 199(3), 943–1014 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hornung, P., Velčić, I.: Derivation of a homogenized von Kármán shell theory from \(3D\) nonlinear elasticity. arXiv, preprint (2012)

  11. Le Dret, H., Raoult, A.: The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. (9) 74(6), 549–578 (1995)

    MathSciNet  MATH  Google Scholar 

  12. Mielke, A., Timofte, A.M.: Two-scale homogenization for evolutionary variational inequalities via the energetic formulation. SIAM J. Math. Anal. 39(2), 642–668 (2007). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  13. Neukamm, S.: Homogenization, Linearization and Dimension Reduction in Elasticity with Variational Methods. PhD thesis, Technische Universität München (2010)

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Neukamm, S., Velčić, I.: Derivation of a homogenized von-Kármán plate theory from \(3D\) nonlinear elasticity. Accepted in Math. Models Methods Appl. Sci. doi:10.1142/S0218202513500449

  16. Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3), 608–623 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Reza Pakzad, M.: On the Sobolev space of isometric immersions. J. Differ. Geom. 66(1), 47–69 (2004)

    MATH  Google Scholar 

  18. Schmidt, B.: Plate theory for stressed heterogeneous multilayers of finite bending energy. J. Math. Pures Appl. (9) 88(1), 107–122 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Visintin, A.: Towards a two-scale calculus. ESAIM Control Optim. Calc. Var. 12(3), 371–397 (2006). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  20. Visintin, A.: Two-scale convergence of some integral functionals. Calc. Var. Partial Differ. Equ. 29(2), 239–265 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Hornung and Velčić were supported by Deutsche Forschungsgemeinschaft grant no. HO-4697/1-1. Work on this project began while Velčić was affiliated with BCAM. Most of it was completed while all authors were affiliated with MPI Leipzig.

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Authors and Affiliations

  1. Institute for Applied Mathematics, Endenicher Allee 60, 53115 , Bonn, Germany

    Peter Hornung

  2. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 , Berlin, Germany

    Stefan Neukamm

  3. Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 , Leipzig, Germany

    Igor Velčić

  4. Basque Center of Applied Mathematics (BCAM), Alameda de Mazarredo 14, 48009 , Bilbao, Spain

    Igor Velčić

Authors
  1. Peter Hornung
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  2. Stefan Neukamm
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  3. Igor Velčić
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Corresponding author

Correspondence to Peter Hornung.

Additional information

Communicated by J. Ball.

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Hornung, P., Neukamm, S. & Velčić, I. Derivation of a homogenized nonlinear plate theory from 3d elasticity. Calc. Var. 51, 677–699 (2014). https://doi.org/10.1007/s00526-013-0691-8

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  • DOI: https://doi.org/10.1007/s00526-013-0691-8

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