Advertisement

The Infinite Hierarchy of Elastic Shell Models: Some Recent Results and a Conjecture

  • Marta Lewicka
  • Mohammad Reza Pakzad
Chapter
Part of the Fields Institute Communications book series (FIC, volume 64)

Abstract

We summarize some recent results of the authors and their collaborators, regarding the derivation of thin elastic shell models (for shells with mid-surface of arbitrary geometry) from the variational theory of 3d nonlinear elasticity. We also formulate a conjecture on the form and validity of infinitely many limiting 2d models, each corresponding to its proper scaling range of the body forces in terms of the shell thickness.

Notes

Acknowledgements

The first author was partially supported by the NSF grants DMS-0707275 and DMS-0846996, and by the Polish MN grant N N201 547438. The second author was partially supported by the University of Pittsburgh grant CRDF-9003034 and by the NSF grant DMS-0907844.

Received 7/12/2009; Accepted 9/12/2010

References

  1. 1.
    J.M. Ball, Some open problems in elasticity, in Geometry, dynamics and mechanics (Marsden Festschrift), (Springer, New York, 2002), pp. 3–59CrossRefGoogle Scholar
  2. 2.
    P.G. Ciarlet, Mathematical Elasticity, Vol I-III Theory of Shells (North-Holland, Amsterdam, 2000)zbMATHGoogle Scholar
  3. 3.
    S. Conti, G. Dolzmann, Υ-convergence for incompressible elastic plates. Calc.Var. PDE 34, 531–551 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    S. Conti, F. Maggi, Confining thin sheets and folding paper. Arch. Ration. Mech. Anal. 187(1), 1–48 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    S. Conti, F. Maggi, S. Müller, Rigorous derivation of Föppl’s theory for clamped elastic membranes leads to relaxation. SIAM J. Math. Anal. 38(2), 657–680 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    G. Dal Maso, An Introduction to Υ-Convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8 (Birkhäuser, MA, 1993)Google Scholar
  7. 7.
    G. Friesecke, R. James, M.G. Mora, S. Müller, Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence. C. R. Math. Acad. Sci. Paris. 336(8), 697–702 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    G. Friesecke, R. James, S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm. Pure. Appl. Math. 55, 1461–1506 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    G. Friesecke, R. James, S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. 180(2), 183–236 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    G. Geymonat, É. Sanchez-Palencia, On the rigidity of certain surfaces with folds and applications to shell theory. Arch. Ration. Mech. Anal. 129(1), 11–45 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    T. von Kármán, Festigkeitsprobleme im Maschinenbau, in Encyclopädie der Mathematischen Wissenschaften, vol. 4 (Leipzig, 1910), pp. 311–385Google Scholar
  12. 12.
    H. LeDret, A. Raoult, The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 73, 549–578 (1995)MathSciNetzbMATHGoogle Scholar
  13. 13.
    H. LeDret, A. Raoult, The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci. 6, 59–84 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Lewicka, A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry, ESAIM: Control, Optimisation and Calculus of Variations 17, 493–505 (2011)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M. Lewicka, M.G. Mora, M.R. Pakzad, Shell theories arising as low energy Υ-limit of 3d nonlinear elasticity, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5, vol. IX, 1–43 (2010)Google Scholar
  16. 16.
    M. Lewicka, M.G. Mora, M.R. Pakzad, A nonlinear theory for shells with slowly varying thickness, C.R. Acad. Sci. Paris, Ser I 347, 211–216 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Lewicka, M.G. Mora, and M. Pakzad, The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells, Arch. Rational Mech. Anal. (3) Vol. 200, 1023–1050 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    M. Lewicka, and M. Pakzad, Scaling laws for non-Euclidean plates and the W2, 2 isometric immersions of Riemannian metrics, ESAIM: Control, Optimisation and Calculus of Variations doi:10.1051/cocv/2010039CrossRefGoogle Scholar
  19. 19.
    M. Lewicka, and S. Müller, The uniform Korn-Poincaré inequality in thin domains, Annales de l’Institut Henri Poincare (C) Non Linear Analysis Vol. 28(3), May-June, 443–469 (2011)Google Scholar
  20. 20.
    A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn. (Cambridge University Press, Cambridge, 1927)zbMATHGoogle Scholar
  21. 21.
    M.G. Mora, and L. Scardia, Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density, J. Differential Equations 252, 35–55 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    S. Müller, M.R. Pakzad, Regularity properties of isometric immersions. Math. Z. 251(2), 313–331 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    S. Müller, M.R. Pakzad, Convergence of equilibria of thin elastic plates – the von Kármán case. Comm. Partial Differ. Equat. 33(4–6), 1018–1032 (2008)CrossRefGoogle Scholar
  24. 24.
    M.R. Pakzad, On the Sobolev space of isometric immersions. J. Differ. Geom. 66(1), 47–69 (2004)MathSciNetCrossRefGoogle Scholar
  25. 25.
    É. Sanchez-Palencia, Statique et dynamique des coques minces. II. Cas de flexion pure inhibeé. Approximation membranaire. C. R. Acad. Sci. Paris Sér. I Math. 309(7), 531–537 (1989)zbMATHGoogle Scholar
  26. 26.
    M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. V, 2nd edn., (Publish or Perish Inc. 1979)Google Scholar
  27. 27.
    S. Venkataramani, Lower bounds for the energy in a crumpled elastic sheet – a minimal ridge, Nonlinearity 17(1), 301–312 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of MathematicsUniversity of PittsburghPittsburghUSA

Personalised recommendations