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Decomposition of the Deformations of a Thin Shell. Asymptotic Behavior of the Green-St Venant’s Strain Tensor

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Abstract

We investigate the behavior of the deformations of a thin shell, whose thickness δ tends to zero, through a decomposition technique of these deformations. The terms of the decomposition of a deformation v are estimated in terms of the L 2-norm of the distance from ∇ v to SO(3). This permits in particular to derive accurate nonlinear Korn’s inequalities for shells (or plates). Then we use this decomposition technique and estimates to give the asymptotic behavior of the Green-St Venant’s strain tensor when the “strain energy” is of order less than δ 3/2.

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References

  1. Blanchard, D., Griso, G.: Microscopic effects in the homogenization of the junction of rods and a thin plate. Asymptot. Anal. 56(1), 1–36 (2008)

    MATH  MathSciNet  Google Scholar 

  2. Blanchard, D., Griso, G.: Decomposition of deformations of thin rods. Application to nonlinear elasticity. Anal. Appl. 7(1), 21–71 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Blanchard, D., Griso, G.: Decomposition of shell deformations—asymptotic behavior of the Green-St Venant strain tensor. C. R. Acad. Sci. Paris, Ser. I 347 (2009)

  4. Blanchard, D., Gaudiello, A., Griso, G.: Junction of a periodic family of elastic rods with a 3d plate. I. J. Math. Pures Appl. 88(1), 149–190 (2007)

    MATH  MathSciNet  Google Scholar 

  5. Blanchard, D., Gaudiello, A., Griso, G.: Junction of a periodic family of elastic rods with a thin plate. II. J. Math. Pures Appl. 88(2), 1–33 (2007)

    MATH  MathSciNet  Google Scholar 

  6. Ciarlet, P.G.: Mathematical Elasticity, Vol. II. Theory of Plates. North-Holland, Amsterdam (1997)

    Google Scholar 

  7. Ciarlet, P.G.: Mathematical Elasticity, Vol. III. Theory of Shells. North-Holland, Amsterdam (2000)

    Google Scholar 

  8. Ciarlet, P.G.: Un modèle bi-dimentionnel non linéaire de coques analogue à celui de W.T. Koiter. C. R. Acad. Sci. Paris, Sér. I 331, 405–410 (2000)

    MATH  MathSciNet  ADS  Google Scholar 

  9. Ciarlet, P.G., Destuynder, P.: A justification of a nonlinear model in plate theory. Comput. Methods Appl. Mech. Eng. 17/18, 227–258 (1979)

    Article  Google Scholar 

  10. Ciarlet, P.G., Mardare, C.: Continuity of a deformation in H 1 as a function of its Cauchy-Green tensor in L 1. J. Nonlinear Sci. 14(5), 415–427 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Ciarlet, P.G., Mardare, C.: An introduction to shell theory. In: Ciarlet, P.G., Li, T.T. (eds.) Differential Geometry: Theory and Applications, pp. 94–184. World Scientific, Singapore (2008)

    Google Scholar 

  12. Ciarlet, P.G., Gratie, L., Mardare, C.: A nonlinear Korn inequality on a surface. J. Math. Pures Appl. 85, 2–16 (2006)

    MATH  MathSciNet  Google Scholar 

  13. Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from the three-dimensional elasticity. Commun. Pure Appl. Math. LV, 1461–1506 (2002)

    Article  Google Scholar 

  14. Friesecke, G., James, R.D., Mora, M.G., Müller, S.: Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma convergence. C. R. Acad. Sci. Paris, Ser. I 336 (2003)

  15. Friesecke, G., James, R., Müller, S.: A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch. Ration. Mech. Anal. 180, 183–236 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Griso, G.: Asymptotic behavior of curved rods by the unfolding method. Math. Methods Appl. Sci. 27, 2081–2110 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Griso, G.: Asymptotic behavior of structures made of plates. Anal. Appl. 3(4), 325–356 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Griso, G.: Decomposition of displacements of thin structures. J. Math. Pures Appl. 89, 199–233 (2008)

    MATH  MathSciNet  Google Scholar 

  19. Le Dret, H., Raoult, A.: The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 75, 551–580 (1995)

    Google Scholar 

  20. Le Dret, H., Raoult, A.: The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function. Proc. R. Soc. Edinb. A 125, 1179–1192 (1995)

    MATH  Google Scholar 

  21. Lewicka, M., Mora, M.G., Pakzad, M.R.: The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells. 0811.2238 (2008)

  22. Lewicka, M., Mora, M.G., Pakzad, M.R.: Shell theories arising as low energy Gamma-limit of 3d nonlinear elasticity. Preprint 23/2008 Max-Planck Institute, Leipzig, 0803.0358v1

  23. Pantz, O.: On the justification of the nonlinear inextensional plate model. C. R. Acad. Sci. Paris Sér. I Math. 332(6), 587–592 (2001)

    MATH  MathSciNet  ADS  Google Scholar 

  24. Pantz, O.: On the justification of the nonlinear inextensional plate model. Arch. Ration. Mech. Anal. 167(3), 179–209 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Université de Rouen, UMR 6085, 76801, Saint Etienne du Rouvray Cedex, France

    Dominique Blanchard

  2. Laboratoire d’Analyse Numérique, Université P. et M. Curie, Case Courrier 187, 75252, Paris Cédex 05, France

    Georges Griso

Authors
  1. Dominique Blanchard
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  2. Georges Griso
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Correspondence to Dominique Blanchard.

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Blanchard, D., Griso, G. Decomposition of the Deformations of a Thin Shell. Asymptotic Behavior of the Green-St Venant’s Strain Tensor. J Elast 101, 179–205 (2010). https://doi.org/10.1007/s10659-010-9255-8

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  • DOI: https://doi.org/10.1007/s10659-010-9255-8

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