Journal of Elasticity

, Volume 117, Issue 2, pp 163–188 | Cite as

A New Linear Shell Model for Shells with Little Regularity

Article

Abstract

In this paper, a new model of linearly elastic shell is formulated. The model is of Koiter’s type capturing the membrane and bending effects. However, the model is well formulated for shells with little regularity, namely the shells whose middle surface is parameterized by a W1,∞ function.

Unknowns in the model are the displacement \(\tilde{\boldsymbol {u}}\) of the middle surface of the shell and the infinitesimal rotation \(\tilde{\boldsymbol {\omega}}\) of the shell cross-section. The existence and uniqueness of the solution of the model has been proved for \((\tilde{\boldsymbol {u}}, \tilde{\boldsymbol {\omega}}) \in H^{1} \times H^{1}\) with the help of the Lax-Milgram lemma.

The model is analyzed asymptotically with respect to the small thickness of the shell. It is shown that the model asymptotically behaves just as the membrane model, the flexural model, and the generalized membrane model in each regime. In this way the model is justified.

Since the model is well formulated for shells whose middle surface is with corners, we compare the model with Le Dret’s model of folded plates. It turns out that in the regime of the flexural shells the models are the same. The differential equations of the model are derived. They imply that the model is as a special case of the Cosserat shell model with a single director for a particular constitutive law.

Keywords

Linear elasticity Shell model Koiter model Little regularity Justification Junctions Folded shells 

Mathematics Subject Classification

74K25 74K30 74K15 74B05 

References

  1. 1.
    Akian, J.-L.: Asymptotic analysis of bending-dominated shell junctions. Ann. Math. Pures Appl. 84(6), 667–716 (2005) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anicic, S.: Du modèle de Kirchhoff-Love exact à un modèle de coque mince et à un modèle de coque pliée. Doctoral dissertation, Université Joseph Fourier (2001) Google Scholar
  3. 3.
    Anicic, S.: Mesure des variations infinitésimales des courbures principales d’une surface. C. R. Math. Acad. Sci. Paris, Sér. I 335, 301–306 (2002) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Anicic, S., Le Dret, H., Raoult, A.: The infinitesimal rigid displacement lemma in Lipschitz co-ordinates and application to shells with minimal regularity. Math. Methods Appl. Sci. 27(11), 1283–1299 (2004) MathSciNetCrossRefMATHADSGoogle Scholar
  5. 5.
    Antman, S.S.: Nonlinear Problems of Elasticity. Applied Mathematical Sciences, vol. 107. Springer, New York (1995) MATHGoogle Scholar
  6. 6.
    Blouza, A., Le Dret, H.: Existence and uniqueness for the linear Koiter model for shells with little regularity. Q. Appl. Math. 57, 317–337 (1999) MATHGoogle Scholar
  7. 7.
    Blouza, A., Le Dret, H.: Naghdi’s shell model: existence, uniqueness and continuous dependence on the midsurface. J. Elast. 64(2–3), 199–216 (2001) CrossRefMATHGoogle Scholar
  8. 8.
    Budiansky, B., Sanders, J.L. Jr.: On the “Best” First-Order Linear Shell Theory. Progress in Applied Mechanics, pp. 129–140. Macmillan, New York (1963) Google Scholar
  9. 9.
    Ciarlet, P.G.: Mathematical Elasticity. Vol. III. Theory of Shells. Studies in Mathematics and Its Applications, vol. 29. North-Holland, Amsterdam (2000) Google Scholar
  10. 10.
    Ciarlet, P.G., Lods, V.: Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations. Arch. Ration. Mech. Anal. 136(2), 119–161 (1996) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ciarlet, P.G., Lods, V.: Asymptotic analysis of linearly elastic shells. III. Justification of Koiter’s shell equations. Arch. Ration. Mech. Anal. 136(2), 191–200 (1996) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ciarlet, P.G., Lods, V.: Asymptotic analysis of linearly elastic shells: “generalized membrane shells”. J. Elast. 43(2), 147–188 (1996) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ciarlet, P.G., Lods, V., Miara, B.: Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations. Arch. Ration. Mech. Anal. 136(2), 163–190 (1996) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Griso, G.: Asymptotic behavior of structures made of plates. Anal. Appl. (Singap.) 3(4), 325–356 (2005) MathSciNetCrossRefMATHADSGoogle Scholar
  15. 15.
    Koiter, W.T.: On the foundations of the linear theory of thin elastic shells. I, II. Proc. K. Ned. Akad. Wet., Ser. B, Phys. Sci. 73, 169–182 (1970). Ibid. 73, 183–195 (1970) MathSciNetMATHGoogle Scholar
  16. 16.
    Le Dret, H.: Folded plates revisited. Comput. Mech. 5(5), 345–365 (1990) CrossRefGoogle Scholar
  17. 17.
    Le Dret, H.: Modeling of a folded plate. Comput. Mech. 5(6), 401–416 (1990) CrossRefMATHGoogle Scholar
  18. 18.
    Le Dret, H.: Problèmes Variationnels dans les Multi-Domaines. Modélisation des Jonctions et Applications. Masson, Paris (1991) MATHGoogle Scholar
  19. 19.
    Le Dret, H.: Vibrations of a folded plate. RAIRO Modél. Math. Anal. Numér. 24(4), 501–521 (1990) MATHGoogle Scholar
  20. 20.
    Le Dret, H.: Well-posedness for Koiter and Naghdi shells with a G1-midsurface. Anal. Appl. (Singap.) 2(4), 365–388 (2004) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lods, V., Mardare, C.: Asymptotic justification of the Kirchhoff-Love assumptions for a linearly elastic clamped shell. J. Elast. 58(2), 105–154 (2000) MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Naghdi, P.M.: The Theory of Shells and Plates, Handbuch der Physik, vol. VIa/2, pp. 425–640. Springer, New York (1972) Google Scholar
  23. 23.
    Nardinocchi, P.: Modelling junctions of thin plates. Eur. J. Mech. A, Solids 21(3), 523–534 (2002) MathSciNetCrossRefMATHADSGoogle Scholar
  24. 24.
    Nardinocchi, P., Podio-Guidugli, P.: Angle plates. J. Elast. 63(1), 19–53 (2001) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Percivale, D.: Folded shells: a variational approach. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 19(2), 207–221 (1992) MathSciNetMATHGoogle Scholar
  26. 26.
    Tambača, J.: A model of irregular curved rods. In: Applied Mathematics and Scientific Computing, Dubrovnik, 2001, pp. 289–299. Plenum, New York (2003) Google Scholar
  27. 27.
    Tambača, J.: A note on the “flexural” shell model for shells with little regularity. Adv. Math. Sci. Appl. 16, 45–55 (2006) MathSciNetMATHGoogle Scholar
  28. 28.
    Titeux, I., Sanchez-Palencia, E.: Junction of thin plates. Eur. J. Mech. A, Solids 19(3), 377–400 (2000) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ZagrebZagrebCroatia

Personalised recommendations