Asymptotic Behaviors of Shell Models

  • Dominique Chapelle
  • Klaus-Jürgen Bathe
Part of the Computational Fluid and Solid Mechanics book series (COMPFLUID)


Implicit in the concept of a “shell” is the idea that the thickness is “small” compared to the other two dimensions. In practice, it is not unusual to deal with structures for which the thickness is smaller by several orders of magnitude, in which case the shell is said to be “thin” (consider, for example, the shell body of a motor car). Considering the role of the thickness parameter t in the shell models that we presented in the previous chapter (see for example Eqs. (4.36) and (4.51)), with different powers of t in the bilinear terms on the left-hand side, it is essential to determine how the mathematical properties and physical behaviors of the models are affected when this parameter becomes small.


Asymptotic Behavior Mode Shape Shell Model Asymptotic Analysis Limit Problem 
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  1. Ciarlet, P.G. (2000). Mathematical Elasticity - Volume III: Theory of Shells. Amsterdam: North-Holland.Google Scholar
  2. Kim, D.N., & Bathe, K.J. (2008). A 4-node 3D-shell element to model shell surface tractions and incompressible behavior. Comput. & Structures, 86(21-22), 2027-2041.CrossRefGoogle Scholar
  3. Pitkäranta, J., & Sanchez-Palencia, E. (1997). On the asymptotic behaviour of sensitive shells with small thickness. C. R. Acad. Sci. Paris, Série IIb, 325, 127-134.Google Scholar
  4. Sanchez-Hubert, J., & Sanchez-Palencia, E. (1997). Coques Elastiques Minces - Propriétés Asymptotiques. Paris: Masson.zbMATHGoogle Scholar
  5. Bathe, K.J., Iosilevich, A., & Chapelle, D. (2000b). An inf-sup test for shell finite elements. Comput. & Structures, 75(5), 439-456.CrossRefMathSciNetGoogle Scholar
  6. Pitkäranta, J., Matache, A.M., & Schwab, C. (2001). Fourier mode analysis of layers in shallow shell deformations. Comput. Methods Appl. Mech. Engrg., 190, 2943-2975.CrossRefzbMATHGoogle Scholar
  7. Chenais, D., & Paumier, J.-C. (1994). On the locking phenomenon for a class of elliptic problems. Numer. Math., 67, 427-440.CrossRefzbMATHMathSciNetGoogle Scholar
  8. Leissa, A.W. (1973). Vibration of Shells, vol. SP-288. NASA.Google Scholar
  9. Lions, J.L., & Sanchez-Palencia, E. (1994). Problèmes aux limites sensitifs. C. R. Acad. Sci. Paris, Série I, 319, 1021-1026.zbMATHMathSciNetGoogle Scholar
  10. Bathe, K.J. (1996). Finite Element Procedures. Englewood Cliffs: Prentice Hall.Google Scholar
  11. Kikuchi, F. (1982). Accuracy of some finite element models for arch problems. Comput. Methods Appl. Mech. Engrg., 35, 315-345.CrossRefzbMATHMathSciNetGoogle Scholar
  12. Pitkäranta, J., Leino, Y., Ovaskainen, O., & Piila, J. (1995). Shell deformation states and the finite element method: a benchmark study of cylindrical shells. Comput. Methods Appl. Mech. Engrg., 128, 81-121.CrossRefzbMATHMathSciNetGoogle Scholar
  13. Sanchez-Hubert, J., & Sanchez-Palencia, E. (2001). Anisotropic finite element estimates and local locking for shells: parabolic case. C. R. Acad. Sci. Paris, Série IIb, 329, 153-159.zbMATHGoogle Scholar
  14. Karamian, P., Sanchez-Hubert, J., & Sanchez-Palencia, E. (2000). A model problem for boundary layers of thin elastic shells. M2AN Math. Model. Numer. Anal., 34(1), 1-30.CrossRefzbMATHMathSciNetGoogle Scholar
  15. Sanchez-Palencia, E. (1992). Asymptotic and spectral properties of a class of singular-stiff problems. J. Math. Pures Appl., 71, 379-406.zbMATHMathSciNetGoogle Scholar
  16. Soedel, W. (2004). Vibrations of Shells and Plates. Marcel Dekker, 3rd ed.Google Scholar
  17. Lee, P.S., & Bathe, K.J. (2005). Insight into finite element shell discretizations by use of the “basic shell mathematical model”. Comput. & Structures, 83, 69-90.CrossRefGoogle Scholar
  18. Blouza, A., Brezzi, F., & Lovadina, C. (1999). Sur la classification des coques linéairement élastiques. C. R. Acad. Sci. Paris, Série I, 328, 831-836.zbMATHMathSciNetGoogle Scholar
  19. Pitkäranta, J. (1992). The problem of membrane locking in finite element analysis of cylindrical shells. Numer. Math., 61, 523-542.CrossRefzbMATHMathSciNetGoogle Scholar
  20. Bathe, K.J., Hiller, J.F., & Zhang, H. (2002). On the finite element analysis of shells and their full interaction with Navier-Stokes fluid flows. In B. Topping, & Z. Bittnar (Eds.) Computational Structures Technology. Edinburgh: Civil-Comp Press.Google Scholar
  21. Arnold, D.N., & Falk, R.S. (1996). Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model. SIAM J. Math. Anal., 27(2), 486-514.CrossRefzbMATHMathSciNetGoogle Scholar
  22. Dauge, M., & Yosibash, Z. (2000). Boundary layer realization in thin elastic 3D domains and 2D hierarchic plate models. Internat. J. Solids Structures, 37, 2443-2471.CrossRefzbMATHGoogle Scholar
  23. Lions, J.L. (1973). Perturbations Singulières dans les Problèmes aux Limites et en Contrôle Optimal. Berlin, New York: Springer-Verlag.zbMATHGoogle Scholar
  24. Baiocchi, C., & Lovadina, C. (2002). A shell classi_cation by interpolation. Math. Models Methods Appl. Sci., 12(10), 1359-1380.CrossRefzbMATHMathSciNetGoogle Scholar
  25. Bathe, K.J., Bucalem, M.L., & Brezzi, F. (1990). Displacement and stress convergence of our MITC plate bending elements. Eng. Comput., 7, 291-302.CrossRefGoogle Scholar
  26. Kirmse, A. (1993). Bending-dominated deformation of thin spherical shells: analysis and finite element approximation. SIAM J. Numer. Anal., 30(4), 1015-1040.CrossRefzbMATHMathSciNetGoogle Scholar
  27. Artioli, E., Beirão da Veiga, L., Hakula, H., & Lovadina, C. (2008). Free vibrations for some Koiter shells of revolution. Appl. Math. Lett., 21, 1245—1248.CrossRefzbMATHMathSciNetGoogle Scholar
  28. Delfour, M.C. (1999). Intrinsic P(2,1) thin shell model and Naghdi’s models without a priori assumption on the stress tensor. In K. Hoffmann, G. Leugering, & F. Tröltzsch (Eds.) Optimal Control of Partial Diérential Equations, (pp. 99-113). Basel: Birkhäuser.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Dominique Chapelle
    • 1
  • Klaus-Jürgen Bathe
    • 2
  1. 1.INRIA Paris-RocquencourtLe ChesnayFrance
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

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