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A brief introduction to mathematical shell theory

  • Philippe G. Ciarlet
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 503)

Abstract

In the first chapter, we study basic notions about surfaces, such as their two fundamental forms, the Gaussian curvature and covariant derivatives. We also state the fundamental theorem of surface theory, which asserts that the Gauß and Codazzi-Mainardi equations constitute sufficient conditions for two matrix fields defined in a simply-connected open subset of ℝ2 to be the two fundamental forms of a surface in a three-dimensional Euclidean space. We also state the corresponding rigidity theorem.

The second chapter, which heavily relies on Chapter 1, begins by a detailed description of the nonlinear and linear equations proposed by W.T. Koiter for modeling thin elastic shells. These equations are “two-dimensional”, in the sense that they are expressed in terms of two curvilinear coordinates used for defining the middle surface of the shell. The existence, uniqueness, and regularity of solutions to the linear Koiter equations is then established, thanks this time to a fundamental “Korn inequality on a surface” and to an “infinitesimal rigid displacement lemma on a surface”.

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Bibliography

  1. Agmon, S.; Douglis, A.; Nirenberg, L. [1964]: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17, 35–92.Google Scholar
  2. Akian, J.L. [2003]: A simple proof of the ellipticity of Koiter’s model, Analysis and Applications 1, 1–16.Google Scholar
  3. Alexandrescu, O. [1994]: Théorème d’existence pour le modèle bidimensionnel de coque non linéaire de W.T. Koiter, C. R. Acad. Sci. Paris, Sér. I,319, 899–902.Google Scholar
  4. Amrouche, C.; Girault, V. [1994]: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czech. Math. J. 44, 109–140.Google Scholar
  5. Anicic, S.; Le Dret, H.; Raoult, A. [2005]: The infinitesimal rigid displacement lemma in Lipschitz coordinates and application to shells with minimal regularity, Math. Methods Appl. Sci. 27, 1283–1299.Google Scholar
  6. Bamberger, Y. [1981]: Mécanique de l’Ingénieur, Volume II, Hermann, Paris.Google Scholar
  7. Berger M.; Gostiaux, B. [1987]: Géométrie Différentielle: Variétés, Courbes et Surfaces, Presses Universitaires de France, Paris.Google Scholar
  8. Bernadou, M.; Ciarlet, P.G. [1976]: Sur l’ellipticité du modèle linéaire de coques de W.T. Koiter, in Computing Methods in Applied Sciences and Engineering (R. Glowinski & J.L. Lions, Editors), pp. 89–136, Lecture Notes in Economics and Mathematical Systems, 134, Springer-Verlag, Heidelberg.Google Scholar
  9. Bernadou, M.; Ciarlet, P.G.; Miara, B. [1994]: Existence theorems for two-dimensional linear shell theories, J. Elasticity 34, 111–138.Google Scholar
  10. Blouza, A.; Le Dret, H. [1999]: Existence and uniqueness for the linear Koiter model for shells with little regularity, Quart. Appl. Math. 57, 317–337.Google Scholar
  11. Bolley, P.; Camus, J. [1976]: Régularité pour une classe de problèmes aux limites elliptiques dégénérés variationnels, C.R. Acad. Sci. Paris, Sér. A 282, 45–47.Google Scholar
  12. Borchers, W.; Sohr, H. [1990]: On the equations rot v=g and div u=f with zero boundary conditions, Hokkaido Math. J. 19, 67–87.Google Scholar
  13. Do Carmo, M.P. [1976]: Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs.Google Scholar
  14. Ciarlet, P.G. [1982]: Introduction à l’Analyse Numérique Matricielle et à l’Optimisation, Masson, Paris (English translation: Introduction to Numerical Linear Algebra and Optimisation, Cambridge University Press, Cambridge, 1989).Google Scholar
  15. Ciarlet, P.G. [1988]: Mathematical Elasticity, Volume I: Three-Dimensional Elasticity, North-Holland, Amsterdam.Google Scholar
  16. Ciarlet, P.G. [1977]: Mathematical Elasticity, Volume II: Theory of Plates, North-Holland, Amsterdam.Google Scholar
  17. Ciarlet, P.G. [2000a]: Mathematical Elasticity, Volume III: Theory of Shells, North-Holland, Amsterdam.Google Scholar
  18. Ciarlet, P.G. [2000b]: Un modèle bi-dimensionnel non linéaire de coque analogue à celui de W.T. Koiter, C.R. Acad. Sci. Paris, Sér. I 331, 405–410.zbMATHMathSciNetGoogle Scholar
  19. Ciarlet, P.G. [2005]: An Introduction to Differential Geometry with Applications to Geometry, Springer, Dordrecht.Google Scholar
  20. Ciarlet, P.G.; Larsonneur, F. [2001]: On the recovery of a surface with prescribed first and second fundamental forms, J. Math. Pures Appl. 81, 167–185.Google Scholar
  21. Ciarlet, P.G.; Lods, V. [1996]: On the ellipticity of linear membrane shell equations, J. Math. Pures Appl. 75, 107–124.Google Scholar
  22. Ciarlet, P.G.; Mardare, S. [2001]: On Korn’s inequalities in curvilinear coordinates, Math. Models Methods Appl. Sci. 11, 1379–1391.Google Scholar
  23. Ciarlet, P.G.; Miara, B. [1992]: On the ellipticity of linear shell models, Z. angew. Math. Phys. 43, 243–253.Google Scholar
  24. Ciarlet, P.G.; Roquefort, A. [2001]: Justification of a two-dimensional nonlinear shell model of Koiter’s type, Chinese Ann. Math. 22B, 129–244.Google Scholar
  25. Ciarlet, P.G.; Sanchez-Palencia, E. [1996]: An existence and uniqueness theorem for the two-dimensional linear membrane shell equations. J. Math. Pures Appl. 75, 51–67.Google Scholar
  26. Dacorogna, B. [1989]: Direct Methods in the Calculus of Variations, Springer, Berlin.Google Scholar
  27. Destuynder, P. [1985]: A classification of thin shell theories, Acta Applicandae Mathematicae 4, 15–63.Google Scholar
  28. Duvaut, G.; Lions, J.L. [1972]: Les Inéquations en Mécanique et en Physique, Dunod, Paris (English translation: Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976).Google Scholar
  29. Friesecke, G.; James, R.D.; Mora, M.G.; Müller, S. [2003]: Derivation of nonlinear bending theory for shells from three dimensional nonlinear elasticity by Gamma-convergence, C.R. Acad. Sci. Paris, Sér. I, 336, 697–702.Google Scholar
  30. Gauss, C.F. [1828]: Disquisitiones generales circas superficies curvas, Commentationes societatis regiae scientiarum Gottingensis recentiores 6, Gttingen.Google Scholar
  31. Geymonat, G. [1965]: Sui problemi ai limiti per i sistemi lineari ellittici, Ann. Mat. Pura Appl. 69, 207–284.Google Scholar
  32. Geymonat, G.; Gilardi, G. [1998]: Contre-exemples à l’inégalité de Korn et au lemme de Lions dans des domaines irréguliers, in Equations aux Dérivées Partielles et Applications. Articles Dédiés à Jacques-Louis Lions, pp. 541–548, Gauthier-Villars, Paris.Google Scholar
  33. Geymonat, G.; Suquet, P. [1986]: Functional spaces for Norton-Hoff materials, Math. Methods Appl. Sci. 8, 206–222.Google Scholar
  34. Hartman, P.; Wintner, A. [1950]: On the embedding problem in differential geometry, Amer. J. Math. 72, 553–564.Google Scholar
  35. John, F. [1965]: Estimates for the derivatives of the stresses in a thin shell and interior shell equations, Comm. Pure Appl. Math. 18, 235–267.Google Scholar
  36. John, F. [1971]: Refined interior equations for thin elastic shells, Comm. Pure Appl. Math. 18, 235–267.Google Scholar
  37. Klingenberg, W. [1973]: Eine Vorlesung über Differentialgeometrie, Springer-Verlag, Berlin (English translation: A Course in Differential Geometry, Springer-Verlag, Berlin, 1978).Google Scholar
  38. Koiter, W.T. [1966]: On the nonlinear theory of thin elastic shells, Proc. Kon. Ned. Akad. Wetensch. B69, 1–54.Google Scholar
  39. Koiter, W.T. [1970]: On the foundations of the linear theory of thin elastic shells, Proc. Kon. Ned. Akad. Wetensch. B73, 169–195.Google Scholar
  40. Kühnel, W. [2002]: Differentialgeometrie, Fried. Vieweg & Sohn, Wiesbaden (English translation: Differential Geometry: Curves-Surfaces-Manifolds, American Mathematical Society, Providence, 2002).Google Scholar
  41. Le Dret, H. [2004]: Well-posedness for Koiter and Naghdi shells with a G 1-midsurface, Analysis and Applications 2, 365–388.Google Scholar
  42. Le Dret, H.; Raoult, A. [1996]: The membrane shell model in nonlinear elasticity: A variational asymptotic derivation, J. Nonlinear Sci. 6, 59–84.Google Scholar
  43. Lods, V. & Miara, B. [1998]: Nonlinearly elastic shell models. II. The flexural model, Arch. Rational Mech. Anal. 142, 355–374.Google Scholar
  44. Magenes, E.; Stampacchia, G. [1958]: I problemi al contorno per le equazioni differenziali di tipo ellittico, Ann. Scuola Norm. Sup. Pisa 12, 247–358.Google Scholar
  45. Mardare, S. [2003a]: Inequality of Korn’s type on compact surfaces without boundary, Chinese Annals Math. 24B, 191–204.CrossRefMathSciNetGoogle Scholar
  46. Mardare, S. [2003b]: The fundamental theorem of surface theory for surfaces with little regularity, J. Elasticity 73, 251–290.zbMATHCrossRefMathSciNetGoogle Scholar
  47. Mardare, S. [2005]: On Pfaff systems with L p coefficients and their applications in differential geometry, J. Math. Pures Appl., to appear.Google Scholar
  48. Miara, B. [1998]: Nonlinearly elastic shell models. I. The membrane model, Arch. Rational Mech. Anal. 142, 331–353.Google Scholar
  49. Morrey, Jr., C.B. [1952]: Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2, 25–53.Google Scholar
  50. Nečas, J. [1967]: Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris.Google Scholar
  51. Schwartz, L. [1966]: Théorie des Distributions, Hermann, Paris.Google Scholar
  52. Schwartz, L. [1992]: Analyse II: Calcul Différentiel et Equations Différentielles, Hermann, Paris.Google Scholar
  53. Spivak, M. [1999]: A Comprehensive Introduction to Differential Geometry, Volumes I to V, Third Edition, Publish or Perish, Berkeley.Google Scholar
  54. Stein, E. [1970]: Singular Integrals and Differentiability Properties of Functions, Princeton University Press.Google Scholar
  55. Stoker, J.J. [1969]: Differential Geometry, John Wiley, New York.Google Scholar
  56. Tartar, L. [1978]: Topics in Nonlinear Analysis, Publications Mathématiques d’Orsay No. 78.13, Université de Paris-Sud, Orsay.Google Scholar

Copyright information

© CISM, Udine 2008

Authors and Affiliations

  • Philippe G. Ciarlet
    • 1
  1. 1.City University of Hong KongHong Kong

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