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Journal of Elasticity

, Volume 85, Issue 2, pp 125–152 | Cite as

Local Symmetry Group in the General Theory of Elastic Shells

  • Victor A. Eremeyev
  • Wojciech Pietraszkiewicz
Article

Abstract

We establish the local symmetry group of the dynamically and kinematically exact theory of elastic shells. The group consists of an ordered triple of tensors which make the shell strain energy density invariant under change of the reference placement. Definitions of the fluid shell, the solid shell, and the membrane shell are introduced in terms of members of the symmetry group. Within solid shells we discuss in more detail the isotropic, hemitropic, and orthotropic shells and corresponding invariant properties of the strain energy density. For the physically linear shells, when the density becomes a quadratic function of the shell strain and bending tensors, reduced representations of the density are established for orthotropic, cubic-symmetric, and isotropic shells. The reduced representations contain much less independent material constants to be found from experiments.

Key words

shell nonlinear six-field theory elastic constitutive equations material symmetry solid shell isotropy 

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References

  1. 1.
    Adeleke, S.A.: On possible symmetry of shells. In: Dafermos, C.M., Joseph D.D., Leslie, F.M. (eds.) The Breadth and Depth of Continuum Mechanics. A Collection of Papers Dedicated to J.L. Ericksen on his 60th Birthday, pp. 745–757. Springer, Berlin Heidelberg New York (1986)Google Scholar
  2. 2.
    Altenbach, H., Zhilin, P.A.: The theory of elastic thin shells (in Russian). Adv. Mech. 11, 107–148 (1988)MathSciNetGoogle Scholar
  3. 3.
    Altenbach, H., Zhilin, P.A.: The theory of simple elastic shells. In: Kienzler, R., Altenbach, H., Ott, I. (eds.) Theories of Plates and Shells: Critical Review and New Applications, pp. 1–12. Springer, Berlin Heidelberg New York (2004)Google Scholar
  4. 4.
    Altenbach, H., Naumenko, K., Zhilin, P.A.: A direct approach to the formulation of constitutive equations for rods and shells. In: Pietraszkiewicz, W., Szymczak, C. (eds.) Shell Structures: Theory and Applications, pp. 87–90. Taylor & Francis, London (2005)Google Scholar
  5. 5.
    Arfken G.B., Weber, H.J.: Mathematical Methods for Physicists. Springer, Berlin Heidelberg New York (2000)Google Scholar
  6. 6.
    Carrol, M.M., Naghdi, P.M.: The influence of the reference geometry on the response of elastic shells. Arch. Ration. Mech. Anal. 48, 302–318 (1972)CrossRefGoogle Scholar
  7. 7.
    Chróścielewski, J., Makowski, J., Stumpf, H.: Genuinely Resultant Shell Finite Elements Accounting For Geometric And Material Non-Linearity. Int. J. Numer. Methods Eng. 35 (1992) 63–94CrossRefzbMATHGoogle Scholar
  8. 8.
    Chróścielewski, J., Makowski, J., Pietraszkiewicz, W.: Statics and Dynamics of Multifold Shells: Nonlinear Theory and Finite Element Method (in Polish). Wydawnictwo IPPT PAN, Warszawa (2004)Google Scholar
  9. 9.
    Cohen, H., Wang, C.-C.: A mathematical analysis of the simplest direct models for rods and shells. Arch. Rational Mech. Anal. 108, 35–81 (1989)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Cosserat, E., Cosserat, F.: Théorie des corps deformables. Herman et Flis, Paris (1909); English translation: NASA TT F-11, 561, NASA, Washington, District of Columbia (1968)Google Scholar
  11. 11.
    de Gennes, P.G.: The Physics of Liquid Crystals. Clarendon, Oxford (1974)Google Scholar
  12. 12.
    Eremeyev, V.A.: Nonlinear micropolar shells: Theory and applications. In: Pietraszkiewicz, W., Szymczak, C. (eds.) Shell Structures: Theory and Applications, pp. 11–18. Taylor & Francis, London (2005)Google Scholar
  13. 13.
    Eremeyev, V.A., Pietraszkiewicz, W.: The nonlinear theory of elastic shells with phase transitions. J. Elasticity 74, 67–86 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Eremeyev, V.A., Zubov, L.M.: The theory of elastic and viscoelastic micropolar liquids. J. Appl. Math. Mech. 63, 755–767 (1999)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Eremeyev, V.A., Zubov, L.M.: The general nonlinear theory of elastic micropolar shells (in Russian). Izvestiya VUZov, Sev.-Kakavk. Region, Estestv. Nauki, Special issue: Nonlinear Problems of Continuum Mechanics, pp. 124–169 (2003)Google Scholar
  16. 16.
    Ericksen, J.L.: Symmetry transformations for thin elastic shells. Arch. Ration. Mech. Anal. 47, 1–14 (1972)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Ericksen, J.L.: Apparent symmetry of certain thin elastic shells. J. Meć. 12, 12–18 (1973)MathSciNetGoogle Scholar
  18. 18.
    Ericksen, J.L., Truesdell, C.: Exact theory of stress and strain in rods and shells. Arch. Ration. Mech. Anal. 1, 295–323 (1957)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Eringen, A.C., Kafadar, C.B.: Polar field theories. In. Eringen A.C. (ed.) Continuum Physics, vol. 4, pp. 1–75. Academic, New York (1976)Google Scholar
  20. 20.
    Eringen, A.C.: Microcontinuum Field Theory. I. Foundations and Solids. Springer, Berlin Heidelberg New York (1999)Google Scholar
  21. 21.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Konopińska, V., Pietraszkiewicz, W.: Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells. Int. J. Solids Struct. (2006) (in press). Available online 29 April 2006. DOI: 10.1016/j.ijsolstr.2006.04.030
  23. 23.
    Libai, A., Simmonds, J.G.: Nonlinear elastic shell theory. Adv. Appl. Mech. 23, 271–371 (1983)zbMATHCrossRefGoogle Scholar
  24. 24.
    Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd ed. Cambridge, UK (1998)Google Scholar
  25. 25.
    Makowski, J., Pietraszkiewicz, W., Stumpf, H.: Jump conditions in the nonlinear theory of thin irregular shells. J. Elast. 54, 1–26 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Makowski, J., Stumpf, H.: Buckling equations for elastic shells with rotational degrees of freedom undergoing finite strain deformation. Int. J. Solids Struct. 26, 353–368 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Man, C.-S., Cohen, H.: A coordinate-free approach to the kinematics of membranes. J. Elast. 16, 97–104 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Murdoch, A.I.: A coordinate free approach to surface kinematics. Glasg. Mat. J. 32, 299–307 (1990)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Murdoch, A.I.: A thermodynamical theory of elastic material interfaces. Q. J. Mech. Appl. Math. XXIX, 245–274 (1976)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Murdoch, A.I., Cohen, H.: Symmetry considerations for material surfaces. Arch. Ration. Mech. Anal. 72, 61–98 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Murdoch, A.I., Cohen, H.: Symmetry considerations for material surfaces. Addendum. Arch. Ration. Mech. Anal. 76, 393–400 (1981)CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Naghdi, P.M.: The theory of plates and shells. In: Flügge, S. (ed.) Handbuch der Physik, vol. VIa/2, pp. 425–640. Springer, Berlin Heidelberg New York (1972)Google Scholar
  33. 33.
    Nowacki, W.: Theory of Asymmetric Elasticity. Pergamon, Oxford (1986)zbMATHGoogle Scholar
  34. 34.
    Pietraszkiewicz, W.: Nonlinear theories of shells (in Polish). In: Woźniak, C. (ed.) Mechanics of Elastic Plates and Shells (in Polish), pp. 424–497. Wydawnictwo Naukowe PWN, Warszawa (2001)Google Scholar
  35. 35.
    Pietraszkiewicz, W., Chróścielewski, J., Makowski, J.: On dynamically and kinematically exact theory of shells. In: Pietraszkiewicz, W., Szymczak, C. (eds.) Shell Structures: Theory and Applications, pp. 163–167. Taylor & Francis, London (2005)Google Scholar
  36. 36.
    Reissner, E.: Linear and nonlinear theory of shells. In: Fung, Y.C., Sechler, E.E. (eds.) Thin Shell Structures, pp. 29–44. Prentice-Hall, Englewood Cliffs, NJ (1974)Google Scholar
  37. 37.
    Rubin, M.B.: Cosserat Theories: Shells, Rods and Points. Kluwer, Dordrecht (2000)zbMATHGoogle Scholar
  38. 38.
    Šilhavý, M.: The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin Heidelberg New York (1997)zbMATHGoogle Scholar
  39. 39.
    Spencer, A.J.M.: Theory of Invariants. In: Eringen A.C. (ed.) Continuum Physics, vol. 1, pp. 292–307. Academic, New York (1971)Google Scholar
  40. 40.
    Steigmann, D.J.: Elements of the theory of elastic surfaces. In: Fu, I.B., Ogden, R.W. (eds.) Nonlinear Elasticity: Theory and Applications, pp. 268–304. Cambridge University Press, UK (2001)Google Scholar
  41. 41.
    Steigmann, D.J., Ogden, R.W.: Elastic surface-substrate interaction. Proc. R. Soc. Lond. A 455, 437–474 (1999)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Truesdell, C., Noll, W.: The nonlinear field theories of mechanics. In: Flügge, S. (ed.) Handbuch der Physik, vol. III/3, pp. 1–602. Springer, Berlin Heidelberg New York (1965)Google Scholar
  43. 43.
    Truesdell, C.: Rational Thermodynamics. Springer, Berlin Heidelberg New York (1984)zbMATHGoogle Scholar
  44. 44.
    Truesdell, C.: A First Course in Rational Continuum Mechanics. Academic, New York (1977)zbMATHGoogle Scholar
  45. 45.
    Wang, C.-C.: On the response functions of isotropic elastic shells. Arch. Ration. Mech. Anal. 50, 81–98 (1973)CrossRefzbMATHGoogle Scholar
  46. 46.
    Wang C.-C., Truesdell, C.: Introduction to Rational Elasticity. Noordhoff, Leyden (1973)zbMATHGoogle Scholar
  47. 47.
    Zhilin, P.A.: Basic equations of the non-classical shell theory (in Russian). Tr. LPI 386, 29–46 (1982)Google Scholar
  48. 48.
    Zubov, L.M.: Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, Berlin Heidelberg New York (1997)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Rostov State University and South Scientific Center of the Russian Academy of SciencesRostov on DonRussia
  2. 2.Institute of Fluid-Flow Machinery of the Polish Academy of SciencesGdańskPoland

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