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.
Similar content being viewed by others
References
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)
Altenbach, H., Zhilin, P.A.: The theory of elastic thin shells (in Russian). Adv. Mech. 11, 107–148 (1988)
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)
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)
Arfken G.B., Weber, H.J.: Mathematical Methods for Physicists. Springer, Berlin Heidelberg New York (2000)
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)
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) 63–94
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)
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)
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)
de Gennes, P.G.: The Physics of Liquid Crystals. Clarendon, Oxford (1974)
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)
Eremeyev, V.A., Pietraszkiewicz, W.: The nonlinear theory of elastic shells with phase transitions. J. Elasticity 74, 67–86 (2004)
Eremeyev, V.A., Zubov, L.M.: The theory of elastic and viscoelastic micropolar liquids. J. Appl. Math. Mech. 63, 755–767 (1999)
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)
Ericksen, J.L.: Symmetry transformations for thin elastic shells. Arch. Ration. Mech. Anal. 47, 1–14 (1972)
Ericksen, J.L.: Apparent symmetry of certain thin elastic shells. J. Meć. 12, 12–18 (1973)
Ericksen, J.L., Truesdell, C.: Exact theory of stress and strain in rods and shells. Arch. Ration. Mech. Anal. 1, 295–323 (1957)
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)
Eringen, A.C.: Microcontinuum Field Theory. I. Foundations and Solids. Springer, Berlin Heidelberg New York (1999)
Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)
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
Libai, A., Simmonds, J.G.: Nonlinear elastic shell theory. Adv. Appl. Mech. 23, 271–371 (1983)
Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd ed. Cambridge, UK (1998)
Makowski, J., Pietraszkiewicz, W., Stumpf, H.: Jump conditions in the nonlinear theory of thin irregular shells. J. Elast. 54, 1–26 (1999)
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)
Man, C.-S., Cohen, H.: A coordinate-free approach to the kinematics of membranes. J. Elast. 16, 97–104 (1986)
Murdoch, A.I.: A coordinate free approach to surface kinematics. Glasg. Mat. J. 32, 299–307 (1990)
Murdoch, A.I.: A thermodynamical theory of elastic material interfaces. Q. J. Mech. Appl. Math. XXIX, 245–274 (1976)
Murdoch, A.I., Cohen, H.: Symmetry considerations for material surfaces. Arch. Ration. Mech. Anal. 72, 61–98 (1979)
Murdoch, A.I., Cohen, H.: Symmetry considerations for material surfaces. Addendum. Arch. Ration. Mech. Anal. 76, 393–400 (1981)
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)
Nowacki, W.: Theory of Asymmetric Elasticity. Pergamon, Oxford (1986)
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)
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)
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)
Rubin, M.B.: Cosserat Theories: Shells, Rods and Points. Kluwer, Dordrecht (2000)
Šilhavý, M.: The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin Heidelberg New York (1997)
Spencer, A.J.M.: Theory of Invariants. In: Eringen A.C. (ed.) Continuum Physics, vol. 1, pp. 292–307. Academic, New York (1971)
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)
Steigmann, D.J., Ogden, R.W.: Elastic surface-substrate interaction. Proc. R. Soc. Lond. A 455, 437–474 (1999)
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)
Truesdell, C.: Rational Thermodynamics. Springer, Berlin Heidelberg New York (1984)
Truesdell, C.: A First Course in Rational Continuum Mechanics. Academic, New York (1977)
Wang, C.-C.: On the response functions of isotropic elastic shells. Arch. Ration. Mech. Anal. 50, 81–98 (1973)
Wang C.-C., Truesdell, C.: Introduction to Rational Elasticity. Noordhoff, Leyden (1973)
Zhilin, P.A.: Basic equations of the non-classical shell theory (in Russian). Tr. LPI 386, 29–46 (1982)
Zubov, L.M.: Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, Berlin Heidelberg New York (1997)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Eremeyev, V.A., Pietraszkiewicz, W. Local Symmetry Group in the General Theory of Elastic Shells. J Elasticity 85, 125–152 (2006). https://doi.org/10.1007/s10659-006-9075-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-006-9075-z