On Universal Deformations of Nonlinear Isotropic Elastic Shells

  • Leonid M. ZubovEmail author
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 15)


A two-dimensional nonlinear model of the elastic shell is considered. It is assumed that the shell is the deformable surface with kinematically independent fields of translations and rotations. Within this model, several sets of finite non-uniform deformations are found, which for any isotropic shell satisfy equilibrium equations without surface loads. Universal solutions are obtained for six families of deformations, which are characterized by certain fields of surface translations. Each of these families consists of several subfamilies, which differ by the rotation field. It is found out that the closed isotropic spherical shell without external loads has four different equilibrium states, and the shell remains spherical in each of these states. The general expression for the rotation field of the isotropic micropolar plate is found, using the equilibrium equations and the distorsion tensor for uniform deformations. Equilibrium of the Cosserat membrane, which has the form of minimal surface, is studied.


Universal solution Nonlinear elasticity Cosserat shell Micropolar shell 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Altenbach, H., Zhilin, P.A.: General theory of elastic simple shells. Uspekhi Mekhaniki (Advances of Mechanics) 11(4), 107–148 (1988) (in Russian)Google Scholar
  2. 2.
    Chróścielewski, J., Makowski, J., Pietraszkiewicz, W.: Statyka i dynamika pow\({\l}\)ok wielop\({\l}\)atowych: Nieliniowa teoria i metoda elementów skończonych. Wydawnictwo IPPT PAN, Warszawa (2004)Google Scholar
  3. 3.
    Eremeyev, V.A., Zubov, L.M.: Mechanics of Elastic Shells. Nauka, Moscow (2008) (in Russian)Google Scholar
  4. 4.
    Libai, A., Simmonds, J.G.: Nonlinear Theory of Elastic Shells. 2nd edn. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  5. 5.
    Zhilin, P.A.: Mechanics of deformable directed surfaces. Int. J. Solid. Struct. 12(9–10), 635–648 (1976)CrossRefGoogle Scholar
  6. 6.
    Zubov, L.M.: Nonlinear Theory of Dislocations and Disclinations in Elastic Bodies. Springer, Berlin (1997)Google Scholar
  7. 7.
    Zubov, L.M.: The representation of the displacement gradient of isotropic elastic body in terms of the Piola stress tensor. J. Appl. Math. Mech. 40(6), 1012–1019 (1976)CrossRefGoogle Scholar
  8. 8.
    Zubov, L.M.: Methods of Nonlinear Elasticity in Shell Theory. Izdatelstvo Rostovskogo Universiteta, Rostov-on-Don (1982) (in Russian)Google Scholar
  9. 9.
    Zubov, L.M.: Universal solutions for isotropic incompressible micropolar solids. Doklady Physics 55(11), 551–555 (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.South Federal UniversityRostov on DonRussia

Personalised recommendations