Skip to main content

Elastic Shells, Resultant Nonlinear Theory

  • Living reference work entry
  • First Online:
Encyclopedia of Continuum Mechanics

Synonyms

Six-field theory of shells; Six-parameter shell theory

Definition

Two-dimensional (2D) equilibrium conditions of this general shell model are exact resultant implications of corresponding equilibrium conditions of continuum mechanics. The 2D unique shell kinematics consists of six scalar finite translation and rotation fields. Two surface drilling couples and two work-conjugate surface drilling bendings appear in the description of the shell stress and strain state. These additional fields and the shell drilling rotation become of primary importance when formulating and analyzing problems of irregular shell structures with branchings, intersections, or junctions along singular surface curves and/or having connections with beams or columns.

Introduction

In classical linear and nonlinear models of shells, the reduction of 3D mechanical problem of a thin solid body to the 2D problem of the shell is usually achieved applying some simplifying assumptions about kinematics of...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

References

  • Altenbach H, Zhilin PA (1988) Theory of elastic simple shells (in Russian). Adv Mech 11:107–148

    Google Scholar 

  • Chróścielewski J, Makowski J, Stumpf H (1992) Genuinely resultant shell finite elements accounting for geometric and material non-linearity. Int J Numer Methods Eng 35:63–94

    Article  Google Scholar 

  • Chróścielewski J, Makowski J, Pietraszkiewicz W (2004) Statics and dynamics of multi-shells: nonlinear theory and finite element method (in Polish). IFTR PASci Press, Warsaw

    Google Scholar 

  • Cosserat E, Cosserat F (1909) Théorie des corps deformables. Hermann et Fils, Paris

    MATH  Google Scholar 

  • Eremeyev VA, Pietraszkiewicz W (2006) Local symmetry group in the general theory of thin shells. J Elast 85:125–152

    Article  Google Scholar 

  • Eremeyev VA, Zubov LM (2008) Mechanics of elastic shells (in Russian). Nauka, Moscow

    MATH  Google Scholar 

  • John F (1965) Estimates for the derivatives of the stresses in a thin shell and interior shell equations. Commun Pure Appl Math 18(1):235–267

    Article  MathSciNet  Google Scholar 

  • Koiter WT (1960) A consistent first approximation in the general theory of thin elastic shells. In: Proceedings of the IUTAM symposium on the theory of thin elastic shells. North-Holland, Amsterdam, pp 12–32

    Google Scholar 

  • Konopińska V, Pietraszkiewicz W (2007). Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells. Int J Solids Struct 44:352–367.

    Article  Google Scholar 

  • Libai A, Simmonds JG (1983) Nonlinear elastic shell theory. Adv Appl Mech 23:271–371

    Article  Google Scholar 

  • Libai A, Simmonds JG (1998) The nonlinear theory of elastic shells, 2nd edn. Cambridge University Press, Cambridge, UK

    Book  Google Scholar 

  • Naghdi PM (1963) Foundations of elastic shell theory. In: Sneddon IN, Hill R (eds) Progress in solid mechanics IV. North-Holland, Amsterdam, pp 1–90

    Google Scholar 

  • Naghdi PM (1972) The theory of shells and plates. In: Truesdell C (ed) Handbuch der Physik, Band VIa/2. Springer, Berlin et al, pp 425–640

    Google Scholar 

  • Pietraszkiewicz W (2018) Surface geometry, elements. In: Altenbach H, Őchsner A (eds) Encyclopedia of continuum mechanics. Springer, Berlin (in print)

    Google Scholar 

  • Pietraszkiewicz W, Chróścielewski J, Makowski J (2006). On dynamically and kinematically exact theory of shells. In: Pietraszkiewicz W, Szymczak C (eds) Shell structures: theory and applications. Taylor & Francis Group, London, pp. 163–167.

    Google Scholar 

  • Pietraszkiewicz W, Badur J (1983) Finite rotations in the description of continuum deformation. Int J Eng Sci 21(9):1097–1115

    Article  MathSciNet  Google Scholar 

  • Pietraszkiewicz W, Konopińska V (2014) Drilling couples and refined constitutive equations in the resultant geometrically non-linear theory of elastic shells. Int J Solids Struct 51:2133–2143

    Article  Google Scholar 

  • Reissner E (1974) Linear and nonlinear theory of shells. In: Fung YC, Sechler EE (eds) Thin shell structures. Prentice-Hall, Englewood Cliffs, pp 29–44

    Google Scholar 

  • Rubin MB (2000) Cosserat theories: shells, rods and points. Kluwer, Dordrecht

    Book  Google Scholar 

  • Rychter Z (1988) Global error estimates in Reissner theory of thin elastic shells. Int J Eng Sci 26:787–795

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wojciech Pietraszkiewicz .

Editor information

Editors and Affiliations

Section Editor information

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer-Verlag GmbH Germany

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Pietraszkiewicz, W. (2018). Elastic Shells, Resultant Nonlinear Theory. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_189-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-53605-6_189-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53605-6

  • Online ISBN: 978-3-662-53605-6

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

Publish with us

Policies and ethics