Skip to main content
SpringerLink
Log in

Derivation of a dual-mixed hp-finite element model for axisymmetrically loaded cylindrical shells

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

The two-field dual-mixed Fraeijs de Veubeke variational formulation of three-dimensional elasticity serves as the starting point of the derivation of a dimensionally reduced shell model presented in this paper. The fundamental variables of this complementary energy-based variational principle are the not a priori symmetric stress tensor and the skew-symmetric rotation tensor. The tensor of first-order stress functions is applied to satisfy translational equilibrium, while the rotation tensor plays the role of a Lagrange multiplier to ensure rotational equilibrium. The volumetric locking-free shell model uses unmodified three-dimensional constitutive equations, and no classical kinematical hypotheses are employed during the derivation. The numerical performance of the related low-order h-, and higher-order p-version finite elements developed for axisymmetrically loaded cylindrical shells is investigated by two representative model problems. It is numerically proven that no negative effect can be experienced when the thickness is small and tends to zero.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amara M., Thomas J.M.: Equilibrium finite elements for the linear elastic problem. Numer. Math. 33(4), 367–383 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold D.N., Falk R.S.: A new mixed formulation for elasticity. Numer. Math. 53(1–2), 13–30 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold D.N., Brezzi F., Douglas J. Jr.: PEERS: a new mixed finite element for plane elasticity. Jpn. J. Appl. Math. 1(2), 347–367 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnold D.N., Falk R.S., Winther R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comput. 76, 1699–1723 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Atluri S.N.: Alternate stress and conjugate strain measures, and mixed variational formulations involving rigid rotations, for computational analyses of finitely deformed solids, with application to plates and shells–I: theory. Comput. Struct. 18(1), 93–116 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bertóti E.: Indeterminacy of first order stress functions and the stress- and rotation-based formulation of linear elasticity. Comput. Mech. 14(3), 249–265 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bertóti E.: Dual-mixed hp finite element methods using first-order stress functions and rotations. Comput. Mech. 26(1), 39–51 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bertóti E.: Dual-mixed p and hp finite elements for elastic membrane problems. Int. J. Num. Meth. Eng. 53(1), 3–29 (2002)

    Article  MATH  Google Scholar 

  9. Bertóti, E.: Stress-based and locking-free hp finite elements for thin elastic plates. In: Mang, H.A., Rammerstorfer, F.G., JE (eds.) Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V), pp. 1–10. Vienna University of Technology, Austria (2002b)

  10. Bertóti, E.: Derivation of plate and shell models using the Fraeijs de Veubeke variational principle. In: Ramm, E., Wall, W.A., Bletzinger, K.U., Bischoff, M. (eds.) Proceedings of the Fifth International Conference on Computation of Shell and Spatial Structures, pp. 1–4. Austria (2005)

  11. Bertóti E.: On the stress function approach in three-dimensional elasticity. Acta Mech 190(1–4), 197–204 (2007)

    Article  MATH  Google Scholar 

  12. Brezzi F., Fortin M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    Book  MATH  Google Scholar 

  13. Cazzani A., Atluri S.N.: Four-noded mixed finite elements, using unsymmetric stresses, for linear analysis of membranes. Comput. Mech. 11(4), 229–251 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chapelle D., Bathe K.J.: Fundamental considerations for the finite element analysis of shell structures. Comput. Struct. 66(1), 19–36 (1998)

    Article  MATH  Google Scholar 

  15. Farhloul M., Fortin M.: Dual hybrid methods for the elasticity and stokes problems: a unifed approach. Numer. Math. 76(4), 419–440 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fraeijs de Veubeke B.M.: A new variational principle for finite elastic displacements. Int. J. Eng. Sci. 10(9), 745–763 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fraeijs de Veubeke, B.M.: Stress function approach. In: Proceedings of the World Congress on Finite Element Methods in Structural Mechanics, pp. J.1–J.51. Bournemouth (1975)

  18. Fraeijsde Veubeke B.M., Millard A.: Discretization of stress fields in the finite element method. J Frankl. Inst. 302(5–6), 389–412 (1976)

    Article  MathSciNet  Google Scholar 

  19. Gatica G.N., Meddahi S., Ma A.: A new dual-mixed finite element method for the plane linear elasticity problem with pure traction boundary conditions. Comput. Methods Appl. Mech. Eng. 197, 1115–1130 (2008)

    Article  MATH  Google Scholar 

  20. Hughes T.J.R.: The Finite Element Method—Linear Static and Dynamic Finite Element Analysis. Dover Publications, Inc., New York (2000)

    MATH  Google Scholar 

  21. Iura M., Atluri S.N.: Formulation of a membrane finite element with drilling degrees of freedom. Comput. Mech. 9(6), 417–428 (1992)

    Article  MATH  Google Scholar 

  22. Klaas O., Schröder J., Stein E., Miehe C.: A regularized dual mixed element for plane elasticity implementation and performance of the BDM element. Comput. Methods Appl. Mech. Eng. 121(1–4), 201–209 (1995)

    Article  MATH  Google Scholar 

  23. Kocsán L.G.: Analytical solution using first-order stress functions for axisymmetrically loaded cylindrical shells. GÉP 60(6), 21–31 (2009) in Hungarian

    Google Scholar 

  24. Naghdi P.: Foundations of elastic shell theory. In: Sneddon, I., Hill, R. (eds) Progress in Solid Mechanics, vol. 4, chap 1, pp. 3–90. North-Holland Publishing Co., Amsterdam (1963)

    Google Scholar 

  25. Punch E.F., Atluri S.N.: Large displacement analysis of plates by a stress-based finite element approach. Comput. Struct. 24(1), 107–117 (1986)

    Article  MATH  Google Scholar 

  26. Reissner E.: A note on variational principles in elasticity. Int. J. Solids. Struct. 1(1), 93–95 (1965)

    Article  MathSciNet  Google Scholar 

  27. Roberts J., Thomas J.M.: Mixed and hybrid methods. In: Ciarlet, P.G., Lions, J.L. (eds) Finite Element Methods (Part 1), Handbook of Numerical Analysis, vol. 2., pp. 523–639. Elsevier Science Publishers B.V., North-Holland (1991)

    Google Scholar 

  28. Schröder J., Klaas O., Stein E., Miehe C.: A physically nonlinear dual mixed finite element formulation. Comput. Methods Appl. Mech. Eng. 144(1–2), 77–92 (1997)

    Article  MATH  Google Scholar 

  29. Stein E., Rolfes R.: Mechanical conditions for stability and optimal convergence of mixed finite elements for linear plane elasticity. Comput. Methods Appl. Mech. Eng. 84(1), 77–95 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  30. Stenberg R.: On the construction of optimal mixed finite element methods for the linear elasticity problem. Numer. Math. 48(4), 447–462 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  31. Stenberg R.: A family of mixed finite elements for the elasticity problem. Numer. Math. 53(5), 513–538 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  32. Stenberg R., Suri M.: Mixed hp finite element methods for problems in elasticity and stokes flow. Numer. Math. 72(3), 367–389 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Stolarski H., Belytschko T.: Membrane locking and reduced integration for curved elements. J. Appl. Mech. 49(1), 172–176 (1982)

    Article  MATH  Google Scholar 

  34. Stolarski H., Belytschko T.: Shear and membrane locking in curved c 0 elements. Comput. Methods Appl. Mech. Eng. 41(3), 279–296 (1983)

    Article  MATH  Google Scholar 

  35. Suetake Y., Iura M., Atluri S.N.: A note on variational principles in elasticity. Comput. Model. Simul. Eng. 4(1), 42–49 (1999)

    Google Scholar 

  36. Szabó B., Babuška I.: Finite Element Analysis. Wiley, New York (1991)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanics, University of Miskolc, H-3515, Miskolc-Egyetemváros, Hungary

    Lajos György Kocsán

Authors
  1. Lajos György Kocsán
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lajos György Kocsán.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kocsán, L.G. Derivation of a dual-mixed hp-finite element model for axisymmetrically loaded cylindrical shells. Arch Appl Mech 81, 1953–1971 (2011). https://doi.org/10.1007/s00419-011-0530-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-011-0530-3

Keywords

Search

Navigation