Skip to main content
SpringerLink
Log in

A macro-element for incompressible finite deformations based on a volume averaged deformation gradient

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

A three-dimensional 8-node brick continuum finite element formulation for incompressible finite elasticity is presented. The core idea is to introduce a substructure consisting of eight sub-elements inside each finite element, further referred to as macro-element. For each of the sub-elements, the deformation is averaged. The weak form for each sub-element is based on the Hu-Washizu principle. The response of each sub-element is assembled and projected onto the eight external nodes of the macro-element. The introduction of deformable sub-elements in case of incompressible elasticity has two major advantages. Firstly, it is possible to suppress locking by evaluating the volumetric part of the response only in the macro-element instead of in each of the sub-elements. Secondly, no integration is necessary due to the use of averaged deformations on the sub-element level. The idea originates from the Cosserat point element developed in Nadler and Rubin (Int J Solids Struct 40:4585–4614, 2003). A consistent transition between the Cosserat point macro-element and a displacement macro-element formulation using a kinematical description from the enhanced strain element formulation (Flanagan, Belytschko in Int J Numer Methods Eng 17:679–706, 1981) or (Belytschko et al. in Comput Methods Appl Mech Eng 43:251–276, 1984) and the principle of Hu-Washizu is presented. The performance is examined by means of numerical examples.

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. Nadler B, Rubin MB (2003) A new 3-d finite element for nonlinear elasticity using the theory of a Cosserat point. Int J Solids Struct 40: 4585–4614

    Article  MATH  Google Scholar 

  2. Flanagan DP, Belytschko T (1981) A uniform strain hexahedron and quadrilateral with orthogonal hourglass control. Int J Numer Methods Eng 17: 679–706

    Article  MATH  Google Scholar 

  3. Belytschko T, Ong JSJ, Liu WK, Kennedy JM (1984) Hourglass control in linear and nonlinear problems. Comput Methods Appl Mech Eng 43: 251–276

    Article  MATH  Google Scholar 

  4. Hughes TJR (1987) The finite element method. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  5. Malkus DS, Hughes TJR (1978) Mixed finite element methods - reduced and selective integration techniques: a unification of concepts. Comput Methods Appl Mech Eng 15: 63–81

    Article  MATH  Google Scholar 

  6. Pian THH, Sumihara K (1984) Rational Approach for assumed stress finite elements. Int J Numer Methods Eng 20: 1685–1695

    Article  MATH  Google Scholar 

  7. Simo JC, Rifai MS (1990) A class of assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29: 1595–1638

    Article  MATH  MathSciNet  Google Scholar 

  8. Belytschko T, Bindemann LP (1986) Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems. Comput Methods Appl Mech Eng 54: 279–301

    Article  MATH  Google Scholar 

  9. Reese S, Wriggers P, Reddy BD (2000) A new locking-free brick element technique for large deformation problems in elasticity. Comput Struct 75: 291–304

    Article  MathSciNet  Google Scholar 

  10. Reese S, Wriggers P (2000) A new stabilization concept for finite elements in large deformation problems. Int J Numer Methods Eng 48: 79–110

    Article  MATH  Google Scholar 

  11. Simo JC, Armero F (1992) Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int J Numer Methods Eng 33: 1413–1449

    Article  MATH  MathSciNet  Google Scholar 

  12. Taylor RL, Beresford PJ, Wilson EL (1976) A non-conforming element for stress analysis. Int J Numer Methods Eng 10: 1211–1219

    Article  MATH  Google Scholar 

  13. Hueck U, Wriggers P (1995) A formulation for the four-node quadrilateral element, part i: Plane element. Int J Numer Methods Eng 38: 3007–3037

    Article  MATH  MathSciNet  Google Scholar 

  14. Wriggers P, Hueck U (1996) A formulation of the enhanced qs6-element for large elastic deformations. Int J Numer Methods Eng 39: 3039–3053

    Article  MathSciNet  Google Scholar 

  15. Loehnert S, Boerner EFI, Rubin MB, Wriggers P (2005) Response of a nonlinear elastic general Cosserat brick element in simulations typically exhibiting locking and hourglassing. Comput Mech 36: 255–265

    Article  MATH  Google Scholar 

  16. Boerner EFI, Loehnert S, Wriggers P (2007) A new finite element based on the theory of a Cosserat point - Extension to initially distorted elements for 2D plane strain. Int J Numer Methods Eng 71: 454–472

    Article  MathSciNet  Google Scholar 

  17. Rubin MB, Jabareen M (2007) An improved 3-D brick Cosserat point element for irregular shaped elements. Comput Mech 40(6): 979–1004

    Article  MathSciNet  MATH  Google Scholar 

  18. Camacho GT, Ortiz M (1996) Computational modeling of impact damage in brittle materials. Int J Solids Struct 33: 2899–2938

    Article  MATH  Google Scholar 

  19. Guo Y, Ortiz M, Belytschko T, Repetto EA (2000) Triangular composite finite elements. Int J Numer Methods Eng 47: 287–316

    Article  MATH  MathSciNet  Google Scholar 

  20. Thoutireddy P, Molinari JF, Repetto EA, Ortiz M (2002) Tetrahedral composite finite elements. Int J Numer Methods Eng 53: 1337–1351

    Article  MATH  Google Scholar 

  21. Rubin MB, Jabareen M (2007) A 3D brick macro-element for nonlinear elasticity. Comput Mech (Submitted)

  22. Simo JC, Hughes TJR (1986) On the variational foundations of assumed strain methods. J Appl Mech 53: 51–54

    Article  MATH  MathSciNet  Google Scholar 

  23. Flory PJ (1961) Thermodynamic relations for high elastic materials. Trans Faraday Soc 57: 829–838

    Article  MathSciNet  Google Scholar 

  24. Simo JC, Armero F, Taylor RL (1993) Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems. Comput Meth Appl Mech Eng 110: 359–386

    Article  MATH  MathSciNet  Google Scholar 

  25. Crisfield MA, Moita GF, Jelenic G, Lyons LPR (1995) Enhanced lower-order element formulations for large strains. Comput Mech 17: 62–73

    Article  MATH  Google Scholar 

  26. Crisfield MA, Moita GF (1996) A co-rotational formulation for 2-D continua including incompatible modes. Int J Numer Methods Eng 39: 2619–2633

    Article  MATH  MathSciNet  Google Scholar 

  27. Moita GF, Crisfield MA (1996) A finite element formulation for 3-D continua using the co-rotational technique. Int J Numer Methods Eng 39: 3775–3792

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mechanics and Computational Mechanics, Leibniz University of Hannover, 30167, Hannover, Germany

    E. F. I. Boerner & P. Wriggers

Authors
  1. E. F. I. Boerner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Wriggers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. F. I. Boerner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Boerner, E.F.I., Wriggers, P. A macro-element for incompressible finite deformations based on a volume averaged deformation gradient. Comput Mech 42, 407–416 (2008). https://doi.org/10.1007/s00466-008-0250-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-008-0250-x

Keywords

Search

Navigation