Advertisement

Computational Mechanics

, Volume 19, Issue 1, pp 105–119 | Cite as

On some mixed finite element methods for incompressible and nearly incompressible finite elasticity

  • U. Brink
  • E. Stein
Originals

Abstract

We compare some mixed methods based on different variational formulations, namely a displacement-pressure formulation employed by de Borst and coworkers, the three-field formulation investigated by Simo and Taylor and a two-field formulation which is directly based on an energy functional. It emerges that all these yield the same discrete results if the stored energy function contains a volumetric contribution 1/2k(J−1)2 whereJ is the volume dilatation, i.e., the Jacobian determinant of the deformation, andk is the bulk modulus. The equivalence holds for arbitrary 3D and plane strain elements. In the numerical examples the mixed formulations are discretized by the quadrilateral Q1/P0 and Q2/P1 elements and the triangular Crouzeix-Raviart P2+/P1 element. We also compare with standard displacement elements and the enhanced strain Q1/E4 element.

Keywords

Mixed Formulation Load Increment Incompressible Material Saddle Point Problem Finite Element Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Atluri, S. N.;Reissner, E. (1989): On the formulation of variational theorems involving volume constraints. Comput. Mech. 5: 337–344.zbMATHCrossRefGoogle Scholar
  2. van den Bogert, P. A. J.;de Borst, R. (1994): On the behaviour of rubberlike materials in compression and shear. Arch. Appl. Mech. 64: 136–146zbMATHGoogle Scholar
  3. van den Bogert, P. A. J.;de Borst, R.;Luiten, G. T.;Zeilmaker, J. (1991): Robust finite elements for 3D-analysis of rubber-like materials. Engng. Comput. 8: 3–17Google Scholar
  4. de Borst, R.;van den Bogert, P. A. J.;Zeilmaker, J. (1988): Modelling and analysis of rubberlike materials. HERON 33: 1–57Google Scholar
  5. Brezzi, F. (1974): On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrangian multipliers. RAIRO Anal. Numer. 8: 129–151MathSciNetGoogle Scholar
  6. Brezzi, F.;Fortin, M. (1991): Mixed and Hybrid Finite Element Methods. Berlin, Heidelberg, New York: SpringerzbMATHGoogle Scholar
  7. Chang, T. Y. P.;Saleeb, A. F.;Li, G. (1991): Large strain analysis of rubber-like materials based on a perturbed Lagrangian variational principle. Comput. Mech. 8: 221–233zbMATHCrossRefGoogle Scholar
  8. Chapelle, D.;Bathe, K.-J. (1993): The inf-sup test. Comp. & Struct. 47: 537–545zbMATHCrossRefMathSciNetGoogle Scholar
  9. Chen, J. S.;Satyamurthy, K.;Hirschfelt, L. R. (1994): Consistent finite element procedures for nonlinear rubber elasticity with a higher order strain energy function. Comp. & Struct. 50: 715–727zbMATHCrossRefGoogle Scholar
  10. Crouzeix, M.;Raviart, P.-A. (1973): Conforming and non conforming finite element methods for solving the stationary Stokes equations. RAIRO R3: 33–76MathSciNetGoogle Scholar
  11. Engelman, M. S.;Sani, R. L.;Gresho, P. M.;Bercovier, M. (1982): Consistent vs. reduced integration penalty methods for incompressible media using several old and new elements. Int. J. Numer. Meth. Fluids, 2: 25–42zbMATHCrossRefMathSciNetGoogle Scholar
  12. Fortin, M. (1981): Old and new finite elements for incompressible flows. Int. J. Numer. Meth. Fluids. 1: 347–364zbMATHCrossRefMathSciNetGoogle Scholar
  13. Girault, V.;Raviart, P.-A. (1986): Finite Element Methods for Navier-Stokes Equations. Berlin, Heidelberg, New York: SpringerzbMATHGoogle Scholar
  14. Le Tallec, P. (1981): Compatibility condition and existence results in discrete finite incompressible elasticity. Comp. Meth. Appl. Mech. Engrg. 27: 239–259zbMATHCrossRefGoogle Scholar
  15. Le Tallec, P. (1994): Numerical methods for nonlinear three-dimensional elasticity. In: Ciarlet, P. G.; Lions, J. L. (eds.): Handbook of Numerical Analysis, vol. 3, Amsterdam: ElsevierGoogle Scholar
  16. Liu, C. H.;Hofstetter, G.;Mang, H. A. (1994): 3D finite element analysis of rubber-like materials at finite strains. Engng. Comput. 11: 111–128Google Scholar
  17. Miehe, C. (1993): Computation of isotropic tensor functions. Comm. Numer. Meth. Engrg. 9: 889–896zbMATHCrossRefGoogle Scholar
  18. Miehe, C. (1994): Aspects of the formulation and finite element implementation of large strain isotropic elasticity. Int. J. Numer. Meth. Engrg 37: 1981–2004zbMATHCrossRefMathSciNetGoogle Scholar
  19. Ogden, R. W. (1984): Non-Linear Elastic Deformations. Chicester: Ellis Horwood-John WileyGoogle Scholar
  20. Reddy, B. D.;Simo, J. C. (1995): Stability and convergence of a class of enhanced strain methods. SIAM J. Numer. Anal. 32: 1705–1728zbMATHCrossRefMathSciNetGoogle Scholar
  21. Seki, W.;Atluri, S. N. (1994): Analysis of strain localization in strainsoftening hyperelastic materials, using assumed stress hybrid elements. Comput. Mech. 14: 549–585zbMATHCrossRefMathSciNetGoogle Scholar
  22. Seki, W.;Atluri, S. N. (1995): On newly developed assumed stress finite element formulations for geometrically and materially nonlinear problems. Finite Elements in Analysis and Design 21: 75–110zbMATHCrossRefMathSciNetGoogle Scholar
  23. Simo, J.;Armero, F. (1992): Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes. Int. J. Numer. Meth. Engrg. 33: 1413–1449zbMATHCrossRefMathSciNetGoogle Scholar
  24. Simo, J.;Armero, F.;Taylor, R. L. (1993): Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems. Comp. Meth. Appl. Mech. Engrg. 110: 359–386zbMATHCrossRefMathSciNetGoogle Scholar
  25. Simo, J.;Taylor, R. L. (1991): Quasi-incompressible finite elasticity in principle stretches. Continuum basis and numerical algorithms. Comp. Meth. Appl. Mech. Engrg. 85: 273–310zbMATHCrossRefMathSciNetGoogle Scholar
  26. Simo, J.;Taylor, R. L.;Pister, K. S. (1985): Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comp. Meth. Appl. Mech. Engrg. 51: 177–208zbMATHCrossRefMathSciNetGoogle Scholar
  27. Stein, E.;Müller-Hoeppe, N. (1987): Finite element analysis and algorithms for large elastic strains. In: Pande, G. N.; Middleton, J. (eds.): Numerical Techniques for Engineering Analysis and Design: Proceedings of NUMETA 87, D4, pp. 1–8 Dordrecht: Nijhoff Publ.Google Scholar
  28. Sussman, T.;Bathe, K.-J. (1987): A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Comp. & Struct. 26: 357–409.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • U. Brink
    • 1
  • E. Stein
    • 1
  1. 1.Institute for Structural and Computational Mechanics (IBNM)Universität HannoverHannoverGermany

Personalised recommendations