Skip to main content
SpringerLink
Log in

One point quadrature shell elements for sheet metal forming analysis

  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Summary

Numerical simulation of sheet metal forming processes is overviewed in this work. Accurate and efficient elements, material modelling and contact procedures are three major considerations for a reliable numerical analysis of plastic forming processes. Two new quadrilaterals with reduced integration scheme are introduced for shell analysis in order to improve computational efficiency without sacryfying accuracy: the first one is formulated for plane stress condition and the second designed to include through-thickness effects with the consideration of the normal stress along thickness direction. Barlat’s yield criterion, which was reported to be adequate to model anisotropy of aluminum alloy sheets, is used together with a multi-stage return mapping method to account for plastic anisotropy of the rolled sheet. A brief revision of contact algorithms is included, specially the computational aspects related to their numerical implementation within sheet metal forming context. Various examples are given to demonstrate the accuracy and robustness of the proposed formulations.

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. N. El-Abbasi and S.A. Meguid (2000). A new shell element accounting for through-thickness deformation.Computer Methods in Applied Mechanics and Engineering,189, 841.

    Article  MATH  Google Scholar 

  2. N. El-Abbasi and S.A. Meguid (1998). Large deformation analysis of contact in degenerate shell elements.International Journal for Numerical Methods in Engineering 43, 1127.

    Article  MATH  Google Scholar 

  3. S. Ahmad, B.M. Irons and O.C. Zienkiewicz (1970). Analysis of thick and thin shell structure by curved finite elements.International Journal for Numerical Methods in Engineering,2, 419.

    Article  Google Scholar 

  4. U. Andelfinger and E. Ramm (1993). EAS-elements for two-dimensional, three-dimensional, plate and shell structures and their equivalence to HR-elements.International Journal for Numerical Methods in Engineering,36, 1311.

    Article  MATH  Google Scholar 

  5. F. Barlat and J. Lian (1989). Plastic behavior and stretchability of sheet metals. Part I: a yield function for orthotropic sheets under plane stress conditions.International Journal of Plasticity,5, 51.

    Article  Google Scholar 

  6. F. Barlat, D.J. Lege and J.C. Brem (1991). A six-component yield function for anisotropic metals.International Journal of Plasticity,7, 693.

    Article  Google Scholar 

  7. F. Barlat, K. Chung and O. Richmond (1993). Plastic dual potential for FCC metals and application to minimum plastic work path calculation.International Journal of Plasticity,9, 1.

    Article  Google Scholar 

  8. F. Barlat, Y. Maeda, K. Chung, M. Yanagawa, J.C. Brem, Y. Hayashida, D.J. Lege, K. Matsui, S.J. Murtha, S. Hattori, R.C. Becker and S. Makosey (1997). Yield function development for aluminum alloy sheet.J. Mech. Phys. Solids,45, 1727.

    Article  Google Scholar 

  9. K.J. Bathe (1996).Finite element procedures. Prentice-Hall International Editions.

  10. J.L. Batoz and M. Ben Tahar. Formulation et evaluation d’un nouvel elment quadrilatŕal á 12 D.L. pour la flexion des plaques minces. Dép. de Génie Méc., Univ. de Tech., Compiégne, France.

  11. T. Belytschko, W.K. Lin and B. Moran (2000).Nonlinear finite elements for continua and structures. John Wiley & Sons Ltd., Baffins Lane. Chichester, England.

    MATH  Google Scholar 

  12. T. Belytschko and C.S. Tsay (1983). A stabilization procedure for the quadrilateral plate element with one-point quadrature.International Journal for Numerical Methods in Engineering,19, 405.

    Article  MATH  Google Scholar 

  13. T. Belytschko, J.I. Lin and C. Tsay (1984). Explicit algorithms for the nonlinear dynamics of shells.Computer Methods in Applied Mechanics and Engineering,42, 225.

    Article  Google Scholar 

  14. T. Belytschko, J.S.J. Ong, W.K. Liu and J.M. Kennedy (1984). Hourglass control in linear and nonlinear problems.Computer Methods in Applied Mechanics and Engineering,43, 251.

    Article  MATH  Google Scholar 

  15. T. Belytschko and W. Bachrach (1986). Efficient implementation of quadrilaterals with high coarse-mesh accuracy.Computer Methods in Applied Mechanics and Engineering,54, 279.

    Article  MATH  MathSciNet  Google Scholar 

  16. T. Belytschko and L.P. Bindeman (1991). Assumed strain stabilization of the 4-node quadrilateral with 1-point quadrature for nonlinear problems.Computer Methods in Applied Mechanics and Engineering,88, 311.

    Article  MATH  MathSciNet  Google Scholar 

  17. T. Belytschko and I. Leviathan (1994). Physical stabilization of the 4-node shell element with one point quadrature.Computer Methods in Applied Mechanics and Engineering,113, 321.

    Article  MATH  MathSciNet  Google Scholar 

  18. T. Belytschko and M.O. Neal (1991). Contact-impact by the pinball algorithm with penalty and Lagrangian methods.International Journal for Numerical Methods in Engineering,31, 547.

    Article  MATH  Google Scholar 

  19. T. Belytschko and I. Leviathan (1994). Projection schemes for one-point quadrature shell elements.Computer Methods in Applied Mechanics and Engineering,115, 277.

    Article  MathSciNet  Google Scholar 

  20. D.J. Benson and J.O. Hallquist (1987). A single surface contact algorithm for the postbuckling analysis of shell structures. University of California, San Diego, La Jolla.

    Google Scholar 

  21. P. Betsch, F. Gruttmann and E. Stein (1996). A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains.Computer Methods in Applied Mechanics and Engineering 130, 57.

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Bischoff and E. Ramm (1997). Shear deformable shell elements for large strains and rotations.International Journal for Numerical Methods in Engineering,40, 4427.

    Article  MATH  Google Scholar 

  23. N. Buchter, E. Ramm and D. Roehl (1994). Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept.International Journal for Numerical Methods in Engineering,37, 2551.

    Article  Google Scholar 

  24. R.P.R. Cardoso, J.W. Yoon, J.J. Grácio, F. Barlat and J.M.A. César de Sá (2002). Development of a one point quadrature shell element for nonlinear applications with contact and anisotropy.Computer Methods in Applied Mechanics and Engineering,191, 5177.

    Article  MATH  Google Scholar 

  25. R.P.R. Cardoso (2002). Development of one point quadrature shell elements with anisotropic material models for sheet metal forming analysis.PhD thesis, University of Aveiro, Portugal.

    Google Scholar 

  26. R.P.R. Cardoso and J.W. Yoon (2005). One point quadrature shell element with throughthickness stretch.Computer Methods in Applied Mechanics and Engineering,194, 1161.

    Article  MATH  Google Scholar 

  27. J.M.A. César de Sá, R.M. Natal Jorge, R.A.F. Valente and P.M.A. Areias (2002). Development of shear locking-free shell elements using an enhanced assumed strain formulation.International Journal for Numerical Methods in Engineering,53, 1721.

    Article  MATH  Google Scholar 

  28. K. Chung and K.N. Shah (1992). Finite element simulation of sheet metal forming for planar anisotropic metals.International Journal of Plasticity,8, 453.

    Article  Google Scholar 

  29. M.A. Crisfield (1991),Non-linear Finite element analysis of solids and structures. Vol. 1. John Wiley & Sons Ltd., Baffins Lane, Chichester, England.

    Google Scholar 

  30. E.N. Dvorkin and K.J. Bathe (1984). A continuum mechanics based four-node shell element for general nonlinear analysis.Engineering and Computations,1, 77.

    Google Scholar 

  31. D.P. Flanagan and T. Belytschko (1981). A uniform strain hexaedron and quadrilateral with orthogonal hourglass control.International Journal for Numerical Methods in Engineering,17, 679.

    Article  MATH  Google Scholar 

  32. R. Frisch-Fay (1962),Flexible bars. Butterworths, London.

    MATH  Google Scholar 

  33. M. Goth and F. Ishise (1978). A finite element analysis of rigid-plastic deformation of the flange in a deep-drawing process based on a fourth-degree yield function.International Journal of Mechanical Science,20, 423.

    Article  Google Scholar 

  34. J.O. Hallquist, P.J. Benson and G.L. Gougrean (1986). Implementation of a modified Hughes-Liu shell into a fully vectorized explicit finite element code. In P. Berganet al, Finite Element Methods for Nonlinear Problems, Springer, Berlin),283.

    Google Scholar 

  35. R. Hauptmann and K. Schweizerhof (1998). A systematic development of solid-shell element formulations for linear and non-linear analyses employing only displacement degrees of freedom.International Journal for Numerical Methods in Engineering,42, 49.

    Article  MATH  Google Scholar 

  36. R. Hill (1948). A theory of the yielding and plastic flow of anisotropic metals.Proc. Roy. Soc. London A193, 281.

    MATH  Google Scholar 

  37. R. Hill (1979). Theoretical plasticity of textured aggregates.Math. Proc. Cambridge Philos. Soc.,85, 179.

    Article  MATH  MathSciNet  Google Scholar 

  38. R. Hill (1987). Constitutive dual potential in classical plasticity.J. Mech. Phys. Solids,35, 23.

    Article  MATH  MathSciNet  Google Scholar 

  39. R. Hill (1990). Constitutive modelling of orthotropic plasticity in sheet metals.J. Mech. Phys. Solids,38, 405.

    Article  MATH  MathSciNet  Google Scholar 

  40. R. Hill (1993). A user-friendly theory of orthotropic plasticity in sheet metals.International Journal of Mechanical Sciences,35, 19.

    Article  MATH  Google Scholar 

  41. H.C. Huang and E. Hinton (1984). A nine-node Lagrangian plate element with enhanced shear interpolation.Engineering and Computations,1, 369.

    Google Scholar 

  42. T.J.R. Hughes, R.L. Taylor and W. Kanoknukulchai (1977). A simple and efficient element for plate bending.International Journal for Numerical Methods in Engineering,11, 1529.

    Article  MATH  Google Scholar 

  43. T.J.R. Hughes, M. Cohen and M. Haroum (1978). Reduced selective integration techniques in finite element analysis of plates.Nuclear Engineering,46, 203.

    Article  Google Scholar 

  44. T.J.R. Hughes (1980). Generalization of selective integration procedures to anisotropic and nonlinear media.International Journal for Numerical Methods in Engineering,15, 1413.

    Article  MATH  MathSciNet  Google Scholar 

  45. T.J.R. Hughes and W.K. Liu (1981). Nonlinear finite element analysis of shell: Part I. Three-dimensional shells.Computer Methods in Applied Mechanics and Engineering,26, 331.

    Article  MATH  MathSciNet  Google Scholar 

  46. T.J.R. Hughes and W.K. Liu (1981). Nonlinear finite element analysis of shell: Part II. Two-dimensional shells.Computer Methods in Applied Mechanics and Engineering,27, 167.

    Article  MATH  MathSciNet  Google Scholar 

  47. T.J.R. Hughes and T.E. Tezduyar (1981). Finite elements based upon mindlin plate theory with particular reference to the four-node bilinear isoparametric element.Journal of Applied Mechanics,48, 587.

    Article  MATH  Google Scholar 

  48. T.J.R. Hughes and E. Carnoy (1983). Nonlinear finite element shell formulation accounting for large membrane strains.Computer Methods in Applied Mechanics and Engineering,39, 69.

    Article  MATH  Google Scholar 

  49. B.M. Irons (1976). The semiloof shell element.Finite Element Thin Shells Curved Members,11, 197.

    Google Scholar 

  50. N. Kikuchi and J.T. Oden (1988). Contact problems in elasticity: A study of variational inequalities and finite element methods.SIAM Publication, Philadelphia.

  51. A.P. Karafillis and M.C. Boyce (1993). A general anisotropic yield criterion using bounds and a transformation weighting tensor.J. Mech. Phys. Solids,41, 1859.

    Article  MATH  Google Scholar 

  52. T.A. Laursen and J.C. Simo (1993). Algorithmic symmetrization of Coulomb frictional problems using augmented Lagrangians.Computer Methods in Applied Mechanics and Engineering,108, 133.

    Article  MATH  MathSciNet  Google Scholar 

  53. T.A. Laursen (2002).Computational contact and impact mechanics: fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. Springer-Verlag.

  54. W.K. Liu, E.S. Law, D. Lam and T. Belytschko (1986). Resultant-stress degenerated shell element.Computer Methods in Applied Mechanics and Engineering,55, 259.

    Article  MATH  Google Scholar 

  55. L.E. Malvern (1969).Introduction to the mechanics of a continuous medium. Prentice-Hall, Englewood Cliffs, NJ.

    Google Scholar 

  56. J.E. Marsden and T.J.R. Hughes (1983).Mathematical foundations of elasticity. Prentice-Hall, Englewood Cliffs, NJ.

    MATH  Google Scholar 

  57. L.S.D. Morley (1963).Skew Plates and Structures. Pergamon Press, Oxford.

    MATH  Google Scholar 

  58. NUMISHEET’93 organizing committe (1993).Proc. of the 2nd International Conference NUMISHEET’93. In A. Machinouchiet al. (ed.), Isehara, Japan.

  59. M. Oldenburg and L. Nilsson (1994). The position code algorithm for contact searching.International Journal for Numerical Methods in Engineering,37, 359.

    Article  MATH  Google Scholar 

  60. J. Oliver and E. Oñate (1984). A total lagrangian formulation for the geometrically nonlinear analysis of structures using finite elements. Part I. Two-dimensional problems: shell and plate structures.International Journal for Numerical Methods in Engineering,20, 2253.

    Article  MATH  Google Scholar 

  61. M. Ortiz and P.M. Pinsky (1981). Global analysis methods for the solution of elastoplastic and viscoplastic dynamic problems.Rep. UCB/SESM 81/08, Dept. Civil Engrg., Univ. California, Berkeley.

    Google Scholar 

  62. M. Ortiz, P.M. Pinsky and R.L. Taylor (1983). Operator split methods for the numerical solution of the elastoplastic dynamic problem.Computer Methods in Applied Mechanics and Engineering,39, 137.

    Article  MATH  MathSciNet  Google Scholar 

  63. M. Ortiz and J.C. Simo (1986). An analysis of a new class of integration algorithms for elastoplastic relations.International Journal for Numerical Methods in Engineering,23, 353.

    Article  MATH  MathSciNet  Google Scholar 

  64. H. Parisch (1979). A critical survey of the 9-node degenerated shell element with special emphasis on thin shell application and reduced integration.Computer Methods in Applied Mechanics and Engineering,20, 323.

    Article  MATH  Google Scholar 

  65. H. Parisch (1981). Large displacements of shells including material nonlinearities.Computer Methods in Applied Mechanics and Engineering,27, 183.

    Article  MATH  Google Scholar 

  66. H. Parisch (1981). Nonlinear analysis of shells using isoparametric elements. In T.J.R. Hugheset al. (Eds.),ASME,48, ASME, New York.

    Google Scholar 

  67. H. Parisch (1991). An investigation of a finite rotation four node assumed strain shell element.International Journal for Numerical Methods in Engineering,31, 127.

    Article  MATH  Google Scholar 

  68. H. Parisch (1995). A continuum-based shell theory for non-linear applications.International Journal for Numerical Methods in Engineering,38, 1855.

    Article  MATH  Google Scholar 

  69. K.C. Park and G. Stanley (1986). A curvedC o shell element based on assumed natural-coordinate strains.Journal of Applied Mechanics,53, 278.

    MATH  Google Scholar 

  70. D. Peric and D.R.J. Owen (1992). Computational model for three dimensional contact problems with friction based on the penalty method.International Journal for Numerical Methods in Engineering,35, 1289.

    Article  MATH  Google Scholar 

  71. T.H.H. Pian and K. Sumihara (1984). Rational approach for assumed stress finite elements.International Journal for Numerical Methods in Engineering,20, 1685.

    Article  MATH  Google Scholar 

  72. E. Ramm (1977). A plate/shell element for large deflection and rotations. In K.J. Bathe, J.T. Oden and W. Wunderlish (Eds.),Formulation and Computational Algorithms in Finite Element Analysis, MIT Press, Cambridge, MA.

    Google Scholar 

  73. E. Ramm and A. Matzenmiller (1986). Large deformation shell analysis based on the degeneration concept. In T.J.R. Hughes, E. Hinton (Eds.),Finite Element Methods for Plate and Shell Structures, Pineridge Press, Swansea, 365.

    Google Scholar 

  74. D. Roehl and E. Ramm (1996). Large elasto-plastic finite element analysis of solids and shells with the enhanced assumed strain concept.International Journal of Solids and Structures,33, 3215.

    Article  MATH  MathSciNet  Google Scholar 

  75. J.C. Simo and R.L. Taylor (1986). A return mapping algorithm for plane stress elastoplasticity.International Journal for Numerical Methods in Engineering,22, 649.

    Article  MATH  MathSciNet  Google Scholar 

  76. J.C. Simo, D.D. Fox and M.S. Rifai (1988). Formulation and computational aspects of a stress resultant geometrically exact shell model.Proc. of the Int. Conf. on Computational Engineering Science, Atlanta, USA.

  77. J.C. Simo and D.D. Fox. On a stress resultant geometrically exact shell model. Part I: formulation and optimal parametrization.Computer Methods in Applied Mechanics and Engineering,72, 267.

  78. J.C. Simo, D.D. Fox and M.S. Rifai (1990). On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory.Computer Methods in Applied Mechanics and Engineering,79, 21.

    Article  MATH  MathSciNet  Google Scholar 

  79. J.C. Simo, M.S. Rifai and D.D. Fox (1990). On a stress resultant geometrically exact shell model. Part IV: variable thickness shells with through-thickness stretching.Computer Methods in Applied Mechanics and Engineering,81, 91.

    Article  MATH  MathSciNet  Google Scholar 

  80. J.C. Simo, F. Armero and R.L. Taylor (1993). Improved versions of assumed enhanced strain tri-linear elements or 3D finite deformation problems.Computer Methods in Applied Mechanics and Engineering,110, 359.

    Article  MATH  MathSciNet  Google Scholar 

  81. J.C. Simo, P. Wriggers and R.L. Taylor (1985). A perturbed Lagrangian formulation for the finite element solution of contact problems.Computer Methods in Applied Mechanics and Engineering,50, 163.

    Article  MATH  MathSciNet  Google Scholar 

  82. J.C. Simo and T.A. Laursen (1992). An augmented Lagrangian treatment of contact problems involving friction.Computers and Structures,42, 97.

    Article  MATH  MathSciNet  Google Scholar 

  83. G.M. Stanley (1985). Continuum-based shell elements.Ph.D. Dissertation, Applied Mechanics Division, Stanford University, Stanford, CA.

    Google Scholar 

  84. G.M. Stanley, K.C. Park and T.J.R. Hughes (1986). Continuum-based resultant shell. In T.J.R. Hughes, E. Hinton (Eds.),Finite Element Methods for Plate and Shell Structures, Pineridge Press, Swansea, 1.

    Google Scholar 

  85. J.A. Stricklin, W. Haisler, P. Tisdale and R. Gunderson (1969). A rapidly converging triangular plate element.J. AIAA,7, 180.

    Article  MATH  Google Scholar 

  86. C. Truesdell and W. Noll (1965). The non-linear field theories of mechanics. InEncyclopedia of Physics, S Flugge (Ed.), 3/3, Springer-Verlag, Berlin.

    Google Scholar 

  87. A. Vila, A.R. Ferran and A. Huerta (1995). Nonlinear finite element techniques using an objectoriented code.Monograph CIMNE 31, Int. Center for Num. Methods in Engrg., Barcelona, Spain.

  88. D.Y. Yang and Y.J. Kim (1986). A rigid-plastic finite element calculation for the analysis of general deformation of planar anisotropic sheet metals and its application.International Journal of Mechanical Science,28, 825.

    Article  MATH  Google Scholar 

  89. J.W. Yoon, I.S. Song, D.Y. Yang, K. Chung and F. Barlat (1995). Finite element method for sheet forming based on an anisotropic strain-rate potential and the convected coordinate system.International Journal of Mechanical Science,37, 733.

    Article  MATH  Google Scholar 

  90. J.W. Yoon (1997). Finite element formulation based on incremental deformation theory for sheet metal forming of planar anisotropic materials,Ph.D. Dissertation, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Korea.

    Google Scholar 

  91. J.W. Yoon, D.Y. Yang and K. Chung (1999). Elasto-plastic finite element method based on incremental deformation theory and continuum based shell elements for planar anisotropic sheet materials.Computer Methods in Applied Mechanics and Engineering,174, 23.

    Article  MATH  Google Scholar 

  92. J.W. Yoon, D.Y. Yang, K. Chung and F. Barlat (1999). A general elasto-plastic finite element formulation based on incremental deformation theory for planar anisotropy and its application to sheet metal forming.International Journal of Plasticity,15, 35.

    Article  MATH  Google Scholar 

  93. J.W. Yoon, F. Barlat, K. Chung, F. Pourboghrat and D.Y. Yang (2000). Earing predictions based on asymmetric nonquaratic yield function.International Journal of Plasticity,16, 1075.

    Article  MATH  Google Scholar 

  94. M.L. Wilkins (1964). Calculation of Elastic-Plastic flow,in Methods of Comp. Physics, B. Alderet al. (Eds.),3, Academic Press, New York.

    Google Scholar 

  95. E.L. Wilson, R.L. Taylor, W.P. Doherty and J. Ghaboussi (1973). Incompatible displacement models. InNumerical and Computer Models in Structural Mechanics, Fenves SJ, Perrone N, Robinson AR, Schnobrich WC (Eds.), Academic Press, New York,43.

    Google Scholar 

  96. P. Wriggers and G. Zavarise (1993). On the application of augmented Lagrangian techniques for nonlinear constitutive laws in contact interfaces.Communications in Numererical Methods in Engineering,9, 815.

    Article  MATH  Google Scholar 

  97. Z.H. Zhong (1993). Finite element procedures for contact-impact problems. New York: Oxford University Press.

    Google Scholar 

  98. O.C. Zienkiewicz, R.L. Taylor and J.M. Too (1977). Reduced integration technique in general analysis of plates and shells.International Journal for Numerical Methods in Engineering,3, 275.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Mechanical Technology and Automation, Univ. of Aveiro, Campus Univ. Santiago, 3810-193, Portugal

    Rui P. R. Cardoso

  2. Alcoa Technical Center, Alcoa Center, 100 Technical Drive, 15069, PA, USA

    Jeong-Whan Yoon

Authors
  1. Rui P. R. Cardoso
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeong-Whan Yoon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rui P. R. Cardoso.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cardoso, R.P.R., Yoon, JW. One point quadrature shell elements for sheet metal forming analysis. ARCO 12, 3–66 (2005). https://doi.org/10.1007/BF02736172

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02736172

Keywords

Search

Navigation