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Finite Element Methods for Rolling Contact

  • P. Wriggers
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 411)

Abstract

This contribution is concerned with finite-element-formulation of rolling contact problems. For this, first the theoretical background of continuum mechanics and contact kinematics is given for steady and non-steady rolling processes. This includes remarks on the implementation of time-dependent materials.

Next the basic finite element formulation for large deformation processes is presented for fixed and moving reference frames. The development of the discretization of contact contributions follows. Here standard approaches and new C 1-continuous contact elements are discussed for the case of frictional and frictionless contact. Examples show the performance of the different formulations and discretizations.

Keywords

Finite Element Method Contact Problem Reference Configuration Frictional Contact Contact Element 
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.

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References

  1. [AC91]
    P. Alart and A. Curnier. A mixed formulation for frictional contact problems prone to newton like solution methods. Computer Methods in Applied Mechanics and Engineering, 92: 353–375, 1991.ADSCrossRefMATHMathSciNetGoogle Scholar
  2. [Bat80]
    R. C. Batra. Quasistatic indentation of a rubber—covered roll by a rigid roll. International Journal for Numerical Methods in Engineering, 17: 1823–1833, 1980.ADSCrossRefGoogle Scholar
  3. [Bat82]
    K. J. Bathe. Finite Element Procedures in Engineering Analysis. Prentice-Hall, New Jersey, 1982.Google Scholar
  4. [Bat86]
    K. J. Bathe. Finite-Elemente-Methoden, Matrizen und lineare Algebra. Die Methode der finiten Elemente. L“osung von Gleichgewichtsbedingungen und Bewegungsgleichungen; Deutsche ”Ubersetzung von P. Zimmermann. Springer-Verlag, Berlin-Heidelberg-New York, 1986.Google Scholar
  5. [BC85]
    K. J. Bathe and A. B. Chaudhary. A solution method for planar and axisymmetric contact problems. International Journal for Numerical Methods in Engineering, 21: 65–88, 1985.ADSCrossRefMATHGoogle Scholar
  6. [BC98]
    E. Bittencourt and G. J. Creus. Finite element analysis of three-dimensional contact and impact in large deformation problems. Computers and Structures, 69: 219–234, 1998.CrossRefMATHGoogle Scholar
  7. [Ber84]
    D. P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York, 1984.Google Scholar
  8. [CA88]
    A. Curnier and P. Alart. A generalized newton method for contact problems with friction. J. Mec. Theor. Appl., 7: 67–82, 1988.MATHMathSciNetGoogle Scholar
  9. [Chr80]
    R. M. Christensen. A nonlinear theory of viscoelastocity for application to elastomers. Journal of Applied Mechanics, 47: 762–768, 1980.ADSCrossRefMATHGoogle Scholar
  10. [CHT92]
    A. Curnier, Q. C. He, and J. J. Telega. Formulation of unilateral contact between two elastic bodies undergoing finite deformation. C. R. Acad. Sci. Paris, 314: 1–6, 1992.MATHGoogle Scholar
  11. [Cia88]
    P. G. Ciarlet. Mathematical Elasticity I: Three-dimensional Elasticity.North-Holland, Amsterdam, 1988.Google Scholar
  12. [Cri91]
    M. A. Crisfield. Non-linear Finite Element Analysis of Solids and Structures, volume 1. J. Wiley, Chichester, 1991.Google Scholar
  13. [CSW99]
    C. Carstensen, O. Scherf, and P. Wriggers. Adaptive finite elements for elastic bodies in contact. SIAM Journal Scientific Computing, 20: 1605 1626, 1999.MathSciNetGoogle Scholar
  14. [DL76]
    G. Duvaut and J. L. Lions. Inequalities in Mechanics and Physics. Springer Verlag, Berlin, 1976.CrossRefMATHGoogle Scholar
  15. [DT85]
    G. Dhatt and G. Touzot. The Finite Element Method Displayed. J. Wiley, Chichester, 1985.Google Scholar
  16. [Eri67]
    A.C. Eringen. Mechanics of Continua. J. Wiley and Sons, New York, London, Sidney, 1967.Google Scholar
  17. [Far93]
    G. Farin. Curves and Surfaces for Computer Aided Geometric Design. A Practical Guide. Academic Press, Boston, third edition, 1993.Google Scholar
  18. [FCM99]
    L. Fourment, J. L. Chenot, and K. Mocellin. Numerical formulations and algorithms for solving contact problems in metal forming simulation. International Journal for Numerical Methods in Engineering, 46: 1435–1463, 1999.ADSCrossRefMATHGoogle Scholar
  19. [Fre76]
    B. Fredriksson. Finite element solution of surface nonlinearities in structural mechanics with special emphasis to contact and fracture mechanics problems. Computers and Structures, 6: 281–290, 1976.CrossRefMATHGoogle Scholar
  20. [FZ75]
    A. Francavilla and O. C. Zienkiewicz. A note on numerical computation of elastic contact problems. International Journal for Numerical Methods in Engineering, 9: 913–924, 1975.ADSCrossRefGoogle Scholar
  21. [GHSW99]
    D. Gross, W. Hauger, W. Schnell, and P. Wriggers. Technische Mechanik 4. Springer, Berlin, dritte edition, 1999.Google Scholar
  22. [Gia89]
    A. E. Giannokopoulos. The return mapping method for the integration of friction constitutive relations. Computers and Structures, 32: 157–168, 1989.CrossRefGoogle Scholar
  23. [GLT84]
    R. Glowinski and P. Le Tallec. Finite element analysis in nonlinear incompressible elasticity. In Finite Element, ’Vol. V: Special Problems in Solid Mechanics. Prentice—Hall, Englewood Cliffs, New Jersey, 1984.Google Scholar
  24. [GM99]
    S. Govindjee and P. A. Mihalic. Viscoelastic constitutive relations for the steady spinning of a cylinder. International Journal for Numerical Methods in Engineering, 1999.Google Scholar
  25. [HA96]
    A. Heege and P. Alart. A frictional contact element for strongly curved contact problems. International Journal for Numerical Methods in Engineering, 39: 165–184, 1996.ADSCrossRefMATHMathSciNetGoogle Scholar
  26. [HC93]
    J.-H. Heegaard and A. Curnier. An augmented lagrangian method for discrete large-slip contact problems. International Journal for Numerical Methods in Engineering, 36: 569–593, 1993.ADSCrossRefMATHMathSciNetGoogle Scholar
  27. [HGB85]
    J. O. Hallquist, G. L. Goudreau, and D. J. Benson. Sliding interfaces with contact—impact in large—scale lagrangian computations. Computer Methods in Applied Mechanics and Engineering, 51: 107–137, 1985.ADSCrossRefMATHMathSciNetGoogle Scholar
  28. [HK90]
    E. Hansson and A. Klarbring. Rigid contact modelled by cad surface. Engineering Computations, 7: 344–348, 1990.CrossRefGoogle Scholar
  29. [HSS92]
    J. O. Hallquist, K. Schweizerhof, and D. Stillman. Efficiency refinements of contact strategies and algorithms in explicit fe programming. In D. R. J. Owen, E. Hinton, and E. Onate, editors, Proceedings of COMPLAS III, pages 359–384. Pineridge Press, 1992.Google Scholar
  30. [HTK77]
    T. R. J. Hughes, R. L. Taylor, and W. Kanoknukulchai. A finite element method for large displacement contact and impact problems. In K. J. Bathe, editor, Formulations and Computational Algorithms in FE Analysis, pages 468–495, Boston, 1977. MIT-Press.Google Scholar
  31. [HW00]
    G. D. Hu and P. Wriggers. On the adaptive finite element method of steady-state rolling contact for hyperelasticity in finite deformations. Computer Methods in Applied Mechanics and Engineering, 2000.Google Scholar
  32. [Joh87]
    C. Johnson. Numerical solution of partial differential equations by the finite element method. Cambridge University Press, 1987.Google Scholar
  33. [JT88]
    W. Ju and R. L. Taylor. A perturbed lagrangian formulation for the finite element solution of nonlinear frictional contact problems. Journal of Theoretical and Applied Mechanics, 7: 1–14, 1988.MATHGoogle Scholar
  34. [Ka190]
    J. J.. Kalker. Three-dimensional Elastic Bodies in Rolling Contact. Kluwer Academic Publishers, Dordrecht, 1990.CrossRefGoogle Scholar
  35. [K1a86]
    A. Klarbring. A mathematical programming approach to three-dimensional contact problems with friction. Computer Methods in Applied Mechanics and Engineering, 58: 175–200, 1986.ADSCrossRefMATHMathSciNetGoogle Scholar
  36. [KO88]
    N. Kikuchi and J. T. Oden. Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia, 1988.CrossRefMATHGoogle Scholar
  37. [Kor97]
    J. Korelc. Automatic generation of finite-element code by simultaneous optimization of expressions. Theoretical Computer Science, 187: 231–248, 1997.CrossRefMATHGoogle Scholar
  38. [LMM99]
    W. N. Liu, G. Meschke, and H. A. Mang. A note on the algorithmic stabilization of 2d contact analyses. In L. Gaul and C. A. Brebbia, editors, Computational Methods in Contact Mechanics IV, pages 231–240, Southhampton, 1999. Wessex Institute.Google Scholar
  39. [L593]
    T. A. Laursen and J. C. Simo. A continuum-based finite element formulation for the implicit solution of multibody, large deformation frictional contact problems. International Journal for Numerical Methods in Engineering, 36: 3451–3485, 1993.ADSCrossRefMATHMathSciNetGoogle Scholar
  40. [Lue84]
    D. G. Luenberger. Linear and Nonlinear Programming. Addison-Wesley Publishing Company, second edition, 1984.Google Scholar
  41. [Ma169]
    L. E. Malvern. Introduction to the Mechanics of a Continous Medium. Prentice-Hall, Inc., Englewood Cliffs, 1969.Google Scholar
  42. [MM78]
    R. Michalowski and Z. Mroz. Associated and non—associated sliding rules in contact friction problems. Archives of Mechanics, 30: 259–276, 1978.MATHGoogle Scholar
  43. [Nac93]
    U. Nackenhorst. On the finite element analysis of steady state rolling contact. In M. H. Aliabadi and C. A. Brebbia, editors, Contact Mechanics, pages 53–60, Southampton, 1993. Computational Mechanics Publications.Google Scholar
  44. [Nac95]
    U. Nackenhorst. An adaptive finite element method to analyse contact problems. In M. H. Aliabadi and C. Alessandri, editors, Contact Mechanics II, Southampton, 1995. Computational Mechanics Publications.Google Scholar
  45. [Ode81]
    J. T. Oden. Exterior penalty methods for contact problems in elasticity. In W. Wunderlich, E. Stein, and K. J. Bathe, editors, Nonlinear Finite Element Analysis in Structural Mechanics, Berlin, 1981. Springer.Google Scholar
  46. [Ogd84]
    R. W. Ogden. Non-Linear Elastic Deformations. Ellis Horwood and John Wiley, Chichester, 1984.Google Scholar
  47. [OL86]
    J. T. Oden and T. L. Lin. On the general rolling contact problem for finite deformations of a viscoelastic cylinder. Computer Methods in Applied Mechanics and Engineering, 52: 297–367, 1986.ADSCrossRefMathSciNetGoogle Scholar
  48. [OM86]
    J. T. Oden and J. A. C. Martins. Models and computational methods for dynamic friction phenomena. Compûter Methods in Applied Mechanics and Engineering, 52: 527–634, 1986.ADSCrossRefMathSciNetGoogle Scholar
  49. [PC97]
    G. Pietrzak and A. Curnier. Continuum mechanics modeling and augmented lagrangian formulation of multibody, large deformation frictional contact problems. In D. R.J. Owen, E. Hinton, and E. Onate, editors, Proceedings of COMPLAS 5, pages 878–883, Barcelona, 1997. CIMNE.Google Scholar
  50. [SH98]
    J. C. Simo and T. J. R. Hughes. Computational Inelasticity. Springer, New York, Berlin, 1998.Google Scholar
  51. [Pie97]
    G. Pietrzak. Continuum mechanics modelling and augmented lagrangian formulation of large deformation frictional contact problems. Technical Report 1656, EPFL, Lausanne, 1997.Google Scholar
  52. [PT92]
    P. Papadopoulos and R. L. Taylor. A mixed formulation for the finite element solution of contact problems. Computer Methods in Applied Mechanics and Engineering, 94: 373–389, 1992.ADSCrossRefMATHGoogle Scholar
  53. [PZ84]
    J. Padovan and I. Zeid. Finite element analysis of steadily moving contact fields. Computers and Structures, 2: 111–200, 1984.Google Scholar
  54. [SL92]
    J. C. Simo and T. A. Laursen. An augmented lagrangian treatment of contact problems involving friction. Computers and Structures, 42: 97–116, 1992.CrossRefMATHMathSciNetGoogle Scholar
  55. [ST85]
    J. C. Simo and R. L. Taylor. Consistent tangent operators for rate independent elastoplasticity. Computer Methods in Applied Mechanics and Engineering, 48: 101–118, 1985.ADSCrossRefMATHGoogle Scholar
  56. [SW79]
    J. T. Stadter and R. O. Weiss. Analysis of contact through finite element gaps. Computers and Structures, 10: 867–873, 1979.CrossRefMATHGoogle Scholar
  57. [SWT85]
    J. C. Simo, P. Wriggers, and R. L. Taylor. A perturbed lagrangian formulation for the finite element solution of contact problems. Computer Methods in Applied Mechanics and Engineering, 50: 163–180, 1985.ADSCrossRefMATHMathSciNetGoogle Scholar
  58. [TN65]
    C. Truesdell and W. Noll. The nonlinear field theories of mechanics. In S. Flügge, editor, Handbuch der Physik 111/3. Springer, Berlin, Heidelberg, Wien, 1965.Google Scholar
  59. [TR94]
    P. Le Tallec and C. Rahier. Numerical methods of steady rolling for non-linear viscoelastic structures in finite deformations. International Journal for Numerical Methods in Engineering, 37: 1159–1186, 1994.ADSCrossRefMATHGoogle Scholar
  60. [TT60]
    C. Truesdell and R. Toupin. The classical field theories. In Handbuch der Physik III/1. Springer, Berlin, Heidelberg, Wien, 1960.Google Scholar
  61. [TW99]
    R. L. Taylor and P. Wriggers. Smooth surface discretization for large deformation frictionless contact. Technical report, University of California, Berkeley, February 1999. Report No. UCB/SEMM-99–04.Google Scholar
  62. [WI93]
    P. Wriggers and M. Imhof. On the treatment of nonlinear unilateral contact problems. Ing. Archiv, 63: 116–129, 1993.MATHGoogle Scholar
  63. [WK000]
    P. Wriggers and L. Krstulovic-Opara. On smooth finite element discretization for frictional contact problems. Zeitschrift für angewandte Mathematik und Mechanik, 2000.Google Scholar
  64. [WKOK99]
    P. Wriggers, L. Krstulovic-Opara, and J. Korelc. Development of 2d smooth polynomial frictional contact element based on a symbolic approach. In Wunderlich, Stein, Ramm, and Wriggers, editors, Proceedings of ECCM, München, 1999.Google Scholar
  65. [WM92]
    P. Wriggers and C. Miehe. On the treatment of contact contraints within coupled thermomechanical analysis. In D. Besdo and E. Stein, editors, Proc. of EUROMECH, Finite Inelastic Deformations, Berlin, 1992. Springer.Google Scholar
  66. [WM94]
    P. Wriggers and C. Miehe. Contact constraints within coupled thermomechanical analysis–a finite element model. Computer Methods in Applied Mechanics and Engineering, 113: 301–319, 1994.ADSCrossRefMATHMathSciNetGoogle Scholar
  67. [WN99]
    S. P. Wang and E. Nakamachi. The inside-outside contact search algorithm for finite element analysis. International Journal for Numerical Methods in Engineering, 40: 3665–3685, 1999.ADSCrossRefMathSciNetGoogle Scholar
  68. [WP92]
    J. R. Williams and A. P. Pentland. Superquadrics and modal dynamics for discrete elements in interactive design. Engineering Computations, 9: 115–127, 1992.CrossRefGoogle Scholar
  69. [Wri87]
    P. Wriggers. On consistent tangent matrices for frictional contact problems. In G.N. Pande and J. Middleton, editors, Proceedings of NUMETA 87, Dordrecht, 1987. M. Nijhoff Publishers.Google Scholar
  70. [Wri95]
    P. Wriggers. Finite element algorithms for contact problems. Archive of Computational Methods in Engineering, 2: 1–49, 1995.CrossRefMathSciNetGoogle Scholar
  71. [WS85]
    P. Wriggers and J.C. Simo. A note on tangent stiffnesses for fully nonlinear contact problems. Communications in Applied Numerical Methods, 1: 199–203, 1985.CrossRefMATHGoogle Scholar
  72. [WST85]
    P. Wriggers, J.C. Simo, and R.L. Taylor. Penalty and augmented lagrangian formulations for contact problems. In J. Middleton and G.N. Pande, editors, Proceedings of NUMETA Conference, Rotterdam, 1985. Balkema.Google Scholar
  73. [WVS90]
    P. Wriggers, T. Vu Van, and E. Stein. Finite-element-formulation of large deformation impact-contact -problems with friction. Computers and Structures, 37: 319–333, 1990.CrossRefMATHGoogle Scholar
  74. [WZ93]
    P. Wriggers and G. Zavarise. On the application of augmented lagrangian techniques for nonlinear constitutive laws in contact interfaces. Communications in Applied Numerical Methods, 9: 815–824, 1993.CrossRefMATHGoogle Scholar
  75. [ZT89]
    O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method, 4rd Ed., volume 1. McGraw Hill, London, 1989.Google Scholar
  76. [ZT91]
    O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method, 4rd Ed., volume 2. McGraw Hill, London, 1991.Google Scholar
  77. [ZWS95]
    G. Zavarise, P. Wriggers, and B. A. Schrefler. On augmented lagrangian algorithms for thermomechanical contact problems with friction. Interna- tional Journal for Numerical Methods in Engineering, 38: 2929–2949, 1995.ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2000

Authors and Affiliations

  • P. Wriggers
    • 1
  1. 1.University of HannoverHannoverGermany

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