Advertisement

Computational Mechanics

, Volume 64, Issue 4, pp 951–970 | Cite as

A concise frictional contact formulation based on surface potentials and isogeometric discretization

  • Thang X. DuongEmail author
  • Roger A. Sauer
Original Paper
  • 133 Downloads

Abstract

This work presents a concise theoretical and computational framework for the finite element formulation of frictional contact problems with arbitrarily large deformation and sliding. The aim of this work is to extend the contact theory based on surface potentials (Sauer and De Lorenzis in Comput Methods Appl Mech Eng 253:369–395, 2013) to account for friction. Coulomb friction under isothermal conditions is considered here. For a consistent friction formulation, we start with the first and second laws of thermodynamics and derive the governing equations at the contact interface. A so-called interacting gap can then be defined as a kinematic variable unifying both sliding/sticking and normal/tangential contact. A variational principle for the frictional system can then be formulated based on a purely kinematical constraint. The direct elimination approach applied to the tangential part of this constraint leads to the so-called moving friction cone approach of Wriggers and Haraldsson (Commun Numer Methods Eng 19:285–295, 2003). Compared with existing friction formulations, our approach reduces the theoretical and computational complexity. Several numerical examples are presented to demonstrate the accuracy and robustness of the proposed friction formulation.

Keywords

Contact mechanics Isogeometric analysis Moving friction cone Nonlinear finite element methods Sliding friction Thermodynamical consistency. 

Notes

Acknowledgements

The authors are grateful to the German Research Foundation (DFG) for supporting this research under Grants GSC 111 and SA1822/8-1.

References

  1. 1.
    Argento C, Jagota A, Carter WC (1997) Surface formulation for molecular interactions of macroscopic bodies. J Mech Phys Solids 45(7):1161–1183MathSciNetCrossRefGoogle Scholar
  2. 2.
    Borden MJ, Scott MA, Evans JA, Hughes TJR (2011) Isogeometric finite element data structures based on bezier extraction of NURBS. Int J Numer Methods Eng 87:15–47CrossRefzbMATHGoogle Scholar
  3. 3.
    Brivadis E, Buffa A, Wohlmuth B, Wunderlich L (2015) Isogeometric mortar methods. Comput Methods Appl Mech Eng 284(Supplement C):292–319MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Corbett CJ, Sauer RA (2014) NURBS-enriched contact finite elements. Comput Methods Appl Mech Eng 275:55–75MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Corbett CJ, Sauer RA (2015) Three-dimensional isogeometrically enriched finite elements for mixed-mode contact and debonding. Comput Methods Appl Mech Eng 284:781–806CrossRefzbMATHGoogle Scholar
  6. 6.
    De Lorenzis L, Wriggers P, Hughes TJR (2014) Isogeometric contact: a review. GAMM Mitteilungen 37:85–123MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Lorenzis L, Temizer I, Wriggers P, Zavarise G (2011) A large deformation frictional contact formulation using NURBS-based isogeometric analysis. Int J Numer Methods Eng 87:1278–1300MathSciNetzbMATHGoogle Scholar
  8. 8.
    De Lorenzis L, Wriggers P, Zavarise G (2012) A mortar formulation for 3D large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method. Comput Mech 49:1–20MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Del Piero G, Raous M (2010) A unified model for adhesive interfaces with damage, viscosity, and friction. Eur J Mech A Solid 29:496–507MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dimitri R, Zavarise G (2017) Isogeometric treatment of frictional contact and mixed mode debonding problems. Comput Mech 60(2):315–332MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dittmann M, Franke M, Temizer I, Hesch C (2014) Isogeometric analysis and thermomechanical mortar contact problems. Comput Methods Appl Mech Eng 274:192–212MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Duong TX, De Lorenzis L, Sauer RA (2018) A segmentation-free isogeometric extended mortar contact method. Comput Mech.  https://doi.org/10.1007/s00466-018-1599-0 zbMATHGoogle Scholar
  13. 13.
    Fischer KA, Wriggers P (2006) Mortar based frictional contact formulation for higher order interpolations using the moving friction cone. Comput Methods Appl Mech Eng 195:5020–5036MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gitterle M, Popp A, Gee MW, Wall WA (2010) Finite deformation frictional mortar contact using a semi-smooth Newton method with consistent linearization. Int J Numer Methods Eng 84(5):543–571MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hiermeier M, Wall WA, Popp A (2018) A truly variationally consistent and symmetric mortar-based contact formulation for finite deformation solid mechanics. Comput Methods Appl Mech Eng 342:532–540MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Khiêm VN, Itskov M (2017) An averaging based tube model for deformation induced anisotropic stress softening of filled elastomers. Int J Plast 90:96–115CrossRefGoogle Scholar
  18. 18.
    Kiliç K, Temizer I (2016) Tuning macroscopic sliding friction at soft contact interfaces: interaction of bulk and surface heterogeneities. Tribol Int 104:83–97CrossRefGoogle Scholar
  19. 19.
    Kim J-Y, Youn S-K (2012) Isogeometric contact analysis using mortar method. Int J Numer Methods Eng 89(12):1559–1581MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Krstulovic-Opara L, Wriggers P, Korelc J (2002) A \(C^1\)-continuous formulation for 3D finite deformation friction contact. Comput Mech 29:27–42MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Laursen TA (2002) Computational contact and impact mechanics: fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. Springer, BerlinzbMATHGoogle Scholar
  22. 22.
    Laursen TA, Simo JC (1993) A continuum-based finite element formulation for the implicit solution of multibody, large deformation frictional contact problems. Int J Numer Methods Eng 36:3451–3485MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lu J (2011) Isogeometric contact analysis: geometric basis and formulation for frictionless contact. Comput Methods Appl Mech Eng 200:726–741MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mergel JC, Sahli R, Scheibert J, Sauer RA (2018) Continuum contact models for coupled adhesion and friction. J Adhes 94:1–33CrossRefGoogle Scholar
  25. 25.
    Neto D, Oliveira M, Menezes L, Alves J (2016) A contact smoothing method for arbitrary surface meshes using Nagata patches. Comput Methods Appl Mech Eng 299:283–315MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ogden RW (1987) Non-linear elastic deformations. Dover Edition, MineolaGoogle Scholar
  27. 27.
    Persson BNJ (2000) Sliding friction: physical principles and application, 2nd edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
  28. 28.
    Popp A, Wohlmuth BI, Gee MW, Wall WA (2012) Dual quadratic mortar finite element methods for 3D finite deformation contact. SIAM J Sci Comput 34:B421–B446MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Puso MA, Laursen TA (2004) A mortar segment-to-segment contact method for large deformation solid mechanics. Comput Methods Appl Mech Eng 193:601–629CrossRefzbMATHGoogle Scholar
  30. 30.
    Raous M, Cangémi L, Cocu M (1999) A consistent model coupling adhesion, friction, and unilateral contact. Comput Methods Appl Mech Eng 177:383–399MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Sauer RA (2006) An atomic interaction based continuum model for computational multiscale contact mechanics. Ph.D. thesis, University of California, Berkeley, USAGoogle Scholar
  32. 32.
    Sauer RA (2011) Enriched contact finite elements for stable peeling computations. Int J Numer Methods Eng 87:593–616MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Sauer RA (2013) Local finite element enrichment strategies for 2D contact computations and a corresponding postprocessing scheme. Comput Mech 52(2):301–319MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sauer RA, De Lorenzis L (2013) A computational contact formulation based on surface potentials. Comput Methods Appl Mech Eng 253:369–395MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Sauer RA, De Lorenzis L (2015) An unbiased computational contact formulation for 3D friction. Int J Numer Methods Eng 101:251–280MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Sauer RA, Duong TX, Corbett CJ (2014) A computational formulation for constrained solid and liquid membranes considering isogeometric finite elements. Comput Methods Appl Mech Eng 271:48–68MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sauer RA, Li S (2007) An atomic interaction-based continuum model for adhesive contact mechanics. Finite Elem Anal Des 43(5):384–396MathSciNetCrossRefGoogle Scholar
  38. 38.
    Sauer RA, Li S (2008) An atomistically enriched continuum model for nanoscale contact mechanics and its application to contact scaling. J Nanosci Nanotech 8(7):3757–3773CrossRefGoogle Scholar
  39. 39.
    Seitz A, Farah P, Kremheller J, Wohlmuth BI, Wall WA, Popp A (2016) Isogeometric dual mortar methods for computational contact mechanics. Comput Methods Appl Mech Eng 301:259–280MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Shadowitz A (1988) The electromagnetic field. Dover Publications, New YorkGoogle Scholar
  41. 41.
    Simo J, Ju J (1987) Strain- and stress-based continuum damage models-I. Formulation. Int J Solids Struct 23(7):821–840CrossRefzbMATHGoogle Scholar
  42. 42.
    Temizer I (2013) A mixed formulation of mortar-based contact with friction. Comput Methods Appl Mech Eng 255:183–195MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Temizer I (2016) Sliding friction across the scales: thermomechanical interactions and dissipation partitioning. J Mech Phys Solids 89:126–148MathSciNetCrossRefGoogle Scholar
  44. 44.
    Temizer I, Wriggers P, Hughes T (2011) Contact treatment in isogeometric analysis with NURBS. Comput Methods Appl Mech Eng 200:1100–1112MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Temizer I, Wriggers P, Hughes TJR (2012) Three-dimensional mortar-based frictional contact treatment in isogeometric analysis with NURBS. Comput Methods Appl Mech Eng 209–212:115–128MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Weeger O, Narayanan B, Dunn ML (2018) Isogeometric collocation for nonlinear dynamic analysis of Cosserat rods with frictional contact. Nonlinear Dyn 91(2):1213–1227CrossRefGoogle Scholar
  47. 47.
    Wriggers P (2006) Computational contact mechanics, 2nd edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
  48. 48.
    Wriggers P, Haraldsson A (2003) A simple formulation for two-dimensional contact problems using a moving friction cone. Commun Numer Methods Eng 19:285–295MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Wriggers P, Krstulovic-Opara L (2004) The moving friction cone approach for three-dimensional contact simulations. Int J Comput Methods 01(01):105–119CrossRefzbMATHGoogle Scholar
  50. 50.
    Yang B, Laursen TA, Meng X (2005) Two dimensional mortar contact methods for large deformation frictional sliding. Int J Numer Methods Eng 62:1183–1225MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Aachen Institute for Advanced Study in Computational Engineering Science (AICES)RWTH Aachen UniversityAachenGermany

Personalised recommendations