Advertisement

A multi-scale FEM-BEM formulation for contact mechanics between rough surfaces

  • Jacopo BonariEmail author
  • Maria R. Marulli
  • Nora Hagmeyer
  • Matthias Mayr
  • Alexander Popp
  • Marco Paggi
Original Paper
  • 153 Downloads

Abstract

A novel multi-scale finite element formulation for contact mechanics between nominally smooth but microscopically rough surfaces is herein proposed. The approach integrates the interface finite element method (FEM) for modelling interface interactions at the macro-scale with a boundary element method (BEM) for the solution of the contact problem at the micro-scale. The BEM is used at each integration point to determine the normal contact traction and the normal contact stiffness, allowing to take into account any desirable kind of rough topology, either real, e.g. obtained from profilometric data, or artificial, evaluated with the most suitable numerical or analytical approach. Different numerical strategies to accelerate coupling between FEM and BEM are discussed in relation to a selected benchmark test.

Keywords

Contact mechanics Roughness Finite element method Boundary element method Multi-scale method 

Notes

Acknowledgements

The authors would like to acknowledge funding from the MIUR-DAAD Joint Mobility Program 2017 to the project “Multi-scale modelling of friction for large scale engineering problems”. The project has been granted by the Italian Ministry of Education, University and Research (MIUR) and by the Deutscher Akademischer Austausch Dienst (DAAD) through funds of the German Federal Ministry of Education and Research (BMBF).

References

  1. 1.
    Luan B, Robbins M (2005) The breakdown of continuum models for mechanical contacts. Nature 435:929–932CrossRefGoogle Scholar
  2. 2.
    Raja J, Muralikrishnan B, Fu S (2002) Recent advances in separation of roughness, waviness and form. Precis Eng 26:222–235CrossRefGoogle Scholar
  3. 3.
    Vakis A, Yastrebov V, Scheibert J, Nicola L, Dini D, Minfray C, Almqvist A, Paggi M, Lee S, Limbert G, Molinari J, Anciaux G, Echeverri Restrepo S, Papangelo A, Cammarata A, Nicolini P, Aghababaei R, Putignano C, Stupkiewicz S, Lengiewicz J, Costagliola G, Bosia F, Guarino R, Pugno N, Carbone G, Mueser M, Ciavarella M (2018) Modeling and simulation in tribology across scales: an overview. Tribol Int 125:169–199.  https://doi.org/10.1016/j.riboint.2018.02.005 CrossRefGoogle Scholar
  4. 4.
    Rabinowicz E (1965) Friction and wear of materials. Wiley, New YorkGoogle Scholar
  5. 5.
    Johnson K (1985) Contact mechanics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  6. 6.
    Goryacheva I (1998) Contact mechanics in tribology, vol 61. Springer, NetherlandsCrossRefGoogle Scholar
  7. 7.
    Persson B (2000) Sliding friction, physical principles and applications. Springer, BerlinCrossRefGoogle Scholar
  8. 8.
    Popov V (2010) Contact mechanics and friction. Springer, BerlinCrossRefGoogle Scholar
  9. 9.
    Popov V, Hess M (2015) Method of dimensionality reduction in contact mechanics and friction. Springer, BerlinCrossRefGoogle Scholar
  10. 10.
    Barber J (2018) Contact mechanics. Springer International Publishing, BerlinCrossRefGoogle Scholar
  11. 11.
    McCool J (1986) Comparison of models for the contact of rough surfaces. Wear 107:37–60CrossRefGoogle Scholar
  12. 12.
    Zavarise G, Borri-Brunetto M, Paggi M (2004) On the reliability of microscopical contact models. Wear 257:229–245CrossRefGoogle Scholar
  13. 13.
    Greenwood J, Williamson J (1966) Contact of nominally flat surfaces. Proc R Soc Lond A Math Phys Eng Sci 295:300–319.  https://doi.org/10.1098/rspa.1966.0242 CrossRefGoogle Scholar
  14. 14.
    Ciavarella M, Delfine V, Demelio G (2006) A “re-vitalized” greenwood and williamson model of elastic contact between fractal surfaces. J Mech Phys Solids 54:2569–2591.  https://doi.org/10.1016/j.jmps.2006.05.006 CrossRefzbMATHGoogle Scholar
  15. 15.
    Greenwood J (2006) A simplified elliptic model of rough surface contact. Wear 261:191–200CrossRefGoogle Scholar
  16. 16.
    Ciavarella M, Greenwood J, Paggi M (2008) Inclusion of “interaction” in the greenwood and williamson contact theory. Wear 265:729–734.  https://doi.org/10.1016/j.wear.2008.01.019 CrossRefGoogle Scholar
  17. 17.
    Majumdar A, Bhushan B (1990) Role of fractal geometry in roughness characterization and contact mechanics of surfaces. ASME J Tribol 112:205–216CrossRefGoogle Scholar
  18. 18.
    Borri-Brunetto M, Carpinteri A, Chiaia B (1999) Scaling phenomena due to fractal contact in concrete and rock fractures. Int J Fract 95:221–238.  https://doi.org/10.1023/A:1018656403170 CrossRefGoogle Scholar
  19. 19.
    Persson B, Albohr O, Tartaglino U, Volokitin A, Tosatti E (2005) On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion. J Phys Condens Matter 17:R1CrossRefGoogle Scholar
  20. 20.
    Andersson T (1981) The boundary element method applied to two-dimensional contact problems with friction. In: Boundary element methods, Vol 3, pp 239–258Google Scholar
  21. 21.
    Man K (1994) Contact mechanics using boundary elements, topics in engineering, vol 22. Southampton, BostonGoogle Scholar
  22. 22.
    Wriggers P, Reinelt J (2009) Multi-scale approach for frictional contact of elastomers on rough rigid surfaces. Comput Methods Appl Mech Eng 198:1996–2008.  https://doi.org/10.1016/j.cma.2008.12.021 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Leroux J, Fulleringer B, Nélias D (2010) Contact analysis in presence of spherical inhomogeneities within a half-space. Int J Solids Struct 47:3034–3049.  https://doi.org/10.1016/j.ijsolstr.2010.07.006 CrossRefzbMATHGoogle Scholar
  24. 24.
    Putignano C, Carbone G, Dini D (2015) Mechanics of rough contacts in elastic and viscoelastic thin layers. Int J Solids Struct 69–70:507–517.  https://doi.org/10.1016/j.ijsolstr.2015.04.034 CrossRefGoogle Scholar
  25. 25.
    Pei L, Hyun S, Molinari J, Robbins M (2005) Finite element modeling of elasto-plastic contact between rough surfaces. J Mech Phys Solids 53:2385–2409.  https://doi.org/10.1016/j.jmps.2005.06.008 CrossRefzbMATHGoogle Scholar
  26. 26.
    Hyun S, Pei L, Molinari J-F, Robbins M (2004) Finite-element analysis of contact between elastic self-affine surfaces. Phys Rev E 70:026117.  https://doi.org/10.1103/PhysRevE.70.026117 CrossRefGoogle Scholar
  27. 27.
    Paggi M, Reinoso J (2018) A variational approach with embedded roughness for adhesive contact problems. Mech Adv Mater Struct 10(1080/15376494):1525454Google Scholar
  28. 28.
    Zavarise G, Wriggers P, Stein E, Schrefler B (1992) Real contact mechanisms and finite element formulation-a coupled thermomechanical approach. Int J Numer Methods Eng 35:767–785CrossRefGoogle Scholar
  29. 29.
    Lenarda P, Gizzi A, Paggi M (2018) A modeling framework for electro-mechanical interaction between excitable deformable cells. Eur J Mech A/Solids 72:374–392MathSciNetCrossRefGoogle Scholar
  30. 30.
    Popp A, Wriggers P (2018) Contact modeling for solids and particles, CISM international centre for mechanical sciences. Springer International Publishing, BerlinCrossRefGoogle Scholar
  31. 31.
    Seitz A, Wall WA, Popp A (2019) Nitsche’s method for finite deformation thermomechanical contact problems. Comput Mech 63(6):1091–1110.  https://doi.org/10.1007/s00466-018-1638-x MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Seitz A, Wall WA, Popp A (2018) A computational approach for thermo-elasto-plastic frictional contact based on a monolithic formulation using non-smooth nonlinear complementarity functions. Adv Model Simul Eng Sci 5(1):5.  https://doi.org/10.1186/s40323-018-0098-3 CrossRefGoogle Scholar
  33. 33.
    Farah P, Wall WA, Popp A (2017) An implicit finite wear contact formulation based on dual mortar methods. Int J Numer Methods Eng 111(4):325–353.  https://doi.org/10.1002/nme.5464 MathSciNetCrossRefGoogle Scholar
  34. 34.
    Barber J (2003) Bounds on the electrical resistance between contacting elastic rough bodies. Proc R Soc Lond A 459:53–66.  https://doi.org/10.1098/rspa.2002.1038 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Barber J (2010) Elasticity, 3rd edn. Springer, DordrechtCrossRefGoogle Scholar
  36. 36.
    Bemporad A, Paggi M (2015) Optimization algorithms for the solution of the frictionless normal contact between rough surfaces. Int J Solids Struct 69–70:94–105CrossRefGoogle Scholar
  37. 37.
    Paggi M, Barber J (2011) Contact conductance of rough surfaces composed of modified rmd patches. Int J Heat Mass Transf 54:4664–4672.  https://doi.org/10.1016/j.ijheatmasstransfer.2011.06.011 CrossRefzbMATHGoogle Scholar
  38. 38.
    Conway H, Farnham K (1968) The relationship between load and penetration for a rigid, flat-ended punch of arbitrary cross section. Int J Eng Sci 6(9):489–496.  https://doi.org/10.1016/0020-7225(68)90001-3 CrossRefGoogle Scholar
  39. 39.
    Nakamura M (1993) Constriction resistance of conducting spots in an electric contact surface. WIT Trans Modell Simul 3:10.  https://doi.org/10.2495/BT930121 CrossRefGoogle Scholar
  40. 40.
    Peitgen H, Saupe D, Barnsley M (1988) The science of fractal images. Springer, BerlinzbMATHGoogle Scholar
  41. 41.
    Yovanivich MM, Devaal J, Hegazy AH (1982) A statistical model to predict thermal gap conductance between conforming rough surfaces, In: 3rd joint thermophysics, fluids, plasma and heat transfer conference, American Institute of Aeronautics and Astronautics, Reston, Virigina.  https://doi.org/10.2514/6.1982-888
  42. 42.
    Yovanivich MM, Hegazy AH, Devaal J (1982) Surface hardness distribution effects upon contact, gap and joint conductances, In: 3rd Joint thermophysics, fluids, plasma and heat transfer conference, American Institute of Aeronautics and Astronautics, Reston, Virigina.  https://doi.org/10.2514/6.1982-887

Copyright information

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

Authors and Affiliations

  1. 1.IMT School for Advanced Studies LuccaLuccaItaly
  2. 2.Institute for Mathematics and Computer-Based SimulationUniversity of the Bundeswehr MunichNeubibergGermany

Personalised recommendations