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Complete boundary element formulation for normal and tangential contact problems

Abstract

The boundary element method as a numerical tool in contact mechanics is widely used and allows for surface roughness to be investigated with very fine grids. However, for every two grid points, influence coefficients have to be employed for every force-displacement combination. In this paper, we derive the matrixes of influence coefficients for the deformation of an elastic half space, starting from the classical solutions of Boussinesq and Cerruti. We show how to overcome complexity problems by using FFT-based fast convolution. A comprehensive algorithm is given for solving the case of dry Coulomb friction with partial slip. The resulting computer program can be used effectively in iterative schemes also in similar problems, such as mixed lubrication and notably improves the applicability of the boundary element method in contact mechanics.

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References

  1. 1.

    Johnson, K.L., Contact Mechanics, Cambridge: Cambrigde University Press, 2003.

    Google Scholar 

  2. 2.

    Popov, V.L., Contact Mechanics and Friction. Physical Principles and Applications, Berlin: Springer, 2009.

    Google Scholar 

  3. 3.

    Zavarise, G., Borri-Brunetto, M., and Paggi, M., On the Resolution Dependence of Micromechanical Contact Models, Wear, 2007, vol. 262, pp. 42–54.

    Article  Google Scholar 

  4. 4.

    Pohrt, R. and Popov, V., Contact Stiffness of Randomly Rough Surfaces, Sci. Rep., 2013, vol. 3, p. 3293.

    ADS  Article  Google Scholar 

  5. 5.

    Hamrock, B.J. and Dowson, D., Isothermal Elastohydro-dynamic Lubrication of Point Contacts. Part I. Theoretical Formulation, ASME J. Lubr. Tech., 1976, vol. 98, pp. 223–229.

    Article  Google Scholar 

  6. 6.

    Spikes, H.A., Sixty Years of EHL, Lubr. Sci., 2006, vol. 18, pp. 265–291.

    Article  Google Scholar 

  7. 7.

    Love, H.A.E., The Stress Produced in a Semi-Infinite Solid by Pressure on Part of the Boundary, Philos. Trans. Roy. Soc. London, 1929, vol. 228, pp. 377–420.

    ADS  Article  MATH  Google Scholar 

  8. 8.

    Li, J. and Berger, E.J., A Boussinesq-Cerruti Solution Set for Constant and Linear Distribution of Normal and Tangential Load over a Triangular Area, J. Elasticity, 2001, vol. 63, pp. 137–151.

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Boussinesq, V.J., Application des Potentiels I’Etude de I’Equilibre et du. S. 1, Gautier-Villar, 1882.

    Google Scholar 

  10. 10.

    Hertz, H., Über die Berührung fester elastischer Körper, J. für die reine und angewandte Mathematik, 1881, vol. 92, pp.156–171.

    Google Scholar 

  11. 11.

    Pohrt, R., Popov, V.L., and Filippov, A.E., Normal Contact Stiffness of Elastic Solids with Fractal Rough Surfaces for One- and Three-Dimensional Systems, Phys. Rev. E, 2012, vol. 86, pp. 026710.

    ADS  Article  Google Scholar 

  12. 12.

    Grzemba, B., Pohrt, R., Teidelt, E., and Popov, V.L., Maximum Micro-Slip in Tangential Contact of Randomly Rough Self-affine Surfaces, Wear, 2014, vol. 309, no. 1–2, pp. 256–258.

    Article  Google Scholar 

  13. 13.

    Lubrecht, A.A. and Ioannides, E., A Fast Solution of the Dry Contact Problem and the Associated Sub-Surface Stress Field, Using Multilevel Techniques, ASME J. Tribo-logy, 1991, vol. 113, pp. 128–133.

    Article  Google Scholar 

  14. 14.

    Venner, C.H. and Lubrecht, A.A., Multilevel Methods in Lubrication, Amsterdam: Elsevier, 2000.

    Google Scholar 

  15. 15.

    Cho, Y.-J., Koo, Y.-P., and Kim, T.-W., A New FFT Technique for the Analysis of Contact Pressure and Subsurface Stress in a Semi-Infinite Solid, KSME Int. J., 2000, vol. 14/3, pp. 331–337.

    Google Scholar 

  16. 16.

    Stanley, H.M. and Kato, T., An FFT-Based Method for Rough Surface Contact, J. Tribology, 1984, vol. 119(3), pp. 481–485.

    Google Scholar 

  17. 17.

    Wang, W.Z., Wang, H., Liu, Y.C., and Hu, Y.Z., A Comparative Study of the Methods for Calculation of Surface Elastic Deformation, J. Eng. Tribology: Proc. Inst. of Mechanical Engineers, 2003, vol. 217,part J, pp. 145–153.

    Google Scholar 

  18. 18.

    Polonsky, I.A. and Keer, L.M., A Numerical Method for Solving Rough Contact Problems Based on the Multi-Level Multi-Summation and Conjugate Gradient Techniques, Wear, 1999, vol. 231, pp. 206–219.

    Article  Google Scholar 

  19. 19.

    Allwooda, J. and Ciftci, H., An Incremental Solution Method for Rough Contact Problems, Wear, 2004, vol. 258, pp.1601–1615.

    Article  Google Scholar 

  20. 20.

    Mindlin, R., Compliance of Elastic Bodies in Contact, ASME J. Appl. Mech., 1949, vol. 16, pp. 259–262.

    MATH  MathSciNet  Google Scholar 

  21. 21.

    Wang, F.S., Block, J.M., Chen, W.W., Martini, A., Zhou, K., Keer, L.M., and Wang, Q.J., A Multilevel Model for Elastic-Plastic Contact Between a Sphere and a Flat Rough Surface, J. Tribology, 2009, vol. 131, pp. 021409.

    Article  Google Scholar 

  22. 22.

    Zhu, D. and Wang, J., Mixed Elastohydrodynamic Lubrication in Finite Roller Contacts Involving Realistic Geometry and Surface Roughness, J. Tribology, 2012, vol. 134, p. 011504.

    Article  Google Scholar 

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Correspondence to R. Pohrt.

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Original Text © R. Pohrt, Q. Li, 2014, published in Fizicheskaya Mezomekhanika, 2014, Vol. 17, No. 3, pp. 118–123.

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Pohrt, R., Li, Q. Complete boundary element formulation for normal and tangential contact problems. Phys Mesomech 17, 334–340 (2014). https://doi.org/10.1134/S1029959914040109

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Keywords

  • boundary element method
  • influence matrix
  • elastic deformation
  • contact mechanics
  • dry friction
  • mixed lubrication