Advertisement

Computational Mechanics

, Volume 61, Issue 3, pp 319–329 | Cite as

Boundary element method for normal non-adhesive and adhesive contacts of power-law graded elastic materials

  • Qiang Li
  • Valentin L. Popov
Original Paper

Abstract

Recently proposed formulation of the boundary element method for adhesive contacts has been generalized for contacts of power-law graded materials with and without adhesion. Proceeding from the fundamental solution for single force acting on the surface of an elastic half space, first the influence matrix is obtained for a rectangular grid. The inverse problem for the calculation of required stress in the contact area from a known surface displacement is solved using the conjugate-gradient technique. For the transformation between the stresses and displacements, the Fast Fourier Transformation is used. For the adhesive contact of graded material, the detachment criterion based on the energy balance is proposed. The method is validated by comparison with known exact analytical solutions as well as by proving the independence of the mesh size and the grid orientation.

Keywords

Boundary element method Functionally graded materials Adhesive contact 

References

  1. 1.
    Bhushan B, Gupta BK (1991) Handbook of tribology: materials, coatings, and surface treatments. McGraw-Hill, New YorkGoogle Scholar
  2. 2.
    Roos JR, Celis JP, Fransaer J, Buelens C (1990) The development of composite plating for advanced materials. JOM 42:60–63. doi: 10.1007/BF03220440 CrossRefGoogle Scholar
  3. 3.
    Erdogan F (1995) Fracture mechanics of functionally graded materials. Compos Eng 5:753–770. doi: 10.1016/0961-9526(95)00029-M CrossRefGoogle Scholar
  4. 4.
    Jha DK, Kant T, Singh RK (2013) A critical review of recent research on functionally graded plates. Compos Struct 96:833–849. doi: 10.1016/j.compstruct.2012.09.001 CrossRefGoogle Scholar
  5. 5.
    Birman V, Keil T, Hosder S (2013) Functionally graded materials in engineering. Structural interfaces and attachments in biology. Springer, New York, pp 19–41CrossRefGoogle Scholar
  6. 6.
    Udupa G, Rao SS, Gangadharan KV (2014) Functionally graded composite materials: an overview. Procedia Mater Sci 5:1291–1299. doi: 10.1016/j.mspro.2014.07.442 CrossRefGoogle Scholar
  7. 7.
    Selvadurai AP (2007) The analytical method in geomechanics. Appl Mech Rev 60:87–106CrossRefGoogle Scholar
  8. 8.
    Borowicka H (1943) Die Druckausbreitung im Halbraum bei linear zunehmendem Elastizitätsmodul. Ingenieur-Archiv 14:75–82. doi: 10.1007/BF02084171 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Taylor DW (1948) Fundamentals of soil mechanics. Wiley, New YorkGoogle Scholar
  10. 10.
    Gibson RE (1967) Some results concerning displacements and stresses in a non-homogeneous elastic half-space. Géotechnique 17:58–67. doi: 10.1680/geot.1967.17.1.58 CrossRefGoogle Scholar
  11. 11.
    Booker JR, Balaam NP, Davis EH (1985) The behaviour of an elastic non-homogeneous half-space. Part I-line and point loads. Int J Numer Anal Methods Geomech 9:353–367. doi: 10.1002/nag.1610090405 CrossRefzbMATHGoogle Scholar
  12. 12.
    Booker JR, Balaam NP, Davis EH (1985) The behaviour of an elastic non-homogeneous half-space. Part II-circular and strip footings. Int J Numer Anal Methods Geomech 9:369–381. doi: 10.1002/nag.1610090406 CrossRefzbMATHGoogle Scholar
  13. 13.
    Giannakopoulos AE, Suresh S (1997) Indentation of solids with gradients in elastic properties: Part I. Point force. Int J Solids Struct 34:2357–2392. doi: 10.1016/S0020-7683(96)00171-0 CrossRefzbMATHGoogle Scholar
  14. 14.
    Giannakopoulos AE, Suresh S (1997) Indentation of solids with gradients in elastic properties: Part II. Axisymmetric indentors. Int J Solids Struct 34:2393–2428. doi: 10.1016/S0020-7683(96)00172-2 CrossRefzbMATHGoogle Scholar
  15. 15.
    Heß M (2016) A simple method for solving adhesive and non-adhesive axisymmetric contact problems of elastically graded materials. Int J Eng Sci 104:20–33. doi: 10.1016/j.ijengsci.2016.04.009 MathSciNetCrossRefGoogle Scholar
  16. 16.
    Paggi M, Zavarise G (2011) Contact mechanics of microscopically rough surfaces with graded elasticity. Eur J Mech A Solids 30:696–704. doi: 10.1016/j.euromechsol.2011.04.007 CrossRefzbMATHGoogle Scholar
  17. 17.
    Hyun S, Pei L, Robbins MO (2004) Finite-element analysis of contact between elastic self-affine surfaces. Phys Rev E 70:26117. doi: 10.1103/PhysRevE.70.026117 CrossRefGoogle Scholar
  18. 18.
    Abali BE, Völlmecke C, Woodward B, Kashtalyan M, Guz I, Müller WH (2012) Numerical modeling of functionally graded materials using a variational formulation. Contin Mech Thermodyn 24:377–390. doi: 10.1007/s00161-012-0244-y MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Putignano C, Afferrante L, Carbone G, Demelio G (2012) A new efficient numerical method for contact mechanics of rough surfaces. Int J Solids Struct 49:338–343. doi: 10.1016/j.ijsolstr.2011.10.009 CrossRefGoogle Scholar
  20. 20.
    Paggi M, Ciavarella M (2010) The coefficient of proportionality \(\kappa \) between real contact area and load, with new asperity models. Wear 268:1020–1029. doi: 10.1016/j.wear.2009.12.038 CrossRefGoogle Scholar
  21. 21.
    Pohrt R, Popov VL (2012) Normal contact stiffness of elastic solids with fractal rough surfaces. Phys Rev Lett 108:104301. doi: 10.1103/PhysRevLett.108.104301 CrossRefGoogle Scholar
  22. 22.
    Aleynikov S (2010) Spatial contact problems in geotechnics: boundary-element method. Springer, BerlinCrossRefzbMATHGoogle Scholar
  23. 23.
    Bemporad A, Paggi M (2015) Optimization algorithms for the solution of the frictionless normal contact between rough surfaces. Int J Solids Struct 69–70:94–105. doi: 10.1016/j.ijsolstr.2015.06.005 CrossRefGoogle Scholar
  24. 24.
    Yang C, Persson BNJ (2008) Molecular dynamics study of contact mechanics: contact area and interfacial separation from small to full contact. Phys Rev Lett 100:24303. doi: 10.1103/PhysRevLett.100.024303 CrossRefGoogle Scholar
  25. 25.
    Campañá C, Müser MH (2007) Contact mechanics of real vs randomly rough surfaces: a Green’s function molecular dynamics study. EPL Europhys Lett 77:38005CrossRefGoogle Scholar
  26. 26.
    Kusche S (2016) The boundary element method for viscoelastic material applied to the oblique impact of spheres. Facta Univ Ser Mech Eng 14:293–300. doi: 10.22190/FUME1603293K CrossRefGoogle Scholar
  27. 27.
    Kusche S (2016) Frictional force between a rotationally symmetric indenter and a viscoelastic half-space. ZAMM J Appl Math Mech 239:226–239. doi: 10.1002/zamm.201500169 MathSciNetGoogle Scholar
  28. 28.
    Pohrt R, Li Q (2014) Complete boundary element formulation for normal and tangential contact problems. Phys Mesomech 17:334–340. doi: 10.1134/S1029959914040109 CrossRefGoogle Scholar
  29. 29.
    Pohrt R, Popov VL (2015) Adhesive contact simulation of elastic solids using local mesh-dependent detachment criterion in boundary elements method. Facta Univ Ser Mech Eng 13:3–10Google Scholar
  30. 30.
    Jin F, Guo X (2013) Mechanics of axisymmetric adhesive contact of rough surfaces involving power-law graded materials. Int J Solids Struct 50:3375–3386. doi: 10.1016/j.ijsolstr.2013.06.007 CrossRefGoogle Scholar
  31. 31.
    Jin F, Guo X, Zhang W (2013) A unified treatment of axisymmetric adhesive contact on a power-law graded elastic half-space. J Appl Mech 80:61024. doi: 10.1115/1.4023980 CrossRefGoogle Scholar
  32. 32.
    Jin F, Zhang W, Wan Q, Guo X (2016) Adhesive contact of a power-law graded elastic half-space with a randomly rough rigid surface. Int J Solids Struct 81:244–249. doi: 10.1016/j.ijsolstr.2015.12.001 CrossRefGoogle Scholar
  33. 33.
    Burland JB, Longworth TI, Moore JFA (1977) A study of ground movement and progressive failure caused by a deep excavation in Oxford Clay. Géotechnique 27:557–591. doi: 10.1680/geot.1977.27.4.557 CrossRefGoogle Scholar
  34. 34.
    Lee D, Barber JR, Thouless MD (2009) Indentation of an elastic half space with material properties varying with depth. Int J Eng Sci 47:1274–1283. doi: 10.1016/j.ijengsci.2008.08.005 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Love HAE (1929) The Stress produced in a semi-infinite solid by pressure on part of the boundary. Philos Trans R Soc Lond 228:337–420CrossRefzbMATHGoogle Scholar
  36. 36.
    Johnson KL (1987) Contact mechanics. Cambridge University Press, CambridgezbMATHGoogle Scholar
  37. 37.
    Venner CH, Lubrecht AA (2000) Multilevel methods in lubrication. Elsevier, New YorkzbMATHGoogle Scholar
  38. 38.
    Stanley HM, Kato T (1997) An FFT-based method for rough surface contact. J Tribol 119:481–485CrossRefGoogle Scholar
  39. 39.
    Polonsky IA, Keer LM (1999) A numerical method for solving rough contact problems based on the multi-level multi-summation and conjugate gradient techniques. Wear 231:206–219CrossRefGoogle Scholar
  40. 40.
    Borodich FM, Galanov BA, Suarez-Alvarez MM (2014) The JKR-type adhesive contact problems for power-law shaped axisymmetric punches. J Mech Phys Solids 68:14–32. doi: 10.1016/j.jmps.2014.03.003 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Argatov I, Li Q, Pohrt R, Popov VL (2016) Johnson-Kendall-Roberts adhesive contact for a toroidal indenter. Proc R Soc Lond A Math Phys Eng Sci. doi: 10.1098/rspa.2016.0218 zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Institut für MechanikTechnische Universität BerlinBerlinGermany

Personalised recommendations