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


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11


  1. 1.

    Bhushan B, Gupta BK (1991) Handbook of tribology: materials, coatings, and surface treatments. McGraw-Hill, New York

    Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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 19–41

    Google 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

    Article  Google Scholar 

  7. 7.

    Selvadurai AP (2007) The analytical method in geomechanics. Appl Mech Rev 60:87–106

    Article  Google Scholar 

  8. 8.

    Borowicka H (1943) Die Druckausbreitung im Halbraum bei linear zunehmendem Elastizitätsmodul. Ingenieur-Archiv 14:75–82. doi:10.1007/BF02084171

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Taylor DW (1948) Fundamentals of soil mechanics. Wiley, New York

    Google 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

    Article  Google 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

    Article  MATH  Google 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

    Article  MATH  Google 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

    Article  MATH  Google 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

    Article  MATH  Google 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

    MathSciNet  Article  Google 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

    Article  MATH  Google 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

    Article  Google 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

    MathSciNet  Article  MATH  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google Scholar 

  22. 22.

    Aleynikov S (2010) Spatial contact problems in geotechnics: boundary-element method. Springer, Berlin

    Google 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

    Article  Google 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

    Article  Google 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 77:38005

    Article  Google 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

    Article  Google 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

    MathSciNet  Google 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

    Article  Google 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 13:3–10

    Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    MathSciNet  Article  MATH  Google 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 228:337–420

    Article  MATH  Google Scholar 

  36. 36.

    Johnson KL (1987) Contact mechanics. Cambridge University Press, Cambridge

    Google Scholar 

  37. 37.

    Venner CH, Lubrecht AA (2000) Multilevel methods in lubrication. Elsevier, New York

    Google Scholar 

  38. 38.

    Stanley HM, Kato T (1997) An FFT-based method for rough surface contact. J Tribol 119:481–485

    Article  Google 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 231:206–219

    Article  Google 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

    MathSciNet  Article  MATH  Google 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

    MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Valentin L. Popov.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, Q., Popov, V.L. Boundary element method for normal non-adhesive and adhesive contacts of power-law graded elastic materials. Comput Mech 61, 319–329 (2018).

Download citation


  • Boundary element method
  • Functionally graded materials
  • Adhesive contact