, Volume 5, Issue 3, pp 308–325 | Cite as

Strength of adhesive contacts: Influence of contact geometry and material gradients

Open Access
Research Article


The strength of an adhesive contact between two bodies can strongly depend on the macroscopic and microscopic shape of the surfaces. In the past, the influence of roughness has been investigated thoroughly. However, even in the presence of perfectly smooth surfaces, geometry can come into play in form of the macroscopic shape of the contacting region. Here we present numerical and experimental results for contacts of rigid punches with flat but oddly shaped face contacting a soft, adhesive counterpart. When it is carefully pulled off, we find that in contrast to circular shapes, detachment occurs not instantaneously but detachment fronts start at pointed corners and travel inwards, until the final configuration is reached which for macroscopically isotropic shapes is almost circular. For elongated indenters, the final shape resembles the original one with rounded corners. We describe the influence of the shape of the stamp both experimentally and numerically.

Numerical simulations are performed using a new formulation of the boundary element method for simulation of adhesive contacts suggested by Pohrt and Popov. It is based on a local, mesh dependent detachment criterion which is derived from the Griffith principle of balance of released elastic energy and the work of adhesion. The validation of the suggested method is made both by comparison with known analytical solutions and with experiments. The method is applied for simulating the detachment of flat-ended indenters with square, triangle or rectangular shape of cross-section as well as shapes with various kinds of faults and to “brushes”. The method is extended for describing power-law gradient media.


adhesion boundary element method (BEM) flat-ended indenters gradient media 



Authors acknowledge the aßsistance of C. Jahnke in conduction of experiments and very useful discussions of adhesion with gradient media with M. Heß and E. Willert. This work has been conducted under partial financial support from DFG (Grant number PO 810/22-1).

Contributions of authors: R. Pohrt built the experimental setup and processed the experimental data. R. Pohrt and Q. Li executed the numerical simulations. Theoretical analysis was carried our primarily by V. L. Popov. All authors contributed equally to the writing of the manuscript.


  1. [1]
    Lee L H (Ed.). Fundamentals of Adhesion. New York: Springer Science & Business Media, 1991CrossRefGoogle Scholar
  2. [2]
    Dzyaloshinskii I E, Lifshitz E M, Pitaevskii L P. General Theory of van der Waals’ Forces. Soviet Physics Uspekhi 4: 153–176 (1961)CrossRefMATHGoogle Scholar
  3. [3]
    Landau L D, Lifshitz E M. Statistical Physics, Pt. 2, (Volume 9 of the Course of Theoretical Physics). Oxford: Pergamon Press, 1980Google Scholar
  4. [4]
    Afferrante L, Carbone G. The ultratough peeling of elastic tapes from viscoelastic substrates. Journal of the Mechanics and Physics of Solids 96: 223–234 (2016)CrossRefMathSciNetGoogle Scholar
  5. [5]
    Popov V L, Filippov A E, Gorb S N. Biological microstructures with high adhesion and friction. Numerical approach. Physics-Uspekhi 59(9): 829–845 (2016)CrossRefGoogle Scholar
  6. [6]
    Autumn K, Liang Y A, Tonia Hsieh S, Zesch W, Chan W P, Kenny T W, Fearing R, Full R J. Adhesive force of a single gecko foot-hair. Nature 405: 681–685 (2000)CrossRefGoogle Scholar
  7. [7]
    Köster S, Janshoff A. Editorial–Special issue on mechanobiology. Biochimica et Biophysica Acta (BBA) - Molecular Cell Research 1853(11, Part B): 2975–2976 (2015)CrossRefGoogle Scholar
  8. [8]
    Popov V L. Contact Mechanics and Friction─Physical Principles and Applications. Berlin: Springer-Verlag Berlin Heidelberg, 2010CrossRefMATHGoogle Scholar
  9. [9]
    Kendall K. Molecular Adhesion and Its Applications. New York (US): Springer Science & Business Media, 2001Google Scholar
  10. [10]
    Luan B, Robbins M O. The breakdown of continuum models for mechanical contacts. Nature 435(7044): 929–932 (2005)CrossRefGoogle Scholar
  11. [11]
    Ciavarella M. On Pastewka and Robbins’ criterion for macroscopic adhesion of rough surfaces. Journal of Tribology 139(3): 031404 (2017)CrossRefGoogle Scholar
  12. [12]
    Guduru P R. Detachment of a rigid solid from an elastic wavy surface: theory. Journal of the Mechanics and Physics of Solids 55(3): 445–472 (2007)CrossRefMATHGoogle Scholar
  13. [13]
    Johnson K L, Kendall K, Roberts A D. Surface energy and the contact of elastic solids. Proc. R. Soc. London A 324: 301–313 (1971)CrossRefGoogle Scholar
  14. [14]
    Maugis D. Adhesion of spheres: The JKR-DMT transition using a Dugdale model. Journal of Colloid and Interface Science 150(1): 243–269 (1992)CrossRefGoogle Scholar
  15. [15]
    Cheng A H-D, Cheng D T. Heritage and early history of the boundary element method. Engineering Analysis with Boundary Elements 29: 268–302 (2005)CrossRefMATHGoogle Scholar
  16. [16]
    Cruse T A. Boundary Element Analysis in Computational Fracture Mechanics. Kluwer, Dordrecht, 1988CrossRefMATHGoogle Scholar
  17. [17]
    Blandford G E, Ingraffea A R, Liggett J A. Two-dimensional stress intensity factor computations using the boundary element method. International Journal for Numerical Methods in Engineering 17(3): 387–404 (1974)CrossRefMATHGoogle Scholar
  18. [18]
    Pohrt R, Popov V L. Adhesive contact simulation of elastic solids using local mesh-dependent detachment criterion in Boundary Elements Method. Facta Universitatis, Series: Mechanical Engineering 13(1): 3–10 (2015)Google Scholar
  19. [19]
    Hulikal S, Bhattacharya K, Lapusta N. A threshold-force model for adhesion and mode I fracture. arXiv:1606.03166.Google Scholar
  20. [20]
    Rey V, Anciaux G, Molinari J-F. Normal adhesive contact on rough surfaces: efficient algorithm for FFT-based BEM resolution. Comput Mech, DOI: 10.1007/s00466-017-1392-5 (2017)Google Scholar
  21. [21]
    Kendall K. The adhesion and surface energy of elastic solids. Journal of Physics D: Applied Physics 4(8): 1186 (1971)CrossRefGoogle Scholar
  22. [22]
    Li Q, Popov V L. Indentation of flat-ended and tapered indenters with polygonal cross-section. Facta Universitatis Series: Mechanial Engineeering 14(3): 241–249 (2016)CrossRefGoogle Scholar
  23. [23]
    Holm R, Holm E. Electric Contacts Handbook. Berlin: Springer-Verlag, 1958MATHGoogle Scholar
  24. [24]
    Griffith A A. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London, A 221: 163–198 (1921)CrossRefGoogle Scholar
  25. [25]
    Pohrt R, Li Q. Complete boundary element formulation for normal and tangential contact problems. Physical Mesomechanics 17(4): 334–340 (2014)CrossRefGoogle Scholar
  26. [26]
    Putignano C, Afferrante L, Carbone G, Demelio G. A new efficient numerical method for contact mechanics of rough surfaces. International Journal of Solids and Structures 49(2): 338–343 (2012)CrossRefGoogle Scholar
  27. [27]
    Maugis D, Barquins M. Adhesive contact of a conical punch on an elastic half-space. Le Journal de Physique Lettres 42(5): 95–97 (1981)CrossRefGoogle Scholar
  28. [28]
    Li Q, Popov V L. Boundary element method for normal non-adhesive and adhesive contacts of power-law graded elastic materials. arXiv:1612.08395 (2016)Google Scholar
  29. [29]
    Argatov I I, Li Q, Pohrt R, Popov V L. Johnson-Kendall- Roberts Adhesive Contact for a Toroidal Indenter. Proceedings of the Royal Society of London, Series A 472(2191): (2016)Google Scholar
  30. [30]
    Popov V L. Basic ideas and applications of the method of reduction of dimensionality in contact mechanics. Physical Mesomechanics 15: 254–263 (2012)CrossRefGoogle Scholar
  31. [31]
    Maugis D. Adhesion of spheres: The JKR-DMT transition using a Dugdale model. Journal of Colloid and Interface Science 150(1): 243–269 (1992)CrossRefGoogle Scholar
  32. [32]
    Suresh S. Graded materials for resistance to contact deformation and damage. Science 292: 2447–2451 (2001)CrossRefGoogle Scholar
  33. [33]
    Jha D K, Kant T, Singh R K. A critical review of recent research on functionally graded plates. Composite Structures 96: 833–849 (2013)CrossRefGoogle Scholar
  34. [34]
    Hess M, Popov V L. Method of dimensionality reduction in contact mechanics and friction: A user’s handbook. II Powerlaw graded materials. Facta Universitatis, Series: Mechanical Engineering 14(3): 251–268 (2016)CrossRefGoogle Scholar
  35. [35]
    Heß M. A simple method for solving adhesive and nonadhesive axisymmetric contact problems of elastically graded materials. International Journal of Engineering Science 104: 20–33 (2016)CrossRefMathSciNetGoogle Scholar

Copyright information

© The author(s) 2017

Open Access: The articles published in this journal are distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MechanicsTechnische Universität BerlinBerlinGermany
  2. 2.National Research Tomsk Polytechnic UniversityTomskRussia

Personalised recommendations