Acta Mechanica

, Volume 230, Issue 12, pp 4311–4320 | Cite as

Analytical solution of elastic deformations inside and outside circular contact area between tilted rigid punch and elastic half space

  • Yoji IguchiEmail author
  • Pasomphone Hemthavy
  • Shigeki Saito
  • Kunio Takahashi
Original Paper


This paper proposes an analytical model for the Boussinesq problem between a tilted rigid punch and an elastic half space to enable the analysis of elastic deformations inside and outside a contact area. Inside the contact area, two types of pressure distributions are applied: one generates a flat elastic deformation, and the other produces a tilted elastic deformation. The projection of this elastic deformation varies depending on the observed horizontal direction because the elastic deformation is non-axisymmetric. To calculate integrals for the non-axisymmetric elastic deformation, we use the polar coordinate system with two angular coordinates, which can enable the calculation of an integral at any arbitrary point. The proposed model can obtain the relationship between the pressure distribution and the elastic deformations inside and outside a contact area from any arbitrary direction. In addition, the normal load and torque applied inside the contact area are obtained, and these parameters are normalized using the contact radius and the elastic modulus. At the zero-pressure point around the contact edge, the elastic deformation is smooth.


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Hertz, H.: On the contact of elastic solids. J. Reine Angew. Math. 92, 156–171 (1881). CrossRefzbMATHGoogle Scholar
  2. 2.
    Johnson, K.L., Kendall, K., Roberts, A.D.: Surface energy and the contact of elastic solids. Proc. R. Soc. Lond. Math. Phys. Eng. Sci. 324, 301–313 (1971). CrossRefGoogle Scholar
  3. 3.
    Liechti, K.M., Schnapp, S.T., Swadener, J.G.: Contact angle and contact mechanics of a glass/epoxy interface. Int. J. Fract. 86, 361–374 (1997). CrossRefGoogle Scholar
  4. 4.
    Myshkin, N.K., Petrokovets, M.I., Kovalev, A.V.: Tribology of polymers: adhesion, friction, wear, and mass-transfer. Tribol. Int. 38, 910–921 (2005). CrossRefGoogle Scholar
  5. 5.
    Momozono, S., Takeuchi, H., Iguchi, Y., Nakamura, K., Kyogoku, K.: Dissipation characteristics of adhesive kinetic friction on amorphous polymer surfaces. Tribol. Int. 48, 122–127 (2012). CrossRefGoogle Scholar
  6. 6.
    Fujiwara, R., Hemthavy, P., Takahashi, K., Saito, S.: The effect of surface conductivity and adhesivity on the electrostatic manipulation condition for dielectric microparticles using a single probe. J. Micromech. Microeng. 26, 055010 (2016). CrossRefGoogle Scholar
  7. 7.
    Baek, D., Saito, S., Takahashi, K.: Estimating work of adhesion using spherical contact between a glass lens and a PDMS block. J. Adhes. Sci. Technol. 32, 158–172 (2018). CrossRefGoogle Scholar
  8. 8.
    Greenwood, J.A., Williamson, J.B.P., Bowden, F.P.: Contact of nominally flat surfaces. Proc. R. Soc. Lond. Ser. Math. Phys. Sci. 295, 300–319 (1966). CrossRefGoogle Scholar
  9. 9.
    Majumdar, A., Bhushan, B.: Fractal model of elastic–plastic contact between rough surfaces. J. Tribol. 113, 1–11 (1991). CrossRefGoogle Scholar
  10. 10.
    Persson, B.N.J.: Contact mechanics for randomly rough surfaces. Surf. Sci. Rep. 61, 201–227 (2006). CrossRefGoogle Scholar
  11. 11.
    Sridhar, I., Johnson, K.L., Fleck, N.A.: Adhesion mechanics of the surface force apparatus. J. Phys. Appl. Phys. 30, 1710–1719 (1997). CrossRefGoogle Scholar
  12. 12.
    Taljat, B., Zacharia, T., Kosel, F.: New analytical procedure to determine stress–strain curve from spherical indentation data. Int. J. Solids Struct. 35, 4411–4426 (1998). CrossRefzbMATHGoogle Scholar
  13. 13.
    Dao, M., Chollacoop, N., Van Vliet, K.J., Venkatesh, T.A., Suresh, S.: Computational modeling of the forward and reverse problems in instrumented sharp indentation. Acta Mater. 49, 3899–3918 (2001). CrossRefGoogle Scholar
  14. 14.
    Pelletier, H.: Predictive model to estimate the stress–strain curves of bulk metals using nanoindentation. Tribol. Int. 39, 593–606 (2006). CrossRefGoogle Scholar
  15. 15.
    Beghini, M., Bertini, L., Fontanari, V.: Evaluation of the stress–strain curve of metallic materials by spherical indentation. Int. J. Solids Struct. 43, 2441–2459 (2006). CrossRefzbMATHGoogle Scholar
  16. 16.
    Saito, S., Ochiai, T., Yoshizawa, F., Dao, M.: Rolling behavior of a micro-cylinder in adhesional contact. Sci. Rep. 6, 34063 (2016). CrossRefGoogle Scholar
  17. 17.
    Green, A.E.: On Boussinesq’s problem and penny-shaped cracks. Math. Proc. Camb. Philos. Soc. 45, 251–257 (1949). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sneddon, I.N.: The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3, 47–57 (1965). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Barber, J.R.: Some polynomial solutions for the non-axisymmetric Boussinesq problem. J. Elast. 14, 217–221 (1984). CrossRefzbMATHGoogle Scholar
  20. 20.
    Menga, N., Carbone, G.: The surface displacements of an elastic half-space subjected to uniform tangential tractions applied on a circular area. Eur. J. Mech. A Solids 73, 137–143 (2019). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Timoshenko, S., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York (1969)zbMATHGoogle Scholar
  22. 22.
    Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1987)Google Scholar
  23. 23.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  24. 24.
    Takahashi, K., Mizuno, R., Onzawa, T.: Influence of the stiffness of the measurement system on the elastic adhesional contact. J. Adhes. Sci. Technol. 9, 1451–1464 (1995). CrossRefGoogle Scholar
  25. 25.
    Sneddon, I.N.: Boussinesq’s problem for a flat-ended cylinder. Math. Proc. Camb. Philos. Soc. 42, 29 (1946). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of International Development EngineeringTokyo Institute of TechnologyMeguro-kuJapan
  2. 2.School of Environment and SocietyTokyo Institute of TechnologyMeguro-kuJapan

Personalised recommendations