Skip to main content
Log in

Analytical solution of axisymmetric indentation of multi-layer coating on elastic substrate body

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

We consider the axisymmetric contact problem of a multi-elastic layer with various elastic constants bonded to an elastic semi-infinite substrate indented by rigid flat-ended cylindrical and spherical indenters. The transfer matrix method is applied to each elastic layer, and dual integral equations are reduced to an infinite system of simultaneous equations by expressing the normal contact stress at the surface elastic layer as an appropriate series with Chebyshev orthogonal polynomials. Numerical results demonstrate the effects of the elastic constant of each elastic layer and the semi-infinite elastic substrate on the radial distribution of the normal contact stress and normal displacement of the free surface of the elastic layer, stress singularity factor at the edge of the cylindrical indenter, and axial load of a rigid indenter which penetrates the multi-layer material to a constant depth. The results of axial load are in good agreement with previously reported results. The numerical results are given for several combinations of the shear modulus of each elastic layer and the substrate. These results will contribute to the establishment of indentation tests for composite materials and serve as guidelines for the design of appropriate mechanical properties of layered materials.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Harding, J.W., Sneddon, I.N.: The elastic stresses produced by the indentation of the plane surface of a semi-infinite elastic solid by a rigid punch. Math. Proc. Camb. Philos. Soc. 41, 16 (1945). https://doi.org/10.1017/S0305004100022325

    Article  MathSciNet  MATH  Google Scholar 

  2. 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). https://doi.org/10.1016/0020-7225(65)90019-4

    Article  MathSciNet  MATH  Google Scholar 

  3. Muki, R.: Asymmetric problems of the theory of elasticity for a semi infinite solid and a thick plate. Prog. Solid Mech. 1, 399–439 (1960)

    MathSciNet  Google Scholar 

  4. Lebedev, N.N., Ufliand, I.S.: Axisymmetric contact problem for an elastic layer. J. Appl. Math. Mech. 22, 442–450 (1958). https://doi.org/10.1016/0021-8928(58)90059-5

    Article  MathSciNet  MATH  Google Scholar 

  5. Hayes, W.C., Keer, L.M., Herrmann, G., Mockros, L.F.: A mathematical analysis for indentation tests of articular cartilage. J. Biomech. 5, 541–551 (1972). https://doi.org/10.1016/0021-9290(72)90010-3

    Article  Google Scholar 

  6. Argatov, I., Daniels, A.U., Mishuris, G., Ronken, S., Wirz, D.: Accounting for the thickness effect in dynamic spherical indentation of a viscoelastic layer: application to non-destructive testing of articular cartilage. Eur. J. Mech. A/Solids 37, 304–317 (2013). https://doi.org/10.1016/j.euromechsol.2012.07.004

    Article  MathSciNet  MATH  Google Scholar 

  7. Argatov, I.I., Sabina, F.J.: Asymptotic analysis of the substrate effect for an arbitrary indenter. Q. J. Mech. Appl. Math. 66, 75–95 (2013). https://doi.org/10.1093/qjmam/hbs020

    Article  MathSciNet  MATH  Google Scholar 

  8. Dhaliwal, R.S.: Punch problem for an elastic layer overlying an elastic foundation. Int. J. Eng. Sci. 8, 273–288 (1970). https://doi.org/10.1016/0020-7225(70)90058-3

    Article  MATH  Google Scholar 

  9. Yu, H.Y., Sanday, S.C., Rath, B.B.: The effect of substrate on the elastic properties of films determined by the indentation test—axisymmetric boussinesq problem. J. Mech. Phys. Solids 38, 745–764 (1990). https://doi.org/10.1016/0022-5096(90)90038-6

    Article  Google Scholar 

  10. Korsunsky, A.M., Constantinescu, A.: The influence of indenter bluntness on the apparent contact stiffness of thin coatings. Thin Solid Films 517, 4835–4844 (2009). https://doi.org/10.1016/j.tsf.2009.03.018

    Article  Google Scholar 

  11. Keer, L.M., Kim, S.H., Eberhardt, A.W., Vithoontien, V.: Compliance of coated elastic bodies in contact. Int. J. Solids Struct. 27, 681–698 (1991). https://doi.org/10.1016/0020-7683(91)90028-E

    Article  MATH  Google Scholar 

  12. Sakamoto, M., Zhu, Q.F., Hara, T.: An axisymmetric contact problem of rigid spheres coated with transversely isotropic layers. Theor. Appl. Mech. Jpn. 44, 51–62 (1995)

    Google Scholar 

  13. Gao, H., Chiu, C.H., Lee, J.: Elastic contact versus indentation modeling of multi-layered materials. Int. J. Solids Struct. 29, 2471–2492 (1992). https://doi.org/10.1016/0020-7683(92)90004-D

    Article  Google Scholar 

  14. Argatov, I.: Frictionless and adhesive nanoindentation: asymptotic modeling of size effects. Mech. Mater. 42, 807–815 (2010). https://doi.org/10.1016/j.mechmat.2010.04.002

    Article  Google Scholar 

  15. Volkov, S., Aizikovich, S., Wang, Y.S., Fedotov, I.: Analytical solution of axisymmetric contact problem about indentation of a circular indenter into a soft functionally graded elastic layer. Acta Mech. Sin. 29, 196–201 (2013). https://doi.org/10.1007/s10409-013-0022-5

    Article  MathSciNet  MATH  Google Scholar 

  16. Selvadurai, A.P.S., Katebi, A.: An adhesive contact problem for an incompressible non-homogeneous elastic halfspace. Acta Mech. 226, 249–265 (2015). https://doi.org/10.1007/s00707-014-1171-8

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, T.J., Wang, Y.S., Xing, Y.M.: Fretting contact of two elastic solids with graded coatings under torsion. Int. J. Solids Struct. 49, 1283–1293 (2012). https://doi.org/10.1016/j.ijsolstr.2012.02.011

    Article  Google Scholar 

  18. Liu, T.J., Wang, Y.S., Xing, Y.M.: The axisymmetric partial slip contact problem of a graded coating. Meccanica 47, 1673–1693 (2012). https://doi.org/10.1007/s11012-012-9547-0

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, T.J., Wang, Y.S., Zhang, C.: Axisymmetric frictionless contact of functionally graded materials. Arch. Appl. Mech. 78, 267–282 (2008). https://doi.org/10.1007/s00419-007-0160-y

    Article  MATH  Google Scholar 

  20. Liu, T.J., Xing, Y.M.: Analysis of graded coatings for resistance to contact deformation and damage based on a new multi-layer model. Int. J. Mech. Sci. 81, 158–164 (2014). https://doi.org/10.1016/j.ijmecsci.2014.02.009

    Article  Google Scholar 

  21. Constantinescu, A., Korsunsky, A.M., Pison, O., Oueslati, A.: Symbolic and numerical solution of the axisymmetric indentation problem for a multilayered elastic coating. Int. J. Solids Struct. 50, 2798–2807 (2013). https://doi.org/10.1016/j.ijsolstr.2013.04.017

    Article  Google Scholar 

  22. Wei, Y., Ji, Y., Weiqiu, C.: A three-dimensional solution for laminated orthotropic rectangular plates with viscoelastic interfaces. Acta Mech. Solida Sin. 19, 181–188 (2006). https://doi.org/10.1007/s10338-006-0622-8

    Article  Google Scholar 

  23. Ai, Z.Y., Zhang, Y.F.: The analysis of a rigid rectangular plate on a transversely isotropic multilayered medium. Appl. Math. Model. 39, 6085–6102 (2015). https://doi.org/10.1016/j.apm.2015.01.054

    Article  MathSciNet  MATH  Google Scholar 

  24. Stan, G., Adams, G.G.: Adhesive contact between a rigid spherical indenter and an elastic multi-layer coated substrate. Int. J. Solids Struct. 87, 1–10 (2016). https://doi.org/10.1016/j.ijsolstr.2016.02.043

    Article  Google Scholar 

  25. Zhang, H., Wang, W., Zhang, S., Zhao, Z.: Semi-analytical solution of three-dimensional steady state thermoelastic contact problem of multilayered material under friction heating. Int. J. Therm. Sci. 127, 384–399 (2018). https://doi.org/10.1016/j.ijthermalsci.2018.02.006

    Article  Google Scholar 

  26. Miura, K., Sakamoto, M., Kobayashi, K.: Analytical solution of axisymmetric indentation of an elastic layer-substrate body. Theor. Appl. Mech. Jpn. 64, 81–101 (2018)

    Google Scholar 

  27. Miura, K., Sakamoto, M., Kobayashi, K., Pramudita, J.A., Tanabe, Y.: An analytical solution for the axisymmetric problem of a penny-shaped crack in an elastic layer sandwiched between dissimilar materials. Mech. Eng. J. 5, 18-00125–18–00125 (2018). https://doi.org/10.1299/mej.18-00125

    Article  Google Scholar 

  28. Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press, Cambridge (1927)

    MATH  Google Scholar 

  29. Bateman, H.: Tables of Integral Transforms. McGraw-Hill, New York (1954)

    Google Scholar 

  30. Moriguchi, S., Udagawa, K., Hitomatsu, S.: Mathematical Formulas, vol. 3. Iwanami, Tokyo (1987). (in Japanese)

    Google Scholar 

Download references

Funding

This research did not receive any specific grants from funding agencies in the public, commercial, or not-for-profit sectors.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kotaro Miura.

Ethics declarations

Conflict of interest

The authors declare that they have no conflicts of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

See Tables 3456, and 7.

Table 3 Comparison of normalized axial load \(P^{*} = P/P_{{1}}\) for various h/a ratios in the range of \(h/a > 2.0\) between the present and previously reported results for a cylindrical indenter
Table 4 Comparison of normalized axial load \(P^{*} = P/P_{{1}}\) for various h/a ratios in the range of \(h/a < 2.0\) between the present and previously reported results for a spherical indenter (\(N = 1\))
Table 5 Comparison of normalized axial load \(P^{*} = P/P_{{1}}\) with various h/a ratios in the range of \(h/a <2.0\) between the present and previously reported results for a spherical indenter (\(N = 2\))
Table 6 Comparison of normalized axial load \(P^{*} = P/P_{{1}}\) with various h/a ratios in the range of \(h/a <2.0\) between the present and previously reported results for a spherical indenter (\(N = 3\))
Table 7 Comparison of normalized axial load \(P^{*} = P/P_{{1}}\) with various h/a ratios in the range of \(h/a < 2.0\) between the present and previously reported results for a spherical indenter (\(N = 4\))

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Miura, K., Sakamoto, M. & Tanabe, Y. Analytical solution of axisymmetric indentation of multi-layer coating on elastic substrate body. Acta Mech 231, 4077–4093 (2020). https://doi.org/10.1007/s00707-020-02752-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-020-02752-1

Navigation