Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T12:46:02.249Z Has data issue: false hasContentIssue false
  • English
  • Français

Combined empirical–analytical method for determining contact radius and indenter displacement during Hertzian indentation on coating/substrate systems

Published online by Cambridge University Press:  03 March 2011

Chun-Hway Hsueh  and
Pedro Miranda
Chun-Hway Hsueh*
Affiliation:
Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
Pedro Miranda
Affiliation:
Departamento Electrónica e Ingeniería Electromecánica, Escuela de Ingenierías Industriales, Universidad de Extremadura, 06071 Badajoz, Spain
*
a) Address all correspondence to this author. e-mail: [email protected]
Get access

Abstract

An analytical model was developed in a previous work to relate the normalized indenter displacement to both the coating-to-substrate Young's modulus ratio and the coating-thickness-to-contact-radius ratio for Hertzian indentation on coating/substrate systems. However, application of this model is contingent upon the determination of the contact radius during indentation. Using the data from finite element analyses, an empirical equation is proposed in this paper to determine the normalized contact radius. Combining this empirical equation with the previous analytical equation, both the contact radius and the indenter displacement for Hertzian indentation on coating/substrate systems are predicted. The predictions obtained by this combined empirical–analytical method are shown to agree with the finite element results in general although the indenter displacement is over-estimated when the coating is stiffer than the substrate. Finally, the potential applications of this method to determine the elastic properties of coatings from the indentation data are envisaged.

Keywords

CoatingIndentationElastic properties
Type
Articles
Information
Journal of Materials Research , Volume 19 , Issue 9 , September 2004 , pp. 2774 - 2781
Copyright
Copyright © Materials Research Society 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Johnson, K.L.Contact Mechanics (Cambridge Univ. Press, Cambridge, U.K., 1985), p. 60Google Scholar
2.Pharr, G.M., Oliver, W.C. andBrotzen, F.R.: On the generality of the relationship among contact stiffness, contact area, and elastic modulus during indentation. J. Mater. Res. 7, 613 (1992).Google Scholar
3.Field, J.S. andSwain, M.V.: A simple predictive model for spherical indentation. J. Mater. Res. 8, 297 (1993).Google Scholar
4.Lawn, B.R.: Indentation of ceramics with spheres: A century after Hertz. J. Am. Ceram. Soc 81, 1977 (1998).Google Scholar
5.Doerner, M.F. andNix, W.D.: A method for interpreting the data from depth-sensing indentation instruments. J. Mater. Res. 1, 601 (1986).Google Scholar
6.Swain, M.V. andMencik, J.: Mechanical property characterization of thin films using spherical tipped indenters. Thin Solid Films 253, 204 (1994).Google Scholar
7.Sawa, T., Akiyama, Y., Shimamoto, A. andTanaka, K.: Nanoindentation of a 10 nm thick thin film. J. Mater. Res. 14, 2228 (1999).CrossRefGoogle Scholar
8.Chudoba, T., Schwarzer, N., Richter, F. andBeck, U.: Determination of mechanical film properties of a bilayer system due to elastic indentation measurements with a spherical indenter. Thin Solid Films 377–378, 366 (2000).Google Scholar
9.van der Zwaag, S. andField, J.E.: The effect of thin hard coatings on the Hertzian stress field. Philos. Mag. A 46, 133 (1982).Google Scholar
10.Djabella, H. andArnell, R.D.: Finite element analysis of the contact stresses in elastic coating/substrate under normal and tangential load. Thin Solid Film s 223, 87 (1993).Google Scholar
11.Sun, Y., Bloyce, A. andBell, T.: Finite element analysis of plastic deformation of various TiN coating/substrate systems under normal contact with a rigid sphere. Thin Solid Films 271, 122 (1995).CrossRefGoogle Scholar
12.Waters, N.E.: The indentation of thin rubber sheets by spherical indenters. Brit. J. Appl. Phys 16, 557 (1965).Google Scholar
13.Hu, X.Z. andLawn, B.R.: A simple indentation stress-strain relation for contacts with spheres on bilayer structure. Thin Solid Films 322, 225 (1998).Google Scholar
14.Jaffar, M.J.: A numerical solution for axisymmetric contact problems involving rigid indenters on elastic layers. J. Mech. Phys. Solids 36, 401 (1988).CrossRefGoogle Scholar
15.Yu, H.Y., Sanday, S.C. andRath, 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 (1990).Google Scholar
16.Yang, F.: Indentation of an incompressible elastic film. Mech. Mater. 30, 275 (1998).CrossRefGoogle Scholar
17.Stone, D.S.: Elastic rebound between an indenter and a layered specimen: Part I. Model. J. Mater. Res. 13, 3207 (1998).Google Scholar
18.Schwarzer, N., Richter, F. andHecht, G.: The elastic field in a coated half-space under Hertzian pressure distribution. Surf. Coat. Technol 114, 292 (1999).CrossRefGoogle Scholar
19.Hsueh, C.H. andMiranda, P.: Master curves for Hertzian indentation on coating/substrate systems. J. Mater. Res. 19, 94 (2004).CrossRefGoogle Scholar
20.Timoshenko, S.P. andGoodier, J.N.: Theory of Elasticity, 3rd ed. (McGraw-Hill, New York, 1951), pp. 401Google Scholar
21.Savageau, M.A.: Growth equations–A general equation and a survey of special case. Math. Biosci. 48, 267 (1980).Google Scholar
22.Tsoularis, A.: Analysis of logistic growth models. Math. Biosci. 179, 21 (2002).Google Scholar
23.Miranda, P., Pajares, A., Guiberteau, F., Deng, Y. andLawn, B.R.: Designing damage-resistant brittle-coating structures: I. Bilayers. Acta Mater 51, 4347 (2003).Google Scholar
24.Hsueh, C.H. andMiranda, P.: Modeling of contact-induced radial cracking in ceramic bilayer coatings on compliant substrates. J. Mater. Res. 18, 1275 (2003).CrossRefGoogle Scholar
25.Sneddon, I.N.: The relation between load and penetration in the axisymmetric Boussinesq problem for punch of arbitrary profile. Int. J. Eng. Sci 3, 47 (1965).Google Scholar