Skip to main content Accessibility help

A general relation for contact stiffness including adhesion in indentation analysis

  • Pin Lu (a1), Yong L. Foo (a1), Lu Shen (a1), Davy W.C. Cheong (a1) and Sean J. O’Shea (a1)...


The Maugis–Barquins (MB) solutions for the adhesive contact between an axisymmetric indenter and an elastic half-space are modified by incorporating the interfacial energy defined by the real area of contact. With the modified MB solutions, general relations for contact stiffness including adhesive effects in indentation analysis are derived. Numerical calculations showed that the difference in expected stiffness for the modified MB model compared to the standard MB results can be significant at low loads of interest in atomic force microscopy measurements and also for indentation tests at high load if the interfacial energy is large (∼0.1 J/m2) and the material is soft (Young’s modulus ≤100 MPa).


Corresponding author

a)Address all correspondence to this author. e-mail:


Hide All
1.Pharr, G.M., Oliver, W.C., and Brotzen, F.R.: On the generality of the relationship among contact stiffness, contact area, and elastic modulus during indentation. J. Mater. Res. 7, 613 (1992).
2.Oliver, W.C. and Pharr, G.M.: An improved technique for determining hardness and elastic-modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564 (1992).
3.Schwarz, U.D.: A generalized analytical model for the elastic deformation of an adhesive contact between a sphere and a flat surface. J. Colloid Interface Sci. 261, 99 (2003).
4.Ebenstein, D.M. and Pruitt, L.A.: Nanoindentation of biological materials. Nano Today. 1, 26 (2006).
5.Cao, Y.F., Yang, D.H., and Soboyejoy, W.: Nanoindentation method for determining the initial contact and adhesion characteristics of soft polydimethylsiloxane. J. Mater. Res. 20, 2004 (2005).
6.Johnson, K.L., Kendall, K., and Roberts, A.D.: Surface energy and the contact of elastic solids. Proc. R. Soc. London, Ser. A 324, 301 (1971).
7.Barthel, E.: Adhesive elastic contacts: JKR and more. J. Phys. D: Appl. Phys. 41, 163001 (2008).
8.Maugis, D. and Barquins, M.: Adhesive contact of sectionally smooth-ended punches on elastic half-spaces: Theory and experiment. J. Phys. D: Appl. Phys. 16, 1843 (1983).
9.Maugis, D.: Adhesion of spheres: The JKR-DMT transition using a Dugdale model. J. Colloid Interface Sci. 150, 243 (1992).
10.Maugis, D.: Contact, Adhesion and Rupture of Elastic Solids (Springer, Heidelberg, 2000).
11.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 (1965).
12.Yang, F.Q.: Effect of adhesion energy on the contact stiffness in nanoindentation. J. Mater. Res. 21, 2683 (2006).
13.Sirghi, L. and Rossi, F.: Adhesion and elasticity in nanoscale indentation. Appl. Phys. Lett. 89, 243118 (2006).
14.Maugis, D. and Barquins, M.: Fracture mechanics and the adherence of viscoelastic bodies. J. Phys. D: Appl. Phys. 11, 1989 (1978).
15.Tabor, D.: Surface forces and surface interactions. J. Colloid Interface Sci. 58, 2 (1977).
16.Lawn, B.: Fracture of Brittle Solids, 2nd ed. (Cambridge University Press, Cambridge, 1993).
17.Vallet, D. and Barquins, M.: Adhesive contact and kinetics of adherence if a rigid conical punch on an elastic half-space (natural rubber). Int. J. Adhes. Adhes. 22, 41 (2002).
18.Kendall, K.: The adhesion and surface energy of elastic solids. J. Phys. D: Appl. Phys. 4, 1186 (1971).
19.Yang, F.Q.: Thickness effect on the indentation of an elastic layer. Mater. Sci. Eng., A 358, 226 (2003).
20.Yang, F.Q.: Adhesive contact between a rigid axisymmetric indenter and an incompressible elastic thin film. J. Phys. D: Appl. Phys. 35, 2614 (2002).
21.Hertz, H.: On the contact of elastic solids, in Miscellaneous Papers by Heinrich Hertz, English translated by Jones, D.E. and Schott, G.A. (Macmillan and Co., Ltd., new York, 1896), p. 146.
22.Fischer-Cripps, A.C.: Review of analysis and interpretation of nanoindentation test data. Surf. Coat. Tech. 200, 4153 (2006).
23.Olah, A. and Vancso, G.J.: Characterization of adhesion at solid surfaces: Development of an adhesion-testing device. Eur. Polym. J. 41, 2803 (2005).
24.Korsunsky, A.M.: The influence of punch blunting on the elastic indentation response. J. Strain Anal. Eng. Des. 36, 391 (2001).
25.Dimitriadis, E.K., Horkay, F., Maresca, J., Kachar, B., and Chadwick, R.S.: Determination of elastic moduli of thin layers of soft material using the atomic force microscope. Biophys. J. 82, 2798 (2002).
26.Withers, J.R. and Aston, D.E.: Nanomechanical measurements with AFM in the elastic limit. Adv. Colloid Interface Sci. 120, 57 (2006).



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed