Skip to main content Accesibility Help

Effective Poisson’s ratio from combined normal and lateral contacts of single crystals

  • J.H. Lee (a1), Y.F. Gao (a2) and G.M. Pharr (a3)

When an elastic half-space is subjected to both normal and tangential contact, the ratio of normal and tangential contact stiffnesses can be measured by various scanning force microscopy techniques. For elastically isotropic solids, this stiffness ratio depends on Poisson’s ratio as given by the Mindlin solution. An anisotropic elastic contact analysis here shows the difference between the effective Poisson’s ratio as defined from the stiffness ratio and its uniaxial counterpart with respect to various crystal structures and various normal/tangential contact directions. Closed-form analytical solutions of effective indentation moduli are derived for materials with at least one plane of transverse isotropy. Since the Sneddon (normal contact) and Mindlin (lateral contact) solutions are derived under different frictional conditions, finite element simulations were performed which show that the effects of elastic dissimilarity and contact shape are generally small but not negligible. The predicted dependence on crystallographic orientation and elastic anisotropy has been compared favorably with previously reported multiaxial contact experiments for a number of cubic single crystals. Implications for atomic force microscopy based experiments are also discussed.

Corresponding author
a)Address all correspondence to these authors. e-mail:
Hide All
1.Carpick, R.W., Ogletree, D.F., and Salmeron, M.: Lateral stiffness: a new nanomechanical measurement for the determination of shear strengths with friction force microscopy. Appl. Phys. Lett. 70, 1548 (1997).
2.Hurley, D.C. and Turner, J.A.: Measurement of Poisson’s ratio with contact-resonance atomic force microscopy. J. Appl. Phys. 102, 033509 (2007).
3.Stan, G. and Cook, R.F.: Mapping the elastic properties of granular Au films by contact resonance atomic force microscopy. Nanotechnology 19, 235701 (2008).
4.Reinstädtler, M., Kasai, T., Rabe, U., Bhushan, B., and Arnold, W.: Imaging and measurement of elasticity and friction using the TR mode. J. Phys. D: Appl. Phys. 38, R269 (2005).
5.Lucas, B.N., Hay, J.C., and Oliver, W.C.: Using multidimensional contact mechanics experiments to measure Poisson’s ratio. J. Mater. Res. 19, 58 (2004).
6.Gao, Y.F., Lucas, B.N., Hay, J.C., Oliver, W.C., and Pharr, G.M.: Nanoscale incipient asperity sliding and interface micro-slip assessed by the measurement of tangential contact stiffness. Scr. Mater. 55, 653 (2006).
7.Gao, Y.F., Xu, H.T., Oliver, W.C., and Pharr, G.M.: A comparison of Coulomb friction and friction stress models based on multidimensional nanocontact experiments. J. Appl. Mech. 75, 034504 (2008).
8.Johnson, K.L.: Contact Mechanics (Cambridge University Press, Cambridge, 1985).
9.Gao, Y.F. and Pharr, G.M.: Multidimensional contact moduli of elastically anisotropic solids. Scr. Mater. 57, 13 (2007).
10.Baughman, R.H., Shacklette, J.M., Zakhidov, A.A., and Stafstroem, S.: Negative Poisson’s ratios as a common feature of cubic metals. Nature 392, 362 (1998).
11.Ting, T.C.T. and Barnett, D.M.: Negative Poisson’s ratios in anisotropic linear elastic media. J. Appl. Mech. 72, 929 (2005).
12.Ting, T.C.T. and Chen, T.Y.: Poisson’s ratio for anisotropic elastic materials can have no bounds. Q. J. Mech. Appl. Math. 58, 73 (2005).
13.Lethbridge, Z.A.D., Walton, R.I., Marmier, A.S.H., Smith, C.W., and Evans, K.E.: Elastic anisotropy and extreme Poisson’s ratios in single crystals. Acta Mater. 58, 6444 (2010).
14.Vlassak, J.J. and Nix, W.D.: Indentation modulus of elastically anisotropic half spaces. Philos. Mag. A 67, 1045 (1993).
15.Vlassak, J.J. and Nix, W.D.: Measuring the elastic properties of anisotropic materials by means of indentation experiments. J. Mech. Phys. Solids 42, 1223 (1994).
16.Bower, A.F.: Applied Mechanics of Solids (CRC Press, Boca Raton, FL, 2009).
17.Espinasse, L., Keer, L., Borodich, F., Yu, H., and Wang, Q.J.: A note on JKR and DMT theories of contact on a transversely isotropic half-space. Mech. Mater. 42, 477 (2010).
18.Gao, Y.F., Xu, H.T., Oliver, W.C., and Pharr, G.M.: Effective elastic modulus of film-on-substrate systems under normal and tangential contact. J. Mech. Phys. Solids 56, 402 (2008).
19.Lee, H., Lee, J.H., and Pharr, G.M.: A numerical approach to spherical indentation techniques for material property evaluation. J. Mech. Phys. Solids 53, 2037 (2005).
20.Lee, J.H., Lee, H., and Kim, D.H.: A numerical approach to evaluation of elastic modulus using conical indenter with finite tip radius. J. Mater. Res. 23, 2528 (2008).
21.Lee, J.H., Kim, T.H., and Lee, H.: A study on robust indentation techniques to evaluate elastic-plastic properties of metals. Int. J. Solids Struct. 47, 647 (2010).
22.Annett, J., Gao, Y.F., Cross, G.L.W., Herbert, E.G., and Lucas, B.N.: Mesoscale friction anisotropy revealed by slidingless tests. J. Mater. Res. 26(18), 2373 (2011).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Materials Research
  • ISSN: 0884-2914
  • EISSN: 2044-5326
  • URL: /core/journals/journal-of-materials-research
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



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