On the measurements of viscoelastic functions of a sphere by nanoindentation

  • Zhong Zhou
  • Hongbing LuEmail author


Nanoindentation using such instruments as instrumented nanoindenter and scanning probe microscope is effective for measurements of viscoelastic functions of a sphere in micron or sub-micron scale. In this paper, we provide methods for nanoindentation measurements of linearly viscoelastic functions in both time- and frequency-domains for a viscoelastic sphere under small deformations. In the time-domain, both relaxation and creep functions are determined from three types of loading histories, namely constant-rate loading, ramp loading, and step loading. In the frequency-domain, methods are given for the calculation of complex modulus, or complex compliance under a small amplitude of sinusoidal load superimposed on a carrier load. The effects of the radius of the viscoelastic sphere relative to the indenter tip radius, as well as the deformation of the sphere induced by contact with the flat substrate supporting the sphere are discussed.


Relaxation modulus Creep compliance Complex modulus Complex compliance Nanoindentation Spherical indenter Time-domain Frequency-domain Viscoelastic sphere 


  1. Arenz, R.J.: Nonlinear shear behavior of Poly(vinyl acetate) material. Mech. Time-Depend. Mater. 2, 287–305 (1999) CrossRefGoogle Scholar
  2. Burnham, N.A., Baker, S.P., Pollock, H.M.: Model for mechanical properties nanoprobes. J. Mater. Res. 15, 2006–2014 (2000) Google Scholar
  3. Cao, Y.: Determination of the creep exponent of a power-law creep solid using indentation tests. Mech. Time-Depend. Mater. 11, 159–172 (2007) CrossRefGoogle Scholar
  4. Cheng, Y.-T., Cheng, C.-M.: General relationship between contact stiffness, contact depth, and mechanical properties for indentation in linear viscoelastic solids using axisymmetric indenters of arbitrary profiles. Appl. Phys. Lett. 87, 111914 (2005) CrossRefGoogle Scholar
  5. Cheng, Y.-T., Cheng, C.-M., Ni, W.: Methods of obtaining instantaneous modulus of viscoelastic solids using displacement-controlled instrumented indentation with axisymmetric indenters of arbitrary smooth profiles. Mater. Sci. Eng. A 423, 2–7 (2006a) CrossRefGoogle Scholar
  6. Cheng, Y.-T., Ni, W., Cheng, C.-M.: Nonlinear analysis of oscillatory indentation in elastic and viscoelastic solids. Phys. Rev. Lett. 97, 075506 (2006b) CrossRefGoogle Scholar
  7. Cheng, L., Xia, X., Yu, W., Scriven, L.E., Gerberich, W.W.: Flat-punch indentation of viscoelastic material. J. Polym. Sci. B Polym. Phys. 38, 10–22 (2000) CrossRefGoogle Scholar
  8. Daphalapurkar, N.P., Dai, C., Gan, R.Z., Lu, H.: Characterization of the linearly viscoelastic behavior of human tympanic membrane by nanoindentation. J. Mech. Behav. Mater. 2, 82–92 (2009) CrossRefGoogle Scholar
  9. Darling, E.M., Zauscher, S., Guilak, F.: Viscoelastic properties of zonal articular chondrocytes measured by atomic force microscopy. Osteoarthr. Cartil. 14, 571–579 (2006) CrossRefGoogle Scholar
  10. Emri, I., Pavsek, V.: On the influence of moisture on the mechanical behavior of polymers. In: Proceedings of VII International Congress on Experimental Mechanics, vol. II, pp. 1429–1437. Society for Experimental Mechanics, Bethel (1992) Google Scholar
  11. Francius, G., Hemmerle, J., Ball, V., Lavalle, P., et al.: Stiffening of soft polyelectrolyte architectures by multilayer capping evidenced by viscoelastic analysis of AFM indentation measurements. J. Phys. Chem. C 111, 8299–8306 (2007) CrossRefGoogle Scholar
  12. Giannakopoulos, A.E.: Strength analysis of spherical indentation of piezoelectric materials. J. Appl. Mech. 67, 409–416 (2000) zbMATHCrossRefGoogle Scholar
  13. Hertz, H.: Uber die beruhrung fester elastischer korper. J. Reine Angew. Math. 92, 156–171 (1881) Google Scholar
  14. Huang, G., Lu, H.: Measurement of two independent viscoelastic functions by nanoindentation. Exp. Mech. 47, 87–98 (2007) CrossRefGoogle Scholar
  15. Huang, G., Wang, B., Lu, H.: Measurement of viscoelastic function of polymers in the frequency-domain using nanoindentation. Mech. Time-Depend. Mater. 8, 345–364 (2004) CrossRefGoogle Scholar
  16. Huang, G., Daphalapurkar, N., Gan, R.Z., Lu, H.: A method for measuring linearly viscoelastic properties of human tympanic membrane using nanoindentation. J. Biomech. Eng. 130, 014501 (2008) CrossRefGoogle Scholar
  17. Keddie, J.L.: Film formation of latex. Mater. Sci. Eng. R 21, 101–170 (1997) CrossRefGoogle Scholar
  18. Knauss, W.G., Zhao, J.: Improved relaxation time coverage in ramp-strain histories. Mech. Time-Depend. Mater. 11, 199–216 (2007) CrossRefGoogle Scholar
  19. Lee, E.H., Radok, J.R.M.: The contact problem for viscoelastic bodies. J. Appl. Mech. 27, 438–444 (1960) zbMATHMathSciNetGoogle Scholar
  20. Lee, S., Knauss, W.G.: A note on the determination of relaxation and creep data from ramp tests. Mech. Time-Depend. Mater. 4, 1–7 (2000) CrossRefGoogle Scholar
  21. Loubet, J.L., Oliver, W.C., Lucas, B.N.: Measurement of the loss tangent of low-density polyethylene with a nanoindentation technique. J. Mater. Res. 15, 1195–1198 (2000) CrossRefGoogle Scholar
  22. Lu, H., Wang, B., Ma, J., Huang, G., Viswanathan, H.: Measurement of creep compliance of solid polymers by nanoindentation. Mech. Time-Depend. Mater. 7, 189–207 (2003) CrossRefGoogle Scholar
  23. Lu, H., Huang, G., Wang, B., Mamedov, A., Gupta, S.: Characterization of the linear viscoelastic behavior of single wall carbon nanotube/polyelectrolyte multilayer nanocomposite film using nanoindentation. Thin Solid Films 500, 197–202 (2006) CrossRefGoogle Scholar
  24. Lu, H., Huang, G., Wang, F.: Measurements of viscoelastic properties of polymers using flat punch indenter. In: Proceedings of the SEM Annual Conference and Exposition on Experimental and Applied Mechanics, vol. 2, pp. 697–704 (2007) Google Scholar
  25. Mahaffy, R.E., Shih, C.K., MacKintosh, F.C., Kas, J.: Scanning probe-based frequency-dependent microrheology of polymer gels and biological cells. Phys. Rev. Lett. 85, 880–883 (2000) CrossRefGoogle Scholar
  26. Mattice, J.M., Lau, A.G., Oyen, M.L., Kent, R.W.: Spherical indentation load-relaxation of soft biological tissues. J. Mater. Res. 21(8), 2003–2010 (2006) CrossRefGoogle Scholar
  27. Oliver, W.C., Pharr, G.M.: An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7(6), 1564–1583 (1992) CrossRefGoogle Scholar
  28. Oyen, M.L.: Spherical indentation creep following ramp loading. J. Mater. Res. 20(8), 2094–2100 (2005) CrossRefGoogle Scholar
  29. Oyen, M.L., Cook, R.F.: Load-displacement behavior during sharp indentation of viscous–elastic–plastic materials. J. Mater. Res. 18(1), 139–150 (2003) CrossRefGoogle Scholar
  30. Sadr, A., Shimada, Y., Lu, H., Tagami, J.: The viscoelastic behavior of dental adhesives: a nanoindentation study. Dent. Mater. 25, 13–19 (2009) CrossRefGoogle Scholar
  31. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  32. Tan, S., Sherman, R.L. Jr., Ford, W.T.: Nanoscale compression of polymer microsphere by atomic force microscopy. Langmuir 20, 7015–7020 (2004) CrossRefGoogle Scholar
  33. Tatara, Y.: On compression of rubber elastic sphere over a large range of displacements–Part 1: Theoretical study. J. Eng. Mater. Technol. 113, 285–291 (1991) CrossRefGoogle Scholar
  34. Ting, T.C.T.: The contact stresses between a rigid indenter and a viscoelastic half-space. J. Appl. Mech. 33, 845–854 (1966) zbMATHGoogle Scholar
  35. VanLandingham, M.R., Villarrubia, J.S., Guthrie, W.F., Meyers, G.F.: Nanoindentation of polymers: an overview. Macromol. Symp. 167, 15–43 (2001) CrossRefGoogle Scholar
  36. VanLandingham, M.R., Chang, N.K., Drzal, P.L., White, C.C., Chang, S.H.: Viscoelastic characterization of polymers using instrumented indentation-1 quasi-static testing. J. Polym. Sci. B Polym. Phys. 43, 1794–1811 (2005) CrossRefGoogle Scholar
  37. Zhao, J., Knauss, W.G., Ravichandran, G.: Applicability of the time-temperature superposition principle in modeling dynamic response of a polyurea. Mech. Time-Depend. Mater. 11, 289–308 (2007) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, B. V. 2009

Authors and Affiliations

  1. 1.School of Mechanical and Aerospace EngineeringOklahoma State UniversityStillwaterUSA
  2. 2.Dept. of Mechanical and Energy EngineeringUniversity of North TexasDentonUSA

Personalised recommendations