Mechanics of Time-Dependent Materials

, Volume 8, Issue 4, pp 345–364 | Cite as

Measurements of Viscoelastic Functions of Polymers in the Frequency-Domain Using Nanoindentation

  • G. HuangEmail author
  • B. Wang
  • H. Lu


A method to measure the complex compliance (or modulus) of linearly viscoelastic materials is presented using nanoindentation with a spherical indenter. The Hertzian solution for an elastic indentation problem, in combination with a hereditary integral operator proposed by Lee and Radok (Journal of Applied Mechanics 27, 1960, 438–444) for the situation of non-decreasing indentation contact area, was used to derive formulas for the complex viscoelastic functions in the frequency-domain. The formulas are most suitable for frequencies lower than a frequency limit such that the condition of non-decreasing contact area holds; they are reasonably good approximation at higher frequencies under which decreasing contact area occurs and the Ting (Journal of Applied Mechanics 33, 1966, 845–854) approach for arbitrary contact area history is needed. Nanoindentation tests were conducted on both polycarbonate and polymethyl methacrylate under a harmonic indentation load superimposed on either step or ramp indentation load, while the resulting displacement under steady state was recorded. The load and displacement data at each frequency were processed using the derived formulas to determine the viscoelastic functions in the frequency-domain. The same materials were also tested using a dynamic mechanical analysis (DMA) apparatus to determine the complex viscoelastic functions. The DMA and nanoindentation results were compared and found in a good agreement, indicating the validity of the new method presented.

Key words

complex compliance frequency-domain nanoindentation polymer spherical indenter viscoelasticity 


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  1. Cheng, L., Xia, X., Yu, W., Scriven, L.E. and Gerberich, W.W., ‘Flat-punch indentation of viscoelastic material’, J. Polym. Sci [B]: Polym. Phys. 38, 2000, 10–22.Google Scholar
  2. Ferry, J.D., ‘Mechanical properties of substances of high molecular weight. VI. Dispersion in concentrated polymer solutions and its dependence on temperature and concentration’, J. Am. Chem. Soc. 72, 1950, 3746–3752.CrossRefGoogle Scholar
  3. Giannakopoulos, A.E. ‘Strength analysis of spherical indentation of piezoelectric materials’, J. Appl. Mech. 67, 2000, 409–416.CrossRefGoogle Scholar
  4. Hertz, H., ‘Über die beruhrung fester elastischer körper’, J. für die Reine und Angewandte Mathematik 92, 1881, 156–171.Google Scholar
  5. Knauss, W.G. and Zhu, W., ‘Nonlinearly viscoelastic behavior of polycarbonate. I. Response under pure shear’, Mech. Time-Depend. Mater. 6, 2002, 231–269.CrossRefGoogle Scholar
  6. Lee, E.H. and Radok, J.R.M., ‘The contact problem for viscoelastic bodies’, J. Appl. Mech. 27, 1960, 438–444.Google Scholar
  7. Li, X. and Bhushan, B., ‘A review of nanoindentation continuous stiffness measurement technique and its applications’, Mater. Char. 48, 2002, 11–36.Google Scholar
  8. Ling, F.F., Lai, W.M. and Lucca, D.A., Fundamental of Surface Mechanics, Springer-Verlag, New York, 2002.Google Scholar
  9. Loubet, J.L., Lucas, B.N. and Oliver, W.C., ‘Some measurements of viscoelastic properties with the help of nanoindentation’, in International Workshop on Instrumental Indentation, San Diego, CA, April 1995, D.T. Smith (ed.), 1995, pp. 31–34.Google Scholar
  10. Lu, H., Zhang, X. and Knauss, W.G., ‘Uniaxial, shear and Poisson relaxation and their conversion to bulk relaxation: studies on poly (methyl methacrylate)’, Polymer Engineering Science 37, 1997, 1053–1064.CrossRefGoogle Scholar
  11. Lu, H., Wang, B., Ma, J., Huang, G. and Viswanathan, H., ‘Measurement of creep compliance of solid polymers by nanoindentation’, Mech. Time-Depend. Mater. 7, 2003, 189–207.CrossRefGoogle Scholar
  12. Lucas, B.N., Oliver, W.C., and Swindeman, J.E., ‘The dynamics of frequency-specific, depth-sensing indentation testing’, Mater. Res. Soc. Symp. Proc. 522, San Francisco, Moody, N.R. et al. (ed.), 1998, 3–14.Google Scholar
  13. 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, 1992, 1564–1583.Google Scholar
  14. Oyen-Tiesma, M., Toivola, Y.A. and Cook, R.F., ‘Load-displacement behavior during sharp indentation of viscous-elastic-plastic materials’, in Fundamentals of Nanoindentation and Nanotribology II, Baker, S.P., Corcoran, S., Moody, G.N.R. and Cook, R.F. (ed.), Warrendale, PA, 2001, Q.5.1.Google Scholar
  15. Pethica, J.B. and Oliver, W.C., ‘Tip surface interactions in STM and AFM’, Phys. Scr. T19, 1987, 61–66.Google Scholar
  16. Pethica, J.B. and Oliver, W.C., ‘Mechanical properties of nanometer volumes of material: Use of the elastic response of small area indentations’, in Materials Research Society Symposium Proceedings 130, Pittsburgh, Bravman, J.C. et al. (ed.), 1989, 13–23.Google Scholar
  17. Sane, S.B. and Knauss, W.G., ‘The time-dependent bulk response of poly(methyl methacrylate)’, Mech. Time-Depend. Mater. 5, 2001, 293–324.CrossRefGoogle Scholar
  18. Sneddon, I.N., ‘The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary punch’, Inter. J. Eng. Sci. 3, 1965, 47–56.CrossRefGoogle Scholar
  19. Syed, S.A., Wahl, K.J. and Colton, R.J., ‘Nanoindentation and contact stiffness measurement using force modulation with a capacitive load-displacement transducer’, Rev. Sci. Instrum. 70(5), 1999, 2408–2413.CrossRefGoogle Scholar
  20. Ting, T.C.T., ‘The contact stresses between a rigid indenter and a viscoelastic half-space’, J. Appl. Mech. 33, 1966, 845–854.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2004

Authors and Affiliations

  1. 1.School of Mechanical and Aerospace EngineeringOklahoma State UniversityStillwaterUSA

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