Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
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.
Giannakopoulos, A.E. ‘Strength analysis of spherical indentation of piezoelectric materials’, J. Appl. Mech. 67, 2000, 409–416.
Hertz, H., ‘Über die beruhrung fester elastischer körper’, J. für die Reine und Angewandte Mathematik 92, 1881, 156–171.
Knauss, W.G. and Zhu, W., ‘Nonlinearly viscoelastic behavior of polycarbonate. I. Response under pure shear’, Mech. Time-Depend. Mater. 6, 2002, 231–269.
Lee, E.H. and Radok, J.R.M., ‘The contact problem for viscoelastic bodies’, J. Appl. Mech. 27, 1960, 438–444.
Li, X. and Bhushan, B., ‘A review of nanoindentation continuous stiffness measurement technique and its applications’, Mater. Char. 48, 2002, 11–36.
Ling, F.F., Lai, W.M. and Lucca, D.A., Fundamental of Surface Mechanics, Springer-Verlag, New York, 2002.
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.
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.
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.
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.
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.
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.
Pethica, J.B. and Oliver, W.C., ‘Tip surface interactions in STM and AFM’, Phys. Scr. T19, 1987, 61–66.
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.
Sane, S.B. and Knauss, W.G., ‘The time-dependent bulk response of poly(methyl methacrylate)’, Mech. Time-Depend. Mater. 5, 2001, 293–324.
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.
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.
Ting, T.C.T., ‘The contact stresses between a rigid indenter and a viscoelastic half-space’, J. Appl. Mech. 33, 1966, 845–854.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huang, G., Wang, B. & Lu, H. Measurements of Viscoelastic Functions of Polymers in the Frequency-Domain Using Nanoindentation. Mech Time-Depend Mater 8, 345–364 (2004). https://doi.org/10.1007/s11043-004-0440-7
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11043-004-0440-7