Mechanics of Time-Dependent Materials

, Volume 10, Issue 3, pp 229–243

Measurement of Young’s relaxation modulus using nanoindentation

Article

Abstract

In a previous paper (Lu et al., Mechanics of Time-Dependent Materials, 7, 2003, 189–207), we described methods to measure the creep compliance of polymers using Berkovich and spherical indenters by nanoindentation. However, the relaxation modulus is often needed in stress and deformation analysis. It has been well known that the interconversion between creep compliance and relaxation function presents an ill-posed problem, so that converting the creep compliance function to the relaxation function cannot always give accurate results, especially considering that the creep data at short times in nanoindentation are often not reliable, and the overall nanoindentation time is short, typically a few hundred seconds. In this paper, we present methods to measure Young’s relaxation functions directly using nanoindentation. A constant-rate displacement loading history is usually used in nanoindentations. Using viscoelastic contact mechanics, Young’s relaxation modulus is extracted using nanoindentation load-displacement data. Three bulk polymers, Polymethyl Methacrylate (PMMA), Polycarbonate (PC) and Polyurethane (PU), are used in this study. The Young’s relaxation functions measured from the nanoindentation are compared with data measured from conventional tensile and shear tests to evaluate the precision of the methods. A reasonably good agreement has been reached for all these materials for indentation depth higher than a certain value, providing reassurance for these methods for measuring relaxation functions.

Keywords

Young’s relaxation modulus Nanoindentation Viscoelasticity Polymer Berkovich indenter Spherical indenter Conical indenter 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Cheng, L., Xia, X., Yu, W., Scriven, L.E., Gerberich, W.W.: Flat-punch indentation of viscoelastic material. J. Polym. Sci., Part B: Polym. Phys. 38, 10–22 (2000)CrossRefGoogle Scholar
  2. 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, Art. No. 111914 (2005)Google Scholar
  3. Cheng, Y.T., Ni, W.Y., Cheng, C.M.: Nonlinear analysis of oscillatory indentation in elastic and viscoelastic solids. Phys. Rev. Lett. 97, Art. No. 075506 (2006)Google Scholar
  4. Doerner, M.F., Nix, W.D.: A method for interpreting the data from depth-sensing indentation instruments. J. Mater. Res. 1, 601–609 (1992)ADSGoogle Scholar
  5. Emri, I., Tschoegl, N.W.: Generating line spectra from experimental responses. Part I: elaxation modulus and creep compliance. Rheol. Acta 32, 311–321 (1993)CrossRefGoogle Scholar
  6. Emri, I., Tschoegl, N.W.: Generating line spectra from experimental responses. Part IV: aplication to experimental data. Rheol. Acta 33, 60–70 (1994)CrossRefGoogle Scholar
  7. Emri, I., Tschoegl, N.W.: Generating line spectra from experimental responses. Part V. Time-dependent viscosity. Rheol. Acta 36, 303–306 (1997)Google Scholar
  8. Fischer-Cripps, A.C.: Nanoindentation, Mechanical Engineering Series. Springer-Verlag, Berlin (2002)Google Scholar
  9. Giannakopoulos, A.E.: Strength analysis of spherical indentation of piezoelectric materials. J. Appl. Mech. 67, 409–416 (2000)CrossRefGoogle Scholar
  10. Hertz, H.: über die Berührung fester elastischer Körper. Journal für die Reine und Angewandte Mathematik 92, 156–171 (1881)Google Scholar
  11. Hopkins, I.L., Hamming, R.W.: On creep and relaxation. J. Appl. Phys. 28, 906–909 (1957)CrossRefGoogle Scholar
  12. Huang, G., Wang, B., Lu, H.: Measurements of viscoelastic functions in frequency-domain by nanoindentation. Mech. Time-Depend. Mater. 8, 345–364 (2004)CrossRefADSGoogle Scholar
  13. Knauss, W.G., Zhu, W.: Nonlinearly viscoelastic behavior of polycarbonate. I. Response under pure shear. Mech. Time-Depend. Mater. 6, 231–169 (2002)CrossRefGoogle Scholar
  14. Lee, E.H., Radok J.R.M.: The contact problem for viscoelastic bodies. J. Appl. Mech. 27, 438–444 (1960)MathSciNetGoogle Scholar
  15. Li, X., Bhushan, B.: A review of nanoindentation continuous stiffness measurement technique and its applications. Mater. Charact. 48, 11–36 (2002)CrossRefGoogle Scholar
  16. Liu, Y., Wang, B., Yoshino, M., Roy, S., Lu, H., Komanduri, R.: Combined numerical simulation and nanoindentation for determining mechanical properties of single crystal copper at mesoscale. J. Mech. Phys. Solids 53, 2718–2741 (2005)CrossRefGoogle Scholar
  17. Lu, H., Cary, P.D.: Deformation measurements by digital image correlation: implementation of a second-order displacement gradient. Exp. Mech. 40, 393–400 (2000)CrossRefGoogle Scholar
  18. 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)CrossRefADSGoogle Scholar
  19. Lu, H., Zhang, X., Knauss, W.G.: Uniaxial, shear, and poisson relaxation and their conversion to bulk relaxation: studies on poly(methyl methacrylate). Polym. Eng. Sci. 37, 1053–1064 (1997)CrossRefGoogle Scholar
  20. Nikonov, A., Davies, A.R., Emri, I.: The determination of creep and relaxation functions from a single experiment. J. Rheol. 49, 1193–1211 (2005)CrossRefGoogle Scholar
  21. 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, 1564–1583 (1992)ADSGoogle Scholar
  22. Odegard, G.M., Gates, T.S., Herring, H.M.: Characterization of viscoelastic properties of polymeric materials through nanoindentation. Exp. Mech. 45, 130–136 (2005)CrossRefGoogle Scholar
  23. 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)CrossRefMathSciNetGoogle Scholar
  24. Sane, S.B., Knauss, W.G.: The time-dependent bulk response of poly(mehyl methacrylate). Mech. Time-Depend. Mater. 5, 293–324 (2001)CrossRefGoogle Scholar
  25. Ting, T.C.T.: The contact stresses between a rigid indenter and a viscoelastic half-space. J. Appl. Mech. 33, 845–854 (1966)Google Scholar
  26. Tschoegl, N.W., Emri, I.: Generating line spectra from experimental responses. Part II. Storage and loss functions. Rheol. Acta 32, 322–327 (1993)CrossRefGoogle Scholar
  27. Tschoegl, N.W., Emri, I.: Generating line spectra from experimental responses. III. Interconversion between relaxation and retardation behavior. Int. J. Polym. Mater. 18, 117–127 (1992)Google Scholar
  28. 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. Part B: Polym. Phys. 43, 1794–1811 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

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

Personalised recommendations