Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-26T08:05:58.684Z Has data issue: false hasContentIssue false
  • English
  • Français

Obtaining shear relaxation modulus and creep compliance of linear viscoelastic materials from instrumented indentation using axisymmetric indenters of power-law profiles

Published online by Cambridge University Press:  31 January 2011

Yang-Tse Cheng  and
Fuqian Yang
Yang-Tse Cheng*
Affiliation:
Department of Chemical and Materials Engineering, University of Kentucky, Lexington, Kentucky 40506
Fuqian Yang*
Affiliation:
Department of Chemical and Materials Engineering, University of Kentucky, Lexington, Kentucky 40506
*
a) Address all correspondence to this author. e-mail: ycheng@engr.uky.edu
This author was an editor of this journal during the review and decision stage. For the JMR policy on review and publication of manuscripts authored by editors, please refer to http://www.mrs.org/jmr_policy
Get access

Abstract

Using Laplace transform, we solve the inverse problem of obtaining the shear relaxation modulus and creep compliance of linear viscoelastic solids from indentation by axisymmetric indenters of power-law profiles. We identify several simple, though nontrivial, loading paths for carrying out indentation measurements such that the inverse problem has analytical solutions. We show that the shear relaxation modulus and creep compliance may be readily obtained using the newly derived analytical expressions together with proposed indentation loading paths.

Keywords

Elastic propertiesHardnessNanoscale
Type
Articles
Information
Journal of Materials Research , Volume 24 , Issue 10 , October 2009 , pp. 3013 - 3017
Copyright
Copyright © Materials Research Society 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Fischer, A.C.-Cripps:Nanoindentation, 2nd ed. (Springer, New York, 2004).CrossRefGoogle Scholar
2.Van Landingham, M.R.: Review of instrumented indentation. J. Res. Nat. Inst. Stand. Technol. 108, 249 (2003).CrossRefGoogle ScholarPubMed
3.Ebenstein, D.M. and Pruitt, L.A.: Nanoindentation of biological materials. Nano Today 1, 26 (2006).CrossRefGoogle Scholar
4.Pharr, G.M., Cheng, Y-T., Hutchings, I.M., Sakai, M., Moody, N.R., Sundararajan, G., and Swain, M.V.: Introduction. J. Mater. Res. 24, 579 (2009).CrossRefGoogle Scholar
5.Lee, E.H.: Stress analysis in visco-elastic bodies. Q. Appl. Math. 13, 183 (1955).CrossRefGoogle Scholar
6.Radok, J.R.M.: Visco-elastic stress analysis. Q. Appl. Math. 15, 198 (1957).CrossRefGoogle Scholar
7.Lee, E.H. and Radok, J.R.M.: The contact problem for viscoelastic bodies. J. Appl. Mech. 27, 438 (1960).CrossRefGoogle Scholar
8.Hunter, S.C.: The Hertz problem for a rigid spherical indenter and a viscoelastic half-space. J. Mech. Phys. Solids 8, 219 (1960).CrossRefGoogle Scholar
9.Graham, G.A.C.: The contact problem in the linear theory of viscoelasticity. Int. J. Eng. Sci. 3, 27 (1965).CrossRefGoogle Scholar
10.Graham, G.A.C.: Contact problem in linear theory of viscoelasticity when time dependent contact area has any number of maxima and minima. Int. J. Eng. Sci. 5, 495 (1967).CrossRefGoogle Scholar
11.Ting, T.C.T.: Contact stresses between a rigid indenter and a viscoelastic half-space. J. Appl. Mech. 33, 845 (1966).CrossRefGoogle Scholar
12.Ting, T.C.T.: Contact problems in linear theory of viscoelasticity. J. Appl. Mech. 35, 248 (1968).CrossRefGoogle Scholar
13.Briscoe, B.J., Fiori, L., and Pelillo, E.: Nano-indentation of polymeric surfaces. J. Phys. D: Appl. Phys. 31, 2395 (1998).CrossRefGoogle Scholar
14.Cheng, L., Xia, X., Yu, W., Scriven, L.E., and Gerberich, W.W.: Flat-punch indentation of viscoelastic material. J. Polym. Sci., Part B: Polym. Phys. 38, 10 (2000).3.0.CO;2-6>CrossRefGoogle Scholar
15.Larrson, P-L. and Carlsson, S.: On microindentation of viscoelastic polymers. Polym. Test 17, 49 (1998).CrossRefGoogle Scholar
16.Shimizu, S., Yanagimoto, T., and Sakai, M.: Pyramidal indentation load-depth curve of viscoelastic materials. J. Mater. Res. 14, 4075 (1999).CrossRefGoogle Scholar
17.Sakai, M. and Shimizu, S.: Indentation rheometry for glassforming materials. J. Non-Cryst. Solids 282, 236 (2001).CrossRefGoogle Scholar
18.Sakai, M.: Time-dependent viscoelastic relation between load and penetration for an axisymmetric indenter. Philos. Mag. A 82, 1841 (2002).CrossRefGoogle Scholar
19.Ngan, A.H.W. and Tang, B.: Viscoelastic effects during unloading in depth-sensing indentation. J. Mater. Res. 17, 2604 (2002).CrossRefGoogle Scholar
20.Oyen, M.L. and Cook, R.F.: Load-displacement behavior during sharp indentation of viscous-elastic-plastic materials. J. Mater. Res. 18, 139 (2003).CrossRefGoogle Scholar
21.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, 189 (2003).CrossRefGoogle Scholar
22.Tang, B. and Ngan, A.H.W.: Investigation of viscoelastic properties of amorphous selenium near class transition using depthsensing indentation. Soft Materials 2, 125 (2004).CrossRefGoogle Scholar
23.Geng, B., Yang, F.Q., Druffel, T., and Grulke, E.A.: Nanoindentation behavior of ultrathin polymeric films. Polymer (Guildf.) 46, 11768 (2005).CrossRefGoogle Scholar
24.Zhang, C.Y., Zhang, Y.W., Zeng, K.Y., and Shen, L.: Nanoindentation of polymers with a sharp indenter. J. Mater. Res. 20, 1597 (2005).CrossRefGoogle Scholar
25.Zhang, C.Y., Zhang, Y.W., Zeng, K.Y., and Shen, L.: Characterization of mechanical properties of polymers by nanoindentation tests. Philos. Mag. 86, 4487 (2006).CrossRefGoogle Scholar
26.Huang, G. and Lu, H.B.: Measurement of Young's relaxation modulus using nanoindentation. Mech. Time-Depend. Mater. 10, 229 (2006).CrossRefGoogle Scholar
27.Tranchida, D., Piccarolo, S., Loos, J., and Alexeev, A.: Accurately evaluating Young's modulus of polymers through nanoindentations: A phenomenological correction factor to the Oliver and Pharr procedure. Appl. Phys. Lett. 89, 171905 (2006).CrossRefGoogle Scholar
28.Tweedie, C.A. and Van Vliet, K.J.: Contact creep compliance of viscoelastic materials via nanoindentation. J. Mater. Res. 21, 1576 (2006).CrossRefGoogle Scholar
29.Vandamme, M. and Ulm, F.J.: Viscoelastic solutions for conical indentation. Int. J. Solids Struct. 43, 3142 (2006).CrossRefGoogle Scholar
30.Liu, C.K., Lee, S., Sung, L.P., and Nguyen, T.: Load-displacement relations for nanoindentation of viscoelastic materials. J. Appl. Phys. 100, 033503 (2006).CrossRefGoogle Scholar
31.Giannakopoulos, A.E.: Elastic and viscoelastic indentation of flat surfaces by pyramid indentors. J. Mech. Phys. Solids 54, 1305 (2006).CrossRefGoogle Scholar
32.Oyen, M.L.: Analytical techniques for indentation of viscoelastic materials. Philos. Mag. 86, 5625 (2006).CrossRefGoogle Scholar
33.Deuschle, J., Enders, S., and Arzt, E.: Surface detection in nanoindentation of soft polymers. J. Mater. Res. 22, 3107 (2007).CrossRefGoogle Scholar
34.Wright, W.J., Maloney, A.R., and Nix, W.D.: An improved analysis for viscoelastic damping in dynamic nanoindentation. Int. J. Surf. Sci. Eng. 1, 274 (2007).CrossRefGoogle Scholar
35.Fujisawa, N. and Swain, M.V.: Nanoindentation-derived elastic modulus of an amorphous polymer and its sensitivity to loadhold period and unloading strain rate. J. Mater. Res. 23, 637 (2008).CrossRefGoogle Scholar
36.Herbert, E.G., Oliver, W.C., and Pharr, G.M.: Nanoindentation and the dynamic characterization of viscoelastic solids. J. Phys. D: Appl. Phys. 41, 074021 (2008).CrossRefGoogle Scholar
37.Pichler, C., Lackner, R., and Ulm, F.J.: Scaling relations for viscoelastic-cohesive conical indentation. Int. J. Mater. Res. 99, 836 (2008).CrossRefGoogle Scholar
38.Cao, Y.P., Ma, D.C., and Raabe, D.: The use of flat punch indentation to determine the viscoelastic properties in the time and frequency domains of a soft layer bonded to a rigid substrate. Acta Biomater. 5, 240 (2009).CrossRefGoogle ScholarPubMed
39.Cheng, Y.T. and Cheng, C.M.: Scaling, dimensional analysis, and indentation measurements. Mater. Sci. Eng., R 44, 91 (2004).CrossRefGoogle Scholar
40.Cheng, Y.T. and 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
41.Cheng, Y.T. and Cheng, C.M.: Relationships between initial unloading slope, contact depth, and mechanical properties for conical indentation in linear viscoelastic solids. J. Mater. Res. 20, 1046 (2005).CrossRefGoogle Scholar
42.Cheng, Y.T., Ni, W.Y., and Cheng, C.M.: Determining the instantaneous modulus of viscoelastic solids using instrumented indentation measurements. J. Mater. Res. 20, 3061 (2005).CrossRefGoogle Scholar
43.Cheng, Y.T. and Cheng, C.M.: Relationships between initial unloading slope, contact depth, and mechanical properties for spherical indentation in linear viscoelastic solids. Mater. Sci. Eng., A 409, 93 (2005).CrossRefGoogle Scholar
44.Cheng, Y.T., Ni, W.Y., and Cheng, C.M.: Nonlinear analysis of oscillatory indentation in elastic and viscoelastic solids. Phys. Rev. Lett. 97, 075506 (2006).CrossRefGoogle ScholarPubMed
45.Findley, W.N., Lai, J.S., and Onaran, K.: Creep and Relaxation of Nonlinear Viscoelastic Materials (Dover Publications, Inc., Mineola, NY, 1989).Google Scholar
46.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 (1965).CrossRefGoogle Scholar