Nonlinear elastic load–displacement relation for spherical indentation on rubberlike materials


Because of the lack of universal contact models for nonlinear strain problems, indentation analysis on rubberlike materials is confined to small deformation in which Hertz’s solution is applied. Recognizing that deep indentation may provide more material information, in this paper we propose a nonlinear elastic model for large spherical indentation of rubberlike materials based on the higher-order approximation of spherical function and Sneddon’s solution. The effect of limiting network stretch is studied on the initial elastic modulus for lightly cross-linked rubbers. With the comparisons of the finite-element simulation and the experimental result, the proposed model is verified to predict the large indentation of rubberlike materials over the indentation depth of 0.8 times the indenter radius.

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  1. 1.

    W.C. Oliver, G.M. Pharr: An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7 1564 (1992)

    CAS  Article  Google Scholar 

  2. 2.

    ISO 14577 Metallic Materials—Instrumented Indentation Test for Hardness and Materials Parameters (International Organization for Standardization, Geneva, Switzerland 2002)

    Google Scholar 

  3. 3.

    G.M. Pharr, Y-T. Cheng, I.M. Hutchings, M. Sakai, N.R. Moody, G. Sundararajan, M.V. Swain: Focus issue on indentation methods in advanced materials research. J. Mater. Res. 24, (3) 579 (2009)

    Article  Google Scholar 

  4. 4.

    K.L. Johnson: Contact Mechanics (Cambridge University Press, Cambridge, UK 1985)

    Book  Google Scholar 

  5. 5.

    J. Tan, Y.J. Chao, J.W. Van Zee, X. Li, X. Wang, M. Yang: Assessment of mechanical properties of fluoroelastomer and EPDM in a simulated PEM fuel cell environment by microindentation test. Mater. Sci. Eng., A 496 464 (2008)

    Article  Google Scholar 

  6. 6.

    D.C. Lin, E.K. Dimitriadis, F. Horkay: Elasticity of rubber-like materials measured by AFM nanoindentation. eXPRESS Polym. Lett. 9 576 (2007)

    Article  Google Scholar 

  7. 7.

    A.C. Fisher-Cripps: Nanoindentation (Springer Press, New York 2002)

    Book  Google Scholar 

  8. 8.

    I.M. Ward, J. Sweeney: An Introduction to the Mechanical Properties of Solid Polymers 2nd ed (Wiley Press, Chichester, UK 2004) 32–36

    Google Scholar 

  9. 9.

    M.R. VanLandingham, J.S. Villarrubia, W.F. Guthrie, G.F. Meyers: Nanoindentation of polymer: An overview. Macromol. Symp. 167 167 (2001)

    Article  Google Scholar 

  10. 10.

    D.C. Lin, F. Horkay: Nanomechanics of polymer gels and biological tissues: A critical review of analytical approaches in the Hertzian regime and beyond. Soft Mater. 4 669 (2008)

    CAS  Article  Google Scholar 

  11. 11.

    R.S. Rivlin: Large elastic deformations of isotropic materials: IV. Future developments of the general theory. Philos. Trans. R. Soc. London, Ser. A 241 379 (1948)

    Article  Google Scholar 

  12. 12.

    G.C.W. Sabin, P.N. Kaloni: Contact problem of a rigid indenter with notational friction in second order elasticity. Int. J. Eng. Sci. 27 203 (1989)

    Article  Google Scholar 

  13. 13.

    A.E. Giannakopoulos, A. Triantafyllou: Spherical indentation of incompressible rubber-like materials. J. Mech. Phys. Solids 55 1196 (2007)

    CAS  Article  Google Scholar 

  14. 14.

    I.N. Sneddon: The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int. J. Eng. Sci. 3 47 (1965)

    Article  Google Scholar 

  15. 15.

    E.M. Arruda, M.C. Boyce: A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 41 389 (1993)

    CAS  Article  Google Scholar 

  16. 16.

    I. Kang, D. Panneerselvam, V.P. Panoskaltsis, S.J. Eppell, R.E. Marchant: Changes in the hyperelastic properties of endothelial cells induced by tumor necrosis factor-α. Biophys. J. 94 3273 (2008)

    CAS  Article  Google Scholar 

  17. 17.

    H. Yin, L.Z. Sun, G. Wang, T. Yamada, J. Wang, M. Vannier: ImageParser: A tool for finite element generation from three-dimensional medical images. Biomed. Eng. Online 3 31 (2004)

    CAS  Article  Google Scholar 

  18. 18.

    G.M. Pharr, W.C. Oliver, F.R. Brotzen: On the generality of the relationship among contact stiffness, contact area, and elastic modulus during indentation. J. Mater. Res. 7 613 (1992)

    CAS  Article  Google Scholar 

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Correspondence to L. Z. Sun.

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Liu, D.X., Zhang, Z.D. & Sun, L.Z. Nonlinear elastic load–displacement relation for spherical indentation on rubberlike materials. Journal of Materials Research 25, 2197–2202 (2010).

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