Abstract
A unilateral frictionless axisymmetric contact problem for an isotropic viscoelastic layer attached to a rigid substrate and loaded with a spherical indenter is considered. It is assumed that the indentation protocol is composed of two stages. In the indentation phase, the layer is subjected to displacement loading, while at the end of the first stage, the load is immediately removed and the second stage, called the recovery phase, lasts for a theoretically indefinite time. Under the assumption of time-independent Poisson’s ratio, we derive closed-form analytical expressions for the contact force (in the indentation phase) and for the indentation displacement (in the the recovery phase). The obtained closed-form analytical solution is valid for the indentation phase with an arbitrary monotonic loading displacement and can be used for evaluation of the rebound indentation test for soft biological tissues and originally suggested for assessment of articular cartilage viability.
Similar content being viewed by others
References
Mattice J.M., Lau A.G., Oyen M.L., Kent R.W.: Spherical indentation load-relaxation of soft biological tissues. J. Mater. Res. 21, 2003–2010 (2006)
Lee E.H., Radok J.R.M.: The contact problem for viscoelastic bodies. J. Appl. Mech. Trans. ASME 27, 438–444 (1960)
Cheng Y.-T., 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, 93–99 (2005)
Giannakopoulos A.E.: Elastic and viscoelastic indentation of flat surfaces by pyramid indentors. J. Mech. Phys. Solids 54, 1305–1332 (2006)
Vandamme M., Ulm F.-J.: Viscoelastic solutions for conical indentation. Int. J. Solids Struct. 43, 3142–3165 (2006)
Jäger A., Lackner R., Eberhardsteiner J.: Identification of viscoelastic properties by means of nanoindentation taking the real tip geometry into account. Meccanica 42, 293–306 (2007)
Cheng L., Xia X., Yu W., Scriven L.E., Gerberich W.W.: Flat-punch indentation of viscoelastic material. J. Polym. Sci. B Polym. Phys. 38, 10–22 (2000)
Cheng L., Xia X., Yu W., Scriven L.E., Gerberich W.W.: Spherical-tip indentation of viscoelastic material. Mech. Mater. 37, 213–226 (2005)
Oyen M.L.: Analytical techniques for indentation of viscoelastic materials. Philos. Mag. 86(33–35), 5625–5641 (2006)
Tweedie C.A., Van Vliet K.J.: Contact creep compliance of viscoelastic materials via nanoindentation. J. Mater. Res. 21, 1576–1589 (2006)
Shimizu S., Yanagimoto T., Sakai M.: Pyramidal indentation load-depth curve of viscoelastic materials. J. Mater. Res. 14, 4075–4086 (1999)
Huang G., Lu H.: Measurement of Young’s relaxation modulus using nanoindentation. Mech. Time-Depend. Mater. 10, 229–243 (2006)
Larsson P.-L., Carlsson S.: On microindentation of viscoelastic polymers. Polym. Test. 17, 49–75 (1998)
Oyen M.L.: Spherical indentation creep following ramp loading. J. Mater. Res. 20, 2094–2100 (2005)
Cheng Y.-T., Yang F.: Obtaining shear relaxation modulus and creep compliance of linear viscoelastic materials from instrumented indentation using axisymmetric indenters of power-law profiles. J. Mater. Res. 24, 3013–3017 (2009)
Brown C.P., Crawford R.W., Oloyede A.: An alternative mechanical parameter for assessing the viability of articular cartilage. Proc. Inst. Mech. Eng. Part H 223, 53–62 (2009)
Hunter S.C.: The Hertz problem for a rigid spherical indenter and a viscoelastic half-space. J. Mech. Phys. Solids 8, 219–234 (1960)
Graham G.A.C.: The contact problem in the linear theory of viscoelasticity. Int. J. Eng. Sci. 3, 27–46 (1965)
Ting T.C.T.: The contact stresses between a rigid indenter and a viscoelastic half-space. J. Appl. Mech. 33, 845–854 (1966)
Ting T.C.T.: Contact problems in the linear theory of viscoelasticity. J. Appl. Mech. 35, 248–254 (1968)
Ramesh Kumar M.V., Narasimhan R.: Analysis of spherical indentation of linear viscoelastic materials. Curr. Sci. 87, 1088–1095 (2004)
Greenwood J.A.: Contact between an axisymmetric indenter and a viscoelastic half-space. Int. J. Mech. Sci. 52, 829–835 (2010)
Hsueh C.H., Miranda P.: Master curves for Hertzian indentation on coating/substrate systems. J. Mater. Res. 19, 94–100 (2004)
Sakai M.: Elastic and viscoelastic contact mechanics of coating/substrate composites in axisymmetric indentation. Philos. Mag. 86(33–35), 5607–5624 (2006)
Choi S.T., Lee S.R., Earmme Y.Y.: Flat indentation of a viscoelastic polymer film on a rigid substrate. Acta Materialia 56, 5377–5387 (2008)
Zhang C.Y., Zhang Y.W.: Extracting the mechanical properties of a viscoelastic polymeric film on a hard elastic substrate. J. Mater. Res. 14, 3053–3061 (2004)
Cao Y.-P., Ji X.-Y., Feng X.-Q.: Geometry independence of the normalized relaxation functions of viscoelastic materials in indentation. Philos. Mag. 90, 1639–1655 (2010)
Argatov I., Mishuris G.: An analytical solution for a linear viscoelastic layer loaded with a cylindrical punch: Evaluation of the rebound indentation test with application for assessing viability of articular cartilage. Mech. Res. Commun. 38, 565–568 (2011)
Lebedev N.N., Ufliand Ia.S.: Axisymmetric contact problem for an elastic layer. J. Appl. Math. Mech. 22, 442–450 (1958)
Hayes W.C., Keer L.M., Herrmann G., Mockros L.F.: A mathematical analysis for indentation tests of articular cartilage. J. Biomech. 5, 541–551 (1972)
Sakamoto M., Li G., Hara T., Chao E.Y.S.: A new method for theoretical analysis of static indentation test. J. Biomech. 29, 679–685 (1996)
Mow V.C., Kuei S.C., Lai W.M., Armstrong C.G.: Biphasic creep and stress relaxation of articular cartilage in compression. J. Biomech. Eng. 102, 73–84 (1980)
Hayes W.C., Mockros L.F.: Viscoelastic properties of human articular cartilage. J. Appl. Physiol. 31, 562–568 (1971)
Parsons J.R., Black J.: The viscoelastic shear behavior of normal rabbit articular cartilage. J. Biomech. 10, 21–29 (1977)
Argatov I., Mishuris G.: Frictionless elliptical contact of thin viscoelastic layers bonded to rigid substrates. Appl. Math. Model. 35, 3201–3212 (2011)
Ateshian G.A., Lai W.M., Zhu W.B., Mow V.C.: An asymptotic solution for the contact of two biphasic cartilage layers. J. Biomech. 27, 1347–1360 (1994)
Argatov, I.: Development of an asymptotic modeling methodology for tibio-femoral contact in multibody dynamic simulations of the human knee joint. Multibody Syst. Dyn. doi:10.1007/s11044-011-9275-6
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Argatov, I. An analytical solution of the rebound indentation problem for an isotropic linear viscoelastic layer loaded with a spherical punch. Acta Mech 223, 1441–1453 (2012). https://doi.org/10.1007/s00707-012-0668-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-012-0668-2