Spherical-tip indentation of viscoelastic material

L. Cheng, X. Xia, L. E. Scriven, W. W. Gerberich

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In this paper, the indentation of viscoelastic solids by a spherical-tip indenter is analyzed. The viscoelastic solids are described by a standard solid model, that is, a three-element model. The analytical solutions are derived for a half-space of the viscoelastic material based on the method of functional equations. These solutions can apply to the response of compressible as well as incompressible coated layers to a spherical-tip indentation. The results include convenient closed-form formulas for approximating both relaxation testing and creep testing. They establish a fundamental basis for probing mechanical properties of polymeric coatings on substrates with nanoindentation or microindentation tests. Comparisons are made on the tests of both bulk polystyrene (PS) and drying semicrystalline polyvinyl alcohol (PVOH) coatings on a silicon wafer to verify the application of analytical solutions to viscoelastic polymers. The moduli obtained for bulk PS correspond with literature values well. While the results from PVOH coatings show good agreement between elastic unloading and viscoelastic models in 10% relative humidity, the viscoelastic model gives a considerably lower value after long-term holds in 70% relative humidity. This is partially attributed to a strain rate effect.

Original languageEnglish (US)
Pages (from-to)213-226
Number of pages14
JournalMechanics of Materials
Issue number1
StatePublished - Jan 2005

Bibliographical note

Funding Information:
The authors would like to gratefully acknowledge the support from Dow Corning and Kodak Corporations, the Center for Interfacial Engineering at the University of Minnesota, and Grant DE-FG02-96ER 45574 for WWG for this research. The computer resources and technical support provided by the University of Minnesota Supercomputing Institute is also gratefully acknowledged.

Copyright 2008 Elsevier B.V., All rights reserved.


  • Creep
  • Indentation
  • Relaxation
  • Spherical-tip indenter
  • Viscoelastic properties


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