Depth-sensing spherical indentation of an elastic sphere on an elastic substrate

Ivan I. Argatov, Xiaoqing Jin*, Leon M. Keer

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


The axisymmetric three-body, double-unilateral contact problem for an elastic sphere is treated by analytical techniques. It is assumed that the elastic sphere is put on the surface of an elastic layer of finite thickness and, afterwards, is indented at the upper pole of the sphere with a rigid spherical punch. Friction is neglected at both contact interfaces. A first-order asymptotic solution for the displacement–force relation, which generalizes the corresponding Hertzian formula, is derived in explicit form in terms of dimensionless asymptotic constants that account for the finite sizes of the elastic sphere and layer. Under the assumption that the sphere/substrate contact diameter is less than the substrate thickness, the presented results can cover a wide range of parameter combinations, which may be used for the purpose of benchmark assessments in the finite element analysis. The constructed asymptotic model is remarkably simple and elegant, and yet can be applied to various contact problems of practical importance (including a very timely problem of indentation of spherical viruses). The underlying theoretical framework is versatile and can be further extended for analysis of multiple contacts involving elastic spheres and elastic layers.

Original languageEnglish (US)
Article number104297
JournalJournal of the Mechanics and Physics of Solids
StatePublished - Apr 2021


  • Asymptotic modeling
  • Depth-sensing indentation
  • Elastic sphere
  • Indentation compliance

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


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