Numerical Analysis of Indentation of an Elastic Hemispherical Shell

Y. F. Jia, F. Z. Xuan, F. Q. Yang

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


Nanoindentation technique has been used to measure the elastic modulus of virus capsids, capsules, vesicles, and tubules. The principle of the nanoindentation technique is based on the elastic solution of an isotropic, homogeneous, semi-infinite material, which limits the use of nanoindentation in measuring the mechanical properties of materials of small scales. To address this limitation, Reissner's thin shell model, which is based on the point indentation on a thin shell, has been used in analyzing the indentation of thin shells. In this work, the indentation of elastic hemispherical shells of various thicknesses by a rigid, spherical indenter is analyzed, using the finite element method. The simulation results reveal the limitation of the Reissner's thin shell model. A semi-analytical relationship between the indentation depth and the indentation load is proposed, which consists of the contributions of the Hertz's local deformation, the Reissner's local flattening, and the Pogorelov's deflection of the shell. This relationship provides an analytical basis of using nanoindentation to determine the nominal contact modulus of spherical shells. The comparison between the numerical results and the experimental results in literature supports the proposed semi-analytical relationship and reveals the effect of viscoelastic characteristic of shell structures on the indentation deformation.

Original languageEnglish
Pages (from-to)245-253
Number of pages9
JournalJournal of Mechanics
Issue number3
StatePublished - Jun 1 2016

Bibliographical note

Publisher Copyright:
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2016.


  • Biomaterials
  • Elastic properties
  • Finite element analysis
  • Indentation (normal, micro and nano)

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanical Engineering
  • Applied Mathematics


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