Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T21:54:37.107Z Has data issue: false hasContentIssue false

Numerical Analysis of Indentation of an Elastic Hemispherical Shell

Published online by Cambridge University Press:  15 July 2015

Y.-F. Jia
Affiliation:
Key Laboratory of Pressure System and Safety (MOE)School of Mechanical and Power EngineeringEast China University of Science and TechnologyShanghai, P.R. China
F.-Z. Xuan
Affiliation:
Key Laboratory of Pressure System and Safety (MOE)School of Mechanical and Power EngineeringEast China University of Science and TechnologyShanghai, P.R. China
F.-Q. Yang*
Affiliation:
Materials ProgramDepartment of Chemical and Materials EngineeringUniversity of Kentucky Lexington, USA
*
*Corresponding author ([email protected])
Get access

Abstract

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.

Type
Research Article
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Oliver, W. C. and Pharr, G. M., “An Improved Technique for Determining Hardness and Elastic-Modulus Using Load and Displacement Sensing Indentation Experiments,” Journal of Materials Research, 7, pp. 15641583 (1992).Google Scholar
2.Carrasco, C., Carreira, A., Schaap, I. A. T., Serena, P. A., Gomez-Herrero, J., Mateu, M. G. and Pablo, P. J., “DNA-Mediated Anisotropic Mechanical Rein forcement of a Virus,” Proceedings of the National Academy of Sciences of the United States of America, 103, pp. 1370613711 (2006).Google Scholar
3.Ivanovska, I. L., de Pablo, P. J., Ibarra, B., Sgalari, G., MacKintosh, F. C., Carrascosa, J. L., Schmidt, C. F. and Wuite, G. J. L., “Bacteriophage Capsids: Tough Nanoshells with Complex Elastic Properties,“ Proceedings of the National Academy of Sciences of the United States of America, 101, pp. 76007605 (2004).Google Scholar
4.Dubreuil, F., Elsner, N. and Fery, A., “Elastic Prop erties of Poly electrolyte Capsules Studied by Atomic Force Microscopy and RICM,” European Physical Journal E, 12, pp. 215221 (2003).Google Scholar
5.Heuvingh, J., Zappa, M. and Fery, A., “Salt Softening of Polyelectrolyte Multilayer Capsules,” Langmuir, 21, pp. 31653171 (2005).Google Scholar
6.Zhang, L. J., D'Acunzi, M., Kappl, M., Imhof, A., van Blaaderen, A., Butt, H. J., Graf, R. and Vollmer, D., “Tuning the Mechanical Properties of Silica Microcapsules,” Physical Chemistry Chemical Physics, 12, pp. 1539215398 (2010).Google Scholar
7.Delorme, N. and Fery, A., “Direct Method to Study Membrane Rigidity of Small Vesicles Based on Atomic Force Microscope Force Spectroscopy,“ Physical Review E, 74, p. 030901 (2006).Google Scholar
8.Delorme, N., Dubois, M., Garnier, S., Laschewsky, A., Weinkamer, R., Zemb, T. and Fery, A., “Surface Immobilization and Mechanical Properties of Catanionic Hollow Faceted Polyhedrons,” Journal of Physical Chemistry B, 110, pp. 17521758 (2006).Google Scholar
9.Minary-Jolandan, M. and Yu, M. F., “Reversible Radial Deformation up to the Complete Flattening of Carbon Nanotubes in Nanoindentation,” Journal of Applied Physics, 103, p. 073516 (2008).Google Scholar
10.Palaci, I., Fedrigo, S., Brune, H., Klinke, C., Chen, M. and Riedo, E., “Radial Elasticity of Multiwalled Carbon Nanotubes,” Physical Review Letters, 94, p. 175502 (2005).Google Scholar
11.Shen, W. D., Jiang, B., Han, B. S. and Xie, S. S., “Investigation of the Radial Compression of Carbon Nanotubes with a Scanning Probe Microscope,“ Physical Review Letters, 84, pp. 36343637 (2000).Google Scholar
12.Yu, M. F., Kowalewski, T. and Ruoff, R. S., “Investigation of the Radial Deformability of Individual Carbon Nanotubes under Controlled Indentation Force,” Physical Review Letters, 85, pp. 14561459 (2000).Google Scholar
13.Zhao, Y., Ge, Z. B. and Fang, J. Y., “Elastic Modulus of Viral Nanotubes,” Physical Review E, 78, p. 031914(2008).Google Scholar
14.Kis, A.et al., “Nanomechanics of Microtubules,“ Physical Review Letters, 89, p. 248101 (2002).Google Scholar
15.Zhao, Y., An, L. A. and Fang, J. Y., “Buckling Instability of Lipid Tubules with Multibilayer Walls un der Local Radial Indentation,” Physical Review E, 80, p. 021911 (2009).Google Scholar
16.Zhao, Y., Tamhane, K., Zhang, X. J., An, L. N. and Fang, J. Y., “Radial Elasticity of Self-Assembled Lipid Tubules,” ACS Nano, 2, pp. 14661472 (2008).Google Scholar
17.Hertz, H., “Über die Berührung Fester Elastischer Körper,” Journal für die Reine und Angewandte Mathematik, 92, pp. 156171 (1882).Google Scholar
18.Laney, D. E., Garcia, R. A., Parsons, S. M. and Hans-ma, H. G., “Changes in the Elastic Properties of Cholinergic Synaptic Vesicles as Measured by Atomic Force Microscopy,” Biophysical Journal, 72, pp. 806813 (1997).Google Scholar
19.Zhang, L., D'Acunzi, M., Kappl, M., Auernhammer, G. K., Vollmer, D., van Kats, C. M. and van Blaaderen, A., “Hollow Silica Spheres: Synthesis and Mechanical Properties,” Langmuir, 25, pp. 27112717(2009).Google Scholar
20.Reissner, E., “Stresses and Small Displacements of Shallow Spherical Shells.2,” Journal of Mathematics and Physics, 25, pp. 279300 (1946).Google Scholar
21.Nasto, A., Ajdari, A., Lazarus, A., Vaziri, A. and Reis, P. M., “Localization of Deformation in Thin Shells under Indentation,” Soft Matter, 9, pp. 67966803 (2013).Google Scholar
22.Pogorelov, A. V., Bendings of Surfaces and Stability of Shells, AMS Bookstore, Providence, RI, (1988).Google Scholar
23.Vaziri, A., “Mechanics of Highly Deformed Elastic Shells,” Thin-Walled Structures, 47, pp. 692700 (2009).Google Scholar
24.Vaziri, A. and Mahadevan, L., “Localized and Extended Deformations of Elastic Shells,” Proceedings of the National Academy of Sciences of the United States of America, 105, pp. 79137918 (2008).Google Scholar
25.Vella, D., Ajdari, A., Vaziri, A. and Boudaoud, A., “The Indentation of Pressurized Elastic Shells: From Polymeric Capsules to Yeast Cells,” Journal of the Royal Society Interface, 9, pp. 448455 (2012).Google Scholar