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Finite indentation of highly curved elastic shells

Pearce, Simon P.; King, John R.; Steinbrecher, Tina; Leubner-Metzger, Gerhard; Everitt, Nicola M.; Holdsworth, Michael J.


Simon P. Pearce

Professor of Theoretical Mechanics

Tina Steinbrecher

Gerhard Leubner-Metzger

Nicola M. Everitt


Experimentally measuring the elastic properties of thin biological surfaces is non-trivial, particularly when they are curved. One technique that may be used is the indentation of a thin sheet of material by a rigid indenter, whilst measuring the applied force and displacement. This gives immediate information on the fracture strength of the material (from the force required to puncture), but it is also theoretically possible to determine the elastic properties by comparing the resulting force-displacement curves with a mathematical model. Existing mathematical studies generally assume that the elastic surface is initially at, which is often not the case for biological membranes. We previously outlined a theory for the indentation of curved isotropic, incompressible, hyperelastic membranes (with no bending stiffness) which breaks down for highly curved surfaces, as the entire membrane becomes wrinkled. Here we introduce the effect of bending stiffness, ensuring that energy is required to change the shell shape without stretching, and find that commonly neglected terms in the shell equilibrium equation must be included. The theory presented here allows for the estimation of shape- and size-independent elastic properties of highly curved surfaces via indentation experiments, and is particularly relevant for biological surfaces.


Pearce, S. P., King, J. R., Steinbrecher, T., Leubner-Metzger, G., Everitt, N. M., & Holdsworth, M. J. (2018). Finite indentation of highly curved elastic shells. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 474(2209),

Journal Article Type Article
Acceptance Date Dec 7, 2017
Publication Date Jan 24, 2018
Deposit Date Dec 19, 2017
Publicly Available Date Jan 24, 2018
Journal Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Print ISSN 1364-5021
Electronic ISSN 1471-2946
Publisher The Royal Society
Peer Reviewed Peer Reviewed
Volume 474
Issue 2209
Public URL
Publisher URL


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