Finite indentation of highly curved elastic shells

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, while measuring the applied force and displacement. This gives immediate i...

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Bibliographic Details
Published in:Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2018-01, Vol.474 (2209), p.20170482-20170482
Main Authors: Pearce, S. P., King, J. R., Steinbrecher, T., Leubner-Metzger, G., Everitt, N. M., Holdsworth, M. J.
Format: Article
Language:English
Subjects:
Biological properties
Computational fluid dynamics
Elastic properties
Elastic Shell
Elastic shells
Endosperm
Equilibrium equations
Fracture strength
Indentation
Mathematical analysis
Membranes
Nonlinear Elasticity
Stiffness
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Summary: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, while 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 flat, 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.
ISSN:1364-5021
1471-2946
DOI:10.1098/rspa.2017.0482