Finite indentation of highly curved elastic shells

TL;DR

This paper extends the theoretical modeling of indentation tests on highly curved elastic shells by incorporating bending stiffness, enabling more accurate estimation of elastic properties of biological surfaces with complex geometries.

Contribution

It introduces a new model that includes bending stiffness in the indentation analysis of highly curved elastic shells, addressing limitations of previous flat-surface assumptions.

Findings

Inclusion of bending stiffness improves model accuracy for curved shells.

The model allows estimation of elastic properties independent of shape and size.

The theory is particularly applicable to biological membrane surfaces.

Abstract

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 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.