  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.
