  Indentation tests are classical tools to determine material properties. For
biological samples such as cysts of cells, however, the observed
force-displacement relation cannot be expected to follow predictions for simple
materials. Here, by solving the Pogorelov problem of a point force indenting an
elastic shell for a purely nonlinear material, we discover that complex
material behaviour can even give rise to new scaling exponents in this
force-displacement relation. In finite-element simulations, we show that these
exponents are surprisingly robust, persisting even for thick shells indented
with a sphere. By scaling arguments, we generalise our results to pressurised
and pre-stressed shells, uncovering additional new scaling exponents. We find
these predicted scaling exponents in the force-displacement relation observed
in cyst indentation experiments. Our results thus form the basis for inferring
the mechanisms that set the mechanical properties of these biological
materials.
