  Pressurized elastic shells are ubiquitous in nature and technology, from the
outer walls of yeast and bacterial cells to artificial pressure vessels.
Indentation measurements simultaneously probe the internal pressure and elastic
properties of thin shells, and serve as a useful tool for strength testing and
for inferring internal biological functions of living cells. We study the
effects of geometry and pressure-induced stress on the indentation stiffness of
ellipsoidal and cylindrical elastic shells using shallow-shell theory. We show
that the linear indentation response reduces to a single integral with two
dimensionless parameters that encode the asphericity and internal pressure.
This integral can be numerically evaluated in all regimes and is used to
generate compact analytical expressions for the indentation stiffness in
various regimes of technological and biological importance. Our results provide
theoretical support for previous scaling and numerical results describing the
stiffness of ellipsoids, reveal a new pressure scale that dictates the
large-pressure response, and give new insights to the linear indentation
response of pressurized cylinders.
