Indentation of ellipsoidal and cylindrical shells: new insights from shallow-shell theory

Pressurized elastic shells are ubiquitous in nature, from pollen grains to the outer walls of yeast and bacterial cells. Indentation measurements provide a means of simultaneously probing the internal pressure and elastic properties of thin shells, which in turn can be relevant to understanding cellular function. We study the effects of geometry and pressure-induced stress on the indentation stiffness of ellipsoidal and cylindrical elastic shells using shallow shell theory. The key advance in our work lies in reducing the linear indentation response to a single integral with two dimensionless parameters which encode the asphericity and internal pressure. This integral can be numerically evaluated in all regimes, and is used to generate analytical expressions in various limits. Our results provide theoretical support for previous scaling and numerical results on the stiffness of ellipsoids, and give new insights to the linear indentation response of pressurized cylinders.