Indentation responses of pressurized ellipsoidal and cylindrical elastic shells: Insights from shallow-shell theory

TL;DR

This paper develops a theoretical framework using shallow-shell theory to analyze how geometry and internal pressure affect the indentation stiffness of pressurized ellipsoidal and cylindrical elastic shells, relevant for biological and engineering applications.

Contribution

It introduces a unified integral formulation capturing the indentation response of shells, providing analytical expressions and new insights into pressure effects on shell stiffness.

Findings

Derived a single integral expression for indentation response

Identified a new pressure scale influencing large-pressure behavior

Provided analytical formulas for various shell geometries

Abstract

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.