A coordinate-free guide to the mechanics of thin shells

TL;DR

This paper presents a coordinate-free derivation of the equilibrium equations for thin shells with shear, introducing tensorial force and couple fields, and applies the theory to analyze vibrations of pressurized spherical shells.

Contribution

It offers a novel coordinate-free framework for thin shell mechanics, simplifying derivation and analysis of shell behavior including shear effects and vibrations.

Findings

Coordinate-free derivation simplifies shell equations.

Application to pressurized spherical shells demonstrates practical utility.

Linearized equations facilitate vibration analysis.

Abstract

In this tutorial, we provide a coordinate-free derivation of the system of equations that govern equilibrium of a thin shell that can undergo shear. This system involves tensorial fields representing the internal force and couple per unit length that adjacent parts of the shell exchange at their common boundary. By an appropriate decomposition of those quantities, we obtain a representation of the internal power in terms of time derivatives of suitable strain measures. Subsequently, we propose constitutive equations that employ these strain measures as independent variables. After specializing the theory to the case of unshearable shells, we linearize the resulting equations. As an application, we study the free vibrations of a pressurized spherical shell, showcasing the advantages of a coordinate-free perspective, which simplifies both the deduction and the solution of the final governing equations.