A coordinate-free guide to the mechanics of thin shells.

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. • Equilibrium equations for thin shells are derived via a coordinate-free approach. • A direct approach has been employed. • Mechanical insight is facilitated by the use of the direct notation. • The effectiveness of the approach is showcased by solving a free-vibration problem. [ABSTRACT FROM AUTHOR]