Optimal exponentials of thickness in Korn's inequalities for parabolic and elliptic shells

TL;DR

This paper derives optimal Korn's inequalities for parabolic and elliptic shells, determining how the best constants scale with shell thickness and removing previous geometric restrictions.

Contribution

It establishes Korn's interpolation inequalities for shells without assuming a single principal coordinate, including closed elliptic shells, and identifies optimal scaling laws.

Findings

Korn's inequality constant scales as h^{3/2} for parabolic shells.

Korn's inequality constant scales as h for elliptic shells.

The results include shells with complex geometries, such as closed elliptic shells.

Abstract

We establish Korn's interpolation inequalities and the rigidity results of the strain tensor of the middle surface for the parabolic and elliptic shells and show that the best constant in Korn's inequalities scales like $h^{3/2}$ for the parabolic shell and $h$ for the elliptic shell, removing the main assumption that the middle surface of the shell is given by one single principal coordinate in the literature and, in particular, including the closed elliptic shell.