A hint on the localization of the buckling deformation at vanishing curvature points on thin elliptic shells

TL;DR

This paper investigates how the presence of points with zero principal curvature on elliptic shells affects buckling behavior, showing that buckling localizes at these points and that the Korn constant scaling changes accordingly.

Contribution

It provides the first analysis of Korn's constant scaling for elliptic shells with finite zero-curvature points and links this to buckling deformation localization.

Findings

Korn's constant scales as h^{-3/2} with zero-curvature points.

Buckling deformation localizes at vanishing curvature points.

First result for non-developable elliptic shells with such points.

Abstract

The general theory of slender structure buckling by Grabovsky and Truskinovsky [\textit{Cont. Mech. Thermodyn.,} 19(3-4):211-243, 2007], (later extended in [\textit{Journal of Nonlinear Science.,} Vol. 26, Iss. 1, pp. 83--119, 2016] by Grabovsky and the author), predicts that the critical buckling load of a thin shell under dead loading is closely related to the Korn's constant (in Korn's first inequality) of the shell under the Dirichlet boundary conditions resulting from the loading program. It is known that under zero Dirichlet boundary conditions on the thin part of the boundary of positive, negative, and zero (one principal curvature vanishing, and one apart from zero) Gaussian curvature shells, the optimal Korn constant in Korn's first inequality scales like the thickness to the power of $-1, -4/3,$ and $-3/2$ respectively. In this work we analyse the scaling of the optimal constant in Korn's first inequality for elliptic shells that contain a finite number of points where both principal curvatures vanish. We prove that the presence of at least one such point on the shell leads to the scaling drop from the thickness to the power of $-1$ to the thickness to the power of $-3/2.$ To our best knowledge, this is the first result in the direction for constant-sign curvature shells, that do not contain a developable region. In addition, under the assumption that a suitable trivial branch exists, we prove that in fact the buckling deformation of such shells under dead loading, should be localized at the vanishing curvature points, as the shell thickness h goes to zero.