Korn inequalities for shells with zero Gaussian curvature

TL;DR

This paper establishes sharp Korn inequalities for shells with zero Gaussian curvature, such as cylindrical shells, revealing how the inequality constant scales with shell thickness and impacting buckling analysis.

Contribution

It introduces a novel 'first-and-a-half Korn inequality' for cross-sections, leading to precise asymptotic bounds for shells with zero Gaussian curvature.

Findings

Korn inequality constant scales as thickness^{3/2}

Methodology involves three 2D inequalities assembling into a 3D result

Implications for buckling load sensitivity to shape imperfections

Abstract

We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a "first-and-a-half Korn inequality" - a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.