Sharp weighted Korn and Korn-like inequalities and an application to washers

TL;DR

This paper establishes sharp weighted Korn inequalities with singular weights, applies them to washers, and determines how the Korn constants scale with thickness, aiding in buckling load calculations.

Contribution

It introduces asymptotically sharp weighted Korn inequalities with singular weights and applies these to analyze washers, revealing their Korn constants' dependence on thickness and geometry.

Findings

Korn constant for washers scales like h^2, depending only on outer radius.

Korn constant for conical shells scales like h^{1.5}.

Korn constants are realized by a Kirchhoff Ansatz.

Abstract

In this paper we prove asymptotically sharp weighted "first-and-a-half" $2D$ Korn and Korn-like inequalities with a singular weight occurring from Cartesian to cylindrical change of variables. We prove some Hardy and the so-called "harmonic function gradient separation" inequalities with the same singular weight. Then we apply the obtained $2D$ inequalities to prove similar inequalities for washers with thickness $h$ subject to vanishing Dirichlet boundary conditions on the inner and outer thin faces of the washer. A washer can be regarded in two ways: As the limit case of a conical shell when the slope goes to zero, or as a very short hollow cylinder. While the optimal Korn constant in the first Korn inequality for a conical shell with thickness $h$ and with a positive slope scales like $h^{1.5}$ e.g. [10], the optimal Korn constant in the first Korn inequality for a washer scales like $h^2$ and depends only on the outer radius of the washer as we show in the present work. The Korn constant in the first and a half inequality scales like $h$ and depends only on $h.$ The optimal Korn constant is realized by a Kirchoff Ansatz. This results can be applied to calculate the critical buckling load of a washer under in plane loads, e.g. [1].