Sharp Weighted Korn and Korn-Like Inequalities and an Application to Washers

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., (Grabovsky and Harutyunyan in arXiv:1602.03601 , 2016), 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 Kirchhoff Ansatz. This results can be applied to calculate the critical buckling load of a washer under in plane loads, e.g., (Antman and Stepanov in J. Elast. 124(2):243–278, 2016).