Sharp Korn Inequalities for Washers
Davit Harutyunyan · arXiv (Cornell University) · 2015
In this paper we prove asymptotically sharp first-and-a-half and second Korn inequalities for washers with a 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 as the slope goes to zero, or as a very short hollow cylinder. While the optimal Korn constant in the second Korn inequality for a conical shell with thickness $h$ and with a positive slope is expected to scale like $h^{1.5},$ the optimal Korn constant in the second 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 $h.$ Letting then the inner radius of the washer go to zero we obtain sharp Korn inequalities for discs as a corollary. 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 radial pressure.