Applied Mathematics Seminar | Davit Harutyunyan, From buckling to rigidity of shells: Recent mathematical progress

MC 6460

Speaker

Dr. Davit Harutyunyan
​Department of Mathematics | Swiss Federal Institute of Technology Lausanne

Title

From buckling to rigidity of shells: Recent mathematical progress

Abstract

It is known that the rigidity of a shell (for instance under compression) is closely related to the optimal Korn's constant in the nonlinear Korn's first inequality (geometric rigidity estimate) for H^1 fields under the appropriate conditions (with no or with Dirichlet type boundary conditions arising from the nature of the compression). In their celebrated work, Frisecke, James and Mueller (2002, 2006) derived an asymptotically sharp nonlinear geometric rigidity estimate for plates, which gave rise to a derivation of a hierarchy of nonlinear plate theories for different scaling regimes of the elastic energy depending on the thickness h of the plate (the optimal constant scales like h^2). Frisecke-James-Mueller type theories have been derived by Gamma-convergence and rely on LpLp compactness arguments and of course the underlying nonlinear Korn's inequality. While plate deformations have been understood almost completely, the rigidity, in particular the buckling of shells is less well understood. This is first of all due to the luck of sharp rigidity estimates for shells. In our recent work we derive linear sharp geometric estimates for shells by classifying them according to the Gaussian curvature. It turns out, that for zero Gaussian curvature (when one principal curvature is zero, the other one never vanishes) the amount of rigidity is h^{3/2}, for negative curvature it is h^{4/3} and for positive curvature it is h. These results represent a breakthrough in both the shell buckling and nonlinear shell theories. All three estimates have completely new optimal constant scaling for any sharp geometric rigidity estimates to have appeared, and are classical. This is partially joint work with Yury Grabovsky (Temple University)