**Contact Info**

Department of Applied Mathematics

University of Waterloo

Waterloo, Ontario

Canada N2L 3G1

Phone: 519-888-4567, ext. 32700

Fax: 519-746-4319

PDF files require Adobe Acrobat Reader

Wednesday, January 25, 2017 — 2:30 PM EST

MC 6460

Dr. Davit Harutyunyan

Department of Mathematics | Swiss Federal Institute of Technology Lausanne

From buckling to rigidity of shells: Recent mathematical progress

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)

**Contact Info**

Department of Applied Mathematics

University of Waterloo

Waterloo, Ontario

Canada N2L 3G1

Phone: 519-888-4567, ext. 32700

Fax: 519-746-4319

PDF files require Adobe Acrobat Reader

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.