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We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a “first-and-a-half Korn inequality”—a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.
@article{AIHPC_2018__35_1_267_0,
author = {Grabovsky, Yury and Harutyunyan, Davit},
title = {Korn inequalities for shells with zero {Gaussian} curvature},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {267--282},
publisher = {Elsevier},
volume = {35},
number = {1},
year = {2018},
doi = {10.1016/j.anihpc.2017.04.004},
mrnumber = {3739933},
zbl = {1395.74056},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.04.004/}
}
TY - JOUR AU - Grabovsky, Yury AU - Harutyunyan, Davit TI - Korn inequalities for shells with zero Gaussian curvature JO - Annales de l'I.H.P. Analyse non linéaire PY - 2018 SP - 267 EP - 282 VL - 35 IS - 1 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.04.004/ DO - 10.1016/j.anihpc.2017.04.004 LA - en ID - AIHPC_2018__35_1_267_0 ER -
%0 Journal Article %A Grabovsky, Yury %A Harutyunyan, Davit %T Korn inequalities for shells with zero Gaussian curvature %J Annales de l'I.H.P. Analyse non linéaire %D 2018 %P 267-282 %V 35 %N 1 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.04.004/ %R 10.1016/j.anihpc.2017.04.004 %G en %F AIHPC_2018__35_1_267_0
Grabovsky, Yury; Harutyunyan, Davit. Korn inequalities for shells with zero Gaussian curvature. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 1, pp. 267-282. doi: 10.1016/j.anihpc.2017.04.004
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