Investigation of solvability of variational problems in the nonlinear theory of thin shells.
type: Item
instance of: scholarly article
title: Investigation of solvability of variational problems in the nonlinear theory of thin shells.
zbMATH Open document ID: 1064.74128
author: S. N. Timergaliev
published in: Russian Mathematics
publication date: 2002-06-09T00:00:00Z
review text: This is an attempt to establish the solvability of a general boundary value problem of equilibrium of a geometrically and physically nonlinear non-shallow shell (Koiter's version) of nonzero Gaussian curvature with clamped edge. The problem is considered as the problem of minimum of total energy of the shell in some Hilbert spaces of three-component vector functions taking the components of displacement vector as unknowns. The author applies the techniques developed by \textit{I. I. Vorovich} [Functional analysis in mechanics. Springer Monographs in Mathematics. New York, NY: Springer. xi (2003; Zbl 1014.46002)] for nonlinear shallow shells. Particular problems for this type of shells were studied by \textit{I. I. Vorovich} and \textit{Sh. M. Shlafman} [J. Appl. Math. Mech. 38, 894--897 (1974); translation from Prikl. Mat. Mekh. 38, 944--946 (1974; Zbl 0311.73051)] and by \textit{I. I. Vorovich, L. P. Lebedev} and \textit{Sh. M. Shlafman} [J. Appl. Math. Mech. 38, 310--319 (1974); translation from Prikl. Math. Mekh. 38, 339--348 (1974; Zbl 0304.73068)]. Unfortunately, the attempt is not successful. The main difficulty of the proof is to show that the total energy functional is growing. Repeating the ideas of paper by \textit{I. I. Vorovich} and \textit{L. P. Lebedev} [J. Appl. Math. Mech. 36, 652--665 (1972); translation from Prikl. Mat. Mekh. 36, 691--704 (1972; Zbl 0282.73056)], the author considers the energy functional on the ellipsoid. The ellipsoid is obtained by the deformation of the components of displacement \((u_1^*,u_2^*,u_3^*)\) of unit sphere in energy space (the author uses other functions but his transformations are equivalent to what is described) by formulas \(u_1=R u_1^*, u_2=R u_2^*, u_3=R^2 u_3^*\). Then the author states that the structure of the total energy functional with respect to \(R\) is a polynomial of 4th order (page 72). This is correct for the shallow shell theory but it is incorrect for the theory considered by the author. In Shlafm…
Mathematics Subject Classification ID: 74K25
Mathematics Subject Classification ID: 74G25
Mathematics Subject Classification ID: 74G65
zbMATH DE Number: 1752707
zbMATH Keywords: Koiter shell
reviewed by: Leonid Lebedev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1752707