A justification of the Marguerre-von Kármán equations
type: Item
instance of: scholarly article
title: A justification of the Marguerre-von Kármán equations
zbMATH Open document ID: 0633.73069
DOI: 10.1007/BF00272623
author: Philippe G. Ciarlet
author: J.-C. Paumier
published in: Computational Mechanics
publication date: 1986-01-01T00:00:00Z
review text: In two particular cases of the mathematical problem of nonlinear thin elastic shells, the authors show, without any a priori assumptions, that the solution is of a Kirchhoff-Love (K-L) type for the displacements, while the stresses are affine in the normal coordinate. These particular cases, which make the demonstrations possible, are those of shallow shells either with a clamped boundary, or with admissible boundary displacements parallel to the given plane of the middle surface boundary, the so-called Marguerre-Von Kármán problem. The method follows globally that one already utilized by the first author and \textit{P. Destuynder} for linear and nonlinear plates [e.g. Comput. Methods Appl. Mech. Engin. 17-18, 227-258 (1979; Zbl 0405.73050) and J. Mec. Paris 18, 315-344 (1979; Zbl 0415.73072)], i.e. an asymptotic method applied to the mixed variational three-dimensional principle of Hellinger-Reissner, the small parameter being the thickness \(\epsilon\) of the shell. The shallowness assumption which insures the orientation preserving condition of the maps, is such that the normal coordinate and normal deflection be of order of \(\epsilon\). The demonstration is given by several theorems, for the two cases under consideration. The first step defines consistent orders of magnitude for the unknowns and data, and provides the leading terms of the variational equations. The next step shows the equivalence of the problem with a two- dimensional (2-D) one for the middle surface. The proof is carried out in the isotropic case by considering successively the equations relative to each unknown component. The results are that the displacement is of a K-L type and the tangential stresses affine in the normal coordinate, the coefficients being the membrane and bending surface stresses, which are thus made conspicuous. Then the other stress components are calculated in terms of the 2-D solution. The last step shows the equivalence to the second boundary value problem of Marguerr…
Mathematics Subject Classification ID: 74K15
Mathematics Subject Classification ID: 74S30
Mathematics Subject Classification ID: 74G30
Mathematics Subject Classification ID: 74H25
Mathematics Subject Classification ID: 46N99
zbMATH DE Number: 4031157
zbMATH Keywords: method of asymptotic expansions
zbMATH Keywords: equilibrium of nonlinearly elastic shells
zbMATH Keywords: Kirchhoff-Love-type solution for displacements
zbMATH Keywords: Marguerre-Von Kármán problem
zbMATH Keywords: asymptotic method
zbMATH Keywords: mixed variational three- dimensional principle of Hellinger-Reissner
zbMATH Keywords: shallowness assumption
zbMATH Keywords: consistent orders of magnitude
zbMATH Keywords: two-dimensional
zbMATH Keywords: middle surface
zbMATH Keywords: isotropic case
zbMATH Keywords: tangential stresses affine in the normal coordinate
zbMATH Keywords: Airy stress function
zbMATH Keywords: global boundary closure conditions
reviewed by: Roger Valid
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4031157