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A justification of the Marguerre-von Kármán equations. (English) Zbl 0633.73069
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 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 Marguerre-Von Kármán equations by the use of a Airy stress function under global boundary closure conditions for the given loads. The equations are not too much complicated and the solution is shown to be unique. Existence is not treated but good references are given.
To summarize, the author has shown that the classical assumptions are confirmed i.e. K-L type for the displacements, linear, quadratic, or cubic variations of the stress components, depending on the case, with respect to the normal coordinate, as in case of plates. In conclusion, this article devoted to a difficult mathematical and attractive problem is particularly clear. The proofs are easily followed with precise assumptions, functional spaces, and details. Having chosen restrictive conditions for sake of rigour, the author provides a mathematical justification for the nonlinear case of the classical thin shell assumptions. May we think that there remains some doubt in more general cases?
Reviewer: R.Valid

##### MSC:
 74K15 Membranes 74S30 Other numerical methods in solid mechanics (MSC2010) 74G30 Uniqueness of solutions of equilibrium problems in solid mechanics 74H25 Uniqueness of solutions of dynamical problems in solid mechanics 46N99 Miscellaneous applications of functional analysis
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