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Une modification du modèle de Mindlin pour les placques minces en flexion présentant un bord libre. (A modification of the Mindlin model for thin plates in bending with a free edge). (French) Zbl 0646.73026
In that important study of functional and numerical analysis, the authors propose a Modified Mindlin Model (M.M.M.) for the analysis of thin plates in bending, taking into account that Mindlin’s model is numerically more interesting - finite elements of class \(C^ 0\)- than the Kirchhoff-Love Model (K-L.M) - class \(C^ 1\)- but less attractive to correct edge effects, particularly on multilayered plates because of non zero edge stresses.
The modification consists in introducing, in a weak form by a Lagrange multiplier, the K-L constraint on the transverse shear strain only on the free edge, which gives a mixed formulation in the deflection, rotation, transverse shear stress and multiplier unknowns. It is shown successively that M.M.M. tends to K-L.M. when the thickness tends to zero; then an alternative formulation is proposed where the transverse shear stress is decomposed onto two supplementary spaces by means of two stress functions, using an equivalent surface Helmholtz theorem. A third part gives existence, uniqueness and convergence results for the discretized M.M.M. using these stress functions under the satisfaction of 3 compatibility conditions of L.B.B. type. Finally a fourth part proposes 4 types of finite elements which satisfy these compatibility conditions. The demonstration is incidentally not very easy. The numerical steps are given, some of them with augmented Lagrangean, and indications are given relative to the penalty parameter.
To conclude, this article is difficult but well motivated and detailed. Let us notice that the convergence result of M.M.M. to K-L.M. is not surprising due to the fact that Mindlin’s model is nothing but a surface linearization of the three-dimension one which is known to tend to K.L.M. after first author’s works. Let us add that the decomposition of the transverse shear is owing as much to a change of variables as to the use of stress functions. It is hoped finally that the proposed model will reveal to be efficient in the computation of edge effects, according to authors objectives.
Reviewer: R.Valid

MSC:
74K20 Plates
74S05 Finite element methods applied to problems in solid mechanics
74B10 Linear elasticity with initial stresses
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References:
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