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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

74K20 Plates
74S05 Finite element methods applied to problems in solid mechanics
74B10 Linear elasticity with initial stresses
Full Text: DOI EuDML
[1] I. RAJU, J. CREWS, Interlaminar stress singularities at the free edge in composites laminates. Comp. Struc. Vol. 14 n^\circ 1, pp. 21-28 (1981).
[2] R. ZWIERS, T. TING, R. SPILKER, On the logarithmic singularity of free-edges stress in laminated composite under uniform extension. J. Appl. Mechan., Vol. 49, pp. 561-569 (1982). Zbl0522.73056 · Zbl 0522.73056 · doi:10.1115/1.3162526
[3] A. S. D. WANG, F. CROSSMAN, Some new results on edge effects in symmetric composite laminates. J. Comp. Mat., Vol. 11, pp. 92-106 (1977).
[4] J. L. DAVET, Ph. DESTUYNDER, Singularités logarithmiques dans les effets de bord d’une plaque en matériaux composites. J.M.T.A., Vol. 4, pp. 357-373 (1985). Zbl0564.73066 MR800024 · Zbl 0564.73066
[5] R. D. MINDLIN, Influence of rotary inertia and shear on flexural motion of isotropic elastic plates. J. Appl. Mech., Vol. 18, pp. 31-38 (1951). Zbl0044.40101 · Zbl 0044.40101
[6] Ph. DESTUYNDER, Thèse université P. M. Curie, Paris (1980).
[7] R. TEMAM, Navier Stokes equations. North Holland Studies in Mathematics and its application. Vol. 2 (1979). Zbl0426.35003 MR603444 · Zbl 0426.35003
[8] F. BREZZI, K. J. BATHE, Article à paraître et communication personnelle de F. Brezzi au colloque d’Analyse Numérique, Puy-Saint-Vincent (1985).
[9] [9] Ph. DESTUYNDER, J. C. NEDELEC, Approximation numérique du cisaillement transverse dans les plaques minces en flexion. Numér. Math., pp. 1-22, n^\circ 876 (1985). Zbl0613.73066 MR826470 · Zbl 0613.73066 · doi:10.1007/BF01389476 · eudml:133070
[10] G. DUVAUT, J. L. LIONS, Les inéquations en mécanique et en physique. Vol. 21. Travaux et recherches mathématiques. Dunod 1972. Zbl0298.73001 MR464857 · Zbl 0298.73001
[11] K. YOSIDA, Functional Analysis. 4th edition. Springer Verlag, Berlin (1975).
[12] P. G. CIARLET, The finite element method for elliptic problems. Stud. Math. Appl. Amsterdam North Holland (1978). Zbl0383.65058 MR520174 · Zbl 0383.65058
[13] D. BERTSEKAS, An penalty and multiplier method for constrained minimization. E.E.S. Departement working paper. Standford university. August (1973). Zbl0324.49029 MR407692 · Zbl 0324.49029 · doi:10.1137/0314017
[14] V. GIRAULT, P. A. RAVIART, Finite Element Methods for Navier-Stokes Equations. Springer-Verlag, 1986. Zbl0585.65077 MR851383 · Zbl 0585.65077
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