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Justification of the boundary conditions of a clamped plate by an asymptotic analysis. (English) Zbl 0699.73011
It was shown in previous articles that a thin linear elastic plate could be joined to a three-dimensional supported elastic structure under the conditions that the plate were partly inserted into a slit of the bulk [the authors and R. Nzengwa, J. Math. Pures Appl., IX. Ser. 68, No.3, 261-295 (1989; Zbl 0661.73013), and C. R. Acad. Sci., Paris, Ser. I 305, 55-58 (1987; Zbl 0632.73015)]. In that case a $$H^ 1$$-convergence was shown through the assumption that the Lamé constants of the plate tend to zero as $$\epsilon^{-3}$$, $$2\epsilon$$ being the plate thickness, those of the supporting structure being independent of $$\epsilon$$.
The objective here is to show that a similar limit analysis may also be obtained for such a plate provided the Lamé constants of the 3-D supporting structure also converge to $$+\infty$$ as $$\epsilon^{-2-s}$$ for some $$s>0$$, when $$\epsilon\to 0$$, that is if their increase be somewhat slower than those of the plate.
The interest of the paper lies in particular in the method which consists in that a change of scaling makes possible to replace the integration domain by a fixed one in order to bring the evolution onto the integrands as usual, but in that case, due to the differences between the elastic constants, the junction is apparently counted twice. However the overlapping is avoided by the introduction of the characteristic function in the 3-D domain itself. The scaling of the different parts of the full structure being independent of each other, is of course applied also to the loads.
The variational problem is then written as a pluri-dimensional one, where the unknowns are the displacement $$\tilde u$$ of the supporting structure $${\tilde \Omega}$$, and the displacement u of the plate $$\Omega$$, such that $$\tilde u\times u\in H^ 1({\tilde \Omega})\times H^ 1(\Omega)$$. An important theorem is shown through 7 technical lemmae, according which when $$\epsilon$$ $$\to 0$$, $$\tilde u(\epsilon)\to 0$$, and $$u(\epsilon)\to 0$$ for the clamped inserted part of the plate, and tends to a Kirchhoff-Love (K-L) field for the rest of it.
All the proof is carefully exposed and the article ends with conclusions and comments which make all the text clear and interesting. Namely it is shown that the limit displacement field of the plate middle surface solves the classical 2-D plate equations, as a classical K-L field without any a priori assumption, the supporting structure becomes rigid in the limit, and above all, in the special case where the whole structure is made of the same material $$(s=1)$$, the tangential load components being null, it is found that the inserted part of the plate becomes arbitrary, for instance null. The problem becomes no more coupled contrary to the preceding studies. It may be thought that theoretical equilibrium difficulties be avoided by appealing a second gradient theory for the bulk.
Reviewer: R.Valid

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74K20 Plates 74B05 Classical linear elasticity 74E30 Composite and mixture properties