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Modeling and justification of eigenvalue problems for junctions between elastic structures. (English) Zbl 0699.73010
This article is concerned with the modeling by an asymptotic procedure of eigenvalue problems for linearly elastic structures comprising parts of different dimensions. Namely, the authors consider the same model family of three-dimensional structures as in P. G. Ciarlet, H. Le Dret and R. Nzengwa [(*) J. Math. Pures Appl. 68, 261-295 (1989; Zbl 0661.73013)], a cube into which is inserted with a constant depth \(\beta >0\), a thin plate of thickness \(2\epsilon\). The Lamé constants of the cube and its mass density are assumed to be constant, while the Lamé constants of the plate behave as \(\epsilon^{-3}\) and its mass density as \(\epsilon^{-1}.\)
The authors combine the techniques introduced in (*) for the modeling of such pluri-dimensional structures in the static case with those of P. G. Ciarlet and S. Kesavan [Comput. Methods Appl. Mech. Eng. 26, 145-172 (1981; Zbl 0489.73057)] for the asymptotics of the eigenvalue problems in a single thin plate as the thickness goes to 0. Letting \(\epsilon\) tend to 0, they prove that the kth eigenvalue of the three- dimensional problem converge toward the kth eigenvalue of a limit 2d-3d eigenvalue problem, which is precisely the eigenvalue problem corresponding to the limit static problem of (*). This problem couples three-dimensional linear elasticity in the cube minus a two-dimensional slit of depth \(\beta >0\), with a two-dimensional plate equation through a set of junction conditions. Likewise, the eigenfunctions are shown to converge strongly in the \(H^ 1\) sense toward displacements that are obtained from the eigenfunctions of the limit problems by constructing in the plate the Kirchhoff-Love displacement corresponding to the two- dimensional eigenfunction on the middle plane of the plate, and in the cube by simply taking the limit three-dimensional eigenfunction.
Reviewer: H.Le Dret

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
49R50 Variational methods for eigenvalues of operators (MSC2000)
74B05 Classical linear elasticity
49S05 Variational principles of physics
74E30 Composite and mixture properties
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