zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[1] Aganovič, C; Tutek, Z, A justification of the one-dimensional model of an elastic beam, Math. methods appl. sci., 8, 1-14, (1986) · Zbl 0603.73056
[2] Aufranc, M, Numerical study of junctions between elastic structures of different dimensions, ()
[3] {\scM. Aufranc}, Modeling of junctions between three-dimensional and two-dimensional nonlinearly elastic structures, to appear. · Zbl 0754.73034
[4] Bermudez, A; Viaño, J.M, Une justification des équations de la thermoélasticité des poutres à section variable par des méthodes asymptotiques, RAIRO modél. math. anal. numér., 18, 347-376, (1984) · Zbl 0572.73053
[5] Blanchard, D; Ciarlet, P.G, A remark on the von Kármán equations, Comput. methods appl. mech. engrg., 37, 79-92, (1983) · Zbl 0486.73051
[6] {\scF. Bourquin}, Modal synthesis by substructuring methods for junctions between three-dimensional and two-dimensional elastic structures, to appear.
[7] Brezis, H, Analyse fonctionnelle, théorie et applications, (1983), Masson Paris · Zbl 0511.46001
[8] Caillerie, D, The effect of a thin inclusion of high rigidity in an elastic body, Math. methods appl. sci., 2, 251-270, (1980) · Zbl 0446.73014
[9] Ciarlet, P.G, A justification of the von Kármán equations, Arch. rational mech. anal., 73, 349-389, (1980) · Zbl 0443.73034
[10] Ciarlet, P.G, Recent progresses in the two-dimensional approximation of the three-dimensional plate models in nonlinear elasticity, (), 3-19
[11] Ciarlet, P.G, Modeling and numerical analysis of junctions between elastic structures, () · Zbl 0706.73046
[12] Ciarlet, P.G, Mathematical elaslicity: vol. I three-dimensional elasticity, (1988), North-Holland Amsterdam
[13] Ciarlet, P.G, Mathematical elasticity, vol. II: lower-dimensional theories of plates and rods, (1990), North-Holland Amsterdam
[14] {\scP. G. Ciarlet}, Junctions between plates and rods, to appear.
[15] Ciarlet, P.G; Destuynder, P, A justification of the two-dimensional plate model, J. Méc., 18, 315-344, (1979) · Zbl 0415.73072
[16] Ciarlet, P.G; Destuynder, P, A justification of a nonlinear model in plate theory, Comput. methods appl. mech. engrg., 17/18, 227-258, (1979) · Zbl 0405.73050
[17] Ciarlet, P.G; Kesavan, S, Two-dimensional approximation of three-dimensional eigenvalue problems in plate theory, Comput. methods appl. mech. engrg., 26, 149-172, (1981) · Zbl 0489.73057
[18] {\scP. G. Ciarlet and H. Le Dret}, Justification of the boundary conditions of a clamped plate by an asymptotic analysis, Asymptotic Analysis, in press. · Zbl 0699.73011
[19] Ciarlet, P.G; Le Dret, H; Nzengwa, R, Modélisation de la jonction entre un corps élastique tridimensionnel et une plaque, C. R. acad. sci. Paris Sér. I, 305, 55-58, (1987) · Zbl 0632.73015
[20] {\scP. G. Ciarlet, H. Le Dret, and R. Nzengwa}, Junctions between three-dimensional and two-dimensional linearly elastic structures, J. Math. Pures Appl., in press. · Zbl 0661.73013
[21] Ciarlet, P.G; Paumier, J.C, A justification of the Marguerre-von Kármán equations, Comput. mech., 1, 177-202, (1986) · Zbl 0633.73069
[22] Cimetiere, A; Geymonat, G; Le Dret, H; Raoult, A; Tutek, Z, Asymptotic theory and analysis for displacements and stress distribution in nonlinear elastic straight slender rods, J. elasticity, 19, 111-161, (1988) · Zbl 0653.73010
[23] Courant, R; Hilbert, D, ()
[24] Dautray, R; Lions, J.L, ()
[25] Destuynder, P, Comparaison entre LES modèles tridimensionnels et bidimensionnels de plaques en élasticité, RAIRO anal. numér., 15, 331-369, (1981) · Zbl 0479.73042
[26] Destuynder, P, A classification of thin shell theories, Acta. appl. math., 4, 15-63, (1985) · Zbl 0531.73044
[27] Destuynder, P, Une théorie asymptotique des plaques minces en elasticité linéaire, (1986), Masson Paris · Zbl 0627.73064
[28] Friedrichs, K.O; Dressler, R.F, A boundary-layer theory for elastic plates, Comm. pure appl. math., 14, 1-33, (1961) · Zbl 0096.40001
[29] {\scP. Germain}, “Mécanique des Milieux Continus,” Tome I, Masson, Paris.
[30] Germain, P, ()
[31] Glowinski, R; Li, C.H; Lions, J.L, A numerical approach to the exact boundary controllability of the wave equations. I. Dirichlet controls: description of the numerical methods, ()
[32] Goldenveizer, A.L, Derivation of an approximate theory of bending of a plate by the method of asymptotic integration of the equations of the theory of elasticity, Prikl. mat. mech., 26, 668-686, (1964)
[33] John, F, Refined interior equations for thin elastic shells, Comm. pure appl. math., 24, 583-615, (1971) · Zbl 0299.73037
[34] Kesavan, S, Homogenization of eigenvalue problems, part 1, Appl. math. optim., 5, 153-167, (1979) · Zbl 0415.35061
[35] Lagnese, J.E; Lions, J.L, Modelling, analysis and control of thin plates, (1988), Masson Paris · Zbl 0662.73039
[36] Le Dret, H, Modélisation d’une plaque pliée, C. R. acad. sci. Paris Sér. I, 305, 571-573, (1987) · Zbl 0634.73047
[37] {\scH. Le Dret}, Modeling of a folded plate, to appear. · Zbl 0741.73026
[38] {\scH. Le Dret}, Folded plates revisited, to appear.
[39] {\scH. Le Dret}, Modeling of the junction between two rods, to appear.
[40] Lions, J.-L, Perturbations singulières dans LES problèmes aux limites et en contrôle optimal, () · Zbl 0268.49001
[41] Lions, J.-L, Remarques sur LES problèmes d’homogénéisation dans LES milieux à structure périodique et sur quelques problèmes raides, ()
[42] Lions, J.-L, Contrôlabilité exacte et perturbations singulières. II. la méthode de dualité, (), 223-237
[43] Lions, J.-L, Exact controllability, stabilization and perturbations for distributed systems, SIAM rev., 30, 1-68, (1988) · Zbl 0644.49028
[44] Miara, B, Numerical assessment of the validity of two-dimensional plate models, ()
[45] Panasenko, G.P, Asymptotic behavior of solutions and eigenvalues of elliptic equations with strongly varying coefficients, Dokl. akad. nauk. SSSR, 252, 1320-1325, (1980)
[46] Poincaré, H, Sur LES équations aux dérivées partielles de la physique mathématique, Amer. J. math., 12, 211-294, (1890) · JFM 22.0977.03
[47] Raoult, A, Construction d’un modèle d’évolution de plaques avec terme d’inertie de rotation, Ann. mat. pura appl., 139, 361-400, (1985), (4) · Zbl 0596.73033
[48] {\scA. Raoult}, Asymptotic theory of nonlinearly elastic dynamic plates, to appear.
[49] Rigolot, A, Sur une théorie asymptotique des poutres, J. Méc., 11, 673-703, (1972) · Zbl 0257.73013
[50] Rigolot, A, Sur une théorie asymptotique des poutres, () · Zbl 0257.73013
[51] Rigolot, A, Approximation asymptotique des vibrations de flexion des poutres droites élastiques, J. Mécan., 16, 493-529, (1977) · Zbl 0403.73067
[52] Taylor, A.E, Introduction to functional analysis, (1958), Wiley New York · Zbl 0081.10202
[53] Trabucho, L; Viaño, J.M, Derivation of generalized models for linear elastic beams by asymptotic expansion methods, () · Zbl 0646.73024
[54] {\scL. Trabucho and J. M. Viaño}, Existence and characterization of higher order terms in an asymptotic expansion method for linearized elastic beams, to appear.
[55] {\scL. Trabucho and J. M. Viaño}, A derivation of generalized Saint-Venant’s torsion theory from three-dimensional elasticity by asymptotic expansion methods, to appear.
[56] Weinberger, H, Variational methods for eigenvalue approximations, (1974), SIAM Philadelphia
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.