Asymptotic analysis of torsional and stretching modes of thin rods
Authors:
H. Irago, N. Kerdid and J. M. Viaño
Journal:
Quart. Appl. Math. 58 (2000), 495-510
MSC:
Primary 74K10; Secondary 74B05, 74G10, 74H45
DOI:
https://doi.org/10.1090/qam/1770651
MathSciNet review:
MR1770651
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Abstract: In this article, we show that a class of high frequencies of the three-dimensional linearized elasticity system in a thin rod and their associated eigenfunctions converge in a precise sense, as the area of the cross section of the rod goes to zero. The limit model is a coupled one-dimensional problem giving the classical equations for torsion and stretching modes in rods.
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I. Aganovic and Z. Tutek, A justification of the one-dimensional linear model of elastic beams, Math. Meth. Appl. Sci. 8, 1–14 (1986)
J. A. Álvarez-Dios and J. M. Viaño, Mathematical justification of a one-dimensional model for general elastic shallow arches, Math. Meth. Appl. Sci. 21, 281–325 (1997)
L. J. Álvarez-Vázquez and J. M. Viaño, Asymptotic justification of an evolution linear thermoelastic model for rods, Comput. Methods Appl. Mech. Engrg. 115, 93–109 (1994)
A. Bermúdez and J. M. Viaño, Une justification des équations de la thermo-élasticité de poutres à section variable par des méthodes asymptotiques, RAIRO Anal. Numér. 18, 347–376 (1984)
F. Bourquin and P. G. Ciarlet, Modelling and justification of eigenvalue problems for junctions between elastic structures, J. Funct. Anal. 87, 392–427 (1989)
C. Castro and E. Zuazua, Une remarque sur l’analyse asymptotique spectrale en homogénéisation, C. R. Acad. Sci. Paris Sér. I Math. 322, 1043–1047 (1996)
P. G. Ciarlet and S. Kesavan, Two-dimensional approximation of three-dimensional eigenvalue problems in plate theory, Comput. Methods Appl. Mech. Engrg. 26, 145–172 (1981)
J. L. Davet, Correction du second ordre pour le calcul des fréquences propres d’une plaque en flexion, C. R. Acad. Sci. Paris Sér. II Math. 303, 521–524 (1986)
V. Girault and P. A. Raviart, Finite Element Approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, Vol. 749, Springer, Berlin, 1981
I. Gruais, Modélisation de la jonction entre une plaque et une poutre en élasticité linéarisée, RAIRO Modél. Math. Anal. Numér. 27, 77–105 (1993)
H. Irago, Comparación numérica de vibraciones 3D-1D en vigas elásticas, Tesina de Licenciatura de la Universidad de Santiago de Compostela, España, 1995
H. Irago and J. M. Viaño, Second-order asymptotic approximation of flexural vibrations in elastic rods, Math. Models Methods Appl. Sci. 8, 1343–1362 (1998)
N. Kerdid, Comportement asymptotique quand l’épaisseur tend vers zéro du problème de valeurs propres pour une poutre mince encastrée en élasticité linéaire, C. R. Acad. Sci. Paris Sér. I Math. 316, 755–758 (1993)
N. Kerdid, Modélisation des vibrations d’une multi-structure formée de deux poutres, C. R. Acad. Sci. Paris Série I Math. 321, 1641–1646 (1995)
N. Kerdid, Étude de problèmes de jonctions de poutres en élasticité linéaire, Thèse de Doctorat Université Pierre et Marie Curie, Paris, 1995
N. Kerdid, Modeling the vibrations of a multi-rod structure, RAIRO Módel. Math. Anal. Numér. 31, 1–34 (1997)
H. Le Dret, Modelling of the junction between two rods, J. Math. Pures Appl. 68, 365–397 (1989)
H. Le Dret, Vibrations of a folded plate, RAIRO Módel. Math. Anal. Numér. 24, 501–521 (1990)
H. Le Dret, Problèmes variationnels dans les multi-domaines. Modélisation des jonctions et applications, Recherches en Mathématiques Appliquées, Vol. 19, Masson, Paris, 1991
V. Lods, Modélisation et justification d’un problème aux valeurs propres pour une plaque insérée dans un support tridimensional, C. R. Acad. Sci. Paris Sér. I Math. 320, 391–396 (1995)
P. A. Raviart and J. M. Thomas, Introduction à l’Analyse Numérique des Équations aux Dérivées Partielles, Masson, Paris, 1983
J. M. Rodríguez-Seijo and J. M. Viaño, Asymptotic derivation of a general linear model for thin-walled elastic rods, Comput. Methods Appl. Mech. Engrg. 147, 287–321 (1997)
M. Roseau, Vibrations des systèmes méchaniques, Masson, Paris, 1984
J. Sanchez-Hubert and E. Sanchez-Palencia, Vibration and coupling of continuous systems. Asymptotic methods, Springer-Verlag, Berlin, 1989
L. Trabucho and J. M. Viaño, Dérivation de modèles généralisés de poutres en élasticité par méthode asymptotique, C. R. Acad. Sci. Paris Sér. I Math. 304, 303–306 (1987)
L. Trabucho and J. M. Viaño, Existence and characterization of higher-order terms in an asymptotic expansion method for linearized elastic beams, Asymptotic Analysis 2, 223–255 (1989)
L. Trabucho and J. M. Viaño, A new approach of Timoshenko’s beam theory by the asymptotic expansion method, RAIRO Módel. Math. Anal. Numér. 24, 651–680 (1990)
L. Trabucho and J. M. Viaño, Mathematical modelling of rods, In Handbook of Numerical Analysis, Vol. IV, Ciarlet, P. G. and Lions, J. L., Editors, North-Holland, Amsterdam, 1996
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© Copyright 2000
American Mathematical Society