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
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
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.
- Z. Tutek and I. Aganović, A justification of the one-dimensional linear model of elastic beam, Math. Methods Appl. Sci. 8 (1986), no. 4, 502–515. MR 870989, DOI https://doi.org/10.1002/mma.1670080133
- J. A. Álvarez-Dios and J. M. Viaño, Mathematical justification of a one-dimensional model for general elastic shallow arches, Math. Methods Appl. Sci. 21 (1998), no. 4, 281–325. MR 1605351, DOI https://doi.org/10.1002/%28SICI%291099-1476%2819980310%2921%3A4%3C281%3A%3AAID-MMA951%3E3.0.CO%3B2-O
- L. Álvarez Vázquez and J. M. Viaño, Asymptotic justification of an evolution linear thermoelastic model for rods, Comput. Methods Appl. Mech. Engrg. 115 (1994), no. 1-2, 93–109. MR 1278812, DOI https://doi.org/10.1016/0045-7825%2894%2990189-9
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, Modeling and justification of eigenvalue problems for junctions between elastic structures, J. Funct. Anal. 87 (1989), no. 2, 392–427. MR 1026860, DOI https://doi.org/10.1016/0022-1236%2889%2990017-7
- Carlos Castro and Enrique Zuazua, Une remarque sur l’analyse asymptotique spectrale en homogénéisation, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 11, 1043–1047 (French, with English and French summaries). MR 1396637
- P. G. Ciarlet and S. Kesavan, Two-dimensional approximations of three-dimensional eigenvalue problems in plate theory, Comput. Methods Appl. Mech. Engrg. 26 (1981), no. 2, 145–172. MR 626720, DOI https://doi.org/10.1016/0045-7825%2881%2990091-8
- Jean-Louis Davet, Correction du second ordre pour le calcul des fréquences propres d’une plaque en flexion, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 303 (1986), no. 7, 521–524 (French, with English summary). MR 977379
- V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR 548867
- Isabelle Gruais, Modélisation de la jonction entre une plaque et une poutre en élasticité linéarisée, RAIRO Modél. Math. Anal. Numér. 27 (1993), no. 1, 77–105 (French, with English and French summaries). MR 1204630, DOI https://doi.org/10.1051/m2an/1993270100771
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 (1998), no. 8, 1343–1362. MR 1663448, DOI https://doi.org/10.1142/S0218202598000639
- Nabil 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 (1993), no. 7, 755–758 (French, with English and French summaries). MR 1214429
- Nabil Kerdid, Modélisation des vibrations d’une multi-structure formée de deux poutres, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 12, 1641–1646 (French, with English and French summaries). MR 1367822
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 Modél. Math. Anal. Numér. 31 (1997), no. 7, 891–925 (English, with English and French summaries). MR 1489177, DOI https://doi.org/10.1051/m2an/1997310708911
- H. Le Dret, Modeling of the junction between two rods, J. Math. Pures Appl. (9) 68 (1989), no. 3, 365–397 (English, with French summary). MR 1025910
- Hervé Le Dret, Vibrations of a folded plate, RAIRO Modél. Math. Anal. Numér. 24 (1990), no. 4, 501–521 (English, with French summary). MR 1070967, DOI https://doi.org/10.1051/m2an/1990240405011
- H. Le Dret, Problèmes variationnels dans les multi-domaines, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 19, Masson, Paris, 1991 (French). Modélisation des jonctions et applications. [Modeling of junctions and applications]. MR 1130395
- Véronique Lods, Modélisation et justification d’un problème aux valeurs propres pour une plaque insérée dans un support tridimensionnel, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 391–396 (French, with English and French summaries). MR 1320391
- P.-A. Raviart and J.-M. Thomas, Introduction à l’analyse numérique des équations aux dérivées partielles, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). MR 773854
- J. M. Rodríguez and J. M. Viaño, Asymptotic derivation of a general linear model for thin-walled elastic rods, Comput. Methods Appl. Mech. Engrg. 147 (1997), no. 3-4, 287–321. MR 1460114, DOI https://doi.org/10.1016/S0045-7825%2897%2900019-4
- Maurice Roseau, Vibrations des systèmes mécaniques, Masson, Paris, 1984 (French). Méthodes analytiques et applications. [Analytical methods and applications]. MR 762434
- J. Sanchez Hubert and E. Sánchez-Palencia, Vibration and coupling of continuous systems, Springer-Verlag, Berlin, 1989. Asymptotic methods. MR 996423
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 Anal. 2 (1989), no. 3, 223–255. MR 1020349
- L. Trabucho and J. M. Viaño, A new approach of Timoshenko’s beam theory by asymptotic expansion method, RAIRO Modél. Math. Anal. Numér. 24 (1990), no. 5, 651–680 (English, with French summary). MR 1076964, DOI https://doi.org/10.1051/m2an/1990240506511
- L. Trabucho and J. M. Viaño, Mathematical modelling of rods, Handbook of numerical analysis, Vol. IV, Handb. Numer. Anal., IV, North-Holland, Amsterdam, 1996, pp. 487–974. MR 1422507
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
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
74K10,
74B05,
74G10,
74H45
Retrieve articles in all journals
with MSC:
74K10,
74B05,
74G10,
74H45
Additional Information
Article copyright:
© Copyright 2000
American Mathematical Society
