Asymptotic analysis of torsional and stretching modes of thin rods

Irago, H.; Kerdid, Nabil; Viaño, J. M.

doi:10.1090/qam/1770651

Cited by 6 publications

(3 citation statements)

References 15 publications

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“…Later, the method was successfully applied by Trabucho and Viaño [29,30] for mathematically derive more complete models as the Saint-Venant, Timoshenko and Vlassov theories for thick and thin profile cross section, isotropic and anisotropic materials and curved beams (see Rodríguez-Seijo and Viaño [25,26], Alvarez-Dios and Viaño [1][2][3], Irago and Viaño [13], Irago, Kerdid and Viaño [14]). The nonlinear case was studied by Cimetière et al [8].…”

Section: Introductionmentioning

confidence: 99%

Mathematical justification of Kelvin–Voigt beam models by asymptotic methods

Rodríguez-Arós

Viaño

2011

Z. Angew. Math. Phys.

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Section: Introductionmentioning

confidence: 99%

Mathematical justification of Kelvin–Voigt beam models by asymptotic methods

Rodríguez-Arós

Viaño

2011

Z. Angew. Math. Phys.

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“…In fact, we get results for eigenvalues and eigenfunctions of (1.1) which are of interest in terms of the associated evolution problems since, from (1.8), we can construct standing waves which approach time-dependent solutions for long times, and these times can be precisely computed in terms of bounds for discrepancies such as that in (1.5) (see [41] and [34] for an abstract framework as well as for applications to very different vibrating systems). It should be emphasized that it seems to be a common fact to many mechanical systems arising in thin structures that the low frequencies give rise to longitudinal vibrations while for other kinds of vibrations such as torsional or stretching vibrations one must look among those associated to the high frequencies: see, for instance, [8], [23], [24], [35] and references therein. Also, we emphasize that, to our knowledge, this is the first work in the literature addressing the asymptotics of the high frequencies for a thin T-like shaped structure.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of the high frequencies for the Laplace operator in a thin T-like shaped structure

Gaudiello

Gómez

Pérez‐Martínez

2020

Journal de Mathématiques Pures et Appliquées

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We consider a spectral problem for the Laplacian operator in a planar T-like shaped thin structure Ω ε , where ε denotes the transversal thickness of both branches. We assume the homogeneous Dirichlet boundary condition on the ends of the branches and the homogeneous Neumann boundary condition on the remaining part of the boundary of Ω ε . We study the asymptotic behavior, as ε tends to zero, of the high frequencies of such a problem. Unlike the asymptotic behavior of the low frequencies where the limit problem involves only longitudinal vibrations along each branch of the T-like shaped thin structure (i.e. 1D limit spectral problems), we obtain a two dimensional limit spectral problem which allows us to capture other kinds of vibrations. We also give a characterization of the asymptotic form of the eigenfunctions originating these vibrations.Résumé: On considère le problème spectral pour le Laplacien, dans une structure mince bidimensionnelle Ω ε en forme de T, où ε désigne l'épaisseur des deux branches du T. Les conditions aux limites sont du type Neumann homogène sur tout le bord sauf aux extrémités des branches où une condition de Dirichlet homogène est imposée. Onétudie le comportement asymptotique des hautes fréquences lorsque ε tend vers zéro. Contrairement au comportement asymptotique des basses fréquences, pour lesquelles le problème limite ne fait apparaître que des vibrations longitudinales le long de chaque branche de la structure (c'est-à-dire, des problèmes spectraux limites 1D ), on obtientà la limite un problème spectral bidimensionnel, qui nous permet de capter d'autres types de vibrations. On donneégalement une caractérisation de la forme asymptotique des fonctions propres qui sontà l'origine de ces vibrations.

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“…Using asymptotic analysis, several classical reduced models have been mathematically developed over the years. The asymptotic method was originally introduced by Lions [22] and since then it has been extensively used to derive and justify reduced models for elastic plates and shells [10][11][12], elastic beams [4,19,21,31,[40][41][42][43], viscoelastic beams [28,29] and also for elastic beams in contact with a foundation (see [20,30,44,47], the last two to justify and generalize contact models found in [17,37]). The success of this method is due to the inherent small geometrical parameters involved (thickness of plates and shells and diameter of the cross-section in beams).…”

Section: Introductionmentioning

confidence: 99%

A model for bending and stretching of piezoelectric rods obtained by asymptotic analysis

Viaño

Figueiredo²,

Ribeiro³

et al. 2014

Z. Angew. Math. Phys.

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The aim of this paper was to obtain a new model for the bending-stretching of an anisotropic heterogeneous linearly piezoelectric cantilever rod when the electric potential is applied on the both ends. The process is assumed to be static, and the piezoelectric material is monoclinic of class 2. To derive the model, we start with the corresponding threedimensional problem, introduce a change of variable together with a scaling of the unknowns and then we use a passage to the limit procedure, based on arguments of asymptotic analysis taking the diameter of the cross-section as small parameter. Finally, we prove a result of strong convergence that justifies both the method and the one-dimensional model obtained. One of the most relevant features of this one-dimensional model is that the stretching is coupled with the electric potential, while the bendings are not.Mathematics Subject Classification (2010). Primary 74K10; Secondary 41A60 · 35C20 · 35Q60.

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Asymptotic analysis of torsional and stretching modes of thin rods

Cited by 6 publications

References 15 publications

Mathematical justification of Kelvin–Voigt beam models by asymptotic methods

Mathematical justification of Kelvin–Voigt beam models by asymptotic methods

Asymptotic analysis of the high frequencies for the Laplace operator in a thin T-like shaped structure

A model for bending and stretching of piezoelectric rods obtained by asymptotic analysis

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