# Nouvelles propriétés des courbes et relation de dispersion en élasticité linéaire

• Volume: 33, Issue: 5, page 1071-1090
• ISSN: 0764-583X

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## Abstract

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In the case of an elastic strip we exhibit two properties of dispersion curves λn,n ≥ 1, that were not pointed out previously. We show cases where λ'n(0) = λ''n(0) = λ'''n(0) = 0 and we point out that these curves are not automatically monotoneous on ${ℝ}_{+}$. The non monotonicity was an open question (see [2], for example) and, for the first time, we give a rigourous answer. Recall the characteristic property of the dispersion curves: {λn(p);n ≥ 1} is the set of eigenvalues of Ap, counted with their multiplicity. The operators Ap, $p\in ℝ$, are the reduced operators deduced from the elastic operator A using a partial Fourier transform. The second goal of this article is the introduction of a dispersion relation D(p,λ) = 0 in a general framework, and not only for a homogeneous situation (in this last case the relation is explicit). Recall that a dispersion relation is an implicit equation the solutions of which are eigenvalues of Ap. The main property of the function D that we build is the following one: the multiplicity of an eigenvalue λ of Ap is equal to the multiplicity it has as a root of D(p,λ) = 0. We give also some applications.

## How to cite

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Bouhennache, Tark, and Dermenjian, Yves. "Nouvelles propriétés des courbes et relation de dispersion en élasticité linéaire." ESAIM: Mathematical Modelling and Numerical Analysis 33.5 (2010): 1071-1090. <http://eudml.org/doc/197408>.

@article{Bouhennache2010,
abstract = { In the case of an elastic strip we exhibit two properties of dispersion curves λn,n ≥ 1, that were not pointed out previously. We show cases where λ'n(0) = λ''n(0) = λ'''n(0) = 0 and we point out that these curves are not automatically monotoneous on $\{\mathbb\{R\}\}_\{+\}$. The non monotonicity was an open question (see [2], for example) and, for the first time, we give a rigourous answer. Recall the characteristic property of the dispersion curves: \{λn(p);n ≥ 1\} is the set of eigenvalues of Ap, counted with their multiplicity. The operators Ap, $p\in\{\mathbb\{R\}\}$, are the reduced operators deduced from the elastic operator A using a partial Fourier transform. The second goal of this article is the introduction of a dispersion relation D(p,λ) = 0 in a general framework, and not only for a homogeneous situation (in this last case the relation is explicit). Recall that a dispersion relation is an implicit equation the solutions of which are eigenvalues of Ap. The main property of the function D that we build is the following one: the multiplicity of an eigenvalue λ of Ap is equal to the multiplicity it has as a root of D(p,λ) = 0. We give also some applications. },
author = {Bouhennache, Tark, Dermenjian, Yves},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Élasticité; relation de dispersion; valeur propre.; elastic strip; dispersion curves; non-monotonicity; eigenvalues; multiplicity; Fourier transform; dispersion relation},
language = {eng},
month = {3},
number = {5},
pages = {1071-1090},
publisher = {EDP Sciences},
title = {Nouvelles propriétés des courbes et relation de dispersion en élasticité linéaire},
url = {http://eudml.org/doc/197408},
volume = {33},
year = {2010},
}

TY - JOUR
AU - Bouhennache, Tark
AU - Dermenjian, Yves
TI - Nouvelles propriétés des courbes et relation de dispersion en élasticité linéaire
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 33
IS - 5
SP - 1071
EP - 1090
AB - In the case of an elastic strip we exhibit two properties of dispersion curves λn,n ≥ 1, that were not pointed out previously. We show cases where λ'n(0) = λ''n(0) = λ'''n(0) = 0 and we point out that these curves are not automatically monotoneous on ${\mathbb{R}}_{+}$. The non monotonicity was an open question (see [2], for example) and, for the first time, we give a rigourous answer. Recall the characteristic property of the dispersion curves: {λn(p);n ≥ 1} is the set of eigenvalues of Ap, counted with their multiplicity. The operators Ap, $p\in{\mathbb{R}}$, are the reduced operators deduced from the elastic operator A using a partial Fourier transform. The second goal of this article is the introduction of a dispersion relation D(p,λ) = 0 in a general framework, and not only for a homogeneous situation (in this last case the relation is explicit). Recall that a dispersion relation is an implicit equation the solutions of which are eigenvalues of Ap. The main property of the function D that we build is the following one: the multiplicity of an eigenvalue λ of Ap is equal to the multiplicity it has as a root of D(p,λ) = 0. We give also some applications.
LA - eng
KW - Élasticité; relation de dispersion; valeur propre.; elastic strip; dispersion curves; non-monotonicity; eigenvalues; multiplicity; Fourier transform; dispersion relation
UR - http://eudml.org/doc/197408
ER -

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