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

Tark Bouhennache; Yves Dermenjian

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

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topBouhennache, 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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