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Nouvelles propriétés des courbes et relation de dispersion en élasticité linéaire

Published online by Cambridge University Press:  15 August 2002

Tark Bouhennache
CMI, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France.
Yves Dermenjian
CMI, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France.
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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 ${\mathbb{R}}_{+}$. 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.

Research Article
© EDP Sciences, SMAI, 1999

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