Viscoélastodynamique monodimensionnelle avec conditions de Signorini

Adrien Petrov; Michelle Schatzman

doi:10.1016/S1631-073X(02)02399-3

Viscoélastodynamique monodimensionnelle avec conditions de Signorini

Adrien Petrov ¹ ; Michelle Schatzman ¹

¹ MAPLY, CNRS et Université Claude Bernard-Lyon 1, Mathématiques, 21 avenue Claude Bernard, 69622 Villeurbanne cedex, France

Comptes Rendus. Mathématique, Volume 334 (2002) no. 11, pp. 983-988.

Résumé (VO)
Abstract

Soit α un nombre strictement positif. Le problème viscoélastique monodimensionnel

u_{tt} - u_{xx} - α u_{xxt} = f, x \in (- \infty, 0], t \in [0, + \infty),

avec les conditions au bord unilatérales

u (0, \cdot) ⩾ 0, (u_{x} + α u_{xt}) (0, \cdot) ⩾ 0, (u (u_{x} + α u_{xt})) (0, \cdot) = 0,

peut être réduit à l'inéquation variationnelle suivante :

λ_{1} * w = g + b, w ⩾ 0, b ⩾ 0, 〈 w, b 〉 = 0 .

Ici

{\hat{λ}}_{1} (ω)

est la détermination causale de

i ω \sqrt{1 + i α ω}

. On démontre que ce problème possède une solution et que les pertes d'énergie sont purement visqueuses ; ce résultat provient de la relation

〈 \dot{w}, b 〉 = 0

, qui n'est pas triviale puisque, a priori, b est une mesure et

\dot{w}

n'est définie que presque partout.

Let α be a positive number. The one-dimensional viscoelastic problem

u_{tt} - u_{xx} - α u_{xxt} = f, x \in (- \infty, 0], t \in [0, + \infty),

with unilateral boundary conditions

u (0, \cdot) ⩾ 0, (u_{x} + α u_{xt}) (0, \cdot) ⩾ 0, (u (u_{x} + α u_{xt})) (0, \cdot) = 0,

can be reduced to the following variational inequality:

λ_{1} * w = g + b, w ⩾ 0, b ⩾ 0, 〈 w, b 〉 = 0 .

Here

{\hat{λ}}_{1} (ω)

is the causal determination of

i ω \sqrt{1 + i α ω}

. We show that the energy losses are purely viscous; this result is a consequence of the relation

〈 \dot{w}, b 〉 = 0

; since a priori, b is a measure and

\dot{w}

is defined only almost everywhere, this relation is not trivial.

Reçu le : 2002-04-08
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02399-3

Affiliations des auteurs :

Adrien Petrov ¹ ; Michelle Schatzman ¹

¹ MAPLY, CNRS et Université Claude Bernard-Lyon 1, Mathématiques, 21 avenue Claude Bernard, 69622 Villeurbanne cedex, France

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     title = {Visco\'elastodynamique monodimensionnelle avec conditions de {Signorini}},
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Adrien Petrov; Michelle Schatzman. Viscoélastodynamique monodimensionnelle avec conditions de Signorini. Comptes Rendus. Mathématique, Volume 334 (2002) no. 11, pp. 983-988. doi : 10.1016/S1631-073X(02)02399-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02399-3/

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