Homogenization of thermo-viscoelastic Kelvin–Voigt model

Zouhair Abdessamad; Ilya Kostin; Grigory Panasenko; Valery P. Smyshlyaev

doi:10.1016/j.crme.2007.05.022

Homogenization of thermo-viscoelastic Kelvin–Voigt model

Presented by: Évariste Sanchez-Palencia

Zouhair Abdessamad ¹; Ilya Kostin ¹; Grigory Panasenko ¹ ; Valery P. Smyshlyaev ²

¹ LaMUSE EA 3989, University of Saint Étienne, 23, rue P. Michelon, 42023 Saint-Étienne, France
² Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

Comptes Rendus. Mécanique, Volume 335 (2007) no. 8, pp. 423-429.

Abstract
Résumé

We consider an ε-periodic composite material, $ε ≪ 1$ , constituted of periodic fibres surrounded by a polymer matrix, solidifying under a heating process. The mechanical behaviour of the material is described by the Kelvin–Voigt visco-elasticity equation with rapidly oscillating space and time dependent coefficients. This time dependence is caused by the dependence of the state of the material on the temperature, that is a solution of a thermo-chemical model studied earlier. The existence and uniqueness of a solution of the Kelvin–Voigt visco-elasticity model are proved, the homogenized model is obtained and the existence and uniqueness of its solution are studied. The estimates for the difference between the solution of the original problem and the homogenized one are obtained.

On considère un matériau composite de structure périodique de période $ε ≪ 1$ , constitué d'un tissu de fibres noyé dans une résine qui se soldifie sous l'effet de la chaleur. Les propriétés mécaniques du matériau sont décrites par l'équation de viscoélasticité de Kelvin–Voigt avec des coefficients oscillants dépendant des variables spatiale et temporaire x et t. Cette dépendance de temps est engendrée par la dépendance de l'état déformé du matériau de la temperature, une solution du problème thermo-chimique étudié précedement. On établit un résultat d'existence et d'unicité de la solution, puis à l'aide de la méthode du développement asymptotique on détermine le problème homogénéisé. On prouve l'existence et l'unicité de la solution du problème homogénéisé, puis on obtient une estimation pour la différence entre la solution du problème de départ et la solution du problème homogénéisé lorsque ε tend vers zéro.

Received: 2007-03-16
Accepted: 2007-05-18
Published online: 2007-07-31

DOI: 10.1016/j.crme.2007.05.022

Keywords: Computational solid mechanics, Homogenization, Visco-elasticity, Time dependent coefficients, Memory effect
Mot clés : Mécanique des solides numérique, Homogénéisation, Visco-élasticité, Coefficients dépendents de temps, Effet de mémoire

Author's affiliations:

Zouhair Abdessamad ¹; Ilya Kostin ¹; Grigory Panasenko ¹; Valery P. Smyshlyaev ²

¹ LaMUSE EA 3989, University of Saint Étienne, 23, rue P. Michelon, 42023 Saint-Étienne, France
² Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

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     author = {Zouhair Abdessamad and Ilya Kostin and Grigory Panasenko and Valery P. Smyshlyaev},
     title = {Homogenization of thermo-viscoelastic {Kelvin{\textendash}Voigt} model},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {423--429},
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     year = {2007},
     doi = {10.1016/j.crme.2007.05.022},
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Zouhair Abdessamad; Ilya Kostin; Grigory Panasenko; Valery P. Smyshlyaev. Homogenization of thermo-viscoelastic Kelvin–Voigt model. Comptes Rendus. Mécanique, Volume 335 (2007) no. 8, pp. 423-429. doi : 10.1016/j.crme.2007.05.022. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.05.022/

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