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On the Viscoelastic Mixtures of Solids

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Abstract

In this paper we analyze an homogeneous and isotropic mixture of viscoelastic solids. We propose conditions to guarantee the coercivity of the internal energy and also of the dissipation, first in dimension two and later in dimension three. We obtain an uniqueness result for the solutions when the dissipation is positive and without any hypothesis over the internal energy. When the internal energy and the dissipation are both positive, we prove the existence of solutions as well as their analyticity. Exponential stability and impossibility of localization of the solutions are immediate consequences.

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Acknowledgements

The work of J. R. Fernández was supported by the Ministerio de Economía and Competitividad under the Research Project MTM2015-66640-P (with the participation of FEDER). The work of A. Magaña and R. Quintanilla is supported by Projects “Análisis Matemático de las Ecuaciones en Derivadas Parciales de la Termomecánica” (MTM2013-42004-P) and “Análisis Matemático de Problemas de la Termomecánica” (MTM2016-74934-P), (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness. We thank Professor D. Ieşan for his useful comments and an anonymous referee for helping us to improve the paper.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada I, Universidade de Vigo, Escola de Enxeñería de Telecomunicación, Campus As Lagoas Marcosende s/n, 36310, Vigo, Spain

    Jose R. Fernández & María Masid

  2. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Escola Superior d’Enginyeries Industrial, Aeroespacial i Audiovisual de Terrassa, C. Colom 11, 08222, Terrassa, Barcelona, Spain

    Antonio Magaña & Ramón Quintanilla

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  1. Jose R. Fernández
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  2. Antonio Magaña
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  3. María Masid
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  4. Ramón Quintanilla
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Correspondence to Antonio Magaña.

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Fernández, J.R., Magaña, A., Masid, M. et al. On the Viscoelastic Mixtures of Solids. Appl Math Optim 79, 309–326 (2019). https://doi.org/10.1007/s00245-017-9439-8

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  • DOI: https://doi.org/10.1007/s00245-017-9439-8

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