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Lord–Shulman Thermoelasticity with Microtemperatures

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Abstract

In this paper we consider the Lord–Shulman thermoelastic theory with porosity and microtemperatures. The new aspect we propose here is to introduce a relaxation parameter in the microtemperatures. Then we obtain an existence theorem for the solutions. In the case that a certain symmetry is satisfied by the constitutive tensors, we prove that the semigroup is dissipative. In fact, an exponential decay of solutions can be shown for the one-dimensional case. In the last section, we restrict our attention to the case where we have an isotropic and homogeneous material without porosity effects and assuming that two of the constitutive parameters have the same sign. We see that the semigroup is dissipative.

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Notes

  1. In the general case the materials responses depend on the loading directions. In the case that the response is invariant under inversion we say that the material is centrosymmetric. It implies that the constitutive tensors of odd order vanish.

  2. It is clear that we could consider a general case when the relaxation parameter for the microstructure is different from the one for the macrostructure. But in this paper we want to restrict our attention to the easier case where both parameters coincide.

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Acknowledgements

The authors are grateful to the anonymous referees for theirs useful comments which allow us to improve the manuscript.

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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

    Noelia Bazarra & José R. Fernández

  2. Departamento de Matemáticas, E.S.E.I.A.A.T.-U.P.C., Colom 11, 08222, Terrassa, Barcelona, Spain

    Ramón Quintanilla

Authors
  1. Noelia Bazarra
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  2. José R. Fernández
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  3. Ramón Quintanilla
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Correspondence to José R. Fernández.

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The work of J.R. Fernández was partially supported by Ministerio de Ciencia, Innovación y Universidades under the research project PGC2018-096696-B-I00 (FEDER, UE). The work of R. Quintanilla was supported by project “Análisis Matemático de Problemas de la Termomecánica” (MTM2016-74934-P), (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness, and it is also part of the project “Análisis Matemático Aplicado a la Termomecánica” which is currently submitted to the Spanish Ministry of Science, Innovation and Universities.

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Bazarra, N., Fernández, J.R. & Quintanilla, R. Lord–Shulman Thermoelasticity with Microtemperatures. Appl Math Optim 84, 1667–1685 (2021). https://doi.org/10.1007/s00245-020-09691-2

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