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Existence, stability and numerical results for a Timoshenko beam with thermodiffusion effects

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Abstract

In this paper, we consider a new Timoshenko beam model with thermal and mass diffusion effects. Heat and mass exchange with the environment during thermodiffusion in Timoshenko beam. Firstly, by the \({C}_{0}\)-semigroup theory, we prove the well posedness of the considered problem with Dirichlet or Neumann boundary conditions. Then we show, without assuming the well-known equal wave speeds condition, the lack of exponential stability for the Neumann problem, meanwhile one linear frictional damping is strong enough to guarantee the exponential stability for the Dirichlet problem. Then, we introduce a finite element approximation and we prove that the associated discrete energy decays. Finally, we obtain some a priori error estimates assuming additional regularity on the solution and we present some numerical results which demonstrate the accuracy of the approximation and the behaviour of the solution.

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Acknowledgements

The authors would like to thank the referees for their critical review and valuable comments which allowed to improve the paper. Part of this work was done when the first author visited the Departamento de Matemática Aplicada I, Universidade de Vigo, ETSI Telecomunicación. He thanks them for their hospitality and financial support. The work of M. Campo and J. R. Fernández has been partially supported by the Ministerio de Economía y Competitividad under the research Project MTM2015-66640-P (with FEDER Funds) and the Ministerio de Ciencia, Innovación y Universidades under the research Project PGC2018-096696-B-I00 (with FEDER Funds). The work of M.I.M. Copetti has been partially supported by the Brazilian institution CNPq (Grant 304709/2017-4).

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Authors and Affiliations

  1. École Nationale d’Ingénieurs de Bizerte, Université de Carthage, BP66, Campus Universitaire, 7035, Menzel Abderrahman, Tunisia

    Moncef Aouadi

  2. Departamento de Matemáticas, ETS de Ingenieros de Caminos, Canales y Puertos, Universidade da Coruña, Campus de Elviña, 15071, A Coruña, Spain

    Marco Campo

  3. Laboratório de Análise Numérica e Astrofísica, Departamento de Matemática, Universidade Federal de Santa Maria, Santa Maria, RS, 97105-900, Brazil

    Maria I. M. Copetti

  4. Departamento de Matemática Aplicada I, Universidade de Vigo, ETSI Telecomunicación, Campus As Lagoas Marcosende s/n, 36310, Vigo, Spain

    José R. Fernández

Authors
  1. Moncef Aouadi
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  2. Marco Campo
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  3. Maria I. M. Copetti
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  4. José R. Fernández
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Correspondence to José R. Fernández.

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Aouadi, M., Campo, M., Copetti, M.I.M. et al. Existence, stability and numerical results for a Timoshenko beam with thermodiffusion effects. Z. Angew. Math. Phys. 70, 117 (2019). https://doi.org/10.1007/s00033-019-1161-8

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  • DOI: https://doi.org/10.1007/s00033-019-1161-8

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