Acta Mechanica

, Volume 230, Issue 1, pp 31–48 | Cite as

A thermoelastic problem with diffusion, microtemperatures, and microconcentrations

  • Noelia Bazarra
  • Marco Campo
  • José R. FernándezEmail author
Original Paper


In this paper, we analyze, from the numerical point of view, a dynamic problem involving a thermoelastic body with diffusion, whose microelements are assumed to possess microtemperatures and microconcentrations. Using the linear theory, the mechanical problem is written as a coupled system of hyperbolic and parabolic partial differential equations for the displacement, temperature, chemical potential, microconcentrations, and microtemperatures fields. The variational formulation is derived, and it leads to a coupled system of parabolic linear variational equations, for which an existence and uniqueness result is stated. Then, using the finite element method and the implicit Euler scheme, fully discrete approximations are introduced. Stability properties and a priori error estimates are obtained, from which the linear convergence of the algorithm is deduced under suitable additional regularity conditions. Finally, some numerical simulations are presented to show the accuracy of the approximation and the behaviour of the solution.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Nowacki, W.: Dynamical problems of thermoelastic diffusion in solids I, II, III. Bull. Acad. Pol. Sci. Ser. Sci. Tech. 22, 55–64 (1974). 129–135, 257–266Google Scholar
  2. 2.
    Sherief, H., Hamza, F., Saleh, H.: The theory of generalized thermoelastic diffusion. Int. J. Eng. Sci. 42, 591–608 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aouadi, M.: A theory of thermoelastic diffusion materials with voids. Z. Angew. Math. Phys. 61, 357–379 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Aouadi, M.: A contact problem of a thermoelastic diffusion rod. Z. Angew. Math. Mech. 90, 278–286 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aouadi, M.: Stability in thermoelastic diffusion theory with voids. Appl. Anal. 91, 121–139 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Aouadi, M., Copetti, M.I.M., Fernández, J.R.: A contact problem in thermoviscoelastic diffusion theory with second sound. Math. Model. Numer. Anal. 51, 759–796 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fernández, J.R., Masid, M.: Numerical analysis of a thermoelastic diffusion problem with voids. Int. J. Numer. Anal. Model. 14(2), 153–174 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Grot, R.: Thermodynamics of a continuum with microstructure. Int. J. Eng. Sci. 7, 801–814 (1969)CrossRefGoogle Scholar
  9. 9.
    Wozniak, Cz: Thermoelasticity of non-simple oriented materials. Int. J. Eng. Sci. 2, 605–612 (1967)CrossRefGoogle Scholar
  10. 10.
    Ieşan, D., Quintanilla, R.: On thermoelastic bodies with inner structure and microtemperatures. J. Therm. Stress. 23, 199–215 (2000)CrossRefGoogle Scholar
  11. 11.
    Ieşan, D.: Thermoelasticity of bodies with microstructure and microtemperatures. Int. J. Solids Struct. 44, 8648–8662 (2007)CrossRefGoogle Scholar
  12. 12.
    Casas, P., Quintanilla, R.: Exponential stability in thermoelasticity with microtemperatures. Int. J. Eng. Sci. 43, 33–47 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pamplona, P.X., Muñoz-Rivera, J.E., Quintanilla, R.: Analyticity in porous-thermoelasticity with microtemperatures. J. Math. Anal. Appl. 31, 645–655 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Scalia, A., Svanadze, M.: On the representation of solutions of the theory of thermoelasticity with microtemperatures. J. Therm. Stress. 29, 849–863 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Svanadze, M.: Fundamental solutions of the equations of the theory of thermoelasticity with microtemperatures. J. Therm. Stress. 27, 151–170 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Aouadi, M., Ciarletta, M., Tibullo, V.: A thermoelastic diffusion theory with microtemperatures and microconcentrations. J. Therm. Stress. 40(4), 486–501 (2017)CrossRefGoogle Scholar
  17. 17.
    Clement, Ph: Approximation by finite element functions using local regularization. RAIRO Math. Model. Numer. Anal. 9(2), 77–84 (1975)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Andrews, K.T., Fernández, J.R., Shillor, M.: Numerical analysis of dynamic thermoviscoelastic contact with damage of a rod. IMA J. Appl. Math. 70(6), 768–795 (2005)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Campo, M., Fernández, J.R., Kuttler, K.L., Shillor, M., Viaño, J.M.: Numerical analysis and simulations of a dynamic frictionless contact problem with damage. Comput. Methods Appl. Mech. Eng. 196(1–3), 476–488 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Münch, A., Pazoto, A.F.: Uniform stabilization of a viscous numerical approximation for a locally damped wave equation. ESAIM Control Optim. Calc. Var. 13, 265–293 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tébou, L.R.T., Zuazua, E.: Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity. Numer. Math. 95, 563–598 (2003)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada IUniversidade de VigoVigoSpain
  2. 2.Departamento de MatemáticasETS de Ingenieros de Caminos, Canales y Puertos Universidade da CoruñaA CoruñaSpain

Personalised recommendations