An a priori error analysis of a Lord–Shulman poro-thermoelastic problem with microtemperatures

Abstract

In this paper, we deal with the numerical analysis of the Lord–Shulman thermoelastic problem with porosity and microtemperatures. The thermomechanical problem leads to a coupled system composed of linear hyperbolic partial differential equations written in terms of transformations of the displacement field and the volume fraction, the temperature and the microtemperatures. An existence and uniqueness result is stated. Then, a fully discrete approximation is introduced using the finite element method and the implicit Euler scheme. A discrete stability property is shown, and an a priori error analysis is provided, from which the linear convergence is derived under suitable regularity conditions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximation, the comparison with the classical Fourier theory and the behavior of the solution in two-dimensional examples.

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Notes

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    This assumption is not relevant in the analysis proposed here. It only allows to simplify the calculations.

References

  1. 1.

    Cattaneo, C.: On a form of heat equation which eliminates the paradox of instantaneous propagation. C. R. Acad. Sci. Paris 247, 431–433 (1958)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Lord, H.W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967)

    MATH  Google Scholar 

  3. 3.

    Eringen, A.C.: Microcontinuum Field Theories. I. Foundations and Solids. Springer, New York (1999)

    Google Scholar 

  4. 4.

    Ieşan, D.: Thermoelastic Models of Continua. Kluwer Academic Publishers, Dordreecht (2004)

    Google Scholar 

  5. 5.

    Grot, R.: Thermodynamics of a continuum with microstructure. Int. J. Eng. Sci. 7, 801–814 (1969)

    MATH  Google Scholar 

  6. 6.

    Riha, P.: On the theory of heat-conducting micropolar fluids with microtemperatures. Acta Mech. 23, 1–8 (1975)

    MATH  Google Scholar 

  7. 7.

    Riha, P.: On the microcontinuum model of heat conduction in materials with inner structure. Int. J. Eng. Sci. 14, 529–535 (1976)

    Google Scholar 

  8. 8.

    Verma, P.D.S., Singh, D.V., Singh, K.: Poiseuille flow of microthermopolar fluids in a circular pipe. Acta Tech. CSAV 24, 402–412 (1979)

    MATH  Google Scholar 

  9. 9.

    Aouadi, M., Ciarletta, M., Passarella, F.: Thermoelastic theory with microtemperatures and dissipative thermodynamics. J. Thermal Stresses 41, 522–542 (2018)

    Google Scholar 

  10. 10.

    Casas, P., Quintanilla, R.: Exponential stability in thermoelasticity with microtemperatures. Int. J. Eng. Sci. 43, 33–47 (2005)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Chirita, S., Ciarletta, M., D’Apice, C.: On the theory of thermoelasticity with microtemperatures. J. Math. Anal. Appl. 397, 349–361 (2013)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Ciarletta, M., Passarella, F., Tibullo, V.: Plane harmonic waves in strongly elliptic thermoelastic materials with microtemperatures. J. Math. Anal. Appl. 424, 1186–1197 (2015)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Ciarletta, M., Straughan, B., Tibullo, V.: Structural stability for a rigid body with thermal microstructure. Int. J. Eng. Sci. 48, 592–598 (2010)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Ieşan, D.: Thermoelasticity of bodies with microstructure and microtemperatures. Int. J. Solids Struct. 44, 8648–8653 (2007)

    MATH  Google Scholar 

  15. 15.

    Ieşan, D., Quintanilla, R.: On a theory of thermoelasticity with microtemperatures. J. Thermal Stresses 23, 195–215 (2000)

    MathSciNet  Google Scholar 

  16. 16.

    Ieşan, D., Quintanilla, R.: Qualitative properties in strain gradient thermoelasticity with microtemperatures. Math. Mech. Solids 23, 240–258 (2018)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Jaiani, G., Bitsadze, L.: On basic problems for elastic prismatic shells with microtemperatures. ZAMM Z. Angew. Math. Mech. 96, 1082–1088 (2016)

    MathSciNet  Google Scholar 

  18. 18.

    Magaña, A., Quintanilla, R.: Exponential stability in type III thermoelasticity with microtemperatures. ZAMP Z. Angew. Math. Phys. 69(5), 129(1)–129(8) (2018)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Magaña, A., Quintanilla, R.: Exponential stability in three-dimensional type III thermo- porous-elasticity with microtemperatures. J. Elast. 139, 153–161 (2020)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Pamplona, P.X., Muñoz-Rivera, J.E., Quintanilla, R.: Analyticity in porous-thermoelasticity with microtemperatures. J. Math. Anal. Appl. 394, 645–655 (2012)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Passarella, F., Tibullo, V., Viccione, G.: Rayleigh waves in isotropic strongly elliptic thermoelastic materials with microtemperatures. Meccanica 52, 3033–3041 (2017)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Quintanilla, R.: On the growth and continuous dependence in thermoelasticity with microtemperatures. J. Thermal Stresses 34, 911–922 (2011)

    Google Scholar 

  23. 23.

    Quintanilla, R.: On the logarithmic convexity in thermoelasticity with microtemperatures. J. Thermal Stresses 36, 378–386 (2013)

    Google Scholar 

  24. 24.

    Cowin, S.C.: The viscoelastic behavior of linear elastic materials with voids. J. Elast. 15, 185–191 (1985)

    MATH  Google Scholar 

  25. 25.

    Cowin, S.C., Nunziato, J.W.: Linear elastic materials with voids. J. Elast. 13, 125–147 (1983)

    MATH  Google Scholar 

  26. 26.

    Nunziato, J.W., Cowin, S.C.: A nonlinear theory of elastic materials with voids. Arch. Rational Mech. Anal. 72, 175–201 (1979)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Casas, P., Quintanilla, R.: Exponential decay in one-dimensional porous-thermoelasticity. Mech. Res. Commun. 32, 652–658 (2005)

    MATH  Google Scholar 

  28. 28.

    Feng, B., Apalara, T.A.: Optimal decay for a porous elasticity system with memory. J. Math. Anal. Appl. 470, 1108–1128 (2019)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Feng, B., Yin, M.: Decay of solutions for a one-dimensional porous elasticity system with memory: the case of non-equal wave speeds. Math. Mech. Solids 24, 2361–2373 (2019)

    MathSciNet  Google Scholar 

  30. 30.

    Ieşan, D., Quintanilla, R.: On a theory of thermoelastic materials with double porosity structure. J. Thermal Stresses 37, 1017–1036 (2014)

    Google Scholar 

  31. 31.

    Kumar, R., Vohra, R.: Effect of hall current in thermoelastic materials with double porosity structure. Int. J. Appl. Mech. Eng. 22, 303–319 (2017)

    Google Scholar 

  32. 32.

    Kumar, R., Vohra, R.: Forced vibrations of a thermoelastic double porous microbeam subjected to a moving load. J. Theor. Appl. Mech. 57, 155–166 (2019)

    Google Scholar 

  33. 33.

    Kumar, R., Vohra, R., Gorla, M.: Reflection of plane waves in thermoelastic medium with double porosity. Multidiscip. Model. Mater. Struct. 12, 748–778 (2016)

    Google Scholar 

  34. 34.

    Leseduarte, M.C., Magaña, A., Quintanilla, R.: On the time decay of solutions in porous-thermo-elasticity of type II. Discrete Contin. Dyn. Syst. B 13, 375–391 (2010)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Magaña, A., Quintanilla, R.: On the spatial behavior of solutions for porous elastic solids with quasi-static microvoids. Math. Comput. Model. 44, 710–716 (2006)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Magaña, A., Quintanilla, R.: On the exponential decay of solutions in one-dimensional generalized porous-thermo-elasticity. Asymptot. Anal. 49, 173–187 (2006)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Magaña, A., Quintanilla, R.: On the time decay of solutions in porous-elasticity with quasi-static microvoids. J. Math. Anal. Appl. 331, 617–630 (2007)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Miranville, A., Quintanilla, R.: Exponential decay in one-dimensional type III thermoelasticity with voids. Appl. Math. Lett. 94, 30–37 (2019)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Miranville, A., Quintanilla, R.: Exponential decay in one-dimensional type II thermoviscoelasticity with voids. J. Comput. Appl. Math. 368, 112573 (2020)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Pamplona, P.X., Muñoz-Rivera, J.E., Quintanilla, R.: On the decay of solutions for porous-elastic systems with history. J. Math. Anal. Appl. 379, 682–705 (2011)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Straughan, B.: Mathematical Aspects of Multi-porosity Continua, Advances in Mechanics and Mathematics, 38. Springer, Cham (2017)

    Google Scholar 

  42. 42.

    Svanadze, M.: On the linear equilibrium theory of elasticity for materials with triple voids. Q. J. Mech. Appl. Math. 71, 329–348 (2018)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Svanadze, M.: Steady vibration problems in the theory of elasticity for materials with double voids. Acta Mech. 229, 1517–1536 (2018)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Bazarra, N., Fernández, J.R., Quintanilla, R.: Lord-Shulman thermoelasticity with microtemperatures. Appl. Math. Optim. (2020). https://doi.org/10.1007/s00245-020-09691-2

    Article  Google Scholar 

  45. 45.

    Lebeau, G., Zuazua, E.: Decay rates for the three-dimensional linear system of thermoelasticity. Arch. Rat. Mech. Anal. 148, 179–231 (1999)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Zhang, X., Zuazua, E.: Decay of solutions of the thermoelasticity of type III. Commun. Contemp. Math. 13, 1–59 (2003)

    MathSciNet  MATH  Google Scholar 

  47. 47.

    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)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    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)

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Clement, Ph: Approximation by finite element functions using local regularization. RAIRO Math. Model. Numer. Anal. 9(2), 77–84 (1975)

    MathSciNet  MATH  Google Scholar 

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Correspondence to José R. Fernández.

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J. Baldonedo acknowledges the funding by Xunta de Galicia (Spain) under the program Axudas á etapa predoutoral with Ref. ED481A-2019/230. The work of J.R. Fernández has been partially supported by Ministerio de Ciencia, Innovación y Universidades under the research Project PGC2018-096696-B-I00 (FEDER, UE). The work of R. Quintanilla has been supported by Ministerio de Economía y Competitividad under the research project “Análisis Matemático de Problemas de la Termomecánica” (MTM2016-74934-P), (AEI/FEDER, UE), and Ministerio de Ciencia, Innovación y Universidades under the research project “Análisis matemático aplicado a la termomecánica” (currently under evaluation).

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Baldonedo, J., Bazarra, N., Fernández, J.R. et al. An a priori error analysis of a Lord–Shulman poro-thermoelastic problem with microtemperatures. Acta Mech 231, 4055–4076 (2020). https://doi.org/10.1007/s00707-020-02738-z

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