Skip to main content

Influence of microstructure on thermoelastic wave propagation


Numerical simulations of the thermoelastic response of a microstructured material on a thermal loading are performed in the one-dimensional setting to examine the influence of temperature gradient effects at the microstructure level predicted by the thermoelastic description of microstructured solids (Berezovski et al. in J. Therm. Stress. 34:413–430, 2011). The system of equations consisting of a hyperbolic equation of motion, a parabolic macroscopic heat conduction equation, and a hyperbolic evolution equation for the microtemperature is solved by a finite-volume numerical scheme. Effects of microtemperature gradients exhibit themselves on the macrolevel due to the coupling of equations of the macromotion and evolution equations for macro- and microtemperatures.

This is a preview of subscription content, access via your institution.


  1. 1

    Nowacki W.: Thermoelasticity. Pergamon Press, Oxford (1962)

    Google Scholar 

  2. 2

    Hetnarski R.B., Reza Eslami M.: Thermal Stresses: Advanced Theory and Applications. Springer, Berlin (2009)

    Google Scholar 

  3. 3

    Chandrasekharaiah D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 705–729 (1998)

    Article  Google Scholar 

  4. 4

    Hetnarski R.B., Ignaczak J.: Generalized thermoelasticity. J. Therm. Stress. 22, 451–476 (1999)

    MathSciNet  Article  Google Scholar 

  5. 5

    Ignaczak J., Ostoja-Starzewski M.: Thermoelasticity with Finite Wave Speeds. Oxford University Press, Oxford (2010)

    MATH  Google Scholar 

  6. 6

    Mindlin R.D.: Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16, 51–78 (1964)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7

    Capriz G.: Continua with Microstructure. Springer, Heidelberg (1989)

    Book  MATH  Google Scholar 

  8. 8

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

    Book  Google Scholar 

  9. 9

    Forest S.: Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 13, 117–131 (2009)

    Article  Google Scholar 

  10. 10

    Cardona J.-M., Forest S., Sievert R.: Towards a theory of second grade thermoelasticity. Extr. Math. 14, 127–140 (1999)

    MathSciNet  MATH  Google Scholar 

  11. 11

    Forest S., Amestoy M.: Hypertemperature in thermoelastic solids. C. R. Mecanique 336, 347–353 (2008)

    Article  MATH  Google Scholar 

  12. 12

    Nguyen Q.-S.: On standard dissipative gradient models. Ann. Solid Struct. Mech. 1, 79–86 (2010)

    Article  Google Scholar 

  13. 13

    Tamma K.K., Zhou X.: Macroscale and microscale thermal transport and thermo-mechanical interactions: some noteworthy perspectives. J. Therm. Stress. 21, 405–449 (1998)

    Article  Google Scholar 

  14. 14

    Berezovski A., Engelbrecht J., Maugin G.A.: Thermoelasticity with dual internal variables. J. Therm. Stress. 34, 413–430 (2011)

    Article  Google Scholar 

  15. 15

    Berezovski, A., Engelbrecht, J.: Waves in microstructured solids: dispersion and thermal effects. In: Bai, Y., Wang, J., Fang, D. (eds.) Proceedings of the 23rd International Congress of Theoretical and Applied Mechanics, August 19–24, 2012, Beijing, China, CD-ROM, SM07-005 (2012)

  16. 16

    Ván P., Berezovski A., Engelbrecht J.: Internal variables and dynamic degrees of freedom. J. Non-Equilib. Thermodyn. 33, 235–254 (2008)

    Article  MATH  Google Scholar 

  17. 17

    Maugin G.A.: Internal variables and dissipative structures. J. Non-Equilib. Thermodyn. 15, 173–192 (1990)

    Article  Google Scholar 

  18. 18

    Joseph D.D., Preziosi L.: Heat waves. Rev. Mod. Phys. 61, 41–73 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19

    Muschik W., Berezovski A.: Thermodynamic interaction between two discrete systems in non-equilibrium. J. Non-Equilib. Thermodyn. 29, 237–255 (2004)

    MATH  Google Scholar 

  20. 20

    Berezovski A., Maugin G.A.: Simulation of thermoelastic wave propagation by means of a composite wave-propagation algorithm. J. Comput. Phys. 168, 249–264 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21

    Berezovski A., Maugin G.A.: Thermoelastic wave and front propagation. J. Therm. Stress. 25, 719–743 (2002)

    MathSciNet  Article  Google Scholar 

  22. 22

    Berezovski A., Engelbrecht J., Maugin G.A.: Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. World Scientific, Singapore (2008)

    MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Arkadi Berezovski.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Berezovski, A., Berezovski, M. Influence of microstructure on thermoelastic wave propagation. Acta Mech 224, 2623–2633 (2013).

Download citation


  • Internal Variable
  • Computational Cell
  • Thermoelastic Wave
  • Thermoelastic Response
  • Induce Stress Wave