Skip to main content
SpringerLink
Log in

Dispersion relations for the hyperbolic thermal conductivity, thermoelasticity and thermoviscoelasticity

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

The Maxwell–Cattaneo heat conduction theory, the Lord–Shulman theory of thermoelasticity and a hyperbolic theory of thermoviscoelasticity are studied. The dispersion relations are analyzed in the case when a solution is represented in the form of an exponential function decreasing in time. Simple formulas that quite accurately approximate the dispersion curves are obtained. Based on the results of analysis of the dispersion relations, an experimental method of determination of the heat flux relaxation time is suggested.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Pop, E., Sinha, S., Kenneth, E.: Goodson heat generation and transport in nanometer-scale transistors. Proc. IEEE 94(8), 1587–1601 (2006)

    Article  Google Scholar 

  2. Jou, D., Casas-Vazquez, J., Lebon, G.: Extended Irreversible Thermodynamics. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  4. Tzou, D.Y.: Macro-to-Microscale Heat Transfer. The Lagging Behaviour. Taylor and Francis, New York (1997)

    Google Scholar 

  5. Shashkov, A. G., Bubnov, V. A., Yanovski, S. Y.: Wave phenomena of heat conductivity: system and structural approach. (1993).(in Russian)

  6. Wang, C.C.: The principle of fading memory. Arch. Ration. Mech. Anal. 18(5), 343–366 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  7. Tzou, D.Y.: On the thermal shock wave induced by a moving heat source. Int. J. Heat Mass Transf. 111, 232–238 (1989)

    Google Scholar 

  8. Qiu, T.Q., Tien, C.L.: Short-pulse laser heating on metals. Int. J. Heat Mass Transf. 35(3), 719–726 (1992)

    Article  ADS  Google Scholar 

  9. Sobolev, S.L.: Transport processes and traveling waves in systems with local nonequilibrium. Sov. Phys. Uspekhi 34(3), 217 (1991)

    Article  ADS  Google Scholar 

  10. Cattaneo, C.: A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Compte Rendus 247, 431–433 (1958)

    MATH  Google Scholar 

  11. Vernotte, P.: Les paradoxes de la theorie continue de lequation de la chaleur. CR Acad. Sci. 246(22), 3154–3155 (1958)

    MATH  Google Scholar 

  12. Lykov, A. V.: Theory of heat conduction. Vysshaya Shkola, Moscow. (1967): 599. (in russian)

  13. Babenkov, M.B., Ivanova, E.A.: Analysis of the wave propagation processes in heat transfer problems of the hyperbolic type. Contin. Mech. Thermodyn. 26(4), 483–502 (2014). doi:10.1007/s00161-013-0315-8

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Peshkov, V.: Second sound in helium II. J. Phys. 8, 381 (1944)

    Google Scholar 

  15. Liu, Y., Mandelis, A.: Laser optical and photothermal thermometry of solids and thin films. Exp. Methods Phys. Sci. 42, 297–336 (2009)

    Article  Google Scholar 

  16. Magunov, A.N.: Laser thermometry of solids: state of the art and problems. Meas. Tech. 45(2), 173–181 (2002)

    Article  Google Scholar 

  17. Magunov, A.N.: Laser Thermometry of Solids. Cambridge International Science Publishing, Cambridge (2003)

    Google Scholar 

  18. Wang, X.: Experimental Micro/nanoscale Thermal Transport. Wiley, Hoboken (2012)

    Book  Google Scholar 

  19. Krilovich, V.I., Bil, G.N., Ivakin, E.V., Rubanov, A.C.: Experimental determination heat velocity. NASB 1, 129–134 (2000). (in Russian)

  20. Ivakin, E.V., Kizak, A.I., Rubanov, A.S.: Active spectroscopy of rayleigh light-scattering in study of heat-transfer. Izvestiya Akademii Nauk SSSR Seriya fizicheskaya 56(12), 130–134 (1992). (in Russian)

    Google Scholar 

  21. Ivakin, E.V., Lazaruk, A.M., Filipov, V.V.: Application of laser induced gratings for thermal diffusivity measurements of solids. Proc. SPIE 2648, 196–206 (1995). (in Russian)

    Article  ADS  Google Scholar 

  22. Xu, F., Tianjian, Lu: Introduction to Skin Biothermomechanics and Thermal Pain, vol. 7. Science Press, New York (2011)

    Book  Google Scholar 

  23. Grassmann, A., Peters, F.: Experimental investigation of heat conduction in wet sand. Heat Mass Transf. 35(4), 289–294 (1999)

    Article  ADS  Google Scholar 

  24. Herwig, H., Beckert, K.: Experimental evidence about the controversy concerning Fourier or non-Fourier heat conduction in materials with a nonhomogeneous inner structure. Heat Mass Transf. 36(5), 387–392 (2000)

    Article  ADS  Google Scholar 

  25. Kaminski, W.: Hyperbolic heat conduction equation for materials with a nonhomogeneous inner structure. J. Heat Transf. 112(3), 555–560 (1990)

    Article  Google Scholar 

  26. Mitra, K., Kumar, S., Vedavarz, A., Moallemi, M.K.: Experimental evidence of hyperbolic heat conduction in processed meat. J. Heat Transf. 117, 568–573 (1995)

    Article  Google Scholar 

  27. Roetzel, W., Putra, N., Das, Sarit K.: Experiment and analysis for non-Fourier conduction in materials with non-homogeneous inner structure. Int. J. Therm. Sci. 42, 541–552 (2003)

    Article  Google Scholar 

  28. Vovnenko, N.V., Zimin, B.A., Sud’enkov, YuV: Nonequilibrium motion of a metal surface exposed to sub-microsecond laser pulses. Zhurnal tekhnicheskoi fiziki 80(7), 41–45 (2010). (in Russian)

    Google Scholar 

  29. Sudenkov, Y.V., Pavlishin, A.I.: Nanosecond pressure pulses propagating at anomalously high velocities in metal foils. Tech. Phys. Lett. 29(6), 491–493 (2003)

    Article  ADS  Google Scholar 

  30. Szekeres, A., Fekete, B.: Continuummechanics-heat conduction-cognition. Period. Polytech. Eng. Mech. Eng. 59(1), 8 (2015)

    Article  Google Scholar 

  31. Tzou, D.Y.: An engineering assessment to the relaxation time in thermal wave propagation. Int. J. Heat Mass Transf. 36(7), 1845–1851 (1993). doi:10.1016/s0017-9310(05)80171-1

    Article  MATH  Google Scholar 

  32. Gembarovic, J., Majernik, V.: Non-Fourier propagation of heat pulses in finite medium. Int. J. Heat Mass Transf. 31(5), 1073–1080 (1988)

    Article  MATH  Google Scholar 

  33. Poletkin, K.V., Gurzadyan, G.G., Shang, J., Kulish, V.: Ultrafast heat transfer on nanoscale in thin gold films. Appl. Phys. B 107, 137–143 (2012)

    Article  ADS  Google Scholar 

  34. Sieniutycz, S.: The variational principles of classical type for non-coupled non-stationary irreversible transport processes with convective motion and relaxation, S. Sieniutycz. Int. J. Heat Mass Transf. 20(11), 1221–1231 (1977)

    Article  ADS  MATH  Google Scholar 

  35. Majumdar, A.: Microscale heat conduction in lelectnc thin films. J. Heat Transf. 115, 7 (1993)

    Article  Google Scholar 

  36. Matsunaga, R. H., dos Santos, I.: Measurement of the thermal relaxation time in agar-gelled water. In: Engineering in Medicine and Biology Society (EMBC), 2012 Annual International Conference of the IEEE. IEEE, pp. 5722–5725 (2012)

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

    Article  ADS  MATH  Google Scholar 

  38. Green, A.E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2, 1–7 (1972)

    Article  MATH  Google Scholar 

  39. Hetnarski, R.B., Ignaczak, J.: Solution-like waves in a low-temperature nonlinear thermoelastic solid. Int. J. Eng. Sci. 34, 1767–1787 (1996)

    Article  MATH  Google Scholar 

  40. Green, A.E., Nagdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ivanova, E.A.: Derivation of theory of thermoviscoelasticity by means of two-component medium. Acta Mech. 215(1–4), 261–286 (2010)

    Article  MATH  Google Scholar 

  42. Ivanova, E.A.: On one model of generalised continuum and its thermodynamical interpretation. In: Altenbach, H., Maugin, G.A., Erofeev, V. (eds.) Mechanics of Generalized Continua, pp. 151–174. Springer, Berlin (2011)

    Chapter  Google Scholar 

  43. Ivanova, E.A.: Derivation of theory of thermoviscoelasticity by means of two-component Cosserat continuum. Tech. Mech. 32(2–5), 273–286 (2012)

    Google Scholar 

  44. Ivanova, E.A.: Description of mechanism of thermal conduction and internal damping by means of two component Cosserat continuum. Acta Mech. 225(3), 757–795 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Niu, T., Dai, W.: A hyperbolic two-step model-based finite-difference method for studying thermal deformation in a 3-D thin film exposed to ultrashort pulsed lasers. Numer. Heat Transf. Part A Appl. 53(12), 1294–1320 (2008)

    Article  ADS  Google Scholar 

  46. Qin, Y.: Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, vol. 184. Springer, Berlin (2008)

    MATH  Google Scholar 

  47. Mondal, S., Mallik, S. H., Kanoria, M.: Fractional order two–temperature dual–phase–lag thermoelasticity with variable thermal conductivity. Int. Sch. Res. Not. vol. 2014, Article ID 646049, 13 pages, (2014). doi:10.1155/2014/646049

  48. Babenkov, M.B.: Analysis of dispersion relations of a coupled thermoelasticity problem with regard to heat flux relaxation. J. Appl. Mech, Tech. Phys. 52(6), 941–949 (2011)

    Article  ADS  MATH  Google Scholar 

  49. Babenkov, M.B.: Propagation of harmonic perturbations in a thermoelastic medium with heat relaxation. J. Appl. Mech. Tech. Phys. 54(2), 277–286 (2013)

    Article  ADS  MATH  Google Scholar 

  50. Nowacki, W.: Dyn. Probl. Thermoelast. Springer, Berlin (1975)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Theoretical Mechanics, Peter the Great Saint-Petersburg Polytechnic University, Polytechnicheskaya, 29, St. Petersburg, Russia, 195251

    Evgeniy Yu. Vitokhin & Elena A. Ivanova

  2. Institute for Problems in Mechanical Engineering of Russian Academy of Sciences, Bolshoy pr. V.O., 61, St. Petersburg, Russia, 199178

    Elena A. Ivanova

Authors
  1. Evgeniy Yu. Vitokhin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elena A. Ivanova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Evgeniy Yu. Vitokhin.

Additional information

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vitokhin, E.Y., Ivanova, E.A. Dispersion relations for the hyperbolic thermal conductivity, thermoelasticity and thermoviscoelasticity. Continuum Mech. Thermodyn. 29, 1219–1240 (2017). https://doi.org/10.1007/s00161-017-0574-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-017-0574-x

Keywords

Search

Navigation