Advertisement

Continuum Mechanics and Thermodynamics

, Volume 29, Issue 6, pp 1219–1240 | Cite as

Dispersion relations for the hyperbolic thermal conductivity, thermoelasticity and thermoviscoelasticity

  • Evgeniy Yu. VitokhinEmail author
  • Elena A. Ivanova
Original Article
First Online:

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.

Keywords

Hyperbolic thermoelasticity Hyperbolic thermoviscoelasticity Maxwell–Cattaneo law Heat flux relaxation Dispersion relations 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Pop, E., Sinha, S., Kenneth, E.: Goodson heat generation and transport in nanometer-scale transistors. Proc. IEEE 94(8), 1587–1601 (2006)CrossRefGoogle Scholar
  2. 2.
    Jou, D., Casas-Vazquez, J., Lebon, G.: Extended Irreversible Thermodynamics. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  3. 3.
    Chandrasekharaiah, D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 705–729 (1998)ADSCrossRefGoogle Scholar
  4. 4.
    Tzou, D.Y.: Macro-to-Microscale Heat Transfer. The Lagging Behaviour. Taylor and Francis, New York (1997)Google Scholar
  5. 5.
    Shashkov, A. G., Bubnov, V. A., Yanovski, S. Y.: Wave phenomena of heat conductivity: system and structural approach. (1993).(in Russian)Google Scholar
  6. 6.
    Wang, C.C.: The principle of fading memory. Arch. Ration. Mech. Anal. 18(5), 343–366 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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. 8.
    Qiu, T.Q., Tien, C.L.: Short-pulse laser heating on metals. Int. J. Heat Mass Transf. 35(3), 719–726 (1992)ADSCrossRefGoogle Scholar
  9. 9.
    Sobolev, S.L.: Transport processes and traveling waves in systems with local nonequilibrium. Sov. Phys. Uspekhi 34(3), 217 (1991)ADSCrossRefGoogle Scholar
  10. 10.
    Cattaneo, C.: A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Compte Rendus 247, 431–433 (1958)zbMATHGoogle Scholar
  11. 11.
    Vernotte, P.: Les paradoxes de la theorie continue de lequation de la chaleur. CR Acad. Sci. 246(22), 3154–3155 (1958)zbMATHGoogle Scholar
  12. 12.
    Lykov, A. V.: Theory of heat conduction. Vysshaya Shkola, Moscow. (1967): 599. (in russian)Google Scholar
  13. 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 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Peshkov, V.: Second sound in helium II. J. Phys. 8, 381 (1944)Google Scholar
  15. 15.
    Liu, Y., Mandelis, A.: Laser optical and photothermal thermometry of solids and thin films. Exp. Methods Phys. Sci. 42, 297–336 (2009)CrossRefGoogle Scholar
  16. 16.
    Magunov, A.N.: Laser thermometry of solids: state of the art and problems. Meas. Tech. 45(2), 173–181 (2002)CrossRefGoogle Scholar
  17. 17.
    Magunov, A.N.: Laser Thermometry of Solids. Cambridge International Science Publishing, Cambridge (2003)Google Scholar
  18. 18.
    Wang, X.: Experimental Micro/nanoscale Thermal Transport. Wiley, Hoboken (2012)CrossRefGoogle Scholar
  19. 19.
    Krilovich, V.I., Bil, G.N., Ivakin, E.V., Rubanov, A.C.: Experimental determination heat velocity. NASB 1, 129–134 (2000). (in Russian)Google Scholar
  20. 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. 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)ADSCrossRefGoogle Scholar
  22. 22.
    Xu, F., Tianjian, Lu: Introduction to Skin Biothermomechanics and Thermal Pain, vol. 7. Science Press, New York (2011)CrossRefGoogle Scholar
  23. 23.
    Grassmann, A., Peters, F.: Experimental investigation of heat conduction in wet sand. Heat Mass Transf. 35(4), 289–294 (1999)ADSCrossRefGoogle Scholar
  24. 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)ADSCrossRefGoogle Scholar
  25. 25.
    Kaminski, W.: Hyperbolic heat conduction equation for materials with a nonhomogeneous inner structure. J. Heat Transf. 112(3), 555–560 (1990)CrossRefGoogle Scholar
  26. 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)CrossRefGoogle Scholar
  27. 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)CrossRefGoogle Scholar
  28. 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. 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)ADSCrossRefGoogle Scholar
  30. 30.
    Szekeres, A., Fekete, B.: Continuummechanics-heat conduction-cognition. Period. Polytech. Eng. Mech. Eng. 59(1), 8 (2015)CrossRefGoogle Scholar
  31. 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 CrossRefzbMATHGoogle Scholar
  32. 32.
    Gembarovic, J., Majernik, V.: Non-Fourier propagation of heat pulses in finite medium. Int. J. Heat Mass Transf. 31(5), 1073–1080 (1988)CrossRefzbMATHGoogle Scholar
  33. 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)ADSCrossRefGoogle Scholar
  34. 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)ADSCrossRefzbMATHGoogle Scholar
  35. 35.
    Majumdar, A.: Microscale heat conduction in lelectnc thin films. J. Heat Transf. 115, 7 (1993)CrossRefGoogle Scholar
  36. 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)Google Scholar
  37. 37.
    Lord, H., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967)ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    Green, A.E., Lindsay, K.A.: Thermoelasticity. J. Elast. 2, 1–7 (1972)CrossRefzbMATHGoogle Scholar
  39. 39.
    Hetnarski, R.B., Ignaczak, J.: Solution-like waves in a low-temperature nonlinear thermoelastic solid. Int. J. Eng. Sci. 34, 1767–1787 (1996)CrossRefzbMATHGoogle Scholar
  40. 40.
    Green, A.E., Nagdi, P.M.: Thermoelasticity without energy dissipation. J. Elast. 31, 189–208 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Ivanova, E.A.: Derivation of theory of thermoviscoelasticity by means of two-component medium. Acta Mech. 215(1–4), 261–286 (2010)CrossRefzbMATHGoogle Scholar
  42. 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)CrossRefGoogle Scholar
  43. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 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)ADSCrossRefGoogle Scholar
  46. 46.
    Qin, Y.: Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors, vol. 184. Springer, Berlin (2008)zbMATHGoogle Scholar
  47. 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. 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)ADSCrossRefzbMATHGoogle Scholar
  49. 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)ADSCrossRefzbMATHGoogle Scholar
  50. 50.
    Nowacki, W.: Dyn. Probl. Thermoelast. Springer, Berlin (1975)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Theoretical MechanicsPeter the Great Saint-Petersburg Polytechnic UniversitySt. PetersburgRussia
  2. 2.Institute for Problems in Mechanical Engineering of Russian Academy of SciencesSt. PetersburgRussia

Personalised recommendations

Cite article

Buy options