Advertisement

Continuum Mechanics and Thermodynamics

, Volume 18, Issue 7–8, pp 395–409 | Cite as

Weak shock waves in isotropic solids at finite temperatures up to the melting point

  • C. Currò
  • M. Sugiyama
  • H. Suzumura
  • G. ValentiEmail author
Original Article

Abstract

Propagation speeds and Rankine–Hugoniot relations for weak shock waves in isotropic solids are derived analytically in order to elucidate mechanical and thermal properties of the waves. In the analysis, we adopt a new continuum model for the solids, which takes into account explicitly microscopic thermal vibration of the constituent atoms. As the model is valid in a wide temperature range up to the melting point, we can discuss the relations at high temperatures even near the melting point. Typical numerical results are also shown and discussed as illustrations.

Keywords

Rankine–Hugoniot condition Weak shock waves Isotropic solids Thermal effects Entropy production 

PACS

46.05.+b 62.30.+d 65.40.-b 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    (1992). High-Pressure Shock Compression of Solids. Springer, Berlin Heidelberg New York Google Scholar
  2. 2.
    Graham R.A. (1992). Solids under High-Pressure Shock Compression, Mechanics, Physics and Chemistry. Springer, Berlin Heidelberg New York Google Scholar
  3. 3.
    (1993). Shock Waves in Materials Science. Springer, Tokyo Google Scholar
  4. 4.
    (1996). High-Pressure Shock Compression of Solids II. Springer, Berlin Heidelbeg New York zbMATHGoogle Scholar
  5. 5.
    (1997). High-Pressure Shock Compression of Solids III. Springer, Berlin Heidelberg New York Google Scholar
  6. 6.
    (1997). High-Pressure Shock Compression of Solids IV. Springer, Berlin Heidelberg New York Google Scholar
  7. 7.
    (2002). High-Pressure Shock Compression of Solids V. Springer, Berlin Heidelberg New York Google Scholar
  8. 8.
    (2002). High-Pressure Shock Compression of Solids VI. Springer, Berlin Heidelberg New York Google Scholar
  9. 9.
    (2003). High-Pressure Shock Compression of Solids VII. Springer, Berlin Heidelberg New York Google Scholar
  10. 10.
    Sugiyama M. (2003). Statistical-thermodynamic study of nonequilibrium phenomena in three-dimensional anharmonic crystal lattices. II. Continuum approximation of the basic equations. J. Phys. Soc. Jpn. 72: 1989–1994 CrossRefGoogle Scholar
  11. 11.
    Valenti G., Currò C. and Sugiyama M. (2004). Acceleration waves analyzed by a new continuum model of solids incorporating microscopic thermal vibrations. Continuum. Mech. Thermodyn. 16: 185–198 CrossRefGoogle Scholar
  12. 12.
    Ruggeri T. and Sugiyama M. (2005). Hyperbolicity convexity and shock waves in one-dimensional crystalline solids. J. Phys. A: Math. Gen. 38: 4337–4347 CrossRefMathSciNetGoogle Scholar
  13. 13.
    Sugiyama M. and Isogai T. (1996). Microscopic approach to shock waves in crystal solids. II. Rankine–Hugoniot relations. Jpn. J. Appl. Phys. 35: 3505–3517 CrossRefGoogle Scholar
  14. 14.
    Sugiyama M. and Goto K. (2003). Statistical-thermodynamic study of nonequilibrium phenomena in three-dimensional anharmonic crystal lattices. I . Microscopic basic equations. J. Phys. Soc. Jpn. 72: 545–550 CrossRefGoogle Scholar
  15. 15.
    Truesdell C. and Noll W. (1992). The non-linear field theories of mechanics. Springer, Berlin Heidelberg New York zbMATHGoogle Scholar
  16. 16.
    Valenti, G., Currò, C., Sugiyama, M.: Wave features for a new continuum model of isotropic solids. In: Proceedings WASCOM 2003, Villasimius 1–7 June 2003, pp. 547–554, Word Scientific, Singapore (2003)Google Scholar
  17. 17.
    Sugiyama M., Goto K., Takada K., Valenti G. and Currò C. (2003). Statistical-thermodynamic study of nonequilibrium phenomena in three-dimensional anharmonic crystal lattices: III. Linear waves. J. Phys. Soc. Jpn. 72: 3132–3141 CrossRefGoogle Scholar
  18. 18.
    Boillat, G.:Non linear hyperbolic fields and waves, in CIME course. In: Ruggeri, T. (ed.) Recent Mathematical Methods in Nonlinear Wave Propagation, Lecture Notes in Mathematics, vol. 1640, pp. 1–47. Springer, Berlin Heidelberg New York (1995)Google Scholar
  19. 19.
    Jeffrey A. (1976). Quasilinear hyperbolic systems and waves. Pitman, London zbMATHGoogle Scholar
  20. 20.
    Sugiyama M. and Suzumura H. (2005). Statistical-thermodynamic study of nonequilibrium phenomena in three-dimensional anharmonic crystal lattices. IV. Elastic constants and free energy of elastic solids. J. Phys. Soc. Jpn. 74: 631–637 CrossRefGoogle Scholar
  21. 21.
    Friedrichs K.O. and Lax P.D. (1971). Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci. USA 68: 1686–1688 CrossRefMathSciNetGoogle Scholar
  22. 22.
    Ruggeri T. and Strumia A. (1981). Main field and convex covariant density for quasi-linear hyperbolic systems. Relat. Fluid Dyn. Annal. Inst. H. Poincaré 34: 65–84 MathSciNetGoogle Scholar
  23. 23.
    Boillat G. and Ruggeri T. (1980). Symmetric form of nonlinear mechanics equations and entropy growth across a shock. Acta Mech. 35: 271–274 CrossRefGoogle Scholar
  24. 24.
    Dafermos, C.M.: Entropy and stability of classical solutions of hyperbolic systems of conservation laws. In: Ruggeri, T. (ed.) Recent Mathematical Methods in Nonlinear Wave Propagation. Lecture Notes in Mathematics, vol. 1640, pp. 48–102. Springer, Berlin Heidelberg New York (1995)Google Scholar
  25. 25.
    Boillat G. (1976). Sur une fonction croissante comme l’entropie et génératrice de chocs dans les systèmes hyperboliques. C.R. Acad. Sci. Paris 283A: 539–542 MathSciNetGoogle Scholar
  26. 26.
    Ruggeri T., Muracchini A. and Seccia L. (1994). Continuum approach to phonon gas and shape changes of second sound via shock waves theory. Il Nuovo Cimento 16: 15–44 CrossRefGoogle Scholar
  27. 27.
    Girifalco L.A. and Weizer V.G. (1959). Application of the Morse potential function to cubic metals. Phys. Rev. 114: 687–690 CrossRefGoogle Scholar
  28. 28.
    Torrens I.M. (1972). Interatomic Potentials. Academic, New York Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • C. Currò
    • 1
  • M. Sugiyama
    • 2
  • H. Suzumura
    • 2
  • G. Valenti
    • 1
    Email author
  1. 1.Department of MathematicsUniversity of MessinaMessinaItaly
  2. 2.Graduate School of EngineeringNagoya Institute of TechnologyNagoyaJapan

Personalised recommendations