Acta Mechanica

, Volume 226, Issue 3, pp 917–930 | Cite as

Multi-scale model of steady-wave shock in medium with relaxation

  • D. A. Indeitsev
  • Yu. I. Meshcheryakov
  • A. Yu. Kuchmin
  • D. S. VavilovEmail author


The propagation of a steady wavefront in a medium with three levels of plastic deformation is considered: (a) dislocation, (b) meso-scale and (c) macro-scale. To take into account collective mechanisms of microplasticity, a relaxation term describing the momentum exchange between meso- and macro scales is incorporated into a dislocation-based constitutive law. This leads to a nonlinear second-order differential equation. Analytical and numerical analyses of the equation are performed. Using as example D16 aluminum alloy, we determine the parameters that provide a satisfactory correspondence between calculated and experimental profiles of particle velocity.


Particle Velocity Impact Velocity Shock Front Heterogeneous Medium Shock Compression 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Johnson J.N., Jones O.E., Michaels T.E.: Dislocation dynamics and single-crystal constitutive relation: shock-wave propagation and precursor decay. J. Appl. Phys. 41, 2230–2239 (1970)Google Scholar
  2. 2.
    Panin V.E., Grinyaev Y.V., Elsukova T.F., Ivanchin A.G.: Structural levels of deformation in solids. Russ. Phys. J. 25, 479–497 (1982)Google Scholar
  3. 3.
    Eshby M.F.: The deformation of plastically non-homogeneous material. Philos. Mag. 21, 399–422 (1970)CrossRefGoogle Scholar
  4. 4.
    Kröner, E.: Initial studies of plasticity theory based upon statistical mechanics. Colloquium on Elastic Behavior of Solids. Battelle Memorial Inst. Columbus, Ohio (1968)Google Scholar
  5. 5.
    Johnson J.N., Barker L.M.: Dislocation dynamics and steady plastic wave profiles in 6061-T6 aluminum. J. Appl. Phys. 40, 4321–4335 (1969)CrossRefGoogle Scholar
  6. 6.
    Prieto F.E., Renero C.: Steady shock profile in solids. J. Appl. Phys. 44, 401–4013 (1973)CrossRefGoogle Scholar
  7. 7.
    Swegle J.W., Grady D.E.: Shock viscosity and the prediction of shock-wave rise times. J. Appl. Phys. 58, 692–701 (1985)CrossRefGoogle Scholar
  8. 8.
    Grady D.E.: Steady rise-time and spall measurement on Uranium (3-16 GPa). In: Murr, L.E., Staudhammer, K.R., Meyers, M.A. (eds.) Metallurgical Applications of Shock-Wave and High-Strain-Rate Phenomena” (Explomet-85), pp. 763–780. Marcel Dekker, NY (1986)Google Scholar
  9. 9.
    Asay J.R., Chhabildas L.C. : Determination of the shear strength of shock compressed 6061-T6 aluminum. In: Meyers, M.A., Murr, L.E. (eds.) Shock Waves and High-Strain Rate Phenomena in Metals, pp. 417–428. Plenum Publishing Co, NY (1981)CrossRefGoogle Scholar
  10. 10.
    Grady D.E., Kipp M.E.: The growth of unstable thermoplastic shear with application to steady-wave shock compression of solids. J. Mech. Phys. Solids 35, 95–119 (1987)CrossRefzbMATHGoogle Scholar
  11. 11.
    Yano K., Horie Y-Y.: Discrete element modeling of shock compression of polycrystalline copper. Phys. Rev. B. 59, 103–122 (1999)CrossRefGoogle Scholar
  12. 12.
    Meshcheryakov, Yu.I.: In: Shock Compression of Condensed Matter-1999. Furnish, M.D., Chhabildas, L.C., Nixon, R.S. (eds) AIP Conference Proceedings-505, pp. 1065–1070. Melville, NY (2000)Google Scholar
  13. 13.
    Meshcheryakov Yu.I., Divakov A.K., Zhigacheva N.I., Makarevich I.P., Barakhtin B.K.: Dynamic structures in shock-loaded copper. Phys. Rev. B. 78, 64301–64316 (2008)CrossRefGoogle Scholar
  14. 14.
    Mesheryakov, Yu.I.: On Evolutionary and Catastrophic Regimes of Mesomacro Energy Exchange in Dynamically Loaded Media, 6, pp. 765–768. Doklady RAN (2005)Google Scholar
  15. 15.
    Hinze, J.O.: Turbulence, pp. 680. Mc. Graw Hill Inc., NY (1959)Google Scholar
  16. 16.
    Indeitsev D.A., Naumov V.N., Semenov B.N.: Dynamic effects in materials of complex structure. Mech. Solids 42, 672–691 (2007)CrossRefGoogle Scholar
  17. 17.
    Taylor J.W.: Dislocation dynamics and dynamic yielding. J. Appl. Phys. 36, 3146–3155 (1965)CrossRefGoogle Scholar
  18. 18.
    Duvall G.E.: Propagation of plane shock waves in a stress-relaxing medium. In: Kolsky, H., Prager, W. (eds.) Stress Waves in Inelastic Solids, pp. 20–32. Springer, Berlin (1964)CrossRefGoogle Scholar
  19. 19.
    Johnston W.G., Gilman J.J.: J. Appl. Phys. 30, 129–141 (1959)CrossRefGoogle Scholar
  20. 20.
    Kumar A., Hauser F.E., Dorn I.E.: Viscous drag of dislocation in aluminum at high strain rates. J. Appl. Phys. 38, 1863–1871 (1967)CrossRefGoogle Scholar
  21. 21.
    Bland D.R.: On shock structure in a solid. J. Inst. Math. Appl. 1, 56–75 (1965)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Gilman J.J.: Dislocation dynamics and the response of materials to impact. Appl. Mech. Rev. 21, 767–783 (1968)Google Scholar
  23. 23.
    Meshcheryakov Yu.I.: Meso-macro energy exchange in shock-deformed and fractured solids. In: Davison, L., Horie, Y-Y. (eds.) High Pressure Shock Compression of Solids-VI, pp. 169–213. Springer, NY (2003)CrossRefGoogle Scholar
  24. 24.
    Meshcheryakov, Yu.I., Divakov, A.K., Barakhtin, B.K., Zhigacheva, N.I.: Interscale momentum exchange in dynamically deformed heterogeneous medium. In: Elert, M.L., Borg, J.P., Jordan, J.L., Volger, T.J. Melville, N.Y. (eds) Shock Compression of Condensed Matter-2011 (2011) AIP Conference Proceedings-1426, pp. 1109–1112 (2012)Google Scholar
  25. 25.
    Louis K., Maab P., Rieder A.: Wavelet Theory and Applications. Wiley, New York (1997)Google Scholar
  26. 26.
    Mallat, S.: A Wavelet Tour of Signal Processing. Wavelet Analysis & Its Application, 2nd edn. Academic Press, New York (1998)Google Scholar
  27. 27.
    Tabak D., Kuo B.C.: Application of mathematical programming in the design of optimal control systems. Int. J. Control 10, 548–552 (1969)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Gill P.E., Murray W., Wright M.A.: Practical Optimization. Academic Press, New York (1982)Google Scholar

Copyright information

© Springer-Verlag Wien 2014

Authors and Affiliations

  • D. A. Indeitsev
    • 1
  • Yu. I. Meshcheryakov
    • 2
  • A. Yu. Kuchmin
    • 2
  • D. S. Vavilov
    • 2
    Email author
  1. 1.Saint-Petersburg State Polytechnic UniversitySaint-PetersburgRussia
  2. 2.Institute of Problems of Mechanical Engineering RASSaint-PetersburgRussia

Personalised recommendations