Acta Mechanica

, Volume 220, Issue 1–4, pp 349–363 | Cite as

Waves in microstructured solids: a unified viewpoint of modeling

  • Arkadi BerezovskiEmail author
  • Jüri Engelbrecht
  • Mihhail Berezovski
Review Article


The basic ideas for describing the dispersive wave motion in microstructured solids are discussed in the one-dimensional setting because then the differences between various microstructure models are clearly visible. An overview of models demonstrates a variety of approaches, but the consistent structure of the theory is best considered from the unified viewpoint of internal variables. It is shown that the unification of microstructure models can be achieved using the concept of dual internal variables.


Dispersion Curve Internal Variable Linear Momentum Free Energy Function Microstructure Model 
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.
    Brillouin L.: Wave Propagation and Group Velocity. Academic Press, New York (1960)zbMATHGoogle Scholar
  2. 2.
    Maugin G.A.: On some generalizations of Boussinesq and KdV systems. Proc. Estonian Acad. Sci. Phys. Mat. 44, 40–55 (1995)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Santosa F., Symes W.W.: A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math. 51, 984–1005 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fish J., Chen W., Nagai G.: Non-local dispersive model for wave propagation in heterogeneous media: one-dimensional case. Int. J. Numer. Meth. Engng. 54, 331–346 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Askes H., Metrikine A.V.: One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure part 1: generic formulation. Eur. J. Mech. A/Solids 21, 555–572 (2002)zbMATHCrossRefGoogle Scholar
  6. 6.
    Erofeyev V.I.: Wave Processes in Solids with Microstructure. World Scientific, Singapore (2003)zbMATHCrossRefGoogle Scholar
  7. 7.
    Love A.E.H.: Mathematical Theory of Elasticity. Dover, New York (1944)zbMATHGoogle Scholar
  8. 8.
    Graff K.F.: Wave Motion in Elastic Solids. Clarendon Press, Oxford (1975)zbMATHGoogle Scholar
  9. 9.
    Maugin G.A.: Nonlinear Waves in Elastic Crystals. Oxford University Press, Oxford (1999)zbMATHGoogle Scholar
  10. 10.
    Wang Z.-P., Sun C.T.: Modeling micro-inertia in heterogeneous materials under dynamic loading. Wave Motion 36, 473–485 (2002)zbMATHCrossRefGoogle Scholar
  11. 11.
    Wang L.-L.: Foundations of Stress Waves. Elsevier, Amsterdam (2007)Google Scholar
  12. 12.
    Metrikine A.V.: On causality of the gradient elasticity models. J. Sound Vibr. 297, 727–742 (2006)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Papargyri-Beskou S., Polyzos D., Beskos D.E.: Wave dispersion in gradient elastic solids and structures: a unified treatment. Int. J. Solids Struct. 46, 3751–3759 (2009)zbMATHCrossRefGoogle Scholar
  14. 14.
    Engelbrecht J., Pastrone F.: Waves in microstructured solids with nonlinearities in microscale. Proc. Estonian Acad. Sci. Phys. Mat. 52, 12–20 (2003)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Mindlin R.D.: Microstructure in linear elasticity. Arch. Rat. Mech. Anal. 16, 51–78 (1964)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Engelbrecht J., Berezovski A., Pastrone F., Braun M.: Waves in microstructured materials and dispersion. Phil. Mag. 85, 4127–4141 (2005)CrossRefGoogle Scholar
  17. 17.
    Horstemeyer M.F., Bammann D.J.: Historical review of internal state variable theory for inelasticity. Int. J. Plasticity 26, 1310–1334 (2010)CrossRefGoogle Scholar
  18. 18.
    Coleman B.D., Gurtin M.E.: Thermodynamics with internal state variables. J. Chem. Phys. 47, 597–613 (1967)CrossRefGoogle Scholar
  19. 19.
    Maugin G.A., Muschik W.: Thermodynamics with internal variables. J. Non-Equilib. Thermodyn. 19, 217–249 (1994)zbMATHCrossRefGoogle Scholar
  20. 20.
    Maugin G.A.: Internal variables and dissipative structures. J. Non-Equilib. Thermodyn. 15, 173–192 (1990)CrossRefGoogle Scholar
  21. 21.
    Maugin G.A.: On the thermomechanics of continuous media with diffusion and/or weak nonlocality. Arch. Appl. Mech. 75, 723–738 (2006)zbMATHCrossRefGoogle Scholar
  22. 22.
    Ván P., Berezovski A., Engelbrecht J.: Internal variables and dynamic degrees of freedom. J. Non-Equilib. Thermodyn. 33, 235–254 (2008)zbMATHCrossRefGoogle Scholar
  23. 23.
    Maugin G.A.: Material Inhomogeneities in Elasticity. Chapman and Hall, London (1993)zbMATHGoogle Scholar
  24. 24.
    Berezovski A., Engelbrecht J., Maugin G.A.: Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. World Scientific, Singapore (2008)zbMATHCrossRefGoogle Scholar
  25. 25.
    Rice J.R.: Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455 (1971)zbMATHCrossRefGoogle Scholar
  26. 26.
    Muschik W.: Aspects of Non-Equilibrium Thermodynamics. World Scientific, Singapore (1990)Google Scholar
  27. 27.
    Askes H., Aifantis E.C.: Gradient elasticity theories in statics and dynamics—a unification of approaches. Int. J. Fract. 139, 297–304 (2006)zbMATHCrossRefGoogle Scholar
  28. 28.
    Askes H., Metrikine A.V., Pichugin A.V., Bennett T.: Four simplified gradient elasticity models for the simulation of dispersive wave propagation. Phil. Mag. 88/28, 3415–3443 (2008)CrossRefGoogle Scholar
  29. 29.
    Engelbrecht J., Cermelli P., Pastrone F.: Wave hierarchy in microstructured solids. In: Maugin, G.A. (eds) Geometry, Continua and Microstructure, pp. 99–111. Hermann Publ., Paris (1999)Google Scholar
  30. 30.
    Berezovski A., Engelbrecht J., Maugin G.A.: One-dimensional microstructure dynamics. In: Ganghoffer, J.-F., Pastrone, F. (eds) Mechanics of Microstructured Solids: Cellular Materials, Fibre Reinforced Solids and Soft Tissues. Series: Lecture Notes in Applied and Computational Mechanics, pp. 21–28. Springer, Berlin (2009)Google Scholar
  31. 31.
    Berezovski A., Engelbrecht J., Maugin G.A.: Generalized thermomechanics with internal variables. Arch. Appl. Mech. 81, 229–240 (2011)CrossRefGoogle Scholar
  32. 32.
    Askes H., Metrikine A.V.: Higher-order continua derived from discrete media: continualisation aspects and boundary conditions. Int. J. Solids Struct. 42, 187–202 (2005)zbMATHCrossRefGoogle Scholar
  33. 33.
    Berezovski A., Engelbrecht J., Peets T.: Multiscale modelling of microstructured solids. Mech. Res. Commun. 37, 531–534 (2010)CrossRefGoogle Scholar
  34. 34.
    Pastrone F., Cermelli P., Porubov A.: Nonlinear waves in 1-D solids with microstructure. Mater. Phys. Mech. 7, 9–16 (2004)Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Arkadi Berezovski
    • 1
    Email author
  • Jüri Engelbrecht
    • 1
  • Mihhail Berezovski
    • 1
  1. 1.Centre for Nonlinear StudiesInstitute of Cybernetics at Tallinn University of TechnologyTallinnEstonia

Personalised recommendations