Deformation Waves in Microstructured Materials: Theory and Numerics

  • Jüri EngelbrechtEmail author
  • Arkadi Berezovski
  • Mihhail Berezovski
Conference paper
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 26)


A linear model of the microstructured continuum based on Mindlin theory is adopted which can be represented in the framework of the internal variable theory. Fully coupled systems of equations for macro-motion and microstructure evolution are represented in the form of conservation laws. A modification of wave propagation algorithm is used for numerical calculations. Results of direct numerical simulations of wave propagation in periodic medium are compared with similar results for the continuous media with the modelled microstructure. It is shown that the proper choice of material constants should be made to match the results obtained by both approaches


Solitary Wave Direct Numerical Simulation Internal Variable Elastic Wave Propagation Dissipation Inequality 
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.
    Maugin, G.A.: Nonlinear Waves in Elastic Crystals, Oxford University Press (1999).Google Scholar
  2. 2.
    Sun, C.T., Achenbach, J.D., Herrmann, G.: Continuum theory for a laminated medium. J. Appl. Mech. 35 467–475 (1968).Google Scholar
  3. 3.
    Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials, Elsevier, Amsterdam (1993).Google Scholar
  4. 4.
    Mindlin, R.D.: Microstructure in linear elasticity. Arch. Rat. Mech. Anal 16 51-78 (1964).CrossRefGoogle Scholar
  5. 5.
    Eringen, A.C., Suhubi, E.S.: Nonlinear theory of micro-elastic solids II. Int. J. Eng. Sci. 2 189-203 (1964).CrossRefGoogle Scholar
  6. 6.
    Maugin, G.A.: On the thermomechanics of continuous media with diffusion and/or weak nonlocality. Arch. Appl. Mech. 75 723-738 (2006).CrossRefGoogle Scholar
  7. 7.
    Ván, P., Berezovski, A., Engelbrecht, J.: Internal variables and dynamic degrees of freedom. J. Non-Equilib. Thermodyn. 33 235-254 (2008).CrossRefGoogle Scholar
  8. 8.
    Nowacki, W.: Thermoelasticity. Pergamon/PWN, Oxford/Warszawa (1962).Google Scholar
  9. 9.
    Maugin, G.A.: Internal variables and dissipative structures. J. Non-Equilib. Thermodyn. 15 173-192 (1990).CrossRefGoogle Scholar
  10. 10.
    Engelbrecht, J., Berezovski, A., Pastrone, F., Braun, M.: Waves in microstructured materials and dispersion. Phil. Mag. 85 4127-4141 (2005).CrossRefGoogle Scholar
  11. 11.
    Engelbrecht, J., Cermelli, P., Pastrone, F.: Wave hierarchy in microstructured solids. In: Maugin, G.A. (ed.) Geometry, Continua and Microstructure, pp. 99-111. Hermann Publ., Paris (1999).Google Scholar
  12. 12.
    Effective Computational Methods for Wave Propagation. Kampanis, N. A., Dougalis, V.A., Ekaterinaris, J.A. (eds.), Chapman & Hall/CRC, Boca Raton (2008).Google Scholar
  13. 13.
    LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002).CrossRefGoogle Scholar
  14. 14.
    Berezovski, A., Engelbrecht, J., Maugin, G.A.: Numerical Simulation of Waves and Fronts in Inhomogeneous Solids. World Scientific, Singapore (2008).CrossRefGoogle Scholar
  15. 15.
    Lakes, R.S.: Experimental method for study of Cosserat elastic solids and other generalized continua. In: Mühlhaus, H.-B. (ed.) Continuum Models for Materials with Microstructure, pp.1-22. Wiley, New York (1995).Google Scholar
  16. 16.
    Berezovski, A., Berezovski, M. and Engelbrecht, J.: Numerical simulation of nonlinear elastic wave propagation in piecewise homogeneous media. Mater. Sci. Engng. A 418 364-369 (2006).CrossRefGoogle Scholar
  17. 17.
    Janno, J., Engelbrecht J.: An inverse solitary wave problem related to microstructural materials. Inverse Problems. 21 2019-2034 (2005).CrossRefGoogle Scholar
  18. 18.
    Engelbrecht, J. Berezovski, A., Salupere, A.: Nonlinear deformation waves in solids and dispersion. Wave Motion. 44 493-500 (2007).CrossRefGoogle Scholar
  19. 19.
    Pastrone, F., Cermelli, P. and Porubov, A.: Nonlinear waves in 1-D solids with microstructure. Mater. Phys. Mech. 7 9-16 (2004).Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

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

Personalised recommendations