eesti teaduste
akadeemia kirjastus
Estonian Journal of Engineering

On modelling of wave propagation in microstructured solids; pp. 171–182

Full article in PDF format | doi: 10.3176/eng.2013.3.01

Jüri Engelbrecht, Arkadi Berezovski


The basic concepts for modelling wave propagation in solids with microstructure are described. It is shown that the Green method, based on postulating the potential energy function, has certain advantages compared with the widely used Cauchy method, which postulates directly the stress-strain relations. Simple examples demonstrate how the Green method together with internal variables permits to determine the microstress and the interactive force between the constituents of solids. The structure of governing equations and possible physical effects captured by such modelling are described. The microstress and interactive force lead to the dispersion of waves at the macrolevel.


  1. Eringen, A. C. and Suhubi, E. S. Nonlinear Theory of Continuous Media. McGraw Hill, New York, 1962.

  2. Bland, D. R. Nonlinear Dynamics Elasticity. Blaisdell, Waltham MA, 1969.

  3. Haupt, P. Continuum Mechanics and Theory of Materials. Springer, Berlin, 2002.

  4. Eringen, A. C. and Suhubi, E. S. Nonlinear theory of simple microelastic solids I & II. Int. J. Eng. Sci., 1964, 2, 189–203, 389–404.

  5. Mindlin, R. D. Microstructure in linear elasticity. Arch. Rat. Mech. Anal., 1964, 16, 51–78.

  6. Capriz, G. Continua with Microstructure. Springer, New York, 1989.

  7. Maugin, G. A. Nonlinear Waves in Elastic Crystals. Oxford University Press, Oxford, 1999.

  8. Eringen, A. C. Microcontinuum Field Theories. I Foundations and Solids}. Springer, New York, 1999.

  9. Maugin, G. A. A historical perspective of generalized continuum mechanics. In Mechanics of Generalized Continua (Altenbach, H. et al., eds). Springer, Berlin, 2011, 3–19.

10. Berezovski, A., Engelbrecht, J. and Maugin, G. A. Generalized thermomechanics with internal variables. Arch. Appl. Mech., 2011, 81, 229–240.

11. Engelbrecht, J. and Berezovski, A. Internal structures and internal variables in solids. J. Mech. Mater. Struct., 2012, 7, 983–996.

12. Kestin, J. Internal variables in the local-equilibrium approximation. J. Non-Equilibr. Thermodyn., 1993, 18, 360–379.

13. Maugin, G. A. and Muschik, W. Thermodynamics with internal variables I: General concepts. J. Non-Equilib. Thermodyn., 1994, 19, 217–249.

14. Ván, P., Berezovski, A. and Engelbrecht, J. Internal variables and dynamic degrees of freedom. J. Non-Equilib. Thermodyn., 2008, 33, 235–254.

15. Maugin, G. A. Material Inhomogeneities in Elasticity. Chapman and Hall, London, 1993.

16. Askes, H. and Metrikine, A. V. One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part I: Generic formulation. Eur. J. Mech. A/Solids, 2002, 21, 555–572.

17. Papargyri-Beskou, S., Polyzos, D. and Beskos, D. E. Wave dispersion in gradient elastic solids and structures: A unified treatment. Int. J. Solids Struct., 2009, 16, 3751–3759.

18. Berezovski, A., Engelbrecht, J. and Berezovski, M. Waves in microstructured solids: a unified view point of modelling. Acta Mech., 2011, 220, 349–363.

19. Engelbrecht, J., Pastrone, F., Braun, M. and Berezovski, A. Hierarchies of waves in nonclassical materials. In Universality in Nonclassical Nonlinearity (Delsanto, P.-P., ed.). Springer, New York, 2007, 29–47.

20. Berezovski, A., Engelbrecht, J. and Peets, T. Multiscale modeling of microstructured solids. Mech. Res. Comm., 2010, 37, 531–534.

21. Berezovski, A., Berezovski, M. and Engelbrecht, J. Two-scale microstructure dynamics. J. Multiscale Modelling, 2011, 3, 177–188.

22. Berezovski, A., Engelbrecht, J. and Maugin, G. A. Thermoelasticity with dual internal variables. J. Thermal Stresses, 2011, 34, 413–430.

23. Berezovski, A. and Engelbrecht, J. Waves in microstructured solids: dispersion and thermal effects. In Proc. 23rd International Congress of Theoretical and Applied Mechanics (Bai, Y., Wang, J. and Fang, D., eds). Beijing, China, 2012, SM07-005.

24. Engelbrecht, J., Berezovski, A., Pastrone, F. and Braun, M. Waves in microstructured materials and dispersion. Phil. Mag., 2005, 85, 4127–4141.

25. Whitham, G. B. Linear and Nonlinear Waves. J. Wiley, New York, 1974.

26. Berezovski, M., Berezovski, A. and Engelbrecht, J. Waves in materials with micro\-structure: numerical simulation. Proc. Estonian Acad. Sci., 2010, 59, 99–107.

27. Berezovski, A., Engelbrecht, J., Salupere, A., Tamm, K., Peets, T. and Berezovski, M. Dispersive waves in microstructured solids. Int. J. Solids Struct., 2013, 50, 1981–1990.

28. Peets, T., Randrüüt, M. and Engelbrecht, J. On modelling dispersion in microstructured solids. Wave Motion, 2008, 45, 471–480.

29. Engelbrecht, J., Peets, T., Tamm, K. and Salupere, A. Deformation waves in microstructured solids and dimensionless parameters. Proc. Estonian Acad. Sci., 2013, 62, 109–115.

30. Christov, C., Maugin, G. A. and Porubov, A. On Boussinesq’s paradigm in nonlinear wave motion. C.R.Mécanique, 2007, 335, 521–535.

31. Berezovski, A., Engelbrecht, J. and Berezovski, M. Dispersive wave equations for solids with microstructure. In Vibration Problems – ICOVP 2011: The 10th International Conference on Vibration Problems} (Naprstek, J. et al., eds). Springer, 2011, 699–705.

32. Engelbrecht, J., Berezovski, A. and Salupere, A. Nonlinear deformation waves in solids and dispersion. Wave Motion, 2007, 44, 493–500.

33. Engelbrecht, J., Salupere, A. and Tamm, K. Waves in microstructured solids and the Boussinesq paradigm. Wave Motion, 2011, 48, 717–726.

34. Randrüüt, M. and Braun, M. On one-dimensional waves in microstructured solids. Wave Motion, 2010, 47, 217–230.

35. Janno, J. and Engelbrecht, J. Microstructured Materials: Inverse Problems. Springer, Berlin, 2011.

Back to Issue