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Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
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Deformation waves in microstructured solids and dimensionless parameters; pp. 109–115
PDF | doi: 10.3176/proc.2013.2.04

Jüri Engelbrecht, Tanel Peets, Kert Tamm, Andrus Salupere

On the basis of the Mindlin-type micromorphic theory for wave motion in microstructured solids the 1D governing equations and corresponding dispersion relations are derived. The leading physical dimensionless parameters are established and their importance for describing dispersion effects is discussed. The general discussion reveals the role of both geometrical and physical dimensionless parameters in mechanics of microstructured materials.


  1. Mindlin, R. D. Microstructure in linear elasticity. Arch. Ration. Mech. An., 1964, 16(1), 51–78.

  2. Askes, H. and Metrikine, A. V. Higher-order continua derived from discrete media: continualisation aspects and boundary conditions. Int. J. Solids Struct., 2005, 42(1), 187–202.

  3. Engelbrecht, J., Berezovski, A., Pastrone, F., and Braun, M. Waves in microstructured materials and dispersion. Philos. Mag., 2005, 85(33–35), 4127–4141.

  4. Polyzos, D. and Fotiadis, D. I. Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models. Int. J. Solids. Struct., 2012, 49(3–4), 470–480.

  5. Askes, H. and Aifantis, E. C. Gradient elasticity theories in statics and dynamics – a unification of approaches. Int. J. Fracture, 2006, 139(2), 297–304.

  6. Askes, H., Metrikine, A. V., Pichugin, A. V., and Bennett, T. Four simplified gradient elasticity models for the simulation of dispersive wave propagation. Philos. Mag., 2008, 88(28), 3415–3443.

  7. Berezovski, A., Engelbrecht, J., and Berezovski, M. Waves in microstructured solids: a unified viewpoint of modeling. Acta Mech., 2011, 220(1–4), 349–363.

  8. Huang, G. L. and Sun, C. T. A higher-order continuum model for elastic media with multiphased microstructure. Mech. Adv. Mater. Struct., 2008, 15(8), 550–557.

  9. 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, 46(21), 3751–3759.

10. Barenblatt, G. I. Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press, 1996.

11. Berezovski, A., Engelbrecht, J., and Maugin, G. A. Thermoelasticity with dual internal variables. J. Therm. Stresses, 2011, 34(5–6), 413–430.

12. Christiansen, P. L., Muto, V., and Rionero, S. Solitary wave solutions to a system of Boussinesq-like equations. Chaos Soliton Fract., 1992, 2(1), 45–50.

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

14. Peets, T., Randrüüt, M., and Engelbrecht, J. On modelling dispersion in microstructured solids. Wave Motion, 2008, 45(4), 471–480.

15. Salupere, A., Tamm, K., and Engelbrecht, J. Numerical simulation of interaction of solitary deformation waves in microstructured solids. Int. J. Nonlinear Mech., 2008, 43(3), 201–208.

16. Tamm, K. Wave Propagation and Interaction in Mindlin-type Microstructured Solids: Numerical Simulation. Tallinn University of Technology, 2011.

17. Peets, T. Dispersion Analysis of Wave Motion in Microstructured Solids. Tallinn University of Technology, 2011.

18. Peets, T. and Tamm, K. Dispersion analysis of wave motion in microstructured solids. In IUTAM Symposium on Recent Advances of Acoustic Waves in Solids (Tsung-Tsong Wu and Chien-Ching Ma, eds). Springer, Berlin, 2010, 349–354.

19. Thomas, C. N., Papargyri-Beskou, S., and Mylonakis, G. Wave dispersion in dry granular materials by the distinct element method. Soil Dyn. Earthq. Eng., 2009, 29(5), 888–897.

20. Exner, H. E. Stereology and 3D microscopy: useful alternatives or competitors in the quantitative analysis of microstructures? Image Anal. Stereol., 2004, 23(2), 73–82.

21. Shearing, P. R., Howard, L. E., Jørgensen, P. S., Brandon, N. P., and Harris, S. J. Characterization of the 3-dimensional microstructure of a graphite negative electrode from a Li-ion battery. Electrochem. Commun., 2010, 12(3), 374–377.

22. Mishnaevsky, L. jr. and Qing, H. Micromechanical modelling of mechanical behaviour and strength of wood: state-of-the-art review. Comp. Mater. Sci., 2008, 44(2), 363–370.

23. Zhu, H. X., Hobdell, J. R., and Windle, A. H. Effects of cell irregularity on the elastic properties of 2D Voronoi honeycombs. J. Mech. Phys. Solids, 2001, 49(4), 857–870.

24. De Bellis, A. C. Computer Modelling of Sintering in Ceramics. University of Pittsburgh, 2002.

25. Agrawal, K. C. and Mittal, R. K. Influence of microstructure on mechanical properties of snow. Science, 1995, 45(2), 93–105.

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

27. Giovine, P. and Oliveri, F. Dynamics and wave propagation in dilatant granular materials. Meccanica, 1995, 30(4), 341–357.


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