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Proceedings of the Estonian Academy of Sciences. Physics. Mathematics

Nonlinear waves in dissipative microstructured two-dimensional solids; 75–83

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Alessia Casasso, Franco Pastrone, Alexander M. Samsonov


Plane Cosserat solids are introduced. It is shown that only one parameter is needed to describe intrinsic rotation. The field equations are simple enough to study the propagation of nonlinear waves and can be reduced to some particular form that admits different nonlinear waves, including solitons and cnoidal waves. We took into account both the matrix–grains and grain–grain interactions, nonlinearity, dispersion, and dissipation. Further examples can be provided of different kinds of analytical approaches, like asymptotical analysis, reduction to the Weierstrass equation, hierarchy of leading equations and waves. Solitary wave solutions and periodic bounded solutions are explicitly obtained.

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

   2. Phillips, R. Crystals, Defects and Microstructures. Modeling Across Scales. Cambridge University Press, Cambridge, 2001.

  3. Pastrone, F. Mathematical models of microstructured solids. Lect. Notes Mech. 4/04, Tallinn Tech. Univ. Tallinn, 2004.

  4. Pastrone, F. Waves in solids with vectorial microstructure. Proc. Estonian Acad. Sci. Phys. Math., 2003, 52, 21–29.

  5. Pastrone, F. Wave propagation in microstructured solids. Math. Mech. Solids, 2005, 10, 349–357.


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


  7. Sillat, T. and Engelbrecht, J. Wave propagation in dissipative microstructured materials. Proc. Estonian Acad. Sci. Phys. Math., 2003, 52, 103–114.

  8. Janno, J. and Engelbrecht, J. Solitary waves in nonlinear microstructured materials. J. Phys. A: Math Gen., 2005, 38, 5159–5172.


  9. Giovine, P. and Oliveri, F. Dynamics and wave propagation in dilatant granular materials. Meccanica, 1995, 30.

10. Casasso, A. and Pastrone, F. Nonlinear waves in microstructured solids and complex structures. In Proceedings of the International Seminar “Days on Diffraction 2004”, Saint Petersburg, Russia. Faculty of Physics, SpbU, 2004, 43–51.

11. Casasso, A. and Pastrone, F. Nonlinear waves motion in complex elastic structures. Rend. Circ. Mat. Palermo, Serie II, Suppl. 78, 2006, 45–58.

12. Pastrone, F. Microstructures and granular media. Rend. Sem. Mat. Univ. Polit. Torino, 2007, 65, 87–95.

13. Casasso, A. and Pastrone, F. Nonlinear waves in plane granular media. In Proceedings of the International Seminar “Days on Diffraction 2005”, Saint Petersburg, Russia. 2005, 30–39.

14. Porubov, A. V. and Pastrone, F. Nonlinear bell-shaped and kink-shaped strain waves in microstructured solids. Int. J. Non-Linear Mech., 2004, 39, 1289–1299.


15. Samsonov, A. M. Strain Soliton in Solids and How to Construct Them. Chapman \& Hall/CRC, New York, 2000.

16. Whittaker, E. T. and Watson, G. N. A Course of Modern Analysis. The MacMillan Company, New York, 1946.

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