eesti teaduste
akadeemia kirjastus
Proceedings of the Estonian Academy of Sciences. Physics. Mathematics
On the interaction of deformation waves in microstructured solids; 93–99

Andrus Salupere, Kert Tamm, Jüri Engelbrecht, Pearu Peterson

The modelling of wave propagation in microstructured materials should be able to account for various scales of microstructure. In the present paper governing equations for 1D waves in microstructured material are presented, based on the Mindlin model and the hierarchical approach. The governing equation under consideration has an analytical solution only in limit cases, therefore numerical analysis is carried out. Numerical solutions are found in the case of localized initial conditions by employing the pseudospectral method. Special attention is paid to the solitonic character of the solution.


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

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

3. Engelbrecht, J. and Pastrone, F. Waves in microstructured solids with strong nonlinearities in microscale. Proc. Estonian Acad. Sci. Phys. Math., 2003, 52, 12–20.

4. Erofeev, V. I. Wave Processes in Solids with Microstructure. World Scientific, Singapore, 2003.

5. Porubov, A. V. Amplification of Nonlinear Strain Waves in Solids. World Scientific, Singapore, 2003.

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

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

8. Janno, J. and Engelbrecht, J. An inverse solitary wave problem related to microstructured materials. Inverse Probl., 2005, 21, 2019–2034.

9. Mindlin, R. D. Micro-structure in linear elasticity. Arch. Rat. Mech. Anal., 1964, 16, 51–78.

10. Fornberg, B. A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge, 1998.

11. Salupere, A., Engelbrecht, J. and Peterson, P. On the long-time behaviour of soliton ensembles. Math. Comput. Simul., 2003, 62, 137–147.

12. Jones, E., Oliphant, T., Peterson, P. et al. SciPy: Open Source Scientific Tools for Python. 2001, available at

13. Frigo, M. and Johnson, S. G. The design and implementation of FFTW3. Proc. IEEE, 2005, 93, 216–231.

14. Peterson, P. Fortran to Python Interface Generator. 2005, available at

15. Hindmarsh, A. C. Odepack, a systematized collection of ODE solvers. In Scientific Computing (Stepleman, R. S. et al., eds). North-Holland, Amsterdam, 1983, 55–64.

16. Christov, C. I. and Maugin, G. A. An implicit difference scheme for the long-time evolution of localized solutions of a generalized Boussinesq system. J. Comput. Phys., 1995, 116, 39–51.

Back to Issue

Back issues