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Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
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On the propagation of solitary waves in Mindlin-type microstructured solids; pp. 118–125

(Full article in PDF format) doi: 10.3176/proc.2010.2.09


Authors

Kert Tamm, Andrus Salupere

Abstract

The Mindlin–Engelbrecht–Pastrone model is applied to simulating 1D wave propagation in microstructured solids. The model takes into account the nonlinearity in micro- and macroscale. Numerical solutions are found for the full system of equations (FSE) and the hierarchical equation (HE). The latter is derived from the FSE by making use of the slaving principle. Analysis of results demonstrates good agreement between the solutions of the FSE and HE in the considered domain of parameters. For numerical integration the pseudospectral method is used.

Keywords

nonlinearity, microstructured solids, solitons, dispersion, pseudospectral methods.

References

  1. Christov , C. I. , Maugin , G. A. , and Porubov , A. V. On Boussinesq’s paradigm in nonlinear wave propagation. C. R. Mecanique , 2007 , 335 , 521–535.
doi:10.1016/j.crme.2007.08.006

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

  3. Engelbrecht , J. , Berezovski , A. , Pastrone , F. , and Braun , M. Waves in microstructured materials and dispersion. Phil. Mag. , 2005 , 85 , 4127–4141.
doi:10.1080/14786430500362769

  4. Eringen , A. C. Microcontinuum Field Theories I: Foundations and Solids. Springer , Berlin , 1999.

  5. Erofeev , V. I. Wave Processes in Solids with Microstructure. World Scientific , Singapore , 2003.
doi:10.1142/9789812794505

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

  7. Frigo , M. and Johnson , S. G. The design and implementation of FFTW3. Proc. IEEE , 2005 , 93 , 216–231.
doi:10.1109/JPROC.2004.840301

  8. 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.

  9. Ilison , L. and Salupere , A. Propagation of sech2-type solitary waves in hierarchical KdV-type systems. Math. Comput. Simulat. , 2009 , 79 , 3314–3327.
doi:10.1016/j.matcom.2009.05.003

10. Ilison , L. , Salupere , A. , and Peterson , P. On the propagation of localized perturbations in media with microstructure. Proc. Estonian Acad. Sci. Phys. Math. , 2007 , 56 , 84–92.

11. Janno , J. and Engelbrecht , J. An inverse solitary wave problem related to microstructured materials. Inverse Problems , 2005 , 21 , 2019–2034.
doi:10.1088/0266-5611/21/6/014

12. Janno , J. and Engelbrecht , J. Solitary waves in nonlinear microstructured materials. J. Phys. A: Math. Gen. , 2005 , 38 , 5159–5172.
doi:10.1088/0305-4470/38/23/006

13. Jones , E. , Oliphant , T. , Peterson , P. et al. SciPy: Open source scientific tools for Python , http://www.scipy.org , 2007.

14. Kreiss , H.-O. and Oliger , J. Comparison of accurate methods for the integration of hyperbolic equations. Tellus , 1972 , 30 , 341–357.

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

16. Metrikine , A. V. On causality of the gradient elasticity models. J. Sound Vib. , 2006 , 297 , 727–742.
doi:10.1016/j.jsv.2006.04.017

17. Mindlin , R. D. Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. , 1964 , 16 , 51–78.
doi:10.1007/BF00248490

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 , 2009 (accepted).

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

20. Peterson , P. F2PY: Fortran to Python interface generator , http://cens.ioc.ee/projects/f2py2e/ (2005).

21. Porubov , A. V. Amplification of Nonlinear Strain Waves in Solids. World Scientific , Hong Kong , 2003.
doi:10.1142/9789812794291

22. Randrüüt , M. and Braun , M. On one-dimensional solitary waves in microstructured solids. Wave Motion , 2010 , 47 , 217–230.
doi:10.1016/j.wavemoti.2009.11.002

23. Randrüüt , M. , Salupere , A. , and Engelbrecht , J. On modelling wave motion in microstructured solids. Proc. Estonian Acad. Sci. , 2009 , 58 , 241–246.
doi:10.3176/proc.2009.4.05

24. Salupere , A. The pseudospectral method and discrete spectral analysis. In Applied Wave Mathematics (Quak , E. and Soomere , T. , eds). Springer , Berlin , 2009 , 301–334.
doi:10.1007/978-3-642-00585-5_16

25. Salupere , A. , Engelbrecht , J. , and Peterson , P. On the long-time behaviour of soliton ensembles. Math. Comput. Simulat. , 2003 , 62 , 137–147.
doi:10.1016/S0378-4754(02)00178-7

26. Salupere , A. , Tamm , K. , Engelbrecht , J. , and Peterson , P. On the interaction of deformation waves in microstructured solids. Proc. Estonian Acad. Sci. Phys. Math. , 2007 , 56 , 93–99.

27. Salupere , A. , Ilison , L. , and Tamm , K. On numerical simulation of propagation of solitons in microstructured media. In Proceedings of the 34th Conference on Applications of Mathematics in Engineering and Economics (AMEE ’08) , Volume 1067 of AIP Conference Proceedings (Todorov , M. D. , ed.). American Institute of Physics , 2008 , 155–165.

28. Salupere , A. , Tamm , K. , and Engelbrecht , J. Numerical simulation of interaction of solitary deformation waves in microstructured solids. Int. J. Non-Linear Mech. , 2008 , 43 , 201–208.
doi:10.1016/j.ijnonlinmec.2007.12.011
 
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Current Issue: Vol. 68, Issue 4, 2019




Publishing schedule:
No. 1: 20 March
No. 2: 20 June
No. 3: 20 September
No. 4: 20 December