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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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Hierarchical structures in complex solids with microscales; pp. 79–86

(Full article in PDF format) doi: 10.3176/proc.2010.2.04


Authors

Franco Pastrone

Abstract

In nonclassical mechanics it is natural to deal with the problem of the propagation of nonlinear waves in solids with different internal structural scales (Engelbrecht, J., Pastrone, F., Braun, M., and Berezovski, A. Hierarchy of waves in nonclassical materials. In Universality of Nonclassical Nonlinearity (Delsanto, P. P., ed.). Springer, 2007, 29–48). The choice of suitable microstrain functions of the microdisplacements, of their time derivatives as strain velocities, allows us to obtain the field equations via a variational principle (see Pastrone, F., Cermelli, P., and Porubov, A. V. Nonlinear waves in 1-D solids with microsctructure. Mater. Phys. Mech., 2004, 7, 9–16; Casasso, A. and Pastrone, F. Wave propagation in solids with vectorial microstructures. Wave Motion, doi: 10.1016/j.wavemoti.2009.12.006; Porubov, A. V., Pastrone, F., and Maugin, G. A. Selection of two-dimensional nonlinear strain waves in micro-structured media. C. R. Acad. Sci. Paris}, 2004, Ser. I 337, 513–518) in three different cases: one-dimensional solids with two different microscales, two-dimensional solids with microstructures, and plane granular media. In all cases the hierarchical structure of equations due to the scales in materials is evident.

Keywords

microstructures, nonlinear elasticity, hierarchical structures, granular media.

References

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

  2. Casasso , A. and Pastrone , F. Wave propagation in solids with vectorial microstructures. Wave Motion ,
doi:10.1016/j.wavemoti.2009.12.006

  3. Engelbrecht , J. Nonlinear Wave Dynamics: Complexity and Simplicity. Kluwer , Dordrecht , 1997.

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

  5. Engelbrecht , J. , Pastrone , F. , Braun , M. , and Berezovski , A. Hierarchy of waves in nonclassical materials. In Universality of Nonclassical Nonlinearity (Delsanto , P. P. , ed.). Springer , 2007 , 29–48.

  6. Ericksen , J. L. Simpler static problems in nonlinear theories of rods. Int. J. Solids Structures , 1970 , 6 , 371–377.
doi:10.1016/0020-7683(70)90045-4

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

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

  9. Pastrone , F. , Cermelli , P. , and Porubov , A. V. Nonlinear waves in 1-D solids with microsctructure. Mater. Phys. Mech. , 2004 , 7 , 9–16.

10. 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.
doi:10.1016/j.ijnonlinmec.2003.09.002

11. Porubov , A. V. , Maugin , G. A. , and Mareev , V. V. Localization of two-dimensional non-linear waves in a plate. Int. J. Non-Linear Mech. , 2004 , 39 , 1359–1370.
doi:10.1016/j.ijnonlinmec.2003.12.002

12. Porubov , A. V. , Pastrone , F. , and Maugin , G. A. Selection of two-dimensional nonlinear strain waves in micro-structured media. C. R. Acad. Sci. Paris , 2004 , Ser. I 337 , 513–518.

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