Bulk Nonlinear Elastic StrainWaves in a Bar with Nanosize Inclusions

  • Igor A. Gula
  • Alexander M. Samsonov (†)
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)


We propose a mathematical model for propagation of the long nonlinearly elastic longitudinal strain waves in a bar, which contains nanoscale structural inclusions. The model is governed by a nonlinear doubly dispersive equation (DDE) with respect to the one unknown longitudinal strain function. We obtained the travelling wave solutions to DDE, and, in particular, the strain solitary wave solution, which was shown to be significantly affected by parameters of the inclusions. Moreover we found some critical inaccuracies, committed in papers by others in the derivation of a constitutive equation for the long strain waves in a microstructured medium, revised them, and showed an importance of improvements for correct estimation of wave parameters.


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  1. Capriz G (1989) Continua with Microstructure. SpringerGoogle Scholar
  2. Casasso A, Pastrone F, Samsonov AM (2010) Travelling waves in microstructure as the exact solutions to the 6th order nonlinear equation. Acoustical Physics 56(6):871–876Google Scholar
  3. Cosserat E, Cosserat F (1909) Théorie des Corps Déformables. Hermann, ParisGoogle Scholar
  4. Engelbrecht J, Berezovski A (2015) Reflections on mathematical models of deformation waves in elastic microstructured solids. Mathematics and Mechanics of Complex Systems 3(1):43–82Google Scholar
  5. Engelbrecht J, Braun M (1998) Nonlinear waves in nonlocal media. Appl Mech Reviews 51(8):475–487Google Scholar
  6. Engelbrecht J, Pastrone F (2003) Waves in microstructured solids with nonlinearities in microscale. Proceedings of the Estonian Academy of Sciences Physics Mathematics 52(1):12–20Google Scholar
  7. Engelbrecht J, Berezovski A, Pastrone F, Braun M (2005) Waves in microstructured materials and dispersion. Philosophical Magazine 85:4127–4141Google Scholar
  8. Eringen A, Suhubi E (1964a) Nonlinear theory of simple micro-elastic solids - I. International Journal of Engineering Science 2(2):189–203Google Scholar
  9. Eringen A, Suhubi E (1964b) Nonlinear theory of simple micro-elastic solids - II. International Journal of Engineering Science 2(4):389–404Google Scholar
  10. Hughes DS, Kelly JL (1953) Second-order elastic deformation of solids. Physical Review 92(5):1145–1149Google Scholar
  11. Khusnutdinova KR, Samsonov AM (2008) Fission of a longitudinal strain solitary wave in a delaminated bar. Physical Review E 77:066,603Google Scholar
  12. Landau LD, Lifshitz EM (1986) Theoretical Physics, vol VII. Theory of Elasticity, 3rd edn. Butterworth-Heinemann, OxfordGoogle Scholar
  13. Maugin GA (2011) A historical perspective of generalized continuum mechanics. In: Altenbach H, Maugin GA, Erofeev VI (eds) Mechanics of Generalized Continua, Springer, Berlin, Heidelberg, Advanced Structured Materials, vol 7, pp 3–19Google Scholar
  14. Mindlin RD (1964) Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis 16:51–78Google Scholar
  15. Murnaghan FD (1951) Finite Deformation of an Elastic Solid. Chapman & HallGoogle Scholar
  16. Pastrone F (2005) Wave propagation in microstructured solids. Mathematics and Mechanics of Solids 10:349–357Google Scholar
  17. Porubov A, Andrianov I, Danishevs’kyy V (2012) Nonlinear strain wave localization in periodic composites. International Journal of Solids and Structures 49(23):3381–3387Google Scholar
  18. Porubov AV, Pastrone F (2004) Non-linear bell-shaped and kink-shaped strain waves in microstructured solids. International Journal of Non-Linear Mechanics 39(8):1289–1299Google Scholar
  19. Porubov AV, Pastrone F, Maugin GA (2004) Selection of two-dimensional nonlinear strain waves in micro-structured media. Comptes Rendus Mécanique 332(7):513–518Google Scholar
  20. Samsonov AM (2001) Strain Solitions in Solids and How to Construct Them. Chapman & Hall/CRCGoogle Scholar
  21. Samsonov AM, Semenova IV, Belashov AV (2017) Direct determination of bulk strain soliton parameters in solid polymeric waveguides. Wave Motion 71:120–126Google Scholar
  22. Semenova IV, Dreiden GV, Samsonov AM (2011) Strain solitary waves in polymeric nanocomposites. In: Proulx T (ed) Dynamic Behavior of Materials, Springer, New York, vol 1, pp 261–267Google Scholar
  23. Weisstein EW (2017) ”Isotropic Tensor”. From MathWorld. A Wolfram Web ResourceGoogle Scholar
  24. Wolfram Research, Inc (2017) Mathematica, Version 11.1. URL, Champaign, IL

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Ioffe Institute of the Russian Academy of SciencesSt. PetersburgRussia
  2. 2.Department of Physics, Chemistry and PharmacyUniversity of Southern Denmark (SDU)OdenseDenmark

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