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Structural Modeling of Nonlinear Localized Strain Waves in Generalized Continua

  • Vladimir I. ErofeevEmail author
  • Anna V. Leontyeva
  • Alexey O. Malkhanov
  • Igor S. Pavlov
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 120)

Abstract

The basic principles of structural modeling for the construction of mathematical models of microstructured media (generalized continua) are given. Here, microstructure means not the smallness of absolute values, but the smallness of some medium scale with respect to other scales, and the particles are considered to be non-deformable and homogeneous, without their own internal structure, presenting realistic materials. A nonlinear dynamically consistent model of a gradient-elastic medium has been elaborated by the method of structural modeling and using the continualization method involving nonlocality of coupling between the displacements of the lattice sites and the obtained continuum. The formation of spatially localized nonlinear strain waves in such media has been investigated.

Keywords

Structural modeling Gradient theory Nonlinear strain waves 

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Notes

Acknowledgements

The research was carried out under the financial support of the Russian Foundation for Basic Research (projects NN 18-29-10073-mk and 19-08-00965-a).

References

  1. Altenbach H, Maugin GA, Erofeev VI (eds) (2011) Mechanics of Generalized Continua, Advanced Structured Materials, vol 7. Springer-Verlag, Berlin, HeidelbergGoogle Scholar
  2. Askes H, Metrikine A (2002) One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 1: Generic formulation. European Journal of Mechanics A/Solids 21(4):573–588CrossRefGoogle Scholar
  3. Berglund K (1982) Structural models of micropolar media. In: Brulin O, Hsieh RKT (eds) Mechanics of Micropolar Media, World Scientific, Singapore, pp 35–86CrossRefGoogle Scholar
  4. Bobrovnitsky YI, Tomilina TM (2018) Sound absorption and metamaterials: A review. Acoustical Physics 64(5):519–526CrossRefGoogle Scholar
  5. Bogoliubov NN, Mitropolsky YA (1961) Asymptotic Methods in the Theory of Non-Linear Oscillations. Gordon and Breach, New YorkGoogle Scholar
  6. Born M, Huang K (1954) Dynamical Theory of Crystal Lattices. Int. Series of Monographs on Physics, Clarendon Press, OxfordGoogle Scholar
  7. Broberg KB (1997) The cell model of materials. Computational Mechanics 19:447–452CrossRefGoogle Scholar
  8. Chang C, Ma L (1994) A micromechanical-based micropolar theory for deformation of granular solids. Intern J Solids and Structures 28(1):67–87CrossRefGoogle Scholar
  9. Christoffersen J, Mehrabadi MM, Nemat-Nasser SA (1981) A micromechanical description of granular material behavior. Trans ASME J Appl Mech 48(2):339–344CrossRefGoogle Scholar
  10. Cleland AN (2003) Foundations of Nanomechanics. Advanced Texts in Physics, Springer-Verlag, BerlinCrossRefGoogle Scholar
  11. Cosserat E, Cosserat F (1909) Theorié des Corps Déformables. Librairie Scientifique A. Hermann et Fils, ParisGoogle Scholar
  12. de Borst R, van der Giessen E (eds) (1998) Material Instabilities in Solids. J. Wiley and Sons, Chichester-New York-Wienheim-Brisbane-Singapore-TorontoGoogle Scholar
  13. Erofeev VI, Mal’khanov AO (2017) Localized strain waves in a nonlinearly elastic conducting medium interacting with a magnetic field. Mechanics of Solids 52(2):224–231CrossRefGoogle Scholar
  14. Erofeev VI, Pavlov IS (2018) Rotational waves in microstructured materials. In: dell’Isola F, Eremeyev A, Porubov A (eds) Advances in Mechanics of Microstructured Media and Structures, Springer, Cham, Advanced Structured Materials, vol 87, pp 103–124Google Scholar
  15. Erofeev VI, Kazhaev VV, Semerikova NP (2002) Volny v sterzhnyakh. Dispersiya. Dissipatsiya (Nelineitost’ (Waves in Pivots. Dispersion. Dissipation. Nonlinearity, in Russ.). Fizmatlit, MoscowGoogle Scholar
  16. Erofeyev VI (2003) Wave Processes in Solids with Microstructure. World Scientific Publishing, New Jersey-London-Singapore-Hong Kong-Bangalore-TaipeiGoogle Scholar
  17. Fedorov VI (1968) Theory of Elastic Waves in Crystals. Plenum Press, New YorkCrossRefGoogle Scholar
  18. Ghoniem NM, Busso EP, Kioussis N, Huang H (2003) Multiscale modelling of nanomechanics and micromechanics: an overview. Phil Magazine 83(31–34):3475–3528Google Scholar
  19. Gulyaev YV, Lagar’kov AN, Nikitov SA (2008) Metamaterials: basic research and potential applications. Herald of the Russian Academy of Sciences 78(3):268–278CrossRefGoogle Scholar
  20. Kolken HMA, Zadpoor AA (2017) Auxetic mechanical metamaterials. RSC Advances 7(9):5111–5129CrossRefGoogle Scholar
  21. Krivtsov AM (2007) Deformirovanie i razrushenie tverdykh tels mikrostrukturoi (Deformation and Fracture of Solids with Microstructure, in Russ.). Fizmatlit Publishers, MoscowGoogle Scholar
  22. Kudryashov NA (2010) Methods of Nonlinear Mathematical Physics (in Russ.). Intellect, DolgoprudnyGoogle Scholar
  23. Li S, Wang G (2018) Introduction to Micromechanics and Nanomechanics, 2nd edn. World Scientific Co.Google Scholar
  24. Lippman H (1995) Cosserat plasticity and plastic spin. ASME Appl Mech Rev 48(11):753–762Google Scholar
  25. Love AEH (1920) A Treatise on the Mathematical Theory of Elasticity, 3rd edn. The University Press, CambridgeGoogle Scholar
  26. Maugin GA (1988) Continuum Mechanics of Electromagnetic Solids. Elsevier Science Publisher, AmsterdamGoogle Scholar
  27. Metrikine A, Askes H (2002) One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure. Part 2: Static and dynamic response. European Journal of Mechanics A/Solids 21(4):589–596Google Scholar
  28. Nikolaevskiy VN (1996) Geomechanics and Fluidodynamics, Theory and Applications of Transport in Porous Media, vol 8. Springer, DordrechtGoogle Scholar
  29. Ostoja-Starzewski M, Sheng Y, Alzebdeh K (1996) Spring network models in elasticity and fracture of composites and polycrystals. Computational Materials Science 7:82–93CrossRefGoogle Scholar
  30. Pavlov IS (2010) Acoustic identification of the anisotropic nanocrystalline medium with non-dense packing of particles. Acoustical Physics 56(6):924–934CrossRefGoogle Scholar
  31. Pavlov IS, Potapov AI (2008) Structural models in mechanics of nanocrystalline media. Doklady Physics 53(7):408–412CrossRefGoogle Scholar
  32. Pavlov IS, Potapov AI,, Maugin GA (2006) A 2D granular medium with rotating particles. Int J of Solids and Structures 43(20):6194–6207CrossRefGoogle Scholar
  33. Potapov AI, Pavlov IS, Maugin GA (1999) Nonlinear wave interactions in 1D crystals with complex lattice. Wave Motion 29(4):297–312CrossRefGoogle Scholar
  34. Potapov AI, Pavlov IS, Lisina SA (2009) Acoustic identification of nanocrystalline media. Journal of Sound and Vibration 322(3):564–580CrossRefGoogle Scholar
  35. Shining Z, Xiang Z (2018) Metamaterials: artificial materials beyond nature. National Science Review 5(2):131CrossRefGoogle Scholar
  36. Suiker ASJ, Metrikine A, de Borst R (2001) Comparison of wave propagation characteristics of the Cosserat continuum model and corresponding discrete lattice models. Int J of Solids and Structures 38:1563–1583CrossRefGoogle Scholar
  37. Vakhnenko A (1996) Diagnosis of the properties of a structurized medium by long nonlinear waves. Journal of Applied Mechanics and Technical Physics 37(5):643–649CrossRefGoogle Scholar
  38. Vardoulakis I, Sulem J (1995) Bifurcation Analysis in Geomechanics. Blackie Academic and Professional, LondonGoogle Scholar
  39. Vasiliev AA, Pavlov IS (2018) Structural and mathematical modeling of Cosserat lattices composed of particles of finite size and with complex connections. IOP Conference Series: Materials Science and Engineering 447:012,079CrossRefGoogle Scholar
  40. Voigt W (1887) Theoretische Studien über die Elasticitätsverhältnisse der Kristalle. Abn der Königl Ges Wiss Göttingen 34Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Vladimir I. Erofeev
    • 1
    Email author
  • Anna V. Leontyeva
    • 1
  • Alexey O. Malkhanov
    • 2
  • Igor S. Pavlov
    • 1
  1. 1.Mechanical Engineering Research Institute of Russian Academy of Sciences, Research Institute for MechanicsNizhny Novgorod Lobachevsky State UniversityNizhny NovgorodRussia
  2. 2.Mechanical Engineering Research Institute of Russian Academy of SciencesNizhny NovgorodRussia

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