Archive of Applied Mechanics

, Volume 84, Issue 9–11, pp 1533–1538 | Cite as

Modeling of strongly nonlinear effects in diatomic lattices

  • Alexey PorubovEmail author
Special Issue


A previously developed strongly nonlinear continuum model for diatomic crystals is examined using continuum limit of a discrete diatomic model. It suggested suitable expression for the forces in the discrete model; however, its continuum limit not only explains the nonlinear terms continuum model but also gives rise to some additional terms. It turns out that one of them supports bell-shaped localized variations in the diatomic material but suppresses kink-shaped variations.


Diatomic lattice Continuum limit Nonlinear equation Traveling wave solution Localized strain wave 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aero E.L.: Micromechanics of a double continuum in a model of a medium with variable periodic structure. J. Eng. Math. 55, 81–95 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aero E.L., Bulygin A.N.: Strongly nonlinear theory of nanostructure formation owing to elastic and nonelastic strains in crystalline solids. Mech. Solids 42, 807–822 (2007)CrossRefGoogle Scholar
  3. 3.
    Porubov A.V., Aero E.L., Maugin G.A.: Two approaches to study essentially nonlinear and dispersive properties of the internal structure of materials. Phys. Rev. E 79, 046608 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Sayadi M.K., Pouget J.: Soliton dynamics in a microstructured lattice model. J. Phys. A: Math. Gen. 24, 2151–2172 (1991)CrossRefGoogle Scholar
  5. 5.
    Maugin G.A., Pouget J., Drouot R., Collet B.: Nonlinear Electromechanical Couplings. Wiley, UK (1992)Google Scholar
  6. 6.
    Maugin, G.A.: Nonlinear Waves in Elastic Crystals. Oxford University Press, UK (1999)Google Scholar
  7. 7.
    dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peri-dynamics, non-local and higher gradient continuum mechanics. An underestimated and still topical contribution of Gabrio Piola, to appear in Mechanics and Mathematics of Solids (MMS) (2013)Google Scholar
  8. 8.
    Alibert J.-J., Seppecher P., dell’Isola F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8, 51–73 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Askar A.: Lattice Dynamical Foundations of Continuum Theories. World Scientific, Singapore (1985)Google Scholar
  10. 10.
    Zabusky N.J., Deem G.S.: Dynamics of nonlinear lattices. I. Localized optical excitations, acoustic radiation, and strong nonlinear behavior. J. Comput. Phys. 2, 126–153 (1967)CrossRefGoogle Scholar
  11. 11.
    Kosevich A.M., Kovalev A.S.: Self-localization of vibrations in a one-dimensional anharmonic chain. Sov. Phys. JETP 40, 891–896 (1975)Google Scholar
  12. 12.
    Manevich L.I. et al.: Solitons in non-degenerated bistable systems. Physics-Uspekhi 37, 859–879 (1994)CrossRefGoogle Scholar
  13. 13.
    Kevrekidis, P.G.: Non-linear waves in lattices: past, present, future. IMA J. Appl. Math. 1–35 (2010). doi: 10.1093/imamat/hxr015
  14. 14.
    Sena S., Hong J., Bangb J., Avalosa E., Doneyd R.: Solitary waves in the granular chain. Phys. Rep. 462, 21–66 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Porter M.A., Daraio C., Herbold E.B., Szelengowicz I., Kevrekidis P.G.: Highly nonlinear solitary waves in periodic dimer granular chains. Phys. Rev. E 77, 015601R (2008)CrossRefGoogle Scholar
  16. 16.
    Andrianov, I.V., Awrejcewicz, J., Weichert, D.: Improved continuous models for discrete media, Mathematical Problems in Engineering (Open Access), Article ID 986242. doi: 10.1155/2010/986242 (2010)
  17. 17.
    Born, M., Kármán, Th.von : Über Schwingungen in Raumgittern. Phys. Zeitschr. 13, 297–309 (1912)Google Scholar
  18. 18.
    Born M., Huang K.: Dynamic Theory of Crystal Lattices. Clarendon Press, Oxford (1954)Google Scholar
  19. 19.
    Brillouin L., Parodi M.: Wave Propagation in Periodic Structures. Dover, New York (1953)zbMATHGoogle Scholar
  20. 20.
    Yajima N., Satsuma J.: Soliton solutions in a diatomic lattice system. Prog. Theor. Phys. 62, 370–378 (1979)CrossRefGoogle Scholar
  21. 21.
    Pnevmatikos St., Remoissenet M., Flytzanis N.: Propagation of acoustic and optical solitons in nonlinear diatomic chains. J. Phys. C: Solid State Phys. 16, L305–L310 (1983)CrossRefGoogle Scholar
  22. 22.
    Collins M.A.: Solitons in the diatomic chain. Phys. Rev. 31, 1754–1762 (1985)CrossRefGoogle Scholar
  23. 23.
    Pnevmatikos St., Flytzanis N., Remoissenet M.: Soliton dynamics of nonlinear diatomic lattices. Phys. Rev. B 33, 2308–2311 (1986)CrossRefGoogle Scholar
  24. 24.
    Landa P.S.: Nonlinear Oscillations and Waves in Dynamical Systems. Kluwer, Dordrecht (1996)CrossRefzbMATHGoogle Scholar
  25. 25.
    Huang G.: Soliton excitations in one-dimensional diatomic lattices. Phys. Rev. B 51, 12347–12360 (1995)CrossRefGoogle Scholar
  26. 26.
    Porubov A.V., Andrianov I.V.: Nonlinear waves in diatomic crystals. Wave Motion 50, 1153–1160 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Luongo A.: Mode localization by structural imperfections in one-dimensional continuous systems. J. Sound Vib. 155(2), 249–271 (1992)CrossRefzbMATHGoogle Scholar
  28. 28.
    Luongo A.: Mode localization in dynamics and buckling of linear imperfect continuous structures. Nonlinear Dyn. 25(1–3), 133–156 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Luongo A.: On the amplitude modulation and localization phenomena in interactive buckling problems. Int. J. Solids Struct. 27(15), 1943–1954 (1991)CrossRefzbMATHGoogle Scholar
  30. 30.
    Luongo A., Paolone A., Di Egidio A.: Multiple timescales analysis for 1:2 and 1:3 resonant Hopf bifurcations. Nonlinear Dyn. 34(3–4), 269–291 (2003)CrossRefzbMATHGoogle Scholar
  31. 31.
    Luongo A., Di Egidio A.: Divergence, Hopf and double-zero bifurcations of a nonlinear planar beam. Comput. Struct. 84(24–25), 1596–1605 (2006)CrossRefGoogle Scholar
  32. 32.
    Di Egidio A., Luongo A., Paolone A.: Linear and non-linear interactions between static and dynamic bifurcations of damped planar beams. Int. J. Non-Linear Mech. 42(1), 88–98 (2007)CrossRefzbMATHGoogle Scholar
  33. 33.
    dell’Isola F., Rosa L., Wozniak C.: A micro-structured continuum modelling compacting fluid-saturated grounds: the effects of pore-size scale parameter. Acta Mech. 127, 165–182 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Carcaterra A., Akay A.: Dissipation in a finite-size bath. Phys. Rev. E 84, 011121 (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute of Problems in Mechanical EngineeringSaint-PetersburgRussia
  2. 2.St. Petersburg State Polytechnical UniversitySt. PetersburgRussia

Personalised recommendations