Skip to main content
SpringerLink
Log in

Continuous model of 2D discrete media based on composite equations

  • Published:
Acoustical Physics Aims and scope Submit manuscript

This article is dedicated to the memory of prof. A.I. Potapov

Abstract

The paper focuses on the development of 2D continuous models for a theoretical prediction of dynamic properties of discrete microstructures. A new continualization procedure, which refers to nonlocal interactions between variables of the discrete media, is proposed and the corresponding continuous model is obtained. The performed study is based on the application of composite equations. The developed approach is suitable for the dynamic analysis of 2D lattices of micro- and nanoparticles oscillating with arbitrary frequencies.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. Gusev, A. A. Rempel, and A. J. Magerl, Disorder and Order in Strongly Nonstoichiometric Compounds: Transition, Metal Carbides, Nitrides, and Oxides (Springer, Berlin, 2001).

    Google Scholar 

  2. X. Blanc, C. Le Bris, and P.-L. Lions, ESAIM: Math. Models Num. Anal. 41, 391 (2007).

    Article  MATH  Google Scholar 

  3. E. H. Dowell and D. Tang, J. Appl. Mech. 70, 328 (2003).

    Article  MATH  Google Scholar 

  4. V. I. Erofeev and V. M. Rodyushkin, Sov. Phys. Acoust. 38, 611 (1992).

    Google Scholar 

  5. S. Pagano and R. Paroni, Q. Appl. Math. 61, 89 (2003).

    MATH  MathSciNet  Google Scholar 

  6. A. I. Potapov, I. S. Pavlov, and G. A. Maugin, Int. J. Solids Struct. 43, 6194 (2006).

    Article  MATH  Google Scholar 

  7. S. A. Lisina, A. I. Potapov, and V. F. Nesterenko, Acoust. Phys. 47, 598 (2001).

    Article  ADS  Google Scholar 

  8. N. A. Fleck and J. W. Hutchinson, J. Mech. Phys. Solids 41, 1825 (1993).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. E. Aero, A. Fradkov, B. Andrievsky, and S. Vakulenko, Phys. Lett. A 353, 24 (2006).

    Article  ADS  Google Scholar 

  10. A. A. Vasiliev, S. V. Dmitriev, and A. E. Miroshnichenko, Int. J. Solids Struct. 47, 510 (2010).

    Article  MATH  Google Scholar 

  11. I. V. Andrianov and J. Awrejcewicz, Eur. J. Mech. A: Solids 24, 532 (2005).

    Article  MATH  ADS  Google Scholar 

  12. I. V. Andrianov and J. Awrejcewicz, Phys. Lett. A 345, 55 (2005).

    Article  MATH  ADS  Google Scholar 

  13. M. A. Collins, Chem. Phys. Lett. 77, 342347 (1981).

    Article  Google Scholar 

  14. A. V. Pichugin, H. Askes, and A. Tyas, J. Sound Vibrat. 313, 858 (2008).

    Article  ADS  Google Scholar 

  15. Ph. Rosenau, Phys. Lett. A 311, 39 (2003).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. P. Rosenau, Phys. Rev. B 36, 5868 (1987).

    Article  MathSciNet  ADS  Google Scholar 

  17. I. V. Andrianov and J. Awrejcewicz, Comp. Struct. 86, 140 (2008).

    Article  Google Scholar 

  18. I. V. Andrianov and J. Awrejcewicz, J. Sound Vibr. 264, 1187 (2003).

    Article  ADS  Google Scholar 

  19. M. Van Dyke, Perturbation Methods (The Parabolic Press, Stanford, CA, 1975).

    MATH  Google Scholar 

  20. I. V. Andrianov, Dokl. Akad. Nauk USSR, Ser. A 2, 13 (1991).

    MathSciNet  Google Scholar 

  21. A. C. Eringen, Res. Mech. 21, 313 (1987).

    Google Scholar 

  22. A. V. Metrikine and H. Askes, Eur. J. Mech. A: Solids 21, 555 (2002).

    Article  MATH  ADS  Google Scholar 

  23. A. C. Eringen, Int. J. Eng. Sci. 30, 1551 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  24. M. Lazar, G. A. Maugin, and E. C. Aifantis, Int. J. Solids Struct. 43, 1404 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  25. A. V. Metrikine and H. Askes, Eur. J. Mech. A 21, 555 (2002).

    Article  MATH  Google Scholar 

  26. A. C. Eringern, Int. J. Eng. Sci. 30, 1551 (1992).

    Article  Google Scholar 

  27. M. Lazar, G. A. Maugin, and E. C. Aifantis, Int. J. Solids Struct. 43, 1404 (2006).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of General Mechanics, RWTH Aachen University, Templergraben 64, Aachen, D-52062, Germany

    I. V. Andrianov & D. Weichert

  2. Prydniprovs’ka State Academy of Civil Engineering and Architecture, Chernishevs’kogo 24a, Dnipropetrovs’k, UA-49600, Ukraine

    V. V. Danishevs’kyy

Authors
  1. I. V. Andrianov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Danishevs’kyy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Weichert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. V. Andrianov.

Additional information

The article is published in the original.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Andrianov, I.V., Danishevs’kyy, V.V. & Weichert, D. Continuous model of 2D discrete media based on composite equations. Acoust. Phys. 56, 807–810 (2010). https://doi.org/10.1134/S1063771010060047

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063771010060047

Keywords

Search

Navigation