Skip to main content
SpringerLink
Log in

Wave Characterisation in a Dynamic Elastic Lattice: Lattice Flux and Circulation

  • Published:
Physical Mesomechanics Aims and scope Submit manuscript

Abstract

A novel characterisation of dispersive waves in a vector elastic lattice is presented in the context of wave polarisation. This proves to be especially important in analysis of dynamic anisotropy and standing waves trapped within the lattice. The operators of lattice flux and lattice circulation provide the required quantitative description, especially in cases of intermediate and high frequency dynamic regimes. Dispersion diagrams are conventionally considered as the ultimate characteristics of dynamic properties of waves in periodic systems. Generally, a waveform in a lattice can be thought of as a combination of pressure-like and shear-like waves. However, a direct analogy with waves in the continuum is not always obvious. We show a coherent way to characterise lattice waveforms in terms of so-called lattice flux and lattice circulation. In the long wavelength limit, this leads to well-known interpretations of pressure and shear waves. For the cases when the wavelength is comparable with the size of the lattice cell, new features are revealed which involve special directions along which either lattice flux or lattice circulation is zero. The cases of high frequency and wavelength comparable to the size of the elementary cell are considered, including dynamic anisotropy and dynamic neutrality in structured solids.

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. Barenblatt, G.I., Dimensional Analysis, New York: Gordon and Breach Science Publishers, 1987.

    Google Scholar 

  2. Barenblatt, G.I., Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge: Cambridge University Press, 1996.

    Book  MATH  Google Scholar 

  3. Barenblatt, G.I., Scaling, Cambridge: Cambridge University Press, 2003.

    Book  MATH  Google Scholar 

  4. Barenblatt, G.I., Flow, Deformation and Fracture, Cambridge: Cambridge University Press, 2014.

    Book  MATH  Google Scholar 

  5. Slepyan, L.I., Models and Phenomena in Fracture Mechanics, Berlin: Springer, 2002.

    Book  MATH  Google Scholar 

  6. Slepyan, L.I., Movchan, A.B., and Mishuris, G.S., Crack in a Lattice Waveguide, Int. J. Fract., 2010, vol. 162, pp. 91–106. doi https://doi.org/10.1007/s10704-009-9389-5

    Article  MATH  Google Scholar 

  7. Nieves, M.J., Mishuris, G.S., and Slepyan, L.I., Analysis of Dynamic Damage Propagation in Discrete Beam Structures, Int. J. Solid. Struct., 2016, vol. 97–98, pp. 699–713. doi https://doi.org/10.1016/j.ijsolstr.2016.02.033

    Article  Google Scholar 

  8. Nieves, M.J., Mishuris, G.S., and Slepyan, L.I., Transient Wave in a Transformable Periodic Exural Structure, Int. J. Solids Struct., 2017, vol. 112, pp. 185–208. doi https://doi.org/10.1016/j.ijsolstr.2016.11.012

    Article  Google Scholar 

  9. Bensoussan, A., Lions, J.L., and Papanicolaou, G., Asymptotic Analysis for Periodic Structures, Amsterdam: North-Holland, 1978.

    MATH  Google Scholar 

  10. Bakhvalov, N.S. and Panasenko, G.P., Homogenization: Averaging Processes in Periodic Media. Mathematics and Its Applications. V. 36, Dordrecht: Kluwer Academic Publishers, 1984.

    Google Scholar 

  11. Zhikov, V.V., Kozlov, S.M., and Oleinik, O.A., Homogenization of Differential Operators and Integral Functionals, Heidelberg: Springer, 1994.

    Google Scholar 

  12. Panasenko, G.P., Multiscale Modelling for Structures and Composites, Dordrecht: Springer, 2005.

    MATH  Google Scholar 

  13. Craster, R.V., Kaplunov, J., and Postnova, J., High-Frequency Asymptotics, Homogenisation and Localisation for Lattices, J. Mech. Appl. Math., 2010, vol. 63, pp. 497–519. doi https://doi.org/10.1093/qjmam/hbq015

    Article  MathSciNet  MATH  Google Scholar 

  14. Antonakakis, T., Craster, R.V., and Guenneau, S., Homogenisation for Elastic Photonic Crystals and Dynamic Anisotropy, J. Mech. Phys. Solids, 2014, vol. 71, pp. 84–96. doi https://doi.org/10.1016/j.jmps.2014.06.006

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Colquitt, D.J., Craster, R.V., and Makwana, M., High Frequency Homogenisation for Elastic Lattices, J. Mech. Appl. Math., 2015, vol. 68, pp. 203–230. doi https://doi.org/10.1093/qjmam/hbv005

    Article  MathSciNet  MATH  Google Scholar 

  16. Ayzenberg-Stepanenko, M. and Slepyan, L.I., Resonant-Frequency Primitive Wave Forms and Star Waves in Lattices, J. Sound Vib., 2008, vol. 313, pp. 812–821. doi https://doi.org/10.1016/j.jsv.2007.11.047

    Article  ADS  Google Scholar 

  17. Colquitt, D.J., Jones, I.S., Movchan, N.V., Movchan, A.B., and McPhedran, R.C., Dynamic Anisotropy and Localization in Elastic Lattice Systems, Waves Random Complex Media, 2012, vol. 22, pp. 143–159. doi https://doi.org/10.1080/17455030.2011.633940

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Movchan, A.B. and Slepyan, L.I., Resonant Waves in Elastic Structured Media: Dynamic Homogenisation Versus Green’s Functions, Int. J. Solids Struct., 2014, vol. 51, pp. 2254–2260. doi https://doi.org/10.1016/j.ijsolstr.2014.03.015

    Article  Google Scholar 

  19. Brun, M., Movchan, A.B., and Movchan, N.V., Shear Polarisation of Elastic Waves by a Structured Interface, Continuum Mech. Thermodyn., 2010, vol. 22, pp. 663–677. doi https://doi.org/10.1007/s00161-010-0143-z

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Brun, M., Jones, I.S., and Movchan, A.B., Vortex-Type Elastic Structured Media and Dynamic Shielding, Proc. R. Soc. A, 2012, vol. 468, pp. 3027–3046. doi https://doi.org/10.1098/rspa.2012.0165

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Carta, G., Brun, M., Movchan, A.B., Movchan, N.V., and Jones, I.S., Dispersion Properties of Vortex-Type Monatomic Lattices, Int. J. Solids Struct., 2014, vol. 51, pp. 2213–2225. doi https://doi.org/10.1016/j.ijsolstr.2014.02.026

    Article  Google Scholar 

  22. Carta, G., Jones, I.S., Movchan, N.V., Movchan, A.B., and Nieves, M.J., “Deflecting Elastic Prism” and Unidirectional Localisation for Waves in Chiral Elastic Systems, Sci. Rep., 2017, vol. 7, p. 26. doi https://doi.org/10.1038/s41598-017-00054-6

    Article  ADS  Google Scholar 

  23. Garau M., Carta G., Nieves M.J., Jones I.S., Movchan N.V., and Movchan A.B., Interfacial Waveforms in Chiral Lattices with Gyroscopic Spinners, Proc. R. Soc. Lond. A, 2018, vol. 474, p. 20180132. doi https://doi.org/10.1098/rspa.2018.0132

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Martinsson, P.G. and Movchan, A.B., Vibrations of Lattice Structures and Phononic Band Gaps, J. Mech. Appl. Math., 2003, vol. 56, pp. 45–64. doi https://doi.org/10.1093/qjmam56.1.45

    Article  MathSciNet  MATH  Google Scholar 

  25. Dietel, J. and Kleinert, H., Triangular Lattice Model of Two-Dimensional Defect Melting, Phys. Rev. B, 2006, vol. 73, p. 024113. doi https://doi.org/10.1103/PhysRevB.73.024113

    Article  ADS  Google Scholar 

  26. Achenbach, J.D., Wave Propagation in Elastic Solids, Amsterdam: North-Holland Publishing Company, 1973.

    MATH  Google Scholar 

  27. Musgrave, M.J.P., Crystal Acoustics, San Francisco: Holden-Day, 1970.

    MATH  Google Scholar 

  28. Carta, G., Jones, I.S., Movchan, N.V., and Movchan, A.B., Waveforms, Video file repository, 2018, https://www.dropbox.com/sh/ae2n3ggsiq2ju21/AAAYJiS1N4JCBNjGFR-4LyyKa?dl=0

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mechanical Engineering and Materials Research Centre, Liverpool John Moores University, Liverpool, L3 3AF, UK

    G. Carta & I. S. Jones

  2. Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK

    N. V. Movchan & A. B. Movchan

Authors
  1. G. Carta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. S. Jones
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. V. Movchan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. B. Movchan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. B. Movchan.

Additional information

In memory of Professor G.I. Barenblatt

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carta, G., Jones, I.S., Movchan, N.V. et al. Wave Characterisation in a Dynamic Elastic Lattice: Lattice Flux and Circulation. Phys Mesomech 22, 152–163 (2019). https://doi.org/10.1134/S102995991902005X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S102995991902005X

Keywords

Search

Navigation