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Dispersive propagation of localized waves in a mass-in-mass metamaterial lattice

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Abstract

Linear localized waves evolution in a discrete mass-in-mass lattice is studied. The presence of the attached mass in the model contributes to dispersion giving rise to appearance of both the acoustic and optic wave modes. Important features of the waves are described using the dispersion relation, which is obtained in a continuum limit of the original discrete equations using a harmonic wave solutions. We study numerically the localized initial perturbations evolution and compare the features of the numerical solutions with those obtained for the analytical harmonic wave ons. We have found differences in the wave dynamics depending on the parameters of the initial conditions. One scenario describes almost permanent shape and velocity counterpart localized waves propagation with oscillating standing wave around the position of the initial pulse. Another wave evolution accounts for a decrease in the moving wave amplitude with the developing oscillating tail behind the localized wave. Contrary to the periodic analytical solution, no evidence of a band gap is found in the simulations of localized waves.

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References

  1. Born, M., Huang, K.: Dynamical Theory of Crystal Lattices. Clarendon Press, Oxford (1954)

    MATH  Google Scholar 

  2. Askar, A.: Lattice Dynamical Foundations of Continuum Theories. World Scientific, Singapore (1985)

    Google Scholar 

  3. Ostoja-Starzewski, M.: Lattice models in micromechanics. Appl. Mech. Rev. 55, 35–60 (2002)

    Article  ADS  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Askes, H., Metrikine, A.V.: Higher-order continua derived from discrete media: continualisation aspects and boundary conditions. Int. J. Solids Struct. 42, 187–202 (2005)

    Article  MATH  Google Scholar 

  6. Krivtsov. A.M.: Deformation and fracture of solids with microstructure. Fizmatlit, Moscow(2007) (in Russian)

  7. Andrianov, I.V., Awrejcewicz, J., Weichert, D.: Improved continuous models for discrete media. Math. Probl. Eng. (Open Access) 986242 (2010)

  8. Michelitsch, T.M., Collet, B., Wang, X.: Nonlocal constitutive laws generated by matrix functions: lattice dynamics models and their continuum limits. Int. J. Eng. Sci. 80, 106–123 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lepri, S. (ed.): Thermal Transport in Low Dimensions: From Statistical Physics to Nanoscale Heat Transfer. Lecture Notes in Physics 921. Springer International Publishing, Switzerland (2016)

  10. Krivtsov, A.M.: The ballistic heat equation for a one-dimensional harmonic crystal. In: Altenbach, H. et al. (eds.), Dynamical Processes in Generalized Continua and Structures, Advanced Structured Materials 103, 345–358, Springer Nature Switzerland AG (2019)

  11. Kuzkin, V.A.: Unsteady ballistic heat transport in harmonic crystals with polyatomic unit cell. Continuum Mech. Thermodyn. 31, 1573–1599 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  12. Panchenko, AYu., Kuzkin, V.A., Berinskii, I.E.: Unsteady ballistic heat transport in two-dimensional harmonic graphene lattice. J. Phys. Cond. Matter. 34, 165402 (2022)

    Article  ADS  Google Scholar 

  13. Cveticanin, L., Cveticanin,D.: Acoustic Metamaterials: Theory and Application. In: Herisanu, N., Marinca, V. (eds.), Acoustics and Vibration of Mechanical Structures-AVMS-2017. Springer Proceedings in Physics 198 (2018)

  14. Huang, H.H., Sun, C.T., Huang, G.L.: On the negative effective mass density in acoustic metamaterials. Int. J. Eng. Sci. 47, 610–617 (2009)

    Article  Google Scholar 

  15. Ma, G., Sheng, P.: Acoustic metamaterials: from local resonances to broad horizons. Sci. Adv. 2(2), 1–16 (2016)

    Article  Google Scholar 

  16. dell’Isola, F., et al.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Continuum Mech. Thermodyn. 31, 851–884 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  17. Eremeyev, V.A., Turco, E.: Enriched buckling for beam-lattice metamaterials. Mech. Res. Commun. 103, 103458 (2020)

    Article  Google Scholar 

  18. Nejadsadeghi, N., Placidi, L., Romeo, V., Misra, A.: Frequency band gaps in dielectric granular metamaterials modulated by electric field. Mech. Res. Commun. 95, 96–103 (2019)

  19. Madeo, A., et al.: First evidence of non-locality in real band-gap metamaterials: determining parameters in the relaxed micromorphic model. Proc. R. Soc. Lond. Ser. A, 47(2), 20160169 (2016)

  20. Kaviany, M.: Heat Transfer Physics, 2nd edn. Cambridge University Press, New York (2014)

    Book  MATH  Google Scholar 

  21. Guo, Y., Wang, M.: Phonon hydrodynamics and its applications in nanoscale heat transport. Phys. Rep. 595, 1–44 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  22. Anufriev, R., Ramiere, A., Maire, J., Nomura, M.: Heat guiding and focusing using ballistic phonon transport in phononic nanostructures. Nat. Commun. 8, 15505 (2017)

    Article  ADS  Google Scholar 

  23. Saito, R., Mizuno, M., Dresselhaus, M.S.: Ballistic and diffusive thermal conductivity of graphene. Phys. Rev. Appl. 9(2), 024017 (2018)

    Article  ADS  Google Scholar 

  24. Zhang, Z., et al.: Size-dependent phononic thermal transport in low-dimensional nanomaterials. Phys. Rep. 860, 1–26 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  25. Kuzkin, V.A., Krivtsov, A.M.: Unsteady ballistic heat transport: linking lattice dynamics and kinetic theory. Acta Mech. 32(5), 1983–1996 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kunin, I.A. : Elastic Media with Microstructure. I. One-dimensional models. Springer Series in Solid-State Sciences, vol. 1, 2. Springer-Verlag, Berlin, Heidelberg (1982)

  27. Krivtsov, A.M.: Dynamics of matter and energy. Z. Angew. Math. Mech. e202100496 (2022)

  28. El Sherbiny, M., Placidi, L.: Discrete and continuous aspects of some metamaterial elastic structures with band gaps. Arch. Appl. Mech. 18, 1725–1742 (2018)

    Article  Google Scholar 

  29. Placidi, L., Galal el Sherbiny,V., Baragatti, P.: Experimental investigation for the existence of frequency band gap in a microstructure model. Mathematics and Mechanics of Complex Systems, 9, 413–422 (2021)

  30. Kuzkin, V.A.: Thermal equilibration in infinite harmonic crystals. Continuum Mech. Thermodyn. 31, 1401–1423 (2019)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by the Russian Science Foundation (Grant No. 22-11-00338).

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Authors and Affiliations

  1. Institute for Problems in Mechanical Engineering, Bolshoy 61, V.O., St. Petersburg, Russia

    A. V. Porubov & A. M. Krivtsov

  2. Peter the Great St. Petersburg Polytechnic University, Polytechnicheskaya St., 29, St. Petersburg, Russia

    A. M. Krivtsov

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  1. A. V. Porubov
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  2. A. M. Krivtsov
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Correspondence to A. V. Porubov.

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Communicated by Andreas Öchsner.

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Porubov, A.V., Krivtsov, A.M. Dispersive propagation of localized waves in a mass-in-mass metamaterial lattice. Continuum Mech. Thermodyn. 34, 1475–1483 (2022). https://doi.org/10.1007/s00161-022-01138-z

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