Abstract
In this paper, we explore the dynamic behaviour of a metamaterial lattice which is composed of elastically connected concentrated masses coupled with local resonators (mass-in-mass lattice also referred to as the Kelvin lattice). This kind of metamaterial lattice is known to be associated with interesting phenomena such as a mass reduction effect, as compared to an equivalent Lagrange lattice composed of elastically connected effective masses. The effective mass of this metamaterial lattice is exactly calculated for finite and infinite metamaterial structures. For finite systems, the effective mass is identified from the eigenfrequencies spectrum of the metamaterial lattice. These eigenfrequencies are obtained for various boundary conditions of the metamaterial lattice, by solving a linear difference eigenvalue problem. For infinite systems, the wave dispersion of the metamaterial lattice is compared to the effective Born–Karman dispersion relation. It is shown for both the finite and the infinite systems, that the effective mass related to each branch (acoustical and optical branches) is proportional to the square of frequency associated with the complementary branch. An asymptotic expansion confirms that the effective mass tends towards the sum of each mass (the primary mass and the mass of the resonator) for the long-wave approximation of the acoustical branch. The effective mass is greater than the sum of each mass for the acoustical branch and is significantly reduced for the optical branch (with eventually a negligible reduced mass). An asymptotic analysis confirms the parametric study for each branch of the lattice spectrum. The effective stiffness is also calculated for both the finite and the infinite systems. A nonlocal continuous approach is developed to approximate the metamaterial lattice response in terms of frequency spectrum or effective mass variation. This study is concluded by a design-oriented methodology for the calibration of the effective mass from the metamaterial lattice properties.
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References
Al Ba’ba’a, H., Nouh, M., Singh, T.: Formation of local resonance band gaps in finite acoustic metamaterials: a closed-form transfer function model. J Sound Vibration 410, 429–446 (2017)
Andrianov, I.V., Awrejcewicz, J., Ivankov, O.: On an elastic dissipation model as continuous approximation for discrete media. Math Problems in Engng (2006). https://doi.org/10.1155/MPE/2006/27373
Andrianov, I.V., Awrejcewicz, J., Weichert, D.: Improved continuous models for discrete media. Math. Probl. Eng. 986242, 1–35 (2010)
Born, M., von Kármán, T.: On fluctuations in spatial grids. Physikalishe Zeitschrift 13, 297–309 (1912)
Born, M.: Lattice dynamics and X-ray scattering. Proc. Physics Soc. 54(4), 362–376 (1942)
Borovnitskii, Y.I.: Impedance theory of wave propagation on infinite periodic structures. J. Sound Vibrat 525, 116801 (2022)
Brillouin, L.: Wave propagation in periodic structures–Electric filters and crystal lattices. Mc Graw-Hill Company, New York (1946)
Brillouin, L., Parodi, M.: Propagation des ondes dans les milieux périodiques. Masson et cie, Paris (1956)
Cauchy A.: Sur les différences finies et les intégrales aux différences des fonctions entières d’une on de plusieurs variables, Exercices de mathématiques, page 155- 159, (1828)
Challamel, N., Wang, C.M., Elishakoff, I.: Nonlocal or gradient elasticity macroscopic models: a question of concentrated or distributed microstructure. Mech. Res. Commun. 71, 25–31 (2016)
Challamel, N., Aydogdu, M., Elishakoff, I.: Statics and dynamics of nanorods embedded in an elastic medium: Nonlocal elasticity and lattice formulations. Eur. J. Mech. A/Solids 67, 1–18 (2018)
Challamel, N., Wang, C.M., Zhang, H., Kitipornchai, S.: Exact and nonlocal solutions for vibration of axial lattices with direct and indirect neighbouring interactions. J. Eng. Mech (2018). https://doi.org/10.1061/(ASCE)EM.1943-7889.0001441
Chan, C.T., Li, J., Fung, K.H.: On extending the concept of double negativity to acoustic waves. J. Zhejiang Univ Sc. A 7, 24–28 (2006)
Chang, I.L., Liang, Z.X., Kao, H.W., Chang, S.H., Yang, C.Y.: The wave attenuation mechanism of the periodic local resonant metamaterial. J. Sound Vibration 412, 349–359 (2018)
Cui, T.J., Smith, D.R., Liu, R.: Metamaterials – Theory, Design and Applications. Springer, Berlin (2010)
Dell’Isola F. and Steigmann D.J. (Eds), Discrete and continuum models for complex metamaterials, Cambridge University Press, (2020)
Elaydi, S.: An introduction to difference equations, 3rd edn. Springer, New-York, NY (2005)
Ghavanloo, E., Fazelzadeh, S.A.: Wave propagation in one-dimensional infinite acoustic metamaterials with long-range interactions. Acta Mech. 230(12), 4453–4461 (2019)
Ghavanloo, E., Fazelzadeh, S.A., Rafii-Tabar, H.: Formulation of an efficient continuum mechanics-based model to study wave propagation in one-dimensional diatomic lattices. Mech. Res. Comm. 103, 103467 (2020)
Ghavanloo, E., Fazelzadeh, S.A.: An analytical approach for calculating natural frequencies of finite one-dimensional acoustic metametarials. Meccanica 56, 1819–1829 (2021)
Goldberg, S.: Introduction to difference equations with illustrative examples from economics, psychology and sociology. Dover Publications, New-York (1958)
He, C., Lim, K.M., Liang, X., Zhang, F., Jiang, J.: Tunable band structures design for elastic wave transmission in tension metamaterial chain. Eur. J. Mech. A/Solids 92, 104481 (2022)
Huang, H.H., Sun, C.T., Huang, G.L.: On the negative effective mass density in acoustic metamaterials. Int. J. Eng. Sc. 47, 610–617 (2009)
Kivshar, Y.S., Zhang, F., Takeno, S.: Nonlinear surface modes in monoatomic and diatomic lattices. Phys. D. 113, 248–260 (1998)
Lagrange J.L.: Recherches sur la nature et la propagation du son, Miscellanea Philosophico-Mathematica Societatis Privatae Taurinensis I, 2rd Pagination, i-112, 1759 (see also Œuvres, Tome 1, 39–148).
LeVeque, R.J.: Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. Society for Industrial and Applied Mathematics, SIAM (2007)
Li, J., Chan, C.T.: Double-negative acoustic metamaterial. Phys. Rev. E 70, 055602 (2004)
Liu, Y., Shen, X., Su, X., Sun, C.T.: Elastic metamaterials with low-frequency passbands based on lattice system with on-site potential. J. Vib. Acoust 138(021011), 1–10 (2016)
Marques, R., Martin, F., Sorolla, M.: Metamaterials with negative parameters. Wiley, Hoborken (2007)
Milton, G.W., Willis, J.R.: On modifications of Newton’s second law and linear continuum elastodynamics. Proc. Royal Soc. A/London 463, 855–880 (2007)
Mostacelli, M., Comi, C., Marigo, J.J.: On the dynamic behavior of discrete metamaterials: From attenuation to energy localization. Wave Motion 104, 102733 (2021)
Pendry, J.B.: Fundamentals and Applications of Negative Refraction in Metamaterials. Princeton University Press, Princeton (2007)
Rosenau, P.: Dynamics of nonlinear mass-spring chains near the continuum limit. Phys. Lett. A 118(5), 222–227 (1986)
Shen, X.H., Sun, C.T., Barnhart, M.V., Huang, G.L.: Analysis of dynamic behaviour of the finite elastic metamaterial-based structure with frequency-dependent properties. J. Vibration Acoustics, ASME 140(031012), 1–11 (2018)
Stein, A., Nouh, M., Singh, T.: Widening, transition and coalescence of local resonance band gaps in multi-resonator acoustic metamaterials: From unit cells to finite chains. J. Sound Vibration 523, 116716 (2022)
Thomson W. (Lord Kelvin): Popular lectures and addresses, Constitution of matter, McMillan and Co, (1889)
Veselago, V.G.: Properties of materials having simultaneously negative values of the dielectric and magnetic susceptibilities. Soviet Phys Solid State USSR 8, 2854–2856 (1967)
Vo, N.H., Pham, T.M., Hao, H., Bi, K., Chen, W.: A reinvestigation of the spring-mass model for metamaterial bandgap prediction. Int. J. Mech. Sc. 221(107219), 1–19 (2022)
Wallis, R.F.: Effect of free ends on the vibration frequencies of one-dimensional lattices. Phys. Rev. 105(2), 540–545 (1957)
Wattis, J.A.D.: Quasi-continuum approximations to lattice equations arising from the discrete non-linear telegraph equation. J. Phys. A: Math. Gen. 33, 5925–5944 (2000)
Zhang, H., Wang, C.M., Challamel, N., Gao, R.P.: Modelling vibrating nano-strings by lattice, finite difference and Eringen’s nonlocal models. J. Sound Vib. 425, 41–52 (2018)
Zhou, Y., Wei, P., Tang, Q.: Continuum model of a one-dimensional lattice of metamaterials. Acta Mech. 227(8), 2361–2376 (2016)
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Li, Y., Challamel, N. & Elishakoff, I. Effective mass and effective stiffness of finite and infinite metamaterial lattices. Arch Appl Mech 93, 301–321 (2023). https://doi.org/10.1007/s00419-022-02250-x
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DOI: https://doi.org/10.1007/s00419-022-02250-x