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Nonlinear wave dispersion in monoatomic chains with lumped and distributed masses: discrete and continuum models
Received date: 2023-12-06
Online published: 2024-04-08
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The main objective of this paper is to investigate the influence of inertia of nonlinear springs on the dispersion behavior of discrete monoatomic chains with lumped and distributed masses. The developed model can represent the wave propagation problem in a non-homogeneous material consisting of heavy inclusions embedded in a matrix. The inclusions are idealized by lumped masses, and the matrix between adjacent inclusions is modeled by a nonlinear spring with distributed masses. Additionally, the model is capable of depicting the wave propagation in bi-material bars, wherein the first material is represented by a rigid particle and the second one is represented by a nonlinear spring with distributed masses. The discrete model of the nonlinear monoatomic chain with lumped and distributed masses is first considered, and a closed-form expression of the dispersion relation is obtained by the second-order Lindstedt-Poincare method (LPM). Next, a continuum model for the nonlinear monoatomic chain is derived directly from its discrete lattice model by a suitable continualization technique. The subsequent use of the second-order method of multiple scales (MMS) facilitates the derivation of the corresponding nonlinear dispersion relation in a closed form. The novelties of the present study consist of (ⅰ) considering the inertia of nonlinear springs on the dispersion behavior of the discrete mass-spring chains; (ⅱ) developing the second-order LPM for the wave propagation in the discrete chains; and (ⅲ) deriving a continuum model for the nonlinear monoatomic chains with lumped and distributed masses. Finally, a parametric study is conducted to examine the effects of the design parameters and the distributed spring mass on the nonlinear dispersion relations and phase velocities obtained from both the discrete and continuum models. These parameters include the ratio of the spring mass to the lumped mass, the nonlinear stiffness coefficient of the spring, and the wave amplitude.
E. GHAVANLOO, S. EL-BORGI . Nonlinear wave dispersion in monoatomic chains with lumped and distributed masses: discrete and continuum models[J]. Applied Mathematics and Mechanics, 2024 , 45(4) : 633 -648 . DOI: 10.1007/s10483-024-3100-9
| 1 | CHOI, C., BANSAL, S., MÜNZENRIEDER, N., and SUBRAMANIAN, S. Fabricating and assembling acoustic metamaterials and phononic crystals. Advanced Engineering Materials, 23 (2), 2000988 (2021) |
| 2 | GHAVANLOO, E., EL-BORGI, S., and FAZELZADEH, S. A. Formation of quasi-static stop band in a new one-dimensional metamaterial. Archive of Applied Mechanics, 93 (1), 287- 299 (2023) |
| 3 | LIU, H., YANG, H., LI, J., WEI, L., and CHEN, H. T. Advances in active and tunable electromagnetic metamaterial devices: principles, realizations, and applications. Frontiers of Physics, 16 (3), 33601 (2021) |
| 4 | CHEN, S., and WU, Y. Recent advances in acoustic and elastic metamaterials. Reports on Progress in Physics, 83 (2), 026001 (2020) |
| 5 | SMITH, D. R., PENDRY, J. B., and WILTSHIRE, M. C. K. Metamaterials and negative refractive index. Science, 305 (5685), 788- 792 (2004) |
| 6 | GARDINER, A., DALY, P., DOMINGO-ROCA, R., WINDMILL, J. F. C., FEENEY, A., and JACKSON-CAMARGO, J. C. Additive manufacture of small-scale metamaterial structures for acoustic and ultrasonic applications. Micromachines, 12 (6), 634 (2021) |
| 7 | MIZUKAMI, K., KAWAGUCHI, T., OGI, K., and KOGA, Y. Three-dimensional printing of locally resonant carbon-fiber composite metastructures for attenuation of broadband vibration. Composite Structures, 255, 112949 (2021) |
| 8 | GÓRA, P., and ŁOPATO, P. Metamaterials' application in sustainable technologies and an introduction to their influence on energy harvesting devices. Applied Sciences, 13 (13), 7742 (2023) |
| 9 | CHALLAMEL, N., ZHANG, Y. P., WANG, C. M., RUTA, G., and DELL'ISOLA, F. Discrete and continuous models of linear elasticity: history and connections. Continuum Mechanics and Thermodynamics, 35 (2), 347- 391 (2023) |
| 10 | DEYMIER, P., and RUNGE, K. One-dimensional mass-spring chains supporting elastic waves with non-conventional topology. Crystals, 6 (4), 44 (2016) |
| 11 | GHAVANLOO, E., FAZELZADEH, S. A., and RAFII-TABAR, H. Formulation of an efficient continuum mechanics-based model to study wave propagation in one-dimensional diatomic lattices. Mechanics Research Communications, 103, 103467 (2020) |
| 12 | XU, S. F., XU, Z. L., and CHUANG, K. C. Hybrid bandgaps in mass-coupled Bragg atomic chains: generation and switching. Frontiers in Materials, 8, 774612 (2021) |
| 13 | ZHAO, P., ZHANG, K., ZHAO, C., and DENG, Z. Multi-resonator coupled metamaterials for broadband vibration suppression. Applied Mathematics and Mechanics (English Edition, 42 (1), 53- 64 (2021) |
| 14 | GRINBERG, I., and MATLACK, K. H. Nonlinear elastic wave propagation in a phononic material with periodic solid-solid contact interface. Wave Motion, 93, 102466 (2020) |
| 15 | HOU, X., DENG, Z., and ZHOU, J. Symplectic analysis for wave propagation in one-dimensional nonlinear periodic structures. Applied Mathematics and Mechanics (English Edition), 31 (11), 1371- 1382 (2010) |
| 16 | MANKTELOW, K. L., LEAMY, M. J., and RUZZENE, M. Analysis and experimental estimation of nonlinear dispersion in a periodic string. Journal of Vibration and Acoustics, 136 (3), 031016 (2014) |
| 17 | PACKO, P., UHL, T., STASZEWSKI, W. J., and LEAMY, M. J. Amplitude-dependent Lamb wave dispersion in nonlinear plates. The Journal of the Acoustical Society of America, 140 (2), 1319- 1331 (2016) |
| 18 | ZHAO, Y., HOU, X., ZHANG, K., and DENG, Z. Symplectic analysis for regulating wave propagation in a one-dimensional nonlinear graded metamaterial. Applied Mathematics and Mechanics (English Edition), 44 (5), 745- 758 (2023) |
| 19 | NARISETTI, R. K., RUZZENE, M., and LEAMY, M. J. Study of wave propagation in strongly nonlinear periodic lattices using a harmonic balance approach. Wave Motion, 49 (2), 394- 410 (2012) |
| 20 | WANG, X., ZHU, W., and LIU, M. Steady-state periodic solutions of the nonlinear wave propagation problem of a one-dimensional lattice using a new methodology with an incremental harmonic balance method that handles time delays. Nonlinear Dynamics, 100, 1457- 1467 (2020) |
| 21 | CAMPANA, M. A., OUISSE, M., SADOULET-REBOUL, E., RUZZENE, M., NEILD, S., and SCARPA, F. Impact of non-linear resonators in periodic structures using a perturbation approach. Mechanical Systems and Signal Processing, 135, 106408 (2020) |
| 22 | WEI, L. S., WANG, Y. Z., and WANG, Y. S. Nonreciprocal transmission of nonlinear elastic wave metamaterials by incremental harmonic balance method. International Journal of Mechanical Sciences, 173, 105433 (2020) |
| 23 | GEORGIEVA, A., KRIECHERBAUER, T., and VENAKIDES, S. Wave propagation and resonance in a one-dimensional nonlinear discrete periodic medium. SIAM Journal on Applied Mathematics, 60 (1), 272- 294 (1999) |
| 24 | CHAKRABORTY, G., and MALLIK, A. K. Dynamics of a weakly non-linear periodic chain. International Journal of Non-Linear Mechanics, 36 (2), 375- 389 (2001) |
| 25 | NARISETTI, R. K., LEAMY, M. J., and RUZZENE, M. A perturbation approach for predicting wave propagation in one-dimensional nonlinear periodic structures. Journal of Vibration and Acoustics, 132, 031001 (2010) |
| 26 | MANKTELOW, K., LEAMY, M. J., and RUZZENE, M. Multiple scales analysis of wave-wave interactions in a cubically nonlinear monoatomic chain. Nonlinear Dynamics, 63, 193- 203 (2011) |
| 27 | WANG, Y. Z., LI, F. M., and WANG, Y. S. Influences of active control on elastic wave propagation in a weakly nonlinear phononic crystal with a monoatomic lattice chain. International Journal of Mechanical Sciences, 106, 357- 362 (2016) |
| 28 | WANG, Y. Z., and WANG, Y. S. Active control of elastic wave propagation in nonlinear phononic crystals consisting of diatomic lattice chain. Wave Motion, 78, 1- 8 (2018) |
| 29 | WANG, J., ZHOU, W., HUANG, Y., LYU, C., CHEN, W., and ZHU, W. Controllable wave propagation in a weakly nonlinear monoatomic lattice chain with nonlocal interaction and active control. Applied Mathematics and Mechanics (English Edition), 39 (8), 1059- 1070 (2018) |
| 30 | PANIGRAHI, S. R., FEENY, B. F., and DIAZ, A. R. Second-order perturbation analysis of low-amplitude traveling waves in a periodic chain with quadratic and cubic nonlinearity. Wave Motion, 69, 1- 15 (2017) |
| 31 | FRONK, M. D., and LEAMY, M. J. Higher-order dispersion, stability, and waveform invariance in nonlinear monoatomic and diatomic systems. Journal of Vibration and Acoustics, 139 (5), 051003 (2017) |
| 32 | ZIVIERI, R., GARESCI, F., AZZERBONI, B., CHIAPPINI, M., and FINOCCHIO, G. Nonlinear dispersion relation in anharmonic periodic mass-spring and mass-in-mass systems. Journal of Sound and Vibration, 462, 114929 (2019) |
| 33 | SEPEHRI, S., MASHHADI, M. M., and FAKHRABADI, M. M. S. Manipulation of wave motion in smart nonlinear phononic crystals made of shape memory alloys. Physica Scripta, 96 (12), 125527 (2021) |
| 34 | SEPEHRI, S., MASHHADI, M. M., and FAKHRABADI, M. M. S. Dispersion curves of electromagnetically actuated nonlinear monoatomic and mass-in-mass lattice chains. International Journal of Mechanical Sciences, 214, 106896 (2022) |
| 35 | FANG, L., and LEAMY, M. J. Perturbation analysis of nonlinear evanescent waves in a one-dimensional monatomic chain. Physical Review E, 105 (1), 014203 (2022) |
| 36 | WATTIS, J. A. D. Approximations to solitary waves on lattices. Ⅲ: the monatomic lattice with second-neighbour interactions. Journal of Physics A: Mathematical and General, 29 (24), 8139 (1996) |
| 37 | ANDRIANOV, I. V., AWREJCEWICZ, J., and WEICHERT, D. Improved continuous models for discrete media. Mathematical Problems in Engineering, 2010, 986242 (2010) |
| 38 | ZHOU, Y., WEI, P., and TANG, Q. Continuum model of a one-dimensional lattice of metamaterials. Acta Mechanica, 227 (8), 2361- 2376 (2016) |
| 39 | HACHE, F., CHALLAMEL, N., ELISHAKOFF, I., and WANG, C. M. Comparison of nonlocal continualization schemes for lattice beams and plates. Archive of Applied Mechanics, 87, 1105- 1138 (2017) |
| 40 | GÓMEZ-SILVA, F., and ZAERA, R. Analysis of low order non-standard continualization methods for enhanced prediction of the dispersive behaviour of a beam lattice. International Journal of Mechanical Sciences, 196, 106296 (2021) |
| 41 | ANDRIANOV, I. V., STARUSHENKO, G. A., and WEICHERT, D. Numerical investigation of 1D continuum dynamical models of discrete chain. Zeitschrift für Angewandte Mathematik und Mechanik, 91 (11-12), 945- 954 (2012) |
| 42 | KEVREKIDIS, P. G., KEVREKIDIS, I. G., BISHOP, A. R., and TITI, E. S. Continuum approach to discreteness. Physical Review E, 65 (4), 046613 (2002) |
| 43 | ZABUSKY, N. J., and DEEM, G. S. Dynamics of nonlinear lattices, I: localized optical excitations, acoustic radiation, and strong nonlinear behavior. Journal of Computational Physics, 2 (2), 126- 153 (1967) |
| 44 | PORUBOV, A. V., and ANDRIANOV, I. V. Nonlinear waves in diatomic crystals. Wave Motion, 50 (7), 1153- 1160 (2013) |
| 45 | ASKAR, A. Lattice Dynamical Foundations of Continuum Theories: Elasticity, Piezoelectricity, Viscoelasticity, Plasticity, Vol. 2, World Scientific, Singapore (1986) |
| 46 | WATTIS, J. A. D. Solitary waves in a diatomic lattice: analytic approximations for a wide range of speeds by quasi-continuum methods. Physics Letters A, 284 (1), 16- 22 (2001) |
| 47 | VILA, J., FERNÁNDEZ-SÁEZ, J., and ZAERA, R. Nonlinear continuum models for the dynamic behavior of 1D microstructured solids. International Journal of Solids and Structures, 117, 111- 122 (2017) |
| 48 | ANDRIANOV, I. V., ZEMLYANUKHIN, A., BOCHKAREV, A., and EROFEEV, V. Steady solitary and periodic waves in a nonlinear nonintegrable lattice. Symmetry, 12 (10), 1608 (2020) |
| 49 | DE DOMENICO, D., ASKES, H., and AIFANTIS, E. C. Gradient elasticity and dispersive wave propagation: model motivation and length scale identification procedures in concrete and composite laminates. International Journal of Solids and Structures, 158, 176- 190 (2019) |
| 50 | ASKES, H., CARAMÉS-SADDLER, M., and RODRÍGUEZ-FERRAN, A Bipenalty method for time domain computational dynamics. Proceedings of the Royal Society A, 466 (2117), 1389- 1408 (2010) |
| 51 | NAYFEH, A. H., and MOOK, D. T. Nonlinear Oscillations, John Wiley & Sons, New York (2008) |
| 52 | HUSSEIN, M. I., LEAMY, M. J., and RUZZENE, M. Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook. Applied Mechanics Reviews, 66 (4), 040802 (2014) |
| 53 | RAHMAN, Z., and BURTON, T. D. On higher order methods of multiple scales in non-linear oscillations-periodic steady state response. Journal of Sound and Vibration, 133 (3), 369- 379 (1989) |
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