Perturbation analysis on the dispersion relations of a two-dimensional nonlinear single atom square lattice

PDF(2048 KB)

Journal of Vibration and Shock ›› 2020, Vol. 39 ›› Issue (9) : 88-96.

WANG Jie1, ZHU Jiang1, HUANG Wenbo1, HE Huan1,2

Author information +

History +

Abstract

The dispersion relation of phononic crystals determines the propagation mode of elastic waves.Starting from the wave equation of a two-dimensional infinite periodic structure, a first-order approximate perturbation method for analyzing the dispersion relation of nonlinear discrete phononic crystals was proposed.Based on the Bloch theory and small parametric perturbation expansion method, the first-order dispersion relations and dispersion curves were obtained to analyze the effects of impedance configuration and nonlinear coefficient on the dispersion and group velocity in different directions.Two-dimensional single-atom lattices were used as examples.Their first-order dispersion curves and group velocity contours were presented.The dispersion results reveal that the band gap and direction of propagation are related to the amplitude of waves.Finally, the response of the lattice to the point harmonic force was simulated to verify the effectiveness of the perturbation analysis.

Key words

sonic crystal / single atom square lattice / weak nonlinear / perturbation analysis / dispersion

Cite this article

EndNote

Ris (Procite)

Bibtex

Download Citations

WANG Jie1, ZHU Jiang1, HUANG Wenbo1, HE Huan1,2. Perturbation analysis on the dispersion relations of a two-dimensional nonlinear single atom square lattice[J]. Journal of Vibration and Shock, 2020, 39(9): 88-96

References

[1] 张研. 声子晶体的计算方法与带隙特性[M]. 北京: 科学出版社, 2015:1-4； Zhang Y. Calculation method and band gap characteristics of phononic crystals[M]. Beijng: Science Press, 2015:1-4. [2] Hussein M I, Leamy M J, Ruzzene M. Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook[J]. Applied Mechanics Reviews, 2014, 66(4):040802. [3] Su X X, Wang Y F, Wang Y S. Effects of Poisson’s ratio on the band gaps and defect states in two-dimensional vacuum/solid porous phononic crystals[J]. Ultrasonics, 2012, 52(2):255-265. [4] Salman A, Kaya O A, Cicek A. Determination of concentration of ethanol in water by a linear waveguide in a 2-dimensional phononic crystal slab[J]. Sensors and Actuators A: Physical, 2014, 208:50-55. [5] Wu L, Chen L. An acoustic bending waveguide designed by graded sonic crystals[J]. Journal of Applied Physics, 2011,110:114507. [6] Zhou X M, Liu X N, Hu G K. Elastic metamaterials with local resonances: an overview[J]. Theoretical and Applied Mechanics Letters, 2012, 2(4):041001. [7] Yablonovitch E. Inhibited spontaneous emission in solid-state physics and electronics[J]. Physical Review Letters, 1987, 58(20): 2059. [8] Sheng P, Zhang X X, Liu Z, et al. Locally resonant sonic materials[J]. Science, 2000, 289(5485): 1734-1736. [9] Phani A S, Woodhouse J, Fleck N A. Wave propagation in two-dimensional periodic lattices[J]. Journal of the Acoustical Society of America, 2006, 119(4): 1995-2005. [10] Scarpa F L, Ruzzene M, Soranna F. Wave beaming effects in bidimensional cellular structures[C]. Smart Structures and Materials 2002: Damping and Isolation, 2002: 63-77. [11] Ruzzene M, Tsopelas P. Control of wave propagation in sandwich plate rows with periodic honeycomb core[J]. Journal of Engineering Mechanics, 2003, 129(9): 975-986. [12] Hussein M I. Theory of damped bloch waves in elastic media[J]. Physical Review B, 2009, 80(21):308-310. [13] Hussein M I, Frazier M J. Band structure of phononic crystals with general damping[J]. Journal of Applied Physics, 2010, 108(9):2022. [14] Langley R S. The response of two-dimensional periodic structures to point harmonic forcing[J]. Journal of Sound & Vibration, 1996, 197(4): 447-469. [15] 温激鸿, 王刚, 刘耀宗,等. 基于集中质量法的一维声子晶体弹性波带隙计算[J]. 物理学报, 2004, 53(10):3384-3388. WEN J H,WANG G,LIU Y Z,et al. Calculation of elastic wave band gap of one-dimensional phononic crystal based on lumped mass method[J]. Acta Physica Sinica, 2004, 53(10):3384-3388. [16] Pankov A. Travelling Waves and Periodic Oscillations in Fermi-Pasta-Ulam Lattices [M]. Imperial College Press, 2005: xvi. [17] Sreelatha K S, Joseph K B. Wave propagation through a 2D lattice[J]. Chaos Solitons & Fractals, 2000, 11(5):711-719. [18] Kartashov Y V, Malomed B A, Torner L. Solitons in nonlinear lattices[J]. Review of Modern Physics, 2011, 83(1):247-306. [19] Narisetti R K, Ruzzene M, Leamy M J. A Perturbation Approach for Analyzing Dispersion and Group Velocities in Two-Dimensional Nonlinear Periodic Lattices[J]. Journal of Vibration and Acoustics, 2011, 133(6):061020. [20] Manktelow K L , Leamy M J , Ruzzene M . Weakly nonlinear wave interactions in multi-degree of freedom periodic structures[J]. Wave Motion, 2014, 51(6):886-904. [21] Narisetti R K . Wave propagation in nonlinear periodic structures[J]. Dissertations & Theses - Gradworks, 2010. [22] Lazarov B S, Jakob Søndergaard Jensen. Low-frequency band gaps in chains with attached non-linear oscillators[J]. International Journal of Non-Linear Mechanics, 2007, 42(10):1186-1193. [23] Manktelow K, Narisetti R K, Leamy M J, et al. Finite-element based perturbation analysis of wave propagation in nonlinear periodic structures[J]. Mechanical Systems & Signal Processing, 2013, 39(1-2):32-46. [24] Fang X, Wen J H, Yin J F, et al. Broadband and tunable one-dimensional strongly nonlinear acoustic metamaterials: Theoretical study[J]. Physical Review E, 2016, 94(5):052206. [25] Pal R K, Vila J, Leamy M, et al. Amplitude-dependent topological edge states in nonlinear phononic lattices[J]. Physical Review E, 2018, 97(3):032209. [26] 陈志远, 张全坤, 谢菊芳. 计及次近邻作用二维单原子正方晶格振动的色散关系[J]. 湖北大学学报(自然科学版), 2013, 35(4): 507-512. CHEN Zhi-yuan, ZHANG Quan-kun, XIE Ju-fang. Dispersion relation of two-dimensional monatomic square Lattice vibration with the secondary neighbor coupling[J]. Journal of Hubei University(Natural Science),2013,35(4):507-512. [27] Duan W S, Shi Y R, Zhang L, et al. Coupled nonlinear waves in two-dimensional lattice[J]. Chaos Solitons & Fractals, 2005, 23(3):957-962. [28] He J H. Modified Lindstedt–Poincare methods for some strongly non-linear oscillations: Part I: expansion of a constant[J]. International Journal of Non-Linear Mechanics, 2002, 37(2):309-314. [29] Wang Y Z, Li F M, Wang Y S. Influences of active control on elastic wave propagation in a weakly nonlinear phononic crystal with a monoatomic lattice chain[J]. International Journal of Mechanical Sciences, 2016, 106:357-362. [30] Fang X, Wen J H, Yin J F, et al. Wave propagation in nonlinear metamaterial multi-atomic chains based on homotopy method[J]. Aip Advances, 2016, 6(12):121706. [31] Diez A, Kakarantzas G, Birks T A, et al. Acoustic stop-bands in periodically microtapered optical fibers[J]. Applied Physics Letters, 2000, 76(23):3481-3483.