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Research Papers

Modeling and Analysis of Nonlinear Wave Propagation in One-Dimensional Phononic Structures

[+] Author and Article Information
M. Liu

Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering,
University of Maryland,
Baltimore County, 1000 Hilltop Circle,
Baltimore, MD 21250

W. D. Zhu

Division of Dynamics and Control,
School of Astronautics,
Harbin Institute of Technology,
Harbin 150001, China;
Department of Mechanical Engineering,
University of Maryland,
Baltimore County, 1000 Hilltop Circle,
Baltimore, MD 21250
e-mail: wzhu@umbc.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 4, 2017; final manuscript received March 6, 2018; published online May 17, 2018. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 140(6), 061010 (May 17, 2018) (12 pages) Paper No: VIB-17-1403; doi: 10.1115/1.4039570 History: Received September 04, 2017; Revised March 06, 2018

Different from elastic waves in linear periodic structures, those in phononic crystals (PCs) with nonlinear properties can exhibit more interesting phenomena. Linear dispersion relations cannot accurately predict band-gap variations under finite-amplitude wave motions; creating nonlinear PCs remains challenging and few examples have been studied. Recent studies in the literature mainly focus on discrete chain-like systems; most studies only consider weakly nonlinear regimes and cannot accurately obtain some relations between wave propagation characteristics and general nonlinearities. This paper presents propagation characteristics of longitudinal elastic waves in a thin rod and coupled longitudinal and transverse waves in an Euler–Bernoulli beam using their exact Green–Lagrange strain relations. We derive band structure relations for a periodic rod and beam and predict their nonlinear wave propagation characteristics using the B-spline wavelet on the interval (BSWI) finite element method. Influences of nonlinearities on wave propagation characteristics are discussed. Numerical examples show that the proposed method is more effective for nonlinear static and band structure problems than the traditional finite element method and illustrate that nonlinearities can cause band-gap width and location changes, which is similar to results reported in the literature for discrete systems. The proposed methodology is not restricted to weakly nonlinear systems and can be used to accurately predict wave propagation characteristics of nonlinear structures. This study can provide good support for engineering applications, such as sound and vibration control using tunable band gaps of nonlinear PCs.

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References

Joannopoulos, J. D. , Villeneuve, P. R. , and Fan, S. H. , 1997, “Photonic Crystals,” Solid State Commun., 102(2–3), pp. 165–173. [CrossRef]
Kushwaha, M. S. , Halevi, P. , Dobrzynski, L. , and Djafari-Rouhani, B. , 1993, “Acoustic Band Structure of Periodic Elastic Composites,” Phys. Rev. Lett., 71(13), pp. 2022–2025. [CrossRef] [PubMed]
Hussein, M. I. , Leamy, M. J. , and Ruzzene, M. , 2014, “Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook,” ASME Appl. Mech. Rev., 66(4), p. 040802. [CrossRef]
Wang, K. , Liu, Y. , and Yang, Q. S. , 2015, “Tuning of Band Structures in Porous Phononic Crystals by Grading Design of Cells,” Ultrasonics, 61, pp. 25–32. [CrossRef] [PubMed]
Ganesh, R. , and Gonella, S. , 2015, “From Modal Mixing to Tunable Functional Switches in Nonlinear Phononic Crystals,” Phys. Rev. Lett., 115(4), p. 054302. [CrossRef]
Cauchy, A. L. , 1830, Exercises De Mathematiques, Mathematics Exercises, Paris, France.
Graff, K. F. , 1991, Wave Motion in Elastic Solids, Dover Publications, Mineola, NY.
Thurston, R. N. , 1984, Waves in Solids, Mechanics of Solids, Vol. 4, Springer-Verlag, Berlin, pp. 109–308.
Nouri, M. B. , and Moradi, M. , 2016, “Presentation and Investigation of a New Two Dimensional Heterostructure Phononic Crystal to Obtain Extended Band Gap,” Physica B, 489, pp. 28–32. [CrossRef]
Anufriev, R. , and Nomura, M. , 2016, “Reduction of Thermal Conductance by Coherent Phonon Scattering in Two-Dimensional Phononic Crystals of Different Lattice Types,” Phys. Rev. B, 93(4), p. 045410. [CrossRef]
Vakakis, A. F. , and King, M. E. , 1995, “Nonlinear Wave Transmission in a Monocoupled Elastic Periodic System,” J. Acoust. Soc. Am., 98(3), pp. 1534–1546. [CrossRef]
Narisetti, R. K. , Leamy, M. J. , and Ruzzene, M. , 2010, “A Perturbation Approach for Predicting Wave Propagation in One-Dimensional Nonlinear Periodic Structures,” ASME J. Vib. Acoust., 132(3), p. 031001. [CrossRef]
Narisetti, R. K. , Ruzzene, M. , and Leamy, M. J. , 2011, “A Perturbation Approach for Analyzing Dispersion and Group Velocities in Two-Dimensional Nonlinear Periodic Lattices,” ASME J. Vib. Acoust., 133(6), p. 061020. [CrossRef]
Sreelatha, K. S. , and Joseph, K. B. , 2000, “Wave Propagation Through a 2D Lattice,” Chaos Solitons Fractals, 11(5), pp. 711–719. [CrossRef]
Lazarov, B. S. , and Jensen, J. S. , 2007, “Low-Frequency Band Gaps in Chains With Attached Non-Linear Oscillators,” Int. J. Non-Linear Mech., 42(10), pp. 1186–1193. [CrossRef]
Duan, W. S. , Shi, Y. , Zhang, L. , Lin, M. M. , and Lv, K. , 2005, “Coupled Nonlinear Waves in Two-Dimensional Lattice,” Chaos Solitons Fractals, 23(3), pp. 957–962. [CrossRef]
Feng, B. , and Kawahara, T. , 2007, “Discrete Breathers in Two-Dimensional Nonlinear Lattices,” Wave Motion, 45(1–2), pp. 68–82. [CrossRef]
Narisetti, R. K. , Ruzzene, M. , and Leamy, M. J. , 2012, “Study of Wave Propagation in Strongly Nonlinear Periodic Lattices Using a Harmonic Balance Approach,” Wave Motion, 49(2), pp. 394–410. [CrossRef]
Spadoni, A. , Ruzzene, M. , Gonella, S. , and Scarpa, F. , 2009, “Phononic Properties of Hexagonal Chiral Lattices,” Wave Motion, 46(7), pp. 435–450. [CrossRef]
Manktelow, K. , Leamy, M. J. , and Ruzzene, M. , 2013, “Comparison of Asymptotic and Transfer Matrix Approaches for Evaluating Intensity-Dependent Dispersion in Nonlinear Photonic and Phononic Crystals,” Wave Motion, 50(3), pp. 494–508. [CrossRef]
Abedinnasab, M. A. , and Hussein, M. I. , 2013, “Wave Dispersion Under Finite Deformation,” Wave Motion, 50(3), pp. 374–388. [CrossRef]
Khajehtourian, R. , and Hussein, M. I. , 2014, “Dispersion Characteristics of a Nonlinear Elastic Metamaterial,” AIP Adv., 4(12), p. 124308. [CrossRef]
Packo, P. , Uhl, T. , Staszewski, W. J. , and Leamy, M. J. , 2016, “Amplitude-Dependent Lamb Wave Dispersion in Nonlinear Plates,” J. Acoust. Soc. Am., 140(2), pp. 1319–1331. [CrossRef] [PubMed]
Xiang, J. W. , Matsumoto, T. , Wang, Y. X. , and Jiang, Z. S. , 2013, “Detect Damages in Conical Shells Using Curvature Mode Shape and Wavelet Finite Element Method,” Int. J. Mech. Sci., 66, pp. 83–93. [CrossRef]
Xiang, J. W. , Chen, X. F. , He, Y. M. , and He, Z. J. , 2006, “The Construction of Plane Elastomechanics and Mindlin Plate Elements of B-Spline Wavelet on the Interval,” Finite Elem. Anal. Des., 42(14–15), pp. 1269–1280. [CrossRef]
Liu, M. , Xiang, J. W. , Gao, H. F. , Jiang, Y. Y. , Zhou, Y. Q. , and Li, F. P. , 2014, “Research on Band Structure of One-Dimensional Phononic Crystals Based on Wavelet Finite Element Method,” Comput. Model. Eng. Sci., 97(5), pp. 425–436. http://www.techscience.com/doi/10.3970/cmes.2014.097.425.pdf
Li, J. , and Zhang, Y. , 2008, “Exact Travelling Wave Solutions in a Nonlinear Elastic Rod Equation,” Appl. Math. Comput., 202(2), pp. 504–510.
Goswami, J. C. , Chen, A. K. , and Chui, C. K. , 1995, “On Solving First-Kind Integral Equations Using Wavelets on a Bounded Interval,” IEEE Trans. Antennas Propag., 43(6), pp. 614–622. [CrossRef]
Kelley, C. T. , 2003, Solving Nonlinear Equations With Newton's Method, SIAM, Philadelphia, PA. [CrossRef]
Reddy, J. N. , 1997, “On Locking-Free Shear Deformable Beam Finite Elements,” Comput. Methods Appl. Mech. Eng., 149(1–4), pp. 113–132. [CrossRef]
Wiebe, R. , and Stanciulescu, I. , 2015, “Inconsistent Stability of Newmark's Method in Structural Dynamics Applications,” J. Comput. Nonlinear Dyn., 10(5), p. 051006. [CrossRef]
Timoshenko, S. P. , and Gere, J. M. , 1961, Theory of Elastic Stability, McGraw-Hill, New York.

Figures

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Fig. 1

Arrangement of the nodal DOF vector of the beam

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Fig. 2

Schematic of a one-dimensional PC structure whose unit cell consists of two materials A and B with the same cross section and thicknesses l1 and l2, respectively

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Fig. 3

Static deflection of the clamped-clamped thin rod

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Fig. 4

Static deflections of the cantilever Euler–Bernoulli beam subjected to a vertical concentrated force at its free end for different values of the force

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Fig. 5

Frequency dispersion curves for a homogeneous infinite rod

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Fig. 6

Band structures of the linear PC rod using BSWI-based rod elements and traditional rod elements: (a) first bands, (b) second bands, and (c) third bands

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Fig. 7

Band structures of linear (dashed lines) and nonlinear PC rods using two nonlinear BSWI-based elements (solid lines), and 20 and 120 nonlinear traditional rod elements

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Fig. 8

Effects of the length fraction on the nonlinear PC-rod band-gap width with B=0.006: (a) the first band-gap width and (b) the second band-gap width

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Fig. 9

Reduced PC band-gap structures with different wave amplitudes: dashed lines correspond to B = 0.004, star lines B = 0.006, and solid lines B = 0.008; f is the dimensional frequency of the PC rod band structure

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Fig. 10

Reduced PC band structures with different elastic moduli: solid lines correspond to Ea=4 N/m2, circle lines Ea=6 N/m2, and dashed lines Ea=8 N/m2

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Fig. 11

Reduced PC band structures with different mass densities: dashed lines correspond to ρa=2 kg/m3, square lines ρa=3 kg/m3, and solid lines ρa=4 kg/m3

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Fig. 12

Band structures of linear (dashed lines) and nonlinear PC beams using two nonlinear BSWI-based elements (solid lines), and 20 and 160 nonlinear traditional beam elements

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Fig. 13

PC beam band structures with different wave amplitudes: solid lines correspond to B=0.004, square lines B=0.006, circle lines B=0.008, and solid lines B=0.01

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Fig. 14

Reduced PC band structures with different values of the Young's modulus: square lines correspond to Ea=4 N/m2, dashed lines Ea=6 N/m2, and solid lines Ea=8 N/m2

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Fig. 15

Reduced PC band structures with different mass densities: solid lines correspond to ρa=4 kg/m3, dashed lines ρa=6 kg/m3, and star lines ρa=8 kg/m3

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