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Research Papers

Wave Directionality in Three-Dimensional Periodic Lattices

[+] Author and Article Information
Alireza Bayat

Department of Civil Engineering,
Johns Hopkins University,
3400 N Charles Street,
Latrobe Hall 205,
Baltimore, MD 21218
e-mail: abayat1@jhu.edu

Stavros Gaitanaros

Department of Civil Engineering,
Johns Hopkins University,
3400 N Charles Street,
Latrobe Hall 201,
Baltimore, MD 21218
e-mail: stavrosg@jhu.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 21, 2017; final manuscript received October 23, 2017; published online November 13, 2017. Assoc. Editor: Yihui Zhang.

J. Appl. Mech 85(1), 011004 (Nov 13, 2017) (17 pages) Paper No: JAM-17-1527; doi: 10.1115/1.4038287 History: Received September 21, 2017; Revised October 23, 2017

This work focuses on elastic wave propagation in three-dimensional (3D) low-density lattices and explores their wave directionality and energy flow characteristics. In particular, we examine the dynamic response of Kelvin foam, a simple-and framed-cubic lattice, as well as the octet lattice, spanning this way a range of average nodal connectivities and both stretching-and bending-dominated behavior. Bloch wave analysis on unit periodic cells is employed and frequency diagrams are constructed. Our results show that in the low relative-density regime analyzed here, only the framed-cubic lattice displays a complete bandgap in its frequency diagram. New representations of iso-frequency contours and group-velocity plots are introduced to further analyze dispersive behavior, wave directionality, and the presence of partial bandgaps in each lattice. Significant wave beaming is observed for the simple-cubic and octet lattices in the low frequency regime, while Kelvin foam exhibits a nearly isotropic behavior in low frequencies for the first propagating mode. Results of Bloch wave analysis are verified by explicit numerical simulations on finite size domains under a harmonic perturbation.

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References

Mead, D. M. , 1996, “ Wave Propagation in Continuous Periodic Structures: Research Contributions From Southampton, 1964–1995,” J. Sound Vib., 190(3), pp. 495–524. [CrossRef]
Langley, R. S. , 1994, “ On the Modal Density and Energy Flow Characteristics of Periodic Structures,” J. Sound Vib., 172(4), pp. 491–511. [CrossRef]
Brillouin, L. , 1953, Wave Propagation in Periodic Structures, 2nd ed., Dover Publications, Mineola, NY.
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]
Sigalas, M. , and Economou, E. N. , 1993, “ Band Structure of Elastic Waves in Two Dimensional Systems,” Solid State Commun., 86(3), pp. 141–143. [CrossRef]
Phani, A. S. , Woodhouse, J. , and Fleck, N. A. , 2006, “ Wave Propagation in Two-Dimensional Periodic Lattices,” J. Acoust. Soc. Am., 119(4), pp. 1995–2005. [CrossRef] [PubMed]
Spadoni, A. , Ruzzene, M. , Gonella, S. , and Scarpa, F. , 2009, “ Phononic Properties of Hexagonal Chiral Lattices,” Wave Motion, 46(7), pp. 435–450. [CrossRef]
Gonella, S. , and Ruzzene, M. , 2008, “ Analysis of in-Plane Wave Propagation in Hexagonal and Re-Entrant Lattices,” J. Sound Vib., 312(1), pp. 125–139. [CrossRef]
Baravelli, E. , and Ruzzene, M. , 2013, “ Internally Resonating Lattices for Bandgap Generation and Low-Frequency Vibration Control,” J. Sound Vib., 332(25), pp. 6562–6579. [CrossRef]
Martinsson, P. G. , and Movchan, A. B. , 2003, “ Vibrations of Lattice Structures and Phononic Band Gaps,” Q. J. Mech. Appl. Math., 56(1), pp. 45–64. [CrossRef]
Langley, R. S. , Bardell, N. S. , and Ruivo, H. M. , 1997, “ The Response of Two-Dimensional Periodic Structures to Harmonic Point Loading: A Theoretical and Experimental Study of a Beam Grillage,” J. Sound Vib., 207(4), pp. 521–535. [CrossRef]
Zelhofer, A. J. , and Kochmann, D. M. , 2017, “ On Acoustic Wave Beaming in Two-Dimensional Structural Lattices,” Int. J. Solids Struct., 115–116, pp. 248–269. [CrossRef]
Wang, Y. F. , Wang, Y. S. , and Zhang, C. , 2014, “ Bandgaps and Directional Properties of Two-Dimensional Square Beam-like Zigzag Lattices,” AIP Adv., 4(12), p. 124403. [CrossRef]
Trainiti, G. , Rimoli, J. J. , and Ruzzene, M. , 2016, “ Wave Propagation in Undulated Structural Lattices,” Int. J. Solids Struct., 97–98, pp. 431–444. [CrossRef]
Ruzzene, M. , Scarpa, F. , and Soranna, F. , 2003, “ Wave Beaming Effects in Two-Dimensional Cellular Structures,” Smart Mater. Struct., 12(3), p. 363. [CrossRef]
Casadei, F. , and Rimoli, J. J. , 2013, “ Anisotropy-Induced Broadband Stress Wave Steering in Periodic Lattices,” Int. J. Solids Struct., 50(9), pp. 1402–1414. [CrossRef]
Messner, M. C. , Barham, M. I. , Kumar, M. , and Barton, N. R. , 2015, “ Wave Propagation in Equivalent Continuums Representing Truss Lattice Materials,” Int. J. Solids Struct., 73–74, pp. 55–66. [CrossRef]
Delpero, T. , Schoenwald, S. , Zemp, A. , and Bergamini, A. , 2016, “ Structural Engineering of Three-Dimensional Phononic Crystals,” J. Sound Vib., 363, pp. 156–165. [CrossRef]
D'Alessandro, L. , Belloni, E. , Ardito, R. , Corigliano, A. , and Braghin, F. , 2016, “ Modeling and Experimental Verification of an Ultra-Wide Bandgap in 3D Phononic Crystal,” Appl. Phys. Lett., 109(22), p. 221907. [CrossRef]
Lucklum, F. , and Vellekoop, M. J. , 2016, “ Realization of Complex 3-D Phononic Crystals With Wide Complete Acoustic Band Gaps,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 63(5), pp. 796–797. [CrossRef]
Matlack, K. H. , Bauhofer, A. , Krödel, S. , Palermo, A. , and Daraio, C. , 2016, “ Composite 3D-Printed Metastructures for Low-Frequency and Broadband Vibration Absorption,” Proc. Natl. Acad. Sci. USA, 113(30), pp. 8386–8390.
Thomson, W. , 1887, “ On the Division of Space With Minimum Partitional Area,” London, Edinburgh, Dublin Philos. Mag. J. Sci., 24(151), pp. 503–514. [CrossRef]
Weaire, D. , and Phelan, R. , 1994, “ A Counter-Example to Kelvin's Conjecture on Minimal Surfaces,” Philos. Mag. Lett., 69(2), pp. 107–110. [CrossRef]
Gong, L. , Kyriakides, S. , and Jang, W. Y. , 2005, “ Compressive Response of Open-Cell Foams. Part I: Morphology and Elastic Properties,” Int. J. Solids Struct., 42(5), pp. 1355–1379. [CrossRef]
Gong, L. , and Kyriakides, S. , 2005, “ Compressive Response of Open Cell Foams—Part II: Initiation and Evolution of Crushing,” Int. J. Solids Struct., 42(5), pp. 1381–1399. [CrossRef]
Jang, W. Y. , and Kyriakides, S. , 2009, “ On the Crushing of Aluminum Open-Cell Foams: Part II Analysis,” Int. J. Solids Struct., 46(3), pp. 635–650. [CrossRef]
Kittel, C. , 2005, Introduction to Solid State Physics, 8th ed., Wiley, Hoboken, NJ.
Åberg, M. , and Gudmundson, P. , 1997, “ The Usage of Standard Finite Element Codes for Computation of Dispersion Relations in Materials With Periodic Microstructure,” J. Acoust. Soc. Am., 102(4), pp. 2007–2013. [CrossRef]
Wang, P. , Casadei, F. , Kang, S. H. , and Bertoldi, K. , 2015, “ Locally Resonant Band Gaps in Periodic Beam Lattices by Tuning Connectivity,” Phys. Rev. B, 91(2), p. 020103(R). [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Schematic representation of (a) Kelvin foam, (b) simple-cubic, (c) octet, and (d) framed-cubic lattices and corresponding unit-cells in canonical form (lower left cells in Fig. 1(a)1(d)). In right column are the unit-cells considered for wave propagation analysis. (e) Reciprocal lattice unit-cell and corresponding IBZ for a cubic lattice.

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Fig. 2

(a) Band diagram, (b) density of states, and (c) the first three mode shapes at high symmetry points of the IBZ for the Kelvin foam with relative density ρr = 15%

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Fig. 3

Iso-frequency contour plots for the first three modes of Kelvin unit-cell with ρr = 15% in ky − kz plane at different levels of the kx component of the wave vector. Different colormaps are assigned for each mode to better illustrate the discrepancy among the modes.

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Fig. 4

Normalized (a) phase and (b) group velocity of Kelvin unit cell with ρr = 15% for the first three modes at different frequencies that correspond to the kx = 0 iso-frequency contours in Fig. 3. The diamond, square, and circle markers represent first, second, and third modes, respectively. (In online version, the red diamond, green square, and blue circle markers represent first, second, and third modes, respectively.)

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Fig. 5

Four-dimensional group velocity plots of the Kelvin foam with density ρr = 15%, representing preferential directions of wave propagation for the first three modes

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Fig. 6

(a) Band diagram, (b) density of states, and (c) the first three mode shapes at high symmetry points of the IBZ for the simple-cubic lattice with relative density ρr = 15%

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Fig. 7

Iso-frequency contour plots for the first three modes of the simple-cubic lattice with ρr = 15% in ky − kz plane at different levels of the kx component of the wave vector

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Fig. 8

Normalized (a) phase and (b) group velocity of the simple-cubic lattice with ρr = 15% for the first three modes at different frequencies that correspond to the kx = 0 iso-frequency contours in Fig. 7. The diamond, square, and circle markers represent first, second, and third modes, respectively. (In online version, the red diamond, green square, and blue circle markers represent first, second, and third modes, respectively.)

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Fig. 9

Four-dimensional group velocity plots of the simple-cubic lattice with density ρr = 15%, representing preferential directions of wave propagation for the first three modes

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Fig. 10

(a) Band diagram, (b) density of states, and (c) the first three mode shapes at high symmetry points of the IBZ for the octet lattice with relative density ρr = 15%

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Fig. 11

Iso-frequency contour plots for the first three modes of the octet lattice with ρr = 15% in ky − kz plane at different levels of the kx component of the wave vector

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Fig. 12

Normalized (a) phase and (b) group velocity of the octet lattice with ρr = 15% for the first three modes at different frequencies that correspond to the kx = 0 iso-frequency contours in Fig. 11. The diamond, square, and circle markers represent first, second, and third modes, respectively. (In online version, the red diamond, green square, and blue circle markers represent first, second, and third modes, respectively.)

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Fig. 13

Four-dimensional group velocity plots of the simple-cubic lattice with density ρr = 15%, representing preferential directions of wave propagation for the first three modes

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Fig. 14

(a) Band diagram and (b) density of states for the framed-cubic lattice with relative density ρr = 15%. The shaded area represents the bandgap.

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Fig. 15

Iso-frequency contour plots for the first three modes of the anisotropic Kelvin foam with ρr = 15% in ky − kz plane at different levels of the kx component of the wave vector

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Fig. 16

Four-dimensional group velocity plots of the anisotropic Kelvin foam with density ρr = 15%, representing preferential directions of wave propagation for the first three modes

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Fig. 17

Anisotropy index versus normalized frequency plots that correspond to the first three propagating modes for (a) Kelvin foam with ρr = 2%, (b) Kelvin foam with ρr = 15%, (c) Anisotropic Kelvin foam with ρr = 15%, (d) simple-cubic lattice with ρr = 2%, (e) simple-cubic lattice with ρr = 15%, (f) anisotropic cubic lattice with ρr = 15%, (g) octet lattice with ρr = 2%, (h) octet lattice with ρr = 15%, and (i) anisotropic octet lattice with ρr = 15%

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Fig. 18

Transient numerical simulation snapshots for a single-point harmonic perturbation at Ω = 0.7 on a 10 × 10 × 10 Kelvin lattice at (a) 0.1 × 10−2, (b) 6.5 × 10−2, and (c) 8.8 × 10−2 s

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Fig. 19

Transient numerical simulation snapshots for a single-point harmonic perturbation atΩ = 0.4 on a 20 × 20 × 20 simple-cubic lattice at (a) 0.3 × 10−2, (b) 2.1 × 10−2, and (c) 3.6 × 10−2 s

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Fig. 20

Transient numerical simulation snapshots for a single-point harmonic perturbation at Ω = 7 on a 10 × 10 × 10 octet lattice at (a) 0.2 × 10−2, (b)1.6 × 10−2, and (c) 4.2 × 10−2 s

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