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Review Article

Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook

[+] Author and Article Information
Mahmoud I. Hussein

Assistant Professor
Mem. ASME
Aerospace Engineering Sciences,
University of Colorado Boulder,
Boulder, CO 80309-0429
e-mail: mih@colorado.edu

Michael J. Leamy

Associate Professor
Mem. ASME
School of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332-0405
e-mail: michael.leamy@me.gatech.edu

Massimo Ruzzene

Professor
ASME Fellow
School of Aerospace Engineering,
School of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332-0150
e-mail: ruzzene@gatech.edu

To the best of the authors' knowledge, this term was first used in the title of a symposium organized by J. S. Jensen at the 2005 U.S. National Congress in Computational Mechanics (Austin, TX, July 24–28, 2005).

Examples include the Annual ASME IMECE Symposium on Phononic Crystals and Acoustic Metamaterials (which started in 2005; http://www.asmeconferences.org/congress20xx), the International Workshop on Phononic Crystals (Nice, France, June 24–26, 2009; iwpc.gatech.edu) and the biannual Phononics 20xx conference series (whose inaugural edition was Phononics 2011: The First International Conference on Phononic Crystals, Metamaterials and Optomechanics, Santa Fe, NM, May 29–June 2, 2011; www.phononics2011.org).

1Corresponding author.

Manuscript received April 17, 2013; final manuscript received December 30, 2013; published online May 2, 2014. Assoc. Editor: Chin An Tan.

Appl. Mech. Rev 66(4), 040802 (May 02, 2014) (38 pages) Paper No: AMR-13-1027; doi: 10.1115/1.4026911 History: Received April 17, 2013; Revised December 30, 2013

The study of phononic materials and structures is an emerging discipline that lies at the crossroads of vibration and acoustics engineering and condensed matter physics. Broadly speaking, a phononic medium is a material or structural system that usually exhibits some form of periodicity, which can be in the constituent material phases, or the internal geometry, or even the boundary conditions. As such, its overall dynamical characteristics are compactly described by a frequency band structure, in analogy to an electronic band diagram. With roots extended to early studies of periodic systems by Newton and Rayleigh, the field has grown to encompass engineering configurations ranging from trusses and ribbed shells to phononic crystals and metamaterials. While applied research in this area has been abundant in recent years, treatment from a fundamental mechanics perspective, and particularly from the standpoint of dynamical systems, is needed to advance the field in new directions. For example, techniques already developed for the incorporation of damping and nonlinearities have recently been applied to wave propagation in phononic materials and structures. Similarly, numerical and experimental approaches originally developed for the characterization of conventional materials and structures are now being employed toward better understanding and exploitation of phononic systems. This article starts with an overview of historical developments and follows with an in-depth literature and technical review of recent progress in the field with special consideration given to aspects pertaining to the fundamentals of dynamics, vibrations, and acoustics. Finally, an outlook is projected onto the future on the basis of the current trajectories of the field.

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Figures

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Fig. 1

Discretization of a rod into a spring-mass lattice

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Fig. 2

Dispersion relation for the 1D spring-mass lattice

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Fig. 3

Dispersion relation (a) and phase velocity (b) of the 1D spring-mass lattice (solid line) and of the continuous rod (dashed line)

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Fig. 4

Dispersion relation for the 1D monatomic spring-mass lattice showing real (solid line) and imaginary (dashed line) parts of the propagation constant

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Fig. 5

Harmonic forced response of a finite lattice with N = 10 masses: lattice configuration (a) and harmonic response of the last mass (b)

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Fig. 6

Dynamic deformed shapes for a lattice of N = 100 masses at Ω = 1.99 (a), Ω = 2.001 (b), Ω = 2.01 (c), Ω = 2.1 (d)

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Fig. 7

Location on the dispersion relation of the natural frequencies of a finite, free–free lattice with N = 4 (“+”), and N = 10 (“o”) masses

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Fig. 8

1D diatomic lattice

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Fig. 9

Dispersion curves of the 1D diatomic spring-mass lattice (m2 = 2m1, m1 = 1, k = 1). The lower branch represents acoustic modes (solid line) and the upper branch represents optical modes (dashed line).

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Fig. 10

Unit-cell interactions at the left (L) and right (R) boundaries

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Fig. 11

Dispersion curves for the 1D diatomic spring-mass lattice (m2 = 2m1, m1 = 1, k = 1) showing real (solid line) and imaginary (dashed line) parts of the propagation constant

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Fig. 12

1D periodic spring-mass chain with internal resonators

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Fig. 13

Dispersion curves for a 1D spring-mass lattice with internal resonators. Comparison between undamped curves obtained for m¯r=0.125,Ω0=1,ΩR=1 (black dashed line), and “S” shaped branches corresponding to a viscously damped configuration (red solid line).

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Fig. 14

2D spring-mass lattice with detail of the unit-cell

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Fig. 15

Dispersion surface of the 2D spring-mass lattice with kx = ky and ax = ay

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Fig. 16

Dispersion surface contours and arrows showing the normal directions to isofrequency contours at Ω = 1 and Ω=2

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Fig. 17

Variation of group velocity at Ω = 1 (black dotted line) and Ω=2 (red solid line)

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Fig. 18

Harmonic response of a finite 2D lattice with 81 × 81 masses: Ω = 1 (a), and Ω=2 (b)

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Fig. 19

Finite element unit-cell denoting “B”ottom, “L”eft, “R”ight, and “T”op nodes. All nodes not on the perimeter of the unit cell are denoted as “i”nternal.

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Fig. 20

(a) Phononic band structure and (b) density of states (DOS) calculated using full model (matrix size: 4050 × 4050) and reduced Bloch mode expansion model (matrix size: 24 × 24). The IBZ, eigenvector selection points, and a schematic of the 2D unit cell are shown as insets in (a). The stiff/dense material phase is shown in black, and the compliant/light material phase is shown in white. (c) Computational efficiency: ratio of RBME model to full model calculation times r versus number of sampled κ-points along the border of the IBZ nκ (for two 2D FE meshes and a two-point expansion scheme). The number of elements is denoted by nel (adapted from Ref. [309] with permission from the Royal Society).

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Fig. 21

Unit cell of a 1D viscously damped diatomic lattice

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Fig. 22

Frequency (top) and damping ratio (bottom) band structures for the 1D viscously damped diatomic lattice depicted in Fig. 21. Acoustic branches are in solid lines and optical branches are in dashed lines (adapted from Refs. [341,342] with permission from Springer).

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Fig. 23

Frequency and wavenumber bandgap maps for the 1D viscously damped diatomic lattice depicted in Fig. 21. The widths of the frequency and wavenumber bandgaps are normalized with respect to the central bandgap frequency, ωdc, and central wavenumber, κc, respectively (from Ref. [341]).

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Fig. 24

Unit cell of a 1D viscously damped periodic spring-mass chain with internal resonators

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Fig. 25

Frequency (top) and damping ratio (bottom) band structures for the 1D viscously damped locally resonant lattice depicted in Fig. 24. Acoustic branches are in solid lines and optical branches are in dashed lines (adapted from Ref. [342] with permission from Springer).

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Fig. 26

Effect of viscous damping on wave dispersion (case 1): (a) frequency dispersion diagram and (b) damping ratio diagram. In both subfigures, the undamped and damped models are represented by black lines and red dashed lines, respectively.

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Fig. 27

Effect of viscous damping on wave propagation directionality: Isofrequency contour lines for model without (black lines) and with (red lines) damping. The topology of the unit cell considered is shown in the inset (white: compliant/low density material; black: stiff/high density material).

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Fig. 28

Central unit cell (colored red) surrounded by neighboring unit cells for 1D, 2D, and 3D periodic systems

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Fig. 29

Linear (dashed lines) and nonlinear (solid lines) dispersion branches for a diatomic chain with parameter space {m1=10,m2=30,k=10,Γ=3,ɛ=0.1,A=2}

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Fig. 30

(a) Depiction of a tunable nonlinear frequency isolator, (b) optical branches near the edge of the Brillouin zone for A = 0.5 (dashed line) and A = 0.75 (solid line), (c) FFT of device output at A = 0.5, (d) and at A = 0.75. The device's parameter space corresponds to {m1=1,m2=2,k=3,Γ=1,ɛ=0.1} [355].

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Fig. 31

(a) Plate with periodic composite (tungsten/silicone rubber) stubs arranged on aluminium plate. (b) Top-view of the plate—showing the lattice principal directions and the PZT transducer (Reprinted with permission from Ref. [397]. Copyright 2012, AIP Publishing LLC).

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Fig. 32

Transmission measurements on plate with resonant stubs show strong evidence of bandgap behavior in the 1300–3500 Hz (Reprinted with permission from Ref. [397]. Copyright 2012, AIP Publishing LLC).

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Fig. 33

Measured steady-state deformed shapes, representing the stationary response of the plate for excitation at 0.79 kHz (below the bandgap) (a), 2.35 kHz (inside the bandgap) (b), and 5.39 kHz (above the bandgap) (c). The red circle represents the source excitation of the acoustic waves (Reprinted with permission from Ref. [397]. Copyright 2012, AIP Publishing LLC).

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Fig. 34

Experimental evidence of waveguiding through a linear defect and plate excitation within the bandgap (Reprinted with permission from Ref. [397]. Copyright 2012, AIP Publishing LLC).

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Fig. 35

Periodic beam with array of shunted piezo patches (a) and unit cell with shunting through an electrical impedance ZSU (b) (Reprinted from Ref. [289] with permission from IOP Publishing & Deutsche Physikalische Gesellschaft. CC BY-NC-SA)

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Fig. 36

Schematic of the experimental beam

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Fig. 37

Experimental space/time response w(x, t) of the beam in space-time domain (a), and space/time response upon removal of the reflection at left boundary (b) (Reprinted from Ref. [289] with permission from IOP Publishing & Deutsche Physikalische Gesellschaft. CC BY-NC-SA).

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Fig. 38

Contour of the of the amplitude of the 2D FT |W(k,x)| outlining the dispersion properties for the beam with open shunts (a) and with shunts tuned at 5000 Hz and R = 33 Ω (b) (Reprinted from Ref. [289] with permission from IOP Publishing & Deutsche Physikalische Gesellschaft. CC BY-NC-SA).

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Fig. 39

Experimentally estimated wavenumbers: open circuit (red dashed line), shunted circuits with tuning at 5000 Hz and resistor R = 2.2 Ω (blue solid line): real (a) and imaginary part of wavenumber (b) (Reprinted from Ref. [289] with permission from IOP Publishing & Deutsche Physikalische Gesellschaft. CC BY-NC-SA).

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Fig. 40

Concept of tunable waveguide with electrical resonating units (Reprinted with permission from Ref. [258]. Copyright 2012, AIP Publishing LLC).

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Fig. 41

Full wavefield measurements demonstrating the waveguiding capability of the phononic crystal plate. (a) Amplitude of the measured wavefield inside the Bragg-type bandgap at 117 kHz. (b) Frequency spectrum of the wavefield averaged along the white dashed line in (a) clearly shows the extent of the bandgap between 100 kHz and 130 kHz generated by the stubbs. (Reprinted with permission from Ref. [258]. Copyright 2012, AIP Publishing LLC).

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Fig. 42

Measured and calculated dispersion properties of the vertical channel with open circuits (Reprinted with permission from Ref. [258]. Copyright 2012, AIP Publishing LLC).

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Fig. 43

Experimental evidence of the tunable resonant-type bandgap. (a) and (d) The contour lines reveal the presence of a bandgap in the response measured in the vertical channel with shunts tuned at two different frequencies. The black line displays the estimated real part of the wavenumber and clearly shows the back-bending of the dispersion curve. (b) and (e) Estimated imaginary part of the wavenumber indicating the strong attenuation at the tuning frequencies. (c) and (f) Real and imaginary part of the effective dynamic modulus of the resonating units (Reprinted with permission from Ref. [258]. Copyright 2012, AIP Publishing LLC).

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