Katie Matlack's blog
https://imechanica.org/blog/10013929
enJournal Club for September 2021: Phononic materials: controlling elastic waves in solids
https://imechanica.org/node/25398
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p class="MsoNormal"><strong><span>Phononic materials: controlling elastic waves in solids</span></strong></p>
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<p class="MsoNormal"><span>Ignacio Arretche (<a href="mailto:ia6@illinois.edu">ia6@illinois.edu</a>), Ganesh Patil (<a href="mailto:gupatil2@illinois.edu">gupatil2@illinois.edu</a>), Kathryn Matlack (<a href="mailto:kmatlack@illinois.edu">kmatlack@illinois.edu</a>)</span></p>
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<p class="MsoNormal"><span><a href="https://matlack.mechanical.illinois.edu/" target="_blank" rel="noopener noreferrer">Wave Propagation and Metamaterials (WPM) Laboratory,</a> Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, USA</span></p>
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<p class="MsoNormal"><strong><span>Introduction</span></strong></p>
<p class="MsoNormal"><span>New wave phenomena have been uncovered through the study of <em>phononic materials</em>, in which micro- and meso-scale features are explicitly engineered to control frequency and spatial properties of acoustic and elastic waves. Specifically, phononic materials are periodic media that cause dispersive propagating modes to form. Under certain conditions, scattering and interference of waves is completely destructive for a range of frequencies, resulting in the prototypical phenomenon of a <em>band gap </em></span><span>[1]</span><span>. This property has important engineering implications in terms of vibration isolation/mitigation and targeted attenuation. In addition, phononic media have been studied for other applications such as energy harvesting </span><span>[2], [3]</span><span>, wave guiding and focusing </span><span>[4], [5]</span><span>, and flow stabilization </span><span>[6]</span><span>, seismic mitigation</span><span>[7], [8]</span><span>, and controlling thermal conductivity </span><span>[9]</span><span>. There has been significant activity in this area of research over the last decade, for example spanning ideas related to tunability</span><span>[10]</span><span>, wave propagation in lattices and other complex geometries </span><span>[11]–[16]</span><span>, topological protection </span><span>[17]</span><span>, nonreciprocity</span><span>[18]</span><span>, and nonlinearity </span><span>[19], [20]</span><span>. In this journal club, we aim to initiate discussion on recent progress in the field of phononic media by sharing some of our group’s recent research: geometry-wave property relations, redefining phononic media when the plane wave assumption does not hold, and exploiting nonlinearities in phononic media.</span></p>
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<p class="MsoNormal"><strong><span>Geometry-wave property relations</span></strong></p>
<p class="MsoNormal"><span>One of the natural questions that arise in phononic media is what is the relationship between the geometry of the unit cell and the wave response. For example, the work of Phani et al </span><span>[11]</span><span> studied the band structures and directionality of 2D lattices with different geometries and symmetries, and this was later extended to 3D lattices </span><span>[15], [21]</span><span>. To push band gaps of lattices to lower frequencies that are more desirable for engineering applications, many researchers have considered <em>internally resonating</em> lattices, including chiral-based structures </span><span>[16], [22], [23]</span><span>, and geometries that amplify the effective inertia of the unit cell </span><span>[24], [25]</span><span>. Some of our recent work has extensively explored a “lattice-resonator” geometry, where the unit cell consists of a solid mass surrounded by a lightweight lattice structure. We showed that with a mechanics understanding of the effective moduli of the lattices, one could qualitatively predict the range of band gap frequencies </span><span>[12], [13]</span><span>.</span></p>
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<p class="MsoNormal"><span>In addition to relating the geometry of lattice structures to band gaps, we recently explored the idea that auxetic lattices can be designed such that the shear wave propagates faster than the longitudinal wave through the lattice. Such “anomalous wave propagation” is atypical, since, in almost all natural materials, the longitudinal wave travels faster than the shear wave. Our work showed that by exploiting an auxetic geometry, we can selectively control the polarization of the faster wave </span><span>[14]</span><span>, which</span><span> can open new avenues for tailoring elastic wave propagation. We studied this phenomenon through two auxetic lattices – a bowtie and star lattice - and demonstrated how they can be used to conserve modes at an interface and to simultaneously accelerate and decelerate waves (Fig. 1).</span></p>
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<p class="MsoNormal"><span><img src="https://imechanica.org/files/APL_fig.jpg" alt="" width="594" height="301" /></span></p>
<p class="MsoNormal"><span><em><span>Fig. 1: Auxetic lattices with anomalous wave polarization (adapted from reference [14]): (a-b) Unit cells of two representative auxetic lattices that exhibit anomalous wave polarization – (a) bowtie and (b) star lattice. (c) By sandwiching an auxetic lattice that exhibits “anomalous polarization” between two isotropic materials, the longitudinal wave decelerates while shear wave simultaneously accelerates.</span></em> </span></p>
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<p class="MsoNormal"><strong><span>Beyond the plane wave assumption</span></strong></p>
<p class="MsoNormal"><span>To analyze phononic materials, one typically applies Bloch theorem to the elastodynamics problem, </span><span>which states that the solutions to the wave equation in a medium that has periodically varying material properties can be written as a sum of periodic functions that repeat spatially. The power of this theorem is that a periodic material can be analyzed simply using a representative unit cell. However, this theory only applies to plane waves and breaks down in the nearfield of point sources, or in the case of cylindrical or spherical waves. So, we must use a different approach to deal with these waves and enable phononic material properties in such systems. These wave types are not only relevant in the near field of point sources but are also typical in rotating machinery such as turbines and compressors where waves propagate outward from a central axis.</span></p>
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<p class="MsoNormal"><span>There were a series of papers published by Torrent and Sanchez-Dehesa </span><span>[26], [27]</span><span> several years ago on this concept, applied to acoustic waves (in fluids and gas, as opposed to elastic waves in solids) and electromagnetic waves. By writing the equations of motion in cylindrical coordinates, they redefined the mass density term to be anisotropic and radially dependent (Fig. 2a) to force the equations of motion to be invariant to radial translation, and enable the application of Bloch theorem. These systems, termed radial wave crystals (RWCs), support dipole-like modes with high quality factors (Fig. 2b) that allow one to detect the frequency and position of acoustic or electromagnetic point sources </span><span>[28]</span><span>. Although anisotropic density is possible using acoustic metamaterials </span><span>[29]</span><span>, so far radial wave crystals have only been realized in the electromagnetic domain </span><span>[28]</span><span>.</span></p>
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<p class="MsoNormal"><span>Our recent work explored similar concepts for elastic waves in solid media </span><span>[30]</span><span>. By writing the equations of motion for elastic waves in cylindrical coordinates and assuming the material properties are any general function of the radial coordinate, one can find specific functions of material parameters that allow the equations of motion to be invariant to translation. We applied this concept to radially-propagating torsional waves and found that modulating the periodic density and shear modulus with an additional radial dependence (Fig. 2c) enables the equations of motion to be invariant to radial translations. What this means is that while this system is not <em>geometrically</em> periodic like typical phononic media, it is, in a sense, <em>mathematically</em> periodic. Thus, we termed these systems <em>effective phononic crystals</em> (EPCs) because they effectively behave like a periodic system. In essence, canonical properties of phononic media such as band gaps (Fig. 2d) or even topologically protected modes (Fig. 2e) can be realized for radially-propagating waves. Further, for torsional elastic waves, both density and modulus can remain isotropic, simplifying physical realization. </span></p>
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<p class="MsoNormal"><span>Thought of another way, the radial dependence of the otherwise periodic material properties counteracts <em>diffraction</em> of the cylindrically propagating wave. Certainly, if one was purely interested in general broad-spectrum damping properties, the purely radial-periodic case may be more beneficial. However, one of the key properties of phononic media is that they allow the propagation of waves at certain frequencies while forbidding the propagation at other frequencies. Our approach allows one to apply this to radially-propagating waves in solids. In addition, energy localization using topological protection is only possible through RWCs or EPCs since these concepts require Bloch wave solutions. </span><span>EPCs could be physically realized using lattice materials or by tailoring impedance of the layers via changes in cross-sectional area. Further, EPCs are not necessarily limited to torsional radial (axisymmetric) waves, and the concept can be extended to circumferentially-propagating waves, other non-Cartesian directions, and other polarizations.</span></p>
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<p class="MsoNormal"><img src="https://imechanica.org/files/Fig1_big-01.jpg" alt="" width="1050" height="172" /></p>
<p class="MsoNormal"><em><span>Figure 2: a) Material properties of a radial wave crystal. b) Dipole-like resonances of two radial wave crystals excited by an external point source, color map displays pressure field. (a-b) adapted from </span></em><span>[26]</span><em><span>. c) Radially-varying shear modulus and density of EPC. d) Dispersion relation and tranmission of EPC, shaded regions show band gaps. e) Topological interface mode in radial tosional waves of an EPC. (c-e) adapted from </span></em><span>[30]</span><em><span>.</span></em></p>
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<p class="MsoNormal"><strong><span>Nonlinear phononic media</span></strong></p>
<p class="MsoNormal"><span>While much of the work in phononic media has been limited to the linear regime, nonlinearity in phononic media opens new avenues for wave control in both time and space. Specifically, nonlinearity enables passive material response, amplitude-dependent phenomena, and even energy transfer between frequencies. Due to these novel behaviors, nonlinear phononic materials hold potential for designing advanced wave tailoring functionalities useful in structural vibration mitigation, signal transmission, imaging, robotics, and sensors.</span></p>
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<p class="MsoNormal"><span>Research in nonlinear phononics, so far, has been primarily focused on two directions: 1) developing new analytical methods to solve wave propagation through both weakly </span><span>[31]</span><span> and strongly </span><span>[32], [33]</span><span> nonlinear phononic materials and 2) exploiting physical sources of nonlinearities to enrich wave dynamics of phononic media, e.g. </span><span>[4], [34]–[36]</span><span>. The incorporation of physical nonlinearities has allowed the realization and experimental validation of novel wave responses. Particularly, classical nonlinearities, such as material and geometric, have been explored by the Gonella group in waveguides </span><span>[37], [38]</span><span> and lattice geometries </span><span>[36]</span><span> to enable energy transfer between frequencies and between different modes or dynamic deformations (Fig. 3a). Hertzian contact nonlinearity of granular crystals has been extensively explored, e.g. by the Daraio group in terms of solitary wave propagation, which are localized waves traveling long distances without dispersion </span><span>[32], [34]</span><span>. Specific architectures of soft materials have revealed propagation of vector solitons - solitary waves of coupled polarization </span><span>[35]</span><span>. Recent efforts by the Bertoldi and Kochmann groups have explored instabilities of structural elements for shape morphing </span><span>[4]</span><span>. The journal club in 2017 by <a href="https://imechanica.org/node/21390" target="_blank" rel="noopener noreferrer">Nicholas Boechler on “Geometrically nonlinear microstructured materials for mechanical wave tailoring”</a> discussed several other representative nonlinear phononic materials based on Hertzian contact, instability, and geometric nonlinearity.</span></p>
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<p class="MsoNormal"><span>Much like the idea of mimicking biological structures in metamaterials </span><span>[39], [40]</span><span>, our recent work in this area is inspired by the structure of naturally-formed geomaterials or rocks, that have complex nonlinear mechanical responses. While numerous studies sought to understand the nonlinear response to interrogate the state of the strength of the rock, we are specifically interested in understanding structures or features that cause nonlinearities. We then aim to successfully merge nonlinear behaviors found in geomaterials with frequency-dependent filtering properties of phononic media.</span></p>
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<p class="MsoNormal"><span>We studied how waves evolve through periodic local microstructures in the form of rough contacts </span><span>[41], [42]</span><span> (Fig. 3b). The nonlinearity in our phononic materials stems from the nonlinear mechanical deformation of the rough surfaces forming contacts, which exhibit power-law nonlinearity. We analyzed these materials analytically by developing an equivalent discrete periodic model </span><span>[41]</span><span> and numerically by considering a continuum between rough contacts </span><span>[42]</span><span>. Our continuum model with discrete nonlinearity revealed that wave self-interaction with rough contacts generates zero- and second-harmonic frequencies, and causes self-demodulation effects to generate very low frequencies (amplitude-frequency plot in Fig. 3b). An interesting finding of the work was that the amplitude of these nonlinearly generated frequencies depends on how the discrete contacts are <em>coupled</em> to the surrounding continuum, and on the contact distribution in the continuum. These studies highlight that the global nonlinear wave response of the phononic material can be manipulated via a spatial distribution of the local microstructural nonlinearity, in this case geomaterial inspired rough contacts, in a continuum.</span></p>
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<p class="MsoNormal"><span>Rough contact-based nonlinear phononic materials are still in their early stages, and a wide range of questions are yet to be investigated. For example, extending the analysis to strong nonlinearity due to surfaces losing contact, the role of friction, or nonlinear scattering at wavelengths comparable to roughness may enable other wave responses yet to be realized. Further, embedded contacts, as demonstrated in </span><span>[42]</span><span>, could be architected to further tailor the nonlinear wave response.</span></p>
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<p class="MsoNormal"><span><img src="https://imechanica.org/files/Blog%20figure_V1%20%281%29.jpg" alt="" width="627" height="525" /></span></p>
<p class="MsoNormal"><em><span>Fig 3: Phononic materials with a) finite strain nonlinearity (image adapted from Ref. </span><span>[38]</span><span>) and b) rough contact nonlinearity (image adapted from Ref. </span><span>[32]</span><span>). Wavefield response in a) reveals that input energy in the form of flexural deformation at fundamental frequency generates axially-dominated second harmonics demonstrating energy transfer between different modes of the system. b) Phononic material with a single row (black) and two rows (blue) of embedded rough contacts. The neighboring continuum influences contact deformation and therefore the amplitudes of zero and second harmonic frequency. A single row of contacts deforms more and therefore exhibits a stronger nonlinear response compared to two rows of contacts.</span></em></p>
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<p class="MsoNormal"><strong><span>Outlook</span></strong></p>
<p class="MsoNormal"><span>In closing, here are a few thoughts on an introspective look and an outlook on the field of phononics:</span></p>
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<p class="MsoNormal"><span>One point is that the rise of 3D printing and additive manufacturing has transformed the field of phononic media. Up until these manufacturing methods took off, experimental realizations of phononic media were mostly limited to simple structures and models. However, the ability to fabricate complex geometries and multi-material structures across many length scales enabled researchers to physically realize and experimentally demonstrate various (and more) behaviors of phononic media. It also opens the door to new designs and applications. For example, one can envision designing a phononic material directly into a structural component, e.g. in an engine or aircraft, for vibration mitigating applications. This would enable a component to passively mitigate damaging vibrations, without the need for additional non-load bearing and degradable dampers or active vibration/noise cancelling devices. Our group has explored these ideas in 3D printed phononic materials, characterizing both the dependence of band gap frequencies and quasi-static properties on the geometry of lattice-resonator metastructures </span><span>[13], [43]</span><span>. Further advances in advanced manufacturing, such as fabricating complex geometries with stimuli-responsive materials, active materials, and multi-material structures (e.g. as discussed in<a href="https://imechanica.org/node/24838" target="_blank" rel="noopener noreferrer"> Qi Ge's journal club on 3D printing of soft materials</a>) would certainly push the state of the art of phononic materials. This may help guide future designs and engineering applications of these concepts, which are mostly limited to academic research as of yet.</span></p>
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<p class="MsoNormal"><span>Further, the integration of phononic media with other fields may help spawn new research directions and novel applications. A recent example of this is exploring how concepts in condensed matter physics, such as topological insulation, can be translated to the classical mechanical domain using phononic media, e.g. </span><span>[5], [44]–[48]</span><span>. For example, a topologically-protected mode is confined to the boundary or edge of a material, and does not penetrate into the bulk. Topological protection in the phononic domain has potential far-reaching applications such as stable wave guiding, delay lines, or acoustic diodes </span><span>[49]</span><span>. As a few other examples, phononic media could enable new and optimized nondestructive evaluation (NDE) techniques </span><span>[50], [51]</span><span>, or offer a new mechanism for flow control </span><span>[6]</span><span>. Another interesting direction is considering deformations beyond the elastic regime, and understanding what role phononic materials play in dynamic damage mechanisms such as fracture and crack propagation.</span></p>
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<p class="MsoNormal"><span>We look forward to discussing with everyone in the comments, and would be interested to hear what others are working on in this area and your thoughts on these ideas.</span></p>
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<p class="MsoNormal"><strong><span>References</span></strong></p>
<p class="MsoNormal"><span>[1] M. I. Hussein, M. J. Leamy, and M. Ruzzene, “Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook,” <em>Appl. Mech. Rev.</em>, vol. 66, no. 4, p. 040802, May 2014, doi: 10.1115/1.4026911.</span></p>
<p class="MsoNormal"><span>[2] M. Carrara, M. R. Cacan, M. J. Leamy, M. Ruzzene, and A. Erturk, “Dramatic enhancement of structure-borne wave energy harvesting using an elliptical acoustic mirror,” <em>Appl. Phys. Lett.</em>, vol. 100, no. 20, pp. 1–4, 2012, doi: 10.1063/1.4719098.</span></p>
<p class="MsoNormal"><span>[3] A. F. Arrieta, T. Delpero, A. E. Bergamini, and P. Ermanni, “Broadband vibration energy harvesting based on cantilevered piezoelectric bi-stable composites,” <em>Appl. Phys. Lett.</em>, vol. 102, no. 17, p. 173904, 2013, doi: 10.1063/1.4803918.</span></p>
<p class="MsoNormal"><span>[4] L. Jin <em>et al.</em>, “Guided transition waves in multistable mechanical metamaterials,” <em>Proc. Natl. Acad. Sci. U. S. A.</em>, vol. 117, no. 5, pp. 2319–2325, 2020, doi: 10.1073/pnas.1913228117.</span></p>
<p class="MsoNormal"><span>[5] K. H. Matlack, M. Serra-Garcia, A. Palermo, S. D. Huber, and C. Daraio, “Designing perturbative metamaterials from discrete models,” <em>Nat. Mater.</em>, vol. 17, no. 4, pp. 323–328, 2018, doi: 10.1038/s41563-017-0003-3.</span></p>
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<p class="MsoNormal"><span>[7] S. Krödel, N. Thomé, and C. Daraio, “Wide band-gap seismic metastructures,” <em>Extrem. Mech. Lett.</em>, vol. 4, pp. 111–117, 2015, doi: 10.1016/j.eml.2015.05.004.</span></p>
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<p class="MsoNormal"><span>[9] M. I. Hussein, C. N. Tsai, and H. Honarvar, “Thermal Conductivity Reduction in a Nanophononic Metamaterial versus a Nanophononic Crystal: A Review and Comparative Analysis,” <em>Adv. Funct. Mater.</em>, vol. 1906718, pp. 1–16, 2019, doi: 10.1002/adfm.201906718.</span></p>
<p class="MsoNormal"><span>[10] Y.-F. Wang, Y.-Z. Wang, B. Wu, W. Chen, and Y.-S. Wang, “Tunable and Active Phononic Crystals and Metamaterials,” <em>Appl. Mech. Rev.</em>, vol. 74, no. 2, p. 040801, 2020, doi: 10.1115/1.4034216.</span></p>
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<p class="MsoNormal"><span>[12] K. H. Matlack, A. Bauhofer, S. Krödel, A. Palermo, and C. Daraio, “Composite 3D-printed meta-structures for low frequency and broadband vibration absorption,” <em>PNAS</em>, vol. 113, no. 30, pp. 8386–8390, 2016, doi: 10.1073/pnas.1600171113.</span></p>
<p class="MsoNormal"><span>[13] I. Arretche and K. H. Matlack, “On the Interrelationship Between Static and Vibration Mitigation Properties of Architected Metastructures,” <em>Front. Mater.</em>, vol. 5, no. 68, pp. 1–16, 2018, doi: 10.3389/fmats.2018.00068.</span></p>
<p class="MsoNormal"><span>[14] G. U. Patil, A. B. Shedge, and K. H. Matlack, “3D auxetic lattice materials for anomalous elastic wave polarization,” <em>Appl. Phys. Lett.</em>, vol. 115, no. 9, p. 091902, 2019, doi: 10.1063/1.5116687.</span></p>
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<p class="MsoNormal"><span>[16] E. Baravelli and M. Ruzzene, “Internally resonating lattices for bandgap generation and low-frequency vibration control,” <em>J. Sound Vib.</em>, vol. 332, no. 25, pp. 6562–6579, 2013, doi: 10.1016/j.jsv.2013.08.014.</span></p>
<p class="MsoNormal"><span>[17] Y. Liu, X. Chen, and Y. Xu, “Topological Phononics: From Fundamental Models to Real Materials,” <em>Adv. Funct. Mater.</em>, vol. 30, no. 8, p. 1904784, 2020, doi: 10.1002/adfm.201904784.</span></p>
<p class="MsoNormal"><span>[18] H. Nassar <em>et al.</em>, “Nonreciprocity in acoustic and elastic materials,” <em>Nat. Rev. Mater.</em>, vol. 5, no. 9, pp. 667–685, 2020, doi: 10.1038/s41578-020-0206-0.</span></p>
<p class="MsoNormal"><span>[19] E. Kim and J. Yang, “Review: Wave propagation in granular metamaterials,” <em>Functional Composites and Structures</em>, vol. 1, no. 1. Web Portal IOP, p. 012002, Mar. 29, 2019, doi: 10.1088/2631-6331/ab0c7e.</span></p>
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<p class="MsoNormal"><span>[21] H. Seiner <em>et al.</em>, “Ultrasonic bandgaps in 3D-printed periodic ceramic microlattices,” <em>Ultrasonics</em>, vol. 82, pp. 91–100, 2018, doi: 10.1016/j.ultras.2017.07.017.</span></p>
<p class="MsoNormal"><span>[22] O. Abdeljaber, O. Avci, and D. J. Inman, “Optimization of chiral lattice based metastructures for broadband vibration suppression using genetic algorithms,” <em>J. Sound Vib.</em>, vol. 369, pp. 50–62, 2016, doi: 10.1016/j.jsv.2015.11.048.</span></p>
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<p class="MsoNormal"><span>[24] C. Yilmaz, G. Hulbert, and N. Kikuchi, “Phononic band gaps induced by inertial amplification in periodic media,” <em>Phys. Rev. B</em>, vol. 76, no. 5, p. 054309, Aug. 2007, doi: 10.1103/PhysRevB.76.054309.</span></p>
<p class="MsoNormal"><span>[25] S. Taniker and C. Yilmaz, “Phononic gaps induced by inertial amplification in BCC and FCC lattices,” <em>Phys. Lett. A</em>, vol. 377, no. 31–33, pp. 1930–1936, 2013, doi: 10.1016/j.physleta.2013.05.022.</span></p>
<p class="MsoNormal"><span>[26] D. Torrent and J. Sanchez-Dehesa, “Radial Wave Crystals : Radially Periodic Structures from Anisotropic Metamaterials for Engineering Acoustic or Electromagnetic Waves,” <em>Phys. Rev. Lett.</em>, vol. 103, no. 064301, pp. 1–4, 2009, doi: 10.1103/PhysRevLett.103.064301.</span></p>
<p class="MsoNormal"><span>[27] D. Torrent and J. Sanchez-Dehesa, “Acoustic resonances in two-dimensional radial sonic crystal shells,” <em>New J. Phys.</em>, vol. 12, 2010, doi: 10.1088/1367-2630/12/7/073034.</span></p>
<p class="MsoNormal"><span>[28] J. Carbonell <em>et al.</em>, “Radial Photonic Crystal for detection of frequency and position of radiation sources,” <em>Sci. Rep.</em>, vol. 2, p. 558, 2012, doi: 10.1038/srep00558.</span></p>
<p class="MsoNormal"><span>[29] D. Torrent and J. Sanchez-Dehesa, “Anisotropic Mass Density by Radially Periodic Fluid Structures,” <em>Phys. Rev. Lett.</em>, vol. 105, no. 174301, pp. 1–4, 2010, doi: 10.1103/PhysRevLett.105.174301.</span></p>
<p class="MsoNormal"><span>[30] I. Arretche and K. H. Matlack, “Effective phononic crystals for non-Cartesian elastic wave propagation,” <em>Phys. Rev. B</em>, vol. 102, no. 13, 2020, doi: 10.1103/PhysRevB.102.134308.</span></p>
<p class="MsoNormal"><span>[31] R. K. Narisetti, M. J. Leamy, and M. Ruzzene, “A Perturbation Approach for Predicting Wave Propagation in One-Dimensional Nonlinear Periodic Structures,” <em>J. Vib. Acoust.</em>, vol. 132, no. 031001, pp. 1–11, 2010, doi: 10.1115/1.4000775.</span></p>
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</div></div></div>Mon, 30 Aug 2021 21:42:39 +0000Katie Matlack25398 at https://imechanica.orghttps://imechanica.org/node/25398#commentshttps://imechanica.org/crss/node/25398