Nanophotonics 2023; 12(4): 659–686 Review Anastasiia O. Krushynska*, Daniel Torrent*, Alejandro M. Aragón, Raffaele Ardito, Osama R. Bilal, Bernard Bonello, Federico Bosia, Yi Chen, Johan Christensen, Andrea Colombi, Steven A. Cummer, Bahram Djafari-Rouhani, Fernando Fraternali, Pavel I. Galich, Pedro David Garcia, Jean-Philippe Groby, Sebastien Guenneau, Michael R. Haberman, Mahmoud I. Hussein, Shahram Janbaz, Noé Jiménez, Abdelkrim Khelif, Vincent Laude, Mohammad J. Mirzaali, Pawel Packo, Antonio Palermo, Yan Pennec, Rubén Picó, María Rosendo López, Stephan Rudykh, Marc Serra-Garcia, Clivia M. Sotomayor Torres, Timothy A. Starkey, Vincent Tournat and Oliver B. Wright Emerging topics in nanophononics and elastic, acoustic, and mechanical metamaterials: an overview https://doi.org/10.1515/nanoph-2022-0671 on these subjects, in contrast to a conventional review arti- Received October 28, 2022; accepted December 6, 2022; cle. The introduction of basic definitions for each topic is fol- published online January 27, 2023 lowed by an outline of design strategies for the media under Abstract: This broad review summarizes recent advances consideration, recently developed analysis and implemen- and “hot” research topics in nanophononics and elastic, tation techniques, and discussions of current challenges and acoustic, and mechanical metamaterials based on results promising applications. This review, while not comprehen- presented by the authors at the EUROMECH 610 Colloquium sive, will be helpful especially for early-career researchers, held on April 25–27, 2022 in Benicássim, Spain. The key goal among others, as it offers a broad view of the current of the colloquium was to highlight important developments state-of-the-art and highlights some unique and flourishing in these areas, particularly new results that emerged during research in the mentioned fields, providing insight into mul- tiple exciting research directions. the last two years. This work thus presents a “snapshot” of the state-of-the-art of different nanophononics- and Keywords: acoustics; additive manufacturing; mechanics; metamaterial-related topics rather than a historical view metamaterials; optomechanics; wave dynamics. *Corresponding authors: Anastasiia O. Krushynska, Engineering and Andrea Colombi, Department of Civil, Environmental and Geomatic Technology Institute Groningen, University of Groningen, Groningen Engineering, ETH Zürich, Zürich 8093, Switzerland 9747AG, The Netherlands, E-mail: a.o.krushynska@rug.nl. Steven A. Cummer, Department of Electrical and Computer Engineering, https://orcid.org/0000-0003-3259-2592; and Daniel Torrent, Duke University, Durham NC 27708, USA GROC-UJI, Institut de Noves Tecnologies de la Imatge, Universitat Bahram Djafari-Rouhani, Institut d’Electronique, de Microléctronique Jaume I, Castelló de la Plana 12071, Spain, E-mail: dtorrent@uji.es. et de Nanotechnologie, UMR CNRS 8520, Université de Lille, Villeneuve https://orcid.org/0000-0001-7092-7360 d’Ascq 59655, France Alejandro M. Aragón, Faculty of Mechanical, Maritime and Materials Fernando Fraternali, Department of Civil Engineering, University of Engineering, Delft University of Technology, Delft 2628 CD, Salerno, Fisciano 84084, Italy The Netherlands Pavel I. Galich, Faculty of Aerospace Engineering, Technion – Israel Raffaele Ardito, Department of Civil and Environmental Engineering, Institute of Technology, Haifa 32000, Israel Politecnico di Milano, Milan 20133, Italy Pedro David Garcia, Catalan Institute of Nanoscience and Nanotechnol- Osama R. Bilal, Department of Mechanical Engineering, University of ogy (ICN2) CSIC and BIST, Barcelona 08193, Spain Connecticut, Storrs, CT 06269, USA. https://orcid.org/0000-0003-3803- Jean-Philippe Groby and Vincent Tournat, Laboratoire d’Acoustique de 5084 l’Université du Mans (LAUM), UMR 6613, Institut d’Acoustique – Graduate Bernard Bonello, Institut des Nanosciences de Paris, Sorbonne School (IA-GS), CNRS, Le Mans Université, Le Mans 72085 Cedex 09, France Université, UMR CNRS 7588, Paris 75005, France Sebastien Guenneau, UMI 2004 Abraham de Moivre-CNRS, Imperial Federico Bosia, DISAT, Politecnico di Torino, Torino 10129, Italy. College London, London SW7 2AZ, UK https://orcid.org/0000-0002-2886-4519 Michael R. Haberman, Walker Department of Mechanical Engineering, Yi Chen, Institute of Nanotechnology, Karlsruhe Institute of Technology The University of Texas at Austin, Austin TX 78712, USA (KIT), 76128 Karlsruhe, Germany Mahmoud I. Hussein, Ann and H.J. Smead Department of Aerospace Johan Christensen, IMDEA Materials Institute, Getafe, Madrid 28906, Engineering Sciences, University of Colorado Boulder, Boulder CO 80303, Spain USA Open Access. © 2023 the author(s), published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 660 — A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials 1 Introduction and young researchers from different parts of the world and provided a platform for a fruitful exchange of Metamaterials—rationally designed composites with prop- ideas–arguably, the most essential ingredient for advancing erties exceeding those of their constituents—has been a research frontiers. This work is aimed at sharing the flour- continuously growing field that initially emerged from ishing collaborative spirit that emerged at the colloquium by the study of materials capable of manipulating electro- overviewing the reported results and summarizing recent magnetic waves [1–3]. Metamaterial concepts have rapidly advances in nanophononics and metamaterials, thus zoom- entered and evolved into the domains of acoustics, mechan- ing out from each specific field to get a more general picture. ics, materials science, thermodynamics, condensed matter It could be of relevance to all specialists and especially physics, seismology, civil engineering, and many others. to those who started their research recently and require This rapid progress has been driven by advanced charac- additional support to gain access to these rapidly developing teristics of metamaterials that enable the creation of novel areas. material systems with previously unforeseen functionali- The paper is structured by topic. We begin by summa- ties or superior performance attractive in multiple appli- rizing the state-of-the-art in nanophononics (Section 2) – cations, and further stimulated by recent developments the research field focused on developing electro- or in (additive) manufacturing and computational analysis optophononic systems and materials that enable the control techniques. of vibrations, heat, and quantum communications on the The variety of investigated metamaterial configura- nanoscale [15]. tions, the ranges of their possible functionalities, and the By switching from the nanoscale to the micro-, number of existing or potential applications are so sub- centimeter- and even meter-scales, we enter the realm of stantial that they have become the topic of tens of compre- ultrasound, sound, and infrasound elastic or acoustic waves hensive review papers, e.g., Refs. [4–16], to mention a few. [7] that can be manipulated by means of elastic metamateri- These and other reviews focus on specific nanophononic als (or “phononic materials”) (Section 3) and acoustic meta- and metamaterial configurations, properties, or applica- materials (Section 4). Currently, there is no clear common tion areas and typically consider relevant developments agreement on terminology in the literature on how to dif- from a historical point of view. This work has a different ferentiate between these two classes of materials. Phononic scope. It presents a snapshot of the research activities in crystals are typically defined as periodic composites with the mentioned fields inspired by lively interactions of the inhomogeneities that have a high impedance contrast with researchers working on various metamaterials topics dur- a base material or as periodic single material systems with ing the EUROMECH Colloquium “Emerging topics in acoustic nonuniform unit-cell geometries [4, 5, 17, 18]. Acoustic meta- and mechanical metamaterials” held on 25–27 April 2022 materials can be non-periodic and contain inhomogeneities in Benicássim, Spain. This event gathered over 35 senior or nonuniform features of subwavelength sizes causing Mahmoud I. Hussein, Ann and H.J. Smead Department of Aerospace Rubén Picó, Instituto de Investigaciön para la Gestiön Integrada de Engineering Sciences, University of Colorado Boulder, Boulder CO 80303, Zonas Costeras, Universitat Politècnica de València, Grau de Gandia USA 46730, Spain Shahram Janbaz, Machine Materials Lab, Institute of Physics, University María Rosendo López, Department of Physics, Universidad Carlos III de of Amsterdam, Amsterdam 1098XH, The Netherlands Madrid, Leganés, Madrid, Spain Noé Jiménez, Consejo Superior de Investigaciones Cientïficas (CSIC), Stephan Rudykh, Department of Mechanical Engineering, University of Instituto de instrumentaciön para Imagen Molecular (i3M), Universitat Wisconsin-Madison, Wisconsin-Madison, WI, USA; and School of Mathe- Politècnica de València, Valencia 46011, Spain matical and Statistical Sciences, University of Galway, Galway, Ireland Abdelkrim Khelif and Vincent Laude, Institut FEMTO-ST, CNRS Marc Serra-Garcia, Hypersmart Matter, AMOLF, Amsterdam 1098XG, UMR 6174, Universitë de Bourgogne Franche-Comtë, Besançon F-25030, The Netherlands France Clivia M. Sotomayor Torres, Catalan Institute of Nanoscience and Nan- Mohammad J. Mirzaali, Department of Biomechanical Engineering, Delft otechnology (ICN2) CSIC and BIST, Barcelona 08193, Spain; and ICREA, University of Technology, Delft 2628CD, The Netherlands Barcelona 08010, Spain Pawel Packo, Department of Robotics and Mechatronics, AGH University Timothy A. Starkey, Centre for Metamaterial Research and Innovation, of Science and Technology, Krakow 30-059, Poland University of Exeter, Exeter EX4 4QL, UK. https://orcid.org/0000-0001- Antonio Palermo, Department of Civil, Chemical, Environmental and 6027-1046 Materials Engineering, University of Bologna, Bologna 40136, Italy Oliver B. Wright, Graduate School of Engineering, Osaka University, Yan Pennec, UMET, UMR 8207, CNRS, Universitë de Lille, Lille F-59000, Yamadaoka 2-1, Suita, Osaka 565-0871, Japan; and Hokkaido University, France Sapporo 060-0808, Japan A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials — 661 exotic effective phenomena such as negative bulk modu- Among possible approaches to controlling phonons, lus and/or negative mass density [4, 10, 19]. Here, for sim- the use of thin structures (metasurfaces) and cavity plicity, we use the term “elastic metamaterial” to refer to optomechanics are particularly promising, although phononic materials1 in general where waves are admitted in many challenges remain to be solved. Metasurfaces are solids and the term “acoustic metamaterials” to exclusively subwavelength-thickness structures that allow efficient describe media engineered to manipulate acoustic waves control and manipulation of reflected and refracted waves in fluids. Therefore, elastic and acoustic metamaterials are by tailoring the geometry of the constituent units. During differentiated through the properties of the medium sup- the last decade, increasing attention on metasurfaces in porting the wave propagation, with some analogy to Refs. acoustics [16, 28, 29] and optics [30–32], in combination with [8–10, 12]. Note that some metamaterials can manipulate advances in nanofabrication techniques, have stimulated both elastic waves and airborne sound waves, simultane- progress in developing structures to control phonons on ously [20, 21]. the micro- and nanoscales [33, 34]. Finally, we overview several research directions in Metasurfaces exploit interactions between surface or mechanical metamaterials—tailored composites with Lamb waves and locally-resonant phononic units. For a unprecedented mechanical characteristics, e.g., auxetic metasurface composed of a one-dimensional (1D) array of behavior (negative Poisson’s ratio), zero or negative cylindrical pillars, for example, the graded variation in the effective volume compressibility, or shape-morphing pillar height enables sub-wavelength focusing that is robust functionality (Section 5). with respect to disorder in geometric parameters or in fre- For each topic, we consider practically relevant config- quency fluctuations [24]. The phase shift of the transmission urations, including bulk materials and metasurfaces, and coefficient associated with each resonance of the pillars discuss recent trends in their design, analysis, and manufac- (e.g., the first or the second bending and compressional turing, and also, when relevant, overview the applications modes) is equal to 𝜋 . In order to achieve a phase shift of metastructures formed from metamaterials or metasur- that spans a range of 2𝜋 , as required for a metasurface, faces. We note that this broad review highlights recent an essential feature of the design is to superimpose two research output of the authors, some related studies, and resonances at the same frequency. In our case, the second brief topical reviews needed to introduce relevant concepts. bending and the first compressional resonance frequencies It is not aimed as a comprehensive review, which given the are made degenerate by tailoring the geometrical param- rapid and ongoing progress of the field is not easily covered eters. Then, the transmission phase shift of the different within a single publication. units in the metasurface is made to span a range of 2𝜋 by gradually varying the height of the pillars as a function of position. For example, silicon pillars that are 120 μm in diameter and with a lattice pitch of 150 μm can focus 2 Nanophononics the flexural plate mode propagating in a silicon (crystalline semiconductor) plate at 8.05 MHz, which corresponds to Phonons forming elastic waves can be considered as pow- about a three times larger wavelength than the diameter of erful low-energy information carriers at the core of mod- ern telecommunication devices. Moreover, phonons are effi- the pillars [24]. The sub-wavelength focusing is achieved as cient carriers of heat; this offers a functional value beyond far as the parameter F ∕D remains below 0.25, where F is electrons, spins, and photons. There has been substantial the focal length and D the length of the metasurface [24]. progress in the study of phonons as heat carriers with rel- An experimental demonstration of focusing functionalities evance to several applications, such as thermoelectrics [22] has been obtained by using 3D-printed polymer materials and heat dissipation [23]. Current research directions also in the mm range [25]. Similar functionality can be achieved include the investigation of robust-to-disorder focusing and by using elliptical pillars that have two degrees of freedom, imaging at broadband frequencies [24, 25], and harnessing to implement the gradual change in phase shifts [35] while nonlinear effects to enable phonon localization, breathers, avoiding the difficulty of height variation in a microfabri- bifurcation, and chaos [26, 27]. cation process. Although the design of metasurfaces based on elliptic scatterers has been already proposed in optics [36, 37], its application to the manipulation of elastic Lamb 1 In some reviews, the term “phononic materials” is used to describe waves in a plate is new. all categories of architected materials for the manipulation of elastic Alternatively, resonator-to-resonator coupling in dual and/or acoustic waves [8, 16]. resonators or in a periodic nanostrip can be exploited to 662 — A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials achieve coherent driving of the mechanical micro- or nano- mechanical Q factors at 2.4 GHz in the resonators fabricated resonators with surface waves and, reciprocally, to con- in nc-Si were found to be higher compared to those struc- trol surface wave propagation at a deep sub-wavelength tures fabricated in SOI, 745 compared to 1680 in SOI and scale. The confinement and enhancement of elastic energy nc-Si, respectively, which led to the question as to the origin in volumes much smaller than the excitation wavelength of the reduced losses, or higher Q, in nc-Si OM cavities in can be demonstrated by using either individually fabricated what otherwise were identically geometrically designed 1D pillars by focused ion beam induced deposition (FIBID) or OM crystals. The study of nc-Si with three different average periodic high-aspect-ratio nanostrips, both integrated on a grain sizes showed that higher thermal diffusivity or more highly coupled piezoelectric substrate [26, 27]. In the case efficient heat transport was associated with the larger mean of dual resonators, optical measurements by laser scanning nc-Si size (290 nm) or the larger volume ratio of nc-Si volume interferometry have been used to enable direct observa- to that of grain boundaries, supported by heat dissipation tions of the frequency splitting response and dipole-like rates being the largest for the 290 nm diameter nanocrys- mode shapes of the coupled resonators. The gap distance tallites. OM crystals and nc-Si membranes have promising and the orientation of the coupled resonators with respect potential for sensing, communication, and coherent quan- to the surface acoustic wave (SAW) wavevector highlight the tum control of mechanical oscillators [41, 42]. vectorial nature arising from the dipole-like mode shape. One of the “hottest” research directions in this field In the case of nanostrip resonators, classical electroacoustic is to realize nano-opto-electromechanical systems (NOEMS) impedance measurements can be achieved at a different enabling the synergistic coexistence of electrical, mechani- stimulation power. The elastic field behavior can be fur- cal, and optical signals on a chip. The major challenge here ther controlled through other types of coupling by reduc- is to make such technology platforms fully compatible with ing the resonator gap and the discrete resonator mode. mainstream CMOS technology. The quest to demonstrate For instance, nonlinear geometry as well as elastic nonlin- an OM circuit based on nc-Si OM resulted in an on-chip earity can be achieved through the control of geometrical NOEMS platform at the proof-of-concept stage, as shown parameters of the unit cell and the resonance mode of a in Figure 1. This is an electrically pumped OM nanobeam single nanostrip finger, leading to a variety of nonlinear by 2 GHz surface waves in a platform based on nc-Si, Si, and AlN with metal contacts that can successfully convert interaction schemes. In the case of a compressional mode microwave signals into optical signals [43]. In these experi- of the nanostrip, reducing the gap distance and increasing ments, the achieved mechanical Q factor is 4000, limited by the SAW amplitude induces a frequency shift of the device losses in the material and the optical Q factor of 10,000 at and allows the implementation of high-frequency phononic- room temperature. The data analysis yielded 200 coherent NEMS (nanoelectromechanical systems) circuits with com- phonons per microWatt, and a peak sensitivity of below plex dynamics. three phonons, which is very promising for a platform Despite the reported progress, studying phonons on their own remains a challenge. A possible solution to over- come this hurdle implies the use of optomechanics (OM) concepts. OM exploits the confinement and interaction of optical and mechanical waves. OM interaction forms a foun- dation for the relationship between phonons, considered as mechanical modes, and photons, which results in the transduction of optical modes in the THz into mechanical modes in the sub-GHz regime and vice versa. An alternative is the transduction of electrically driven mechanical energy from, e.g., SAWs into an optical or electronic signal. Common realizations of OM structures, also known as phoXonic crystals [38], rely on cavity optomechanics that enables coupling of electromagnetic radiation and mechan- ical vibrations enhanced by confining the electromagnetic Figure 1: The NOEMS platform for room temperature operation at radiation in a cavity. The main attention so far has been 2 GHz. The top left inset shows the normalized transmission (black line) on studying 1D cavities realized both in silicon-on-insulator and normalized reflection (red line). Reprinted with permission from D. (SOI) [39] and in nanocrystalline silicon (nc-Si) membranes Navarro-Urrios, M. F. Colombano, G. Arregui, et al. ACS Photonics, 2022, 9, [40] at room temperature and ambient conditions. The 2, 413–419. Copyright 2022 American Chemical Society. A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials — 663 fully compatible with Si technology. The performance of Finally, a promising direction to prevent and minimize this and similar devices critically relies on efficient con- phonon dissipation and losses within the GHz range is to trol of thermal energy transfer and thermal conductivity. engineer the geometry of the nanostructure in order to The analysis of the impact of grain boundaries on thermal open wide frequency gaps in the spectrum inducing destruc- transport in silicon micro- and nano-structures [44] clarified tive interference in all directions [53]. The related plat- that the crystalline state is a key parameter in NEMS and/or form, demonstrated only very recently, enables the creation NOEMS. Similar devices exploit non-linear physical mecha- of waveguides and cavities where the OM coupling can nisms, such as the chaos regime [45] and synchronization be exploited to activate nonlinear dynamics and coherent [46], to unveil rich sets of fundamentally different com- amplification of vibrations at room temperature [54]. plex dynamics and to switch on/off synchronization of the mechanical dynamics in OM crystal cavities. These develop- ments open new horizons for exploiting silicon-based OM 3 Elastic metamaterials crystals in neurocomputational networks and chaos-based applications. The field of elastic metamaterials has largely extended in The extension to two-dimensional (2D) optomechanical the last few decades from studying mass-spring systems platforms enables the realization of Anderson localization [55, 56] and circular periodic heterogeneities or nonuniform of optical modes by using disorder in cavity optomechanics features [5, 17, 18] to developing intricate designs with spi- [47]. For instance, consider a waveguide etched in a sus- ral patterns [57], hierarchical [58], or bio-inspired organi- pended silicon membrane with a slotted line air defect. The zation [59, 60]; from the analysis of Bragg scattering and air slot allows strong confinement of the electromagnetic local resonances of elastic waves [19, 61] to topological opti- field and enhances the ability for light to couple to in- mization of band structures [62], and studying tunable [63], plane mechanical motion, while inherent and unavoidable nonlinear [64], and time-varying metamaterials [65]; from fabrication imperfections are adequate to induce sufficient broadband wave attenuation [5] to futuristic applications back-scattering. The resulting tightly confined Anderson- including speech recognition [66], active thermal cloaking localized modes can be driven to enable mechanical ampli- [67], realistic energy harvesting [68], and seismic wave mit- fication and self-sustained phonon lasing via optomechan- igation [69] systems. This progress has become possible ical back-action. This opens perspectives for studying mul- thanks to advances in computational techniques and manu- tiple scattering and Anderson localization of bosonic exci- facturing approaches that enable the analysis, optimization, tations with parametric coupling to mechanical degrees of and production of complex metastructures at various size freedom. scales. It is worth remembering that open nanophononic In the following, we summarize recent trends in the structures with resonators suffer from radiation loss that design (Section 3.1) and analysis (Section 3.3) of elastic meta- results in a finite lifetime of the modes. Instead of nor- materials as well as in the studies of nonlinear, time-varying, mal modes with pure real eigenfrequencies typical for and externally controlled metamaterial media (Section 3.2). closed systems, open systems possess quasi-normal modes Finally, we consider several emerging applications for elas- (QNMs) with complex-valued eigenfrequencies and diverg- tic metastructures (Section 3.4). ing power at infinity. QNMs have been widely discussed in photonics [48, 49], but the advanced tools required for their analysis of acoustic and elastic waves still require 3.1 Design of elastic metamaterials improvement [50]. Of special interest is the determination of Most of the fascinating functionalities of elastic metamate- complex eigenvalues and eigenmodes, and the definition of rials rely on their ability to generate absolute (directional) an adequate modal volume and the elastic equivalent of the band gaps-frequency ranges with inhibited (directional) Purcell effect [51]. A practical way to obtain all QNMs of an wave propagation. elastic resonating structure is to rely on a stochastic method Band-gap engineering and, on a broader scale, control to evaluate all possible resonances from the response of over wave dispersion, are implemented by designing the a system to a random force [52]. Because the total energy structural topology of a metamaterial that can be done of QNMs is unbounded, the usual normalization relations based on intuition or bio-inspiration, by means of phe- for normal modes are not applicable, while a multipole nomenological approaches or computational optimization, expansion with complex coefficients can be obtained, and or any combinations of these. Such strategies extend pos- a complex modal volume can be defined. sible phononic designs and further expand the range of 664 — A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials elastic metamaterial functionalities or possible devices. compressional, in-plane shear, flexural, and torsional vibra- Below, we present some examples of recent design ideas tions, as proven experimentally [70]. A tapered version of with a focus on lightweight configurations that can sup- this structure has also been demonstrated with a wide- press low-frequency elastic waves. Such structures are band response [71]. Another example is a bio-inspired meta- vitally important for diverse applications that require vibra- rod with a structure resembling that of Turritella Tere- tion isolation and wave manipulation. Furthermore, recent bra shell. 3D-printed prototypes reveal complex resonat- design preferences have been shifting towards single- ing modes determined through resonant ultrasound spec- material configurations to facilitate manufacturing. troscopy and are in good agreement with finite-element Elastic metamaterials can be formally divided into 1D simulations (unpublished work). (metabeams or metarods), 2D (metasurfaces), and 3D (bulk) Elastic metasurfaces – structured materials, typically, configurations according to the directionality of their wave equipped with mechanical resonators – have been open- attenuation performance. ing bright possibilities for the control of surface elastic An interesting example of an intuition-based 1D design waves across different frequency ranges [16]. Hybridization is an aluminum meta-beam with periodic quadruple- between propagating surface waves and localized resonator mode resonators that exhibits a band gap around 1 kHz motion enables probing the wave dispersion [72] and brings (Figure 2a). This structure can efficiently dampen out rise to peculiar phenomena such as surface-to-bulk wave Figure 2: Examples of phononic materials. (a) Meta-beam with quadruple-mode resonators (adapted from Figure 1 published in: Kentaro Fujita; Motonobu Tomoda; Oliver B. Wright; Osamu Matsuda; Appl. Phys. Lett. 115, 081905 (2019), Copyright © 2019); (b) metasurface with aperiodic resonators; (c) octet panel with a highlighted unit cell; (d) 3D phononic material with roton-like dispersion (originally published in Ref. [86]); (e) flexible metamaterial with a “rotating-square” structure (adapted from: B. Deng, P. Wang, Q. He, V. Tournat, K. Bertoldi, Metamaterials with amplitude gaps for elastic solitons, Nat. Commun., Nature Research, 2018); (f–g) self-assembled lattices within 2D magnetic boundaries with square and quasi-crystal configurations. A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials — 665 conversion [73, 74], rainbow trapping [75], wave localization and wave control. Bio-inspired designs have shown inter- [76], and non-reciprocal propagation [77]. Typical metasur- esting wave attenuation properties in hierarchical (diatom- faces consist of a cluster of oscillators arranged in a peri- like) [58], spiral-shaped (cochlea or shell-like), and frame odic fashion over an elastic half-space. A new emerging (spider-web-like) elastic metamaterials [59, 60] (Figure 2c). trend is, rather, the development of aperiodic metasurfaces Band gaps relative to the corresponding non-hierarchical (Figure 2b). The work of Marchal et al. [78] provides exper- structures are mostly preserved in both types of structures, imental data on the dynamics of confinement of elastic but additional hierarchically induced band gaps appear in waves within a cavity in a phononic crystal. A noncontact some cases. Furthermore, hierarchical configurations are laser ultrasonic technique is used to monitor in situ the amenable to a high degree of tunability using a limited displacement field within the cavity as a function of time number of geometrical parameters, and simultaneously and space after a surface acoustic pulse has been excited ensure reduced structural weight. The role of viscoelasticity, outside the cavity. It is shown that the time evolution of which is essential for complex designs, was also considered confinement is distinct according to the symmetry of the [58, 83, 84]. eigenmode. More recently, the same team has investigated A strong source of inspiration for creating new elastic the localization of surface waves after propagation in a meta-designs remains solid-state physics. For instance, the diffusive medium consisting of pillar-shaped mechanical research group of M. Wegener has proposed a 3D roton elas- resonators randomly distributed on a surface (unpublished tic metamaterial for manipulating transversal and longitu- ongoing work). The samples are formed by an assembly of dinal elastic waves (Figure 2d), that exhibits low-frequency several thousand aluminum pillars arranged randomly on dispersion similar to the famous roton dispersion, which a 200 nm thick film, also made of aluminum, deposited on was suggested by Landau around 80 years ago to explain a semi-infinite silica substrate. Both the diameter and the the sharp transition of specific heat in liquid Helium-4 height of the pillars are 300 nm. Depending on the sam- observed at a critical low temperature [85]. The term “roton” ples, the filling ratio (ratio of the total section of the pillars originates from the dependence of the phonon energy on to the surface of the sample) is between 7 and 9%. The the momentum. For small momenta, the phonon energy localization of elastic energy in such a random structure reveals a linear dependence, while a local minimum of results from the interference of the waves bouncing back energy versus momentum occurs for larger momenta. To and forth within the free surface surrounded by pillars. achieve a similar behavior for elastic waves, it was pro- For the surface waves with a spectral content extending posed to introduce beyond-nearest-neighbor (i.e., non-local) up to 2 GHz, the measurements enabled the reconstruction interactions between effective masses that must dominate of the cartography of surface displacements in a purpose- over nearest-neighbor interactions. This idea can be real- built phononic cavity that clearly shows stationary waves, ized in a practical 3D elastic metastructure by connecting which in this particular example last for approximately small cubes, which act as effective masses, by means of 2.5 ns. cylindrical rods, which act as effective springs. The struc- In the realm of 3D elastic metamaterials, representative ture is embedded in an auxiliary frame whose aim is examples are frame-based lattices and shell-like structures to mediate the beyond-nearest-neighbor interactions, real- known for their exotic ability to manipulate waves [79]. The ized from a single polymer by 3D additive manufacturing octet-like lattices (Figure 2c), among others, have a broad- [86]. The designed metamaterials were tested experimen- band gap, in which local-resonance and Bragg scattering tally and revealed the roton dispersion for a transversal naturally coalesce [80]. Shell-based structures (of Schwartz elastic wave at ultrasonic frequencies in micrometer-sized primitive type) with topology based on periodic minimal samples [86]. surfaces also have a promising potential for wave control. Another prominent example of a physics-inspired elas- By changing the geometrical parameters of the lattices, one tic metastructure is a mechanical analog of twisted bilayer can easily tune the band structure and achieve wave focus- graphene [87, 88]. This is a structure composed of two ing [81], thus laying the ground for the development of novel graphene monolayers placed in direct contact with each wave control structures for vibration isolation, focusing, other after rotating one of them by a certain small angle and, possibly, energy harvesting. 𝜃 . This angle is responsible for forming a so-called Moiré Bio-inspiration and structural hierarchy can provide pattern – a periodic arrangement that could alter the elec- powerful tools to design lightweight elastic metamaterials tronic band of the bilayer graphene. At a specific “magic” [82]. Inspiration can be drawn from numerous examples of interlayer rotation angle, twisted bilayer graphene devel- natural structures providing efficient vibration attenuation ops quasi-flat bands through an unconventional mechanism 666 — A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials connected to carrier chirality, which enables a wealth of to extend band-gap tunability by preserving the geometric exotic, correlated-electron phases in the system [89]. The structure is to use 3D-printed photo-responsive polymers, corresponding mechanical realization is represented by which can reversibly change their Young’s modulus by up two vibrating plates patterned with a honeycomb mesh of to 30% upon laser illumination [97]. masses and coupled across a continuum elastic medium. Nonlinear phenomena are another area that continues By connecting the movement of the upper plate to the to attract active research within the field. Material non- movement of the bottom plate by means of an appropriate linearities may be activated by exciting large-amplitude coupling coefficient, it is shown that flexural waves can mechanical waves in highly compliant elastic meta-media, exhibit vanishing group velocity and quasi-flat bands at resulting in a separate class of elastic metamaterials called magic angles are in close correspondence with electrons “flexible mechanical metamaterials” [98]. While the static in graphene models [87]. The strong similarities of spec- character of such structures has been widely studied, elu- tral structure and spatial eigenmodes in the two systems cidating their dynamic properties is still in its early stages, demonstrate the chiral nature of the mechanical flat bands especially with regard to nonlinear dynamics. The nonlin- [88]. Experimental realization of this proposal is also in ear properties can be designed in a rational way, using view [88]. appropriate lump-parameter models and analytical devel- Finally, an alternative design approach aimed at opti- opments [98–100], to enable control of large-amplitude elas- mized elastic metamaterials in terms of target character- tic waves. A prototype of a flexible mechanical metamate- istics (e.g., the band-gap width) is based on computational rial that has been most studied in the context of nonlin- tools. This approach has proven to be fundamental for ear waves is the “rotating square” structure [98]. It con- exploring the vast geometry-property spectrum [90–94]. sists of rigid square masses, connected at their corners to Yet, critical steps in the optimization procedure are often neighboring masses by thin elastic ligaments (Figure 2e). conducted manually. For example, to optimize a metama- Several degrees of freedom in displacement and rotation terial waveguide for wave attenuation and appropriate must be considered for each rigid unit and the thin elas- mechanical performance, one needs first to analyze the tic ligaments can be modeled by longitudinal, shear, and band structure of a representative unit cell and adjust the bending springs. The discrete equations of motion obtained required frequency range. Several configurations may then with this lump-parameter model serve as the basis for be tested in terms of wave transmission and maximum the numerical simulation of nonlinear wave propagation. stress induced by external loading. Next, the best design is Also, using the appropriate assumptions on these discrete selected for further optimization, whereby the design space equations of motion for the wave effect under consider- is surveyed (also performing sensitivity analysis) in pursuit ation, analytical solutions can be found in most cases. In of an optimized configuration. The final design can then particular, it has been shown that rotating square struc- be fabricated and validated experimentally. In contrast, it tures can support vector elastic solitons – large-amplitude is more desirable to design such structures systematically, pulses propagating without changing shape due to the per- e.g., by means of topology optimization – a technique that fect balance between nonlinear and dispersive effects [99, combines finite element analysis with gradient-based opti- 100]. The coupling between rotational and translational mization. One of the recent realizations of a topology opti- degrees of freedom gives the vector character to this soli- mization procedure combines such enriched finite-element formulation in the analysis with a level-set function that is tary wave. Such vector solitary waves in a flexible mechan- used to represent topology [95, 96]. Such computational tools ical metamaterial exist theoretically and experimentally can effectively be used to design multi-functional phononic for a particular range of amplitudes that can be tuned by media that exhibit, e.g., band gaps and required mechanical the specific design of the metamaterial, which highlights performance simultaneously. “amplitude gaps” for vector elastic solitons, by analogy with frequency gaps in dispersive linear media [100]. Various wave or material configurations in this context have been 3.2 Time-varying, nonlinear, and tunable studied [101–104], among which is a “mechanical diode” elastic metamaterials for vector elastic solitons [100]. In the case of multi-stable In parallel to the search for new elastic meta-configurations, designs, so-called transition waves, or topological solitons, substantial attention has been given to harnessing non- can propagate. These waves sequentially switch the units trivial material behavior to generate or enlarge band gaps of the system from one stable state to another, result- and to break the limits imposed by linear base materials ing in the reconfiguration of part or all of the medium on controlling wave dispersion. One of the recent examples [105–107]. A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials — 667 In conventional solid structures, such as rods and long time due to difficulties in the practical implementation beams, nonlinear effects may be introduced through the of this concept. The recent renewed interest in spatiotempo- stress-strain relation and/or the strain-displacement gradi- ral metamaterials is due to the spectacular developments in ent relation. For a homogeneous medium, nonlinear disper- new technologies and artificial intelligence, which promise sion relations may be obtained in exact and analytical form a bright future for this new generation of metamaterials as shown for rods [108, 109] and beams [108, 110]. In these [120–124]. Although this branch of metamaterials is still very works, a theory is given that provides the nonlinear disper- young, studies combining Floquet engineering and meta- sion relations without resorting to perturbation methods, materials show great possibilities for the development of thus being applicable to strongly nonlinear models. In Ref. interesting applications based on the control of the temporal [109], a harmonic dispersion relation for the prediction of properties of waves. Time-varying media are promising can- the spectra of the generated harmonics is also provided. This didates for controlling wave frequency without resorting general approach has also been combined with the standard to nonlinear effects, which pose theoretical and practical transfer-matrix method to obtain approximate nonlinear difficulties. dispersion relations for periodic media. One can consider, On the other side, one of the major limitations of linear for example, longitudinal waves in a continuous thin rod and nonlinear elastic and acoustic metastructures is their with a periodic arrangement of resonators [111] or material fixed topology which results in predefined wave dispersion properties [112]. This allows for the analytical prediction once these structures are fabricated. Despite many theo- of band-gap characteristics. Among the interesting results, retical and computational studies on tunable metamate- for example, is a demonstration that large deformation in a rial configurations, the practical realization of experimental nonlinear elastic metamaterial may cause a pair of isolated platforms remains limited because of challenges in manu- Bragg-scattering and local-resonance band gaps to coalesce facturability, the destructive nature of reprogramming, and to create a much larger band gap [111]. inherent nonlinearities [125–127]. A valid path for func- An alternative approach for elastic metamaterials with tional tunability of metamaterial matter is external stimuli, non-trivial material behavior is time-varying metamateri- e.g., either static or dynamic load, which can effectively als, in which time acts as a new degree of freedom [113]. induce an external field to tune the material. This, however, The concept of time symmetry was first violated by tem- can translate to large shape change in the material with poral modulations in photonic materials. Since then, this the potential rise of instabilities, phase transitions, degrada- phenomenon has attracted much interest leading to the tion, and geometric frustration. This can hinder our ability implementation of novel nonreciprocal wave management to utilize programmable materials [107, 128]. To overcome techniques [114]. Moreover, the need for a fundamental these limitations, the group led by O. R. Bilal has proposed to understanding of wave phenomena in the time domain has obtain programmable dynamical characteristics by utilizing led to the discovery of other surprising physical effects, tunable nonlinear magnetic lattices [129–132]. Such lattices such as frequency conversion [115], temporal aiming [116], can be created by confining self-aligned, free-floating disks or an inverse prism [117]. The properties of time-varying of embedded permanent magnets in a range of symmetries metamaterials can change in time due to temporal varia- within a fixed boundary and a pre-engineered magnetic tions in some of their physical parameters induced by an field (Figure 2f–g). The magnetic repulsive forces between external energy source, thus changing their response over the boundary magnets and the free-floating disks cause time, which can be considered as next-generation metama- the disks to self-align and rest within their surrounding terials [118]. A basic time-varying metamaterial can be rep- magnetic potential wells. Each disk or group of disks can resented by a temporal multi-layer scheme of an unbounded act as a single building block of the lattice. These blocks string, without dissipation, consisting of a cascade of dif- repeat in space, forming a 1D or 2D configuration with ferent speed media over time. An initially transverse wave unique wave propagation characteristics. The geometry of propagating in such a string splits in multiple forward and the confining boundary, disk symmetry, and the number of backward waves at each time interface, due to the change embedded magnets within each disk can be designed such of speed; this complicates the wave dispersion analysis as that the tuning of the boundary’s magnetic field controls further discussed in Section 3.3. the assembly of the disks. The relative positions and orien- In a more general definition, metamaterials that exhibit tation angles between the disks define the corresponding spatial and temporal changes are called spatiotemporal metamaterial phase (ordered, quasi-ordered, or random) metamaterials. Despite these metamaterials being proposed and its stability conditions (mono-stable or multi-stable). By many years ago [119], this topic was hardly explored for a changing the boundary geometry, the number of disks, their 668 — A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials symmetry, and therefore the resulting magnetic couplings and resonators in both 2D and 3D contexts by analysing a within the unit cell, the transmittable frequencies through series of metasurface configurations for wave conversion, the metamaterial can be programmed. The proposed plat- wave trapping, and topological protection of surface edge form can be utilized to engineer self-assembled meta- states. structures capable of manipulating low-frequency waves, Nonlinear and time-modulated metasurfaces require within a relatively small volume, while utilizing negligi- more advanced modeling strategies [77, 138]. For time- ble mass. Moreover, the nonlinear potentials between the modulated scenarios, a metasurface can be composed of res- disks and their boundaries can be harnessed to demonstrate onators with a periodic space-time modulation of stiffness. phenomena with nonlinear parallel such as amplitude- By means of an asymptotic approach, one can obtain an ana- dependent response, bifurcation, chaos, non-reciprocity, lytical solution describing the dispersion properties of such and solitons. In addition, the self-assembling nature of the a structure. The dispersion analysis shows that a spatiotem- disks can be key in creating reprogrammable materials with poral stiffness modulation yields non-reciprocal features in exceptional properties. the Rayleigh wave spectrum where the frequency content of a propagating signal can be filtered and converted to bulk 3.3 Analysis techniques for elastic waves [77]. metamaterials A nonlinear metasurface with an array of Duffing oscil- lators, namely spring-mass resonators featuring a nonlinear Complex metamaterial designs, material or geometric non- force-amplitude response of a cubic type, can be studied by linearity, and strong external stimuli require efficient anal- adopting a leading-order effective medium description. This ysis techniques that can deliver reliable results and useful approach allows the prediction of the amplitude-dependent insights into the physical aspects of the wave propagation dispersive characteristics of Rayleigh waves. It was shown process. that hardening nonlinearity shifts and reduces the attenu- Currently available commercial finite-element soft- ation bandwidth of the metasurface, while softening non- ware allows exploring almost arbitrarily complex designs linearity, in contrast, yields lower and broader frequency of 3D elastic meta-media. The analysis steps and related gaps [138]. This opens new opportunities to realize tunable details remain, however, the responsibility of the software devices for surface wave control. users. For instance, it is important to verify the consistency Nonlinear metaclusters and metasurfaces with wave of the choice of a unit cell and the first irreducible Brillouin scatterers displaying nonlinear properties can, alterna- zone to exclude the possibility of erroneous conclusions. tively, be analyzed by perturbation theory. This approach is This is especially relevant for lattices of a low-symmetry group and when base material(s) behaves non-linearly or illustrated in Ref. [139] for a nonlinear multiple scattering non-elastically [133]. There are two possible shortcomings system represented by a beam with a cluster of scatterers here: (i) the use of non-primitive unit cells for the analy- having a cubic dependency of the attachment forces on sis of directional band gaps and (ii) the limitation of the the displacement. The obtained solution shows that both computations to the reduced Brillouin zone paths. These reflected and transmitted amplitudes depend on the direc- can influence the accuracy of band-gap estimations and tion of incidence. This implies that apart from asymmetric result in fictitious bands. The latter numerical phenomenon reflection, possible with linear scatterers, a cluster of non- is known as spatial aliasing [134]. Low-symmetry designs linear scatterers may also present asymmetric transmission, can be analyzed by using a recently proposed algorithm thus breaking reciprocity. Further, the scattering properties to construct an irreducible Brillouin zone for any crystal of the system depend on the incident wave amplitude. Con- structure in 2D and 3D [135]. sequently, the tunable asymmetric scattering properties can In the realm of metasurfaces, finite-size structures with be used to construct acoustic devices, e.g., tunable filters aperiodic clusters of resonators are nowadays analyzed or frequency converters [139]. Such a frequency converter, numerically. However, it is possible to develop an analyti- which – when hit by a wave of certain frequency – trans- cal framework based on the multiple scattering technique mits only waves at the third harmonic can be built with a to model the wave field in elastic metasurfaces [87]. To cluster of three scatterers. In the cluster, the two outer scat- this purpose, one can utilize the solution of the classical terers are linear and are designed to prevent transmission Lamb’s problem as a Green’s function to describe the mul- of the fundamental frequency and the reflection of the third tiple scattered fields generated by a cluster of mechanical harmonic. The middle scatterer is cubically nonlinear and resonators attached to a surface. Using this approach, Refs. thus generates a third harmonic wave. A significant portion [76, 136, 137] discuss the interplay between surface waves of the incident wave is then reflected, while the transmitted A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials — 669 fundamental-frequency amplitude and the reflected third- to avoid resonance with leading earthquake frequencies. harmonic amplitude both vanish at the nominal working The effective properties of such devices (e.g., the effective frequency, regardless of the incident amplitude. At the same vibration period and the effective damping coefficient) can incident frequency, the cluster transmits the third-harmonic be tuned by changing the geometry of a device, the friction wave. Therefore, the cluster acts as a frequency converter, coefficient of the sliders, the preconditioning, the energy where only the third-harmonic component is transmitted. dissipation capacity of the tendons, the ratio between the This frequency converter also demonstrates the breaking of design value of the re-centering force, and the maximum reciprocity, where incidence from one side, at a specific fre- vertical load [69]. An additional advantage is that the fab- quency, results in transmission of the third harmonic com- rication does not require expensive materials and is par- ponent, whereas incidence from the other side results in tially or fully achievable with ordinary 3D printers. The no transmission [139]. Therefore, the generation of higher- metallic parts can be manufactured using standard lathe harmonic waves and the correction to the fundamental machines and/or fabricated with desktop metallic 3D print- frequency wave can lead to some exceptional scattering ers. Actively controllable versions of such metamaterials behavior when a number of scatterers are clustered on a can be equipped with actuated cables to provide supple- beam. Such systems may facilitate the implementation of mentary re-centering capacity and stiffness control. The lat- elastic analogs of electronic devices, such as isolators and ter is useful to adjust in real-time the fundamental vibra- logic devices. tion period of the device to the earthquake frequency and energy content, as well as to re-center the system at the 3.4 Applications of elastic metamaterials end of a seismic oscillation in case of permanent defor- mations. These metastructures pave the way to a cus- Existing and emerging elastic metamaterials have a promis- tomized approach to protect artworks, houses, and essen- ing potential in multiple applications that require vibra- tial equipment in industrialized and developing countries tion mitigation and waveguiding, wave focusing and col- [143, 146]. limation, wave lensing and demultiplexing, sensing and Vibration energy harvesting (VEH), i.e., the conversion scattering-free transmission, energy conversion, acoustic of mechanical vibrations into electrical energy, is another cloaking, topology-protected states, etc. Elastic metasur- popular application area of elastic metamaterials, which faces open new possibilities to design and realize sim- has approached the proof-of-concept stage [147]. Elastic ilar devices for surface waves. Many of these applica- metamaterials are promising to address a current challenge tions are mentioned above or thoroughly discussed else- in the field – to develop microscale harvesters in the form of where [5, 8, 16, 140–142]. Here we consider several recent developments relative to elastic metamaterials for seismic micro-electromechanical systems (MEMS). The implemen- wave mitigation and energy harvesting and highlight two tation of VEH solutions for efficient energy conversion on more exotic application areas – active cloaking and speech the microscale to meet the decreasing power roadmap of recognition. microelectronics within 5–10 years is now recognized as A new twist to seismic phononic media [143, 144] is given a crucial goal [148]. To reach this goal, recently developed by bio-inspired seismic insulators [69]. These are formed elastic metastructures with endowed piezoelectric inserts by periodic arrays of building blocks that replicate the rely on the idea of creating a parabolic acoustic mirror, mechanics of the human body and/or animal locomotion. introducing point defects in periodic lattices, or an acous- Such blocks are composed of linkages mimicking human tic lens to concentrate narrow-band vibrations [149]. An bones and stretchable cables replicating muscle tendons. alternative approach proposes to exploit graded arrays of The working mechanism is based on the biomechanics of sub-wavelength resonators [150]. animals that can adjust their natural vibration frequencies The basic design can be represented by local resonators to reach a state of resonance with the forces produced on waveguides in the form of beams to create a meta- by the contraction of muscles during locomotion. This fre- frame or meta-lattice controlling the propagation of elas- quency tuning process enables the production of motion tic waves. This idea can be extended to 3D architectures, with low energy consumption [145]. Similarly, human bones as has been done in [151] to achieve selective filtering in legs and arms behave as pendulum systems that are of transverse and longitudinal waves. Here, however, the bent by muscles to tune their resonant frequency, while focus is on planar structures, in order to keep compatibil- the tendons act like nonlinear springs and shock absorbers. ity with the typical industrial processes adopted in indus- The proposed meta-isolators work in an opposite fashion: trial MEMS manufacturing. The actual realization of the they tune the nonlinear stiffness of the tendons in order described metastructure includes square unit cells of a 670 — A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials chiral structure and suitably graded local resonators at the cut manufacturing expenses. For this purpose, a promising corners. The interplay between chirality, local resonance, solution is the soft nanoimprinting lithography technique and grading enables the achievement of a continuous redi- that provides an exciting opportunity for the fabrication rection of the elastic waves, which are finally confined of nanostructures in a scalable, fast, and inexpensive way. and amplified inside the square unit cell. By implement- It uses pre-patterned soft elastomeric stamps to induce a ing the piezoelectric coupling in some specific target res- nanostructure onto a variety of materials such as conduc- onators, the meta-lattice can be tailored for energy har- tive polymers [155] and cellulose [156]. Patterned stamps can vesting applications, minimizing energy losses due to wave also induce the long-range ordering of metal colloids [157, scattering. The performance of the meta-frame is prelim- 158] and perovskite nanocrystals [159] in what is known as inarily checked numerically. The constituents of the unit template-induced self-assembly. In all cases, the resulting cell are modeled using Timoshenko (shear flexible) beams architectures can exhibit a resolution below 100 nm while with six degrees of freedom per node to reduce the high covering an area of 1 cm2 . This fabrication route allows computational burden that would arise when using 3D solid the combination of the desired properties of a pattern with elements. The dynamic response to narrow-band signals those of the original material, resulting in a new gener- confirms that elastic waves are effectively redirected and ation of inexpensive components such as biodegradable strongly interact with the resonators equipped with a piezo- photonic films, highly efficient SERS platforms for sensing, electric material, with the consequence of enhanced energy chiral metamaterials, improved efficiency solar cells, and harvesting with respect to conventional single-resonator more. systems. These results pave the way for the development An exciting newly emerging application domain for of microscale devices by means of standard etching pro- mechanical metamaterials is computation and artificial cesses, with the deposition of lead-free piezoelectric materi- intelligence. Examples of intelligent tasks that have been als such as aluminum nitride. The microscale prototypes are theoretically or experimentally demonstrated in metama- characterized by means of advanced techniques, making terials include the error-correction of communication sig- use of ad-hoc excitation probes and specific measurement nals [160, 161], implementing digital computations [162, procedures [68]. 163] and finite-state machines [164], classifying Braile char- Cloaking represents one of the unusual functional- acters [165] and recognizing spoken commands [66, 160, ities of elastic and acoustic metamaterials, which can- 166]. In these applications, metamaterials have the crucial not be achieved by conventional engineering materials. advantage of requiring zero power. While a conventional Recent developments in thermal cloaking involve using intelligent device such as a Google Home© or Alexa© active sources to cloak objects from thermal measure- speaker consumes energy all the time (even when it is ments instead of exotic materials. Rather than working not being used), a speech-classifying metamaterial will with the heat equation in the time domain as in [152], detect commands by leveraging the acoustic wave physics, the proposed method relies on the frequency domain for- without consuming external power – satisfying a crucial mulation for Helmholtz equation solutions with complex requirement for edge computing applications. Designing wavenumbers, thus, extending the active exterior cloak- elastic metamaterials with artificial intelligence is a prodi- ing for the Helmholtz equation method [153] from positive giously challenging task, due to the large and non-periodic wavenumbers to complex ones. The convergence estimates nature of the required metamaterial geometries. Several for this cloaking approach are given in [154]. This analyti- approaches have emerged to solve this design problem. cal approach allows both gaining a deeper insight into the The first one is to use reservoir learning [161], where the physical problem and establishing useful estimates regard- metamaterial design is taken to be random, and the read- ing the cloaking efficiency in both time (for the parabolic out is optimised to produce the desired functionality. This heat equation) and frequency (for the Helmholtz equation) approach has the advantage of producing convex optimi- domains. A similar approach can be developed for cloaking sation problems. A second approach consists in simplify- in dispersive media. ing the computational problem using a model reduction The current keen interest of the metamaterial commu- technique such as perturbative metamaterials, that reduces nity in the active exterior cloaking theory for the parabolic high-dimensional Finite Element problems to simpler mass- heat equation [152] is illustrated by a fully numerical opti- spring models with a few degrees of freedom [167, 168]. mization route [67] that stimulates practical realizations. Finally, machine learning has also emerged as a tool to Yet, actual production and use of such and other devices solve the hard design problems associated with intelligent at the nanoscale requires low-cost fabrication routes to metamaterials [169]. A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials — 671 4 Acoustic metamaterials standing waves in small enclosures. Traditional sound dif- fusers, such as the quadratic residue diffuser (QRD), are One broad definition of acoustic metamaterials relies on the quasi-random phase gratings attached to reflecting sur- idea of using engineered internal structures, rather than faces, the purpose of which is to augment the spatiotempo- constituent material properties, to control acoustic fields ral incoherence of the acoustic field scattered from reflec- and waves in fluids. Early developments in this area focused tive surfaces [187]. Early designs of a diffuser proposed on creating negative refractive index acoustic metamateri- by Schroeder [188] consisted of acoustic wells of varying als [170, 171] and on finding anisotropic material properties depths, whose performance is limited in practice since scat- needed for transformation acoustics and acoustic cloaking tered energy is unevenly distributed in space owing to [172–175]. A variety of different structural building blocks, minima between grating lobes associated with the peri- including membrane-type metamaterials [176, 177], space- odicity of the acoustic wells. To address this drawback coiling structures [178, 179], and other advanced designs and improve sound diffusion, very recent work [189] pro- [180, 181], have since been developed for a wide range of poses spatiotemporal modulation of a phase grating that is applications that include acoustic beam shaping, impedance inspired by recent advances in spatiotemporal modulation control, and sound absorption [9, 10, 12]. of material properties used as a means to generate nonre- At present, manipulating sound with acoustic meta- ciprocal wave propagation in electromagnetic, acoustic, and materials remains a fast-growing subject with an increas- elastic domains [190–192]. The scattering amplitudes for ing number of research directions, including the above- all sound diffraction modes and the modulation harmonics mentioned airborne acoustics at relatively low frequen- can be obtained by satisfying the impedance relationship cies, the development of acoustic counterparts of phe- between the normal particle velocity and the local sound nomena found in other branches of physics, and novel pressure, and the momentum equation at the diffuser sur- avenues such as the control of ultrasound in water. These face. When the surface impedance varies as a function of topics are overviewed in Section 4.1 with related appli- time and space, the resulting scattered field includes har- cations described in Section 4.3. Subwavelength acoustic monic frequency and wavenumber pairs associated with metastructures have recently formed a distinct class called the spatiotemporal modulation (Figure 4a). As a result, the acoustic metasurfaces and are thus discussed separately in sound field scattered from the modulated diffuser increases Section 4.2. temporal incoherence with the incident field and improves the spatial distribution of acoustic energy when compared with the same design without modulation. Semi-analytical 4.1 From classic acoustics to novel and finite-element simulations show that the diffusion coef- phenomena ficient, a metric for the sound diffusivity, for the modu- Current research on “stopping” sound by using acoustic lated quadratic residual diffuser is significantly improved metamaterials considers complex-structured designs. For in comparison to an unmodulated diffuser with the same instance, an octagonal onion-like structure with intercon- geometry [189]. nected labyrinthine shells can reflect sub-kHz airborne Intense research efforts have also been focused on sound in a cylindrical tube [182]. The 3D-printed samples exploring non-Hermitian fluctuations [193–196], topolog- have been shown to attenuate 67% of the incident sound ical acoustics [197–199], and time-varying modulation on resonance with a volume filling fraction of only 13%, [200–202], acoustic systems with fine-tuned gain and loss, which makes them excellent candidates for sound mitiga- facilitating unidirectional invisibility and exotic character- tion applications. Other examples with similar and supe- istics of exceptional points [121, 203, 204]. Likewise, the surge rior sound attenuation performance include space-coiled in other areas of physics in using topological insulators com- ventilated designs [183], tunable spider-web-inspired struc- prising non-trivial symmetry-protected phases has laid the tures [184], porous metamaterials with reflected phase shifts groundwork in reshaping highly unconventional avenues [185], and metastructures with oblique-section nest res- for robust and reflection-free guiding and steering of both onators [186]. sound and light [205–207]. Promising acoustic metamaterials are sound diffusers, Along these lines, the research team led by Y. Cheng which are structures designed to scatter incident plane and J. Christensen has realized a topological gallery acoustic waves equally in all directions [187]. Sound dif- insulator using sonic crystals made of thermoplastic rods fusers are conventionally used to minimize undesired decorated with carbon nanotube films (Figure 4b), aspects of the sound field, e.g., echoes in large spaces or which act as a sonic gain medium by virtue of 672 — A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials (a) Pinning (b) Increasing amplitude B in the presence of periodicity fin /c(1) 3 /4 Increasing Homogeneous amplitude B Infinitesimal strain Frequency, /c or (change to Finite strain, B = 1/8 dispersion due Introducing /2 to nonlinear periodicity Phononic crystal, a = 1 effects) (change to dispersion due Infinitesimal strain to linear =5 /8 effects) Finite strain, B/a = 1/8 /4 Balance of linear and nonlinear dispersive effects 0 0 /4 /2 3 /4 Wave number, (c) Figure 3: Nonlinear and time-varying effects in elastic metamaterials. (a) Numerically estimated deformation at vertically applied 𝜖 yy = −6% on a flexible metamaterial comprising 21 × 60 squares with localized pinning defects separated by 10 holes. (Adapted from [105]). (b) Wave dispersion in linear and nonlinear homogeneous materials and in linear and nonlinear elastic metamaterials. The intensity of the finite-strain nonlinearity increases with the wave amplitude. (Adapted from Ref. [112] with permission from the Royal Society). (c) A numerical model for right-going surface waves with a right-going modulating wave with the sub-figures showing the time history and frequency content of the point force applied at the source. (Reprinted from Journal of Mechanics and Physics of Solids, vol. 145, A. Palermo, P. Celli, B. Yousefzadeh, et al., “Surface wave non-reciprocity via time-modulated metamaterials”, p. 104181, Copyright © 2020, with permission from Elsevier). electro-thermo-acoustic coupling [208]. The word “gallery” the out-coupling of topological “audio-lasing” modes with refers to so-called whispering-gallery waves that were the desired handedness [208]. discovered by Lord Rayleigh in 1878 as sound waves Other flourishing topics in acoustic metamaterials creeping around the curved gallery of St Paul’s Cathedral in are breaking sound wave reciprocity and Willis coupling. London. By engineering specific non-Hermiticity textures Reciprocity is broken when swapping source and receiver to the activated rods, it is possible to break the chiral have non-identical frequency responses. It has been symmetry of the whispering-gallery modes thus enabling implemented by using spatiotemporal-dependent material A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials — 673 Figure 4: Acoustic metamaterials and metasurfaces. (a) Schematic representation of a diffuser with spatiotemporally modulated input impedance and the associated incident and scattered plane wave fields. (Published in Ref. [189]). (b) Fabricated topological WG insulator with a triangular domain wall. The inset shows the photograph of the cylinder trimers wrapped with a CNT film. (Material from Figure 3 in Hu, B., Zhang, Z., Zhang, H. et al. Non-Hermitian topological whispering gallery, Nature, published 2021, publisher – Nature, ISSN 1476–4687). (c) Thermoacoustic amplifier: (left-top) scaled representation of the unit cell with the temperature distribution indicated by colors; (right-top) the photography of the elements of the thermoacoustic cell and (bottom) a schematic view of the experimental setup for measuring the scattering matrix of the thermoacoustic amplifier. (Reprinted from Physical Review B, vol. 104, C. Olivier, G. Poignand, et al., “Nonreciprocal and even Willis couplings in periodic thermoacoustic amplifiers”, p. 184109, Copyright © 2021, Publisher – APS Physics). properties [209], by combining nonlinear properties with matrix [214]. The latter links the state vectors at both sides an asymmetry [210], or introducing an external bias of the unit cell, and directly provides both even and non- [211]. The external-bias approach, however, often neglects reciprocal Willis coupling terms. In the first-order Taylor thermoacoustic effects that describe thermal interactions expansion of the transfer matrix elements, the odd Willis between acoustic waves and surrounding walls [212]. The coupling is absent, while the even Willis coupling is found Willis coupling parameters couple the potential and kinetic to be only related to the asymmetry of the temperature gra- energies in the acoustic conservation relations, similarly dient that is applied to the unit cell. Even and nonreciprocal to those for their electromagnetic counterparts, which Willis couplings are found of equal modulus but an opposite were first described in the seminal work of Ref. [213]. This sign and are almost purely imaginary at low frequencies, enables the enhancement of the ability to control waves in thus enabling a zero-group-velocity point and the opening of acoustic metamaterials compared to other materials that an amplification band at vanishing frequencies. Hence, the do not exhibit such coupling. system effectively possesses a coalescence point in k-space Nonreciprocal Willis coupling can be beneficial for that resembles a PT-broken phase. The effective parame- developing thermoacoustic amplifiers. Specifically, for a 1D ters and scattering properties of single-unit and 15-unit sys- periodic thermoacoustic amplifier (Figure 4c), it is possible tems have been validated experimentally, thus promoting to derive closed-form expressions for the effective prop- thermoacoustics as an excellent means for designing non- erties by using Padé’s approximation of the total transfer reciprocal systems [215]. 674 — A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials Similarly to the case of elastic metamaterials, one can effect introduced by each HR in an array is often consid- achieve a roton-like dispersion in acoustic metamateri- ered independently of the others, thus neglecting possible als by tailoring beyond-nearest-neighbor interactions [216]. coupling between neighboring HRs. Such a coupling can, Roton-like dispersion inside the first Brillouin zone can be however, be crucial for the ability to absorb and control the obtained if the beyond-nearest-neighbor interactions domi- bandwidth at low sonic frequencies [223]. Therefore, recent nate over the nearest-neighbor interactions. Then, for a cer- studies have carefully analyzed the interaction between tain range of positive wavenumbers, the partial wave medi- twin spherical HRs as a function of their separation and ated by the beyond-nearest-neighbor interactions becomes demonstrated a modal splitting into two resonances of dif- a dominant backward wave that leads to a net negative ferent symmetry [224]. The underlying physical mechanism energy flow represented by a negative slope of the disper- is characteristic of a degeneracy lifting resulting from the sion curve, which is related to the roton minimum. coupling between the two HRs, which leads to the occur- Further developments of acoustic metamaterials are rence of two modes: symmetric and antisymmetric with targeted to control high-frequency ultrasound in water, respect to the mirror plane between the two HRs. The dis- where one of the goals is developing new approaches for tance between the HRs can be adjusted to tune the reso- acoustic tweezers which can trap and manipulate small nance frequencies of the modes, according to a specific aim. particles [217, 218]. Acoustic tweezers typically use multi- The theoretical calculations have been shown to be in good ple sources to create standing wave patterns in the water agreement with experimental measurements [224]. that can trap and move objects in ways constrained by the Other metasurface designs use surface wave phenom- limited complexity of the acoustic wave field, which can be ena to convert a near-field source to an energy focus in generated from an array of simple sources that are generally the far field, above a metasurface. This idea is inspired by removed from the area of trapping and manipulation. A recent analogous devices in electromagnetism [225, 226] and 3D acoustic tweezer proposed recently uses a single trans- elasticity [227], which leverage the process known as Umk- ducer and a PDMS acoustic lens [217], so that 3D trapping lapp diffraction. The acoustic counterpart relies on a similar is achieved by combining the radiation force for trapping physical mechanism: a designed metasurface comprises two in two dimensions with the streaming force to provide levi- regions where guided or trapped surface modes exist, and tation in the third dimension. The experimentally achieved at the interface, between the regions, a scattering process levitation force reaches three orders of magnitude larger occurs whereby energy is diffracted out of the metasurface than for previous 3D trapping. Alternatively, spatially com- states into the far field. The mode dispersion in each region plex particle trapping and manipulation inside a boundary- of the surface is carefully designed; (1) to ensure the scat- free chamber can be obtained by using a single pair of tering process allows the Umklapp-like diffraction of the sources and an engineered structure outside the chamber surface mode, and (2) to maximize the mode-shape overlap called a shadow waveguide [218]. The shadow waveguide and minimize the impedance mismatch between regions. creates a tightly confined, spatially complex acoustic field The analyzed systems are radially symmetric so that the inside the chamber without requiring any interior struc- overlap of diffracted beams in the far field results in an ture that would interfere with net flow or transport. Par- enhanced focal point. ticle trapping, particle manipulation and transport, and an In particular, the acoustic metasurface in this appli- acoustic analog of Thouless pumping have been experimen- cation can be represented by an aluminium plate with tally demonstrated. a periodically modulated thickness profile submerged in water [228]. The periodic structure is designed to modify 4.2 Acoustic metasurfaces the dispersion of the coupled Scholte modes that would Acoustic metasurfaces, the airborne acoustics analogy of be present on an otherwise unpatterned plate in water. elastic metasurfaces, are subwavelength thickness struc- When excited with a point-like (monopole) acoustic source tures allowing the control of sound waves in fluids [29, 219]. at the plate centre, Scholte modes are excited in the first Their realization is a topic of continuous interest because region and undergo diffraction at the transition to the sec- of their compact sizes that enable manipulating large wave- ond region, radiating into the fluid. Finite-element simu- lengths of acoustic waves at audible frequencies that range lation and experiments reveal the existence of frequency- from 20 Hz to 20 kHz. dependent focal spots on both sides of the sample, no matter Many acoustic metasurfaces contain Helmholtz res- which side the source is placed on. It is therefore possible to onators (HR), which are well-known devices for subwave- focus sound on the flat side of the metasurface due to the length manipulation of sound [220–222]. Yet, the phase shift coupling and modulation of the Scholte modes. A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials — 675 4.3 Applications of acoustic metamaterials of complex spatial distribution inside the skull. By using low-cost 3D-printed lenses, ultrasonic beams can be focused Potential applications of acoustic metamaterials and meta- not only on a single point but be made to overlap at one or surfaces have a broad spectrum ranging from sound atten- various target structures simultaneously, e.g., left and right uation [9, 10, 219, 229, 230] to exotic acoustic camouflage hippocampi. Other complex fields such as vortex beams for [172, 174] and analog computing [12]. Employing random trapping small objects can also be obtained. The technique medium theory on acoustic metasurfaces is another promis- was tested to perform localized drug delivery in the brain of ing route for controlling sound fields in a room environ- a small laboratory animal. In addition, acoustic holograms ment (i.e., diffuse sound fields) [189, 231]. Among the designs encode the multiple reflections inside the skull, increasing described above, metastructures with large Willis coupling the angular spectrum of the acoustic image. The technique pave the way for a better physical understanding of Willis was also tested to shape thermal patterns in soft tissue to couplings in non-reciprocal systems, to facilitate engineer- raise the temperature locally [237]. ing applications of Willis materials, and for various appli- These results open new paths for emerging therapeu- cations of such systems to further control acoustic waves at tic ultrasound applications in neurology, including blood- very low frequencies [214, 215]. Recently, topological studies brain barrier opening for localized drug delivery or neuro- of acoustic metasurfaces have been carried out in depth. modulation. On the other hand, increasing the temperature The three key research branches include non-Hermitian, using low-cost and MRI-compatible holographic transduc- nonlinear, and non-Abelian braiding, each of which deliv- ers might be of great interest for ultrasound hyperthermia, ers many interesting results. For instance, non-Abelian physiotherapy, or high-intensity focused ultrasound, where braiding of two or three modes enables the implemen- the control of specific thermal patterns is needed. tation of quantum logic operations [232]. Acoustic tweez- ers allow contact-free, bio-compatible, and precise manip- ulation of particles from millimeter to sub-micrometer 5 Mechanical metamaterials scales [217, 218], while acoustic metasurfaces can be used in underwater sensing or nondestructive testing and Mechanical metamaterials form yet another subcategory of evaluation [228]. architected materials, in this case materials with unprece- Other emerging and very promising application direc- dented mechanical properties or functionalities. Similar to tions are acoustic imaging and medical therapy, e.g., ther- elastic and acoustic metamaterials, these properties origi- apeutic applications of focused ultrasound [233]. Thanks nate from an interplay between the rational design of their to the ability of acoustic metamaterials to accurately engi- 2D or 3D (micro)architectures and their constituent mate- neer wavefronts, one can generate acoustic holograms by rials [13]. Examples include auxetic (also known as nega- synthesizing acoustic images that shape the areas where tive Poisson’s ratio) [238], shape-matching [239], action-at- mechanical waves have high amplitudes and the areas a-distance [240], programmable [241, 242], adaptable [243], where the medium is at rest. Compared to light, acous- tunable elastic properties [244] and crumpled-based meta- tic waves can penetrate deeper into biological tissues, materials [245]. thus enabling the shaping and focusing of ultrasonic fields inside biological media. For instance, accurately designed metastructures can encode complex wavefronts to compen- 5.1 Design approaches and functionalities sate for the phase aberrations produced by stiff layers of tissues, such as skull bones in transcranial propagation, and The architecture of mechanical metamaterials can be rep- allow the creation of arbitrary therapeutic patterns in the resented by either regular repeating unit cells or ran- brain or even vortex beams. dom microstructures, each of which have their own pros Recent works have demonstrated the potential of and cons, and govern unconventional behavior on the acoustic holograms to deliver therapeutical acoustic images macroscale. One of the main challenges in the design of for the non-invasive treatment of neurological disorders mechanical metamaterials is to find a proper combina- [233, 234], and to produce cavitation [235, 236] and ther- tion of microstructural features that can ensure structural mal [237] patterns. When targeting acoustic beams into the integrity and, simultaneously, deliver desired mechanical brain, accurate focusing is mainly limited due to the strong behavior. Other challenges include achieving tunable per- phase aberrations produced by the refraction and attenua- formance, e.g., by varying external loads or environmen- tion of the skull. Acoustic holographic lenses overcome these tal conditions, and multi-functionality (e.g., double aux- limitations by synthesizing aberration-free ultrasonic fields eticity, load- or temperature-dependent response, etc.) that 676 — A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials allows combining several functionalities by using a single the potential of these networks in developing stiff metama- microstructure. terials with extremely negative (or positive) Poisson’s ratios Flexible mechanical metamaterials, previously dis- [249]. Patterned randomness can also be used to indepen- cussed in the context of nonlinear wave propagation dently tailor the elastic modulus and Poisson’s ratio of a (Section 3.1), can be designed to show shape-matching and mechanical metamaterial [251] in addition to the known auxetic behavior in the (quasi-) static regime. These are tuning-by-pruning approach [252]. examples of unusual properties that can be obtained by Other tools to expand the design space of mechanical tailoring mechanical characteristics, e.g., elastic stiffness metamaterials include the use of multiple constituent mate- and Poisson’s ratio, through modifying geometric features rials, harnessing elastic instabilities, and the exploitation at the unit-cell level. For instance, cellular structures formed of nonelastic material behavior. These approaches enable by a purposely designed combination of auxetic and con- properties that are impossible or challenging for single- ventional unit cells can match a predefined shape upon material configurations and break the limits imposed by deformation [239, 246, 247] or exhibit intrinsic curvature elastic regimes. Below, we overview these tools in more (Figure 5a) [248]. Note that despite different geometries, the detail. spatial arrangement of the unit cells remains regular. Multi-material mechanical metamaterials benefit from Alternatively, one can harness randomness as a design rapid advances in multi-material additive manufacturing tool to expand the design space of mechanical meta- that have enabled the production of truly intricate designs materials for achieving even more exotic properties. It at various size scales. Recent works have shown that com- has recently been shown that disordered architectures of bining hard and soft materials in 2D or 3D arrangements lattice-unrestricted networks exhibit a high probability of can be used to circumvent intrinsic limitations (i.e., in terms auxetic and double-auxetic behavior, also in the stretch- of elastic properties) of metamaterials made of a single dominated domain [249, 250]. The latter feature uncovers material. A remarkable example is a random assignment of Figure 5: Mechanical metamaterials. (a) By controlling the values of Poisson’s ratio, one can control the out-of-plane curvature of meta-plates (adapted from M. J. Mirzaali, et al., Advanced Materials, 33(30), p. 2008082 (2021)); (b) a wider range of properties can be achieved by intruding multiple materials in the structure of the mechanical metamaterials (Reprinted from M. J. Mirzaali et al. “Multi-material 3D printed mechanical metamaterials: Rational design of elastic properties through spatial distribution of hard and soft phases”, Appl. Phys. Lett. 113, 241903 (2018) with the permission of AIP Publishing); (c) numerical and experimental images of a multiphase composite with stiff inclusions and voids in an undeformed state and loaded in the x direction (adapted from J. Li, V. Slesarenko and S. Rudykh, Soft Matter, 14, 6171 (2018); (d) viscoelastic bi-beams buckle to left and right in response to the applied strain rate (adapted from S. Janbaz, et al., Science Advances, 6(25), p. eaba0616 (2020)); (e) A viscoelastic micro gripper powered by a bi-beam (adapted from S. Janbaz, et al., arXiv:2206.15168); (f) viscoelastic oligomodal metamaterials exhibit different global buckling modes once compressed slow or fast (adapted from A. Bossart, et al., PNAS, 118(21), p. e2018610118 (2021)). A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials — 677 a hard phase to soft cellular structures that allows the exten- be pre-designed and achieved earlier, at smaller strains, and sion of the ranges in the elastic modulus-Poisson’s ratio distinct post-buckling patterns can be induced through the plane [253]. Toward this aim, artificial neural networks have positioning of stiff inclusions [263]. been considered as a powerful tool that may help design- Other material properties – such as viscoelasticity – ers bypass “trial-and-error” approaches and speed up the can be used to increase the pool of admissible microstruc- process of extracting corresponding (micro-)architectures tures upon instability. For instance, identical samples can and material distributions for any desired target proper- develop different patterns due to different loading histories, ties [254]. On the other hand, patterned designs of the as shown experimentally on 3D-printed laminates loaded at hard phase can be effective for independent tuning of the different strain rates [264]. Viscoelasticity can also be useful elastic properties (Figure 5b) [253]. The deposition of the to control the deformation modes in soft metamaterials that second hard phase in the microstructure of mechanical work on the basis of mechanical instabilities [265, 266]. This metamaterials can also create local non-affine deforma- can be achieved in two ways, both of which imply that tions which can be correlated to the values of the Poisson’s viscoelasticity effectively seeds imperfections required for ratio [255]. switching the response of a metamaterial depending on an Switchable properties in microstructured metamateri- applied strain rate. The first approach relies on the reduc- als can be achieved by harnessing elastic instabilities – a tion of the navigation of instability patterns by designing a phenomenon that plays an important role in pattern forma- simple building block, such as a bi-beam [267]. The bi-beams tions in soft biological systems [256]. Such systems typically are formed by lateral attachment of two flexible beams combine a soft matrix and a stiff phase (such as fibers or made from two polymers with highly different viscoelastic inclusions). In the 2D setting, stiffer layers undergo buck- properties (Figure 5d) so that one of the beams is almost ling when the composite is compressed beyond the criti- strain-rate insensitive, and the other is highly viscoelas- cal level [257]. The larger the layer stiffness contrast and tic. This enables a reliable strain rate-dependent control concentration of the stiffer phase, the larger the critical of buckling direction as a principle to design multifunc- wavelength that allows predesigning switchable metama- tional metamaterials. For example, this idea is applied to terial microstructures. Furthermore, 3D-printed soft lay- construct flexible structures with switchable auxetic-non- ered composites with a non-dilute stiff phase can form auxetic behavior or to show apparent negative viscoelas- a twinning microstructure with finite-size anti-symmetric ticity (i.e., where the apparent stiffness is lower for larger domains by developing long-wave instability upon exceed- loading rates and larger for lower loading rates) [267]. Tak- ing the critical value of the compressive strain along ing into account that the commercial 3D printing of such the layers. The obtained finite-size regulated domain pat- strain-rate-dependent materials might be limited due to tern distinctly differs from a classical wrinkling [258]. The the lack of 3D printable viscoelastic polymers, one can use instability phenomenon can be used to induce a nega- purposeful geometrical imperfections to induce a robust tive group velocity state in non-periodic composites, poten- and highly predictable strain-rate-dependent behavior. For tially enabling left-handed behavior and negative energy example, the curved geometry of a bi-beam (that actuates a transfer. compliant micro-gripper, Figure 5d) can overcome the limits Three-dimensional fiber composites, similarly, undergo of single-material printing using two-photon polymeriza- buckling when compressed, but are more stable and require tion [268]. In this case, once differential curing of a photo- higher strains to trigger buckling [259]. Interestingly, the polymer provides different levels of viscoelasticity at the buckling plane orientation can be controlled by the in-plane two sides of a beam, the low-power cured layer remains periodicity of fiber arrangements, together with material always softer than the one cured with a higher power. The parameters [260]. Composites with periodic stiff inclusions curved geometry of the beam can serve as an imperfec- in a soft elastomer exhibit domain formations and pattern tion that combined with material-based imperfection form transitions under large deformations [261]. Instabilities in strain-rate dependency. multiphase composites with stiff inclusions and voids peri- The second approach uses viscoelasticity to control odically distributed in a soft matrix can induce the rear- minimum energy pathways in combinatorial designs such rangement of new microstructural morphologies (Figure 5c) as oligomodal metamaterials [265, 269]. Oligomodal meta- [262]. Such pattern transformations are accompanied by materials are metamaterials that feature a constant number void collapse that results in auxetic behavior, while the stiff (>1) of soft deformation modes, regardless of system size inclusions regulate the onset of instabilities and the post- [269]. Each of these modes spans the entire design domain buckling transformations. Hence, the auxetic behavior can [269]. With a rational selection of elastic and viscoelastic 678 — A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials joints within their structures they exhibit strain-rate depen- further extending control over buckling modes by means of dent mode selection (Figure 5e) [269]. While comparing with viscoelasticity, one may promote the design of mechanical kinematic systems with many soft modes, viscoelastic oligo- metamaterials in the field of complex strain rate-dependent modal designs transform their properties (e.g., Poisson’s machines. Hence, it can be anticipated that the fabrication ratio) simply by changing the applied strain rate using of even more complex 3D metamaterials and miniaturized a simple loading scheme. Alternatively, one can also use machine-like devices is not far from realization. viscoelasticity to switch between buckling or frustration (Figure 5f) [266]. A similar method, using intermediate strain rates, can also be used to generate extreme shock absorption 6 Conclusions and outlook by delaying both buckling and unbuckling [266]. In general, viscoelasticity elaborates the design space of metamaterials We have seen throughout this broad review that the gen- to exhibit switchable responses that pave the way toward eral domains of nanophononics and elastic, acoustic, and advanced materials and devices. mechanical metamaterials, despite their now relatively mature history, still contain many unsolved problems and 5.2 Applications of mechanical forthcoming challenges whose solution deserves the atten- metamaterials tion and collaborative efforts of the scientific community across multiple disciplines. The majority of the described mechanical metamaterials The broad scalability of the fundamental equations of have complex architectures that can be realized almost mechanics and acoustics, which is relatively rare in other exclusively by means of additive manufacturing technolo- branches of physics, shows that similar phenomena can be gies, e.g., fused deposition modeling [239, 240, 248], two- applied across a broad spectrum of problems ranging from photon polymerization [268], stereolithography [13, 247], control of light–matter interactions with phonons of nano- material jetting [242, 248, 264, 267], etc. Often, a special mate- metric wavelengths to control of seismic waves of several rial or a combination of such materials is required to enable hundred meters in wavelength. For this reason, even in a the designed mechanical properties or functionalities [239, small workshop like the EUROMECH Colloquium summa- 262, 267, 269]. Therefore, further advances in the devel- rized in this review, one can observe the value of interac- opment of additive manufacturing techniques, including tion between scientists from different areas. For example, better resolution and inter-layer adhesion, lower porosity, in this workshop nanochemists working on the fabrication broader ranges of base materials, etc., will promote wider of optomechanical cavities were sharing common inter- applications of mechanical metamaterials. ests with civil engineers working on sound isolation for Mechanical metamaterials have already shown buildings. promising potential for numerous applications, including In this review, we have also shown that new theo- the design and fabrication of programmable matter, retical concepts are still emerging and that more power- soft robotics, biomedical devices, such as implants or ful computational tools are in demand to help elucidate exoskeletal prostheses, wearable electronics, smart the physics of the complex material systems that are con- clothing, shoes, etc. For instance, a proper combination tinuously being proposed or discovered. Equally impor- of conventional and auxetic unit cells can be used to tant, advanced experimental demonstrations continue to design an orthopedic meta-implant that in contrast to a emerge, often inspired by the new observed phenomena. conventional counterpart does not retract from a bone This is all driven by the growing number of applications, under biomechanical loading [270]. The rational design of especially after the recent rise of tunable and reconfigurable programmable metamaterials [242, 271] enables control metastructures and other novel concepts. As extraordinary of the actuation pattern and actuation characteristics in as the current applications are, many more are awaiting in the fabrication of the metamaterials, thereby eliminating the future. the need for networks of distributed actuators, sensors, and controllers that are usually needed to achieve Author contributions: All the authors have accepted complex actuation. Adaptable metamaterials with double responsibility for the entire content of this submitted curvatures can be used to fit a shape curvature required manuscript and approved submission. for soft wearable devices [248]. Meta-plates with cellular Research funding: This work is supported by the DYNAMO [248] or kirigami-based [272] architectures are promising project (101046489) funded by the European Union. Views for the design of adaptive mirrors capable of controlling and opinions expressed are however those of the authors focused laser beams of incoming wave fronts. Finally, by only and do not necessarily reflect those of the European A. O. Krushynska et al.: An overview of nanophononics, elastic, acoustic, and mechanical metamaterials — 679 Union or European Innovation Council. Neither the Euro- [17] M. Sigalas and E. N. 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