# Dynamic characteristics of an acoustic metamaterial with locally resonant microstructures

v T ABLE OF CONTENTS Page LIST OF TABLES................................viii LIST OF FIGURES...............................ix SYMBOLS....................................xiii ABBREVIATIONS................................xiv ABSTRACT...................................xv 1 INTRODUCTION..............................1 1.1 Deﬁnition and Origin of Metamaterials................1 1.2 Acoustic Metamaterials.........................2 1.2.1 Locally resonant acoustic metamaterials and the forerunners 3 1.2.2 Other classes of special materials...............8 1.2.3 Potential applications of acoustic metamaterials.......10 1.3 Chapter Outlines............................12 2 ONTHENEGATIVEEFFECTIVEMASS DENSITYINACOUSTICMETA- MATERIALS.................................15 2.1 Introduction...............................15 2.2 Negative Eﬀective Mass........................15 2.2.1 Single mass-spring unit.....................16 2.2.2 One-dimensional periodic lattice model............18 2.3 Continuum Representation of the Mass-in-Mass System.......23 2.3.1 Representation by a classical elastic solid...........23 2.3.2 Microstructure continuum model...............24 2.3.3 Evaluation of the multi-displacement continuum model...26 2.4 Conclusion................................27 3 WAVE ATTENUATION MECHANISMIN AN ACOUSTIC METAMATE- RIAL WITH NEGATIVE EFFECTIVE MASS DENSITY........31 3.1 Introduction...............................31 3.2 Metamaterial and Equivalent Medium with Negative Eﬀective Mass Density.................................32 3.3 Mechanism of Energy Transfer and Wave Attenuation........35 3.3.1 Single mass-in-mass unit....................38 3.3.2 Mass-in-mass lattice system..................42

vi P age 3.4 Dynamic Response of Metamaterials with Negative Eﬀective Mass Den- sity....................................45 3.4.1 Metamaterials with tailorable material constants.......45 3.4.2 Spatial wave attenuation....................46 3.4.3 Amplitude of internal masses.................49 3.5 Conclusion................................50 4 LOCALLY RESONANT ACOUSTIC METAMATERIALS WITH TWO- DIMENSIONAL ANISOTROPIC EFFECTIVE MASS DENSITY....51 4.1 Introduction...............................51 4.2 2D Mass-in-mass Lattice Model....................52 4.3 Eﬀective System of the 2D Mass-in-mass Model...........55 4.4 Second-order Eﬀective Mass Tensor..................57 4.5 A 2D Continuum with Anisotropic Eﬀective Mass Density.....62 4.5.1 Basic equations.........................62 4.5.2 Determination of eﬀective mass density............63 4.5.3 Wave propagation........................65 4.6 Discussions and Concluding Remarks.................71 5 BEHAVIOR OF WAVE MOTION IN AN ACOUSTIC METAMATERIAL WITH ANISOTROPIC MASS DENSITY.................75 5.1 Introduction...............................75 5.2 Elastic Solid with Anisotropic Eﬀective Mass Density........76 5.2.1 Representative elastic solid...................76 5.2.2 Dispersion curves........................78 5.3 Wave Reﬂection and Transmission...................79 5.3.1 Wave propagation froma ﬂuid-like material to the metamaterial 81 5.3.2 A metamaterial sandwiched between two FL-mat media...86 5.4 Conclusion................................93 6 PREDICTION OF MATERIAL PROPERTIES OF A METACOMPOSITE IN DYNAMIC BEHAVIORS.........................95 6.1 Introduction...............................95 6.2 Metacomposite and the Representative Volume Element (RVE)..96 6.3 Classical Continuum with Anisotropic Eﬀective Mass Density...96 6.4 Eﬀective Elastic Constants.......................99 6.4.1 Normal loading:obtaining E 1 ,E 2 ,ν 12 ,ν 21 ..........99 6.4.2 Shear loading:obtaining G 12 ..................102 6.4.3 Transformation of elastic constants ¯ Q ij ............103 6.5 Eﬀective Mass Density.........................104 6.6 Wave Propagation and Dispersion Relation..............105 6.6.1 Wave propagation in principal directions...........106 6.6.2 Wave propagation along δ = 45 o ................111 6.7 Conclusion................................111

vii P age 7 SUMMARY AND RECOMMENDATION FOR FUTURE RESEARCH.115 7.1 Summary................................115 7.2 Recommendation for Future Research.................118 LIST OF REFERENCES............................119 A 2D MICROSTRUCTURE CONTINUUM THEORY............125 B SUPPLEMENTARY MATERIALS TO CHAPTER 5...........129 B.1 Section 5.2.1 Representative elastic solid...............129 B.2 Section 5.3.1 From a ﬂuid-like material (FL-mat) to a metamaterial 129 VITA.......................................131

viii LIST OF TABLES Table Page 5.1 Mass density in principal directions for various cases..........85 6.1 Material properties of the metacomposite in various cases........107

ix LIST OF FIGURES Figure Page 2.1 Single spring-mass system.........................16 2.2 Eﬀective model of the single spring-mass unit..............17 2.3 Inﬁnite mass-in-mass lattice structure...................18 2.4 Nondimensionalized dispersion curve for the mass-in-mass lattice model 19 2.5 Attenuation factor as a function of dimensionless frequency (ω/ω 0 )..20 2.6 Dimensionless eﬀective mass (m eff /m st ) as a function of dimensionless frequency (ω/ω 0 )..............................22 2.7 Deﬁnition of variables for microstructure continuum model.......25 2.8 Dispersion curve obtained from the multi-displacement continuum model compared with that from the mass-in-mass lattice model........28 2.9 Attenuation factor as a function of frequency for the multi-displacement continuum model compared with that for the lattice model.......29 3.1 (a) Inﬁnite mass-in-mass lattice structure (b) The corresponding eﬀective lattice system................................34 3.2 (a) Dimensionless eﬀective mass m eff /m st and the mass density (ρ eff /ρ st ) as a function of dimensionless frequency (ω/ω 0 );(b) Complex dimension- less wave number qL = α+iβ as a function of ω/ω 0 .In this plot the solid line,β,represents the attenuation factor..................36 3.3 Dispersion curves from equation (3.3) for the mass-in-mass lattice model 37 3.4 A mass-in-mass unit connected to a rigid wall..............38 3.5 The total external work done in time domain by the external excitation for the single mass-in-mass model.Material constants in equation (3.8) are used....................................41 3.6 The energy distribution rate of the mass-spring model (θ = m 2 /m 1 )..43 3.7 The ﬁnite lattice model with boundary conditions............43 3.8 Numerical results of the total external work done by the external excitation for the discrete lattice model........................44

x Figure Page 3.9 A composite with embedded microstructures...............45 3.10 Dimensionless displacement envelopes of propagating wave ﬁelds in various frequencies.ω 0 =

k 2 /m 2 is the local resonance,u (j) γ represents the displacement of mass γ in the (j)th cell..................47 3.11 Snapshot of wave propagation in a metamaterial;each section consists of speciﬁcally selected stop band.......................48 3.12 Frequency domain for each section by FFT................48 3.13 Dimensionless displacement envelope for mass 2 for various dimensionless frequency (ω/ω 0 ).In the plot u (j) γ represents the displacement of mass γ in the (j)th cell...............................49 4.1 (a) 2D mass-in-mass lattice structure;(b) A representative unit cell and the coordinate setup.............................54 4.2 (a) 3D band structure of the acoustic mode;(b) 3D band structure of the optical mode;(c) The directions of dimensionless wave vector for which the 2D band structure is plotted;and (d) 2D band structure of the mass- in-mass lattice model............................56 4.3 Eﬀective system of the mass-in-mass model and the anisotropic eﬀective mass......................................57 4.4 Anisotropic eﬀective masses in principal directions............58 4.5 Coordinate set up for a mass-in-mass unit.................60 4.6 Coordinate set up for an eﬀective mass unit................61 4.7 Approach 1:by matching dispersion curves.The dispersion curves ob- tained fromthe mass-in-mass lattice model (blue lines:Longitudinal mode; green dashed lines:Shear mode) and the anisotropic continuummodel (red dots).....................................66 4.8 Approach 2:by volume average.The dispersion curves obtained from the mass-in-mass lattice model (blue lines:Longitudinal mode;green dashed lines:Shear mode) and the anisotropic continuum model (red dots)...67 4.9 (a) The angle θ between the energy direction and the phase direction with respect to the angle of consideration δ (b) The variation of the wave speeds for diﬀerent modes with respect to δ.”L” refers to longitudinal-dominated wave and ”S” to shear-dominated wave..................69 4.10 Deﬁnition of the angles in ﬁgures 9.....................70

xi Figure Page 4.11 (a) mode 1,and (b) mode 2 of the propagating wave in anisotropic meta- materials at a certain wave frequency;(c) indicates two propagating modes; (d) wave front of each mode in a 2D metamaterial............71 4.12 (a) Indicating only one propagating mode and one attenuated mode (b) the wave front of the only propagating mode,mode 2 is banded,hence, not shown...................................72 4.13 Wave front of (a) ﬁrst propagating mode and (b) second propagating mode, when t = 1e −6 sec.Material constants are chosen as previously shown. Both ﬁgures are plotted in the variation of wave frequency ranging from ω = 9e4 (rad/s) (Color in blue) to ω = 2.5e5 (rad/s) (Color in red)..73 5.1 (a) A 2D metamaterial with microstructures in the form of elastically connected internal masses.(b) Representative mass-in-mass lattice model 77 5.2 Components of the eﬀective mass density in principal directions with re- spect to wave frequency..........................79 5.3 The dispersion curves obtained fromthe mass-in-mass lattice model shown in Figure 5.1(b) (blue lines:Longitudinal mode;green dots:Shear mode) and the anisotropic continuum model (red dotted-lines).........80 5.4 An illustration showing wave propagation from a FL-mat into a meta- material.“A” represents the amplitude of the reﬂected and transmitted waves,“q” is the wavenumber.Subscript “R” means the reﬂected wave, “T” for the transmitted wave,“L” for “extension dominated mode” and “S” for “shear dominated mode”......................81 5.5 Amplitudes of the reﬂected and transmitted waves when an incident wave of unity amplitude strikes the metamaterial from a FL-mat.......87 5.6 A schematic showing wave propagation through a metamaterial.“A” de- notes the amplitude of the reﬂected or transmission wave........88 5.7 The “reﬂection” shows the amplitude of the wave in the ﬁrst medium (a FL-mat) and the “transmission” shows that of the wave in the third medium (another FL-mat).........................92 6.1 A metacomposite and the representative volume element (RVE) with the coordinate set up..............................97 6.2 Boundary conditions for normal loading.................100 6.3 (a) Nodal numbering on boundaries (b) RVE under transverse shear loading 103 6.4 The ﬁnite element mesh of the microstructure for wave propagation in principal directions.A unit cell is shown in the red dashed square...108

xii Figure Page 6.5 Veriﬁcation of dispersion curves for cases 1 ∼ 3 when waves propagate in principal directions.............................110 6.6 The ﬁnite element mesh of the microstructure for wave propagation along δ = 45 o .A unit cell is shown in the red dashed square..........112 6.7 Veriﬁcation of dispersion curves when waves propagate along δ = 45 o .113

xiii SYMBOLS m mass ρ mass density u displacment ˙u,v velocity ¨u,a acceleration t time k internal extensional spring constant K external extensional spring constant G external shear spring constant L lattice spacing V volume Q ij reduced stiﬀness ω wave frequency q wavenumber σ ij stress tensor ε ij strain tensor λ First Lam´e’s constant E Young’s modulus ν Poisson’s ratio

xiv ABBREVIA TIONS NEMD negative eﬀective mass density Metamat metamaterial FL-mat ﬂuid-like material I incidence R reﬂection T transmission AC acoustic mode OP optical mode L-mode extensional dominated mode S-mode shear dominated mode FE ﬁnite element

xv ABSTRA CT Huang,Hsin-Haou Ph.D.,Purdue University,August 2009.Dynamic Characteris- tics of an Acoustic Metamaterial with Locally Resonant Microstructures.Major Professor:C.T.Sun. Wave propagation in acoustic metamaterials with locally resonant-type microstruc- tures was investigated.Because of their unusual forms of microstructures,these meta- materials,if represented by classical elastic continuous solids,would exhibit unusual material properties such as negative mass/mass density in certain frequency range. It was found that the range of frequencies that yield negative mass densities actually correspond to a band gap in which no harmonic wave can propagate in the meta- material without attenuation in amplitude.Moreover,the band gap can be moved by altering the local resonance frequency of the microstructure.This metamaterial can give rise to a signiﬁcant wave attenuation eﬀect near the local resonance fre- quency,and therefore can be used to block waves from passing the metamaterial.In two-dimensional metamaterials,it was shown that the representative classical elastic solid has an anisotropic eﬀective mass density and that the eﬀective mass density assumes the form of a second order tensor.Thus,the propagation directions of en- ergy and phase are diﬀerent and the longitudinal wave and shear wave are coupled in general.This unusual frequency-dependent anisotropic mass density characteristic was studied by examining harmonic wave propagations in arbitrary directions in a two-dimensional acoustic metamaterial.It was found that,for example,a pressure wave impinging on an acoustic metamaterial may be stopped directly by designing the gap frequency.Or alternatively,the impinging pressure wave can be converted to a strong shear-dominated wave mode accompanied by a weak extension-dominated wave.Since the shear-dominated wave motion can hardly be transmitted into a ﬂuid- like material and,thus,the ﬂuid-like material behind the metamaterial can remain

xvi mostly undisturbed.A speciﬁc metamaterial in the form of particulate composites was also considered.In this metamaterial,the resonators were embedded in a contin- uous elastic matrix.An eﬃcient approach was proposed to derive the eﬀective mass density directly from the properties of the actual matrix and microstructure.One great advantage of this method is that the dynamic behavior of the metacomposite can be fairly simply and accurately predicted by using a classical continuum model without performing any wave propagation analysis of the original metamaterial and only static analyses are needed.In addition to the use of the classical continuum model to represent metamaterials,in this study,a non-classical continuum model,a multi-displacement continuum model or microstructure continuum model,was em- ployed to represent the metamaterial.It was found that the characteristic dynamic behavior of the metamaterial could be described without resorting to the use of neg- ative mass/mass density.

1 1. INTRODUCTION “I am still learning” - Michelangelo 1.1 Deﬁnition and Origin of Metamaterials ”Metamaterials”,a name for artiﬁcial materials exhibiting unnatural macroscopic properties,has many attempts to deﬁne it.The ﬁrst deﬁnition was made by Dr. Rodger M.Walser of the University of Texas at Austin:[1] ”Macroscopic composites having a man-made,three dimensional,periodic cellular architecture designed to produce an optimized combination,not available in nature,of two or more responses to a speciﬁc excitation.Each cell contains metaparticles,macroscopic constituents designed with low dimensionality that allow each component of the excitation to be isolated and separately maximized.The metamaterial architecture is selected to strategically recombine local quasi-static responses,or to combine or isolate speciﬁc non-local responses.” Other attempts for deﬁnition of metamaterials can be found,for example,in the review article [2].Generally speaking,metamaterials are materials with man-made microstructures that behave exceptional responses not readily observed in the con- stituent materials and in nature.This term,originally used solely for the ﬁeld of electromagnetic materials,now has been expanded the usage to other branches of physics like,for instance,the ﬁeld of acoustic and elastic materials. Metamaterials with negative refractive index is one of the most active topics among the research community.This type of metamaterials has been called by many

2 names: Veselago medium,negative-index media,negative-refraction media,backward wave media,double-negative media,and even left-handed media [2].The idea of it traces back to 1968 when Veselago [3] postulated a theory (He published this work in 1967 in Russian and a year later in English [4]) for possible materials having neg- ative electric permittivity () and magnetic permeability (µ),and hence resulting in a negative refractive index.However,this concept did not stimulate much interest among researchers at that time.Three decades later,in 2000,Smith et al.[5] pro- posed a design of making an artiﬁcial material that exhibits a frequency region in the microwave regime with simultaneously negative permittivity and negative permeabil- ity.They demonstrated both by numerical simulation and experiment the unusual physical phenomena.In the same year,Pendry [6] proposed a practical application of the left-handed electromagnetic metamaterials:a perfect lens,capable of imaging objects and ﬁne structures that are much smaller than the wavelength of light.Since then,papers started to pour in.Many researchers tried to engage in exploring on this subject with various potentially novel applications [7–10].For a quick glance on this topic,one can ﬁnd some quality review in,for instance,Ref.[11,12]. Other than negative-index metamaterials,researchers have also been trying to discover and investigate diﬀerent types of artiﬁcial metamaterials leading to novel applications.Some exciting applications of general metamaterials include antennae with superior properties and cloaking devices that can make objects invisible [13–17]. However,these topics regarding electromagnetic metamaterials are far beyond the scope of this dissertation and will not be discussed any further. 1.2 Acoustic Metamaterials Motivated by mathematical analogy between acoustic and electromagnetic waves, researchers have recently attempted to ﬁnd the counterpart acoustic metamaterials. This section mentions and reviews particularly the negative-properties acoustic meta-

3 material and its forerunners as well as other classes of special materials that could be also considered as acoustic metamaterials. 1.2.1 Locally resonant acoustic metamaterials and the forerunners The perhaps ﬁrst type of electromagnetic metamaterials that have been discussed is the negative-index metamaterials as we reviewed the literature.It is here natural to drag our attention to the acoustic counterparts immediately.For the negative- index acoustic metamaterials,basically it is to ﬁnd materials possessing negative mass density and negative modulus since these two material properties govern the wave motions in acoustic (or elastic) materials.However,in contrast to the natural negative electric permittivity,no natural material has been found to possess negative mass density nor negative modulus.This diﬃculty brings about some barriers to the realization of acoustic counterparts. In 2000,Liu et al.[18] fabricated and investigated a sonic material based on the idea of localized resonant structures that exhibit spectral gaps with a lattice constant two orders of magnitude smaller than the relevant wavelength.This idea had given a possible solution to the length-scale problemof phononic band-gap materials.In fact, the interest in phononic band-gap materials was inspired from ample researches in photonic ban-gap materials.It is because of the rapid emergence of the industrial ap- plications of photonic band-gap materials such as lasers,antennas,waveguides,etc., researchers believe that the acoustic applications analogous to optics are promising. However,unlike those in photonic ban-gap materials,the wavelength of waves prop- agated in the phononic band-gap materials requires at least two times of that of the phononic lattice constant.This implies gigantic structures for environmental sound and vibration shielding and results in less interest.Based on the solution provided by Liu and his co-worker,no gigantic structure is ever needed when facing low frequen- cies.This has opened up some opportunities of realizing practical applications such

4 as, for example,ambient sound shielding devices.Since then,attention to materials with local resonance has arisen. Sheng et al.[19] continued the work followed by [18].They reported a new paradigm for the realization of robust elastic wave band gaps in the sonic frequency range.By considering the idea of localized sonic resonances,they showed theoreti- cally and experimentally that not only is it possible to realize sonic band-gap crystals with a lattice constant orders of magnitude smaller than the acoustic wavelength, but the band gap can also exist for non-periodic structures.It is noted that the term “metamaterial” was not used,yet,at that time.Same year,Goﬀaux and S´anchez- Dehesa [20] studied the propagation of elastic waves in two-dimensional (2D) periodic systems containing lattices of locally resonant materials.They mainly focus on pre- senting a variational method that oﬀers an alternative procedure to compute the band structure of phononic crystals. In 2004,Li and Chan [21] ﬁrst reported theoretically a possibility of the existence of acoustic metamaterials.They utilized the eﬀective mass density and bulk modu- lus derived by Berryman [22] and showed that both the eﬀective mass density and bulk modulus can be simultaneously negative,in the sense of an eﬀective medium. They claimed that the double negativity in acoustics is derived from low-frequency resonances,as in the case of electromagnetism,but the negative density and modulus are derived from a single resonance structure as distinct from electromagnetism in which the negative permeability and negative permittivity originates from diﬀerent resonance mechanisms.The physical meaning of the double negativity,according to Li and Chan,is that a medium displays an anomalous response at some frequencies such that it expands upon compression (negative bulk modulus) and moves to the left when being pushed to the right (negative mass density) at the same time.However, the latter explanation regarding negative mass density is not true since the negative mass density actually means the acceleration of the body been pushed points in the opposite direction of the acting force,which does not necessarily mean ”moving in the opposite direction”.For this point,we will have further discussion in Chapter 2 of

5 this dissertation.We note that this is the ﬁrst time the term“acoustic metamaterial” oﬃcially appeared,to the writer’s best understanding,in the journal publication.In 2005,Liu et al.[23] presented an analytical model to describe the low-frequency eﬀec- tive mass densities of three-component phononic crystals with local resonances based on eﬀective medium theory.They showed that the eﬀective mass densities can turn negative when close to the local resonances.In their previous work,Liu et al.at- tributed the negative response function to negative elastic constant [18] but later on in this publication they corrected the attribution to negative dynamic mass density. The pace on investigating acoustic metamaterials had increased since 2007.Some publications still discussed theoretically the possibility of realizing negative-index acoustic metamaterials.More shifted their focus toward the applications.The topics had been broadened and diversiﬁed. Sheng et al.[24] discussed the main characters of the dynamic mass density.They showed that the dynamic mass density used in the calculation of (long-wavelength) wave speed can diﬀer signiﬁcantly from the static volume-average value.The physi- cal reason for the diﬀerence between two mass densities is attributed to the relative motion between the components.The implicit assumption - that all components in a composite must move uniformly in the long-wavelength limit - can be violated in the limit of large acoustic impedance contrast between the components.Ambati et al.[25],in the same year,reported that the negative material responses of acoustic metamaterials can lead to a plethora of surface resonant states.Upon their investi- gation,negative eﬀective mass density is the necessary condition for the existence of surface states on acoustic metamaterials.It was found that the surface excitations enhance the transmission of evanescent pressure ﬁelds across the metamaterials,re- sulting in an image with resolution below the diﬀraction limit.This had opened an opportunity to the realization of acoustic superlens. Milton and Willis [26] employed a model modiﬁed from those of Sheng et al.[19] and Liu et al.[23] to show that the eﬀective mass density at a given frequency is not simply the average mass density and can even be negative.Their model consists of

6 the rigid matrix for simplicity so that one does not have to worry about the eﬀect of the elasticity of the matrix.In addition,the model was also modiﬁed to result in a complex anisotropic mass density.It should be noted that the anisotropic mass density,interestingly,is one of the important ingredients of making cloaking device. This part regarding acoustic cloaks will be discussed later in this chapter.Ding et al.[27] claimed that a metamaterial,possessing simultaneously negative bulk modulus and mass density,can be achieved by combining two types of structural units with built-in monopolar and dipolar resonances.The monopolar resonance arises from the fcc structure of bubble-contained-water spheres and the dipolar resonance from another relatively shifted fcc array of rubber-coated-gold spheres in epoxy matrix. However,no followers of this idea have been reported so far. Torrent and S´anchez-Dehesa considered a 2D array of elastic cylinders embedded in a ﬂuid that can be used to form a metamaterial and showed that a great variety of acoustic metamaterials can be possibly designed.They reported that the elastic properties of cylinders rather than rigid ones can broaden the range of acoustic pa- rameters available for designing metamaterials.So as the eﬀect of mixtures of two diﬀerent elastic cylinders.Application-wise speaking,they proposed a sonic Wood lens in which a parabolic variation of the refractive index is achieved by changing the cylinders’ radii in the direction perpendicular to the lens axis.Guenneau et al.[28] proposed a theoretical design for an acoustic metamaterial exhibiting negative refrac- tion eﬀect,as they claimed,in the low frequency regime.Their design consists of interleaving cylindrical double C-shaped cracks and ﬁnite thin stiﬀ sheets in a matrix of silica.They concluded that an acoustic metamaterial with their design can be used for focusing and conﬁning sound. Lazarov and Janson [29] have studied the wave propagation on a locally resonant lattice model which was originally introduced by Vincent [30] for use as a possible mechanical ﬁlter.They mainly focus on the inﬂuence of non-linearities on the ﬁl- tering properties.Based on their analytical and numerical models,they found that in the case with non-linear oscillators,the position of the band gap created by the

7 lo cal resonance can be shifted.The shift depends on the amplitude and the degree of non-linear behavior.Fokin et al.[31] developed a method to extract the eﬀective properties of metamaterials such as mass density and modulus from wave reﬂection and transmission coeﬃcients measured experimentally.Wu et al.[32] developed an eﬀective medium theory which goes beyond the quasistatic limit to predict the un- usual properties of elastic metamaterials in two dimensions.Their theory shows that not only the eﬀective mass density and eﬀective bulk modulus but also the eﬀective shear modulus can turn to negative when near the resonances.Although not familiar with the eﬀective medium theory,the writer believe that all these eﬀective negative properties discussed by Wu and his co-workers correlate closely with the band gap (attenuation mode),which is not really a negative-index metamaterial supposed to behave (propagation mode).For more references regarding eﬀective medium theory on the related topic,one can see Ref.[33–40]. In 2008,Cheng et al.[41] presented a material consists of a 1D array of repeated unit cells with shunted Helmholtz resonators,and showed that this ultrasonic (acous- tic) metamaterial exhibits a band gap where the eﬀective mass density and bulk mod- ulus are simultaneously negative.They admitted that the double-negative region of ultrasonic metamaterials,unlike that of electromagnetic metamaterials give rise to propagating mode,results in a stop band.They further considered the subwave- length resonant units built up by parallel-coupled structures with 1 to 4 Helmholtz resonators having identical resonant frequency [42],and found that the bandwidth of the gaps is dependent on the number of resonators in each unit.Yao et al.[43] conducted experiments using a 1D spring-mass system to demonstrate the eﬀect of negative eﬀective mass.They cited the formula given by Milton and Willis [26] and conﬁrmed the existence of it through the transmission properties of a ﬁnite periodic system composed of such basic units.Low transmissions of the system are observed in the negative mass range. Before ending this section,we note that there is a kind of acoustic metamaterials with negative modulus reported by Fang et al.[44].The ultrasonic metamaterial they

8 in vestigated consists of an array of subwavelength Helmholtz resonators with designed acoustic inductance and capacitance.An eﬀective dynamic modulus is obtained. Negative value is found near the resonance frequency.However,from their study,it is shown that the form of eﬀective modulus was obtained from following the formalism of electromagnetic response in metamaterials (for details see [44]).The writer,in his personal opinion,believe that this negative modulus acoustic metamaterial is one of the locally resonant-type metamaterials.The only diﬀerence is that,like Liu et al.in Ref.[18] had done,Fang and his co-workers attributed the negative response function to negative modulus. The topics regarding the locally resonant acoustic metamaterials are not limited to the references discussed here.They are just about to sail towards promising ap- plications.Aside of the locally resonant acoustic metamaterials,there are still other types of materials that have been published and could be,but yet,categorized as “acoustic metamaterials”,according to the deﬁnition.This leads to the discussion in the next section. 1.2.2 Other classes of special materials We emphasize that it is only,in this section,to brieﬂy introduce some types of special materials other than the existed acoustic metamaterials.Understanding these information broadens the readers’ mind and keeps them thinking out of the box.The readers,however,should be aware of not shifting the attention from premier focus. Phononic band-gap materials should be no doubt categorized as a kind of metama- terials in the writer’s opinion.The attention was drawn fromanalogous investigations of photonic crystals since early 1990s [45,46].Photonic crystals are artiﬁcial com- posites consist of periodic arrays of two transparent dielectrics.Similarly,phononic crystals are periodic structures composed of two ore more elastic freely vibrating materials [47].However,unlike photonic crystals,phononic band-gap materials and structures have yet directly led to industrial applications.Just like the locally res-