Proceedings of Meetings on Acoustics

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1 Proceedings of Meetings on Acoustics Volume 9, 23 ICA 23 Montreal Montreal, Canada 2-7 June 23 Structural Acoustics and Vibration Session psa: Measurement and Modeling of Structures with Attached Noise Control Materials II psa7. Full-band exact homogenization of one-dimensional elastic metamaterials Min Yang*, Zhiyu Yang and Ping Sheng *Corresponding author's address: Physics, The Hong Kong University of Science and Technology, Hong Kong, 852, Kowloon, Hong Kong, Metamaterials extend the realm of materials' properties by carefully designed structural inclusions. By targeting the extraction of effective properties from composite materials, homogenization theory plays an important role for metamaterials in their design and characterization. However, conventional homogenization methods are limited to the long wavelength limit. Here, we introduce an exact homogenization scheme valid for one-dimensional metamaterials over the full frequency band. In this scheme, with the aid of eigenstates's characterization, a set of explicit formulas for effective mass density and effective elastic modulus are obtained by matching the surface responses properties of a metamaterial's single structural unit with a piece of effectively homogenized material. Applying this scheme on a layered structure, the predicted transport properties and displacement fields from the effective parameters show excellent agreement with numerical simulations. Published by the Acoustical Society of America through the American Institute of Physics 23 Acoustical Society of America [DOI:.2/ ] Received 2 Jan 23; published 2 Jun 23 Proceedings of Meetings on Acoustics, Vol. 9, 656 (23) Page

2 INTRODUCTION The emergence of acoustic/elastic metamaterials has significantly broadened the horizon for acoustic/elastic waves. Novel phenomena such as focusing and sub-diffraction imaging [, 2, 3, 4], near field amplification [5], cloaking [6, 7, 8, 9], localization of ultrasound [], one-way transmission [, 2, 3], as well as super absorption [4] have been proposed or experimentally demonstrated. At the core of these phenomena are the resonance-induced effective material characteristics such as the negative effective mass [5, 6, 7, 8], negative effective modulus [9, 2], or negative shear modulus [2, 22], which are frequency-dependent. Constructed from the similar unit by repeating itself, metamaterials effectively perform homogeneity in large scale while the heterogeneity is embedded in the small unit-cells. However, the connection between the effective material characteristics and the under heterogeneous structures is not that straightforward, a mathematical method named homogenization is required to extract the global effective parameters from local structures [2, 23, 24]. In this paper, we propose an exact homogenization scheme for one-dimensional metamaterial, based on the purpose of matching its motions at boundaries of unit-cells by an effective homogeneous material. This requirement is the result of facts that the unit-cells are usually employed as Lego to construct samples of metamaterials thus only their boundaries could interact with environments. The design and characterization for a metamaterial based on this scheme are demonstrated by a simple layered structure. And the scheme s validity is confirmed through the excellent agreement on the transmission process between the homogenization predictions and numerical simulations. HOMOGENIZATION SCHEME The metamaterials periodicity (or quasi-periodicity) enables investigations on only one representative volume unit (RVU) instead of the entire material. The motions of each unit-cell is actually the superpositions of its eigenstates. Consider the eigenfunction expansion of the scalar Green function [24] G(x, x ) = u α (x)u α(x ) α ρ α (ω 2 α + iωβ α ω 2 ), () where angular frequency is denoted as ω. And for each eigenstate u α (x), a generalized mass density is defined as ρ α u α (x)ρ(x)u α(x)dx, (2) Ω resonance angular frequency is given by ω 2 α u α (x) ( C(x) ) u α (x)dx/ρ α ; (3) Ω x x and generalized dissipation coefficient (for β α ω α ) is given by [25] β α u α (x) ( η(x) ) u α (x)dx/ρ α. (4) Ω x x Here ρ(x) is the spatial distributed mass density for the heterogeneous composite, and C(x) and η(x) are its elastic modulus and dissipation coefficients, respectively. Ω = [ x, x ] represent the space occupied by RVU. It is seen from Eq. () that each term in the summation is dominant in the frequency regime around its eigenfrequency ω α. Hence for the purpose of investigation in finite (and low) frequency range, only a few eigenstates are needed. With no volume distributed load assumed, units only interact with each other at boundaries. Hence, we are only interested in boundary evaluated Green functions G(x,±x ), which can always be decomposed into a symmetric component G + and an anti-symmetric component G, where G ± = G(x, x ) ±G(x, x ) = α u α (x )(u α (x ) ± u α ( x )) ρ α (ω 2 α + iωβ α ω 2. (5) ) Proceedings of Meetings on Acoustics, Vol. 9, 656 (23) Page 2

3 : Elastic Matrix; : Inserted Plate; : Boundary of RVU. u(x) u u u 2 FIGURE : Schematics for the layered metamaterial and relevant eigenstates in low frequency regime. The displacement of the inserted plate is distinguished by red point. According to their symmetry, the dipolar modes whose u α (x ) = u α ( x ) only contribute to the symmetric G +, and monopolar modes with u α (x ) = u α ( x ) only affect G. The so called anti-resonance exists in between two adjacent eigenmodes with similar symmetry due to their out-of-phase hybridization. That implies u(x ) + u( x ) = for dipolar ones and u(x ) u( x ) = for monopolar. Apparently, besides the eigenfrequencies, the anti-resonance frequencies also serve as critical points, in which the relevant symmetric Green functions G ± change signs by crossing zero values. Now consider a homogeneous sample with length 2x. The eigenfunctions of such a system are simply given by ( ) απ(x + x ) ū α (x) = cos, α =,2,3,... (6) x 2x with ū (x) = /2x. The relevant eigenfrequencies are given by ω α = απ C/ ρ/(2x ). Here C and ρ are the modulus and mass density of the homogeneous system. It turns out that for the D homogeneous system, the eigenfunction expansion of the Green function can be analytically evaluated, with the results shown in the following: ( cot x ω ) ( ρ/ C tan x ω ) ρ/ C Ḡ + = ω, Ḡ = ρ C ω. (7) ρ C The homogenization condition is then simply G ± = Ḡ±. (8) Substitution of Eq. (7) into Eq. (8) leads to analytical solution for the effective dynamic mass density and effective elastic modulus as ( arctan G / ) G + x ρ(ω) =, C(ω) = ( ω 2 x G+ G arctan G / ). (9) G + G+ G Here the values of G ± at each frequency can be determined from the end displacements u α (±x ) plus the values of the ρ α and β α. The effective wave-vector k and effective impedance Z can also be evaluated as k(ω) = ω ρ(ω) C(ω) = ( arctan G / ) G +, Z(ω) = ρ(ω) C(ω) = x ω. () G + G Apparently, the change of sign for the dipole related G + only reverse the sign of effective mass density ρ hence is mass-density-type (MDT), and the sign change of monopolar G only affecting effective modulus C is elastic-modulus-type (EMT). Therefore, starting from their positive static averaged values at frequency, Proceedings of Meetings on Acoustics, Vol. 9, 656 (23) Page 3

4 5 Im( ρ) 3 (a) Im( C) 2 2 (b) Im( k) π/2x (c) Im( Z) 2 (d) 4 ω 2+ Frequency ω 3 2 ω ω + 3 Re( ρ) 2 2 Re( C) Re( k) π/2x 2 Re( Z) ω + FIGURE 2: Effective parameters of the metamaterial. The black lines represent real parts and the red lines are the imaginary parts. The bandgaps are shaded in gray, the negative-refractive passband is colored in yellow, and the conventional passband is in white. α ω α π π ρ α u α (x ) u α ( x ) TABLE : The values of ω α, ρ α, and the displacement u α (±x ) at two ends for the first 5 eigenstates. negative mass density could be found after its first MDT anti-resonance frequency, and negative modulus could be found after its first EMT eigenfrequency. The first bandgap is, thereby, located in the regime between these two frequencies, in which effective wave-vector k contains imaginary part and only allows evanescent waves existing in the effective medium. EXAMPLE As a demonstration of this homogenization scheme, a simple layered structure will be investigated in the following, which contains a series of stiff plates with negligible thickness periodically inserted in an elastic matrix (as shown in Fig. ). The three dominant eigenstates for low frequency regime have been presented in Fig.. The first one is simple rigid motions for the entire body. In the second mode, the matrix is purely stretched with the center plate keeping still. For the third eigenstate, the plate moves as an oscillator with opposite phase against to the two ends. While the first and third eigenmodes are clearly dipolar in character and hence mass-density-type (MDT), the second mode has the monoplar symmetry and hence elastic-modulus-type (EMT). The frequency of the first eigenstate is always due to its undeformed character, and the second mode is no different from a uniform one s since the center plate is motionless. Unlike the first two eigenstates, whose frequencies are not relevant to the rigid plate, the plate s mass serves as the most important parameter for tuning the third eigenmode s frequency ω 2+. Therefore, by modifying this mass, the width of the first bandgap could be adjusted due to the moving of the first MDT anti-resonance frequency ω + following ω 2+. Choosing the mass density and elastic modulus of the matrix, and the thickness of the RVU as natural units, for the plate with mass as 2, the third eigenmode has frequency ω 2+ = 4.58 and ω + =.723. Hence the bandgap could be expected in the region (ω +,ω ) with width as.49 since ω = π. Proceedings of Meetings on Acoustics, Vol. 9, 656 (23) Page 4

5 5 (a) (b) (c) 4 ω 2+ Frequency ω 3 2 layer 5layers layers ω ω +..3 T. T ω + 4 T FIGURE 3: Transmission intensity coefficients for samples constructed by layer (a), 5 layers (b) and layers (c) units. The black lines are the results predicted from effective parameters, and the red points are numerical simulation results. Excellent agreement is seen. (a) (c) (e) (g) Displacement (b) (d) (f) ω = ω + ω ω 2+ FIGURE 4: The displacement field for the 5 layer units sample under transmission process. The red lines are real fields and the black ones represent effective ones from homogenization. Displacement of inserted plates are marked as red points and the ones of units boundaries are distinguished as black points. Excellent match for the units boundaries motion is seen. Based on these eigenstates, the relevant parameters ω α, ρ α, β α and u α (±x ) are evaluated from their definitions Eq. (2) (3) and (4) as showing in Table.. Where dissipation is ignored thus β α =. Now, the two sets of effective parameters ρ, C, and k, Z are possible to be evaluated from equations (9) and (). The results are showing in Fig. 2. From ρ and C, we can analytically calculate the transmission coefficients T, when the sample is embedded in external radiating medium, by the transfer matrix method [26]. The results for samples constructed by different layers of units are displayed in Fig. 3 as solid curves (in which the material properties of external medium have been set as ρ m =. and C m =.). They are seen to agree remarkably well with the direct simulation results (red points), even beyond the usual long wavelength regime, e.g., around ω + and ω where the half wavelength is on the order of 2x. As we expected, the intensity of transmission is dramatically dropped while increasing of the layers number of units. One of the important features for this homogenization scheme is that it is only attempting to match the motion at boundaries between units by homogeneous effective materials. To confirm this, the effective field has been compared with the real one in Fig. 4 for several representative frequencies under the transmission process of a5layers sample. Apparently, within the sample, only the motions at the units boundaries (the black points) coincide with the effective field (black curves), and at other portions, especially at the inserted plates (red points), the difference between real motions and effective ones becomes significant, when the frequency is relatively high. Nevertheless, despite all these differences inside of materials, excellent Proceedings of Meetings on Acoustics, Vol. 9, 656 (23) Page 5

6 coincidence are seen between the homogenization predicted and real scattering fields at external medium, as we expected. CONCLUTION To conclude, we have implemented an exact homogenization scheme for D metamaterials. By attempting to match unit-cells boundary motions and utilizing the resonance features of eigenstates, this scheme is reliable and easy to use. All of these have been confirmed by a simple example of periodically layered structure, excellent agreements between homogenization predictions and numerical simulations are seen. Benefiting from the connection between effective properties and RVU s eigenstates, which is constructed by this homogenization scheme, the design and characterization of D metamaterials could be facilitated for the applications in, such as, sound blocking or absorptions for certain object via layers of coated thin films on the surface. ACKNOWLEDGMENTS This work is supported by Hong Kong RGC grant HKUST 6427, HKUST 666, and HKUST2/CRF/G. REFERENCES [] J. Li, L. Fok, X. Yin, G. Bartal, and X. Zhang, Experimental demonstration of an acoustic magnifying hyperlens, Nature materials 8, (29). [2] S. Zhang, L. Yin, and N. Fang, Focusing ultrasound with an acoustic metamaterial network, Physical review letters 2, 943 (29). [3] J. Zhu, J. Christensen, J. Jung, L. Martin-Moreno, X. Yin, L. Fok, X. Zhang, and F. Garcia-Vidal, A holey-structured metamaterial for acoustic deep-subwavelength imaging, Nature physics 7, (2). [4] A. Sukhovich, B. Merheb, K. Muralidharan, J. Vasseur, Y. Pennec, P. Deymier, and J. Page, Experimental and theoretical evidence for subwavelength imaging in phononic crystals, Physical review letters 2, 543 (29). [5] C. Park, J. Park, S. Lee, Y. Seo, C. Kim, and S. Lee, Amplification of acoustic evanescent waves using metamaterial slabs, Physical Review Letters 7, 943 (2). [6] S. Zhang, C. Xia, and N. Fang, Broadband acoustic cloak for ultrasound waves, Physical Review Letters 6, 243 (2). [7] B. Popa, L. Zigoneanu, and S. Cummer, Experimental acoustic ground cloak in air, Physical Review Letters 6, 2539 (2). [8] M. Farhat, S. Guenneau, and S. Enoch, Ultrabroadband elastic cloaking in thin plates, Physical review letters 3, 243 (29). [9] N. Stenger, M. Wilhelm, and M. Wegener, Experiments on elastic cloaking in thin plates, Physical Review Letters 8, 43 (22). [] H. Hu, A. Strybulevych, J. Page, S. Skipetrov, and B. van Tiggelen, Localization of ultrasound in a three-dimensional elastic network, Nature Physics 4, (28). [] B. Liang, B. Yuan, and J. Cheng, Acoustic diode: Rectification of acoustic energy flux in one-dimensional systems, Physical review letters 3, 43 (29). Proceedings of Meetings on Acoustics, Vol. 9, 656 (23) Page 6

7 [2] B. Liang, X. Guo, J. Tu, D. Zhang, and J. Cheng, An acoustic rectifier, Nature Materials 9, (2). [3] N. Boechler, G. Theocharis, and C. Daraio, Bifurcation-based acoustic switching and rectification, Nature Materials, (2). [4] J. Mei, G. Ma, M. Yang, Z. Yang, W. Wen, and P. Sheng, Dark acoustic metamaterials as super absorbers for low-frequency sound, Nature Communications 3, 756 (22). [5] Z. Liu, X. Zhang, Y. Mao, Y. Zhu, Z. Yang, C. Chan, and P. Sheng, Locally resonant sonic materials, Science 289, (2). [6] Z. Yang, J. Mei, M. Yang, N. Chan, and P. Sheng, Membrane-type acoustic metamaterial with negative dynamic mass, Physical review letters, 243 (28). [7] S. Lee, C. Park, Y. Seo, Z. Wang, and C. Kim, Acoustic metamaterial with negative density, Physics Letters A 373, (29). [8] S. Yao, X. Zhou, and G. Hu, Investigation of the negative-mass behaviors occurring below a cut-off frequency, New Journal of Physics 2, 325 (2). [9] N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, and X. Zhang, Ultrasonic metamaterials with negative modulus, Nature materials 5, (26). [2] S. Lee, C. Park, Y. Seo, Z. Wang, and C. Kim, Acoustic metamaterial with negative modulus, Journal of Physics: Condensed Matter 2, 7574 (29). [2] Y. Lai, Y. Wu, P. Sheng, and Z. Zhang, Hybrid elastic solids, Nature Materials, (2). [22] Y. Wu, Y. Lai, and Z. Zhang, Elastic metamaterials with simultaneously negative effective shear modulus and mass density, Physical Review Letters 7, 556 (2). [23] I. Andrianov, V. Bolshakov, V. Danishevs kyy, and D. Weichert, Higher order asymptotic homogenization and wave propagation in periodic composite materials, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 464, 8 2 (28). [24] J. R. Willis, Dynamics of composites, in Continuum micromechanics. CISM Lecture Notes, edited by P. Suquet, (997). [25] L. Landau and E. Lifshitz, Theory of elasticity, chapter 5, 35 (Pergamon Press) (989). [26] E. Adler, Matrix methods applied to acoustic waves in multilayers, Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions on 37, (99). Proceedings of Meetings on Acoustics, Vol. 9, 656 (23) Page 7