NOVEL SOUND ABSORPTIVE MATERIALS BASED ON ACOUSTIC METAMATERIAL PRINCIPLES

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1 NOVEL SOUND ABSORPTIVE MATERIALS BASED ON ACOUSTIC METAMATERIAL PRINCIPLES Gopal Mathur Acoustic Metamaterials, Inc., New York, NY, USA Chris Fuller Virginia Tech and National Institute of Aerospace, Hampton, VA, USA The porous sound absorptive materials have been the mainstay of noise control industry. But there are still technical challenges which noise control engineers have to deal with in terms of frequency range (in particular low frequency range), thickness, etc. when working with porous materials. One of the main technical challenge is associated with its high magnitude of the characteristic impedance at low frequencies. Using acoustic metamaterial principles, the authors present a new generation of sound absorbing materials which work efficiently over the broad frequency range and, in particular at low frequencies. Acoustic metamaterials are advanced engineered materials with properties that are hard or impossible to find in natural materials, e.g., negative effective density and/or negative bulk modulus. They thus have shown the potential to be used in exciting new applications, such as invisibility cloaks, high-resolution lenses, efficient and compact antennas, and highly sensitive sensors. Numerical simulation and experimental results are presented to demonstrate that novel sound absorptive materials with much improved acoustic properties can be designed using acoustic metamaterial principles. 1. Introduction Noise and vibration control play an important role in various industries, including aerospace, industrial machines, home appliances, HVAC, automobile vehicles, and buildings. Although a wide range of sound-absorbing materials exist for noise control; most of the materials that have a high value of sound absorption coefficient are usually porous [1]. Porous absorbers, such as foams, carpets, curtains and other soft materials, work due to frictional losses caused by the interaction of the velocity component of the sound wave with the surface of the absorbing material. Common sound absorptive materials are open cell foam or fiberglass. Sound absorption is an energy conversion process. The kinetic energy of the sound (air) is converted to heat energy when the sound strikes the cell walls. However, open-cell foams, are poor sound absorbers at low frequencies as they require a thickness of at least one-quarter of a wavelength to absorb sound. The sound absorbing characteristics of acoustical materials vary significantly with frequency and, in general, low frequency sounds are very difficult to absorb because of their long wavelength. For the vast majority of conventional acoustical materials, the material thickness has the greatest impact on the material s sound absorbing qualities. While the inherent composition of the acoustical material determines the material s acoustical performance, other factors can be brought to bear to improve or influence the acoustical performance. 1

2 In recent years, much work has been devoted to improve performance of sound absorbers in the low frequency range. Numerous studies that have dealt with the absorption of sound in porous materials concluded that: at low frequencies, the sound absorption coefficient is influenced by material thickness. A porous material is effective from the sound absorption point of view when its thickness is approximately one-tenth the wavelength of the incident sound. The maximum absorption occurs at a resonance frequency of a quarter the wavelength of the incident sound. In numerous applications, the micro-perforated panels (MPP) are used as low frequency sound absorbers and have been widely investigated over the last several decades [2]. Combined absorption structures containing the MPP, airspace and porous materials have also been proposed to improve the sound absorption bandwidth of MPP [3,4]. Such combined MPP, porous-airspace system, however, are: (1) not designed using metamaterial principles, (2) not compact and needed more overall space, and (3) didn t have cover broadband frequency range. Fuller and his colleagues have also been recently been investigating novel poro-elastic based acoustic metamaterials using periodic arrays of embedded spheres and MPP sheets [3-5]. These materials have shown significantly increased low frequency absorption and transmission loss as well as a designable capability. Fuller et al have also investigated the physics behind this new class of acoustic metamaterials [3-5]. While preliminary work on poro-elastic based acoustic metamaterials has been carried out, very little investigation of formal design approaches (apart from FEM system arrangement investigations) of these systems has been considered. Recently, Guild et al, used a sonic crystal based acoustic metamaterial approach for designing an acoustic absorber with better performance through improved impedance matching and enhanced absorption [6]. They, however, take a different approach using 2-dimensional sonic crystal and impedance matching method for designing sound absorbing material. Most porous absorbers are both homogenous and isotropic, meaning that their properties are both uniform in space and independent of direction. The uniformity of homogenous and isotropic materials allows their performance to be more easily predicted. Inhomogeneity can arise from a change in composition through space or the inclusion of regions of a second material. This change in the composition often generates a change in the properties, leading to anisotropic behavior. Many fiberreinforced materials show some anisotropy in their properties, which are dependent upon the orientation of the fibers. In this paper, acoustic metamaterial principles are used to design novel sound absorptive materials with much improved properties in the low frequency regime and still be effective over the entire frequency range of interest. Finite element models of foam samples with and without acoustic metamaterial (AMM) were developed and laboratory experiments were conducted to validated predicted results. 2. Application of Acoustic Metamaterial Principles Acoustic metamaterials are artificially fabricated materials designed to control, direct, and manipulate sound waves. Consequently, they open the door for improved or completely new applications. The concept of metamaterials was originally introduced for electromagnetic waves and received significant attention over the last decade, due to unusual interaction of these man-made materials with waves, as in the case of negative refraction materials [9]. More recently, the metamaterial concept has been extended to acoustic waves in a variety of scenarios of interest such as acoustic clocking [9-11], super-lensing [13] and sound focusing and confinement [13]. Several challenges still hold in applying metamaterials to realistic devices, in particular when low losses and wide bandwidths of operation are desired, as most of their exotic features are based on resonant inclusions. Acoustic metamaterials are based on sound mathematical principles and have been demonstrated to work experimentally, though in simple laboratory experiments. Current metamaterial cloaks, hyperlens and other systems, though different from other systems that previous researchers have modeled and conducted experiments so far, but use the same mathematical principles. There are some 2 ICSV23, Athens (Greece), July 2016

3 simpler meta-systems, such as acoustic barriers, resonant poro-elastic foams that have been shown to work are based on the same metamaterial principles. This gives confidence that acoustic metamaterials do work and can be adapted to other applications. Acoustic metamaterials usually gain their properties from their arrangement rather than composition, using the inclusion of small periodically arranged inhomogeneities to enact effective macroscopic behavior. Thus, the use of layered anisotropic arrangement to achieve desired impedance characteristics proposed in this paper takes advantage of its constituent sub wavelength properties rather than its overall material characteristics. Acoustic metamaterials can be generally divided into two main areas. Resonant materials usually consist of a matrix material in which is embedded periodic arrangements of inhomogenities such as rigid spheres or cylinders with spacing less than a wavelength. The embedded structures cause wave scattering and resonant behavior which creates stop band behavior and refraction effects. More recently attention has focused on non-resonant acoustic metamaterials which are designed to control the propagation of acoustic waves through fluids and materials. However, application of acoustic metamaterials for a particular application is not straightforward and requires application of mathematical principles. Recent advances in acoustic metamaterials have made it possible to design engineered metamaterials and associated structures for practical applications. Transformation acoustics provides precise control over acoustic wave propagation and this coupled with metamaterials gives unprecedented control in controlling, manipulating and directing sound waves. Another important aspect of acoustic metamaterials is their acoustic anisotropy which distinguishes metamaterials from normally occurring materials. Therefore, we face the question: how can acoustic anisotropy be achieved in practice. It is not observed in natural acoustic fluids where wave propagation depends on density and compressibility C. At least one more parameter is required; this could be introduced by allowing density or compressibility to be tensors. Schoenberg and Sen showed that the inertia tensor in a layered fluid is transversely isotropic with elements of density <> normal [14]. In recent years, acoustic metamaterial theory has given engineers a design tool in the form of Transformation Acoustics (TA) based on a solid mathematical background to guide and manipulate sound waves [9-11]. It may be mentioned that the TA is based on the invariance of the acoustic wave equation under coordinate transformation. The fluid densities and bulk moduli in real and virtual domains may be obtained using the following TA equations [9-10]: Transformation Acoustics Equations: r det( J )( J = J r κ = det( J ) κ v 1 r v r v Where, and are fluid densities in real and virtual domains, κ and κ are fluid bulk moduli in real and virtual domains, and J is the transformation Jacobian. The concept of TA has been used to realize arbitrary bending of acoustic waves with acoustic metamaterials that generally have anisotropic mass density. Several design methodologies have been proposed in order to obtain this anisotropy and to control the effective material parameters in the desired way. Two metamaterial-based approaches were investigated for designing novel sound absorptive materials in this paper. The concept of anisotropy using a periodic arrangement of multilayer structure, as illustrated by Cheng et al [15], shown in Figure 1, with the incident wave in the r-direction, is first considered. Each layer is composed of isotropic and homogenous materials with mass density A ( B ) and bulk modulus κ A ( κ B ). The thickness of each layer is much smaller than the wavelength so the ) T v ICSV23, Athens (Greece), July

4 whole stack can be treated as a single anisotropic material using effective medium theory. The homogenized density tensor and bulk modulus can be expressed as [15] A + ηb r =, = 1+ η θ 1+ η A = κ 1+ η κ η +, A κ B η + B, where η=d! /d! is ratio of thicknesses for the B and A layers,! and! are the radius and angular components of the effective anisotropic density tensors respectively and κ is the effective bulk modulus. Figure 1: Anisotropic acoustic metamaterial using layers of two homogenous and isotropic natural materials. [15]. Using the Transformation Acoustics approach outlined above, it can be shown that the densities and bulk modulus in two dimensions on a structure can be engineered to be anisotropic. However, for some applications, the fabrication of these materials could be challenging as these material parameters can have values that could be hard to implement in practice without some trade off in performance. In the second approach, in the realization of such a structure, we consider a bulk metamaterial design consisting of a stack of perforated plates made of an acoustically hard material, separated by a sound-supporting fluid (e.g., air). The elementary constituent is a 2D rigid hole array, the transmission properties of which have received considerable attention in connection to novel phenomena such as shielding of sound near the onset of diffraction, Fabry-Pe rot resonances, and acousto-elastic resonances, and as sound absorbers. It has been shown that these metamaterial blocks with perforated stacks exhibit broad-angle negative refraction, unlike fishnet electromagnetic metamaterials, which operate within narrow angular ranges [16]. Thus, such holey anisotropic metamaterial can exert subwavelength control over sound waves beyond what is achieved with naturally occurring materials. Such acoustic metamaterials also do not rely on diffraction to achieve negative refraction, in contrast to phononic crystals. Earlier, Popa et al used similar metamaterial approach of using a stack of perforated plates for their two- and three-dimensional acoustic cloaks [11-12]. Both of these approaches utilize layered and anisotropic acoustic metamaterial methods and can be used to design an acoustic metamaterial device. However, there has been very limited work on such acoustic metamaterials that diffract and refract sound waves and allow control over wave propagation for absorptive material impedance matching and more development is needed in this field. In this paper, using the Transformation Acoustics approach described above, we determined the densities and bulk modulus in two dimensions of a layered structure so that it can engineered to be anisotropic with desired sound absorptive properties, particularly in the low frequency region and also be effective over the entire frequency range of interest. 4 ICSV23, Athens (Greece), July 2016

5 3. Numerical Modelling and Experimental Measurements The COMSOL Multiphysics software was used to develop finite element (FE) numerical model for the acoustic-structure interaction of the acoustic metamaterial (AMM) layered foam samples and to determine acoustic results realized with the metamaterial structure. The porous foam was simulated using the rigid Bio equivalent fluid model in the COMSOL Pressure Acoustics Interface. The Johnson-Champoux-Allard model is one of the equivalent fluid models, which has been widely used in applications. Compared with other empirical models and regarded as a phenomenological one based on the internal flow of pore, this model needs more macroscopic acoustic parameters of materials. Hence, the acoustical characteristics of common porous materials can be accurately depicted by Johnson-Champoux-Allard model. The Johnson-Champoux-Allard model contains five macroscopic acoustic parameters: porosity, static flow resistivity, tortuosity factor, viscous characteristic length and thermal characteristic length. The necessary parameters in the model are given in Table 1: Table 1: Geometric parameters of the skeleton of porous material. Symbol Unit Parameter φ [-] Porosity α! [-] Tortuosity k! [m! ] Viscous permeability! k! [m! ] Thermal permeability Λ [m] Viscous characteristic length Λ! [m] Thermal characteristic length The surface acoustic impedance of the porous absorber sample is given by: Z ω = K exp 2iωl exp 2iωl K + 1 K 1 = - i K cot (ωl!! ), where, and K are effective density and bulk modulus of the porous medium. The acoustic absorption and reflection coefficients are given by: α(ω) = 1 - R(ω)!, R ω =!!!!!!!!!!, where, Z! is the characteristic impedance of poro-fluid. The impedance tube setup for measuring transmission loss and absorption coefficients of porous samples with and without AMM is shown in Figure 2. The impedance tube was calibrated with known sample and procedures. For absorption measurements, the downstream tube and microphones are removed and the system is terminated with a rigid block behind the test sample. Different software is used to estimate transmission loss and absorption from the microphone signals. ICSV23, Athens (Greece), July

6 Figure 2: Impedance tube setup for sound absorption coefficient measurements. 4. Results and Discussion With a view to validate the application of anisotropic metamaterial design approach to AMM sound absorptive materials, a simple foam sample with 2 AMM layers (shown in inset in Figure 3) was constructed and its absorption coefficient compared with that of the baseline foam sample of the same thickness but without AMM layers. Finite element models of the foam sample with and without AMM layers developed using COMSOL MultiPhysics software, as explained in Section 3, were used to predict sound absorption co efficients of the samples. Figure 3 shows a comparison of measured and predicted sound absorption coefficients of the foam samples with and without AMM layers. In addition the AMM system can be seen to have significantly higher sound absorption at low frequencies than the standard poroelastic material ( i.e. without added AMM MPP sheets). It may be observed in Figure 3 that there is reasonably good agreement between measured and predicted results for the foam samples with and without AMM layers. The slight discrepancy between measured and predicted absorption coefficients, in Figure 3, of the baseline foam sample without AMM layers may be attributed to the approximate poro-elastic parameters used for the Johnson-Champoux-Allard model in the COMSOL numerical model. In addition, the construction and installation of the foam sample with AMM layers inside the impedance tube may have also contributed to the discrepancy between the predicted and measured results, particularly above 1000Hz, 1.2 Absorp/on Coefficient α, Foam (Measured) α, Foam w/ AMM-2 Layers (Measured) α, Foam (Predicted) α, Foam w/ AMM-2 Layers (Predicted) 1 α !! Frequency, Hz Figure 3: Measured absorption coefficients of the foam sample with and without AMM system (2-layer AMM porous sample sketch shown in inset in lower right hand corner). 6 ICSV23, Athens (Greece), July 2016

7 for the foam sample with 2 layers of acoustic metamaterial system. The agreement between measured and predicted results for the foam sample with AMM layers thus validates the acoustic metamaterial based design approach for sound absorptive materials for improving sound absorption in the low frequency range. To further enhance the low frequency performance of the foam, a new anisotropic AMM foam sample was designed with four AMM layers and a FE model developed using the COMSOL software. The predicted sound absorption coefficient of the 4-layer AMM foam sample is compared with the baseline 2-inch thick foam sample without AMM layers in Figure 4. It may be observed that the sound absorption coefficient of the 4 layer AMM foam system is enhanced between Hz over the three layer AMM foam sample shown in Figure 3 and also is significantly greater than the standard foam material. Thus, layered AMM systems with much improved sound absorption properties in the lower frequency regime can be designed using acoustic metamaterial principles. α!! Frequency, Hz Figure 4: Predicted Absorption coefficients of 2 inch thick foam sample with and without AMM (4-layer AMM porous sample sketch shown in inset in middle right hand corner. 5. Conclusions Traditional porous materials, in general, exhibit poor low-frequency noise absorption characteristics due to its inherent high acoustic impedance. This study shows that the effective range of sound absorption can be extended to lower frequencies by application of acoustic metamaterial principles to porous materials. Finite element numerical models of porous foam samples with and without acoustic metamaterial layers were developed using COMSOL Multiphysics software. An impedance tube was used to measure sound absorption coefficient of the foam sample to validate numerical modelling and the acoustic metamaterial design approach. Finally, a 4-layer AMM foam system, which is effective over the entire frequency range of interest with enhanced sound absorption between Hz, was designed. Thus, it was shown that layered AMM systems with much improved sound absorption properties in the lower frequency regime can be designed using acoustic metamaterial principles. ICSV23, Athens (Greece), July

8 Acknowledgements The authors acknowledge support from the Acoustic Metamaterial Inc., New York, NY for conducting this research. REFERENCES 1. Arenas, J. P. & Crocker, M. J. Recent Trends in Porous Sound Absorbing Materials, Sound & Vibration, July (2010). 2. Maa, D. Y. General theory and design of micro-perforated panel absorbers, Acta Acoustica, 22(5): , (1997). 3. Tayong, R., Dupont, T., and Leclaire, P. Sound absorption of a micro-perforated plate backed by a porous material under high sound excitation: measurement and prediction, International Journal of Engineering and Technology, 2(4) , (2013). 4. Dengke, LI, CHANG, Daoqing; LIU, Bilong, and TIAN, Jing. Improving sound absorption bandwidth of micro-perforated panel by adding porous materials. Inter-Noise 2014, Melbourne, Australia, November, (2014). 5. Fuller, C.R. and Saux, T.D. Sound Absorption Using Poro-Elastic Acoustic Meta Materials. Proceedings of Inter-Noise 2012, New York, NY, August, (2012). 6. Slagle, A.C. and Fuller, C.R. Low Frequency Noise Reduction Using Poro-Elastic Acoustic Metamaterials. Proceedings of 21 st AIAA/CEAS Aeroacoustics Conference, Dallas, TX, June, (2015). 7. Mitchell, K.R. and Fuller, C.R. Design Optimization of Broadband Acoustic Metamaterial Liners through Finite Element Efficiacy Studies., Proceedings of 21 st AIAA/CEAS Aeroacoustics Conference, Dallas, TX, June, (2015). 8. Guild, M. D., Kan, W., and Sánchez-Dehesa, J. How an acoustic metamaterial can make a better sound absorber? 168th Meeting of the Acoustical Society of America, Indianapolis, IN, USA October, (2014). 9. Smith, D. R., Pendry, J. B. & Wiltshire, M. C. K. Metamaterials and Negative Refractive Index. Science 305, , (2004). 10. Chen, H. and Chan, C. T. Acoustic Cloaking in three dimensions using acoustic metamaterials. Appl. Phys.Lett. 91, , Popa. I. and Cummer S. A. Design and characterization of broadband acoustic composite metamaterials. Physical Review B, 80(17):174303, (2009). 12. Zigoneanu, L. B., Popa, I. Starr, A.F., and Cummer, S. A. Design and measurements of a broadband 2d acoustic metamaterial with anisotropic mass density. Journal of Applied Physics, 109:054906, (2011). 13. Ambati, M., Fang, N., Sun, C. & Zhang, X. Surface resonant states and superlensing in acoustic metamaterials. Phys. Rev. B 75, , (2007). 14. Schoenberg, M. and Sen, P. N. Properties of a periodically stratified acoustic half-space and its relation to a Biot fluid. Journal of the Acoustical Society of America, 73, 61 67, (1983). 15. Cheng, Y., Yang, F., Xu, J.Y. and Liu, X.J. A Multilayer Structured Acoustic Cloak with Homogeneous Isotropic Materials. Applied Physics Letter, 92, , (2008). 16. Christensen, J. and Garcı a de Abajo, F.J. Anisotropic Metamaterials for Full Control of Acoustic Waves. Phys. Rev. Lett. 108, , (2012). 8 ICSV23, Athens (Greece), July 2016