⇦Chinese Physics Letters, 2019, Vol. 36, No. 2, Article code 024301 Physically Realizable Broadband Acoustic Metamaterials with Anisotropic Density * Zhi-Miao Lu (陆智淼), Li Cai (蔡力), Ji-Hong Wen (温激鸿)**, Xing Chen (陈幸) Affiliations Science and Technology on Integrated Logistics Support Laboratory, National University of Defense Technology, Changsha 410073 Received 29 October 2018, online 22 January 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11004250 and 51275519.
**Corresponding author. Email: wenjihong_nudt1@vip.sina.com Citation Text: Lu Z M, Cai L, Wen J H and Chen X 2019 Chin. Phys. Lett. 36 024301    Abstract Transformation acoustics are concentrated for the purpose of designing novel acoustic devices to tailor acoustic waves to achieve desirable characteristics. However, these devices require fluid or fluid-like materials with an anisotropic density that generally does not exist in nature. Therefore, we introduce pentamode metamaterials into an alternating multilayer isotropic medium model to build fluid-like metamaterials with an anisotropic density. A 2D acoustic bending based on transformation acoustics is established and investigated to verify our method. This idea provides a method to design broadband and physically realizable acoustic metamaterials with an anisotropic density and is meaningful for the design of acoustic metamaterials. DOI:10.1088/0256-307X/36/2/024301 PACS:43.35.+d, 43.20.+g, 43.40.+s © 2019 Chinese Physics Society Article Text Considerable attention has recently been devoted to the development of transformation acoustics as a viable approach to designing complex material structures that can control the propagation of acoustic waves.[1-3] A series of novel devices, such as cloak,[4-7] beam shifter,[8] acoustic bending,[9-11] acoustic black hole[12] and acoustic sensor,[13] have been developed and exhibit fantastic phenomenon in acoustics by applying transformation acoustics. Therefore, the research and development of transformation acoustics are very useful in manipulating sound waves. Transformation acoustics interprets the compression and dilation of space as appropriate properties of materials, which generally requires fluid-like materials with an anisotropic density.[14-16] In fact, this derived property is hard to find in nature. Artificial acoustic materials such as phononic crystals[17,18] or acoustic metamaterials[4,19] can achieve the material parameters needed for the transformation acoustics design. An acoustic transmission line has been used to design acoustic cloak.[20] Sonic crystals with asymmetric lattice or scatters have also been discussed to build acoustic bending by transformation acoustics.[11] However, they cannot realize anisotropic density in a broadband because of the strong dispersion inherent to the resonance units. An alternative approach is to use a superlattice approach with alternating isotropic layers to mimic anisotropic density.[16,18] However, realizing this metamaterial is difficult because large changes of parameters of fluid-like materials are needed.[3,15] Norris put forward pentamode metamaterials,[21,22] which are also called metal water because they can mimic the acoustic properties of a fluid by applying solid structures. Their properties are significantly more dependent on the designed structure than on the chemical composition.[23] By adjusting the microstructure of pentamode metamaterials, the effective density and modulus can vary over a wide range.[7,21,24-26] In general, research of pentamode materials mainly focuses on the anisotropic modulus design. However, in this study we first introduce the pentamode materials into an alternating multilayer isotropic medium model to design anisotropic density metamaterials. An effective medium theory analysis is discussed.[18] Without resonant structure units, the design can obtain anisotropic density in a broad frequency band. An acoustic bending is implemented by this method, which performs perfectly in manipulating acoustic wave propagation. This acoustic device can be fabricated by solid hexagonal unit cells and is easier to manufacture. Therefore, this design is meaningful to the development of transformation acoustics. The anisotropic density metamaterial structure with this parameter specification is physically synthesized by a multilayered structure, as shown in Fig. 1(a), which contains two types of isotropic and homogeneous materials that serve a fluid-like material with anisotropic density in the low frequency limit.[18] The acoustic parameters of the anisotropic density metamaterial are given as follows:[19] $$\begin{align} \frac{1}{k}=\,&\frac{1}{d_{1}+d_{2}}\Big({\frac{d_{1}}{k_{1}}+\frac{d_{2}}{k_{2}}}\Big),\\ \rho_{x} =\,&\frac{1}{d_{1}+d_{2}}({d_{1} \rho_{1}+d_{2} \rho_{2}}),\\ \frac{1}{\rho_{y}}=\,&\frac{1}{d_{1}+d_{2}}\Big({\frac{d_{1}}{\rho_{1}}+\frac{d_{2}}{\rho_{2}}}\Big),~~ \tag {1} \end{align} $$ where $k$ is the effective modulus of the anisotropic density metamaterial, $\rho_{x}$ and $\rho_{y}$ are the effective mass densities of two directions, $\rho_{1}$ and $\rho_{2}$ are the mass densities of two isotropic materials, $k_{1}$ and $k_{2}$ are the modulus of two isotropic materials, and $d_{1}$ and $d_{2}$ are the thicknesses of two isotropic materials, respectively. However, the problem of engineering a multilayered structure is hard to solve due to the difficulty of building isotropic and homogeneous materials composed of fluid-fluid or fluid-solid periodic composites.[1,18,19] To overcome this drawback, a pentamode metallic structure is applied to construct an isotropic and homogeneous fluid-like material that replaces these composites, which simplifies the physics realization.
cpl-36-2-024301-fig1.png
Fig. 1. The alternating layered structure and the pentamode metallic structure. (a) Schematic of the alternating layered structure. When the thickness of each layer is significantly smaller than the incident wavelength, the entire layered system can be treated as a single material with anisotropic mass density and an isotropic modulus. (b) Illustration of the pentamode metallic structure. The two sides of the hexagon, the width of the interconnecting arms, and the length and width of the attached mass are $l$, $h$, $t$, $l_{1}$ and $h_{1}$, respectively.
The pentamode metallic structure is designed with the hexagonal unit cell originally proposed by Norris,[21] as shown in Fig. 1(b). The framework is a 2D network of metallic arms that are arranged in a regular honeycomb lattice, with the attached mass located in the middle of the arms. The two sides of the hexagon, the width of the interconnecting arms, and the length and width of the attached mass are $l$, $h$, $t$, $l_{1}$ and $h_{1}$, respectively, as marked in Fig. 1(b). The entire structure is composed of aluminum (density $\rho_{\rm Al} =2700$ kg/m$^{3}$, Young's modulus $E_{\rm Al} =6.9$ GPa, and Poisson's ratio $\nu =0.33$) permeated by air. The effective modulus $k_{\rm effect}$ of the structure is calculated by the Gibson formula[27,28] with $l$, $h$ and $t$, and the effective mass density $\rho_{\rm effect}$ is defined with $l_{1}$ and $h_{1}$ as $$\begin{align} k_{\rm effect} =\,&E_{\rm Al} \Big(\frac{t}{l}\Big)^{3}\frac{\cos \theta}{({\frac{h}{l}+\sin \theta})\sin^{2}\theta}, \\ \rho_{\rm effect} =\,&\rho_{\rm Al} \frac{S+6l_{1} h}{l\cos \theta ({2h+2l\sin \theta})},~~ \tag {2} \end{align} $$ where $S=l\cos \theta ({2h+2l\sin \theta})-({l_{0}^{2} \sin \theta+2h_{0} l_{0} \cos \theta})$, $l_{0} =\frac{l}{2\cos \theta}({2l\cos \theta -t})$, and $h_{0}=2( {l\sin \theta+\frac{h}{2}-l_{0}}$ ${\sin \theta -\frac{t}{2\cos \theta}})$. From Eqs. (1) and (2), the alternating layer structure of the material parameters of the proposed bending waveguide structure can be obtained. To verify the anisotropic density design, the alternating layered structures of two pentamode metallic structures, cells 1 and 2, are built as shown in Fig. 2, where cell 1 is designed with $l=4$ mm, $h=3.98$ mm, $t=0.25$ mm, $\theta =30^{\circ}$, $l_{1}=1.4$ mm and $h_{1}=1.21$ mm as shown in Fig. 2(a) and cell 2 is designed with $l=4$ mm, $h=3.98$ mm, $t=0.31$ mm, $\theta =30^{\circ}$, $l_{1}=0$ mm and $h_{1}=0$ mm as shown in Fig. 2(b). Through Eq. (2), the effective densities, moduli, sound speeds of cells 1 and 2 are 855.6 kg/m$^{3}$, 238.3 kg/m$^{3}$, 1.84 GPa, 1.83 GPa, 1466.5 m/s and 2771.2 m/s, respectively. It means that these two pentamode metallic structures have the same modulus and different densities.
cpl-36-2-024301-fig2.png
Fig. 2. The anisotropic density design. (a) Illustration of cell 1 and the frequency dispersion curve. (b) Illustration of cell 2 and the frequency dispersion curve. (c) The alternating layered pentamode structure. It is made of two pentamode metallic structures with different mass densities and the same modulus. (d) The frequency dispersion curve of a unit of the alternating layered pentamode structure.
The frequency dispersion curves of cells 1 and 2 are shown in Figs. 2(a) and 2(b). Note that the two directional longitudinal wave speeds of cell 1, 1463.1 m/s and 1458.9 m/s, are close to the calculation results from Eq. (2) and the shear wave speeds of cell 1, 118 m/s and 117.9 m/s, are far less than the longitudinal wave speeds. Therefore, cell 1 can be regarded as isotropic fluid-like materials. Cell 2 are in the same situation like cell 1 and can also be viewed as the isotropic fluid-like materials. Through Eq. (1), an anisotropic density metamaterial can be made of two isotropic fluid-like materials, cells 1 and 2, as shown in Fig. 3(c). The effective parameters of the alternating layered pentamode structure are calculated by Eq. (1): $\rho_{x} =338$ kg/m$^{3}$, $\rho_{y} =564$ kg/m$^{3}$, $k=1.83$ GPa, $c_{x}=2326.8$ m/s and $c_{y} =1801.3$ m/s. By analyzing the frequency dispersion curve of the designed metamaterial as shown in Fig. 2(d), the effective sound speeds are $c_{x} =2131.5$ m/s and $c_{y} =1813.8$ m/s. They are close to the effective speeds of the alternating layered pentamode structure by calculating Eq. (1). Hence, this confirms that the two directional effective sound speeds are different. Due to the same modulus of two isotropic fluid-like materials, the alternating layered pentamode structure has an anisotropic density. These results are confirmed that two isotropic fluid-like materials with different densities and the same modulus can be applied to build the metamaterials with an anisotropic density. As an example, we designed a 2D bending waveguide structure by a transformation acoustics that can be used to transfer the original space to change the acoustic wave propagation route,[6,21,22] as shown in Fig. 3(a). We assume the heights of OA and OB to be $a$ and $b$, respectively, and the area ABCD is a rectangle. We define a mapping that maps the rectangle ABCD to the arch A$'$B$'$C$'$D$'$ with an inner radius of $a$ and an outer radius of $b$. The coordinate transformation between the origin space $(x, y)$ and the transformation space $(r', \theta')$ can be expressed as $$\begin{align} x=\,&\frac{L}{\alpha}{\theta}',\\ y=\,&{r}',~~ \tag {3} \end{align} $$ where ($x$, $y$) is an arbitrary point in the original space ABCD, ($r'$, $\theta'$) is the corresponding point in the transformed space A$'$B$'$C$'$D$'$, $L$ is the length of the BC, $\alpha$ is the degree of the transformed space A$'$B$'$C$'$D$'$, and we let $L=\frac{2b}{\pi}\alpha$.
cpl-36-2-024301-fig3.png
Fig. 3. (a) Schematic of the geometric transformation relation. In a Cartesian coordinate system, the original space is the rectangle area ABCD, and the transformation space is the arch area A$'$B$'$C$'$D$'$. (b) The 25$^{\circ}$ fan-shaped structure is divided into four discrete layers, with only one type of pentamode metallic structure in each layer. The inner and outer radii are 0.2 m and 0.3 m, respectively. (c) Pressure map of the free space simulated at 20 kHz. (d) Pressure map of the acoustic bending made with the homogeneous materials simulated at 20 kHz. (d) Pressure map of the acoustic bending made of the pentamode metallic structures simulated at 20 kHz.
The anisotropic material parameters of the acoustic bending can be obtained by calculation of the Jacobi matrices of Eq. (3). We obtain the material parameters expressed in a cylindrical system, $$\begin{align} {\rho}'_{\rm r} =\,&\rho_{0}, \\ {\rho}'_{\theta} =\,&\Big(\frac{L}{\alpha {r}'}\Big)^{2}\rho_{0},\\ {k}'=\,&k_{0},~~ \tag {4} \end{align} $$ where ${\rho}'_{\rm r}$, ${\rho}'_{\theta}$ and ${k}'$ are the radial density, tangential density and scalar modulus of the bending waveguide, and ${\rho}'_{\rm r}$ and ${\rho}'_{\theta}$ are the homogeneous density and modulus of the origin space. We choose the inner and outer radii of the waveguide structure as $a=0.2$ m and $b=0.3$ m and the bending angle as $\theta=25^{\circ}$. The host medium of the origin space is water ($\rho_{0} =1000$ kg/m$^{3}$, $k_{0} =2.25$ GPa). For convenience of design, the bending waveguide is divided into four discrete layers with the same layer thickness, as shown in Fig. 3(b). The same effective modulus of four layers are 2.11 GPa and the different effective densities are 1250 kg/m$^{3}$ for A, 458 kg/m$^{3}$ for B, 1250 kg/m$^{3}$ for C and 263 kg/m$^{3}$ for D. By varying the attached mass, four pentamode metallic structures, units A, B, C and D, with the same effect modulus but different effect densities are designed by Eq. (1), (2) and (4). Unit A is designed with $l=4$ mm, $h=4$ mm, $t=0.345$ mm, $l_{1}=1.8$ mm, $h_{1}=1.8$ mm, unit B is designed with $l=4$ mm, $h=4$ mm, $t=0.345$ mm, $l_{1}=0.8$ mm, $h_{1}=0.8$ mm, unit C is designed with $l=4$ mm, $h=4$ mm, $t=0.345$ mm, $l_{1}=1.8$ mm, $h_{1}=1.8$ mm, and unit D is designed with $l=4$ mm, $h=4$ mm, $t=0.345$ mm, $l_{1}=0$ mm, $h_{1}=0$ mm. To test the effectiveness of the acoustic bending, we employed a full-wave simulation (COMSOL Multiphysics) by launching a plane wave toward the structures at 20 kHz. The total pressure field pattern without the acoustic bending is shown in Fig. 3(c), where the incident wave rectilinearly travels from the left side to the right side. Figures 3(d) and 3(e) show the simulations of the acoustic wave propagation in the bending waveguides with the homogeneous materials and the pentamode metallic structures, respectively, where the wave front rotates 25$^{\circ}$ and some leakage occurs outside the structure. The pressure field distribution in Fig. 3(d) is similar to that in Fig. 3(e), which means that the models have the similar material parameters for bending the wave propagation. The simulation results show that the proposed acoustic bending is valid, which means that the pentamode metallic structure can be applied to design and fabricate the metamaterial with an anisotropic density.
cpl-36-2-024301-fig4.png
Fig. 4. Transmission spectra of the acoustic bending structures for the homogeneous materials (blue line) and the pentamode metallic structures (red line).
To test the wideband property of the metamaterial with an anisotropic density, we calculate the transmission spectra of two cases from 20 kHz to 35 kHz. Due to the small size of the bending waveguide, the wavelength of the sound below 20 kHz is too large that the bending wave effect is not obvious. When the sound wave is greater than 35 kHz, the wavelength is not large enough for the pentamode metallic structure which leads the failure of the anisotropic density approximation. Therefore, we choose the frequency range, 20–35 kHz, as the working frequency range of the bending waveguide. The calculation results are shown in Fig. 4. The blue and red lines denote the transmissions of the bending waveguide structures with the homogeneous materials and the pentamode metallic structures, respectively. From Fig. 4, we know that the proposed acoustic bending structure with the pentamode metallic structures can operate over a broad range of frequencies from 20 kHz to 35 kHz. We can observe that two curves are very consistent, which means that the metamaterial made of the pentamode metallic structure has broadband properties. In summary, we have designed an anisotropic density metamaterial with pentamode metallic structures and we have investigated its effective acoustic parameters. The frequency dispersion curve is applied to verify the anisotropic density. As an example, the acoustic bending with the anisotropic density is established. A good performance of controlling over acoustic wave propagation is confirmed and its operating frequency is broadband. Our results show the effectiveness of the proposed metamaterial, which exerts anisotropic density to control acoustic wave, and that we confirm that its effective acoustic parameters have broadband properties. This proposed metamaterial is a valuable reference for the design of acoustic metamaterials that simultaneously have an anisotropic modulus and density. References Controlling sound with acoustic metamaterialsAcoustic cloaking and transformation acousticsAcoustic cloaking theoryDesign and measurements of a broadband two-dimensional acoustic metamaterial with anisotropic effective mass densityDesign and demonstration of an underwater acoustic carpet cloakExperimental demonstration of three-dimensional broadband underwater acoustic carpet cloakLatticed pentamode acoustic cloakAnalysis and experimental demonstration of an active acoustic metamaterial cellDesign and demonstration of an acoustic right-angle bendDesign of an underwater acoustic bend by pentamode metafluidDesign of an acoustic bending waveguide with acoustic metamaterials via transformation acousticsBroadband energy harvesting using acoustic black hole structural tailoringScattering reduction for an acoustic sensor using a multilayered shell comprising a pair of homogeneous isotropic single-negative mediaDesign of the Coordinate Transformation Function for Cylindrical Acoustic Cloaks with a Quantity of Discrete LayersOne path to acoustic cloakingA multilayer structured acoustic cloak with homogeneous isotropic materialsPhysical Biology Returns to MorphogenesisAcoustic cloaking in two dimensions: a feasible approachAnisotropic Mass Density by Radially Periodic Fluid StructuresBroadband Acoustic Cloak for Ultrasound WavesPeriodic metal structures for acoustic wave controlSpecial transformations for pentamode acoustic cloakingWhich Elasticity Tensors are Realizable?On anisotropic versions of three-dimensional pentamode metamaterialsHighly Anisotropic Elements for Acoustic Pentamode ApplicationsBroadband focusing of underwater sound using a transparent pentamode lensThe Mechanics of Three-Dimensional Cellular Materials
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