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In this study, we design periodic grille structures on a single homogenous thin plate to achieve anisotropic acoustic metamaterials that can control flexural waves. The metamaterials can achieve the bending control of flexural waves in a thin plate at will by designing only one dimension in the thickness direction, which makes it easier to use this metamaterial to design transformation acoustic devices. The numerical simulation results show that the metamaterials can accurately control the bending waves over a wide frequency range. The experimental results verify the validity of the theoretical analysis. This research provides a more practical theoretical method of controlling flexural waves in thin-plate structures.
Acoustic metamaterials are artificial acoustic structural materials that consist of specially designed acoustic structural units that are arranged in a certain period in an elastic medium. This type of structural material has a low-frequency band gap and extraordinary material parameters in the subwavelength band of the structure size, and it can guide the propagation of sound waves in gas, liquid and solid. In recent years, the study of acoustic metamaterials has attracted the attention of scholars worldwide.[1–8]
Coordinate transformation theory is one of the most important theoretical methods applied to the design of acoustic metamaterials, which is an innovative theoretical method proposed in 2006 by Pendry et al.[9] in the field of electromagnetics. The propagation of an electromagnetic wave can be redirected at will with this method. The basic idea of coordinate transformation theory is to consider the arbitrary wave propagation path to be the result of the linear propagation path following the specific coordinate transformation. By using the same coordinate transformation in the material space, the homogeneous medium is compressed into a specific non-homogeneous medium and the medium can achieve the bending control of an arbitrary set of waves. Schurig et al.[10] and Zhang et al.[11] designed electromagnetic stealth cloaking to achieve the diffraction of electromagnetic waves by using this method. Cummer et al.[12,13] compared the acoustic wave equation in fluids with the equation of TE mode electromagnetic wave and found consistency between these two equations. Based on coordinate transformation theory, acoustic cloaking was designed to realize the regulation of acoustic waves. Rahm et al.[14] extended coordinate transformation theory into finite embedded coordinate transformations, which provides more flexibility for coordinate transformation design and can help to design various devices, such as curved waveguides, beam splitters and imaging lenses.[15,16]
Milton et al.[17] explored the feasibility of applying coordinate transformation theory to elastic wave regulation and Farhat et al.[18,19] designed elastic cloaking in a thin plate on the basis of Milton’s research work. The effectiveness of regulating elastic waves by using coordinate transformation theory has been theoretically proven. The homogeneous material in the original coordinate space is transformed into a non-homogeneous anisotropic material. Such an anisotropic non-homogeneous material does not exist in nature. However, it is possible to convert the material into a multi-layer homogeneous material by continuous changing of material parameters by using the equivalent material theory of the material parameters.[20] Nevertheless, it is still difficult to obtain a material with continuously changing material parameters or according to a certain gradient. Therefore, many new electromagnetic devices and acoustic devices can be theoretically designed. However, there are great difficulties in preparing experimental samples, which lead to difficulties of its experimental verification. Few cases have thus been applied to engineering practice. In 2012, Nicolas et al.[21] designed and prepared a type of elastic cloak on the basis of the theoretical research of Farhat. However, the elastic cloaking proposed by Nicolas et al. is composed of 20 layers of a circular composite material and the elastic modulus of each layer varies from 0.002 GPa to 2.22 GPa. Moreover, each layer is made of PVC and PDMS compounds and neither of the compositions of the two materials occurs in the same proportion. Thus, the sample preparation process is complex, which is not conducive to experimental verification and engineering applications.
An anisotropic acoustic metamaterial based on a homogeneous thin plate was designed to achieve a transformation acoustic design that can control flexural wave propagation along an arbitrary curved path. Compared with the metamaterials proposed by Farhat and Nicolas, the metamaterial in this paper is very easy to prepare. The theoretical and experimental results are consistent with each other, which confirms the validity of the theoretical design.
Consider the coordinate transformation shown in Fig.
The transformation equation can be written as[14]
Assuming that the area shown in Fig.
It can be seen from Eq. (
If η = 1, we can obtain
The flexural wave control based on coordinate transformation theory can be realized by using the above-mentioned multi-layer isotropic medium structure. However, as the elastic modulus of each layer is different and continuously varies according to a certain gradient, such materials are difficult to obtain in practice, which causes great difficulties of experimental verification. Therefore, to prepare experimental samples more easily and to reduce the difficulty of experimental verification, we must seek a new theoretical design method.
The flexural wave in a thin plate is a dispersion wave and the wave velocity changes with its frequency. The thickness of the thin plate and the material parameters influence the wave velocity of the flexural wave and its phase velocity expression is Cf = [12(1 − ν 2)]−1/4(ω⋅h)1/2(E/ρ)1/4,[24,25] where ω is the angular frequency, h is the thickness of the plate, E is the elastic modulus, ρ is the mass density and ν is Poisson’s ratio. The phase velocity of the flexural wave that propagates in A and B layered media are
For flexural waves propagating in the coordinate transformation region in the new coordinate system, under the premise that the propagation speed in each layer of A and B layered media in this area is not changed, if we change the respective variables in A and B arbitrarily for all beams having the same initial phase without changing the propagation speed in each layer of A and B in this area, the time for waves to reach a certain position in the region remains constant. Under this condition, the wave front in the region does not change and the mode of vibration of the coordinate transformation region remains unchanged with the change in the variables A and B. That is to say, when CfA and CfB keep constant, even if the size or material parameters of the space region change, the propagation mode of the flexural wave in the new spatial region is still the same as that in the space described in Eq. (
After the deformation according to Eq. (
The finite element model of the metamaterial grille structure is built according to Eq. (
The dimensions of the thin plate are as follows: length of sides D = 400 mm, L = 100 mm, h0 = 1.5 mm, R1 = 200 mm, R2 = 280 mm, θ = π/6, elastic modulus E0 = 72 GPa, and mass density ρ0 = 2700 kg/m3. The anisotropic non-homogeneous medium of the coordinate transformation region is dispersed over 20 layers, each of which contains isotropic homogeneous media A and B.
According to the flexural wave velocity expression, the relationship between the flexural wave wavelength λ and the excitation frequency f can be expressed as λ = [12(1 − ν 2)]−1/4(2πh)1/2(E/ρ)1/4 f −1/2, because the propagation of flexural waves in thin plates must satisfy h0 ≪ λ ≪ D [18,19] and the long wave approximation condition of λ ≫ dA (dB). Thus, we choose a wavelength range of 10h0 ≤ λ ≤ 0.1D, λ ≥ 10dA (dB) to obtain 20 mm≤ λ ≤ 40mm, and the corresponding excitation frequency range should meet 9500 Hz ≤ f ≤ 37000 Hz.
The numerical simulations of the above model are conducted at 9500 Hz and 37000 Hz, respectively. The excitation is between R1 and R2, which is perpendicular to the plane of the thin plate.
As shown in Figs.
According to the grille structure thickness parameters shown in Fig.
In the experimental test system shown in Fig.
As shown in Fig.
Using the coordinate transformation design method and combining thin-plate flexural dynamic analysis with the equivalent medium parameter theory, a new type of metamaterial grille structure is demonstrated. We use a homogenous medium and manipulate the structural dimensions instead of the parameters of the material to achieve anisotropic metamaterials. Then we achieve control of the flexural waves. A finite element model is established and the corresponding experimental sample is prepared. The numerical simulation and experimental results show that the one-dimensional height-tunable grille can guide flexural waves to propagate at will. The theoretical design method of the metamaterial proposed in this paper can be used to regulate the flexural waves in thin-plate structures made of different types of materials and it has a wide range of applications. With this kind of metamaterial, we can achieve the design of vibration and noise reduction in a specific area of a thin plate. The theoretical method is simple and feasible and the corresponding sample is easy to manufacture. Therefore, this method has strong practicality. This paper contributes to the development of microstructural design of metamaterial plates with a tunable anisotropic modulus and density distribution, thus paving the way for the physical realization of transformation acoustic designs for flexural waves.
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