Sage Journals: Discover world-class research

Abstract

The acoustic wave transmission manipulation ability is the most important performance for the acoustic metamaterials. To manipulate the acoustic transmission, the combination acoustic metamaterials structures are involved, and the two-directional acoustic penetration cloaking scheme are constructed. The combination acoustic metamaterials include chiral metamaterials and self-collimation metamaterials, the acoustic wave transmission path are manipulated to bypass from the acoustic scattering target in the penetration cloaking, and the target scattering stealth problem are effectively solved. In the end, the acoustic transmission manipulation performances are verified by numerical analysis with a finite-width acoustic metamaterial structure plates. The results provide technical support for design and application for acoustic transmission manipulation with acoustic metamaterials.

Keywords

Acoustic metamaterials band structure transmission manipulation acoustic cloaking

Introduction

With the development of science and technology, varieties of functional materials have been a topic of wide concern in recent years, and increasing attention has been paid to the acoustic metamaterials or phononic crystals problems.^1–5 Theoretical analysis, numerical calculation, and experimental test were conducted to analyze the distinctive acoustic phenomenon, such as unidirectional transmission, acoustic filtering, acoustic focusing, acoustic negative refraction, acoustic waveguiding, acoustic collimating, and so on.^6,7 As a kind of artificial inhomogeneous material, acoustic metamaterials are synthetic materials formed by a periodic variation in the mechanical properties of materials (i.e., elasticity modulus and mass density), and it can suppress and manipulate acoustic waves transmission.^8–11 Acoustic metamaterials have unique mechanical properties, such as negative equivalent mass density, negative equivalent bulk modulus, and negative shear modulus, which are achieved through topology design of micro-structure and materials distribution design.^12–14

Acoustic scattering is caused by acoustic reflection of the interface, obstacles, or targets in the acoustic transmission process; reducing acoustic scattering is of great significance to improve the acoustic stealth ability.¹⁵ At present, attachment of acoustic absorbing materials layer on the structural surface is an effective method to reduce acoustic scattering, it has made great progress both in theoretical analysis and engineering application. To enhance the acoustic stealth ability, acoustic transformation devices, such as acoustic cloaking device, were recently proposed.^16–18 Two kinds of technology are involved: one is the formation of a hidden region without reflection, and the other is external cloaking by acoustic cancellation. However, double-negative materials such as pentamode material are required to design acoustic cloaking, and this problem encountered a lot of difficulties, such as manufacturing technology, large cost, restrictions for the working frequency range, and so on.^19–21 According to the acoustic manipulation function of the acoustic metamaterials, to study new scattering stealth technology is necessary.

On the basis of the above-mentioned reasons, the main objectives of present work are to construct acoustic stealth cloaking by combination chiral and self-collimation metamaterials.²² Two-directional acoustic target scattering stealth is discussed with symmetry design. The remainder of present work is organized as follows: The basic concept of acoustic penetration cloaking is introduced in Section 2. The band structure problem of chiral and self-collimation metamaterials are investigated in Section 3. The feasibility and effectiveness of acoustic penetration cloaking is illustrated by numerical analysis in Section 4. Major conclusions are summarized in Section 5.

Model description of acoustic penetration cloaking

In the acoustic metamaterials, the acoustic transmission paths are implemented by adjusting the geometric and material parameters of the micro-structure. Through the acoustic transmission manipulation, the incident acoustic wave will follow the specified transmission path in the acoustic field, and the acoustic stealth zones are formed. If an acoustic scattering target is located in the acoustic stealth area, weak acoustic scattering, even silent acoustic scattering, are obtained. The transmission acoustic wave and the incident acoustic wave will be consistent in acoustic pressure amplitude and phase, as if without acoustic scattering target exist, and the so-called acoustic scattering stealth is realized, and a acoustic penetration cloaking “cloaking” is formed. The basic concept of acoustic penetration cloaking in two-dimensional model are shown in Figure 1(b).

Figure 1.

The schematic of acoustic penetration cloaking: (a) sketch of half acoustic penetration cloaking model, and (b) two-directional acoustic penetration cloaking model.

As shown in Figure 1, the acoustic penetration cloaking model consists of two metamaterial structure plates; and the acoustic metamaterial plates have three layers. The core layer is the chiral metamaterials (the C zones), and the skin layers are self-collimation metamaterial (the S zones). The chiral metamaterial layer are symmetrically spliced the chiral metamaterial plates for topological state protection at the boundary (the blue and pink parts in Figure 1) along the x-axis.

The vertical incident acoustic waves, for example, the basic principle of acoustic transmission manipulation are described. As shown in Figure 1(a), a plane acoustic wave beam with a certain width (yellow arrow zone) is incident on the left surface of the self-collimation metamaterial layer (S zone), and the acoustic wave direction is collimated.

When the acoustic wave beam is incident into the chiral metamaterial layer (C zone), the acoustic beam is split into two acoustic beams at the symmetry line x-axis. An angle exists between the direction of the two beams of symmetrical acoustic waves in the chiral metamaterial layer and the direction of the incident acoustic wave. The angle value is determined by the wave vector direction of the micro-structure. The acoustic wave is self-collimated again when it penetrates the self-collimation metamaterial layer (S zone). The split two acoustic wave beams propagate in the acoustic field, and an acoustic stealth zone is obtained. The height of the stealth zone is determined by the angle value and the thickness of the chiral metamaterial layer.

To construct a two-directional acoustic penetration cloaking, two metamaterial structure plates are arranged with parallel, as shown in Figure 1(b). Two acoustic wave beams pass through the acoustic field area and incident on the self-collimation metamaterial layer (S zone) of the right side of the metamaterial structure, then the acoustic wave are self-collimated again. The acoustic wave is incident into the chiral metamaterial layer (C zone) again, and two acoustic wave beams merge. Thus, the amplitude and the phase of the transmission acoustic wave and the incident acoustic wave are consistent. An acoustic stealth zone is formed between the two metamaterial structure plates. If the acoustic scattering target is located in the acoustic stealth zone, then the incident acoustic wave is considered to bypass the scattering target to achieve acoustic stealth, and the goal of acoustic penetration stealth is realized. Bidirectional acoustic stealth is gained according to the symmetry distribution of metamaterial structure.

Band structure analysis of acoustic metamaterials

The band structure, named band gaps, are the key index to evaluate the mechanical performance of acoustic metamaterials, bandgap properties originate from strong scattering and destructive interference of multiple scattered waves

Acoustic metamaterials theory

The acoustic metamaterials is composed of a micro-structured cell in a space infinite period extension, the minimum periodic scale of the periodic micro-structure is called the lattice constant. Acoustic metamaterials are characterized by space periodicity and symmetry, that is, the eigenfield, eigenfrequency, and structural displacement of the acoustic metamaterials are periodic.

In the two-dimensional domain, the periodic boundary conditions are defined to simulate an infinite period of the micro-structured cell, and the x and y directions of the periodic edges for the micro-structured cell are considered. According to the Bloch theory, the displacement vector $u (r, k)$ can be expressed as

u (r, k) = u_{k} (r) e^{i k \cdot r},

(1)

where

u_{k} (r)

is the periodic displacement function, r is the lattice periodicity vector, and

k = (k_{x}, k_{y})

is the Bloch wave vector limited to the first irreducible Brillouin zone (IBZ) in x and y directions, respectively. If the micro-structured cell of the metamaterials is discretized with finite elements, then the motion governing function of the acoustic metamaterials can be written as

(K (k) - ω^{2} M) U = 0

(2)

where K, M, and U denote the stiffness matrix, mass matrix, and vector of displacements, respectively.

Band structure of chiral metamaterials

The band structure properties, named band gaps, are the key index to evaluate the mechanical performance of acoustic metamaterials. The width and position of the band gaps are highly depend on geometric parameters and material parameters of the phononic crystal cell. The wave vector and eigenvalue be defined as the horizontal coordinates and longitudinal coordinates, respectively, the band structure or linear dispersion relationship can be drawn can be obtained. In the band structure, there exist more than one complete bandgap or directional bandgap within the calculated frequency range.

The cell of chiral metamaterial is rectangle, the chiral metamaterial are composed of triangle scatterers (part A), coating (part B), ellipse (part C), and matrix (part D), as shown in Figure 2(a). The triangle scatterers and coating are assumed to be bonded perfectly. The inclination direction of the scatterer indicates the chirality property of the micro-structure. The positions of the three high-symmetry points in irreducible Brillouin zone (IBZ) are as follows: $X (2 π / a, 0)$ , $Γ (0,0)$ , and $M (2 π / a, 2 π / b)$ , as shown in Figure 2(b).

Figure 2.

Diagram of band structure for two-dimensional chiral metamaterials: (a) Microstructure parameter, (b) Irreducible Brillouin zone, and (c) Band-gap diagram.

The chiral metamaterial structure contains four kinds of materials, namely, air, silicon rubber, aluminum, and copper. The material parameters are shown in Table 1.

Table 1.

Mechanical properties of metamaterials.

Component	Materials type	Longitudinal wave velocities (m/s)	Density (kg/m³)
Matrix	Air	340	1.29
Triangle scatterers	Aluminum	6350	2700
Coating	Silicon rubber	240	1260
Ellipse	Copper	4730	8960
Cylinder	Titanium alloy	4510	6060

To obtain the band structure properties of micro-structure, the numerical calculation is carried out. The lattice constant of micro-structure are a = 10.0 mm and b = 8.0 mm, respectively. The parameter of the triangle scatterers and coating are shown in Figure 2(a). In addition, the major axis and minor axis of ellipse are 2.25 mm and 0.75 mm, respectively. With the definition of the lattice constant, geometric modeling, finite element division, material definition, periodic-boundary definition, and physical field definition, band properties numerical analysis are implemented. All finite element method (FEM) predictions are calculated by using the acoustics module on the COMSOL Multiphysics 5.6 software code. Since the geometric parameters and the material parameters of the acoustic metamaterials are determined, then the band structure of the chiral acoustic metamaterial can be calculated by sweeping the wave vector $k = (k_{x}, k_{y})$ in the IBZ. The sweeping direction is X→Г→M→X, and the band structure curve of the chiral acoustic metamaterial are graphed, as shown in Figure 2(c).

As shown in Figure 2(c), there exist 10-band structure, 3 complete bandgaps are shaded with gray, and 10 directional bandgaps are shaded with blue. The first bandgap is located between the first and second bands structure, with corresponding frequency of 15131.2 Hz–18695.1 Hz; The second bandgap is located between the second and third bands structure, with corresponding frequency of 27171.4 Hz–27551.6 Hz; and the third bandgap is located between the fifth and sixth bands structure, with corresponding frequency of 43639.6 Hz–44784.1 Hz.

According to the bandgap structure property, an acoustic propagation mode exists along the ΓX and ΓM directions, while an acoustic bandgap mode is present along the XM direction in the frequency range 20506 Hz–27171 Hz, as shown in Figure 2(c). Thus, the acoustic wave can be guided to propagate along the ΓM direction in the metamaterial structure within frequency range. If the frequency bands of incident acoustic wave are between 20506 Hz and 27171 Hz, then the deflection angle is present between the acoustic wave propagating in the metamaterial structure and the incident acoustic wave.

Band structure of self-collimation metamaterials

The self-collimation metamaterials are composed of cylinder scatterers and matrix, as shown in Figure 3(a). The micro-structure is square and the side length is t; and r is defined as the cylinder scatterers radius. In the numerical analysis, the lattice constant of micro-structure are t = 5.0 mm and r = 0.362 t, respectively.

Figure 3.

Diagram of the self-collimation metamaterials: (a) Microstructure parameter, (b) Irreducible Brillouin zone, (c) Equal frequency diagram, and (d) Band-gap diagram.

The cell of self-collimation metamaterials is square, the micro-structure of self-collimation metamaterials are composed of round scatterers and matrix, as shown in Figure 3(a). In the micro-structure of self-collimation metamaterial, the positions of the three high symmetry points are as follows: $X (2 π / t, 0)$ , $Γ (0,0)$ , and $M (2 π / t, 2 π / t)$ . The sweeping direction is X→Г→M→X. The band structures of the self-collimation acoustic metamaterial are graphed, as shown in Figure 3(d).

In the self-collimation metamaterial, the material of matrix is air, and the material of cylinder scatterers is titanium alloy, respectively. The mechanical properties of titanium air and alloy are shown in Table 1.

As shown in Figure 3(d), the complete bandgaps are shaded with gray, and the directional bandgaps are shaded with blue. The first bandgap is located between the first and second bands structure, with corresponding frequency of 37203 Hz–42125 Hz; the second bandgap is located between the seventh and eighth bands structure, with corresponding frequency of 125790 Hz–128215 Hz.

The equal frequency diagram is obtained, as shown in Figure 3(c). The equal frequency contours are nearly flat at the frequency $f_{e} = 0.372 c / t$ (namely 25305 Hz), where c is the acoustic velocity and t is the lattice constant. The incident acoustic beam propagates along the Г-M direction with minor diffraction, which is the so-called self-collimation phenomenon.

Numerical analysis of penetration cloaking

To verify the design methodology described about the acoustic penetration cloaking, numerical analysis is implemented.

Overview of acoustic penetration cloaking model

Metamaterials are often assumed to be infinite and arranged periodically along a plane in theoretical analysis. In practical engineering, to analyze the transmission properties of finite micro-structure is necessary. As shown in Figure 4, the width of the chiral metamaterial layer and self-collimation metamaterial layers are 12a and 4.8a, respectively. The width of the acoustic field domain is 25a. The height of the acoustic penetration cloaking is 32b. The left and right boundaries are defined as perfectly matched layers to reduce the interference of reflective waves. The width of perfectly matched layers, wave excitation domain, and wave evaluation domain are 1.6a, 5.5a, and 5.5a, respectively. The height of the perfectly matched layers, wave excitation domain, and wave evaluation domain is 7.2b. Moreover, the periodic boundary condition is applied at the top and bottom edges.

Figure 4.

Acoustic penetration cloaking model.

In the numerical analysis, the acoustic wave with unit amplitude is incident perpendicular to the left interface of the penetration cloaking. With the combination of the self-collimation frequency and the directional bandgap frequency, the incident acoustic wave frequency is defined as $f_{a} = 0.372 c / t$ , which corresponds to 25305 Hz.

Acoustic transmission calculation

To demonstrate the acoustic transmission manipulation performance, the acoustic transmission calculation of acoustic penetration cloaking with finite width ate carried out. In the numerical analysis, the free acoustic transmission and two-direction acoustic penetration cloaking are compared. In order to verify the scattering stealth ability, a scattering target is located in the acoustic field. The width and height of the scattering target are 10a and 7.2b, respectively, as shown in Figure 5.

Figure 5.

Comparison of the acoustic pressure distributions in acoustic metamaterials structure: (a) pressure distributions in free acoustic field; (b) pressure distributions in free acoustic field with target scattering; (c) pressure distributions in acoustic penetration cloaking; (d) pressure distributions in acoustic penetration cloaking with target scattering.

Figures 5(a) and (b) show the acoustic pressure distribution of the free acoustic field transmission with/without scattering target. The influence of the acoustic scattering target in the free acoustic field is obvious (Figures 5(a) and (b)).

Figures 5(c) and (d) show the acoustic pressure distribution of the acoustic penetration cloaking with/without scattering target. When the acoustic wave penetrates through the right metamaterial structure plate, the amplitude and phase of the transmission acoustic wave and the incident acoustic wave are consistent. Compared with the case depicted in Figures 5(c) and (d), if the two acoustic metamaterial structure plates are arranged and the acoustic scattering target is located in the acoustic stealth zone, the influence of the acoustic scattering target on the acoustic field can be omitted (Figures 5(c) and (d)). The results show that perfect penetration of acoustic wave can be achieved by arranging acoustic penetration cloaking.

Half acoustic penetration cloaking

To verify the scattering stealth ability, a half acoustic penetration cloaking is discussed, as shown in Figure 1(a). The width of the chiral metamaterial layer and the self-collimation metamaterial layers are 12a and 4.8a, respectively. The width of the acoustic field domain is 25a. The height of the acoustic penetration cloaking structure is 32b. The widths of perfectly matched layers, wave excitation domain are 1.6a and 5.5a, respectively. The height of the perfectly matched layers and wave excitation domain, are 7.2b. The incident acoustic wave frequency is 25305 Hz. Figure 6 shows the acoustic pressure distribution of the half acoustic penetration cloaking in the acoustic field.

Figure 6.

Comparison of the acoustic pressure distributions in the acoustic metamaterials structure: (a) pressure distributions in half acoustic penetration cloaking; (b) pressure distributions in half acoustic penetration cloaking with target scattering.

According to Figure 6, the acoustic stealth zone is formed in the acoustic field, and the influence of scattering target can be reduced. The results show that the acoustic transmission manipulation ability of the penetration cloaking.

Conclusion

Based on the bandgap properties, the acoustic penetration cloaking design with acoustic metamaterial is proposed in present work, in which the acoustic transmission manipulation function is obtained by combination metamaterials structure, and its feasibility and effectiveness are illustrated by numerical analysis. The following conclusion are obtained by numerical analysis: (1) The acoustic transmission path can be effectively manipulated by penetration cloaking with combination metamaterial structures; (2) If the acoustic scattering target is located in the acoustic stealth zone, the parameters of transmission acoustic wave are consistent with the incident acoustic wave, the acoustic stealth is achieved; (3) since the symmetry of the penetration cloaking structure, there is no phase difference problem in acoustic transmission; (4) In the experimental work of the future, the processing technology of the model is very important; (5) Acoustic penetration cloaking based on directional bandgaps has advantages such as simple structure, convenient manipulation, and wide adaptability; (6) Only the vertically incident acoustic wave transmission manipulation problems are investigated in present work, and oblique incident problems will be discussed in future. Some conclusions can provide technical support for design and application of innovative acoustic metamaterials.

Footnotes

Acknowledgements

This support is gratefully acknowledged by the authors.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by the Fund of High-tech Ship Research Projects by MIIT and State Key Laboratory of Ocean Engineering.

ORCID iD

Chen Luyun

References

Liu

Guo

Wang

. A review of acoustic metamaterials and phononic crystals. Crystals 2020; 10: 305.

Liu

Zhang

Mao

et al. Locally resonant sonic materials. Science 2000; 289: 1734–1736.

Liu

Zhong

et al. Research on the band gap characteristics of two-dimensional phononic crystals microcavity with local resonant structure. Shock Vib 2015; 5: 1–8.

Torres

Montero de Espinosa

García-Pablos

et al. Sonic band gaps in finite elastic media: surface states and localization phenomena in linear and point defects. Phys Rev Lett 1999; 82: 3054.

Qiu

Liu

Shi

et al. Directional acoustic source based on the resonant cavity of two-dimensional phononic crystals. Appl Phys Lett 2005; 86: 224105.

Cao

Yang

et al. Deflecting flexural wave with high transmission by using pillared elastic metasurface. Smart Mater Struct 2018; 27: 075051.

Chen

Chang

Huang

et al. Asymmetric transmission of acoustic waves in a layer thickness distribution gradient structure using metamaterials. AIP Adv 2016; 6: 095020.

Wang

Yan

. Configuration effect and bandgap mechanism of quasi-one-dimensional periodic lattice structure. Int J Mech Sci 2021; 190: 106017.

Miao

You

et al. Effects of material parameters on the band gaps of two-dimensional three-component phononic crystals. Appl Phys A-Mater 2019; 106: 014903.

10.

Park

Lee

Rho

. Recent advances in non-traditional elastic wave manipulation by macroscopic artificial structures. Appl Sci 2020; 10: 547.

11.

Cummer

Christensen

Alù

. Controlling sound with acoustic metamaterials. Nat Rev Mater 2016; 1: 16001.

12.

Baravelli

Ruzzene

. Internally resonating lattices for bandgap generation and low-frequency vibration control. J Sound Vib 2013; 332: 6562–6579.

13.

Ebrahimi

Mousanezhad

Nayeb-hashemi

et al. 3D cellular metamaterials with planar anti-chiral topology. Mater Des 2018; 145: 226–231.

14.

Yan

et al. On-chip valley topological materials for elastic wave manipulation. Nat Mater 2018; 17: 993–998.

15.

Zhu

Liang

Kan

et al. Acoustic cloaking by a superlens with single-negative materials. Phys Rev Lett 2011; 106: 014301.

16.

Chen

Liu

. Influences of imperfectness and inner constraints on an acoustic cloak with unideal pentamode materials. J Sound Vib 2019; 458: 62–73.

17.

Jin

Fang

et al. Engineered diffraction gratings for acoustic cloaking. Phys Rev Appl 2019; 11: 011004.

18.

Chen

Fan

Yang

et al. Coiling-up space metasurface for high-efficient and wide-angle acoustic wavefront steering. Front Mater 2021; 8: 790987.

19.

Sun

Fan

Chen

et al. Highly efficient transmissive wavefront steering with acoustic metagrating composed of Helmholtz-resonators. Mater Des 2022; 224: 111352.

20.

Zhu

Chen

Song

et al. Three-dimensional large-scale invisibility cloak with layered metamaterials for underwater operation. Phys Scripta 2019; 94: 115003.

21.

Zhao

Chu

Tao

et al. Acoustic transmissive cloaking using zero-index materials and metasurefaces. APEX 2019; 12: 054004.

22.

Soliveres

Espinosa

Pérez-Arjona

et al. Self collimation of ultrasound in a three-dimensional sonic crystal. Appl Phys Lett 2009; 94: 164101.

Acoustic cloaking design based on penetration manipulation with combination acoustic metamaterials

Abstract

Keywords

Introduction

Model description of acoustic penetration cloaking

Band structure analysis of acoustic metamaterials

Acoustic metamaterials theory

Band structure of chiral metamaterials

Band structure of self-collimation metamaterials

Numerical analysis of penetration cloaking

Overview of acoustic penetration cloaking model

Acoustic transmission calculation

Half acoustic penetration cloaking

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iD

References