1. Introduction
The acoustic stealth performances of submarines are essential in underwater environments, which are mainly determined by the retro-reflected signals detected by active sonars. Similar with stealth technique for radar detecting, acoustic energy absorptive materials such as anechoic coating is generally adopted to reduce the intensity of retro-reflected signals. Traditional anechoic coating made of rubber or polyurethane substrate with cavities are adopted to absorb the incident detecting wave energy or eliminate the radiated noise of the hull, whose stealth performance tends to be very weak at low frequency and higher hydrostatic pressure [
1,
2]. Thick and heavy anechoic coating is demanded to achieve strong absorption at low frequency as limited by laws of mass, which is unimplementable for most underwater vehicles.
Local resonant metamaterial can manipulate long wavelength acoustic waves within subwavelength scale based on resonance of resonators [
3]. The intensive vibrating of the resonators around the resonance frequency would lead to strengthened energy dissipating and strong acoustic absorption consequently. Inspired by this mechanism and introducing viscoelasticity of substrate, low frequency waterborne strong absorption is realized by Wen et al. with a sub-wavelength sample [
4], but the inferiority is that the effective frequency bandwidth is very narrow. A lot of studies have been done to expand the effective frequency bandwidth by introducing multi-resonators [
5,
6,
7,
8,
9], metal spiral and inter-connecting structures [
10,
11,
12], air cavities and combinations of them [
13,
14,
15,
16,
17]. For example, combining two absorption mechanisms of cavity resonance and impedance transition loss, Wang et al. proposed a broadband underwater absorbing metamaterial with gradient cavity array supported by carbon fiber honeycomb [
15], which achieved a broadband sound absorption under a hydrostatic pressure of 3MPa. Zhang et al. proposed a waterborne stealth coating with transversely arranged single-walled carbon nanotubes to broaden the effective absorption range [
16]. Fan et al. also proposed an acoustic absorption–bearing metamaterial consisted by four subunits corresponding to different absorption frequencies and obtained an absorption bandwidth increased by 600% [
17]. These new designs expanded the effective bandwidth greatly, but still suffers from many issues such as hydrostatic resistance, manufacturing techniques and cost, density, thickness restrictions, etc.
Acoustic metasurface is a gradient-index artificial structure capable of manipulating acoustic waves in an extraordinary way within compact sizes [
18,
19,
20,
21], demonstrating promising applications in many practical scenarios such as noise and vibration insulation [
22,
23,
24,
25], underwater acoustic stealth [
26,
27,
28], surveying and imaging [
29,
30,
31]. The acoustic stealth principle of local-resonance method is similar with air cavity-resonance principle, where the incident acoustic energy is absorbed and the intensities of retro-reflected signals is diminished. Instead of acoustic energy absorption, acoustic metasurface could deflect the incident detecting wave into other direction through gradual modulation of phase and amplitude, which also reduce the retro-reflected acoustic signal intensities received by the detecting sonar. Thus, the proposal of acoustic metasurface provides an alternative avenue to eliminate the retro-reflected sound waves. Acoustic metasurface is composted by many sub-units of gradual varying sizes, where both resonant-based and non-resonant based configurations are adopted for sub-units designing. Utilizing thermal loss of air, simultaneous modulations of wavefront and amplitude is achieved for airborne acoustic metasurface [
32,
33]. While for waterborne metasurface, the effective frequency range of resonance-based designs is very small. Thus, non-resonant designs based on pentamode material (PM) is proposed and adopted for metasurface design.
Pentamode material (PM) is a special solid acoustic metamaterial with the merits of broadband efficiency and matched impedance with fluids. For a solid material, none of six eigenvalues of elastic modulus equals zero, while for pentamode material only one eigenvalue of six ones is nonzero [
34]. Benefiting from the arbitrarily tailorable equivalent elastic matrix and mass density [
35], PM can be utilized to construct many different types of acoustic devices especially in acoustic metasurface. Practically, PM acoustic devices are mainly composed by 2D honeycomb-lattice structures [
36,
37]. Chen et al. designed a broad PM acoustic cloaking with a titanium alloy substrate, which is proved to be effective in the frequency range of 9kHz~15kHz [
38]. Zhao et al. conceived a PM device based on a titanium alloy (Ti-6Al-4V) mimicking the acoustic properties of water within the frequency range of 3kHz~30kHz [
39,
40]. Su et al. designed a waterborne pentamode focusing lens, whose broadband focusing effects are experimentally verified within the frequency range of 20~40 kHz [
41]. Chen et al. proposed a broadband and high-transmission metasurface for converting underwater cylindrical waves to plane waves over a broad frequency band of 15~23 kHz [
42]. Sun et al. designed underwater acoustic bend and carpet with PM [
43]. Zhang et al. designed a waterborne acoustic reflective metasurface, which could shift incident waves by an angle of 15° over the frequency range of 6~18 kHz, showing great coincidences of experimental measurements with the finite-element simulations [
44]. Except honeycomb-lattice unit cells, a series of novel PM unit cells with square and triangle lattices are proposed by Dong et al. with a systematic inverse-design strategy [
45]. Based on the optimization procedure, a reflective metasurface capable of absorbing broadband waterborne sound are conceived and numerically validated [
46,
47]. Ren et al. designed a broadband high-efficient gradient lens with square-latticed pentamode metamaterial, which could convert plane waves to subwavelength focusing over an ultra-wide frequency range of 5~33 kHz [
48].
The above PM devices are realized with single-phase substrate, Zhao et al. proposed a multiphase PM configuration composed by metallic supporting lattice, interconnecting phase and mass balancing block [
49]. Then, a multiphase PM structure is fabricated and experimentally verified, which demonstrate the robustness of multiphase PM configuration [
50,
51]. Yet in these studies, hard polymer (E≈1GPa) whose damping coefficient is very small is adopted and the damping of the substrates and structures is ignored.
In this paper, a novel multiphase PM metasurface which could simultaneously manipulate acoustic wavefront and dissipate incident acoustic energy is proposed. Based on the Generalized Snell Law, a directional reflection acoustic metasurface is proposed, whose physical properties are realized with single-phase and multiphase pentamode unit cells furtherly. Damping coefficient is introduced and the corresponding acoustic properties are assessed by COMSOL Multiphysics. The advantages of the multiphase metasurface are illustrated on the viewpoint of acoustic stealth performance, pressure resistance under high hydrostatic pressure. Results of this paper are profit to promote the practical application of underwater acoustic metasurface.
4. Conclusions
In conclusion, a novel underwater multiphase metasurface which could manipulate the wavefront and dissipate the acoustic energy simultaneously was proposed. The study suggested that the proposed multiphase metasurface could improve the acoustic stealth ability significantly, which have great potential in underwater applications. The main conclusions are summarized as follows:
(1) A multiphase pentamode configuration composed of hexagonal latticed microstructures, polymer materials and mass balancing lead columns was proposed to realize the desired physical properties. Compared with the single-phase pentamode unit cell which was mostly designed with metallic materials and the damping coefficient was relatively small, significant damping is introduced in the configuration design. Besides, more degree of freedom was introduced which facilitated the designing of metasurface.
(2) An abnormal directional reflection metasurface with a length of 693.2mm and width of 80mm was proposed and numerically verified. Both the simulation results of scattering acoustic pressure field map and Far-Field Sound Pressure Level (FFSPL) in the frequency range of 3kHz~30kHz revealed that the metasurface could reflect the scattering acoustic wave by an the azimuth angle of 15°, which was in agreement with the original design. It was also shown that the introducing of material damping won’t alter the direction of scattering acoustic wave, but it could abate the scattering acoustic pressure amplitude obviously.
(3) Both multiphase and single-phase metasurfaces were designed for the same theoretical metasurface. It is revealed that both metasurface demonstrated the abilities of altering scattering acoustic wave propagation direction, but the amplitude of the scattering wave couldn’t be abated for single-phase metasurface due to no damping properties of the single-phase unit cell. Utilizing the damping properties of the polymer materials inside the multiphase unit cells, the multiphase metasurface could abate the amplitude of scattering acoustic pressure on the basis of reflecting scattering wave. Quantitative calculations reveal that the average Far-Field Sound Pressure Level for single-phase metasurface decreased by 13.19dB compared with aluminum block within the frequency range of 3kHz~30kHz, while that of multiphase metasurface decreased by 4.82dB compared with single-phase metasurface.
(4) The pressure resistance capabilities of both two metasurfaces were studied and compared. It was illustrated that under the same hydrostatic pressure the linearized mean stress for multiphase metasurface is only about 1/3 of that of single-phase metasurface, which suggested that the metasurface designed with multiphase configuration could withstand three times of hydrostatic pressure than the one designed with single-phase unit cell.
Figure 1.
Schematics of normal incident wave and directional reflected wave under the manipulation of a metasurface and backwall.
Figure 1.
Schematics of normal incident wave and directional reflected wave under the manipulation of a metasurface and backwall.
Figure 2.
General design procedure for pentamode metasurface devices. (a) Continual physical parameters results obtained from analytical solution. (b) Discretized physical parameters. (c) Unit cell design and the construction of gradient latticed device. (d) Fabrication of latticed pentamode device.
Figure 2.
General design procedure for pentamode metasurface devices. (a) Continual physical parameters results obtained from analytical solution. (b) Discretized physical parameters. (c) Unit cell design and the construction of gradient latticed device. (d) Fabrication of latticed pentamode device.
Figure 3.
Schematics of two-dimensional pentamode microstructures. (a) Single-phase (SP) pentamode configuration. (b) Dual-phase pentamode configuration. (c) Triple-phase (TP) pentamode configuration.
Figure 3.
Schematics of two-dimensional pentamode microstructures. (a) Single-phase (SP) pentamode configuration. (b) Dual-phase pentamode configuration. (c) Triple-phase (TP) pentamode configuration.
Figure 4.
(a) Typical dispersion curve for acoustic metamaterials. (b) The flow chart of SA optimization [
51].
Figure 4.
(a) Typical dispersion curve for acoustic metamaterials. (b) The flow chart of SA optimization [
51].
Figure 5.
The scattering acoustic pressure field map of the metasurface at 10kHz, 20kHz and 30 kHz. (a) ~ (c): Aluminum block. (d) ~ (f): Continual metasurface. (g) ~ (i): Discretized metasurface. (j) ~ (l): Discretized metasurface with damping coefficient of 0.1. (m) ~ (o): Discretized metasurface with damping coefficient of 0.2.
Figure 5.
The scattering acoustic pressure field map of the metasurface at 10kHz, 20kHz and 30 kHz. (a) ~ (c): Aluminum block. (d) ~ (f): Continual metasurface. (g) ~ (i): Discretized metasurface. (j) ~ (l): Discretized metasurface with damping coefficient of 0.1. (m) ~ (o): Discretized metasurface with damping coefficient of 0.2.
Figure 6.
The polar plot of Far-Field Sound Pressure Level (FFSPL) of aluminum block and metasurface at (a) 10kHz, (b) 20kHz and (c) 30 kHz.
Figure 6.
The polar plot of Far-Field Sound Pressure Level (FFSPL) of aluminum block and metasurface at (a) 10kHz, (b) 20kHz and (c) 30 kHz.
Figure 7.
The Far-Field Sound Pressure Level (FFSPL) of different frequencies for aluminum block and metasurface at the incident direction.
Figure 7.
The Far-Field Sound Pressure Level (FFSPL) of different frequencies for aluminum block and metasurface at the incident direction.
Figure 8.
The dispersion curves of four typical single-phase unit cells. (a)No.1, (b) No.5, (c)No.13, (d)No.20.
Figure 8.
The dispersion curves of four typical single-phase unit cells. (a)No.1, (b) No.5, (c)No.13, (d)No.20.
Figure 9.
The schematic picture of single-phase metasurface and gradual variation of the unit cells.
Figure 9.
The schematic picture of single-phase metasurface and gradual variation of the unit cells.
Figure 10.
The influence of damping coefficient on dispersion curves of unit cells. (a) and (b) correspond to the dispersion curve of No. 5 unit cell where a damping coefficient (0.2) of TPU substrate is considered in (b). (c) and (d) correspond to the dispersion curve of No. 16 unit cell where a damping coefficient (0.2) of TPU substrate is considered in (d).
Figure 10.
The influence of damping coefficient on dispersion curves of unit cells. (a) and (b) correspond to the dispersion curve of No. 5 unit cell where a damping coefficient (0.2) of TPU substrate is considered in (b). (c) and (d) correspond to the dispersion curve of No. 16 unit cell where a damping coefficient (0.2) of TPU substrate is considered in (d).
Figure 11.
The schematic picture of multiphase metasurface and gradual variation of the unit cells.
Figure 11.
The schematic picture of multiphase metasurface and gradual variation of the unit cells.
Figure 12.
The scattering acoustic pressure field map of the metasurface at 10kHz, 20kHz and 30 kHz. (a) ~ (c): Single-phase metasurface. (d) ~ (f): Multiphase metasurface without damping. (g) ~ (i): Multiphase metasurface with damping coefficient of 0.1. (j) ~ (l): Multiphase metasurface with damping coefficient of 0.2.
Figure 12.
The scattering acoustic pressure field map of the metasurface at 10kHz, 20kHz and 30 kHz. (a) ~ (c): Single-phase metasurface. (d) ~ (f): Multiphase metasurface without damping. (g) ~ (i): Multiphase metasurface with damping coefficient of 0.1. (j) ~ (l): Multiphase metasurface with damping coefficient of 0.2.
Figure 13.
The Far-Field Sound Pressure Level of different frequencies at the incident angle. (a) Comparison of FFSPL for single-phase metasurface with theoretical results. (b) Comparison of FFSPL for multiphase metasurface with theoretical results. (c) Comparison of FFSPL between multiphase metasurface and single-phase metasurface. (d) difference of multiphase metasurface and discretized metasurface with damping.
Figure 13.
The Far-Field Sound Pressure Level of different frequencies at the incident angle. (a) Comparison of FFSPL for single-phase metasurface with theoretical results. (b) Comparison of FFSPL for multiphase metasurface with theoretical results. (c) Comparison of FFSPL between multiphase metasurface and single-phase metasurface. (d) difference of multiphase metasurface and discretized metasurface with damping.
Figure 14.
The stress distribution of the proposed metasurface at a hydrostatic pressure of 5 MPa. (a) Single-phase metasurface. (b) Multiphase metasurface. (c) Enlargement for the most dangerous part of the latticed metasurface structure and compassion of the linearized mean stress (251 MPa for single-phase metasurface and 74.2 MPa for multiphase metasurface).
Figure 14.
The stress distribution of the proposed metasurface at a hydrostatic pressure of 5 MPa. (a) Single-phase metasurface. (b) Multiphase metasurface. (c) Enlargement for the most dangerous part of the latticed metasurface structure and compassion of the linearized mean stress (251 MPa for single-phase metasurface and 74.2 MPa for multiphase metasurface).
Table 1.
Equivalent properties of the discretized metasurface and corresponding microstructure parameters of the single-phase and multiphase configurations.
Table 1.
Equivalent properties of the discretized metasurface and corresponding microstructure parameters of the single-phase and multiphase configurations.
Cell No. |
Equivalent properties |
Single-phase |
Multiphase |
X coordinate (m) |
Density (ρ0) |
Modulus (κ0) |
t (mm) |
b(mm) |
m (mm) |
r (mm) |
t' (mm) |
R (mm) |
1 |
-0.3291 |
0.5677 |
1.7616 |
1.000 |
2.5 |
0.13 |
0 |
1.55 |
7.75 |
2 |
-0.2944 |
0.6237 |
1.6033 |
0.900 |
2.5 |
0.40 |
0 |
1.45 |
7.30 |
3 |
-0.2598 |
0.6797 |
1.4711 |
0.820 |
2.5 |
0.66 |
0 |
1.35 |
6.90 |
4 |
-0.2252 |
0.7358 |
1.3591 |
0.760 |
2.5 |
0.89 |
0 |
1.25 |
6.45 |
5 |
-0.1905 |
0.7918 |
1.2629 |
0.680 |
2.5 |
1.15 |
0 |
1.17 |
5.95 |
6 |
-0.1559 |
0.8478 |
1.1795 |
0.630 |
2.5 |
1.38 |
0 |
1.10 |
5.50 |
7 |
-0.1212 |
0.9039 |
1.1063 |
0.545 |
2.5 |
1.64 |
0 |
1.03 |
4.90 |
8 |
-0.0866 |
0.9599 |
1.0418 |
0.530 |
2.5 |
1.84 |
0 |
0.98 |
4.35 |
9 |
-0.0520 |
1.0159 |
0.9843 |
0.515 |
2.5 |
2.03 |
0 |
0.93 |
3.70 |
10 |
-0.0173 |
1.0720 |
0.9329 |
0.490 |
2.5 |
2.23 |
0 |
0.88 |
2.90 |
11 |
0.0173 |
1.1280 |
0.8865 |
0.465 |
2.5 |
2.44 |
0 |
0.82 |
1.70 |
12 |
0.0520 |
1.1841 |
0.8446 |
0.445 |
2.5 |
2.64 |
0 |
0.77 |
0.55 |
13 |
0.0866 |
1.2401 |
0.8064 |
0.425 |
2.5 |
2.70 |
0.27 |
0.74 |
0.90 |
14 |
0.1212 |
1.2961 |
0.7715 |
0.405 |
2.5 |
2.70 |
0.67 |
0.71 |
1.13 |
15 |
0.1559 |
1.3522 |
0.7396 |
0.380 |
2.5 |
2.70 |
1.08 |
0.68 |
1.33 |
16 |
0.1905 |
1.4082 |
0.7101 |
0.360 |
2.5 |
2.70 |
1.48 |
0.66 |
1.50 |
17 |
0.2252 |
1.4642 |
0.6830 |
0.350 |
2.5 |
2.70 |
1.85 |
0.64 |
1.65 |
18 |
0.2598 |
1.5203 |
0.6578 |
0.340 |
2.5 |
2.70 |
2.23 |
0.62 |
1.80 |
19 |
0.2944 |
1.5763 |
0.6344 |
0.330 |
2.5 |
2.70 |
2.62 |
0.60 |
1.92 |
20 |
0.3291 |
1.6323 |
0.6126 |
0.300 |
2.58 |
2.70 |
2.68 |
0.58 |
2.04 |