# Broadband Waterborne Multiphase Pentamode Metastructure with Simultaneous Wavefront Manipulation and Energy Absorption Capabilities

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}).

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Generalized Snell Law (GSL)

_{i}is the refraction index of the incident medium, θ

_{i}and θ

_{r}denotes the incident and reflected angles, λ

_{i}is the wavelength. x is the horizontal coordinate along the metastructure, Φ(x) denotes the phase variation accumulation and $\frac{\mathrm{d}\Phi \left(x\right)}{\mathrm{d}x}$ is the phase gradient of reflected wave. m is the order of the diffraction peak.

_{i}= 0°. The phase accumulation of the wave along the transversal direction of metastructure is [44]:

#### 2.2. Configurations of Unit Cells

_{i}, ρ

_{i}), as shown in Figure 2b. The physical parameters (K

_{i}, ρ

_{i}) have specified combinations, which are generally unavailable from natural materials. Thus, artificial structures with effective modulus and density are desired in acoustic metastructure realizations. Honeycomb configurations with mass-balancing blocks are widely adopted as shown in Figure 3a, where the thickness of struts determines the desired modulus K and the additional weights were adopted to adjust the desired density ρ. Thus, PM acoustic metastructure designed with single-phase configuration are gradient structures with varying strut thickness and mass-balancing blocks, as shown in Figure 2c, and the typical fabricated sample is shown in Figure 2d. It can be seen that strut thickness changes gradually according to the distribution of the physical parameters.

**Figure 3.**Illustrations of 2D pentamode unit cells. (

**a**) Single-phase (SP) pentamode configuration. (

**b**) Dual-phase pentamode configuration. (

**c**) Triple-phase (TP) pentamode configuration.

#### 2.3. Method for Equivalent Physical Properties Calculation

_{equ}could be deduced by the areas and densities of each material in the configuration. Homogenization theory is employed to obtain the equivalent modulus κ

_{equ}firstly, but the studies revealed that the calculated results are only accurate in the long wavelength condition (low frequency range), while the deviation would be huge at higher frequency due to dynamic effect. Thus, dispersion curves are preferred for deriving the equivalent dynamic modulus κ

_{equ}, which is accepted and adopted by most researchers. The wave velocities can be deduced through the slopes of the dispersion curves (c = ω/k) For the acoustic metamaterials, the equivalent compression wave velocity, c

_{L}, and equivalent shear wave velocity, c

_{T}, are given by:

^{3}) is applied as the matrix material, thermoplastic polyurethanes (TPU, E = 100 MPa, v = 0.4, and ρ = 1000 kg/m

^{3}) is applied as the connecting material, while lead columns (E = 16 GPa, v = 0.42, and ρ = 11,300 kg/m

^{3}) is chosen as the balancing material. Several parameters are fixed: H = L = 10 mm, θ = 30°, tH = tL. Typical dispersion curves of the pentamode cells calculated with COMSOL are presented in Figure 4a. The dashed fitting lines correspond to the compression and shear waves in the long-wavelength limit, respectively.

_{i}, ρ

_{i}) is an inverse problem. Due to the numerous combinations of design space, optimization methods are urgent to pursue the optimal design variables. A gradient-free method, Simulated Annealing (SA) technique, is adopted in this study. The SA optimization method is inspired by the annealing process of metals, which was conceived by Metropolis et al. and developed by Kirkpatrick et al. [24,51,52]. A methodology integrating dispersion curve with SA optimization technique is developed by the authors. The optimization process is carried out by COMSOL (v5.4, COMSOL Inc., Stockholm, Sweden) with MATLAB (R2018a, MathWorks Inc., Natick, MA, USA). For any given set of design variables, the equivalent properties are derived through dispersion curve. The SA algorithm integrated with COMSOL was developed and the flow chart is illustrated in Figure 4b, and a detailed description was introduced in a previous work [51].

## 3. Results and Discussions

#### 3.1. Influences of Damping Coefficient on Acoustic Properties of Ideal Metastructure

_{b}= 70 mm is utilized as the rigid wall, while the thickness of the metastructure is D = 80 mm. The angle of the incident wave is set as θ

_{i}= 0°, while the reflection angle is set as θ

_{r}= 15°. The properties of the metastructure are derived from Equation (4), where the densities vary from 0.5677ρ

_{0}to 1.6323ρ

_{0}and the bulk modulus vary from 1.7616κ

_{0}to 0.6126κ

_{0}. The average equivalent density of the metastructure is about 1.10ρ

_{0}, which is acceptable in underwater applications. The acoustic properties are investigated by COMSOL Multiphysics. A full 2D model simulation is performed in the “Acoustics-solid, Frequency Domain” module. The isotropic structural damping coefficient η

_{s}is endowed to the material via the subnode of “Linear Elastic Material”. Perfectly matched layers are set on the outer boundaries of simulation domains to eliminate reflections. The metastructure is set on the upper surface of the aluminum block, while the incident wave impinges on the models from up to down.

#### 3.2. Microstructure Design of Single-Phase and Multiphase Metastructure

#### 3.3. Formatting of Mathematical Components

#### 3.4. Advantages for Withstanding External Pressure

## 4. Conclusions

- (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. Additionally, more degrees of freedom were introduced, which facilitated the designing of the metastructure.
- (2)
- An abnormal directional reflection metastructure with a length of 693.2 mm and width of 80 mm was proposed and numerically verified. Both the simulation results of the scattering acoustic pressure field map and the Far-Field Sound Pressure Level (FFSPL) in the frequency range of 3 kHz~30 kHz revealed that the metastructure could reflect the scattering acoustic wave by an azimuth angle of 15°, which was in agreement with the original design. It was also shown that the introducing of material damping will not alter the direction of the scattering acoustic wave, but it could abate the scattering acoustic pressure amplitude obviously.
- (3)
- Both multiphase and single-phase metastructures were designed for the same theoretical metastructure. It is revealed that both metastructures demonstrated the abilities of changing the propagation direction of scattering acoustic wave, but the amplitude of the scattering wave could not be abated for the single-phase metastructure due to the lack of damping properties of the single-phase unit cell. Utilizing the damping properties of the polymer materials inside the multiphase unit cells, the multiphase metastructure could abate the amplitude of the scattering acoustic pressure on the basis of reflecting the scattering wave. Quantitative calculations reveal that the average Far-Field Sound Pressure Level for single-phase metastructure decreased by 13.19 dB compared to the aluminum block within the frequency range of 3 kHz~30 kHz, while that of the multiphase metastructure decreased by 4.82 dB compared to the single-phase metastructure.
- (4)
- The pressure resistance capabilities of both metastructures were studied and compared. It was illustrated that the linearized mean stress for the multiphase metastructure is only about 1/3 of that of the single-phase metastructure under the same hydrostatic pressure, which suggested that the metastructure designed with a multiphase configuration could withstand three times the hydrostatic pressure than the one designed with a single-phase unit cell.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematics of normal incident wave and directional reflected wave under the manipulation of a metastructure and backwall.

**Figure 2.**General design procedure for pentamode metastructure 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 4.**(

**a**) Typical dispersion curve for acoustic metamaterials. The solid lines are dispersion curves corresponding to different vibrating modes, and dashed lines are the fitting curve of the first two vibrating modes in the long wavelength regimes (

**b**) The flow chart of SA optimization [51].

**Figure 5.**The scattering acoustic pressure field map of the metastructure at 10 kHz, 20 kHz and 30 kHz. (

**a**–

**c**) Aluminum block. (

**d**–

**f**) Continual metastructure. (

**g**–

**i**) Discretized metastructure. (

**j**–

**l**) Discretized metastructure with damping coefficient of 0.1. (

**m**–

**o**) Discretized metastructure with damping coefficient of 0.2.

**Figure 6.**The polar plot of Far-Field Sound Pressure Level (FFSPL) of aluminum block and metastructure at (

**a**) 10 kHz, (

**b**) 20 kHz and (

**c**) 30 kHz.

**Figure 7.**The Far-Field Sound Pressure Level (FFSPL) of different frequencies for aluminum block and metastructure 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. The solid lines correspond to the first six vibration modes.

**Figure 9.**The schematic picture of single-phase metastructure and gradual variation of the unit cells.

**Figure 10.**The influence of damping coefficient on dispersion curves of unit cells. (

**a**,

**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**,

**d**) correspond to the dispersion curve of No. 16 unit cell where a damping coefficient (0.2) of TPU substrate is considered in (

**d**). The solid lines correspond to the first six vibration modes.

**Figure 11.**The schematic picture of multiphase metastructure and gradual variation of the unit cells.

**Figure 12.**The scattering acoustic pressure field map of the metastructure at 10 kHz, 20 kHz and 30 kHz. (

**a**–

**c**): Single-phase metastructure. (

**d**–

**f**): Multiphase metastructure without damping. (

**g**–

**i**): Multiphase metastructure with damping coefficient of 0.1. (

**j**–

**l**): Multiphase metastructure 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 metastructure with theoretical results. (

**b**) Comparison of FFSPL for multiphase metastructure with theoretical results. (

**c**) Comparison of FFSPL between multiphase metastructure and single-phase metastructure. (

**d**) Difference of multiphase metastructure and discretized metastructure with damping.

**Figure 14.**The stress distribution of the proposed metastructure at a hydrostatic pressure of 5 MPa (corresponding to a water depth of 500 m). (

**a**) Single-phase metastructure. (

**b**) Multiphase metastructure. (

**c**) Enlargement for the most dangerous part of the latticed metastructure structure and compassion of the linearized mean stress (251 MPa for single-phase metastructure and 74.2 MPa for multiphase metastructure).

**Table 1.**Equivalent properties of the discretized metastructure and corresponding microstructure parameters of the single-phase and multiphase configurations.

Cell No. | Equivalent Properties | Single-Phase | Multiphase | ||||||
---|---|---|---|---|---|---|---|---|---|

X Coordinate (m) | Density (ρ _{0}) | Modulus (κ _{0}) | 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 |

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**MDPI and ACS Style**

An, Y.; Zou, H.; Zhao, A.
Broadband Waterborne Multiphase Pentamode Metastructure with Simultaneous Wavefront Manipulation and Energy Absorption Capabilities. *Materials* **2023**, *16*, 5051.
https://doi.org/10.3390/ma16145051

**AMA Style**

An Y, Zou H, Zhao A.
Broadband Waterborne Multiphase Pentamode Metastructure with Simultaneous Wavefront Manipulation and Energy Absorption Capabilities. *Materials*. 2023; 16(14):5051.
https://doi.org/10.3390/ma16145051

**Chicago/Turabian Style**

An, Yi, Han Zou, and Aiguo Zhao.
2023. "Broadband Waterborne Multiphase Pentamode Metastructure with Simultaneous Wavefront Manipulation and Energy Absorption Capabilities" *Materials* 16, no. 14: 5051.
https://doi.org/10.3390/ma16145051