The Ultra-low Frequency and Broad Band Gaps Characteristics of Multilayer Composite Cylindrical Three-dimensional pentamode metamaterials

: For three-dimensional pentamode metamaterials, it is of great significance to realize underwater ultra-low frequency acoustic wave control. Therefore, two types multilayer composite cylindrical three-dimensional pentamode metamaterials with ultra-low frequency and broad band gaps are proposed in this paper. By using pentamode metamaterials with lattice constants on the order of centimeters, the phononic band gaps below 60 Hz and the single-mode area below 30Hz can be obtained. Compared with asymmetrical double-cone locally resonant pentamode metamaterials, the lower edge frequency, relative bandwidth and figure of merit of the first phononic band gap can be reduced by up to 61.4%, 10.3% and 40.6%, respectively. It will provide reference and guidance for the engineering application of pentamode metamaterials in controlling the ultra-low frequency broadband acoustic waves, vibration and noise reduction.

Abstract: For three-dimensional pentamode metamaterials, it is of great significance to realize underwater ultra-low frequency acoustic wave control. Therefore, two types multilayer composite cylindrical three-dimensional pentamode metamaterials with ultra-low frequency and broad band gaps are proposed in this paper. By using pentamode metamaterials with lattice constants on the order of centimeters, the phononic band gaps below 60 Hz and the single-mode area below 30Hz can be obtained.
Compared with asymmetrical double-cone locally resonant pentamode metamaterials, the lower edge frequency, relative bandwidth and figure of merit of the first phononic band gap can be reduced by up to 61.4%, 10.3% and 40.6%, respectively. It will provide reference and guidance for the engineering application of pentamode metamaterials in controlling the ultra-low frequency broadband acoustic waves, vibration and noise reduction.

1.Introduction
The concept of pentamode metamaterials (PMs) was first proposed by Milton and Cherkaev in 1995 [1] . The equivalent elastic properties of PMs can be expressed as in the six-dimensional stress space, only the eigenvalue of the volume compression mode is not zero, and the other five characteristic values of the shear mode are zero. Such a solid structure will show the mechanical properties of traditional fluids as a whole, it is a complex "fluid" with solid characteristics. In theory, the acoustic metamaterials composed of such a periodic arrangement of structural units can achieve a perfect match with water. Therefore, the characteristics of adjustable modulus anisotropy, solid characteristics and wide frequency endow the PMs with excellent sound wave control capabilities, which have important potential applications in the fields of sound wave control, earthquake protection, vibration and noise reduction [2,[4][5][6][7][8][9][10][11][12][13][14] .
In 2006, Milton introduced the feasibility of using PMs for elastic wave stealth through the change law of traditional elastic dynamics equations under curve transformation [15] . In 2007, Chen Huanyang et al. used the invariance of the direct current conductance equation under coordinate transformation to establish a one-to-one correspondence between the acoustic wave equation and the direct current conductance equation, and derived the general three-dimensional transformation acoustic equation for the first time [16] . In 2008, Norris systematically analyzed the inertial acoustic stealth and the PMs acoustic cloak, and proposed the possibility of applying the transform acoustic theory to the PMs [2] . In 2010, Scandrett et al. proposed a method of designing acoustic cloaks using layered PMs [17][18] . In 2012, Kadic et al. used laser direct writing technology and artificial additive manufacturing technology to produce threedimensional PMs samples of micrometer and millimeter order for the first time, and conducted mechanical and acoustic properties research [19,20] . In 2014, N. Aravations-Zafiris et al. designed a three-dimensional layered column structure with pentamode characteristic, which broadened the array structure of PMs units [21] . This has also aroused the interest of researchers in the different lattice configurations of PMs [22] . In 2018, Liu Xiaozhou et al. designed a carpet-style PMs acoustic cloak suitable for underwater broad by adjusting the microstructure geometric parameters of the twodimensional PMs, provided a new choice for PMs in acoustic stealth devices [10] . In 2020, Huang Guoliang et al. designed a two-dimensional elastic wave stealth cloak using the extreme characteristics of similar PMs, which will provide a useful reference for exploring the underwater acoustic wave control of PMs [13][14] .
At present, low-frequency detection sonars with operating frequencies below 300 Hz have appeared, which requires PMs to have the ability to control underwater lowfrequency sound waves. For the PMs composed of simple materials, they generally belong to the Bragg scattering PMs, and their lattice size is generally the same order of magnitude as the acoustic wavelength corresponding to the operating frequency. If the Bragg scattering PMs is used to control low-frequency sound waves below 300 Hz, the structural size of the sound wave control device needs to be at least tens of meters, which brings great difficulties in engineering application. Therefore, studying the method and mechanism of small-scale PMs to control underwater low-frequency sound waves and establishing a three-dimensional control method for working frequency bands are the core issues that need to be solved urgently.
Generally, the wavelength corresponding to the acoustic/elastic wave phononic band gap (PBG) formed by Bragg scattering is in the same order of magnitude of the lattice constant. Due to the limitation of the lattice size, it is difficult to apply Bragg scattering PMs at low frequencies (especially below 100 Hz). In order to overcome the shortcomings of Bragg scattering PMs, we propose two cylindrical three-dimensional PMs with ultra-low frequency and broad band gaps. The primitive of multilayer composite cylindrical is composed of composite materials, one part provides the quality required by the structural unit, that is, a hard material. The other part provides the elasticity required by the resonance of the structural unit, that is, a soft material.
Furthermore, the finite element simulation software COMSOL Multiphysics is used to carry out the numerical analysis of its band structure, single-mode area and pentamodal performance. Furtherly, the effects of asymmetry on the PBG, single-mode area and figure of merit (FOM) of the three samples are discussed. The lower frequency and broad band gaps will endow the structure with better potential for underwater sound wave control.

2.Structure design and energy band characteristics of multilayer composite cylindrical three-dimensional PMs
The unit cell structure of multilayer composite cylindrical three-dimensional PMs is shown in figure 1(a). It consists of 16 primitives connected at the narrow end to form a face-centered cubic structure with a lattice constant of a. Three types primitives are shown in figure 1(b) and composed by two materials. In order to comparative analysis conveniently, we define them as S1, S2 and S3 respectively. For S1, the primitive structure is the composite asymmetric double-cone element. The soft material is at the narrow diameter at both ends ( 1 and 2 ), and the length is ℎ 1 . For S2, it is one of multilayer composite cylindrical three-dimensional PMs. The soft material is added at both ends, the length is ℎ 1 , and the diameters from the top to the bottom of the primitive structure are 1 、 1 、 3 、 3 、 2 、 2 , the heights of the corresponding cylinders are ℎ 1 、ℎ 2 、ℎ 3 、ℎ 3 、ℎ 2 、ℎ 1 , and satisfy the equivalent relationship of ℎ 1 +ℎ 2 +ℎ 3 =1/2H. For S3, the soft material is added to the middle position of the primitive, and the structural parameters are the same as S2. Here, we define the ratio of narrow diameter of soft materials as asymmetry degree     In addition, the bandwidth of the first PBG can be extended, which is 10.3% higher than the relative bandwidth of asymmetric double-cone locally resonant PMs. This means that the using locally resonant PMs formed by S2 to control ultra-low frequency acoustic/elastic waves will produce more excellent effects. However, it is clear that locally resonant PMs formed by S3 are more stable than S2.

3.The influence of asymmetry degrees on the first PBG
The Bragg scattering mechanism emphasizes the influence of periodic structure on waves, while the locally resonant mechanism emphasizes the interaction between the resonance characteristics of the scatterer unit and the waves in the matrix. Since the lower and upper frequencies of the PBG of the locally resonant PMs can be equivalent to the principle of the "spring-mass" system, by analyzing the equivalent parameters in the simplified model, the change trend of the PBG is analyzed. The relationship between the lower edge frequency of the first PBG and the equivalent stiffness and the equivalent mass of the three types locally resonant PMs can be expressed in the following proportional relationship:  Comparison of the influence of asymmetry degrees on the relative bandwidth of the first PBG of three samples (S1, S2, S3).
Any structural parameter that can cause changes in the equivalent parameters of the locally resonant will have an impact on the lower and upper frequencies of the first PBGs. In order to further study the locally resonant characteristics of the three samples, the influence of the asymmetry degrees ( 1 , 2 , and 3 )on the first PBG and relative bandwidth was studied and some results were shown in figure 3.
The influence of N1 on the first PBGs of S1 was shown in figure 3(a), and the vibration mode of the lower edge frequency of the first band gap is given when N1=0.4.
It can be seen that when 1 ranges from 0.1 to 0.9, 2 is more easily to deform. With the increase of 1 , the narrow diameter 2 increases, the equivalent stiffness 1 produced by its deformation also is increased, while the change of the equivalent mass  Therefore, compared with the S1 type locally PMs, the S2 type locally PMs can not only reduce the lower edge frequency of the first PBG by 43.8% to 50.9%, but also broaden the relative bandwidth of the first PBG.

4.Single-mode region
The single-mode area is the frequency region that limits the decoupling of compression waves and shear waves, and it is an important factor to measure the pentamode characters of locally PMs, Therefore, it is necessary to further study the single-mode performance of the three samples. Figure 4 shows the effects of asymmetry degrees 1 , 2 , and 3 on the upper and lower edge frequencies and relative bandwidths of the single-mode regions of the three samples. Obviously, the singlemode lower edge frequency of S1 and S2 increases slowly with the increase of asymmetry, and S3 shows a trend of first rising and then decreasing slightly with the increase of asymmetry. Among them, the single-mode lower edge frequency corresponding to S1 is the lowest. With the increase of asymmetry, the single-mode upper edge frequency of S1 and S2 showed a sharp increase and a gentle increase trend, respectively, while S3 first slowly increased and then decreased slightly. Therefore, the relative bandwidth of the single-mode area of the S1 is the largest, which is basically maintained at about 1.46. while S2 is second, showing a trend of first rising and then decreasing. The relative bandwidth of the single-mode area of S3 gradually increases with the increase of asymmetry 3 , and reaches the maximum value of 0.95 when 3 is equal to 2.0.

5.Pentamodal performance
The FOM is defined as the ratio of the equivalent bulk modulus (B) to the equivalent shear modulus (G). The larger the FOM, the easier it is to decouple the compression and shear waves, thereby obtaining good pentamodal performance. The FOM has the following proportional relationship with the ratio of compression and shear waves phase velocity: While, the compression wave phase velocity ( ) and shear wave phase velocity ( ) of locally resonant PMs are determined by the slope of the compression and shear waves of phononic band. Figure 5 shows the influence of asymmetry degrees on the FOM of three samples.
Both S1 and S2 showed a trend of first increasing and then decreasing with the increase of asymmetry degrees. The FOM of S1 reached the maximum value of 514.29 when 1 =0.9, and the FOM of S2 reached the maximum value of 723.1 when 2 =0.8. For S3, it increases slowly with the increase of 3 , and FOM reaches the maximum value of 97.05 when 3 =2.0. The above analysis shows that, compared with S1, the FOM of S2 can be increased by up to 40.6%.

6.Conclusion
For Bragg scattering PMs, using the PMs to achieve underwater sound insulation below 100Hz requires a material with a several meters thick lattice size. In this paper, two types multilayer composite cylindrical three-dimensional PMs are proposed. In order to compare and analyze with the asymmetrical double-cone locally resonant PMs, a large number of numerical calculations have been carried out on the influence of asymmetry 1 , 2 , 3 on the phononic band gap, single-mode area, and FOM of the three samples (S1, S2, S3). Numerical results show that compared with S1 type locally resonant PMs, S2 type locally resonant PMs proposed in this paper can reduce the lower edge frequency of the first PBG by up to 61.4%, increase the relative bandwidth of the first PBG by up to 10.3%, and increase the FOM by up to 40.6%. This research will provide a theoretical basis and reference for further promoting the application of threedimensional PMs in underwater low-frequency acoustic wave control devices.