Design

Design and optimization of three-dimensional composite multilayer cylindrical pentamode metamaterials for controlling low frequency acoustic waves

The unit cell structure of multilayer composite cylindrical three-dimensional PMs is shown in Fig. 1a. 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 Fig. 1b 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 (({d}_{1} mathrm{and} {d}_{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 ({d}_{1}, {D}_{1}, {D}_{3}, {D}_{3}, {D}_{2}, {d}_{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:

Figure 1

(a) The unit cell structure diagram of the multilayer composite cylindrical three-dimensional PMs. (b) Primitive structure diagrams of three locally resonance PMs.

$$left{begin{array}{c}frac{{d}_{2}}{{d}_{1}}={N}_{1} for S1\ frac{{ d}_{2}}{{d}_{1}}={N}_{2} for S2\ frac{{D}_{2}}{{D}_{1}}={ N}_{3} for S3end{array}right.$$

(One)

In order to facilitate comparative analysis, the hard and soft materials of three samples are polymers and silicone rubber, respectively. The material parameters are shown in Table 1.

Table 1 Material parameters.

For calculating the phononic band structure, the Bloch boundary conditions are applied on the primitive unit cells of the three locally resonant PMs in the finite element simulation software COMSOL Multiphysics. The fixed structure parameters are S1: a= 37.3 mm, H= 16.15 mm,(D)=3 mm,({d}_{1})=0.55 mm,({h}_{1})=(0.1) H. S2:({D}_{3})=3 mm,({D}_{1})=({D}_{2})=1.5 mm,({d}_{1})=0.55 mm,({h}_{1})=(0.1) H,({h}_{2})=(0.3) H,({h}_{3})=(0.1) H. S3:({D}_{3})=3 mm,({D}_{1})=1.5 mm,({d}_{1})=({d}_{2})=0.55 mm,({h}_{1})=(0.1) H,({h}_{2})=(0.3) H , ({h}_{3})=(0.1) H . Select the asymmetry degrees ({N}_{1})=({N}_{2})=({N}_{3})=0.6 as a reference, and the calculated band structures of the three samples are shown in Fig. 2.

Figure 2
figure 2

The phononic band structure of locally resonant PMs composed by (a) S1, (b) S2, (c) S3.

Existing pentamode metamaterials mainly consider cloak from external sound sources, that is, to study their single-mode properties (to realize decoupling of compressional and shear waves). However, it is equally important to study how to control noise inside acoustic cloaks, that is, the effect of structural parameters on the complete phononic band gap (PBG). In the PBG region, the acoustic waves that fall within the frequency range of the PBG emitted by the sound source inside the cloaked object are all confined inside the PMs, preventing the internal sound source from propagating outward. The phononic band structure of the S1 is shown in Fig. 2a. Not only are there two PBG in the phononic band structure diagram, but also the relatively flat energy bands appear near PBG. These flat energy bands mean the existence of resonance modes. The frequencies of the lower edge (({f}_{l})) and upper egde (({f}_{u})) of the first PBG are 132.42 Hz and 198.19 Hz, respectively. The relative bandwidth of the first PBG (Δω/({omega }_{g}) =(frac{Delta omega }{({f}_{u}+{f}_{l})/2})) is 0.398. The lower and upper edge frequencies of the second PBG are 240.06 Hz and 253 Hz, respectively. The relative bandwidth of the second PBG is 0.052. The phononic band structure of the S2 is shown in Fig. 2b. It can be seen that two complete PBGs can be opened in addition to the single-mode area. The lower and upper edge frequencies of the first PBG are 51.16 Hz and 79.95 Hz, respectively. The relative bandwidth of the first PBG is 0.439. The lower and upper edge frequencies of the second PBG are 81.9 Hz and 83.97 Hz, respectively. The relative bandwidth of the second PBG is 0.025. The phononic band structure of the S3 is shown in Fig. 2c. It is obvious that there is only one PBG in the phononic band structure diagram. The lower and upper edge frequencies of the first PBG are 78.99 Hz and 97.06 Hz, respectively. The relative bandwidth of the first PBG is 0.205.

From the above analysis, it can be seen that under the same structural parameters, the phononic band gap of S3 is narrower than S1 and S2, however, the stability of the S3 structure is higher. Elastic Stability of Stress rods was used to analyze the stability of three locally resonance PMs. Considering the length coefficient and the constraints of the rod, the three structures mentioned in this paper can be approximately equivalent to the structure of the pressure rod with fixed ends. The stability of the three structures is compared by calculating the critical force of the compression rod (({P}_{cr})) respectively.

$$P_{cr} { = }frac{{pi^{2} EI}}{{(mu l)^{2} }},mu = 0.5,l = H$$

(2)

Here, E is the elastic model of the materials, l is the length of the rod, and I is the moment of inertia of the cross-section of the rod to the main axis of the row center.

Consider that the Young’s modulus of soft materials is smaller than hard materials. And the deformation generally occurs in the silicone rubber part first. Here we only compare and analyze the corresponding stability of the soft material of the structures, and the obtained results are also applicable to the analysis of the hard material part.

For the three structures with the same structural and material parameters, only I differs in Eq. (2). And I can be expressed as (I = frac{{pi D^{4} }}{64}) for the cross-section of the rod is a circle, Dis the diameter of the section circle. Obviously, Iin the formula ({P}_{cr}) of S3 is larger than S1 and S2, so the ({P}_{cr}) of S3 is larger than S1 and S2. Thus, the structure of S3 is more stable than S1 and S2.

To sum up, compared with the asymmetric double-cone locally resonant PMs, the two types multilayer cylindrical locally resonant PMs can not only obtain the complete PBG, also can greatly reduce the frequency of the first PBG. For S2 and S3, they can reduce the lower edge frequency of the first PBGs by 61.4% and 40.3%, respectively. 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 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.

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