Acoustic black hole (ABH) structures are widely used for vibration and acoustic waves control due to their ability to guide the zero reflection of elastic waves and the concentration of wave energy. However, ABH can hardly suppress the ultra-low-frequency waves. We propose the acoustic cloister to break the low-frequency limit of the cutoff frequency and realize the perfect ABH effect while suppressing the ultra-low frequency waves. Thus, the waves can be localized within this structure and realize the ultra-low frequency ultra-broadband bandgap. We theoretically elaborate the bandgap mechanism of the acoustic cloister and demonstrate the good robustness of the acoustic cloister, which is beneficial for generating stable ultra-low frequency nonlinear bandgaps. Nonlinear buckling theory has been applied to explain the ultra-low frequency nonlinear bandgaps of 3–22 and 24–28 Hz that appear in the experiments, which reduces the wave transmission by 20–40 dB, and it has been demonstrated that the bending stresses appeared in the experiments can generate and greatly extend ultra-low frequency bandgaps. In torsional excitation experiments, the acoustic cloister structure attenuates wave transmission in the 3–100 Hz range by 10–80 dB. Our work makes a significant contribution to advances in vibration and acoustic wave control.

## I. INTRODUCTION

ABH allows elastic waves to propagate with significantly lower phase and group velocities by reducing the wall thickness in a power-law relationship. Under ideal conditions, when the wall thickness is reduced to zero, the ABH structure can achieve reflection-free aggregation of waves,^{1–5} showing great theoretical research value and practical application potential.^{1–20} Currently, the ABH has been widely used in vibration suppression and damping,^{6–11} acoustic radiation control,^{12–17} and energy harvesting.^{5,18–20} However, in practice, the existence of non-zero residual thickness of the ABH structure makes the ABH effect appear only when the frequency of the wave matches the impedance of the structure,^{12,17,21} which seriously hinders the development of the ABH effect in controlling subwavelengths. In addition, the ABH effect generated by the conventional ABH structure is strongly dependent on the thickness distribution,^{1} resulting in poor ABH robustness, where a slight change in geometric thickness can cause a large change in the ABH effect.^{22–24}

In order to overcome the two barriers mentioned above, most of the studies have been extended to combine with Bragg’s scattering and Local Resonance (LR) mechanisms. For example, the Bragg scattering effect of periodic ABH structures can be enhanced by exploiting the energy aggregation phenomenon within the ABH cell,^{25,26} which effectively extends the cut-off frequency and overcomes the limitation that the Bragg bandgap requires a sufficient number of periods to maintain the bandgap.^{27} The periodically arranged ABH beams can also be used in the low and middle frequencies LR bandgap.^{28–30} However, in the ultra-low frequency band, the ABH structure still cannot generate a bandgap and cannot suppress ultra-low-frequency waves.^{31}

Here, we report an acoustic cloister, a combination of stellate ABH and soft material, which activates ultra-low frequency ultra-wide bandgap. The principle of the acoustic cloister is introduced in Sec. II. Then, the bandgap mechanism is elaborated, and the influence of structural parameters on the starting and ending frequencies and width of the bandgap is analyzed in Secs. III and IV, respectively. In Sec. V, we demonstrate the superior robustness of the acoustic cloister at ultra-low frequencies. In Sec. VI, experimental methods have been applied to study the acoustic transmission loss of the acoustic cloister and to explain the mechanism of ultra-low frequency ultra-broadband nonlinear bandgap.

## II. ACOUSTIC CLOISTER PRINCIPLE

The structure and schematic diagrams of the acoustic cloister are shown in Fig. 1. The structure is made of steel fabricated stellate ABH crystals as the scattering body, and the rectangular silicone rubber beam as the “cloister” structure whose ends are glued to the steel to form a “hard-soft-hard” composite. The ABH crystal is symmetrical about x, y, and z axes, the total width of the ABH crystal is *b*, the radius of the circular surface in the ABH crystal is *r*, the angle of the center of the circle is 90°, the width of the ABH crystal tip is *p*, and the elongation of the ABH tip is *d*, where *p* = *b* − 2 * *r* − 2 * *d*. The length of the silicone rubber is *l*, the thickness is *h*, and the lattice constant of the unit structure is *a* = 2 * *b* + *l*, see Fig. 1(a).

For waves above the cutoff frequency, the wavelength will be compressed and the amplitude will increase when propagating to the ABH tip, while no such ABH effect will occur for waves below the cutoff frequency. Therefore, we connect a soft material between two ABH tips, so that Young's modulus of the material on the connection surface changes abruptly, low-frequency and sub-low-frequency waves can realize the ABH effect in the soft material, and make the waves continuously reflect and scatter in the “cloister” formed by the soft material but cannot propagate out. Therefore, the low-frequency wave is focused and localized in the acoustic cloister, as shown in Fig. 1(b).

^{32}

*E*(

*x*) is the Young modulus of the structure,

*I*(

*x*) is the moment of inertia of the cross section,

*ρ*(

*x*) is the density,

*S*(

*x*) is the cross-sectional area, and

*ω*is the transverse vibration displacement.

*x*, the displacement of the wave propagation can be expressed as

*A*(

*x*) is a function with the same period and Φ(

*x*) is the cumulative phase. Assuming that the medium is homogeneous,

*I*(

*x*) can be expressed as

*υ*is the Poisson ratio of the material and

*h*(

*x*) is the thickness of the structure. The cumulative phase Φ(

*x*) can be expressed as

*k*=

_{p}*ω*/

*c*,

_{p}*k*is the wave number of the uniform plate,

_{p}*c*is the phase velocity of the bending wave in the medium, and

_{p}*ω*is the angular frequency. Thus, when the thickness of a structure varies exponentially to 0, the thickness of the structure can be expressed as

Bringing Eq. (5) into Eq. (4), we can see that when the index *m* ≥ 2, the accumulated phase Φ(*x*) will tend to infinity, at which point the wave will never reach the boundary, and reflection will never occur. The energy of the bending wave will be completely gathered at the position where the thickness of the structure tends to be close to 0.

*k*is

*c*of the bending wave can be expressed as

From Eq. (8), *c* is proportional to the square root of the thickness *h*(*x*) and proportional to the Young modulus *E*(*x*) of the material and the quarter power of the cross-sectional moment of inertia *I*(*x*). When the thickness of the structure is 0, the phase velocity *c* of the wave will also be reduced to 0, which is the ABH effect.

*h*

_{0}in

*h*(

*x*), the minimum thickness of the structure, making the expression for the thickness of the medium become

*x*) must be convergent at this point, and the phase velocity

*c*of the wave will not be equal to zero. Therefore, the actual resulting ABH effect does not completely capture the wave energy, and the wave always reaches the boundary of the ABH, which is one of the defects of the ABH structure. However, it is worth mentioning that since the thickness

*h*(

*x*) of the ABH structure decreases with a power function law of more than two times, the output phase velocity

*c*

_{out}from the ABH structure will decrease proportionally with respect to the input phase velocity

*c*

_{in}, and the wavelength of the bending wave can be obtained according to

**=**

*λ**c*/

*ω*as

*E*(

*x*) of the medium, the decrease of the moment of inertia

*I*(

*x*) of the cross section, and the decrease of the thickness

*h*(

*x*). According to the interference condition of Bragg’s law

^{33}

From Eqs. (11) and (12), the scattering angle *θ* will become smaller as the wavelength ** λ** becomes shorter, and the bending wave will be more prone to the Bragg scattering. When the low frequency wave enters the acoustic cloister, the wave can easily propagate from the hard medium to the soft medium, then in the soft medium, the wavelength

**of the wave is reduced because the inertia moment**

*λ**I*(

*x*) and the stiffness

*E*(

*x*) are reduced sharply, which produces the ABH effect and decreases the Bragg scattering angle

*θ*. However, when a wave is incident from a soft medium to a hard medium, the wave will be very prone to the Bragg scattering and difficult to propagate to the hard medium, so the wave can only be reflected back and confined to the soft medium until the energy is exhausted.

## III. FINITE ELEMENT ANALYSIS

The bandgap mechanism of the acoustic cloister is analyzed by using the finite element method and solid energy band theory. The dimensional and material parameters of the acoustic cloister structure are shown in Tables I and II.

Parameters . | b
. | r
. | p
. | D
. | l
. | h
. | a
. |
---|---|---|---|---|---|---|---|

Values (mm) | 106 | 46 | 8 | 3 | 22 | 8 | 234 |

Parameters . | b
. | r
. | p
. | D
. | l
. | h
. | a
. |
---|---|---|---|---|---|---|---|

Values (mm) | 106 | 46 | 8 | 3 | 22 | 8 | 234 |

. | Mass density (kg/m^{3})
. | Young's modulus (Pa) . | Poisson's ratio . |
---|---|---|---|

Structural steel | 7850 | 200 × 10^{9} | 0.30 |

Silicone rubber | 1300 | 7.7 × 10^{6} | 0.47 |

. | Mass density (kg/m^{3})
. | Young's modulus (Pa) . | Poisson's ratio . |
---|---|---|---|

Structural steel | 7850 | 200 × 10^{9} | 0.30 |

Silicone rubber | 1300 | 7.7 × 10^{6} | 0.47 |

^{34,35}

*ρ*is the material density and

**and**

*λ**μ*are Lamé coefficients.

*u*denotes the displacement vector and $ r \u2192$ is the position vector. $ k \u2192$ denotes the Bloch wave vector, and

*a*denotes the lattice vector.

*k*only needs to be scanned along the boundary of the one-dimensional (1D) integrable Brillouin zone, i.e., Γ → X to take different values of the wave vector

*k*. Then, using the finite element method, the single cell is meshed to obtain the dispersion curve. The equation of the original cell eigenvalue after discretization is

^{35}

*K*is the stiffness matrix of the structure,

*M*is the mass matrix, and

*U*is the displacement matrix.

*K*of silicone rubber can be expressed in the form of complex stiffness

_{c}^{36}

*η*= 0.2 is the damping loss factor of silicone rubber.

In addition, in order to verify the energy band calculation of the phononic crystal structure, the transmission loss spectra of the acoustic cloister at one and two periods are calculated using the finite element method, and the calculated model of the acoustic cloister at two periods is shown in Fig. 2.

*P*is the displacement at the excitation end and

_{in}*P*is the displacement at the response end.

_{out}Figure 3 shows the dispersion diagram of the acoustic cloister, the corresponding eigenmode diagram, and the transmission loss spectra of 1 periodic and 2 periodic acoustic cloisters.

From Fig. 3(a), it can be seen that the acoustic cloister generates multiple bandgaps within 1000 Hz. The first complete bandgap is in the range of 38.25–517.2 Hz with a bandwidth of 478.95 Hz, and the bandgap coverage at 1000 Hz has reached 94.8%. In Fig. 3(c), transmission loss spectra, the energy decay frequency intervals of the 1 periodic and 2 periodic acoustic cloister are exactly the same as the Bragg bandgap distribution range in Fig. 3(a) dispersion diagram, which verifies the accuracy of the energy band structure calculation. In addition, two nonlinear bandgaps also appear in the range of 13.97–15.98 and 35.01–37.01 Hz in Fig. 3(c), which are caused by damping of silicone rubber.

Figure 3(b) shows the eigenmodes of points A–D in Fig. 3(a). Points A and B correspond to the transverse and longitudinal wave modes at the lower boundary of the Bragg bandgap, respectively. The motion directions of the stellate ABH crystals at both ends of the two modes are opposite, corresponding to the peak and valley of the wave at that frequency, respectively, implying that the wavelength * λ* at this point has been compressed by the ABH effect to twice lattice constant, i.e.,

*= 2*

**λ***a*. According to the interference condition of Bragg's law, the wavelength at this point has satisfied Eqs. (11) and (12), proving that the frequency band of the bandgap is the Bragg bandgap.

Point C in Fig. 3(c) is the torsional resonance mode of the stellate ABH scatterer, which can be neglected because the band corresponding to point C is a flatband and there is no large perturbation at the corresponding position in the transmission loss spectrum. Point D is the upper boundary of the second Bragg bandgap, which is the internal resonant mode of silicone rubber, corresponding to the flat motion in the z-direction, and the corresponding perturbation is larger here as can be seen from the transmission loss spectrum in Fig. 3(c).

## IV. BANDGAP ANALYSIS

To further reveal the structural characteristics of the acoustic cloister, the parameters of the material, and dimensions (length of silicone rubber, width, and width of the stellate ABH crystal angle) of the structure are varied to investigate the effect of these parameters on the bandgap. The results are shown in Fig. 4.

We analyze the effects of the variation of the dimensional parameters and Young's modulus of the silicone rubber on the bandgap in Figs. 4(a)–4(c), respectively. From Fig. 4(a) and Eq. (12), it can be seen that since the length *l* of the silicone rubber is related to the lattice constant *a* of the acoustic cloister, thus, when *l* increases, *a* also increases and the corresponding Bragg scattering angle *θ* decreases, and, therefore, the onset frequency of the first complete bandgap of the structure decreases, proving that the first bandgap of the stellate ABH is generated by the Bragg scattering.

In Figs. 4(b) and 4(c), it can be seen from Eq. (11) that decreasing the value of both the width *h* and Young's modulus *E* of the silicone rubber leads to shorter wavelengths, however, in Fig. 4(b), the change in this parameter has a small effect on the bandgap width and onset frequency due to the small change in the width *h* of the structure. In contrast, in Fig. 4(c), the variation of Young's modulus *E* is significant; hence, the effect on the bandgap width and onset frequency is also significant.

In addition, from Figs. 3(a) and 3(b), the cutoff frequency of the first complete bandgap is caused by the torsional resonance of the stellate ABH scatterer and is not related to the silicone rubber, so in Figs. 4(a) and 4(b), the change of size of the silicone rubber has almost no effect on the bandgap cutoff frequency of the acoustic cloister. However, in Fig. 4(c), the significant decrease in Young's modulus *E* of the silicone rubber causes the resonant frequency of the silicone rubber to decrease as well. Therefore, when the resonant frequency of the silicone rubber is lower than that of the stellate ABH scatterer, the change in Young's modulus *E* of the silicone rubber will affect the cutoff frequency of the first complete bandgap. So, when Young's modulus *E* is 7.7 × 10^{4} Pa, the first complete bandgap of the structure can be generated in the range of 3.83–82.5 Hz, and its bandgap width is reduced to 78.67 Hz, and its energy band structure is shown in Fig. 1 in supplementary material.

Figures 4(d)–4(f) show the effect of the variation of the dimensional parameters and material of the stellate ABH scatterer on the bandgap, respectively. The mechanism of the effect of the variation of the parameters on the bandgap is similar to the explanation in Figs. 4(a)–4(c), which are due to the fact that the variation of the parameters in them is too small to have an effect on the Bragg scattering angle *θ*.

## V. ROBUSTNESS COMPARISON

Robustness refers to the ability of the structure to maintain its bandgap characteristics when the dimensional parameters of the structure are changed. From Figs. 4(a)–4(f) and Eqs. (11) and (12), it can be seen that the onset frequency of the first complete bandgap of the acoustic cloister is mainly related to the length and Young's modulus *E* of the silicone rubber. Therefore, the structure will have superior robustness at low frequencies.

To verify this, we changed the geometry of the stellate ABH scatterers at the ends of the acoustic cloister structure and replaced them with square-block scatterers, which have the same mass and structural parameters as the stellate ABH scatterers, while the silicone rubber in the middle remained unchanged. The structure is shown in Fig. 5(a). The obtained results include the width of the square scatterer *b'* = 47.636 mm and the lattice constant *a*′ = 117.27 mm for the block-square column acoustic cloister.

We then replaced the middle square-column silicone rubber with a cylindrical silicone rubber with the same mass, length, and material parameters. Thus, the diameter of this cylindrical silicone rubber is *D* = 9.027 mm and the lattice constant is *a*′ = 117.27 mm for the block-cylinder acoustic cloister.

The bandgap calculations, modal analysis, and transmission loss spectra of 1 and 2 periods can be performed for the two acoustic cloister structures after the change, and the results are shown in Figs. 5(b)–5(e).

Comparing Figs. 5(b) and 5(c) with Fig. 3(b), it can be seen that the onset frequencies of the first bandgap of the three acoustic cloister structures are basically the same, and the vibration patterns of the eigenmodes at the first bandgap of all three acoustic cloister structures correspond exactly. In Figs. 5(d) and 5(e), the transmission loss spectra of the three acoustic cloister structures of 1 and 2 periods are almost identical, which proves that the structures have superior robust performance in the low and medium frequency bands. At high frequencies, since the cutoff frequency of the first bandgap of the acoustic cloister structure is mainly related to the scatterers at both ends, the cutoff frequencies of the first bandgaps of the two deformed structures will deviate significantly from those of the first bandgaps of the acoustic cloister structures.

## VI. EXPERIMENT

In order to study the nonlinear bandgap of the acoustic cloister appearing in the ultra-low frequency band, the structural parameters and material parameters in Tables I and II have been used to fabricate the acoustic cloister structure shown in Fig. 6(a). The axial vibration experiment and torsional vibration experiment have been carried out, and the schematic diagram of the vibration test is shown in Fig. 6(b). As shown in Fig. 6(c), an elastic rope is used to lift the acoustic cloister horizontally to simulate the state under the free boundary. The input displacement transducer is bolted to the shaker, and the output one is glued to the other end of the acoustic cloister structure. A PC controls the shaker, which applies a simple harmonic vibration with amplitude *A* = 0.5 mm and displacement equation *u*(*t*) = *A *sin(*ωt*) to the x and z points of the acoustic cloister structure [see Fig. 6(b)], with frequency set to 3–100 Hz and sampling time of 100 ms.

As shown in Figs. 6(d) and 6(e), the experimental results are in general agreement with the corresponding simulation results when subjected to axial excitation from 28 to 100 Hz [Fig. 6(d)], and torsional excitation from 3 to 100 Hz [Fig. 6(e)]. However, the large wave energy attenuation at 3–22 and 25–28 Hz in Fig. 6(d) is due to the nonlinear buckling phenomenon of the silicone rubber.

If the compression bar is ideal, the axial load *P* on the bar will not buckle until the critical load *P*_{cr} is reached, see Fig. 6(f). But, due to the inhomogeneity of the material inside the compression bar and the presence of initial geometrical defects, the compression bar flexes laterally even at an axial load *P* < *P*_{cr}.^{37}

*u*(

*t*) =

*A*sin(

*ωt*). The vibration equation of the flexural beam can be derived according to Hamilton's principle as

^{38}

*w*(

*x*,

*t*) is the disturbance of the beam at position

*x*and moment

*t. P*is the axial force.

*η*is the damping coefficient.

*S*is the cross-sectional area of the beam.

Substituting Eq. (21) into Eq. (20) gives the equation for the vibration of silicone rubber after buckling.

From Eq. (20), it can be seen that in the axial excitation experiment, the nonlinear buckling phenomenon occurs due to the axial pressure on the silicone rubber, which makes part of the internal energy transformed into the nonlinear recovery force after buckling, thus leading to the nonlinear bandgap at 3–22 and 25–28 Hz in Fig. 6(d). Since the buckling phenomenon occurs only in the low frequency band, the lateral perturbation of the silicone rubber will gradually decrease with the increase of the excitation frequency and, finally, stabilize after 28 Hz.

In the torsional experiment of the acoustic cloister structure, the silicone rubber was not subjected to axial pressure, so the experimental results in Fig. 6(e) are basically consistent with the simulation results.

To further investigate the nonlinear flexural bandgap from 3 to 22 Hz in Fig. 6(d), the axial vibration in the first-order flexural mode of the stellate ABH has been simulated using the finite element method, as shown in Fig. 6(g). We assume that the silicone rubber is ideal, and for an ideal compression bar with solid support at both ends, *l _{e}* = 0.5

*l*is the equivalent length in the first-order flexural mode,

^{37}i.e., the bending moment at both end points is equal to 0.

Therefore, we apply a resonant excitation on the x boundary of the acoustic cloister structure along with a specified displacement *w* on area *S* = *h ** *l _{e}* at

*l*in Fig. 6(g) to simulate the first-order flexural mode of the silicone rubber.

_{e}*h*=

*8 mm is the thickness of the silicone rubber. The simulated results correspond exactly to the experimental results of 3–28 Hz in Fig. 6(d) when the equivalent stiffness of the silicone rubber is*

*E*= 0.818 × 10

_{e}^{3}Pa, the equivalent length under nonlinear flexion is

*l*= 9 mm, and the deflection is

_{e}*w*= 6 mm after the nonlinear flexure is measured by the simulation, as shown in Fig. 6(h).

It can be seen that the axial nonlinear buckling phenomenon occurring in the experiment can open the ultra-low frequency bandgap and realize the ultra-low frequency ultra-wide bandgap, which provides a new idea for the ultra-low frequency ultra-wide band vibration and noise reduction technology.

## VII. CONCLUSION AND OUTLOOK

We proposed the acoustic cloister, which enables the perfect ABH effect, i.e., the generation of an ultra-wide bandgap at ultra-low frequencies, breaking the limit of the cutoff frequency of the ABH. The soft material “cloister” at the tip of the acoustic cloister can focus the wave energy so that the wave into the soft structure, due to the sudden change in stiffness at both ends, the wave can only be reflected and scattered back and forth in the “cloister” and cannot propagate out. We demonstrated both theoretically and experimentally that the acoustic cloister can produce ultra-low frequency and ultra-wide Bragg bandgap and nonlinear bandgap at ultra-low frequencies, which reduces the wave transmission by 20–40 dB, and the energy band structure and characteristics of the Bragg bandgap are consistent with the theoretical results. In addition, we demonstrate the superior robustness of the structure by theoretical and finite element methods, which perfectly overcomes the two major limitations of the conventional ABH structure of poor robustness and difficulty in obtaining the ABH effect in the ultra-low frequency range. We also explain the cause of the experimental 3–22 and 24–28 Hz ultra-low frequency bandgap due to the nonlinear buckling phenomenon, which reduces the wave transmission by 20–40 dB and shows that the application of appropriate bending stress can generate a new ultra-low frequency ultra-wide bandgap.

In addition, by studying the effect of the parameters of the “cloister” structure on the Bragg bandgap, we can also adjust Young's modulus *E* of the soft material to achieve an ultra-low frequency ultra-broadband bandgap. As shown in Fig. 1 in supplementary material, we use a soft material with Young's modulus of 7.7 × 10^{4}, the Bragg bandgap can reach 3.83–82.5 Hz, but the first bandgap will be relatively narrow, which can be seen in Fig. 4(c).

If a wider bandgap is desired, the one-dimensional acoustic cloister can be extended to three dimensions, as shown in Figs. 2(a)–2(c) in supplementary material. A 2D acoustic cloister phononic crystal can produce the first bandgap in the range of 40.16–247.46 Hz, and a 3D acoustic cloister phononic crystal can produce the first bandgap in the range of 42.25–810.9 Hz, as shown in Figs. 2(d) and 2(e) in supplementary material. As can be seen from Fig. 2(f) in supplementary material, the vibration modes of the 2D or 3D at the upper and lower boundaries of the first bandgap are exactly the same as those at points A to D in Fig. 3(b). While in the 3D acoustic cloister phononic crystal, the z-direction is also constrained by the periodic boundary condition, making the degrees of freedom of the stellate ABH scatterers restricted and becoming less susceptible to resonance. Therefore, the end frequency Point I of the first bandgap becomes a resonant mode in the z-direction of silicone rubber, which slightly increases the onset frequency of the first bandgap, but greatly widens the bandgap.

The proposed acoustic cloister that generates perfect ABH effects will facilitate engineering applications of ABH, as well as open new ideas for vibration and noise control, energy harvesting, and elastic wave manipulation.

## SUPPLEMENTARY MATERIAL

By studying the effect of the parameters of the “cloister” structure on the Bragg bandgap, we can also adjust Young's modulus E of the soft material to achieve an ultra-low frequency ultra-broadband bandgap. As shown in Fig. 1, we use a soft material with Young's modulus of 7.7 × 104, the Bragg bandgap can reach 3.83–82.5 Hz, but the first bandgap will be relatively narrow, which can be seen in Fig. 4(c). If a wider bandgap is desired, the one-dimensional acoustic cloister can be extended to three dimensions, as shown in Figs. 2(a)2(c). A 2D acoustic cloister phononic crystal can produce the first bandgap in the range of 40.16–247.46 Hz, and a 3D acoustic cloister phononic crystal can produce the first bandgap in the range of 42.25–810.9 Hz, as shown in Figs. 2(d) and 2(e). As can be seen from Fig. 2(f), the vibration modes of the 2D or 3D at the upper and lower boundaries of the first bandgap are exactly the same as those at points A to D in Fig. 3(b). While in the 3D acoustic cloister phononic crystal, the z-direction is also constrained by the periodic boundary condition, making the degrees of freedom of the stellate ABH scatterers restricted and becoming less susceptible to resonance. Therefore, the end frequency Point I of the first bandgap becomes a resonant mode in the z-direction of silicone rubber, which slightly increases the onset frequency of the first bandgap, but greatly widens the bandgap.

## ACKNOWLEDGMENTS

This study was supported by the National Science Foundation of China (NSFC) under Grant No. 12104386, the Guangdong Basic and Applied Basic Research Fund Regional Joint Fund Youth Fund Project under Grant No. 2019A1515111118, the Science and Technology Innovation Program of Hunan Province under Grant No. 2021RC2097, China Postdoctoral Science Foundation under Grant No. 2022M711354, and Natural Science Foundation of Hunan Province Youth Project under Grant No. 2022JJ40421.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Xiao Liang:** Conceptualization (equal); Funding acquisition (equal); Writing – original draft (equal); Writing – review & editing (equal). **Daxiang Meng:** Data curation (equal); Writing – original draft (equal). **Zhen Yang:** Formal analysis (equal); Investigation (equal). **Jiaming Chu:** Methodology (equal); Project administration (equal). **Haofeng Liang:** Project administration (equal); Resources (equal). **Zhi Zhang:** Software (equal); Supervision (equal). **Jiangxia Luo:** Validation (equal). **Zhuo Zhou:** Visualization (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

## REFERENCES

*Stability of Structures*