1 Introduction

Artificial metamaterials [1,2,3,4] composed of periodically assembled microstructure units can produce important performance that natural honeycomb materials do not possess, such as auxetic compressibility [5, 6], negative effective modulus [7, 8], negative refraction [9, 10], directional reflection [11], and acoustic cloaking [12]. The above unusual achievements are mainly derived from the ingenious topology design of periodic architected unit cells, rather than the material properties. Therefore, topological optimization of microstructures according to different engineering requirements is the basis for the successful application of metamaterials [13,14,15,16].

The vibration and noise reduction characteristics of metamaterials have been the focus of scholars in recent years, especially the low-frequency wave attenuation performance [17, 18]. Researchers have conducted specific studies on the wave attenuation properties of acoustic metamaterials and have achieved a series of results. Liu et al. [19] designed two-dimensional and three-dimensional phononic crystals based on the basic principles and the Mie scattering effect of matrix and scattering, breaking through the limitations of Bragg scattering theory and realizing the attenuation of low-frequency waves by small-scale units. Inspired by related studies, Xu et al. [20] transformed the traditional hierarchical hexagonal honeycomb and filled the resonator to produce hierarchical metamaterials containing local resonators. They established a mass-spring system model and obtained the wave dispersion diagrams of the microstructure from the perspective of numerical analysis. Their research proved that the proposed metamaterial has low-frequency wave attenuation properties numerically and experimentally. Wang et al. [21] proposed a novel quasi-zero-stiffness (QZS) metamaterial and explored its vibration suppression bands. The basic units of QZS metamaterial consist of two parts: negative stiffness element and positive stiffness element, and QZS was modeled as a lumped-mass-spring chain.

Auxetic lattice materials with advanced mechanical and physical properties are a kind of important metamaterial, which have attracted the interest of researchers due to their application potentials and remarkable stiffness in smart components [22,23,24,25,26]. Zhang and Liu et al. [27, 28] studied the band gap and wave regulation performance of chiral substructures and successfully generated multiple band gaps in the low-frequency range by adding mass inclusions into the structures. The frequency range of band gaps can also be controlled by changing parameters such as angle and length of the ligaments. Attard et al. [29] invented a novel hexachiral metastructure composed of single-phase materials and tested the behavior of this new class of chiral structure using numerical and experimental techniques. The result indicates that this chiral structure exhibits a unique auxetic deformation mechanism. Qi et al. [30] combined the reentrant structures with anti-chiral structures and filled them with masses wrapped in rubber coating to achieve elastic wave attenuation. It was suggested numerically that the geometric topology and elastic wrapped mass inclusion distribution schemes have obvious effects on the band gaps of the metamaterials. He et al. [31,32,33] presented a shape optimization and single-phase chiral elastic metamaterials (EMMs)-based effective method for designing the EMMs with broadband double negativity. Numerical simulations related to double negativity including negative refraction and imaging were conducted using the optimized model. In addition, an uncertainty model based on the change of variable perturbation stochastic finite element method was formulated to study the types of physical responses including the band structure, mode shapes and frequency response function of acoustic metamaterials. Kumar and Pal [34] proposed a type of hybrid auxetic honeycomb metamaterial with divergent-shaped star units, which gave rise to low-frequency and wide band gaps.

Although a large number of artificial auxetic honeycomb metamaterials have been manufactured, most previous studies used elastomeric coating and resonating mass embedded in periodically arranged cells. However, the low-frequency band gaps of these elements depend on local resonances. Obtaining resonance in local resonance metamaterials (LRMs) essentially requires multiphase materials. Therefore, LRMs possess complex and bulky structures, and manufacturing involves complexity, which limits its practical application. We urgently need to produce single-phase honeycomb metamaterials with simple architected microstructures, lightweight, and excellent vibration isolation performance [35, 36]. In this paper, we introduced two types of elastic ligaments, connected them with rigid nodes and then constructed auxetic hexastarchiral metamaterials through the sixfold rotational symmetry of the synthetic ligaments. Through the FE method, band gap properties of the hexastarchiral metamaterials, effects of geometric parameters of periodic unit cells, and vibration isolation characteristics of finite sandwich structures were systematically explored. Compared with the previous acoustic metamaterials, the 2D hexastarchiral metamaterials have the advantages of simple structure and excellent structural integrity and need no multi-component installation. Previous studies [29] have shown that there are regions in the parameter space where Poisson’s ratio is quite insensitive to the changes of parameters. This not only makes Poisson’s ratio and manufacturing defects elastic, but also reduces the quality while maintaining a certain Poisson’s ratio. Moreover, hexastarchiral metamaterials can generate omnidirectional band gaps in multiple frequency ranges, and the evolution of band gaps can be regulated by the geometric parameters of microstructure to meet different engineering needs. Research of this work provides an innovative method for engineering applications of the auxetic honeycomb metamaterials in sound and vibration control.

2 The Dynamic Model and Numerical Simulation Method of Hexastarchiral Lattice

2.1 Mechanical Design of the Hexastarchiral Metamaterials

In this study, we introduced a periodic artificial metastructure assembled by pinwheel-shaped chiral unit cells. The basic unit of the structure is mainly composed of two parts: the star ligaments (SLs) and the connecting ligaments (CLs). As shown in Fig. 1, the length and width of the SLs are \(L_{\mathrm {s}}\) and \(t_{\mathrm {s}}\), respectively, and the length and width of the CLs are \(L_{\mathrm {c}}\) and \(t_{\mathrm {c}}\), respectively. The distance between the central nodes is considered R. In addition, \(\beta \) and \(\theta \) are introduced to represent the angle between two SLs and the angle between the SLs and the CLs, respectively.

Fig. 1
figure 1

a Hexachiral lattice and b geometry of unit cell of hexachiral lattice

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Fig. 2
figure 2

a Representative sections of hexastarchiral lattice and Cartesian basis and b Brillouin zone and the reciprocal basis

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By applying Bloch’s theorem to the edge of the unit cell, we can investigate characteristics of wave propagation in the whole periodic lattice. Any nodes of the lattice structure can be regarded as lattice points, which can be defined by a set of basic vectors. The whole hexastarchiral lattice can be obtained by defining a unit cell on the sites determined by all linear combination of the lattice vectors \(n_{{1}}{{\varvec{e}}}_{{1}}+n_{{2}}{{\varvec{e}}}_{{2}}\), where \(n_{{1}}\) and \(n_{{2}}\) are integers. The basic lattice vectors e\(_{{i}}\) can be written by the orthogonal Cartesian vector basis and constants as:

$$\begin{aligned} {\varvec{e}}_{1}= & {} (\sqrt{3} \text {i}+\text {j})R/2 \nonumber \\ {\varvec{e}}_{2}= & {} (-\sqrt{3} \text {i}+\text {j})R/2 \end{aligned}$$
(1)

The position of point P in cell (\(n_{{1}}\), \(n_{{2}})\) can be expressed as:

$$\begin{aligned} {\varvec{r}}_{\mathrm{p}} =(n_{1} ,n_{2} )={\varvec{\hat{{r}}}}_{\mathrm{p}} +n_{{1}} {\varvec{e}}_{1} +n_{2} {\varvec{e}}_{{2}} \end{aligned}$$
(2)

where \({\hat{\varvec{r}}}_{\mathrm{p}}\) is the position of the corresponding point in the reference cell. As shown in Fig. 2, the reciprocal lattice is identified by the reciprocal lattice vector (\({{\varvec{e}}}_{{1}}^{{*}}\), \({{\varvec{e}}}_{{2}}^{{*}})\), and its relationship with the basic lattice vector can be expressed as follows:

$$\begin{aligned} {\varvec{e}}_{{i}} \cdot {\varvec{e}}_{{j}}^{{*}} =2\pi \delta _{{ij}} \end{aligned}$$
(3)

where \(\delta _{{ij}}\) is the Kronecker increment. The reciprocal lattice vector of the hexastarchiral metamaterials can be expressed as:

$$\begin{aligned} {\varvec{e}}_{\text {1}}^{{*}}= & {} \frac{\text {2}\pi }{R}\left( \frac{\text {i}}{\sqrt{3} }+\text {j}\right) \nonumber \\ {{\varvec{e}}}_{\text {2}}^{{*}}= & {} \frac{\text {2}\pi }{R}\left( -\frac{\text {i}}{\sqrt{3} }+\text {j}\right) \end{aligned}$$
(4)

2.2 Finite Element Procedure of Free Wave Propagation of the Hexastarchiral Metamaterials

Bloch’s theorem has been utilized for the finite element analysis of wave propagation of the proposed metamaterials [34]. The displacement field u at location P on the periodic lattice with frequency \(\omega \) can be presented as:

$$\begin{aligned} {\varvec{u}}({\hat{\mathbf {r}}}_\mathrm{p})=u_{\text {p}} \text {e}^{\mathrm{i}(\omega t-\kappa \cdot ({\hat{\mathbf {r}}}_\mathrm{p}) )} \end{aligned}$$
(5)

where i denotes the imaginary number, \(\varvec{\kappa }\) is the wave vector, \(\omega \) indicates the angular frequency, and \(u_{\mathrm {p}}\) is the magnitude of the lattice periodic displacement field.

Table 1 The irreducible Brillouin zone points of the lattice
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In the present work, as described in Eq. (5), due to the periodicity of hexastarchiral lattice, the characteristic of wave propagation can be analyzed in a representative unit cell. Correspondingly, the irreducible Brillouin zone, marked by the black area, is presented in Fig. 2b, and the dispersion relations are obtained by sweeping the wave vector along the edges of irreducible Brillouin zone. Besides, Table 1 shows the reference points used to describe the coordinates of the Brillouin zone.

With the application of finite element method, the eigenvalue equations in the unit cell can be expressed as:

$$\begin{aligned} ({\varvec{K}}-\omega ^{2}{{\varvec{M}}}){\varvec{u}}={\varvec{f}} \end{aligned}$$
(6)

where K and M are the global stiffness and mass matrices of the unit cell, respectively; and u and f are the displacement vectors and force vectors, respectively.

3 Numerical Simulation

The commercial finite element software COMSOL Multiphysics5.5 is utilized to study the band gap structure, the mode shape effect of geometric parameters on band gaps and vibration transmissions of the hexastarchiral structures. In addition, in this study, we only consider the fluctuations of plane wave in the x–y plane. The periodic lattice is made of Visijet M3 \(\hbox {Crystal}^{\mathrm {TM}}\) (MJP), which is a kind of 3D printable material. The reference configuration of materials, including geometric parameters and material parameters, is shown in Table 2.

Table 2 Geometric and material parameters of the hexastarchiral lattice
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Table 3 Geometric parameters of different microstructures
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3.1 Band Structure

The band structure of hexastarchiral honeycomb metamaterials is drawn in Fig. 3. One characteristic of the dispersion curves of hexastarchiral honeycomb is the repulsion of dispersion branches (veering of frequencies), which can be observed in the band structure. As shown in the detailed schematic diagram, the ninth and tenth dispersion curves and the fifteenth and sixteenth dispersion curves do not cross each other in the veering zone, but deviate from each other. This phenomenon of dispersion repulsion that often occurs in weak coupling systems is called mode veering [37,38,39]. When the mode veering turns, the wave mode shapes will change rapidly near the critical frequency, and the two dispersion curves will suddenly deviate from each other. However, after the sudden change in slope, each dispersion curve continues to follow the previous path. Obviously, this mode veering phenomenon is closely related to the generation and disappearance of band gaps.

Fig. 3
figure 3

Band structure of the hexastarchiral metamaterials No. 1

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As shown in Fig. 3, five omnidirectional band gaps are located at [684.8, 786.3] Hz, [2478, 3015.3] Hz, [3634.8, 4801.7] Hz, [5828.1, 7699.1] Hz, and [8650.5–11242] Hz, indicating that the proposed metastructures have excellent plane wave attenuation effects in multiple frequency ranges. Furthermore, the mode shapes are conducted to further reveal the formation mechanism of band gaps. The mode shapes corresponding to Points O, A, and B on the lower boundary of the first band gap (marked with green circles in Fig. 3) are plotted in Fig. 4a–c. We can conclude that the SLs remain stationary, while the bending deformations of the CLs suppress the degeneration of the energy band caused by the rotational symmetry of the metastructures, thereby opening the first band gap. Similar mode shapes characteristics are found at Points O, A, and B on the lower boundary of the second band gap (marked with yellow circles in Fig. 3), which are plotted in Fig. 4d–f. However, the first and second band gaps have different positions of bending deformation. The bending deformation of the former is mainly concentrated at the end of the CLs, and the bending deformation of the second band gap occurs in the middle of the CLs. The mode shapes corresponding to Points O, A, and B on the lower boundary of the third band gap are shown in Fig. 4g–i. The overall rotational resonance characteristics of the unit are reflected by the coordinated bending deformations of CLs and SLs. On the other hand, the deformation mechanism of high-frequency band gaps, i.e., the four band gaps where the black circles locate, is the same as that of the third band gap. Obviously, it can be concluded that the width and position of the band gaps are related to the position and form of ligament bending when the resonance of the structure occurs.

Fig. 4
figure 4

Mode shapes of the hexastarchiral structures: a–c represent Points O, A, and B on the lower boundary of the first band gap; d–f represent Points O, A, and B on the lower boundary of the second band gap; g–i represent Points O, A, and B on the lower boundary of the third band gap; and j–l represent Points O, A, and B on the lower boundary of the fourth band gap. The red in the picture represents large deformation, and the blue and green represent small deformation

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3.2 Effects of Microstructure on Band Structure

Previous studies have proved that the geometric parameters of metamaterials have significant effects on the evolution of band structure. In this section, different geometric parameters of hexastarchiral structures are determined as design parameters to explore their influences on the band gap structure. The specific geometric parameters of different microstructures are listed in Table 3.

Fig. 5
figure 5

Band structures of the auxetic hexastarchiral metamaterials a No. 2, b No. 3, c No. 4, d No. 5, e No. 6, and f No. 7

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3.2.1 Effect of \({t}_{{{\varvec{s}}}}\) on the Band Structure of Hexastarchiral Metamaterials

Figure 5a, b shows the band structures of two hexachiral metamaterials with different ligament thicknesses \({t}_{\mathrm {s}}\). By comparing with Fig. 3, it can be found that the band structure shows high sensitivity to changes in ligament thickness.

Fig. 6
figure 6

Dependence of the cutoff frequency of the band gaps on \(t_{\mathrm{s}}\)

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Fig. 7
figure 7

Vibration modes at the marked points C and D in Fig. 3

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The dependence of band structure on \(t_{\mathrm {s}}\) is shown in Fig. 6. When \(t_{\mathrm {s}}\) varies from 1 to 2.6 mm, five omnidirectional band gaps between [0, 14,000] Hz are observed. With the increase of \(t_{\mathrm {s}}\), the upper and lower boundaries of the second, third, and fourth band gaps develop upward, and the frequencies of the upper boundaries increase faster than those of the lower boundaries. Thus, the widths of the band gaps decrease gradually. The second band gap is closed when parameter \(t_{\mathrm {s}}= 1.9\) mm, and the third band gap disappears when the thickness is 2.5 mm. Meanwhile, there are only two complete band gaps in frequency range [0, 10,000] Hz. Obviously, the initial frequency and cutoff frequency of the band gap increase with \(t_{\mathrm{s}}\). In addition, the width of the first band gap is almost invariant.

It can be found that the change of \(t_{\mathrm {s}}\) leads to mode veering between the ninth and tenth dispersion curves to prevent the generation of band gap. We use points M and N in Fig. 5c, d to represent the mode points, respectively. In order to further identify the relationship between mode points and the vibration forms, we draw the mode shapes of Points C and D in Fig. 3, as shown in Fig. 7. We notice that the mode shapes of higher frequencies (C2 or D2) can be obtained by rotating the mode shapes of lower frequencies (C1 or D1) by \(60^{\circ }\) .

3.2.2 Effect of \({t}_{{{\varvec{{c}}}}}\) on the Band Structure of Hexastarchiral Metamaterials

Figure 8 shows the relationship between band gaps (red areas) and the parameter \(t_{\mathrm {c}}\). In addition, the effective band gap width (dashed line) is drawn in the same figure to prove the band gap characteristics of hexastarchiral metamaterial cells. The effective band gap width is extracted by Eq. (7), where \(\omega ^{i}_{\mathrm {u}}\) and \(\omega ^{i}_{\mathrm {l}}\) are the upper and lower frequencies of the i-th band gap, respectively. This phenomenon is similar to the effect of \(t_{\mathrm {s}}\) on band gaps. Interestingly, when the parameter \(t_{\mathrm{c}}\) increases, the width of the first band gap is almost constant, but the width of the second band gap gradually narrows and eventually disappears. As the third band gap moves toward high frequencies, the width continues to increase. The maximum width of the fourth band gap appears when \(t_{\mathrm {c}}=1.4\) mm. In addition, the effective width of the band gap is greatly reduced as \(t_{\mathrm {c}}\) increases.

$$\begin{aligned} f(\Delta \omega )=\sum _{i={1}}^{n} (\omega _{\text {u}}^{i} -\omega _{\mathrm{l}}^{i}) \end{aligned}$$
(7)
Fig. 8
figure 8

Effective band gap width and variation of band gaps vs. \(t_{\mathrm {c}}\)

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3.2.3 Effect of \(\theta \) on the Band Structure of Hexastarchiral Metamaterials

The band structures of metamaterial unit cells with \(\theta = 80^{\circ }\) and \(120^{\circ }\) are shown in Fig. 5e, f, respectively. It can be seen that there are five complete band gaps in the calculated frequency range, which demonstrates that the evolution of \(\theta \) does not change the number of band gaps. The main reason for this is that the value of \(\theta \) has nothing to do with the stiffness of the structures. Furthermore, the variation of the first band gap is usually of concern to scholars. As shown in Fig. 9, it can be found that the frequencies of upper boundary and lower boundary decrease with the increase of the angle. The frequency of upper boundary decreases faster than that of lower boundary, which causes the width of the first band gap to gradually increase.

Fig. 9
figure 9

Dependence of cutoff and starting frequencies of the first band gap on the angle between the SLs and the CLs

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The above analysis shows that the position and width of the band gaps are highly sensitive to the changes in ligament thickness, but the band gaps do not respond significantly to the development of angle between the SLs and the CLs. In addition, due to the existence of mode points, the vibration modes play a key role in the generation and disappearance of band gaps.

3.3 Vibration Isolation Properties of the Finite Hexastarchiral Lattice

We have investigated the wave propagation characteristics of hexastarchiral metamaterials in the previous studies. In this part, in order to verify the above analyses and study the vibration isolation performance of the finite structure, the frequency response diagram is drawn by calculating the steady-state dynamic response of the hexastarchiral core sandwich panel, which contains \(5\times 16\) basic units. As shown in Fig. 10, the excitation is enforced by applying a displacement in the horizontal direction on the boundary of the hexastarchiral core sandwich panel and collecting the response on the other side of the lattice. Except for the excitation point, other boundaries are free boundaries. The transmission coefficient X on the abscissa is a function of the dimensionless frequency on the ordinate, which can be expressed as:

$$\begin{aligned} X=\text {20lg}\left( \frac{X_{\text {rec}} }{X_{\text {exc}} }\right) \end{aligned}$$
(8)

where \(X_{\mathrm {exc}}\) is the displacement of the excitation point, and \(X_{\mathrm {rec}}\) is the displacement of the response collection point.

Fig. 10
figure 10

Schematic diagram of hexastarchiral lattice subjected to imposed displacement

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Fig. 11
figure 11

Frequency response diagrams vs. band structure of No. 1

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Fig. 12
figure 12

Vibration distribution of sandwich panel in a pass band at 2000 Hz and b band gap at 3000 Hz

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Figure 11a shows the frequency response function (FRF) of the finite structure, and we plot the band structure of the units in Fig. 11b as a comparison. It can be found from Fig. 11a that there are several frequency ranges with very low transmittance, which correspond to four frequency ranges of band gaps of the metamaterial, respectively. The position and width of the band gaps correspond well to the results in Fig. 11b, thus verifying the results of band structure simulation. On the other hand, it can be summarized from Fig. 12 that within the band gap region, the elastic wave only propagates in the first few periods, and then the wave propagation is completely isolated. In summary, the sandwich panel composed of hexastarchiral units has excellent vibration attenuation effect.

4 Conclusions

In this work, numerical simulation process based on the finite element method and Bloch’s theorem was used to calculate the band gap characteristics of hexastarchiral lattice metamaterials, the mode shapes, the relationship between geometric parameters and band gap evolution, and the transmission characteristics of the finite structure. We proved numerically that the proposed metamaterials could produce omnidirectional band gaps in multiple frequency ranges, and the evolution of the band gaps has a strong correlation with the geometric parameters of the microstructures. The systematic analysis of mode shapes revealed the mechanism of band gaps and the unusual behavior of dispersion curves. Our research results could provide a valuable reference for the design of simple wave filtering and vibration isolation materials.