## 1 Introduction

Phononic crystals (PnCs), whose material properties or lattice parameters distribute in a periodic way, are engineered periodic composite materials or structures with band gaps [1], in which the elastic wave propagation is prohibited [2]. In recent years, PnCs have become a research hotspot in the field of elastic and acoustic waves due to adjustable band structures and strong manipulability in attenuating mechanical waves at various frequencies [36]. Nevertheless, it still exists many challenges for the industrial applications of phononic crystals, one of which is that traditional PnCs tend to manipulate elastic waves only in a limited frequency range, which limits their engineering applications to a great extent [7,8].

Phononic crystals include Bragg scattering PnCs [9,10] and locally resonant phononic crystals [11] based on the mechanism for the creation of band gaps. The wavelength of the elastic wave in the Bragg scattering band gap is about the same order of magnitude as the periodic lattice size, so the low-frequency band gap is usually realized at the cost of a very large size [12]. To solve this problem, many approaches have been proposed. For example, it is found that using negative stiffness in a unit cell leads to decreasing the frequency range of band gaps [13]. Besides, Bragg scattering phononic crystals have good broadband performance compared to locally resonant phononic crystals [14]. It is expected that the Bragg scattering phonon crystals can achieve the ultra-broadband vibration isolation and noise reduction for engineering applications.

According to previous studies, the simplification of some fluctuation modes at high symmetry points in the Brillouin region can be eliminated by reducing the structural symmetry, thus opening new band gaps, or shifting the band gaps toward the low-frequency region [3739]. With graded supercell configuration, the Bragg scattering phononic crystal can retain most of the band gaps of the nongraded structures to broaden the band gaps well [40]. With a wider original bandwidth, it is expected that Bragg scattering phononic crystals can achieve an extreme broad band gap easier, so as to satisfy the demands of the broadband vibration reduction in engineering applications.

## 2 Models and Methods

Folded beam phononic crystals have been shown to have a low broadband gap [41]. In this article, we take a folded beam-type phononic crystal as an example and further broaden its band gap by introducing a graded structural configuration.

In Fig. 1, the graded folded beam phononic crystals model is shown. The two-dimensional graded structure is shown in Fig. 1(a). It consists of supercells arranged periodically in the x-direction and z-direction. The grades are introduced through the increment of the thickness hi of beams in the z-direction. Different unit cells own different thicknesses of beams. The graded value is defined as follows: dh = hi+1hi. The graded order g refers to the number of cells in a supercell. Note that the grades only work in the z-direction, so only the wave conduction in the z-direction is considered this article. A supercell consists of several unit cells. The unit cell model is shown in Fig. 1(b). Each unit cell is a symmetrical folding beam structure. As shown in Fig. 1(c), the axisymmetric model is taken to introduce the composition of each unit cell. The part marked by dotted lines is the axisymmetric model of the nth unit cell. It is composed of two beams I and II with the same thickness, two mass blocks M1 connecting the adjacent unit cells in the center, and two mass blocks M2 connecting the beams on the left and right. The distance between the central layers of the two beams is kept constant, which is denoted as s. The lattice constant of the unit cell is a0. The thickness of the beam is hi.

Fig. 1
Fig. 1
Close modal

Before proceeding to the solution, we make the following assumptions:

1. The material used in the structure is homogeneous, isotropic, and linearly elastic.

2. The mechanic model is simplified as a two-dimensional plane strain problem.

3. The mass blocks M1 and M2 are assumed to be rigid bodies.

4. Because only the propagation of elastic wave in the z-direction is concerned, the degree-of-freedom in the z-direction of M1 and M2 is considered, while the x directional and rotational degree-of-freedom are ignored.

5. Considering the symmetry of the structure, beam I and II will only be excited in the symmetric mode, and its antisymmetric mode will be ignored.

The transfer matrix method is mostly used to calculate the dispersion relations and the frequency response function of the one-dimensional phononic crystal. In the process of applying the transfer matrix method, it is only necessary to assume the displacement field as a linear combination of trigonometric functions, hyperbolic trigonometric functions. Since the structure proposed in this article is left-right symmetric, only its left-right symmetric part is taken for calculation in the analysis.

The coordinate axes are shown in Fig. 1, and the direction of the beam axis is taken as x-axis, the direction perpendicular to x-axis in the working plane is taken as z-axis. The system of coupled differential equations for transverse vibration of the uniform Timoshenko beam without external force is given by
$−∂Q(x,t)∂x+ρS∂2u(x,t)∂t2=0−∂M(x,t)∂x+Q(x,t)−Iρ∂2θ(x,t)∂t2=0$
(1)
where I, ρ, and S are, respectively, the second moment of the section, the density, and the cross-sectional area of the beam. Q(x, t) = κGSγ(x, t), κ is the Timoshenko shear coefficient depending on the cross section of the beam [42,43], in rectangular section, κ = 5/6, and γ(x, t) is the shear angle.
According to the Fourier method, each function u(x, t) and θ(x, t) can be written as u(x, t) = U(x)T(t), θ(x, t) = Y(x)T(t). With such an assumption, for $ω<κGS/(Iρ)$, $Ujn(x)$ can be written as follows:
$Ujn(x)=Ajncosh(λ1x)+Bjnsinh(λ1x)+Cjncos(λ2x)+Djnsin(λ2x)$
(2)
where j = I, II, indicating different beams in the same unit cell, n denotes the nth unit cell. $λ12=(−d+Δ)/2$, $λ22=(d+Δ)/2$, d = ω2ρI(1 + E/)/EI, Δ = ω4ρ2I2(1 − E/)2 + 4EIω2ρS, $Ajn$, $Bjn$, $Cjn$, and $Djn$ are the coefficient to be determined.
Since we only study the one-dimensional vibration of the structure along the z-axis direction, we can assume that the mass block and the connecting block are rigid bodies with only translational motion and without rotation; thus, the rotation angle of the beam at x1 and x2 is described as follows:
${U′(x1)=0U′(x2)=0$
(3)
By substituting Eq. (2) into Eq. (3), we can obtain that
$[AjnBjn]T=[P][CjnDjn]T$
(4)
where P is the coefficient matrix associated with x1, x2, λ1, and λ2. The superscript T stands for matrix transpose.
Based on the compatibility condition of displacement and force, the continuity conditions between two adjacent cells can be written as follows:
${uIIn−1(x1,t)=u1n(x1,t)QIIn−1(x1,t)+QIn(x1,t)=−m1∂2uM1n(x1,t)∂t2$
(5)
where $ujn$ denotes the displacement, $uj′n$ denotes the rotation angle, and the z-directional displacement of the intermediate mass block $uM1n(x1,t)=uIn(x1,t)$.
By writing the aforementioned continuity conditions in the matrix form, we obtain
$K1φIIn−1=H1φIn$
(6)
where $φjn=[CjnDjn]T$.
Similarly, the continuity condition in a single cell is expressed as follows:
${uIIn(x2,t)=uIn(x2,t)QIIn(x2,t)+QIn(x2,t)=m2∂2uM2n(x2,t)∂t2$
(7)
where the z-directional displacement of the mass blocks on both sides $uM2n(x2,t)=uIIn(x2,t)$.
Writing the aforementioned continuity conditions in the matrix form, we obtain
$K2φIIn=H2φIn$
(8)
Substituting Eq. (6) with Eq. (8), then we can obtain the relationship of beam II in adjacent unit cells
$φIIn=TφIIn−1$
(9)
where $T=K2−1H2H1−1K1$. According to Bloch's theorem, $φIIn=eika0φIIn−1$, where k is the z-directional Bloch wave vector, and then the standard matrix eigenvalue problem can be obtained as follows:
$|T−eika0I|=0$
(10)

The dispersion relation between the wave vector k and the frequency f can be obtained by solving the eigenvalues of the matrix T [44].

The Timoshenko beam model considers the shearing force effect and rotary motion effect. When the length of the beam is more than five times greater than the height of the section, the shearing force effect and rotary motion effect can be ignored. Then the Euler—Bernoulli beam theory can be used to describe the model earlier. According to the classical Euler–Bernoulli beam theory, the bending vibrations of a beam are described by the following equation [45]:
$EI∂4u(x,t)∂x2+ρA∂2u(x,t)∂t2=0$
(11)

## 3 Results and Discussion

### 3.1 The Numerical Results of Bad Gap.

All the structures in this article are made of polylactic acid (PLA), and the specific material parameters are presented in Table 1. The geometric parameters are presented in Table 2.

Table 1

PLA material parameters

NameValueDescription
E3.13E9 PaYoung's modulus
ρ1240 kg/m3Density
ν0.18Poisson's ratio
NameValueDescription
E3.13E9 PaYoung's modulus
ρ1240 kg/m3Density
ν0.18Poisson's ratio
Table 2

Main structural parameters of the model

NameValue (mm)Description
a016Lattice constants
l140Width
s8Distance between the middle layers of the two crossbeams
oh2Initial thickness of beam
hioh + dh*iThickness of beam
b15Width of M2
b210Width of M1
di(a0mhi)/2Half height of M1
N1Number of supercells in z-direction
th153D model depth
l1l/2 − b2/2 − b1Length of beam
ldl1/hiLength-to-diameter ratio
NameValue (mm)Description
a016Lattice constants
l140Width
s8Distance between the middle layers of the two crossbeams
oh2Initial thickness of beam
hioh + dh*iThickness of beam
b15Width of M2
b210Width of M1
di(a0mhi)/2Half height of M1
N1Number of supercells in z-direction
th153D model depth
l1l/2 − b2/2 − b1Length of beam
ldl1/hiLength-to-diameter ratio

To verify the theoretical model and determine the applicable length-to-diameter ratio (ld) range for Euler—Bernoulli beam, the first ordered band gap of the nongraded PnCs of the following three models is calculated and compared: (i) the theoretical Euler—Bernoulli beam model; (ii) the theoretical Timoshenko model; and (iii) the finite element method (FEM) with the comsol multiphysics software, and the result is presented in Fig. 2. The result shows that the Timoshenko beam model fits well with the FEM method. When ld > 10, the Euler—Bernoulli beam can fit well, too. So in the case that ld < 10, it is better to use the Timoshenko beam model. Once ld > 10, the Euler—Bernoulli beam model can be used.

Fig. 2
Fig. 2
Close modal

To obtain band gaps with low frequency, the slender beam is selected here to discuss the influence of grades (ld = 30). Thus, the Euler—Bernoulli beam is used to do the following calculation. The band structure of the graded structure was calculated, and the results are plotted in Fig. 3. In Fig. 3(a), as a Bragg scattering phononic crystal, the first-order bandgap onset frequency of the original structure reaches to 289 Hz, which is comparable to the lowest frequency that can be reached in the bandgap of some local resonant phononic crystals [46]. In Figs. 3(a)3(c), there exists a wide passband between the first- and second-order bandgaps in the graded-free structure. Observing Fig. 3(d), it can be found that the graded structure can broaden the bandgap frequency range without increasing the starting frequency of the structural bandgap. The fifth-order graded structure has many flat bands in the frequency range of 225.5–3000 Hz, and in these flat bands, the elastic wave cannot propagate. So the band gap frequency range of the fifth-order graded structure is broadened to 225.5–3000 Hz, which reduces the band gap starting frequency while greatly broadening the bandgap frequency range of the structure without grades.

Fig. 3
Fig. 3
Close modal

Further, the frequency response function of the finite-period graded structure is calculated. A fifth-order graded two-period structure is considered, and its vibration transmission loss is calculated. In acoustic and vibration engineering, the amplitude–frequency response function is generally defined as follows: TL = 20log 10X/X0, where X and X0 are the displacements, velocities or accelerations at the output and excitation ends, respectively. The displacements at both ends are taken as X and X0. Given the free boundary conditions, the broadening of the band gap by the graded structure is investigated while ensuring that the geometric and material parameters of the finite-period graded structure are the same as those of the infinite-period graded structure. The frequency response curves of the fifth-order graded structure with an initial thickness of 2 mm and a grade of 0.5 mm are shown in Fig. 4. The blue ranges indicate the band gaps. These bandgap ranges are in consistent with those in the band structure of Fig. 3(d). It can be observed from the figure that the input vibration cannot propagate in several frequency ranges, and the broadening effect of the graded structure on the band gap is confirmed.

Fig. 4
Fig. 4
Close modal

### 3.2 Mechanism of Band Gaps Broadening.

To investigate the mechanism for the graded structure to broaden the band gap, we take the two-order graded structure with Euler—Bernoulli beam (ld = 30) as an example and performed calculations. The taken graded structure consists of two unit cells. They have the same material parameters but different beam thicknesses with lattice constants 2a0, and other parameters are the same as presented in Table 2. Figure 5 shows the band structure corresponding to the supercell structures with dh = 0, 0.1, 0.5, and 1 mm. The blue area is the part of the band gap, where the band gaps of the graded structure overlap with the graded-free structure, and the yellow area is the new band gap generated by the graded structure. The comparison of Fig. 5(a) with Figs. 5(b)5(d) reveals two features of the graded structure in the band gap broadening: (i) the retention of the band gap of the nongraded structure and (ii) the generation of new band gaps in the middle- and high-frequency ranges.

Fig. 5
Fig. 5
Close modal

To explain the mechanism for the broadening of band gap in the graded structure, we first analyze the mechanism for the formation of band gap in the original, nongraded structure. In the analysis of the dispersion curves, the propagation of the vertically incident elastic wave in the graded structure corresponds to its wave propagation along the z-direction of the first Brillouin zone. In Figs. 5(e)5(h), each mode corresponds to each ordinary bandgap boundary point. Looking at the mode diagram, it can be found that the lower boundary points of the band gap in Figs. 5(f) and 5(h) own an antisymmetric mode of beam I and II, while the upper boundary points of the band gap in Figs. 5(e) and 5(g) own an symmetric mode of beam I and II. In the vibration process, the blocks M1 and M2 vibrate as rigid bodies, which lead to the interference of the scattered transverse waves by the phase extinction of the structure in the process of elastic wave propagation. In simple terms, the blocks M1 and M2 have infinite bending stiffness compared to the beams. This also leads to a sudden change in bending stiffness experienced periodically during elastic wave propagation, resulting in a strong wave reflection that cancels out the incident wave, thus inhibiting wave propagation and forming a band gap.

The graded structure retains the band gap of the nongraded structure to a large extent. Observing the vibrational modes of the overlapping part of the nongraded structure and the band gap in Figs. 5(e)5(h), we can see that the vibrational modes corresponding to the onset frequency and the cutoff frequency of the band gap are the same. So it can be inferred that the band gap of the graded structure, whose frequency is overlapped with the nongraded structure, is a superposition of the Bragg bandgap in the graded-free structure.

Another feature of the graded structure is the generation of new band gaps in the middle- and high-frequency range. When dh = 0, the structure has mirror symmetry, so the supercell structure can degenerate into a unit cell phononic crystal with a lattice constant a0. At the boundary of the first Brillouin zone, k = π/a0 in Fig. 5, as shown in S1 and S2, there will be double degeneracy due to the symmetry of the structure, and the Dirac cone of the dispersion is created at the same time. This symmetry of the structure is reflected in the mode as local modal symmetry. Due to the periodicity of PnCs, there is no sequential distinction between the two unit cells in the modal diagram. Taking the modes in Figs. 5(k) and 5(l) as an example, the upper unit cell's mode of Fig. 5(k) and the lower cell's mode of (l) are symmetric modes, while the remaining other layer is antisymmetric.

To broaden the band gap as much as possible, many scholars have proposed methods to remove the simplicity in recent years, such as introducing material anisotropy [47] and changing the structure to reduce the symmetry of the structure [47,48]. When dh ≠ 0, as shown in Fig. 5(d), dh = 1 mm, the mirror symmetry of the structure is broken, the double degeneracy disappears, and the Dirac cone opens, resulting in a new bandgap. To some extent, the frequency of the original bandgap is also affected. For example, when dh increases from 0 to 0.1 mm, the frequency range of the second band gap increases from 1221.9–2385.6 Hz to 1259.3–2426.4 Hz. The original double degeneracy, such as the dispersion curve S1 and S2, gradually opens into a flat band, forming a new band gap in the passband range of Fig. 5(a). Thus, the introduction of the graded structure breaks the symmetry and generates a new band gap, broadening the band gap. At the same time, some low-order dispersion curves are shifted to the high-frequency region, thus reducing the bandgap frequency to a certain extent.

The aforementioned phenomena are similar to those observed in one-dimensional layered composites with hierarchical structures [49] and in two-dimensional solid–liquid systems with fractal structures [50]. The difference is that the graded structure proposed in this article is assembled by a simple variation of structural dimensions in one direction, whereas the hierarchical and fractal structures have differences in internal and external gradations and shapes.

Qualitatively, the introduction of the graded structure brings dimensional diversity in the super cell, and this diversity remains most of the bandgap corresponding to the unit cell nongraded structure without increasing the bandgap onset frequency, and so the overall bandwidth is broadened.

### 3.3 Analysis of Structural Parameters on Band Gap Broadening.

The effects of the main structural factors related to graded design on band gap broadening are studied in this section, which includes the length-to-diameter ratio ld and the design parameters of graded structure: the graded order g and the graded value dh. The graded supercell configuration with Euler—Bernoulli beam is taken as the example to analyze the influence of the aforementioned parameters on bandgap broadening.

#### 3.3.1 Structural Parameter ld.

Figure 6 shows the change of band gap with different parameters ld in the unit cell. The black part in the figure is the band gap, and the solid line and dotted line, respectively, represent the upper and lower boundaries of the band gap. Here, the band gap between the second and third band is defined as the first band gap, and the band gap between the fourth and fifth band is defined as the second band gap. With the increase of ld, the upper and lower boundaries of the band gap decrease gradually, the frequency range of the band gap decreases, and the width of the band gap also decreases. Because as ld increases, the stiffness of the beam decreases and the band gap frequency decreases.

Fig. 6
Fig. 6
Close modal

For graded structures, ld increasing in the arithmetic sequence has a great influence on the band gap broadening effect. Next, the influence of graded order g on the band gap broadening effect is discussed.

Fig. 7
Fig. 7
Close modal

Therefore, it can be concluded that the graded order has a step change in the band gap broadening, and the larger the order is, the better the band gap broadens.

Fig. 8
Fig. 8
Close modal

It can be concluded that a moderate value of grades can extremely broaden the bandgap, while when the graded value is too large or too small, it can be difficult for the graded structure to broaden the bandgap.

## 4 Experimental Verification

The band gap broadening of the graded PnCs has been verified by the computational simulation of the band structure and the frequency response function mentioned earlier. In this section, the band gap broadening of the graded PnCs will be further verified by the experimental test. In Fig. 9, the schematic diagram of the experiment is shown. The following experimental equipment was used: lms test.lab 17 software, ASUS VM520U laptop (signal generator), SIEMENS SCM202 signal processor, Brüel & Kjær 2716C power amplifier, Brüel & Kjær 4825 shaker, two acceleration transducers, and an impedance head. The excitation signal is given from the PC, and it passed through the signal processor and power amplifier, and transmitted to the shaker, where the shaker impedance head is placed against one end of the specimen to give the initial excitation. The signal processor collects the signals from the two sensors and feeds them back to the terminal. The signal analysis software LMS is used to process the signals and finally obtain the frequency response function of the specimen at each initial excitation frequency. The sample is placed on a glass plate with a very small coefficient of friction, which realizes the side restraint to restrain the rotational mode of the mass block. The experimental specimen is made of the same PLA material as in the simulation, and the material parameters are presented in Table 1. The geometric parameters are the same as those in the simulation, as shown in Table 2.

Fig. 9
Fig. 9
Close modal

Figure 10 shows the transmission comparison of experiments and simulations. The results obtained from the simulation show the existence of band gaps (blue part) in the frequency ranges of 282–868 Hz, 1212–2370 Hz, and 2831–3000 Hz. As a comparison, the experiment takes the sample with the same material and geometric parameters, and the result is shown in Fig. 10(b). The black part is the overlap of the experimental and simulation band gap. Comparing the experimental and simulation results, the band gap frequency range is not much different and the values are within the acceptable range. The passband obtained from the experimental test is narrower and the attenuation is smaller. This is because of the difference between the simulation and experimental test conditions, i.e., the simulation calculation does not consider damping at all, while the experimental test has unavoidable damp. Second, there exists uncertainty in the experiment itself and the incomplete balance of the experimental setup.

Fig. 10
Fig. 10
Close modal

Figure 11 shows the transmission comparison of experimentally measured graded structure and graded-free structure. It is not difficult to find that the graded-free structure has several band gaps in the frequency range of 147–968 Hz, 1010–2555 Hz, and 2625–3000 Hz, while the graded structure merges adjacent multiple band gaps into an extremely broad one in the frequency range of 214–3000 Hz. This once again demonstrates that the graded design can have a great broadening effect on the band gap, merging adjacent multiple band gaps into an extremely broad one.

Fig. 11
Fig. 11
Close modal

## 5 Conclusion

A new graded phononic crystal to broaden the Bragg scattering band gaps is proposed. The ability of graded PnCs to broaden the bandgap is demonstrated by theory, simulation, and experiment. The effects of the main structural factors related to graded design on band gap broadening are studied. The main conclusions are as follows:

1. The graded design can merge adjacent multiple band gaps into an extremely broad one, broaden the band gaps of Bragg scattering phononic crystals extremely.

2. The graded PnCs achieve the broadening of bandgaps mainly by (i) preserving the bandgap of the graded-free structure and (ii) breaking the mirror symmetry of the structure by introducing a graded change in the size of the structure, thus opening the Dirac cone and forming a new bandgap.

3. Structural parameters related to graded design have a significant influence on band gap broadening: (i) the larger the graded order is, the better the band gap broadens; (ii) a moderate graded value of an extremely broaden the bandgap. While the graded value is too large or too small, it can be difficult to broaden the bandgap.

## Funding Data

• National Natural Science Foundation of China (Grant Nos. 51775201 and 52175095).

• The Young Top-notch Talent Cultivation Program of Hubei Province, China.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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