† Corresponding author. E-mail:

Project supported by the China Scholarship Council.

A two-dimensional phononic crystal (PC) structure possessing a relatively low frequency range of complete bandgap is presented. The structure is composed of periodic spindle-shaped plumbum inclusions in a rubber matrix which forms a square lattice. The dispersion relation, transmission spectrum and displacement field are studied using the finite element method in conjunction with the Bloch theorem. Numerical results show that the present PC structure can achieve a large complete bandgap in a relatively low frequency range compared with two inclusions of different materials, which is useful in low-frequency noise and vibration control and can be designed as a low frequency acoustic filter and waveguides. Moreover, the transmission spectrum and effective mass are evaluated to validate the obtained band structure. It is interesting to see that within the band gap the effective mass becomes negative, resulting in an imaginary wave speed and wave exponential attenuation. Finally, sensitivity analysis of the effect of geometrical parameters of the presented PC structure on the lowest bandgap is performed to investigate the variations of the bandgap width and frequency.

Phononic crystals(PCs) are artificial periodic composite structures which can efficiently control the propagations of sound waves with frequencies in a range from a few hertz to several megahertz.^{[1,2]} In such a medium, elastic waves within some specific range of frequency can be completely prohibited from propagating in the periodic structure and this forbidden frequency range is known as the phononic bandgap. Over the past two decades, a great deal of attention has been paid to the achievement of complete low-frequency bandgaps of PCs by improving their periodic microstructures,^{[3–7]} because of their unique physical properties and potential engineering applications such as vibration isolator, frequency filter and acoustic mirrors.^{[8–12]} Studying the bandgap characteristics of PC could help better understand the propagations of sound and vibration in such a heterogeneous medium.

To apply PCs to noise and vibration control, tuneable complete bandgaps in the low-frequency range are the major concern. Many of the studies have been carried out with various periodic structures of PCs. For example, Yao *et al*. investigated the Lamb waves in a two-dimensional (2D) PC plate with anisotropic inclusions.^{[13]} They found that the complete bandgap can be tuned by changing the orientation of anisotropic inclusion. Yu *et al*. studied the influence of microstructure on the bandgap using a 2D PC composed of periodic slotted tubes with the internal rib structure in air matrix and revealed that the size of the internal rib structure has significant influence on both the lower edge and the upper edge of the lowest bandgap.^{[14]} Subsequently, they investigated the bandgap properties of a 2D PC with periodic neck structures, which could generate several complete bandgaps within a frequency range from 2 kHz to 17 kHz.^{[15]} Xu *et al*.^{[16]} considered a 2D hierarchical PC consisting of periodic square lattice with multiple scatterers. Their results showed that the hierarchical structure possesses tuneable complete bandgap features and these interesting features are favourable for the applications of insulation and vibration attenuation. Li *et al*.^{[17]} presented a 2D PC with periodic Jerusalem slot in air matrix, and a complete bandgap in the frequency range from 2 kHz to 11 kHz is observed. More studies of the effects of structure parameters on bandgaps of different PCs can be found in Refs. [12], [18]–[23]. It should be noted that most studies of bandgaps of the PCs concentrated on the high frequency range from kHz to MHz, while in the field of vibration isolation and noise reduction, bandgaps in the low-frequency range are more meaningful. Moreover, lots of previous studies were devoted to the configuration in which the axes of the inclusions are perpendicular to the surface of the PC, while studies on the configuration with axes parallel to the surface as indicated in Ref. [24] were seldom reported, although PCs with this type of configuration have more potential applications in structural construction inspection, reliable non-destructive evaluation and sensing systems.^{[25]}

In the present paper, a 2D PC possessing a complete bandgap of low frequency is presented. The PC is composed of spindle-shaped inclusions embedded periodically in a host rubber material. The axes of the spindle-shaped inclusions are placed parallelly to the surfaces of the crystal. Making use of the finite element method (FEM) in conjunction with the Bloch theorem,^{[26,27]} the propagation properties including its dispersion relations, transmission spectrum, effective mass and displacements fields are assessed. Meanwhile, sensitivity analysis of the effect of geometry parameters of the spindle-shaped inclusions on the lowest bandgap is conducted. This study also includes their influences on the band structure and the displacement field distributions of the Lamb modes so that relationships can be revealed between the band structure and the geometrical parameters.

As shown in Fig. *z* direction so that a plane strain model can be used in our analysis. According to the periodicity of the PC structure, a 2D representative unit cell displayed in Fig. *x* direction and *y* direction separately. In the representative unit cell, the spindle-shaped inclusion is located in the centre of the square lattice with side height *b*, central height *c*, and half width *d*. The materials P and R (see Fig. ^{[18,28]}

plumbum: *ρ*_{pb} = 11600 kg/m^{3}, *E*_{pb} = 40.8 GPa, *μ*_{pb} = 0.369;

rubber: *ρ*_{ru} = 1300 kg/m^{3}, *E*_{rb} = 0.117 MPa, *μ*_{rb} = 0.47.

The governing equations for elastic wave propagation in homogeneous solids are given by

*ρ*is the mass density,

*t*is the time,

*u*is the

_{i}*i*-th component of the displacement vector

*(*u

*) with position vector*r

*,*r

*c*are the elastic constants, and

_{ijkl}*x*(

_{j}*j*= 1,2,3) are the corresponding coordinate variables

*x*,

*y*, and

*z*, respectively.

Making use of the Bloch theorem, the displacement vector of a periodic PC system can be expressed as

*(*u

_{k}*) is a periodic vector with the same periodicity as the crystal lattice,*r

*= (*k

*k*,

_{x}*k*) is the wave vector limited to the first Brillouin zone of the repeated lattice.

_{y}With the FEM, the discrete form of the eigenvalue equations in the representative unit cell can be written as

*is the displacement vector at the discrete node,*U

*and*K

*are respectively the stiffness and mass matrices of the unit cell, and*M

*ω*is the circular frequency.

Considering the periodicity of the structure, the following Bloch periodic boundary condition should be applied to the unit cell based on Bloch theorem:

*= (*a

*a*,

_{x}*a*) is the lattice basis vector, with

_{y}*a*=

_{x}*a*=

_{y}*a*.

In this study, we solve the eigenvalue equation (* k*, a group of eigenfrequencies and eigenmodes can be obtained by solving the eigenvalue problem. Sweeping the value of Bloch wave vector

*along the boundary of the first reduced Brillouin zone and repeating the calculation, we can obtain the dispersion relation curves of the PC.*k

Besides, to verify the results of the dispersion curves and bandgaps, the transmission spectrum of a finite periodic PC system including multiple unit cells along the *x* direction is also calculated with COMSOL simulation. In the system shown in Fig. *x*-direction. In the *y* direction, the Bloch periodic boundary condition is still imposed on the top and bottom surfaces of the system to represent infinity along the *y* direction. The corresponding transmitted displacement can be recorded on the right boundary of the output domain to evaluate the transmission spectrum defined as follows:

*u*

_{in}and

*u*

_{out}are the values of the incident displacement and transmitted displacement, respectively. By varying the excitation frequency of the incident displacement, the transmission spectrum can be obtained.

In this subsection, numerical results are presented to demonstrate the bandgap characteristics of the present PC structure, which has a spindle-shaped plumbum inclusion embedded in the rubber matrix to form a square lattice. Figure *a* = 30 mm. Based on the lattice constant, a set of moderate parameters for the inclusion is chosen to be *b* = 18 mm, *c* = 16 mm, and *d* = 11 mm. Since the asymmetry of the spindle inclusion is out of 90° degree, the wave vector should sweep the first Brillion zone in two different ways, i.e., *Γ*–*X*–*M*–*Γ* and *Γ*–*X′*–*M*–*Γ*. It can be seen that the proposed PC possesses a complete low-frequency bandgap from 163 Hz to 222 Hz (between the second and fourth band), in which waves from different directions cannot propagate through this structure. The relative gap width of the complete bandgap is nearly 15%. Also, it is obvious that there exists an oriented band gap ranging from 357 Hz to 393 Hz along the *Γ*–*X* direction of the irreducible Brillouin zone.

To validate the band structure, figure *Γ*–*X* direction and the corresponding transmission coefficient spectrum for plane wave propagating in the finite system given in Fig.

In order to better understand the physical formation mechanism of the bandgaps in the proposed PC structure, the eigenmodes marked as *A*, *B*, *C*, *D* in Fig. *A*, which corresponds to the lower edge of the complete bandgap, denote the motion of the material in the upper and down regions along the same *x*-negative direction, then turn around on the left side of the inclusion, get through the inclusion, and finally disperse on the right side of the inclusion. The vibration mainly concentrates at the inlet and the outlet as the arrows get through the inclusion. Unlike mode *A*, mode *B* in Fig. *x*-positive direction. So it can be concluded that the formation of the complete band gap is mainly attributed to the recycling motion of the matrix around the inclusion and the shearing motion of the upside and downside matrix outside the inclusion. Figure *Γ*–*X* direction. This vibration concentrates on two specific sectional matrix sides: upside and downside of the inclusion. In Fig. *D* corresponding to the upper edge of the orientational bandgap is the torsional movement of the matrix around the inclusion. Therefore, the occurrence of the orientational bandgap is attributed to the local energy centralization and the torsional motion of the matrix.

To better understand the wave propagation through the PC, the stress field of the finite system is shown in Fig. *x* direction and deform in the plane, and then propagate through the whole system. With this knowledge, we could better understand the wave propagation in and out of the complete band gap and design the wave filter as desired.

The analysis frequency range is focused on low frequency, so the wavelength, compared with the unit cell, is very large. If the periodic model is treated as an equivalent elastic solid, the effective material parameters based on the homogenization concept can be used to characterize the properties of the PC. The average of local fields, including the local stress *σ*_{αβ}, strain *ε*_{αβ}, forces *F*_{α}, and displacement *u*_{α}, is imposed on the external boundary of the representative volume element as^{[29]}

*α*,

*β*= 1,2 and

*ρ*

_{eff}is the effective mass density, which can be calculated from Newton’s second law as follows:

It can be seen that the effective mass changes dramatically as frequency increases. At 166 Hz, the effective mass approaches to the positive infinity and then falls down to the negative infinity, which means that there is a step change at this frequency. Moreover, the effective mass becomes negative in a frequency range of 166 Hz–213 Hz, which coincides with the band gap shown in Fig. *u _{α}* =

*U*e

^{j(k·r − ωt)}being incident on the PC along the

*Γ*–

*X*direction, with

*and*k

*being the wave vector and position vector, respectively, the eigenvalue can be obtained from the following equation:*r

*u*,

_{α}*u*}

_{β}^{T}, we set the determinant of the coefficient matrix to be zero. And the wave speed can be obtained from the following expressions:

To investigate the lowest bandgap in the proposed PC more deeply, effects of the geometry parameters of the unit cell on the variations of the bandgap width and frequency are studied. From Subsection 3.1 it follows that for the *Γ*–*X*–*M*–*Γ* road and the *Γ*–*X′*–*M*–*Γ* road, their low frequency complete band gaps are the same. And the crucial route is the *Γ*–*X*(*X′*) part. So the geometry parameter investigation is conducted along the *Γ*–*X* part.

Firstly, assuming the spindle’s centre height *c* = 16 mm and the side height *b* = 18 mm, the effect of the spindle width d on the first bandgap is shown in Fig. *d* = 7 mm while the shape of the lower edge is like a groove as the spindle width becomes large. The location of the first bandgap drops into a lower frequency range and then climbs up to a higher frequency range. The gap width of the first bandgap reaches a minimum value when *d* = 14 mm. The occurrence of the dip of the upper edge is interesting, so it is essential to see what happens when the width *d* is equal to 7 mm. The dispersion relations of the band structure with different *d* values are given in Fig. *d* changes from 3 mm to 7 mm, the fourth band climbs up to a higher frequency and the second band comes down to a lower frequency range, and finally they deviate from each other. As a result, a lower bandgap in the lower frequency range appears, owing to the interaction between the second band and the fourth band. The lower bandgap has a significant meaning in aspects of low-frequency noise and vibration control.

Next, the influence of the side height b is investigated when *c* = 16 mm and *d* = 11 mm. Figure *b*. It is observed that as side height *b* increases, both the upper edge and the lower edge of the first bandgap shift to the higher frequency range and the bandgap width increases to some extent.

Finally, figure *c* for the cases of the lower and upper edges of the first bandgap to investigate the effect of *c* on the first bandgap. It is assumed that the central height *c* varies from 6 mm to 16 mm, while the spindle side height *b* = 18 mm and the half width *d* = 11 mm keep unchanged. It can be seen that the upper edge of the first bandgap goes up to the high frequency range, while the lower edge comes down with the increase of central height *c*. As a result, the width of the first bandgap increases. However, the location of the first bandgap looks unmovable.

When the material property of the inclusion changes, wave propagation through the crystal can be different. Here, we provide two other materials to show the dispersion relation, air inclusion and aluminium inclusion as shown in Fig.

It can be seen that there are several bands in Fig.

In this paper, the band characteristics in a novel 2D PC structure composed of periodic spindle-shaped plumbum inclusions embedded in a host rubber material with a square lattice are investigated. Numerical results from the FEM in conjunction with the Bloch theorem show that the proposed PC structure can yield a large complete bandgap in a lower-frequency range from 163 Hz to 222 Hz, which is useful in low-frequency noise and vibration control. The attenuation frequency ranges in the transmission spectrum show good agreement with the bandgaps along the *Γ*–*X* direction. The effective mass is calculated at the same time. The results show that it changes with the increase of frequency, and becomes negative in a certain frequency range which coincides with the band gap. Moreover, the eigenmode analysis is carried out to understand the formation mechanism of the bandgap. Results indicate that the complete bandgap is mainly attributed to the recycling motion of the matrix around the inclusion and the shearing motion of the matrix in the upside and downside regions of the inclusion. Finally, the effect of the geometry parameters of the inclusion on the bandgap is investigated. Numerical results show that the gap width and the location of the first bandgap can be extremely modulated in a large frequency range by the spindle central height and width. These band properties of the proposed PC can potentially be used to optimize bandgaps and generate lower-frequency filters and waveguides.

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