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Band structures in two-dimensional phononic crystals with periodic Jerusalem cross slot
Physica B 456 (2015) 261–266
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Physica B journal homepage: www.elsevier.com/locate/physb
Band structures in two-dimensional phononic crystals with periodic Jerusalem cross slot Yinggang Li a,b, Tianning Chen a,b,n, Xiaopeng Wang a,b, Kunpeng Yu a,b, Ruifang Song a,b a b
School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, PR China State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi'an 710049, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 28 September 2013 Received in revised form 25 July 2014 Accepted 26 August 2014 Available online 6 September 2014
In this paper, a novel two-dimensional phononic crystal composed of periodic Jerusalem cross slot in air matrix with a square lattice is presented. The dispersion relations and the transmission coefficient spectra are calculated by using the finite element method based on the Bloch theorem. The formation mechanisms of the band gaps are analyzed based on the acoustic mode analysis. Numerical results show that the proposed phononic crystal structure can yield large band gaps in the low-frequency range. The formation mechanism of opening the acoustic band gaps is mainly attributed to the resonance modes of the cavities inside the Jerusalem cross slot structure. Furthermore, the effects of the geometrical parameters on the band gaps are further explored numerically. Results show that the band gaps can be modulated in an extremely large frequency range by the geometry parameters such as the slot length and width. These properties of acoustic waves in the proposed phononic crystals can potentially be applied to optimize band gaps and generate low-frequency filters and waveguides. & 2014 Elsevier B.V. All rights reserved.
Keywords: Band gaps Jerusalem cross slot Phononic crystals Finite element method
1. Introduction Over the last two decades, the propagation of acoustic and elastic waves in periodic composite materials, known as phononic crystals (PCs) has attracted considerable attention for their abundant physics and potential engineering applications [1–4]. Phononic crystals are periodic artificial composite materials made of two or more materials with different elastic constants, and they can demonstrate various novel physical properties; in particular, the existence of phononic band gaps (BGs), in which the propagation of elastic waves is prohibited [5–8]. With the existence of BGs, phononic crystals possess extensive potential applications, such as vibration and noise reduction [9–11], sound filters [12,13] and waveguides [14,15]. Earlier studies have demonstrated that the occurrence of the BGs is attributed to Bragg scattering and localized resonances. For the first mechanism, the BGs are attributed to the destructive interference between incident acoustic waves and reflections from the periodic scatterers. When wavelengths are of the order of the lattice constants, the phononic crystals can yield complete phononic BGs [16–18]. For the second mechanism, the resonances of
n Corresponding author at: School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, PR China. Tel./fax: þ86 29 82665376. E-mail address: [email protected] (T. Chen).
http://dx.doi.org/10.1016/j.physb.2014.08.035 0921-4526/& 2014 Elsevier B.V. All rights reserved.
scattering units play a major role in the BGs which are less dependent on the periodicity and symmetry of the structure. The frequency range of the gap could be almost two orders of magnitude lower than the usual Bragg gap [19,20]. In order to promote the engineering application of PCs, the acquisition of large and tunable BGs at low frequencies is of extremely importance. Kushwaha and Djafari-Rouhani [21] computed extensive band structures for periodic arrays of rigid metallic rods in air and obtained multiple complete acoustic stop bands. Li et al. [22] investigated the effects of orientations of square rods on the acoustic band gaps in two-dimensional periodic arrays of rigid solid rods embedded in air and concluded that the acoustic band gaps can be opened and enlarged greatly by increasing the rotation angle. Cheng et al. [23] demonstrated that a class of ultrasonic metamaterial, which was composed of subwavelength resonant units, built up by parallel-coupled Helmholtz resonators with identical resonant frequency, possessed broad locally resonant forbidden bands and the bandwidths were strongly dependent on the number of resonators in each unit. Li et al. [24] studied phononic band structure with periodic elliptic inclusions for the square lattice based on the plane wave expansion method and the numerical results showed the systems composed of tungsten elliptic rods embedded in silicon matrix can exhibit a larger complete band gap than the conventional circular phononic crystal slabs. Cui et al. [25] presented a new band gap structure composed of a square array of parallel steel
262
Y. Li et al. / Physica B 456 (2015) 261–266
tubes with narrow slits and obtained large band gap and low starting frequency by arranging different width of slits embedded in the tubes. Lin et al. [26] presented a theoretical study on the tunability of phononic band gaps in two-dimensional phononic crystals consisting of various anisotropic cylinders in an isotropic host. Phononic band gaps for bulk acoustic waves propagating in the phononic crystal can be opened, modulated, and closed by reorienting the anisotropic cylinders. Yu et al. [27] studied the band gap properties of a two-dimensional phononic crystal with neck structure and showed that the band gaps were significantly dependent upon the geometrical parameters. Jerusalem cross slot structures are widely used in planar microwave photonic crystal and the absorption frequency band can be flexibly adjusted by the slot parameters [28–30]. However, as far as we know, the band gap properties in phononic crystals with Jerusalem cross slot structures have not yet been carefully investigated. In this paper, we investigate the band structures in a novel two-dimensional phononic crystal composed of periodic Jerusalem cross slot in air matrix by using the finite element method (FEM) [31]. The formation mechanisms of the band gap are analyzed based on the acoustic modal analysis. Furthermore, the effects of the geometry parameters on the band gaps are discussed. Numerical results show that the band gaps can be tuned in a wide frequency range by the geometry parameters such as the slot length and width.
2. Model and methods of calculation In this work, we consider a novel two-dimensional phononic crystal composed of periodic Jerusalem cross slot in air matrix with a square lattice. Fig. 1(a) and (b) shows the cross section of the proposed PC structure. The geometrical parameters of the Jerusalem cross slot structure are defined as follows: the parameters of the slot length are l and m respectively, and the parameters of the slot width are n and d respectively. The infinite system of the two-dimensional PC is formed by repeating the unit cell periodically along the x- and y-directions simultaneously. In the unit cell, the lattice constant a¼ 36 mm, the slot length l ¼28 mm and the slot width n ¼2 mm. These three parameters remain unchanged in all the calculations below. In order to theoretically investigate the band gap properties of the proposed PC structure, the finite element method based on the
Bloch theory is applied to calculate the dispersion relations and the transmission coefficient spectra. Since the infinite system is periodic along the x- and y-directions simultaneously, only the unit cell shown in Fig. 1(b) needs to be considered. As the unit cell is composed of air and steel materials, the calculation area can be divided into the fluid and solid domains. In the fluid domain, the governing equation of the acoustic waves can be simplified as frequency-domain Helmholtz equation: 1 ω2 p ð1Þ ∇ ∇p ¼ ρ0 ρ0 c s where p is the acoustic pressure, ρ0 is the density, ω is the angular frequency and cs is the speed of sound. As the acoustic impedance of air is much smaller than that of steel, one knows that the longitudinal waves propagating in air will be almost reflected by the steel inclusions and the wave propagation in the proposed PC is predominant in the air domain. So the transverse waves in steel inclusions can be neglected for the sake of simplicity, and we can consider the steel inclusions as fluid with very high stiffness and specific mass. Based on the Bloch theorem, periodic boundary conditions are applied at the boundaries between the unit cell and its four adjacent cells: pðr þ aÞ ¼ pðrÞeiKa
ð2Þ
where r is the position vector, a denotes the basis vector of the periodic structure and the parameter K is defined as a twodimension Bloch wave vector. We solved the eigenvalue equations (Eqs. (1) and (2)) with COMSOL Multiphysics 3.5a software [32]. The Acoustic Module operating under the two-dimensional pressure acoustics Application Mode (acpr) is chosen for the calculations. The constant boundary condition is imposed on the boundary between the air and steel boards, and the Bloch boundary conditions are imposed on the two opposite boundaries of the unit cell. The unit cell is meshed by using a triangular mesh with the Lagrange quadratic elements provided. One knows that with a given value of Bloch wave vector K, a group of eigenvalues and eigenmodes can be calculated by solving the eigenvalue problem. Letting the value of Bloch wave vector K along the boundary of the first Brillouin zone and repeating the calculation and we can obtain the dispersion relations of the PCs. For the transmission spectrum, the acpr mode of COMSOL Multiphysics 3.5a software is applied to solve the transmission spectra problem. We consider a finite array structure composed of
Fig. 1. (a) Schematics of the gap structure composed of a periodic square array of Jerusalem cross slot in air matrix. The lattice constant is a. (b) Schematics of the unit cell of the gap structure, in which l and m are the parameters of the slot length; n and d are the parameters of the slot width.
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N units to calculate the transmission coefficient spectrum. The finite array structure includes N units in the x-direction, while the periodic boundary conditions are still applied in the y-direction to represent the infinite units. As mentioned above, the Jerusalem cross slot structures can be seen as a fluid with very high stiffness and specific mass, so the boundaries between the air and Jerusalem cross slot structures are treated as a constant boundary. The governing equation for the acoustic waves is the same as Eq. (2). The radiation boundary conditions, which allow an outgoing wave to leave the modeling domain with no or minimal reflections, are applied on the left and right boundaries of the air domain, yielding 1 k i i p nU ∇p þ i p þ ΔT p ¼ ΔT p0 þ ðik iðK U nÞÞ 0 e iðkrÞ ; 2k ρ0 ρ0 2k ρ0 K ¼ kn;
ð3Þ
where p is the pressure, n is the inward normal vector of the structure, k is the wave number, ΔT denotes the Laplace operator in the tangent plane at a certain point on the boundary and p0 is the amplitude of the plane wave sound source. We chose the left boundary as the sound source and define p0 ¼1 Pa. As the right boundary is just a radiation condition with no source, p0 ¼0 is defined accordingly. The plane waves with single-frequency from the left side of the finite array propagate along the x-direction. The transmission spectra are defined as the ratio of the transmitted power through the N layered finite system to the incident power. By varying the excitation frequency of the incident waves, the transmission spectra can be obtained.
3. Results and discussion 3.1. The band gaps of the PCs with periodic Jerusalem cross slot To investigate the band gap characteristics in the proposed PC structure, some numerical calculations are carried out by using the FEM. Fig. 2(a) shows the band structure for the infinite PC composed of periodic Jerusalem cross slot in air matrix with a square lattice. The material parameters of the PCs are chosen as follows: the density ρ0 ¼ 7850 kg/m3 and the sound speed of the
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longitudinal wave cs0 ¼6100 m/s for steel; ρ1 ¼1.25 kg/m3and cs1 ¼343 m/s for air. The geometrical parameters are defined as follows: the lattice constant a ¼36 mm, the slot length l ¼28 mm and m ¼24 mm, the slot width n¼ 2 mm and d¼ 2 mm. One can observe that there are 10 bands in the frequency range 0– 14,000 Hz, where four complete band gaps (red regions in Fig. 2 (a)) are involved. The lowest one ranges from 1825 to 2968 Hz (between the first and second band), while the other three band gaps are extended respectively from 3029 to 4998 Hz for the second one (between the third and fourth band), 6128 to 7668 Hz for the third one (between the fifth and sixth band) and 10,414 to 11,294 Hz for the fourth one (between the sixth and seventh band). Moreover, three incomplete band gaps (green regions in Fig. 2(a)) can be found in the Γ–X direction of the irreducible Brillouin zone. In order to validate the band structure, the transmission coefficient spectrum is calculated by using the FEM. Fig. 2(b) represents the calculated transmission coefficient spectrum for plane acoustic waves propagating in the proposed PC structure along Γ–X direction of the finite array PC structure with 10 1 unit cells. It can be observed that there are four frequency ranges with large attenuation in the transmission coefficient spectrum, which are in reasonable consistent with the band gaps in Fig. 2(a). Furthermore, the width of the first and third attenuation frequency ranges in the spectrum is larger than the complete band gaps frequency range, which is attributed to the effect of the incomplete band gaps in the Γ–X direction. As a comparison, we investigate the band structures of another two traditional PC structures: one is composed of cross board in air matrix and the other is steel square inclusions in air matrix. During the calculations, the material parameters and the lattice constant are still the same as those used in Fig. 2. The band structure of the PC structure with cross board is shown in Fig. 3(a). We can see that there are only three complete band gaps in the frequency range of 0– 14,000 Hz and the lowest one is extended from 3431 to 5026 Hz, which is much higher than that of the proposed PCs. Fig. 3 (b) illustrates the band structure of the PC structure with steel square inclusions. One can observe obviously that no complete band gap is contained in the frequency range of 0–14,000 Hz. Through comparison and analysis, we can conclude that the proposed PCs with
Fig. 2. (a) Band structure in the two-dimension PC composed of a periodic square array of Jerusalem cross slot in air matrix. The inset of the second gap is the reduced Brillouin zone. (b) The transmission spectrum along the ΓX direction of a finite structure composed of 10 1 unit cells. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
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Fig. 3. (a) Band structure in the two-dimension PC composed of cross board in air matrix. The inset is the reduced Brillouin zone. (b) Band structure in the two-dimension PC composed of steel square inclusion in air matrix. The inset is the reduced Brillouin zone. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Fig. 4. Pressure field at a unit cell of the modes marked A (a), B (b), C (c), D (d), E (e), F (f), G (g), H (h) in Fig. 2(a).
periodic Jerusalem cross slot can obtain much larger and lower band gaps than the two traditional PCs. In order to further intuitively illustrate the formation mechanism of the band gaps in the proposed PC structures, we calculate the acoustic pressure field at the edges of the four complete band gaps marked A–H in Fig. 2(a). We can see from Fig. 4(a) that the acoustic pressure field of mode A is mainly concentrated in the four cavities of Jerusalem cross slot structure. The diagonal cavities are in the same phase, while the adjacent cavities are with the reverse phase. Fig. 4(b) shows that the acoustic pressure field of mode B is still mainly distributed in the four cavities. However, the
acoustic pressure in one of the diagonal cavities is quite larger than the other. Fig. 4(c) indicates that the acoustic pressure field of mode C is extremely similar to that of mode A. The main difference between mode A and mode C is concentrated on the phase relation between the cavities and acoustic field outside the cross slot structure. It can be conducted that modes A–C are approximate to the three different resonance modes of the four cavities, and the first and second band gaps are mainly attributed to the resonance of the inner cavities inside the Jerusalem cross slot structure. The first band gap can be tuned by changing the geometry parameters of the Jerusalem cross slot.
Y. Li et al. / Physica B 456 (2015) 261–266
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Fig. 4(d) shows that the acoustic pressure field of mode D covers the proposed PC, whereas the acoustic pressure inside and outside the Jerusalem cross slot structure are with the reverse phase. Fig. 4 (e) indicates that the acoustic pressure field of mode E is almost distributed outside the Jerusalem cross slot structure and the acoustic pressure inside is zero. It can be concluded that the lower edge of the third band gap corresponding with mode E, is only connected with the external structure. For the modes F, G and H in Fig. 4(f), (g) and (h), we can see that the acoustic pressure fields are distributed both inside and outside the Jerusalem cross slot structure, whereas the acoustic pressure inside is much smaller compared with the acoustic pressure outside. So we can conclude that modes D, F, G and H are produced by the interaction of acoustic pressure field inside and outside the Jerusalem cross slot structure.
3.2. The effects of geometry parameters on the first band gap According to the study of the acoustic modes of the band gaps, we can see that the geometry parameters of the Jerusalem cross slot structure play important influences on the band gaps of the proposed PCs, especially on the first band gap. From the acoustic pressure field of modes A and B, it can be observed that the lower and upper edges of the first band gap are attributed to the resonance of the four cavities. Consequently, it is of great significance to investigate the effects of geometry parameters such as the slot length m and width d on the first band gap. In order to investigate the effect of the slot length m on the first band gap, we calculate the band structures of the proposed PCs with different values of the slot length ranging from 4 mm to 28 mm, while the slot width d¼ 2 mm remained unchanged in these calculations. Fig. 5 illustrates the evolution of the band gap as a function of the slot length m. One can observe that, with the increase of the slot length, both the lower edge and the upper edge of the first band gap shift to low frequency range, and the band gap width reduces with the increase of the slot length. It can be concluded that the slot length can tune the first band gap in an extremely large frequency range. To investigate the effect of the slot width d on the first band gap, the band structures of the proposed PCs with different values of the slot width ranging from 2 mm to 22 mm are calculated, while the slot length m¼24 mm remained unchanged in these calculations. Fig. 6 illustrates the evolution of the band gap as a function of the slot width d. With the increase of the slot width, both the lower edge and the upper edge of the first band gap move to high frequency range and the band gap width increase to some extent. It can be concluded that the
Fig. 6. The effect of the slot width d on the lower edge and the upper edge of the first band gap with the slot length m¼ 24 mm. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
first band gap can be modulated by the slot width in an extremely large frequency range.
4. Conclusions In this paper, we theoretically investigate the band gap characteristics in a novel two-dimensional phononic crystal composed of periodic Jerusalem cross slot in air matrix with a square lattice. The band gap structures and the transmission coefficient spectra are calculated by using the finite element method based on the Bloch theorem. The attenuation frequency ranges in the transmission coefficient spectra are in reasonable agreement with the band gaps. Numerical results show that the proposed PC structure can yield large band gap in the low-frequency range. Afterwards, the acoustic modal analysis is applied to investigate the formation mechanisms of the band gap. Results indicate that the first band gap is mainly attributed to the resonance modes of the cavities inside the Jerusalem cross slot structure. Furthermore, the effect of the geometry parameters on the band gaps is investigated. Numerical results show that the location and gap width of the band gaps can be modulated in an extremely large frequency range by the slot length and width. These properties of acoustic waves in the proposed phononic crystals with periodic Jerusalem cross slot can potentially be applied to optimize band gaps and generate low-frequency filters and waveguides. Acknowledgment The authors gratefully acknowledge financial support from the National Basic Research Program of China (No. 2011CB610306), the Project of National Science Foundation of China (No. 51275377), and the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT1172). References
Fig. 5. The effect of the slot length m on the lower edge and the upper edge of the first band gap with the slot width d ¼ 2 mm. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
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Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Band structures in two-dimensional phononic crystals with periodic Jerusalem cross slot Yinggang Li a,b, Tianning Chen a,b,n, Xiaopeng Wang a,b, Kunpeng Yu a,b, Ruifang Song a,b a b
School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, PR China State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi'an 710049, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 28 September 2013 Received in revised form 25 July 2014 Accepted 26 August 2014 Available online 6 September 2014
In this paper, a novel two-dimensional phononic crystal composed of periodic Jerusalem cross slot in air matrix with a square lattice is presented. The dispersion relations and the transmission coefficient spectra are calculated by using the finite element method based on the Bloch theorem. The formation mechanisms of the band gaps are analyzed based on the acoustic mode analysis. Numerical results show that the proposed phononic crystal structure can yield large band gaps in the low-frequency range. The formation mechanism of opening the acoustic band gaps is mainly attributed to the resonance modes of the cavities inside the Jerusalem cross slot structure. Furthermore, the effects of the geometrical parameters on the band gaps are further explored numerically. Results show that the band gaps can be modulated in an extremely large frequency range by the geometry parameters such as the slot length and width. These properties of acoustic waves in the proposed phononic crystals can potentially be applied to optimize band gaps and generate low-frequency filters and waveguides. & 2014 Elsevier B.V. All rights reserved.
Keywords: Band gaps Jerusalem cross slot Phononic crystals Finite element method
1. Introduction Over the last two decades, the propagation of acoustic and elastic waves in periodic composite materials, known as phononic crystals (PCs) has attracted considerable attention for their abundant physics and potential engineering applications [1–4]. Phononic crystals are periodic artificial composite materials made of two or more materials with different elastic constants, and they can demonstrate various novel physical properties; in particular, the existence of phononic band gaps (BGs), in which the propagation of elastic waves is prohibited [5–8]. With the existence of BGs, phononic crystals possess extensive potential applications, such as vibration and noise reduction [9–11], sound filters [12,13] and waveguides [14,15]. Earlier studies have demonstrated that the occurrence of the BGs is attributed to Bragg scattering and localized resonances. For the first mechanism, the BGs are attributed to the destructive interference between incident acoustic waves and reflections from the periodic scatterers. When wavelengths are of the order of the lattice constants, the phononic crystals can yield complete phononic BGs [16–18]. For the second mechanism, the resonances of
n Corresponding author at: School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, PR China. Tel./fax: þ86 29 82665376. E-mail address: [email protected] (T. Chen).
http://dx.doi.org/10.1016/j.physb.2014.08.035 0921-4526/& 2014 Elsevier B.V. All rights reserved.
scattering units play a major role in the BGs which are less dependent on the periodicity and symmetry of the structure. The frequency range of the gap could be almost two orders of magnitude lower than the usual Bragg gap [19,20]. In order to promote the engineering application of PCs, the acquisition of large and tunable BGs at low frequencies is of extremely importance. Kushwaha and Djafari-Rouhani [21] computed extensive band structures for periodic arrays of rigid metallic rods in air and obtained multiple complete acoustic stop bands. Li et al. [22] investigated the effects of orientations of square rods on the acoustic band gaps in two-dimensional periodic arrays of rigid solid rods embedded in air and concluded that the acoustic band gaps can be opened and enlarged greatly by increasing the rotation angle. Cheng et al. [23] demonstrated that a class of ultrasonic metamaterial, which was composed of subwavelength resonant units, built up by parallel-coupled Helmholtz resonators with identical resonant frequency, possessed broad locally resonant forbidden bands and the bandwidths were strongly dependent on the number of resonators in each unit. Li et al. [24] studied phononic band structure with periodic elliptic inclusions for the square lattice based on the plane wave expansion method and the numerical results showed the systems composed of tungsten elliptic rods embedded in silicon matrix can exhibit a larger complete band gap than the conventional circular phononic crystal slabs. Cui et al. [25] presented a new band gap structure composed of a square array of parallel steel
262
Y. Li et al. / Physica B 456 (2015) 261–266
tubes with narrow slits and obtained large band gap and low starting frequency by arranging different width of slits embedded in the tubes. Lin et al. [26] presented a theoretical study on the tunability of phononic band gaps in two-dimensional phononic crystals consisting of various anisotropic cylinders in an isotropic host. Phononic band gaps for bulk acoustic waves propagating in the phononic crystal can be opened, modulated, and closed by reorienting the anisotropic cylinders. Yu et al. [27] studied the band gap properties of a two-dimensional phononic crystal with neck structure and showed that the band gaps were significantly dependent upon the geometrical parameters. Jerusalem cross slot structures are widely used in planar microwave photonic crystal and the absorption frequency band can be flexibly adjusted by the slot parameters [28–30]. However, as far as we know, the band gap properties in phononic crystals with Jerusalem cross slot structures have not yet been carefully investigated. In this paper, we investigate the band structures in a novel two-dimensional phononic crystal composed of periodic Jerusalem cross slot in air matrix by using the finite element method (FEM) [31]. The formation mechanisms of the band gap are analyzed based on the acoustic modal analysis. Furthermore, the effects of the geometry parameters on the band gaps are discussed. Numerical results show that the band gaps can be tuned in a wide frequency range by the geometry parameters such as the slot length and width.
2. Model and methods of calculation In this work, we consider a novel two-dimensional phononic crystal composed of periodic Jerusalem cross slot in air matrix with a square lattice. Fig. 1(a) and (b) shows the cross section of the proposed PC structure. The geometrical parameters of the Jerusalem cross slot structure are defined as follows: the parameters of the slot length are l and m respectively, and the parameters of the slot width are n and d respectively. The infinite system of the two-dimensional PC is formed by repeating the unit cell periodically along the x- and y-directions simultaneously. In the unit cell, the lattice constant a¼ 36 mm, the slot length l ¼28 mm and the slot width n ¼2 mm. These three parameters remain unchanged in all the calculations below. In order to theoretically investigate the band gap properties of the proposed PC structure, the finite element method based on the
Bloch theory is applied to calculate the dispersion relations and the transmission coefficient spectra. Since the infinite system is periodic along the x- and y-directions simultaneously, only the unit cell shown in Fig. 1(b) needs to be considered. As the unit cell is composed of air and steel materials, the calculation area can be divided into the fluid and solid domains. In the fluid domain, the governing equation of the acoustic waves can be simplified as frequency-domain Helmholtz equation: 1 ω2 p ð1Þ ∇ ∇p ¼ ρ0 ρ0 c s where p is the acoustic pressure, ρ0 is the density, ω is the angular frequency and cs is the speed of sound. As the acoustic impedance of air is much smaller than that of steel, one knows that the longitudinal waves propagating in air will be almost reflected by the steel inclusions and the wave propagation in the proposed PC is predominant in the air domain. So the transverse waves in steel inclusions can be neglected for the sake of simplicity, and we can consider the steel inclusions as fluid with very high stiffness and specific mass. Based on the Bloch theorem, periodic boundary conditions are applied at the boundaries between the unit cell and its four adjacent cells: pðr þ aÞ ¼ pðrÞeiKa
ð2Þ
where r is the position vector, a denotes the basis vector of the periodic structure and the parameter K is defined as a twodimension Bloch wave vector. We solved the eigenvalue equations (Eqs. (1) and (2)) with COMSOL Multiphysics 3.5a software [32]. The Acoustic Module operating under the two-dimensional pressure acoustics Application Mode (acpr) is chosen for the calculations. The constant boundary condition is imposed on the boundary between the air and steel boards, and the Bloch boundary conditions are imposed on the two opposite boundaries of the unit cell. The unit cell is meshed by using a triangular mesh with the Lagrange quadratic elements provided. One knows that with a given value of Bloch wave vector K, a group of eigenvalues and eigenmodes can be calculated by solving the eigenvalue problem. Letting the value of Bloch wave vector K along the boundary of the first Brillouin zone and repeating the calculation and we can obtain the dispersion relations of the PCs. For the transmission spectrum, the acpr mode of COMSOL Multiphysics 3.5a software is applied to solve the transmission spectra problem. We consider a finite array structure composed of
Fig. 1. (a) Schematics of the gap structure composed of a periodic square array of Jerusalem cross slot in air matrix. The lattice constant is a. (b) Schematics of the unit cell of the gap structure, in which l and m are the parameters of the slot length; n and d are the parameters of the slot width.
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N units to calculate the transmission coefficient spectrum. The finite array structure includes N units in the x-direction, while the periodic boundary conditions are still applied in the y-direction to represent the infinite units. As mentioned above, the Jerusalem cross slot structures can be seen as a fluid with very high stiffness and specific mass, so the boundaries between the air and Jerusalem cross slot structures are treated as a constant boundary. The governing equation for the acoustic waves is the same as Eq. (2). The radiation boundary conditions, which allow an outgoing wave to leave the modeling domain with no or minimal reflections, are applied on the left and right boundaries of the air domain, yielding 1 k i i p nU ∇p þ i p þ ΔT p ¼ ΔT p0 þ ðik iðK U nÞÞ 0 e iðkrÞ ; 2k ρ0 ρ0 2k ρ0 K ¼ kn;
ð3Þ
where p is the pressure, n is the inward normal vector of the structure, k is the wave number, ΔT denotes the Laplace operator in the tangent plane at a certain point on the boundary and p0 is the amplitude of the plane wave sound source. We chose the left boundary as the sound source and define p0 ¼1 Pa. As the right boundary is just a radiation condition with no source, p0 ¼0 is defined accordingly. The plane waves with single-frequency from the left side of the finite array propagate along the x-direction. The transmission spectra are defined as the ratio of the transmitted power through the N layered finite system to the incident power. By varying the excitation frequency of the incident waves, the transmission spectra can be obtained.
3. Results and discussion 3.1. The band gaps of the PCs with periodic Jerusalem cross slot To investigate the band gap characteristics in the proposed PC structure, some numerical calculations are carried out by using the FEM. Fig. 2(a) shows the band structure for the infinite PC composed of periodic Jerusalem cross slot in air matrix with a square lattice. The material parameters of the PCs are chosen as follows: the density ρ0 ¼ 7850 kg/m3 and the sound speed of the
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longitudinal wave cs0 ¼6100 m/s for steel; ρ1 ¼1.25 kg/m3and cs1 ¼343 m/s for air. The geometrical parameters are defined as follows: the lattice constant a ¼36 mm, the slot length l ¼28 mm and m ¼24 mm, the slot width n¼ 2 mm and d¼ 2 mm. One can observe that there are 10 bands in the frequency range 0– 14,000 Hz, where four complete band gaps (red regions in Fig. 2 (a)) are involved. The lowest one ranges from 1825 to 2968 Hz (between the first and second band), while the other three band gaps are extended respectively from 3029 to 4998 Hz for the second one (between the third and fourth band), 6128 to 7668 Hz for the third one (between the fifth and sixth band) and 10,414 to 11,294 Hz for the fourth one (between the sixth and seventh band). Moreover, three incomplete band gaps (green regions in Fig. 2(a)) can be found in the Γ–X direction of the irreducible Brillouin zone. In order to validate the band structure, the transmission coefficient spectrum is calculated by using the FEM. Fig. 2(b) represents the calculated transmission coefficient spectrum for plane acoustic waves propagating in the proposed PC structure along Γ–X direction of the finite array PC structure with 10 1 unit cells. It can be observed that there are four frequency ranges with large attenuation in the transmission coefficient spectrum, which are in reasonable consistent with the band gaps in Fig. 2(a). Furthermore, the width of the first and third attenuation frequency ranges in the spectrum is larger than the complete band gaps frequency range, which is attributed to the effect of the incomplete band gaps in the Γ–X direction. As a comparison, we investigate the band structures of another two traditional PC structures: one is composed of cross board in air matrix and the other is steel square inclusions in air matrix. During the calculations, the material parameters and the lattice constant are still the same as those used in Fig. 2. The band structure of the PC structure with cross board is shown in Fig. 3(a). We can see that there are only three complete band gaps in the frequency range of 0– 14,000 Hz and the lowest one is extended from 3431 to 5026 Hz, which is much higher than that of the proposed PCs. Fig. 3 (b) illustrates the band structure of the PC structure with steel square inclusions. One can observe obviously that no complete band gap is contained in the frequency range of 0–14,000 Hz. Through comparison and analysis, we can conclude that the proposed PCs with
Fig. 2. (a) Band structure in the two-dimension PC composed of a periodic square array of Jerusalem cross slot in air matrix. The inset of the second gap is the reduced Brillouin zone. (b) The transmission spectrum along the ΓX direction of a finite structure composed of 10 1 unit cells. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
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Fig. 3. (a) Band structure in the two-dimension PC composed of cross board in air matrix. The inset is the reduced Brillouin zone. (b) Band structure in the two-dimension PC composed of steel square inclusion in air matrix. The inset is the reduced Brillouin zone. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Fig. 4. Pressure field at a unit cell of the modes marked A (a), B (b), C (c), D (d), E (e), F (f), G (g), H (h) in Fig. 2(a).
periodic Jerusalem cross slot can obtain much larger and lower band gaps than the two traditional PCs. In order to further intuitively illustrate the formation mechanism of the band gaps in the proposed PC structures, we calculate the acoustic pressure field at the edges of the four complete band gaps marked A–H in Fig. 2(a). We can see from Fig. 4(a) that the acoustic pressure field of mode A is mainly concentrated in the four cavities of Jerusalem cross slot structure. The diagonal cavities are in the same phase, while the adjacent cavities are with the reverse phase. Fig. 4(b) shows that the acoustic pressure field of mode B is still mainly distributed in the four cavities. However, the
acoustic pressure in one of the diagonal cavities is quite larger than the other. Fig. 4(c) indicates that the acoustic pressure field of mode C is extremely similar to that of mode A. The main difference between mode A and mode C is concentrated on the phase relation between the cavities and acoustic field outside the cross slot structure. It can be conducted that modes A–C are approximate to the three different resonance modes of the four cavities, and the first and second band gaps are mainly attributed to the resonance of the inner cavities inside the Jerusalem cross slot structure. The first band gap can be tuned by changing the geometry parameters of the Jerusalem cross slot.
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Fig. 4(d) shows that the acoustic pressure field of mode D covers the proposed PC, whereas the acoustic pressure inside and outside the Jerusalem cross slot structure are with the reverse phase. Fig. 4 (e) indicates that the acoustic pressure field of mode E is almost distributed outside the Jerusalem cross slot structure and the acoustic pressure inside is zero. It can be concluded that the lower edge of the third band gap corresponding with mode E, is only connected with the external structure. For the modes F, G and H in Fig. 4(f), (g) and (h), we can see that the acoustic pressure fields are distributed both inside and outside the Jerusalem cross slot structure, whereas the acoustic pressure inside is much smaller compared with the acoustic pressure outside. So we can conclude that modes D, F, G and H are produced by the interaction of acoustic pressure field inside and outside the Jerusalem cross slot structure.
3.2. The effects of geometry parameters on the first band gap According to the study of the acoustic modes of the band gaps, we can see that the geometry parameters of the Jerusalem cross slot structure play important influences on the band gaps of the proposed PCs, especially on the first band gap. From the acoustic pressure field of modes A and B, it can be observed that the lower and upper edges of the first band gap are attributed to the resonance of the four cavities. Consequently, it is of great significance to investigate the effects of geometry parameters such as the slot length m and width d on the first band gap. In order to investigate the effect of the slot length m on the first band gap, we calculate the band structures of the proposed PCs with different values of the slot length ranging from 4 mm to 28 mm, while the slot width d¼ 2 mm remained unchanged in these calculations. Fig. 5 illustrates the evolution of the band gap as a function of the slot length m. One can observe that, with the increase of the slot length, both the lower edge and the upper edge of the first band gap shift to low frequency range, and the band gap width reduces with the increase of the slot length. It can be concluded that the slot length can tune the first band gap in an extremely large frequency range. To investigate the effect of the slot width d on the first band gap, the band structures of the proposed PCs with different values of the slot width ranging from 2 mm to 22 mm are calculated, while the slot length m¼24 mm remained unchanged in these calculations. Fig. 6 illustrates the evolution of the band gap as a function of the slot width d. With the increase of the slot width, both the lower edge and the upper edge of the first band gap move to high frequency range and the band gap width increase to some extent. It can be concluded that the
Fig. 6. The effect of the slot width d on the lower edge and the upper edge of the first band gap with the slot length m¼ 24 mm. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
first band gap can be modulated by the slot width in an extremely large frequency range.
4. Conclusions In this paper, we theoretically investigate the band gap characteristics in a novel two-dimensional phononic crystal composed of periodic Jerusalem cross slot in air matrix with a square lattice. The band gap structures and the transmission coefficient spectra are calculated by using the finite element method based on the Bloch theorem. The attenuation frequency ranges in the transmission coefficient spectra are in reasonable agreement with the band gaps. Numerical results show that the proposed PC structure can yield large band gap in the low-frequency range. Afterwards, the acoustic modal analysis is applied to investigate the formation mechanisms of the band gap. Results indicate that the first band gap is mainly attributed to the resonance modes of the cavities inside the Jerusalem cross slot structure. Furthermore, the effect of the geometry parameters on the band gaps is investigated. Numerical results show that the location and gap width of the band gaps can be modulated in an extremely large frequency range by the slot length and width. These properties of acoustic waves in the proposed phononic crystals with periodic Jerusalem cross slot can potentially be applied to optimize band gaps and generate low-frequency filters and waveguides. Acknowledgment The authors gratefully acknowledge financial support from the National Basic Research Program of China (No. 2011CB610306), the Project of National Science Foundation of China (No. 51275377), and the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT1172). References
Fig. 5. The effect of the slot length m on the lower edge and the upper edge of the first band gap with the slot width d ¼ 2 mm. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
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