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Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs

Opening complete band gaps in two dimensional locally resonant phononic crystals Xiaoling Zhou a, *, Longqi Wang b a b

Shanghai Institute of Aerospace System Engineering, Shanghai 201109, China School of Civil & Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore

A R T I C L E I N F O

A B S T R A C T

Keywords: Complete band gap Two dimensional LRPC Tunable properties Geometrical parameter

Locally resonant phononic crystals (LRPCs) which have low frequency band gaps attract a growing attention in both scientiﬁc and engineering ﬁeld recently. Wide complete locally resonant band gaps are the goal for researchers. In this paper, complete band gaps are achieved by carefully designing the geometrical properties of the inclusions in two dimensional LRPCs. The band structures and mechanisms of different types of models are investigated by the ﬁnite element method. The translational vibration patterns in both the in-plane and out-ofplane directions contribute to the full band gaps. The frequency response of the ﬁnite periodic structures demonstrate the attenuation effects in the complete band gaps. Moreover, it is found that the complete band gaps can be further widened and lowered by increasing the height of the inclusions. The tunable properties by changing the geometrical parameters provide a good way to open wide locally resonant band gaps.

1. Introduction Phononic crystal or acoustic band gap materials are materials with periodic arrangement [1–3]. The band gap properties which in certain frequency elastic wave cannot transmit through the periodic materials have potential applications in vibration and noise control. Thus, they attracted a growing interest in recent years, especially when the concept of locally resonant phononic crystal (LRPC) is proposed [4–8]. The wave length in the locally resonant band gap is two order smaller than the lattice size. This low frequency band gap property is of great importance in engineering applications. Liu et al. [5] ﬁrst reported the LRPC which consists of a hard core with rubber coating layer embedding into the matrix periodically. The three dimensional LRPC possessed a band gap with frequency as low as 400 Hz. And the lattice size is as small as 1.55 cm. Then Hirsekorn [9] and Wang [10] investigated two dimensional LRPC with three components and gave simpliﬁed models to predict the band gap boundaries. Zhou and Chen [11] utilized electric ﬁeld and the initial stress to tune the locally resonant band gap frequencies. They mainly considered the two dimensional LRPCs with inﬁnite size in the out-of-plane direction. Lamb waves in locally resonant phononic slabs are also studied because the thin plate structures are useful in engineering [12–15]. Hsu and Wu [12] reported ﬂexural-dominated resonant band gaps which signiﬁcantly depend on both the radius of the circular rubber ﬁller and the plate thickness. * Corresponding author. E-mail address: [email protected] (X. Zhou). https://doi.org/10.1016/j.jpcs.2018.01.025 Received 26 November 2017; Accepted 14 January 2018 0022-3697/© 2018 Elsevier Ltd. All rights reserved.

Usually the two dimensional LRPCs have wide band gaps in the periodic directions and they can hardly have large full band gaps. Some researchers proposed PC with resonant stubs deposited on a plate [16–19]. Spring-mass resonators are attached on a thin plate [20] or between double plates [21] to obtain ﬂexural wave band gaps. Wang et al. [22] studied the tuning properties and waveguide of multi-stub LRPC plate. Assouar et al. [23] used double-sided arrangement of pillars in the phononic crystal plates to lower and widen the locally resonant band gaps. Zhang et al. [24] showed low frequency locally resonant band gaps in the out of plane direction by cutting a spiral groove from the phononic crystal plates with periodic spiral resonators. The emphasis of these works is to get wide and low frequency band gaps, and most of the band gaps are in the out-of-plane direction. Another type of LRPC is slab with resonators obtained by etching holes in the structures [25]. Yu and Lesieutre [26] used 3D printing to fabricate an acoustic band gap structure consisting of a honeycomb matrix with spherical cores attached inside and they found that the materials can reduce the peak dynamic response. Very interesting, complete band gaps have been achieved in two dimensional phononic crystal slab due to the local rotational or translational resonance [27]. The geometry parameters can be design to tune the band gaps. Ma et al. [28] also reported to open a large full phononic band gap in thin plate with three layered spherical resonators. The vibration degree of freedoms which are translating and rotating motion patterns appeared in both the in-plane and out-of-plane

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Journal of Physics and Chemistry of Solids 116 (2018) 174–179

directions and contribute to large full band gaps. These works indicated that large complete band gaps can be achieved in two dimensional locally resonant phononic crystals by carefully designing the resonators. In this paper, two dimensional locally resonant phononic crystals which are composed of thin plates and cylindrical resonators are investigated. By adjusting the geometrical property of the cylindrical resonators, complete band gaps are opened. The band structures, band gap mechanisms and transmission properties are revealed. Also, the tunable properties of the band gaps are discussed.

Table 1 Material constants of the three components.

Aluminum Rubber Steel

Lame constants (GPa)

ρ

λ

G

2600 1300 7780

40.4 60.5e-5 121.4

26.9 4e-5 81

and its second order derivative of time, respectively. For the LRPCs, the displacement ﬁeld satisfy the Bloch theory which is described as,

2. Models and method

uðrÞ ¼ uðrÞ⋅eiðk⋅rωtÞ

The unit cell of the two dimensional locally resonant phononic crystal models are shown in Fig. 1. Three different types of LRPC models are investigated in this paper. As shown in Fig. 1(a–c), the embedded inclusions of the three models have different height and coating layers. As shown in Fig. 1(a), the model I is a typical two dimensional three components ﬁlled-in LRPC with the inclusion and the plate having the same height. The component materials are steel, rubber and aluminum, respectively. Model II in Fig. 1(b) is a kind of LRPC with a higher inclusion embedding in the thin plate. The materials of Model II are the same with Model I. For the Model III which shown in Fig. 1(c), the inclusion consists of a higher steel core with rubber-steel-rubber coating layers and then embedding into the thin aluminum plate. The unit cells arrange periodically in a square lattice manner and the Brillouin zone is illustrated in Fig. 1(d). The lattice size of the LRPC is a ¼ 0.04 m and the radius of the inclusion is r1 ¼ 0.018 m. The coating layers have the same thickness of Δt ¼ 0.002 m in the radial direction. For model I in Fig. 1(a), the inclusion and the plate have the same height of 0.002 m. In Fig. 1(b) and (c), the height of the inclusions is h ¼ 0.008 m while the height of the plate is e ¼ 0.002 m. The materials constants are given in Table 1. To elucidate the band gap properties of the three types of LRPCs, the band structures and the frequency response are simulated by the ﬁnite element method (FEM). For the structures, the elastic wave motion equation is as follows, r½½λðrÞ þ 2GðrÞðr⋅uÞ r ½GðrÞr u ¼ ρu€

Density (kg/m3)

(2)

where t is the time and k is the wave vector. ω refers to the angular frequency. For the LRPC models, the displacement amplitudes have the form of, uðrÞ ¼ uðr þ n1 a1 þ n2 a2 Þ

(3)

where n1 and n2 refer to integers; a1 and a2 represent the lattice vectors. For the LRPC models, the linear triangular elements are utilized to simulate the band gap properties by the ﬁnite element method (FEM). Elastic wave in three directions are taken into account in the simulations. 3. Results and discussions 3.1. Band structures and mechanisms The band structures and band gap mechanisms of the three types of models shown in Fig. 1(a–c) are discussed in this section. The band structures of the two dimensional LRPCs are shown in Figs. 2, 4 and 6. The band gap boundaries are identiﬁed in the band structures and the corresponding displacements are given in Figs. 3, 5 and 7. Combined with the band structures and the displacement ﬁelds, the band gaps in the in-plane and out-of-plane directions can be revealed. As shown in Fig. 2, it can be found that there are no obvious complete band gaps in the band structure of Model I. The band gap from 90 Hz (see Point 1) to 201 Hz (see Point 4) is for the ﬂexural wave in the out-of-plane direction. One can see from Fig. 3 that the steel core moves as a whole

(1)

where λ and G are the Lame constants of the materials, and ρ is the density. r is the coordinate vector. u and ü are the displacement vector

Fig. 1. The unit cells of the two dimensional LRPCs with (a) Model I, (b) Model II and (c) Model III. (d) The irreducible ﬁrst Brillouin zone of the square lattices.

175

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Journal of Physics and Chemistry of Solids 116 (2018) 174–179

there are two band gaps of which the boundaries are labeled by Points 1, 4, 7 and 10 with the frequencies being 71 Hz, 169 Hz, 237 Hz and 361 Hz. In the XY plane, there are also two wide band gaps which are 129–389 Hz and 499–609 Hz. The displacement ﬁelds at Points 1, 2 and 3 are the steel core moves in different directions. At Points 4, 5 and 6, the plate and the core move oppositely in the Z, X and Y directions. Due to the multi-layers, the core and the steel coating layer move oppositely in the translational patterns in three directions to create another wide band gaps. At the upper boundaries (see Points 10, 11, 12), The core, steel coating layers and the plate move as masses, and the rubber layers act as springs. These mechanisms agree well with the results in Ref. [11]. The multi-layer inclusions open two complete band gaps in the LRPC. 3.2. Transmission spectra To further investigate the complete band gap properties of the two dimensional LRPCs, the transmission spectra of the ﬁnite periodic plates with 5 5 and 8 8 unit cells are investigated. Prescribed displacements Si in both X and Z directions are imposed on the one side of the periodic plates. Then the displacement amplitudes Ax and Az in the X and Z directions are picked on the other side. The frequency response is calculated by 20 log (A/Si). Both the frequency responses in the X and Z directions are take into consideration. In the simulations, the frequency interval is taken as 5 Hz. Fig. 8(a) shows the transmission spectra of the periodic plate with different number of unit cells shown in Fig. 1(b). It can be seen from the curves that there is a complete band gap in which the vibrations in both X and Z directions reduce. The frequency range is from 140 Hz to 315 Hz

Fig. 2. The simulated band structure of the Model I shown in Fig. 1. Points 1–6 refer to the band gap boundaries in different directions.

part in the Z direction at Point 1 while the core and the thin plate move to the opposite direction at Point 4. The band gap in the XY plane is from 189 Hz to 414 Hz. The displacement modes are the translational vibration patterns of the core and the plate in the XY plane as shown in Fig. 3. For Model II, it is interesting to see that there is a wide complete band gap appearing in the band structure which shown in Fig. 4. The band gap is from Point 2 to Point 4 with the frequency range of 130 Hz–315 Hz. The band gap in the out-of-plane direction is from 80 Hz to 315 Hz while the frequency range of the band gap in the in-plane direction is 130–517 Hz. The displacement ﬁelds at the band gap boundaries which labeled by Points 1–6 in Fig. 4 are shown in Fig. 5. It is found from Fig. 5 that the steel core move in the translational patterns in different directions at the beginning of the band gaps. For the upper boundaries at Points 4, 5 and 6, the core and the plate move oppositely in the Z, X and Y directions, respectively. Comparing Model II to Model I, it can be found that by increasing the height of the inclusion, the band gaps in both the in-plane and out-ofplane directions are lowered and extended. The lower boundaries reduce to smaller frequencies while the upper boundaries increase to higher frequencies. And hence a wide complete band gap is opened in this two dimensional LRPC. That is of great importance for vibration control in engineering. Figs. 6 and 7 display the band structure and displacement ﬁeld of the Model III shown in Fig. 1. Compared with Model II, Model III has rubbersteel-rubber multi-layers coating in the steel core. It can be seen from Fig. 6 that there are two obvious complete band gaps in the band structure. The frequency ranges of the two complete band gaps are 129–169 Hz and 237–361 Hz, respectively. In the out-of-plane direction,

Fig. 4. The simulated band structure of the Model II shown in Fig. 1. Points 1–6 refer to the band gap boundaries in different directions.

Fig. 3. The displacement ﬁelds of the Model I at the band gap boundaries which labeled by points 1–6 in Fig. 2. 176

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Journal of Physics and Chemistry of Solids 116 (2018) 174–179

Fig. 5. The displacement ﬁelds of the Model II at the band gap boundaries which labeled by points 1–6 in Fig. 4.

spectra curves shows a positive transmission. That is because there is a straight line in the band structure in Fig. 6 which represent the rotation mode at the frequency of 270 Hz. 3.3. Tunable properties by changing the geometric parameters From the above analysis, it is known that by increasing the height of the inclusions, complete band gaps can be opened in the two dimensional LRPC. The inﬂuence of the inclusion height on the band gap boundaries is discussed in Fig. 9. As shown in Fig. 9(a), when the height of the inclusion is the same with the thin plate, there is no obvious complete band gap. As the increasing of the height, the lower boundary of the complete band gap (see the shadow area) decrease and the upper boundary increase, which widen the complete band gap. For Model III, it can be found from Fig. 9(b) that there are two complete band gaps. The band gap frequencies become lower and wider as improving the height of the inclusions. It provides a good way to open large complete low frequency band gaps in two dimensional LRPC. By changing the geometrical parameters carefully, these periodic structures can be used widely in engineering.

Fig. 6. The simulated band structure of the Model III shown in Fig. 1. Points 1–12 refer to the band gap boundaries in different directions.

4. Conclusions which agree well with the result in Fig. 4. The attenuation effects in the Z direction is much better than those in the X direction. The transmission spectra of the plate for Model III is given in Fig. 8(b). Again, the complete band gaps agree well with the results shown in Fig. 6. The particular point at 270 Hz (marked by the circle in Fig. 8(b)) in the transmission

In conclusion, the band gap properties of three types of two dimensional LRPCs are investigated by the FEM. It is found that complete band gaps can be opened by changing the geometrical properties of the inclusions which are embedded into the thin plate. The translational

Fig. 7. The displacement ﬁelds of the Model III at the band gap boundaries which labeled by points 1–12 in Fig. 6. 177

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Journal of Physics and Chemistry of Solids 116 (2018) 174–179

Fig. 8. Transmission spectra of the periodic plates with different numbers of unit cells with (a) Model II and (b) Model III.

Fig. 9. Dependence of band gap boundaries on the inclusion height of the LPRC with (a) Model II and (b) Model III.

vibration modes in both the in-plane and out-of-plane directions contribute to the complete band gaps. Moreover, the complete band gaps can be lowered and widened by increasing the height of the inclusions. These results can provide good guidance for applications of LRPC in engineering.

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