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Experimental evidence of large complete bandgaps in zig-zag lattice structures
Accepted Manuscript Short communication Experimental evidence of large complete bandgaps in zig-zag lattice structures Cheng-Lin Yang, Sheng-Dong Zhao, Yue-Sheng Wang PII: DOI: Reference:
S0041-624X(16)30208-6 http://dx.doi.org/10.1016/j.ultras.2016.10.004 ULTRAS 5392
To appear in:
Ultrasonics
Received Date: Revised Date: Accepted Date:
1 July 2016 7 October 2016 7 October 2016
Please cite this article as: C-L. Yang, S-D. Zhao, Y-S. Wang, Experimental evidence of large complete bandgaps in zig-zag lattice structures, Ultrasonics (2016), doi: http://dx.doi.org/10.1016/j.ultras.2016.10.004
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Experimental evidence of large complete bandgaps in zig-zag lattice structures Cheng-Lin Yang, Sheng-Dong Zhao, Yue-Sheng Wang* Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, PR China Abstract In this paper, experimental evidence of large complete bandgaps in a kind of light-weighted zig-zag lattice structure (ZLS) is presented. Ultrasonic experiments are conducted on the stainless steel slab designed with ZLS to detect the complete bandgaps. Also, the numerical simulations of the experiments by the finite element method are carried out. For comparison, we conduct the same experiments and numerical simulations on the stainless steel slab with straight lattice structure (SLS). Good agreement is obtained between the experimental and numerical results. The complete bandgaps of ZLS are successfully tested and no complete bandgap is found in SLS. The band structures and vibration modes of both ZLS and SLS are calculated via the finite element method to understand the experimental data. The effects of the geometry parameters of ZLS on the complete bandgaps are discussed in detail. Keywords: elastic wave, zig-zag lattice structure, complete bandgaps, finite element method. 1. Introduction Phononic crystals are periodic composite materials. The internal periodic structures of these materials lead to the formation of elastic wave bandgaps, i.e. frequency ranges where the elastic wave propagation is prohibited [1]. Phononic crystals have received considerable attention for their potential applications, such as sound insulation [2], wave filters [3] and other acoustic devices [4]. Lattices with periodic structures may be viewed as a kind of two-dimensional (2D) phononic crystals and have already been studied extensively these years
[5-7]
. Gonella
[8]
analyzed the wave
propagation characteristics of the hexagonal lattices with various internal angles including the re-entrant configurations. And together with Celli
[9]
, he developed a method based on a
microstructure-induced relaxation of the cell symmetry for spatial wavefield manipulation via phononic crystals. Liu et al.
[10]
introduced elaborately designed cantilever beams into
conventional square lattices and found that the local resonance bandgaps appeared due to the natural resonances of the auxiliary cantilever beams. Lim et al. [11] found that hybrid bandgaps can be achieved in fractal-inspired self-similar beam lattices without embedding additional resonators. Mousanezhad et al.
[12]
investigated the influence of the structural hierarchy and imposed
* Corresponding author. Tel.:+86 10 51688417; fax: +86 10 51682094. E-mail address: [email protected] (Y.-S. Wang).
deformation on band structures of self-similar hierarchical honeycombs. They discovered that the hierarchy tends to shift the existing bandgaps to lower frequencies while opening up new bandgaps. Xu and Chen
[13]
compared the wave characteristics of 2D second-order hierarchy
hexagonal lattice structures with those of first-order traditional hexagonal lattice structures and concluded that the structure hierarchy is favorable to forming bandgaps because the deformation or vibration of the second-order structure elements can localize some energy of waves. Wang et al. [14]
found that the beams of highly connected lattices act as mechanical resonators and cause the
generation of locally resonant bandgaps whose presence and width depend on the average lattice connectivity. In addition to above motioned situations, the combination of different materials in lattices also helps the generation of bandgaps. Wen et al. [15] analyzed the propagation characteristics of flexural waves in a 2D periodic grid-like structure consisting of two different materials. Liu et al. [16]
integrated a 2D chiral lattice with inclusions for low-frequency bandgaps. Besides, the
topology of the unit cell in lattices plays a significant role in wave propagation of such a kind of 2D phononic crystals. Dong et al. [17] used the non-dominated sorting-based genetic algorithm II to perform the multi-objective optimization of 2D porous phononic crystals. Their results show that the optimized structures with large bandgaps have solid lumps with narrow connections which lead to local resonance. In Ref. [18] they argued that the symmetry reduction of the unit cell can make the lattices have a bigger bandgap with a smaller average mass density. The numerical research of Wang et al.
[19, 20]
shows that the lattice structures with straight
arms are almost incapable of generating the complete bandgaps, while those with zig-zag arms can exhibit multiple complete bandgaps. However, no experimental support to this conclusion has been reported yet. In this paper, we will present experimental evidence of large complete bandgaps in a properly designed zig-zag lattice slabs. The numerical simulations of the experiments by the finite element method are also conducted. Good agreement is obtained between the experimental and numerical results. Finally, we discuss the effects of the geometric parameters on the bandgaps via finite element calculations. 2. Design of the slabs with zig-zag lattice and straight lattice structures A square slab with the zig-zag lattice structure (ZLS) was fabricated by laser cutting technique and is shown in Fig. 1a. The material of the slab is stainless steel whose Young’s 2
modulus, Poisson’s ratio and density are, respectively, E = 2.0×1011 Pa, ν = 0.27, ρ = 7900 kg/m3. The size of the slab is 700×860 mm2, and the thickness is 2.66 mm. There is a 30×30 mm2 homogenous square plate in the center of the slab which is surrounded by 8 rows of the ZLS. The outer region is homogenous stainless steel plate again. The square unit cell of the ZLS is displayed in green in Fig. 1b. The length of the unit cell (i.e. the lattice constant) a=20 mm. It has the 90-degree rotational symmetry with its geometry determined by the central lines (black solid lines) of the bending arms. These lines meet in the center of the unit cell. The cross parts of the beam are named as the main arms, and the left parts as the secondary arms. The geometrical parameters h = 8 mm and v = 3 mm represent the horizontal and vertical distances between the bending point of the arm and the center, respectively. The semi-lengths and widths of the main and secondary arms are
lm v 2 h 2 8.544 mm , ls v 2 (a / 2 h)2 3.606 mm ,
(1)
wm hw0 / v 2 h 2 3.745 mm , ws (a / 2 h)w0 / v 2 (a / 2 h)2 2.219 mm .
(2)
The right-handed coordinate system with the origin at the center of the model is set up. The directions of the x and y axis are shown in Fig. 1b and the z axis is perpendicular to the slab. For comparison, another stainless steel slab with straight lattice structure (SLS) was also fabricated by laser cutting technique with the same size, thickness and arrangement (see Fig. 1c). The unit cell of the SLS is depicted in gray in Fig. 1b. The arms’ thickness is w0 = 4 mm. Moreover, the porosities of these two lattice structures are the same (63.5%). The irreducible Brillouin zones corresponding to the ZLS and SLS is shown in Fig. 1d. 3. Results of experiments and numerical simulations We conducted ultrasonic experiments
[21]
on the slabs with ZLSs to test the existence of the
transmission attenuation of the elastic wave in a certain frequency range passing through the lattice structures. The flow chart of the experiment is shown in Fig. 2. A signal was generated by a Tektronix AFG 3102 arbitrary function generator and amplified by an AG 1006 LF amplifier generator (T&C Power Conversion). It was then transferred into out-of-plane vibration through a Noliac NCE41 disk PZT transducer (8 mm diameter, 0.53 mm thickness) which was affixed at the central point of the slab surface. The vibration signals were received by two same PZT transducers; one of which (transducer ①) was attached on the slab surface 20 mm away from the lattice area 3
along the Γ-X direction, and the other (transducer ②) at 28.28 mm away from the corner along Γ-M direction, see Fig. 2 for details. Both the input and output signals were transmitted to a computer for analysis through Tektronix TDS digital phosphor oscilloscopes. It should be noticed that our measurement which follows the technique used in Ref. [21] is not for the directional bandgap testing. The exciting PZT transducer excites elastic waves propagating in all direction, part of which propagates through the finite lattice region. The receiving transducer receives the scattered wave signals from other directions besides Г-X direction. Therefore, although the PZT receiving transducers are attached along the Г-X and Г-M directions, they are incapable of measuring the directional bandgaps. The input signal is the continuous modulated signal which is given by
U (t ) A0 [1 cos(2 t )]sin(2 f ct ) , where U , A0 , t and
f c represent respectively voltage, amplitude, time and central
frequency. Because the frequency of modulation wave is selected as 1 kHz which is unchangeable when f c changes from 10 kHz to 110 kHz, the number of the cycles of the carrying wave within a wave packet of the waveform equals to the value of f c /(1kHz). As an example, the input signal with f c =30 kHz is shown in Fig. 3. It has 30 cycles within a wave packet. Both the time and frequency domains of the input and output signals are illustrated in Fig.4. It is seen that the energy of both the input and output signals focuses on 30 kHz. This implies that the frequency of 30 kHz locates in the pass band. However, for the input signal with the frequency of 60 kHz, no energy focus of the output signal appears, see Fig. 5. This indicates that the propagation of the wave at the frequency of 60 kHz is prohibited. With the aim of detecting the frequency range of the bandgap, the sweep frequency method with the frequency ranging from 10 kHz to 110 kHz by an increment of 1 kHz was used. The ratio of the amplitude in voltage spectrum (the spectrum of the voltage amplitude density, i.e. the FFTs of the time domain signals) of the output signal to the input signal at each sampling frequency is recorded. The experimental results are presented by the 20 times of the logarithm of this ratio. This ratio also reflects the ratio of the amplitude of the out-of-plane displacement of the receiving point to the exciting point. 4
At the same time, finite element numerical simulations of the experiments were conducted with COMSOL Multiphysics 3.5a. The harmonic out-of-plane displacement with the unit amplitude is applied on the excitation source, the point at which the exciting PZT transducer is attached, and then the displacements at the points where the two receiving PZT transducers are attached are computed. The wave transmission is measured by the 20 times of the logarithm of the ratio of the amplitudes of out-of-plane displacements of the receiving points to the exciting point. The same experiments and numerical simulations are carried out on the slab with SLSs (Fig. 1c). Figs. 6a, 6c, 7a and 7c display the numerical (the red lines) and experimental (the black lines) results (), showing a good agreement between them. For the slab with ZLSs, the signal attenuation in two frequency ranges of (50.7 kHz, 75.5 kHz) and (35.8 kHz, 47.6 kHz) is shown in both the Γ-X and Γ-M directions. For the slab with SLSs, no signal attenuation exists, indicating no bandgap in the SLSs. 4. Analysis of band structures and wave modes To further understand the experimental data, we calculate the band structures and wave modes using the finite element method. The software COMSOL Multiphysics 3.5a is used. The Acoustic Module operating under the three-dimensional (3D) solid, stress-strain Application Mode (acsld) is applied, and Eigenfrequency analysis is utilized as the solver mode [22]. The 3D unit cell is meshed by using the default tetrahedral mesh with 11264 Lagrange quadratic elements provided by COMSOL. And we apply Bloch boundary conditions on the opposite boundaries of the unit cell. With the wave vector sweeping the edges of the irreducible Brillouin zones (see Fig. 1d), the band structures are obtained and shown in Fig. 6b. In order to manifest the in-plane or out-of-plane wave modes represented by the points on the curves, we define a polarization factor p as p
u s
s
2 x
2
uz ds 2
u y uz
2
ds
,
(3)
where s is the volume of the unit cell; and u x , u y , u z are, respectively, the displacement components along x, y and z axes. With p varying from 0 to 1, we apply a color to every
p=
0.0001 of this range where blue represents 0 and red represents 1, making the color change from blue to red gradually. Therefore, the colors close to red mean that the vibration modes are 5
dominantly out-of-plane polarized, while the colors close to blue signifies that the vibration modes are dominantly in-plane polarized. From Fig. 6b, a complete bandgap of the full vibration modes is found, whose upper and lower boundaries are determined by points A and B, respectively. The frequency range of this bandgap is between 50.7 kHz and 75.5 kHz, with the relative bandgap width (gap-to-midgap ratio, i.e. the ratio of the width to central frequency of the bandgap) being 0.393. Besides, a complete bandgap for the out-of-plane vibration mode is also observed with a frequency range between 35.8 Hz and 47.6 kHz. The vibration modes of points C and D are out-of-plane polarized, determining the boundaries of the bandgap; and those of the points within the bandgap (e.g. point E) are in-plane polarized. The complete bandgaps in the band structures of ZLS are covered by the light gray rectangles (Fig. 6b) whose extensions also coincide well with the regions of wave propagation attenuation in both Γ-M and Γ-M directions (see Figs. 6a and 6c). And it is seen that the complete bandgaps for full vibration modes and out-of-plane vibration mode are experimentally demonstrated. Meanwhile, the band structures of SLS are also calculated, as shown in Fig. 7b. No complete bandgap exists, corresponding to the fact that no wave propagation attenuation appears in the experiment (see Figs. 7a and 7c). 5. Effects of geometric parameters on bandgaps Both above experiments and numerical calculations demonstrate that the lattice structures indeed can exhibit multiple wide complete bandgaps if the straight arms are replaced with zig-zag arms. It should be worthwhile to present further discussion on turning bandgap properties by adjusting the geometric parameters of ZLS. To this end, we have calculated the complete bandgaps of full vibration modes for different values of the geometric parameters of ZLS, h and v, i.e. the horizontal and vertical distances between the bending point of the arms and the center of the unit cell, see Fig. 1b. In addition, the porosity of the structure was assumed unchanged with the value of 63.5%. Fig. 8a displays the variation of the bandgap edges (light color) and the corresponding relative bandgap width (dark color) with v increasing from 0 to 8 mm for h = 8 mm. When 2 mm < v < 8 mm, multiple complete bandgaps appear: the first one between the 8th and 9th bands, the second one between 9th and 10th bands, the third one between 11th and 12th bands and the forth 6
one between the 13th and 14th bands. As v increases, the width of the first and third bandgaps increases and that of the second and fourth bandgaps first increases and then decreases. The widest bandgap is the second one. When v is close to 5 mm, it has the biggest relative bandwidth of 0.455. In addition, with the increase of v, multiple bandgaps appears at low frequencies and are close each other. When v varies from 1 to 8 mm with h unchanged, the rotation angle of the main arms increases. The mass of the main and secondary arms remains the same, respectively. The lengths of both main and secondary arms increase while the widths decrease. The secondary arm can be modeled as a spring, and its effective stiffness is determined by the tensile rigidity which is proportional to the cross section area and inversely proportional to the length, and the bending rigidity which is proportional to the section inertia and inversely proportional to the length [23, 24]. The decrease of the width leads to the decrease of the cross section area and section inertia, which together with the increase of the length reduces the effective stiffness of the secondary arms, and therefore results in the appearance of more complete bandgaps for different vibration modes at lower frequencies, as shown in Fig. 8a. The bandgaps with h varying from 0 to 9 mm for v = 5 mm are shown in Fig. 8b. When 2 mm < h < 9 mm, three complete bandgaps appear between 8th and 9th bands, 9th and 10th bands, and 13th and 14th bands in order. With the increase of h, the width of the first and the third bandgaps first increases and then decreases, and that of the second bandgap increases monotonically. The second bandgap is the widest and has the biggest relative width of 0.507 when h is close to 9 mm. When h varies from 1 to 9 mm with v unchanged, the rotation angel of the main arms decreases. Both the length and width of the main arms increases and those of the secondary arms decreases; the mass of the main arms becomes bigger while that of the secondary arms smaller. It is seen in Fig. 8b that the higher mass of the main arms is conducive to the generation of the complete bandgaps with bigger relative bandgap widths at lower frequencies. The above results provide a way of tuning bandgaps by adjusting the geometric parameters of ZLSs. 6. Conclusion In this work, a slab with zig-zag lattice structures has been designed. Ultrasonic experiments 7
have been conducted on the slab to measure the bandgaps, followed by the numerical simulations by the finite element method. The band structures and wave modes are calculated to understand the experimental results. Finally, the effects of the geometric parameters on the bandgaps are discussed. For comparison, we have also carried out experiments and numerical calculations on a slab with straight lattice structures. From the experimental and numerical results, we can draw the conclusions as follows: 1)
The introduction of the bending arms in lattice structures is helpful for the generation of the complete bandgaps: No complete bandgap exists in the straight-arm lattice. However, multiple complete bandgaps appear in the zig-zag lattice structure. Good consistency between the experimental and numerical results is shown.
2)
The geometric parameters of the zig-zag lattice have significant influences on the bandgaps. The main arms with bigger mass are favorable for generation of a complete bandgap with a bigger relative width at lower frequencies; and the secondary arms with lower stiffness are favorable for the generation of complete bandgaps for more different vibration modes at lower frequencies. Acknowledgements
Support by National Natural Science Foundation of China under Grant Number 11272041 is gratefully acknowledged. The authors are grateful to Yan-Feng Wang and Tian-Xue Ma for assistance with the numerical calculations.
Reference [1] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Acoustic band structure of periodic elastic composites, Phys. Rev. Lett. 71 (1993) 2022-2025. [2] K.M. Ho, C.K. Cheng, Z. Yang, X.X. Zhang, P. Sheng, Broadband locally resonant sonic shields, Appl. Phys. Lett. 83 (2003) 5566. [3] Y. Pennec, B. Djafari-Rouhani, J.O. Vasseur, A. Khelif, P.A. Deymier, Tunable filtering and demultiplexing in phononic crystals with hollow cylinders, Phys. Rev. E 69 (2004) 046608. [4] M.I. Hussein, M.J. Leamy, M. Ruzzene, Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook, Appl. Mech. Rev. 66 (2014) 040802. [5] D.J. Mead, Wave propagation in continuous periodic structures: research contributions from Southampton, J. Sound. Vib. 190 (1996) 495-524. 8
[6] R.S. Langley, N.S. Bardell, The response of two-dimensional periodic structures to harmonic point loading: a theoretical and experimental study of a beam grillage, J. Sound. Vib. 207 (1997) 521-535. [7] C. Chesnais, C. Boutin, S. Hans, Effects of the local resonance on the wave propagation in periodic frame structures: generalized Newtonian mechanics, J. Acoust. Soc. Am. 132 (2012) 2873-2886. [8] S. Gonella, M. Ruzzene, Analysis of in-plane wave propagation in hexagonal and re-entrant lattices, J. Sound. Vib. 312 (2008) 125-139. [9] P. Celli, S. Gonella, Low-frequency spatial wave manipulation via phononic crystals with relaxed cell symmetry, J. Appl. Phys. 115 (2014) 103502. [10] W. Liu, J.W. Chen, X.Y. Su, Local resonance phononic band gaps in modified two-dimensional lattice materials, Acta. Mech. Sin. 28 (2012) 659-669. [11] Q.J. Lim, P. Wang, S.J.A. Koh, E.H. Khoo, K. Bertoldi, Wave propagation in fractal-inspired self-similar beam lattices, Appl. Phys. Lett. 107 (2015) 221911. [12] D. Mousanezhad, S. Babaee, R. Ghosh, E. Mahdi, K. Bertoldi, A. Vaziri, Honeycomb phononic crystals with self-similar hierarchy, Phys. Rev. B 92 (2015) 104304. [13] Y.L. Xu, C.Q. Chen, X.G. Tian, Wave characteristics of two-dimensional hierarchical hexagonal lattice structures, J. Vib. Acoust. 136 (2014) 011011. [14] P. Wang, F. Casadei, S.H. Kang, K. Bertoldi, Locally resonant band gaps in periodic beam lattices by tuning connectivity, Phys. Rev. B 91 (2015) 020103. [15] J. Wen, D. Yu, G. Wang, X. Wen, Directional propagation characteristics of flexural wave in two-dimensional periodic grid-like structures, J. Phys. D: Appl. Phys. 41 (2008) 135505. [16] X.N. Liu, G.K. Hu, C.T. Sun, G.L. Huang, Wave propagation characterization and design of two-dimensional elastic chiral metacomposite, J. Sound. Vib. 330 (2011) 2536-2553. [17] H.W. Dong, X.X. Su, Y.S. Wang, Multi-objective optimization of two-dimensional porous phononic crystals, J. Phys. D: Appl. Phys. 47 (2014) 155301. [18] H.W. Dong, Y.S. Wang, Y.F. Wang, Ch. Zhang, Reducing symmetry in topology optimization of two-dimensional porous phononic crystals, AIP Adv. 5 (2015) 117149. [19] Y.F. Wang, Y.S. Wang, Ch. Zhang, Bandgaps and directional propagation of elastic waves in 2D square zigzag lattice structures, J. Phys. D: Appl. Phys. 47 (2014) 485102. [20] Y.F. Wang, Y.S. Wang, Ch. Zhang, Bandgaps and directional properties of two-dimensional square beam-like zigzag Lattices, AIP Adv. 4 (2014) 124403. 9
[21] M. Miniaci, A. Marzani, N. Testoni, L.D. Marchi, Complete band gaps in a polyvinyl chloride (PVC) phononic plate with cross-like holes: numerical design and experimental verification, Ultrasonics 56 (2015) 251-259. [22] Y.F. Wang, Y.S. Wang, X.X. S, Large bandgaps of two-dimensional phononic crystals with cross-like holes, J. Appl. Phys. 110 (2011) 113520. [23] Y.F. Wang, Y.S. Wang, Complete bandgaps in two-dimensional phononic crystal slabs with resonators, J. Appl. Phys. 114 (2013) 043509. [24] W.T. Thomson, Theory of Vibration with Applications, Prentice-Hall, USA, 1972.
10
Figure 1. (a) The designed slab with ZLS and (b) unit cell of ZLS. (c) The designed slab with SLS. (d) Irreducible Brillouin zones. Figure 2. Flow chart of ultrasonic experiments. Figure 3. The input signal with the central frequency of 30 kHz. Figure 4. The input signals with the central frequency of 30 kHz in time (a) and frequency (b) domains, and the associated output signals in time (c) and frequency (d) domains. Figure 5. The input signals with the central frequency of 60 kHz in time (a) and frequency (b) domains, and the associated output signals in time (c) and frequency (d) domains. Figure 6. Experimental (a) and numerical (c) results along Γ-X and Γ-Y directions in the slab of ZLSs, and the band structure of the ZLS (b). Figure 7. Experimental (a) and numerical (c) results along Γ-X and Γ-Y directions in the slab of SLSs, and the band structure of SLS (b). Figure 8. The band edges and the corresponding relative bandgap widths with the increase of geometric parameters v (a) and h (b).
11
12
13
14
15
Highlights
Ultrasonic tests and numerical calculations are conducted on slabs with lattices. Zig-zag lattices are helpful for the generation of complete bandgaps. Straight lattices are difficult to generate complete bandgaps. The effects of geometric parameters of zig-zag lattices on bandgaps are discussed.
16
S0041-624X(16)30208-6 http://dx.doi.org/10.1016/j.ultras.2016.10.004 ULTRAS 5392
To appear in:
Ultrasonics
Received Date: Revised Date: Accepted Date:
1 July 2016 7 October 2016 7 October 2016
Please cite this article as: C-L. Yang, S-D. Zhao, Y-S. Wang, Experimental evidence of large complete bandgaps in zig-zag lattice structures, Ultrasonics (2016), doi: http://dx.doi.org/10.1016/j.ultras.2016.10.004
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Experimental evidence of large complete bandgaps in zig-zag lattice structures Cheng-Lin Yang, Sheng-Dong Zhao, Yue-Sheng Wang* Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, PR China Abstract In this paper, experimental evidence of large complete bandgaps in a kind of light-weighted zig-zag lattice structure (ZLS) is presented. Ultrasonic experiments are conducted on the stainless steel slab designed with ZLS to detect the complete bandgaps. Also, the numerical simulations of the experiments by the finite element method are carried out. For comparison, we conduct the same experiments and numerical simulations on the stainless steel slab with straight lattice structure (SLS). Good agreement is obtained between the experimental and numerical results. The complete bandgaps of ZLS are successfully tested and no complete bandgap is found in SLS. The band structures and vibration modes of both ZLS and SLS are calculated via the finite element method to understand the experimental data. The effects of the geometry parameters of ZLS on the complete bandgaps are discussed in detail. Keywords: elastic wave, zig-zag lattice structure, complete bandgaps, finite element method. 1. Introduction Phononic crystals are periodic composite materials. The internal periodic structures of these materials lead to the formation of elastic wave bandgaps, i.e. frequency ranges where the elastic wave propagation is prohibited [1]. Phononic crystals have received considerable attention for their potential applications, such as sound insulation [2], wave filters [3] and other acoustic devices [4]. Lattices with periodic structures may be viewed as a kind of two-dimensional (2D) phononic crystals and have already been studied extensively these years
[5-7]
. Gonella
[8]
analyzed the wave
propagation characteristics of the hexagonal lattices with various internal angles including the re-entrant configurations. And together with Celli
[9]
, he developed a method based on a
microstructure-induced relaxation of the cell symmetry for spatial wavefield manipulation via phononic crystals. Liu et al.
[10]
introduced elaborately designed cantilever beams into
conventional square lattices and found that the local resonance bandgaps appeared due to the natural resonances of the auxiliary cantilever beams. Lim et al. [11] found that hybrid bandgaps can be achieved in fractal-inspired self-similar beam lattices without embedding additional resonators. Mousanezhad et al.
[12]
investigated the influence of the structural hierarchy and imposed
* Corresponding author. Tel.:+86 10 51688417; fax: +86 10 51682094. E-mail address: [email protected] (Y.-S. Wang).
deformation on band structures of self-similar hierarchical honeycombs. They discovered that the hierarchy tends to shift the existing bandgaps to lower frequencies while opening up new bandgaps. Xu and Chen
[13]
compared the wave characteristics of 2D second-order hierarchy
hexagonal lattice structures with those of first-order traditional hexagonal lattice structures and concluded that the structure hierarchy is favorable to forming bandgaps because the deformation or vibration of the second-order structure elements can localize some energy of waves. Wang et al. [14]
found that the beams of highly connected lattices act as mechanical resonators and cause the
generation of locally resonant bandgaps whose presence and width depend on the average lattice connectivity. In addition to above motioned situations, the combination of different materials in lattices also helps the generation of bandgaps. Wen et al. [15] analyzed the propagation characteristics of flexural waves in a 2D periodic grid-like structure consisting of two different materials. Liu et al. [16]
integrated a 2D chiral lattice with inclusions for low-frequency bandgaps. Besides, the
topology of the unit cell in lattices plays a significant role in wave propagation of such a kind of 2D phononic crystals. Dong et al. [17] used the non-dominated sorting-based genetic algorithm II to perform the multi-objective optimization of 2D porous phononic crystals. Their results show that the optimized structures with large bandgaps have solid lumps with narrow connections which lead to local resonance. In Ref. [18] they argued that the symmetry reduction of the unit cell can make the lattices have a bigger bandgap with a smaller average mass density. The numerical research of Wang et al.
[19, 20]
shows that the lattice structures with straight
arms are almost incapable of generating the complete bandgaps, while those with zig-zag arms can exhibit multiple complete bandgaps. However, no experimental support to this conclusion has been reported yet. In this paper, we will present experimental evidence of large complete bandgaps in a properly designed zig-zag lattice slabs. The numerical simulations of the experiments by the finite element method are also conducted. Good agreement is obtained between the experimental and numerical results. Finally, we discuss the effects of the geometric parameters on the bandgaps via finite element calculations. 2. Design of the slabs with zig-zag lattice and straight lattice structures A square slab with the zig-zag lattice structure (ZLS) was fabricated by laser cutting technique and is shown in Fig. 1a. The material of the slab is stainless steel whose Young’s 2
modulus, Poisson’s ratio and density are, respectively, E = 2.0×1011 Pa, ν = 0.27, ρ = 7900 kg/m3. The size of the slab is 700×860 mm2, and the thickness is 2.66 mm. There is a 30×30 mm2 homogenous square plate in the center of the slab which is surrounded by 8 rows of the ZLS. The outer region is homogenous stainless steel plate again. The square unit cell of the ZLS is displayed in green in Fig. 1b. The length of the unit cell (i.e. the lattice constant) a=20 mm. It has the 90-degree rotational symmetry with its geometry determined by the central lines (black solid lines) of the bending arms. These lines meet in the center of the unit cell. The cross parts of the beam are named as the main arms, and the left parts as the secondary arms. The geometrical parameters h = 8 mm and v = 3 mm represent the horizontal and vertical distances between the bending point of the arm and the center, respectively. The semi-lengths and widths of the main and secondary arms are
lm v 2 h 2 8.544 mm , ls v 2 (a / 2 h)2 3.606 mm ,
(1)
wm hw0 / v 2 h 2 3.745 mm , ws (a / 2 h)w0 / v 2 (a / 2 h)2 2.219 mm .
(2)
The right-handed coordinate system with the origin at the center of the model is set up. The directions of the x and y axis are shown in Fig. 1b and the z axis is perpendicular to the slab. For comparison, another stainless steel slab with straight lattice structure (SLS) was also fabricated by laser cutting technique with the same size, thickness and arrangement (see Fig. 1c). The unit cell of the SLS is depicted in gray in Fig. 1b. The arms’ thickness is w0 = 4 mm. Moreover, the porosities of these two lattice structures are the same (63.5%). The irreducible Brillouin zones corresponding to the ZLS and SLS is shown in Fig. 1d. 3. Results of experiments and numerical simulations We conducted ultrasonic experiments
[21]
on the slabs with ZLSs to test the existence of the
transmission attenuation of the elastic wave in a certain frequency range passing through the lattice structures. The flow chart of the experiment is shown in Fig. 2. A signal was generated by a Tektronix AFG 3102 arbitrary function generator and amplified by an AG 1006 LF amplifier generator (T&C Power Conversion). It was then transferred into out-of-plane vibration through a Noliac NCE41 disk PZT transducer (8 mm diameter, 0.53 mm thickness) which was affixed at the central point of the slab surface. The vibration signals were received by two same PZT transducers; one of which (transducer ①) was attached on the slab surface 20 mm away from the lattice area 3
along the Γ-X direction, and the other (transducer ②) at 28.28 mm away from the corner along Γ-M direction, see Fig. 2 for details. Both the input and output signals were transmitted to a computer for analysis through Tektronix TDS digital phosphor oscilloscopes. It should be noticed that our measurement which follows the technique used in Ref. [21] is not for the directional bandgap testing. The exciting PZT transducer excites elastic waves propagating in all direction, part of which propagates through the finite lattice region. The receiving transducer receives the scattered wave signals from other directions besides Г-X direction. Therefore, although the PZT receiving transducers are attached along the Г-X and Г-M directions, they are incapable of measuring the directional bandgaps. The input signal is the continuous modulated signal which is given by
U (t ) A0 [1 cos(2 t )]sin(2 f ct ) , where U , A0 , t and
f c represent respectively voltage, amplitude, time and central
frequency. Because the frequency of modulation wave is selected as 1 kHz which is unchangeable when f c changes from 10 kHz to 110 kHz, the number of the cycles of the carrying wave within a wave packet of the waveform equals to the value of f c /(1kHz). As an example, the input signal with f c =30 kHz is shown in Fig. 3. It has 30 cycles within a wave packet. Both the time and frequency domains of the input and output signals are illustrated in Fig.4. It is seen that the energy of both the input and output signals focuses on 30 kHz. This implies that the frequency of 30 kHz locates in the pass band. However, for the input signal with the frequency of 60 kHz, no energy focus of the output signal appears, see Fig. 5. This indicates that the propagation of the wave at the frequency of 60 kHz is prohibited. With the aim of detecting the frequency range of the bandgap, the sweep frequency method with the frequency ranging from 10 kHz to 110 kHz by an increment of 1 kHz was used. The ratio of the amplitude in voltage spectrum (the spectrum of the voltage amplitude density, i.e. the FFTs of the time domain signals) of the output signal to the input signal at each sampling frequency is recorded. The experimental results are presented by the 20 times of the logarithm of this ratio. This ratio also reflects the ratio of the amplitude of the out-of-plane displacement of the receiving point to the exciting point. 4
At the same time, finite element numerical simulations of the experiments were conducted with COMSOL Multiphysics 3.5a. The harmonic out-of-plane displacement with the unit amplitude is applied on the excitation source, the point at which the exciting PZT transducer is attached, and then the displacements at the points where the two receiving PZT transducers are attached are computed. The wave transmission is measured by the 20 times of the logarithm of the ratio of the amplitudes of out-of-plane displacements of the receiving points to the exciting point. The same experiments and numerical simulations are carried out on the slab with SLSs (Fig. 1c). Figs. 6a, 6c, 7a and 7c display the numerical (the red lines) and experimental (the black lines) results (), showing a good agreement between them. For the slab with ZLSs, the signal attenuation in two frequency ranges of (50.7 kHz, 75.5 kHz) and (35.8 kHz, 47.6 kHz) is shown in both the Γ-X and Γ-M directions. For the slab with SLSs, no signal attenuation exists, indicating no bandgap in the SLSs. 4. Analysis of band structures and wave modes To further understand the experimental data, we calculate the band structures and wave modes using the finite element method. The software COMSOL Multiphysics 3.5a is used. The Acoustic Module operating under the three-dimensional (3D) solid, stress-strain Application Mode (acsld) is applied, and Eigenfrequency analysis is utilized as the solver mode [22]. The 3D unit cell is meshed by using the default tetrahedral mesh with 11264 Lagrange quadratic elements provided by COMSOL. And we apply Bloch boundary conditions on the opposite boundaries of the unit cell. With the wave vector sweeping the edges of the irreducible Brillouin zones (see Fig. 1d), the band structures are obtained and shown in Fig. 6b. In order to manifest the in-plane or out-of-plane wave modes represented by the points on the curves, we define a polarization factor p as p
u s
s
2 x
2
uz ds 2
u y uz
2
ds
,
(3)
where s is the volume of the unit cell; and u x , u y , u z are, respectively, the displacement components along x, y and z axes. With p varying from 0 to 1, we apply a color to every
p=
0.0001 of this range where blue represents 0 and red represents 1, making the color change from blue to red gradually. Therefore, the colors close to red mean that the vibration modes are 5
dominantly out-of-plane polarized, while the colors close to blue signifies that the vibration modes are dominantly in-plane polarized. From Fig. 6b, a complete bandgap of the full vibration modes is found, whose upper and lower boundaries are determined by points A and B, respectively. The frequency range of this bandgap is between 50.7 kHz and 75.5 kHz, with the relative bandgap width (gap-to-midgap ratio, i.e. the ratio of the width to central frequency of the bandgap) being 0.393. Besides, a complete bandgap for the out-of-plane vibration mode is also observed with a frequency range between 35.8 Hz and 47.6 kHz. The vibration modes of points C and D are out-of-plane polarized, determining the boundaries of the bandgap; and those of the points within the bandgap (e.g. point E) are in-plane polarized. The complete bandgaps in the band structures of ZLS are covered by the light gray rectangles (Fig. 6b) whose extensions also coincide well with the regions of wave propagation attenuation in both Γ-M and Γ-M directions (see Figs. 6a and 6c). And it is seen that the complete bandgaps for full vibration modes and out-of-plane vibration mode are experimentally demonstrated. Meanwhile, the band structures of SLS are also calculated, as shown in Fig. 7b. No complete bandgap exists, corresponding to the fact that no wave propagation attenuation appears in the experiment (see Figs. 7a and 7c). 5. Effects of geometric parameters on bandgaps Both above experiments and numerical calculations demonstrate that the lattice structures indeed can exhibit multiple wide complete bandgaps if the straight arms are replaced with zig-zag arms. It should be worthwhile to present further discussion on turning bandgap properties by adjusting the geometric parameters of ZLS. To this end, we have calculated the complete bandgaps of full vibration modes for different values of the geometric parameters of ZLS, h and v, i.e. the horizontal and vertical distances between the bending point of the arms and the center of the unit cell, see Fig. 1b. In addition, the porosity of the structure was assumed unchanged with the value of 63.5%. Fig. 8a displays the variation of the bandgap edges (light color) and the corresponding relative bandgap width (dark color) with v increasing from 0 to 8 mm for h = 8 mm. When 2 mm < v < 8 mm, multiple complete bandgaps appear: the first one between the 8th and 9th bands, the second one between 9th and 10th bands, the third one between 11th and 12th bands and the forth 6
one between the 13th and 14th bands. As v increases, the width of the first and third bandgaps increases and that of the second and fourth bandgaps first increases and then decreases. The widest bandgap is the second one. When v is close to 5 mm, it has the biggest relative bandwidth of 0.455. In addition, with the increase of v, multiple bandgaps appears at low frequencies and are close each other. When v varies from 1 to 8 mm with h unchanged, the rotation angle of the main arms increases. The mass of the main and secondary arms remains the same, respectively. The lengths of both main and secondary arms increase while the widths decrease. The secondary arm can be modeled as a spring, and its effective stiffness is determined by the tensile rigidity which is proportional to the cross section area and inversely proportional to the length, and the bending rigidity which is proportional to the section inertia and inversely proportional to the length [23, 24]. The decrease of the width leads to the decrease of the cross section area and section inertia, which together with the increase of the length reduces the effective stiffness of the secondary arms, and therefore results in the appearance of more complete bandgaps for different vibration modes at lower frequencies, as shown in Fig. 8a. The bandgaps with h varying from 0 to 9 mm for v = 5 mm are shown in Fig. 8b. When 2 mm < h < 9 mm, three complete bandgaps appear between 8th and 9th bands, 9th and 10th bands, and 13th and 14th bands in order. With the increase of h, the width of the first and the third bandgaps first increases and then decreases, and that of the second bandgap increases monotonically. The second bandgap is the widest and has the biggest relative width of 0.507 when h is close to 9 mm. When h varies from 1 to 9 mm with v unchanged, the rotation angel of the main arms decreases. Both the length and width of the main arms increases and those of the secondary arms decreases; the mass of the main arms becomes bigger while that of the secondary arms smaller. It is seen in Fig. 8b that the higher mass of the main arms is conducive to the generation of the complete bandgaps with bigger relative bandgap widths at lower frequencies. The above results provide a way of tuning bandgaps by adjusting the geometric parameters of ZLSs. 6. Conclusion In this work, a slab with zig-zag lattice structures has been designed. Ultrasonic experiments 7
have been conducted on the slab to measure the bandgaps, followed by the numerical simulations by the finite element method. The band structures and wave modes are calculated to understand the experimental results. Finally, the effects of the geometric parameters on the bandgaps are discussed. For comparison, we have also carried out experiments and numerical calculations on a slab with straight lattice structures. From the experimental and numerical results, we can draw the conclusions as follows: 1)
The introduction of the bending arms in lattice structures is helpful for the generation of the complete bandgaps: No complete bandgap exists in the straight-arm lattice. However, multiple complete bandgaps appear in the zig-zag lattice structure. Good consistency between the experimental and numerical results is shown.
2)
The geometric parameters of the zig-zag lattice have significant influences on the bandgaps. The main arms with bigger mass are favorable for generation of a complete bandgap with a bigger relative width at lower frequencies; and the secondary arms with lower stiffness are favorable for the generation of complete bandgaps for more different vibration modes at lower frequencies. Acknowledgements
Support by National Natural Science Foundation of China under Grant Number 11272041 is gratefully acknowledged. The authors are grateful to Yan-Feng Wang and Tian-Xue Ma for assistance with the numerical calculations.
Reference [1] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Acoustic band structure of periodic elastic composites, Phys. Rev. Lett. 71 (1993) 2022-2025. [2] K.M. Ho, C.K. Cheng, Z. Yang, X.X. Zhang, P. Sheng, Broadband locally resonant sonic shields, Appl. Phys. Lett. 83 (2003) 5566. [3] Y. Pennec, B. Djafari-Rouhani, J.O. Vasseur, A. Khelif, P.A. Deymier, Tunable filtering and demultiplexing in phononic crystals with hollow cylinders, Phys. Rev. E 69 (2004) 046608. [4] M.I. Hussein, M.J. Leamy, M. Ruzzene, Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook, Appl. Mech. Rev. 66 (2014) 040802. [5] D.J. Mead, Wave propagation in continuous periodic structures: research contributions from Southampton, J. Sound. Vib. 190 (1996) 495-524. 8
[6] R.S. Langley, N.S. Bardell, The response of two-dimensional periodic structures to harmonic point loading: a theoretical and experimental study of a beam grillage, J. Sound. Vib. 207 (1997) 521-535. [7] C. Chesnais, C. Boutin, S. Hans, Effects of the local resonance on the wave propagation in periodic frame structures: generalized Newtonian mechanics, J. Acoust. Soc. Am. 132 (2012) 2873-2886. [8] S. Gonella, M. Ruzzene, Analysis of in-plane wave propagation in hexagonal and re-entrant lattices, J. Sound. Vib. 312 (2008) 125-139. [9] P. Celli, S. Gonella, Low-frequency spatial wave manipulation via phononic crystals with relaxed cell symmetry, J. Appl. Phys. 115 (2014) 103502. [10] W. Liu, J.W. Chen, X.Y. Su, Local resonance phononic band gaps in modified two-dimensional lattice materials, Acta. Mech. Sin. 28 (2012) 659-669. [11] Q.J. Lim, P. Wang, S.J.A. Koh, E.H. Khoo, K. Bertoldi, Wave propagation in fractal-inspired self-similar beam lattices, Appl. Phys. Lett. 107 (2015) 221911. [12] D. Mousanezhad, S. Babaee, R. Ghosh, E. Mahdi, K. Bertoldi, A. Vaziri, Honeycomb phononic crystals with self-similar hierarchy, Phys. Rev. B 92 (2015) 104304. [13] Y.L. Xu, C.Q. Chen, X.G. Tian, Wave characteristics of two-dimensional hierarchical hexagonal lattice structures, J. Vib. Acoust. 136 (2014) 011011. [14] P. Wang, F. Casadei, S.H. Kang, K. Bertoldi, Locally resonant band gaps in periodic beam lattices by tuning connectivity, Phys. Rev. B 91 (2015) 020103. [15] J. Wen, D. Yu, G. Wang, X. Wen, Directional propagation characteristics of flexural wave in two-dimensional periodic grid-like structures, J. Phys. D: Appl. Phys. 41 (2008) 135505. [16] X.N. Liu, G.K. Hu, C.T. Sun, G.L. Huang, Wave propagation characterization and design of two-dimensional elastic chiral metacomposite, J. Sound. Vib. 330 (2011) 2536-2553. [17] H.W. Dong, X.X. Su, Y.S. Wang, Multi-objective optimization of two-dimensional porous phononic crystals, J. Phys. D: Appl. Phys. 47 (2014) 155301. [18] H.W. Dong, Y.S. Wang, Y.F. Wang, Ch. Zhang, Reducing symmetry in topology optimization of two-dimensional porous phononic crystals, AIP Adv. 5 (2015) 117149. [19] Y.F. Wang, Y.S. Wang, Ch. Zhang, Bandgaps and directional propagation of elastic waves in 2D square zigzag lattice structures, J. Phys. D: Appl. Phys. 47 (2014) 485102. [20] Y.F. Wang, Y.S. Wang, Ch. Zhang, Bandgaps and directional properties of two-dimensional square beam-like zigzag Lattices, AIP Adv. 4 (2014) 124403. 9
[21] M. Miniaci, A. Marzani, N. Testoni, L.D. Marchi, Complete band gaps in a polyvinyl chloride (PVC) phononic plate with cross-like holes: numerical design and experimental verification, Ultrasonics 56 (2015) 251-259. [22] Y.F. Wang, Y.S. Wang, X.X. S, Large bandgaps of two-dimensional phononic crystals with cross-like holes, J. Appl. Phys. 110 (2011) 113520. [23] Y.F. Wang, Y.S. Wang, Complete bandgaps in two-dimensional phononic crystal slabs with resonators, J. Appl. Phys. 114 (2013) 043509. [24] W.T. Thomson, Theory of Vibration with Applications, Prentice-Hall, USA, 1972.
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Figure 1. (a) The designed slab with ZLS and (b) unit cell of ZLS. (c) The designed slab with SLS. (d) Irreducible Brillouin zones. Figure 2. Flow chart of ultrasonic experiments. Figure 3. The input signal with the central frequency of 30 kHz. Figure 4. The input signals with the central frequency of 30 kHz in time (a) and frequency (b) domains, and the associated output signals in time (c) and frequency (d) domains. Figure 5. The input signals with the central frequency of 60 kHz in time (a) and frequency (b) domains, and the associated output signals in time (c) and frequency (d) domains. Figure 6. Experimental (a) and numerical (c) results along Γ-X and Γ-Y directions in the slab of ZLSs, and the band structure of the ZLS (b). Figure 7. Experimental (a) and numerical (c) results along Γ-X and Γ-Y directions in the slab of SLSs, and the band structure of SLS (b). Figure 8. The band edges and the corresponding relative bandgap widths with the increase of geometric parameters v (a) and h (b).
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Highlights
Ultrasonic tests and numerical calculations are conducted on slabs with lattices. Zig-zag lattices are helpful for the generation of complete bandgaps. Straight lattices are difficult to generate complete bandgaps. The effects of geometric parameters of zig-zag lattices on bandgaps are discussed.
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