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Ultra-wide band gap in a two-dimensional phononic crystal with hexagonal lattices
Journal Pre-proof Ultra-wide band gap in a two-dimensional phononic crystal with hexagonal lattices Guilin Wen (Project administration) (Supervision) (Writing - review and editing) (Validation), Haifeng Ou (Formal analysis) (Software) (Writing - original draft) (Validation), Jie Liu (Supervision) (Writing review and editing) (Validation)
PII:
S2352-4928(20)30076-3
DOI:
https://doi.org/10.1016/j.mtcomm.2020.100977
Reference:
MTCOMM 100977
To appear in:
Materials Today Communications
Received Date:
7 January 2020
Revised Date:
3 February 2020
Accepted Date:
3 February 2020
Please cite this article as: Wen G, Ou H, Liu J, Ultra-wide band gap in a two-dimensional phononic crystal with hexagonal lattices, Materials Today Communications (2020), doi: https://doi.org/10.1016/j.mtcomm.2020.100977
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Ultra-wide band gap in a two-dimensional phononic crystal with hexagonal lattices
Guilin Wen*, Haifeng Ou, Jie Liu
Center for Research on Leading Technology of Special Equipment, School of Mechanical and Electric Engineering, Guangzhou University, Guangzhou 510006, P. R. China
G.L.W. (email: [email protected])
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Graphical abstracts
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*Correspondence and requests for materials should be addressed to
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0.4
Mode A
Mode B
B
1.2
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A phononic crystal constituted of the periodic hexagonal lattices
f(kHz)
1.6
0 Γ
143.6%
A M X Wave vector
Γ
0 -200 Transmission(dB)
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An ultra-wide band gap (a BG% of 165%) in the low-mid frequency range (from 250 Hz to 2500 Hz) can be achieved by the phononic crystal, which is generated by the rigid body resonances.
Highlights:
Inspired by the scissor-like geodesic circle of the Hoberman sphere, a novel two-dimensional phononic crystal with periodic hexagonal lattices is proposed.
Based on the finite element method and the Bloch theorem, we verify that the proposed phononic crystal can achieve an ultra-wide band gap in the low-mid frequency range.
Parametric studies are performed to find the most key design geometric parameter
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effecting the band gap.
Abstract: A new two-dimensional (2D) phononic crystal (PC) constituted by the periodic hexagonal lattices, which is inspired by the scissor-like geodesic circle of the Hoberman sphere, is
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proposed in this study. The band structures, transmission spectrum as well as displacement field of the proposed PC are analyzed by employing the finite element method (FEM) with the Bloch
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theorem. Furthermore, the formation mechanism of the band gaps (BGs) is investigated according to the eigenmodes at the BG edges analysis. Results show that the first BG of the proposed PC can
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be achieved in the range of 244.6 Hz to 1490.0 Hz with a gap-mid gap ratio (BG%) of 143.6%, and it is produced via the rigid body resonance. Also, a BG span from 244.6 Hz to 2543.9 Hz with
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a BG% of 164.9% is obtained when an approximate flat band between the first and second BGs is neglected, showing an ultra-wide BG in the low-mid frequency range. We further show that the position and width of the BGs can be adjusted within a broad frequency range by altering the key
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geometric parameters.
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Keywords: Phononic crystals; Band gaps; Transmission spectrum; Hexagonal lattices
1. Introduction In analogy to the method for controlling electromagnetic waves by the photonic crystals [1], PCs constituted by the materials of mass densities and elastic constants with periodic distribution, can greatly affect the propagation of the elastic or sound waves [2]. Since PCs can be efficiently forbidden elastic or sound waves propagation within a specific frequency range, called BG, so the
PCs are also called BG materials [3]. Therefore, the new functional materials and devices based on PCs have potential application prospects in various fields, such as vibration or sound minimization [4-6], acoustic superlens [7], wave guides [8] and frequency filter [9]. Earlier studies have demonstrated that the BGs are formed by three different mechanisms: Bragg scattering, localized resonance, and inertial amplification [10]. The Bragg scattering [11] is attributed to the periodicity of the structure and the coupling between the unit cells. The localized resonance [12,13], which is provided by the resonances of scattering unit cells, is independent on the structural symmetry and periodicity. The wave length of BG in the PCs is two orders greater than
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the lattice constant.
The location and width of BGs are significantly important for the practical application of PCs.
Several studies have been devoted to achieving an ultra-wide BG in different frequency ranges by
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varying the material and geometric parameters of the structure [14-35]. Liu and his co-workers
[14,15] analyzed the effect of pore shapes on the BGs in PCs. Their results show that the cut-off
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bands are more easily formed in triangular pores than square ones. The BGs vary in accordance with the size of the adjustable pores, but there is a crucial porosity for the opening. A novel 2D PC
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constituted by the periodic Jerusalem cross slots with a square lattice-based air matrix was presented; its BGs can be adjusted over an extremely broad frequency range by altering the length
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and width of slots [16]. The method presented in [17] can enlarge the BGs by optimizing geometrical parameters that affect the eigenmodes at the edges of BGs, ultimately achieving a BG% of 138%. Topology optimization method was also extended to engineer the 2D solid/solid
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six-fold symmetric PCs with hexagonal lattices to maximize the specific BGs [18]. Gao et al. [19] put forward a novel PC composed of hollow inclusions with self-similarity, with the capacity to
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alter the width, location and number of BGs. Jiang et al. [20] presented a PC constituted of four square bumps on the sides or a square slab with a cross hole, which achieved an ultra-wide BG with BG% of 156.0%. Li et al. [21] demonstrated that the randomly specified in-plane BGs can be obtained by using a certain amount of hollow or solid round rods periodically arranged in a base. Most of studies on the BGs pay more attention to the high frequency range, but the BGs within the low-mid frequency range are more meaningful for vibration and noise control in practical engineering. In recent years, the studies of ultra-wide BGs within the low-mid frequency range
have attracted more and more attention. Representatively, Wang et al. [22] developed a novel PC constituted
by
periodic
spindle-shaped
plumbum
inclusions,
which
can
achieve
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relatively wide BG in the low-mid frequency range. An elastic metamaterial plate was also employed to realize two complete BGs, whose total width was 94.45 Hz below 200 Hz [23]. Sun et al. [24] proposed a BG within the low frequency range of 153 Hz to 196 Hz by simply combining and improving the traditional PCs. Meanwhile, some researchers have designed PCs with self-similar hierarchies to achieve an ultra-wide BG within the low-mid frequency range. Mousanezhad et al. [25] employed the topological hierarchical architecture and structural
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instability-induced pattern deformations in the PCs with self-similar hierarchies to adjust the width and directionality of BGs. Wang et al. [26] introduced a fractal grading triangle PC and found that
it was easier to obtain complete BGs than traditional ones with the same porosity. PCs with higher fractal levels are beneficial to multiple broader complete BGs. Liu et al. [27] developed a series of
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hierarchical labyrinthine PCs with periodic z-shaped channels, extremely broadening the overall BGs and lowering the first BG frequency. PCs with innovative topological structures have become
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an effective means to explore ultra-wide BGs over the low-mid frequency. Although the lager of the difference of the component elastic constants, the easier to open a broad BG, the final designs
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can be quite complicated or overweight, which brings challenges to structural stability and fabrication. In general, two PC structures with similar BG, we prefer the one which is more
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lightweight and easier to fabricate.
Herein, in order to broaden the BGs within the low-mid frequency range and obtain the
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lightweight PC, a novel 2D PC constituted by periodic hexagonal lattices with an ultra-wide BG is proposed. The band structures, transmission spectrum as well as displacement field of the
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proposed PC are firstly computed by employing the FEM with the Bloch theorem. Then the formation mechanisms of the first and second BGs are analyzed based on the eigenmodes at the BG edges analysis. The starting, cut-off frequency, and BG% of the proposed PC related to the key geometrical parameters, such as the thick beam thickness t1, the angle θ, and the thickness ratio of the thin beam to thick beam t2/t1, are systematically analyzed and discussed.
2. Models and methods
The design concept is inspired by the scissor-like geodesic circle of Hoberman sphere shown in Figure 1(a), which is a centrosymmetric scissor-like structure that resembles a geodesic dome. Referring to its geometric features, a 2D PC with an ultra-wide BG in the low-mid frequency range is presented. We employ equilateral hexagonal lattices instead of all quadrilateral lattices and change hinges into fixed joints (Figure 1b). The 16 equilateral hexagonal lattices are horizontally connected end to end to form a similar circular unit cell, and the beams of the four hexagonal lattices on the diagonal lines of each quadrant are refined. It should be noted that the hexagonal periodic lattice, presented in this study, have to be an equilateral hexagon in order to
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guarantee the unit cell with high symmetry. The infinite system is periodic along the x- and y-directions simultaneously. As shown in Figure 1(c), the proposed PC is characterized by the
following geometrical parameters: the number of hexagonal lattices N, the length of the beams l,
a can be expressed as l s in (1 a 2
1 cos(2 / N )
l c o s t1
(1)
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ta n ( 2 / N )
)
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the angle θ, the thick beam thickness t1, and the thickness of the thin beam t2, so its lattice constant
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Figure 1. (a) The scissor-like geodesic circle of Hoberman sphere; (b) the proposed 2D PC; and (c) its geometrical
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parameters.
The governing equation for the elastic wave propagation within the 2D PC is as follows [36]
2
u
E 2 (1 )
2
u
E 2 (1 )( 1 2 )
( u )
(2)
where u represents the displacement vector; ρ, E, μ are the material density, Young’s modulus, and Poisson’s ratio, respectively; represents the angular frequency; and
represents the
differential operator. Based on the Bloch theorem [37], the displacement vector of the PC can be calculated as
ui (r a ) e
i ( k r wt )
(3)
ui (r )
where r represents the position vector; k is the Block wave vector; and a represents the lattice constant. The Bloch theorem described in Eq. (3) is imposed on the unit cell boundary along the periodic direction, yielding ui (r a ) e
ik a
(4)
ui (r )
Combining the governing equation in the unit cell Eq. (2) and the boundary condition Eq. (4)
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gives the eigenvalue equation ( K - M )U 0 2
(5)
where M and K represent the global mass and stiffness matrices in the unit cell, respectively; U
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represents the displacement vector of the discrete node.
In order to capture the band structure characteristics of the PCs, the eigenvalue equation Eq. (5) is
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solved by using the general-purpose commercial package COMSOL Multiphysics v5.4, Solid Mechanics Module in this study. Bloch’s periodic boundary conditions are imposed on the unit
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cell boundaries. The band structure is calculated by scanning the Bloch wave vector along the first
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irreducible Brillouin zone boundary.
In addition to describing band structures, the BG characteristics are also characterized by the transmission spectrum (frequency response function). For verifying the analysis on the band
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structures of PCs, the transmission spectrum is also calculated with the frequency response analysis module within the COMSOL. A PC is set to a finite number of periods in the x-direction,
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while Bloch’s periodic boundary conditions are imposed on both sides of the y-direction. In the calculation, an external single-frequency plane wave condition such as normal acceleration incentive is exerted on the left side of PC to act as an input source for elastic or sound waves, as shown in Figure 2. Their corresponding transmission acceleration values are collected on the right side. The evaluation of the transmission spectrum is defined as follows T L 2 0 lo g ( a o u t / a in )
(6)
where ain and aout are the incident acceleration and transmitted acceleration, respectively. The transmission spectrum can represent the attenuation ability of elastic or sound wave transmission, i.e. the smaller the transmission spectrum coefficient, the greater the ability to elastic or sound
Figure 2. The schematic setting of the transmission spectrum computation.
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3. Results and discussions
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wave attenuation.
3.1. Band structures and mechanisms
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In this subsection, the band structures and transmission spectrum of the proposed PC are presented, and the mechanism of the BGs formation is investigated. The material parameters used in the
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simulations are as follows: Young’s modulus E=210.6 GPa, shear modulus G=81 GPa, and mass density ρ=7780 kg/m3, and they are kept constant during the discussion below. The geometrical
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parameters of the proposed PC are: N=16, l=15 mm, θ=45°, t1=5 mm, t2/t1=0.1.
Figure 3 presents the band structure and frequency transmission spectrum in the proposed PC.
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From the left of Figure 3, there are 10 bands that are visible within the frequency range of 0-2600 Hz, where two absolute BGs (light green shaded areas) are contributing. The first BG is produced
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between the sixth and seventh bands, ranging from 244.6 Hz to 1490.0 Hz with a BG width of 1245.4 Hz. The relative BG width can be expressed using the BG%: BG%=2(ftop-fdop)/(ftop+fdop), where fdop and ftop represent the lower and upper edge frequencies of the BG, respectively. So the relative BG width of the first BG defined as BG% is 143.6%. The second BG produced by the eighth and ninth bands is between 1495.2 Hz and 2543.9 Hz. The BG width and the BG% are 1048.7 Hz and 51.9%, respectively. A tiny pass band with a band width of only 5.2 Hz lies between the first and second BGs. If this tiny pass band is neglected, an ultra-wide BG span from
244.5 Hz to 2543.9 Hz with a BG% of 164.9% is obtained. In the right picture of the transmission spectrum, there are two clearly frequency ranges, from 250 Hz to 1520 Hz and from 1600 Hz to 2550 Hz, where the attenuation is large. The tiny pass band around 1500 Hz located between two BGs is moved to the higher frequency relative to the band structure in the left figure, but the overall consistency with the band structure diagram is still obvious. The results of the transmission
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spectra and band structure show good agreement, indicating the accuracy of the calculation.
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Figure 3. Comparison between the band structure and the transmission spectrum.
For further analyzing the mechanism of the BGs formation, the eigenmodes that are denoted as A,
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B, C, and D at the edges of the first and second BGs in Figure 3 are presented in Figure 4. The color scale in Figure 4 represents the magnitude of the overall displacement field, where the blue
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and red stand for the minimum and maximum values, respectively. The red arrows represent the vibration displacement direction. The proposed PC can be considered composed of four lumps
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(the hexagonal lattices constructed by thick beams) and four connectors (the hexagonal lattices constructed by thin beams), i.e. the lump-connector system. According to the four eigenmodes, the thick beams vibrate mainly as the rigid bodies, because they withstand much less deformation compared with the thin beams. The lump-connector system can be served as a mass-spring system.
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Figure 4. The eigenmodes of the modes A, B, C, and D at the lower and upper edges of the first and second BGs in
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the proposed PC.
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From Figure 4, for the lower edge of the first BG (marked as FLE), mode A, all beams vibrate outward around their centroids, and the arrows are consistent, indicating that these vibration directions are continuous. Mode B, corresponding to the upper edge of the first BG (marked as
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FUE), is mainly the torsional motion of the thick beams of the x-direction, while the thick beams of the y-direction stand still. Moreover, two thin beams connectors move outward around the
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center of the unit cell, while the other two connectors vibrate inward. Due to the interaction between the local resonance of the thick beams of the x-direction and the in-plane Lamb wave
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modes, the first BG occurs within the lower frequency range. Similar to mode B, mode C, which corresponds to the lower edge of the second BG (marked as SLE), is also dominated by the torsional motion of the thick beams of the x-direction, but a slight torsional motion is also occurring on the thick beam of the y-direction. The vibration is mainly concentrated on the flexural motion of the thin beams in the upper edge of the second BG (marked as SUE), mode D, while the thick beams remain almost stationary. Therefore, the formation mechanism of the first and second BGs is largely due to the translational or flexural motion and the torsional motion
(local energy centralization), i.e. the existence of the localized resonance modes.
In order to further intuitively understand the propagation of plane waves through the proposed PC, Figure 5 displays the displacement field of a PC with five periods at 201 Hz and 301 Hz. It can be obviously seen that the plane wave propagates through the finite PC and the PC deforms at the right end at 201 Hz. However, it attenuates quickly at the first unit cell at 301 Hz, and almost no response is found at the right end. It is easy to understand that at 201Hz, within the pass band frequency range, the plane wave is rarely attenuated, while at 301Hz, just within the BG frequency
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range, the plane wave attenuation is obvious and even totally. These results show that the
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propagation of the elastic or sound waves can be effectively prevented by the proposed PC.
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Figure 5. (a) The displacement vector fields at a frequency of 201 Hz, and (b) 301 Hz.
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3.2 Effects of the geometric parameters
The adjustment of BG is closely related to the practical application of PCs. In order to further broaden the BGs of the proposed PC within the low-mid frequency range, the effects of the key
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geometric parameters on the variation of the starting, cut-off frequency, and the BG% are analyzed. Figures 6-8 show the BGs alteration with the change of the angle θ, the thick beam thickness t1,
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and the thickness ratio of the thin beam to thick beam t2/t1, respectively.
By varying the angle θ from 25° to 65° and with other basic parameters unchanged, it can be computed from Eq.(1) that the lattice constant a is within the range of (171.34, 229.78 mm). Figure 6 shows that the lower edge of the first BG (light yellow area) drops first and slightly increases later as the angle θ increases, while the upper edge of the second BG (light gray area) has almost an opposite variation trend. Both the FUE and SLE shift to lower frequency ranges. A
tiny pass band between the first and second BG is always an approximate flat band. Therefore, the relative width of the first BG (FBG%) and the relative width of the two BGs overlap which is neglected the tiny pass band (NBG%) firstly increased and then decreased. It is worth noting that a FBG% of 150.3% in the frequency range of 247.8 Hz to 1475.8 Hz and a larger NBG% of 165.9%
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in the frequency range of 247.8 Hz to 2663.2 Hz can be obtained when θ=35°, i.e. a=191.49 mm.
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Figure 6. Effects of the angle θ on the FLE, FUE, SLE, SUE, FBG%, and NBG% with N=16, l=15 mm, t1=5 mm,
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t2/t1=0.1.
Figure 7 demonstrates that the FLE, FUE, SLE, and SUE show a linearly increasing trend with the
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thick beam thickness t1 increasing by 1 mm. But the increasing amplitude of the upper edge is larger compared with the lower edge, which broadens the total BGs width. The equivalent mass of the local resonance increases with the increasing of the t1, but at the same time, the equivalent
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stiffness of the local resonance enlarges, therefore leading to the increase of the eigenfrequency. Similarly, the tiny pass band in the middle is also an almost flat band. Unlike the width of BGs,
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the FBG% and NBG% slowly decrease from 143.6% to 115.4% and from 164.9% to 147.0% as t1 increases, respectively. It is interesting to note that a larger FBG% of 143.6% and NBG% of
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164.9% within the lower frequency range (244.6-1490.0 Hz) appears, meaning that t1 has a dominant effect on the width and location within the low-frequency BGs.
Figure 7. Effects of the thick beam thickness t1 on the FLE, FUE, SLE, SUE, FBG%, and NBG% with N=16, l=15
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mm, θ=45°, t2/t1=0.1.
Figure 8(a) presents the influence of the thickness ratio of the thin beam to thick beam t2/t1 on the BGs. The curve of the FLE, FUE, SLE, and SUE has a rise at t2/t1=0.3 and a dip at t2/t1=0.4, and
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the shape is like a bump curve as the t2/t1 becomes large. The FLE will continue to increase by
enlarging the effective stiffness, which is same for the FUE, SLE, and SUE. Nevertheless, a tiny
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BG with BG % of about 4% appears at the lower frequency when t2/t1 from 0.3 to 0.5, so a groove will appear in the curve at t2/t1=0.4. As shown in Figure 8(b), the first BG is no longer produced
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between the sixth and seventh bands as previously analyzed in Figure 3 but is opened between the third and fourth bands, ranging from 1083.3 to 1122.1 Hz with a BG% of 3.5% when t2/t1=0.4.
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The frequency width between the FUE and SLE is gradually broadened as the equivalent stiffness provided by the thin beams magnifies. The FBG% decreases sharply (from 143.6% to 3.5%) when
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t2/t1 increases from 0.1 to 0.4, but is almost unchanged between 0.4 and 0.5.
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Figure 8. (a) Effects of the thickness ratio of the thin beam to thick beam t2/t1 on FLE, FUE, SLE, SUE, and FBG% with N=16, l=15 mm, θ=45°, t1=5 mm. (b) The band structure when t2/t1=0.4.
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4. Conclusions
In summary, a new 2D PC constituted by periodic hexagonal lattices is proposed to achieve an
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ultra-wide BG within the low-mid frequency range, inspired by the scissor-like geodesic circle of the Hoberman sphere. The band structures, transmission spectrum, as well as displacement field
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are analyzed by employing the FEM with the Bloch theorem. Results indicate that the lowest BG of the proposed PC can be opened in the range of 244.6 Hz to 1490.0 Hz with a BG% of 143.6%. Besides, an ultra-wide BG span from 244.6 Hz to 2543.9 Hz with a BG% of 164.9% is obtained
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when an approximate flat band is neglected. Moreover, the attenuation frequency ranges of the transmission spectrum and band structure show good agreement. The formation mechanism of the
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BGs is investigated according to the eigenmodes at the BG edges analysis, revealing that the BGs are primarily generated by the localized resonance modes. We further show that the position and
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width of the BGs can be adjusted within a broad frequency range by altering the key geometric parameters such as the thick beam thickness t1, the angle θ, and the thickness ratio of the thin beam to thick beam t2/t1. It is worth pointing out that a wider band gap can be achieved if structural optimization [38-41] is performed to our present PC structure, which is our on-going work.
Competing Interests: The authors declare no competing interests.
Data availability statement: The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Credit Author Statement:
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G.L.W.: Project administration; Supervision; Writing - review & editing; Validation. H.F.O: Formal analysis; Software; Roles/Writing - original draft; Validation. J.L.: Supervision; Writing review & editing; Validation.
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Acknowledgements
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This research was financially supported by the Key Program of National Natural Science Foundation of China (No.11832009) and the National Natural Science Foundation of China
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comments and suggestions.
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(No.11902085). Meanwhile, the authors are very grateful to the reviewers for their valuable
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PII:
S2352-4928(20)30076-3
DOI:
https://doi.org/10.1016/j.mtcomm.2020.100977
Reference:
MTCOMM 100977
To appear in:
Materials Today Communications
Received Date:
7 January 2020
Revised Date:
3 February 2020
Accepted Date:
3 February 2020
Please cite this article as: Wen G, Ou H, Liu J, Ultra-wide band gap in a two-dimensional phononic crystal with hexagonal lattices, Materials Today Communications (2020), doi: https://doi.org/10.1016/j.mtcomm.2020.100977
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2020 Published by Elsevier.
Ultra-wide band gap in a two-dimensional phononic crystal with hexagonal lattices
Guilin Wen*, Haifeng Ou, Jie Liu
Center for Research on Leading Technology of Special Equipment, School of Mechanical and Electric Engineering, Guangzhou University, Guangzhou 510006, P. R. China
G.L.W. (email: [email protected])
2.4 a
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Graphical abstracts
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*Correspondence and requests for materials should be addressed to
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0.4
Mode A
Mode B
B
1.2
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A phononic crystal constituted of the periodic hexagonal lattices
f(kHz)
1.6
0 Γ
143.6%
A M X Wave vector
Γ
0 -200 Transmission(dB)
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An ultra-wide band gap (a BG% of 165%) in the low-mid frequency range (from 250 Hz to 2500 Hz) can be achieved by the phononic crystal, which is generated by the rigid body resonances.
Highlights:
Inspired by the scissor-like geodesic circle of the Hoberman sphere, a novel two-dimensional phononic crystal with periodic hexagonal lattices is proposed.
Based on the finite element method and the Bloch theorem, we verify that the proposed phononic crystal can achieve an ultra-wide band gap in the low-mid frequency range.
Parametric studies are performed to find the most key design geometric parameter
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effecting the band gap.
Abstract: A new two-dimensional (2D) phononic crystal (PC) constituted by the periodic hexagonal lattices, which is inspired by the scissor-like geodesic circle of the Hoberman sphere, is
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proposed in this study. The band structures, transmission spectrum as well as displacement field of the proposed PC are analyzed by employing the finite element method (FEM) with the Bloch
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theorem. Furthermore, the formation mechanism of the band gaps (BGs) is investigated according to the eigenmodes at the BG edges analysis. Results show that the first BG of the proposed PC can
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be achieved in the range of 244.6 Hz to 1490.0 Hz with a gap-mid gap ratio (BG%) of 143.6%, and it is produced via the rigid body resonance. Also, a BG span from 244.6 Hz to 2543.9 Hz with
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a BG% of 164.9% is obtained when an approximate flat band between the first and second BGs is neglected, showing an ultra-wide BG in the low-mid frequency range. We further show that the position and width of the BGs can be adjusted within a broad frequency range by altering the key
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geometric parameters.
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Keywords: Phononic crystals; Band gaps; Transmission spectrum; Hexagonal lattices
1. Introduction In analogy to the method for controlling electromagnetic waves by the photonic crystals [1], PCs constituted by the materials of mass densities and elastic constants with periodic distribution, can greatly affect the propagation of the elastic or sound waves [2]. Since PCs can be efficiently forbidden elastic or sound waves propagation within a specific frequency range, called BG, so the
PCs are also called BG materials [3]. Therefore, the new functional materials and devices based on PCs have potential application prospects in various fields, such as vibration or sound minimization [4-6], acoustic superlens [7], wave guides [8] and frequency filter [9]. Earlier studies have demonstrated that the BGs are formed by three different mechanisms: Bragg scattering, localized resonance, and inertial amplification [10]. The Bragg scattering [11] is attributed to the periodicity of the structure and the coupling between the unit cells. The localized resonance [12,13], which is provided by the resonances of scattering unit cells, is independent on the structural symmetry and periodicity. The wave length of BG in the PCs is two orders greater than
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the lattice constant.
The location and width of BGs are significantly important for the practical application of PCs.
Several studies have been devoted to achieving an ultra-wide BG in different frequency ranges by
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varying the material and geometric parameters of the structure [14-35]. Liu and his co-workers
[14,15] analyzed the effect of pore shapes on the BGs in PCs. Their results show that the cut-off
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bands are more easily formed in triangular pores than square ones. The BGs vary in accordance with the size of the adjustable pores, but there is a crucial porosity for the opening. A novel 2D PC
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constituted by the periodic Jerusalem cross slots with a square lattice-based air matrix was presented; its BGs can be adjusted over an extremely broad frequency range by altering the length
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and width of slots [16]. The method presented in [17] can enlarge the BGs by optimizing geometrical parameters that affect the eigenmodes at the edges of BGs, ultimately achieving a BG% of 138%. Topology optimization method was also extended to engineer the 2D solid/solid
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six-fold symmetric PCs with hexagonal lattices to maximize the specific BGs [18]. Gao et al. [19] put forward a novel PC composed of hollow inclusions with self-similarity, with the capacity to
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alter the width, location and number of BGs. Jiang et al. [20] presented a PC constituted of four square bumps on the sides or a square slab with a cross hole, which achieved an ultra-wide BG with BG% of 156.0%. Li et al. [21] demonstrated that the randomly specified in-plane BGs can be obtained by using a certain amount of hollow or solid round rods periodically arranged in a base. Most of studies on the BGs pay more attention to the high frequency range, but the BGs within the low-mid frequency range are more meaningful for vibration and noise control in practical engineering. In recent years, the studies of ultra-wide BGs within the low-mid frequency range
have attracted more and more attention. Representatively, Wang et al. [22] developed a novel PC constituted
by
periodic
spindle-shaped
plumbum
inclusions,
which
can
achieve
a
relatively wide BG in the low-mid frequency range. An elastic metamaterial plate was also employed to realize two complete BGs, whose total width was 94.45 Hz below 200 Hz [23]. Sun et al. [24] proposed a BG within the low frequency range of 153 Hz to 196 Hz by simply combining and improving the traditional PCs. Meanwhile, some researchers have designed PCs with self-similar hierarchies to achieve an ultra-wide BG within the low-mid frequency range. Mousanezhad et al. [25] employed the topological hierarchical architecture and structural
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instability-induced pattern deformations in the PCs with self-similar hierarchies to adjust the width and directionality of BGs. Wang et al. [26] introduced a fractal grading triangle PC and found that
it was easier to obtain complete BGs than traditional ones with the same porosity. PCs with higher fractal levels are beneficial to multiple broader complete BGs. Liu et al. [27] developed a series of
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hierarchical labyrinthine PCs with periodic z-shaped channels, extremely broadening the overall BGs and lowering the first BG frequency. PCs with innovative topological structures have become
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an effective means to explore ultra-wide BGs over the low-mid frequency. Although the lager of the difference of the component elastic constants, the easier to open a broad BG, the final designs
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can be quite complicated or overweight, which brings challenges to structural stability and fabrication. In general, two PC structures with similar BG, we prefer the one which is more
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lightweight and easier to fabricate.
Herein, in order to broaden the BGs within the low-mid frequency range and obtain the
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lightweight PC, a novel 2D PC constituted by periodic hexagonal lattices with an ultra-wide BG is proposed. The band structures, transmission spectrum as well as displacement field of the
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proposed PC are firstly computed by employing the FEM with the Bloch theorem. Then the formation mechanisms of the first and second BGs are analyzed based on the eigenmodes at the BG edges analysis. The starting, cut-off frequency, and BG% of the proposed PC related to the key geometrical parameters, such as the thick beam thickness t1, the angle θ, and the thickness ratio of the thin beam to thick beam t2/t1, are systematically analyzed and discussed.
2. Models and methods
The design concept is inspired by the scissor-like geodesic circle of Hoberman sphere shown in Figure 1(a), which is a centrosymmetric scissor-like structure that resembles a geodesic dome. Referring to its geometric features, a 2D PC with an ultra-wide BG in the low-mid frequency range is presented. We employ equilateral hexagonal lattices instead of all quadrilateral lattices and change hinges into fixed joints (Figure 1b). The 16 equilateral hexagonal lattices are horizontally connected end to end to form a similar circular unit cell, and the beams of the four hexagonal lattices on the diagonal lines of each quadrant are refined. It should be noted that the hexagonal periodic lattice, presented in this study, have to be an equilateral hexagon in order to
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guarantee the unit cell with high symmetry. The infinite system is periodic along the x- and y-directions simultaneously. As shown in Figure 1(c), the proposed PC is characterized by the
following geometrical parameters: the number of hexagonal lattices N, the length of the beams l,
a can be expressed as l s in (1 a 2
1 cos(2 / N )
l c o s t1
(1)
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ta n ( 2 / N )
)
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the angle θ, the thick beam thickness t1, and the thickness of the thin beam t2, so its lattice constant
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Figure 1. (a) The scissor-like geodesic circle of Hoberman sphere; (b) the proposed 2D PC; and (c) its geometrical
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parameters.
The governing equation for the elastic wave propagation within the 2D PC is as follows [36]
2
u
E 2 (1 )
2
u
E 2 (1 )( 1 2 )
( u )
(2)
where u represents the displacement vector; ρ, E, μ are the material density, Young’s modulus, and Poisson’s ratio, respectively; represents the angular frequency; and
represents the
differential operator. Based on the Bloch theorem [37], the displacement vector of the PC can be calculated as
ui (r a ) e
i ( k r wt )
(3)
ui (r )
where r represents the position vector; k is the Block wave vector; and a represents the lattice constant. The Bloch theorem described in Eq. (3) is imposed on the unit cell boundary along the periodic direction, yielding ui (r a ) e
ik a
(4)
ui (r )
Combining the governing equation in the unit cell Eq. (2) and the boundary condition Eq. (4)
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gives the eigenvalue equation ( K - M )U 0 2
(5)
where M and K represent the global mass and stiffness matrices in the unit cell, respectively; U
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represents the displacement vector of the discrete node.
In order to capture the band structure characteristics of the PCs, the eigenvalue equation Eq. (5) is
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solved by using the general-purpose commercial package COMSOL Multiphysics v5.4, Solid Mechanics Module in this study. Bloch’s periodic boundary conditions are imposed on the unit
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cell boundaries. The band structure is calculated by scanning the Bloch wave vector along the first
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irreducible Brillouin zone boundary.
In addition to describing band structures, the BG characteristics are also characterized by the transmission spectrum (frequency response function). For verifying the analysis on the band
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structures of PCs, the transmission spectrum is also calculated with the frequency response analysis module within the COMSOL. A PC is set to a finite number of periods in the x-direction,
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while Bloch’s periodic boundary conditions are imposed on both sides of the y-direction. In the calculation, an external single-frequency plane wave condition such as normal acceleration incentive is exerted on the left side of PC to act as an input source for elastic or sound waves, as shown in Figure 2. Their corresponding transmission acceleration values are collected on the right side. The evaluation of the transmission spectrum is defined as follows T L 2 0 lo g ( a o u t / a in )
(6)
where ain and aout are the incident acceleration and transmitted acceleration, respectively. The transmission spectrum can represent the attenuation ability of elastic or sound wave transmission, i.e. the smaller the transmission spectrum coefficient, the greater the ability to elastic or sound
Figure 2. The schematic setting of the transmission spectrum computation.
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3. Results and discussions
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wave attenuation.
3.1. Band structures and mechanisms
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In this subsection, the band structures and transmission spectrum of the proposed PC are presented, and the mechanism of the BGs formation is investigated. The material parameters used in the
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simulations are as follows: Young’s modulus E=210.6 GPa, shear modulus G=81 GPa, and mass density ρ=7780 kg/m3, and they are kept constant during the discussion below. The geometrical
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parameters of the proposed PC are: N=16, l=15 mm, θ=45°, t1=5 mm, t2/t1=0.1.
Figure 3 presents the band structure and frequency transmission spectrum in the proposed PC.
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From the left of Figure 3, there are 10 bands that are visible within the frequency range of 0-2600 Hz, where two absolute BGs (light green shaded areas) are contributing. The first BG is produced
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between the sixth and seventh bands, ranging from 244.6 Hz to 1490.0 Hz with a BG width of 1245.4 Hz. The relative BG width can be expressed using the BG%: BG%=2(ftop-fdop)/(ftop+fdop), where fdop and ftop represent the lower and upper edge frequencies of the BG, respectively. So the relative BG width of the first BG defined as BG% is 143.6%. The second BG produced by the eighth and ninth bands is between 1495.2 Hz and 2543.9 Hz. The BG width and the BG% are 1048.7 Hz and 51.9%, respectively. A tiny pass band with a band width of only 5.2 Hz lies between the first and second BGs. If this tiny pass band is neglected, an ultra-wide BG span from
244.5 Hz to 2543.9 Hz with a BG% of 164.9% is obtained. In the right picture of the transmission spectrum, there are two clearly frequency ranges, from 250 Hz to 1520 Hz and from 1600 Hz to 2550 Hz, where the attenuation is large. The tiny pass band around 1500 Hz located between two BGs is moved to the higher frequency relative to the band structure in the left figure, but the overall consistency with the band structure diagram is still obvious. The results of the transmission
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spectra and band structure show good agreement, indicating the accuracy of the calculation.
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Figure 3. Comparison between the band structure and the transmission spectrum.
For further analyzing the mechanism of the BGs formation, the eigenmodes that are denoted as A,
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B, C, and D at the edges of the first and second BGs in Figure 3 are presented in Figure 4. The color scale in Figure 4 represents the magnitude of the overall displacement field, where the blue
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and red stand for the minimum and maximum values, respectively. The red arrows represent the vibration displacement direction. The proposed PC can be considered composed of four lumps
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(the hexagonal lattices constructed by thick beams) and four connectors (the hexagonal lattices constructed by thin beams), i.e. the lump-connector system. According to the four eigenmodes, the thick beams vibrate mainly as the rigid bodies, because they withstand much less deformation compared with the thin beams. The lump-connector system can be served as a mass-spring system.
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Figure 4. The eigenmodes of the modes A, B, C, and D at the lower and upper edges of the first and second BGs in
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the proposed PC.
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From Figure 4, for the lower edge of the first BG (marked as FLE), mode A, all beams vibrate outward around their centroids, and the arrows are consistent, indicating that these vibration directions are continuous. Mode B, corresponding to the upper edge of the first BG (marked as
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FUE), is mainly the torsional motion of the thick beams of the x-direction, while the thick beams of the y-direction stand still. Moreover, two thin beams connectors move outward around the
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center of the unit cell, while the other two connectors vibrate inward. Due to the interaction between the local resonance of the thick beams of the x-direction and the in-plane Lamb wave
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modes, the first BG occurs within the lower frequency range. Similar to mode B, mode C, which corresponds to the lower edge of the second BG (marked as SLE), is also dominated by the torsional motion of the thick beams of the x-direction, but a slight torsional motion is also occurring on the thick beam of the y-direction. The vibration is mainly concentrated on the flexural motion of the thin beams in the upper edge of the second BG (marked as SUE), mode D, while the thick beams remain almost stationary. Therefore, the formation mechanism of the first and second BGs is largely due to the translational or flexural motion and the torsional motion
(local energy centralization), i.e. the existence of the localized resonance modes.
In order to further intuitively understand the propagation of plane waves through the proposed PC, Figure 5 displays the displacement field of a PC with five periods at 201 Hz and 301 Hz. It can be obviously seen that the plane wave propagates through the finite PC and the PC deforms at the right end at 201 Hz. However, it attenuates quickly at the first unit cell at 301 Hz, and almost no response is found at the right end. It is easy to understand that at 201Hz, within the pass band frequency range, the plane wave is rarely attenuated, while at 301Hz, just within the BG frequency
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range, the plane wave attenuation is obvious and even totally. These results show that the
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propagation of the elastic or sound waves can be effectively prevented by the proposed PC.
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Figure 5. (a) The displacement vector fields at a frequency of 201 Hz, and (b) 301 Hz.
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3.2 Effects of the geometric parameters
The adjustment of BG is closely related to the practical application of PCs. In order to further broaden the BGs of the proposed PC within the low-mid frequency range, the effects of the key
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geometric parameters on the variation of the starting, cut-off frequency, and the BG% are analyzed. Figures 6-8 show the BGs alteration with the change of the angle θ, the thick beam thickness t1,
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and the thickness ratio of the thin beam to thick beam t2/t1, respectively.
By varying the angle θ from 25° to 65° and with other basic parameters unchanged, it can be computed from Eq.(1) that the lattice constant a is within the range of (171.34, 229.78 mm). Figure 6 shows that the lower edge of the first BG (light yellow area) drops first and slightly increases later as the angle θ increases, while the upper edge of the second BG (light gray area) has almost an opposite variation trend. Both the FUE and SLE shift to lower frequency ranges. A
tiny pass band between the first and second BG is always an approximate flat band. Therefore, the relative width of the first BG (FBG%) and the relative width of the two BGs overlap which is neglected the tiny pass band (NBG%) firstly increased and then decreased. It is worth noting that a FBG% of 150.3% in the frequency range of 247.8 Hz to 1475.8 Hz and a larger NBG% of 165.9%
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in the frequency range of 247.8 Hz to 2663.2 Hz can be obtained when θ=35°, i.e. a=191.49 mm.
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Figure 6. Effects of the angle θ on the FLE, FUE, SLE, SUE, FBG%, and NBG% with N=16, l=15 mm, t1=5 mm,
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t2/t1=0.1.
Figure 7 demonstrates that the FLE, FUE, SLE, and SUE show a linearly increasing trend with the
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thick beam thickness t1 increasing by 1 mm. But the increasing amplitude of the upper edge is larger compared with the lower edge, which broadens the total BGs width. The equivalent mass of the local resonance increases with the increasing of the t1, but at the same time, the equivalent
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stiffness of the local resonance enlarges, therefore leading to the increase of the eigenfrequency. Similarly, the tiny pass band in the middle is also an almost flat band. Unlike the width of BGs,
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the FBG% and NBG% slowly decrease from 143.6% to 115.4% and from 164.9% to 147.0% as t1 increases, respectively. It is interesting to note that a larger FBG% of 143.6% and NBG% of
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164.9% within the lower frequency range (244.6-1490.0 Hz) appears, meaning that t1 has a dominant effect on the width and location within the low-frequency BGs.
Figure 7. Effects of the thick beam thickness t1 on the FLE, FUE, SLE, SUE, FBG%, and NBG% with N=16, l=15
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mm, θ=45°, t2/t1=0.1.
Figure 8(a) presents the influence of the thickness ratio of the thin beam to thick beam t2/t1 on the BGs. The curve of the FLE, FUE, SLE, and SUE has a rise at t2/t1=0.3 and a dip at t2/t1=0.4, and
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the shape is like a bump curve as the t2/t1 becomes large. The FLE will continue to increase by
enlarging the effective stiffness, which is same for the FUE, SLE, and SUE. Nevertheless, a tiny
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BG with BG % of about 4% appears at the lower frequency when t2/t1 from 0.3 to 0.5, so a groove will appear in the curve at t2/t1=0.4. As shown in Figure 8(b), the first BG is no longer produced
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between the sixth and seventh bands as previously analyzed in Figure 3 but is opened between the third and fourth bands, ranging from 1083.3 to 1122.1 Hz with a BG% of 3.5% when t2/t1=0.4.
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The frequency width between the FUE and SLE is gradually broadened as the equivalent stiffness provided by the thin beams magnifies. The FBG% decreases sharply (from 143.6% to 3.5%) when
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t2/t1 increases from 0.1 to 0.4, but is almost unchanged between 0.4 and 0.5.
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Figure 8. (a) Effects of the thickness ratio of the thin beam to thick beam t2/t1 on FLE, FUE, SLE, SUE, and FBG% with N=16, l=15 mm, θ=45°, t1=5 mm. (b) The band structure when t2/t1=0.4.
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4. Conclusions
In summary, a new 2D PC constituted by periodic hexagonal lattices is proposed to achieve an
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ultra-wide BG within the low-mid frequency range, inspired by the scissor-like geodesic circle of the Hoberman sphere. The band structures, transmission spectrum, as well as displacement field
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are analyzed by employing the FEM with the Bloch theorem. Results indicate that the lowest BG of the proposed PC can be opened in the range of 244.6 Hz to 1490.0 Hz with a BG% of 143.6%. Besides, an ultra-wide BG span from 244.6 Hz to 2543.9 Hz with a BG% of 164.9% is obtained
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when an approximate flat band is neglected. Moreover, the attenuation frequency ranges of the transmission spectrum and band structure show good agreement. The formation mechanism of the
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BGs is investigated according to the eigenmodes at the BG edges analysis, revealing that the BGs are primarily generated by the localized resonance modes. We further show that the position and
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width of the BGs can be adjusted within a broad frequency range by altering the key geometric parameters such as the thick beam thickness t1, the angle θ, and the thickness ratio of the thin beam to thick beam t2/t1. It is worth pointing out that a wider band gap can be achieved if structural optimization [38-41] is performed to our present PC structure, which is our on-going work.
Competing Interests: The authors declare no competing interests.
Data availability statement: The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
Declaration of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Credit Author Statement:
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G.L.W.: Project administration; Supervision; Writing - review & editing; Validation. H.F.O: Formal analysis; Software; Roles/Writing - original draft; Validation. J.L.: Supervision; Writing review & editing; Validation.
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Acknowledgements
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This research was financially supported by the Key Program of National Natural Science Foundation of China (No.11832009) and the National Natural Science Foundation of China
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comments and suggestions.
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(No.11902085). Meanwhile, the authors are very grateful to the reviewers for their valuable
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