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Multiple wide complete bandgaps of two-dimensional phononic crystal slabs with cross-like holes Yan-Feng Wang, Yue-Sheng Wang n Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, China

a r t i c l e i n f o

abstract

Article history: Received 11 July 2012 Received in revised form 16 November 2012 Accepted 17 November 2012 Handling Editor: L.G. Tham Available online 3 January 2013

This paper investigates the bandgap properties of two-dimensional phononic crystal slabs with cross-like holes by using the ﬁnite element method. The effects of the slab thickness and the geometry parameters of the holes on the bandgaps are discussed. No bandgap appears in the system with the square holes if the symmetry of the holes is the same as that of the lattice. However if the square holes are replaced with the crosslike holes, multiple wide bandgaps at lower frequencies are generated. The bandgaps are signiﬁcantly dependent on the slab thickness and the geometry (including the size, shape and rotation) of the cross-like holes. The vibration modes at the bandgap edges are computed and analyzed to clarify the mechanism of the bandgap generation. It is found that the generation of the bandgap is due to the ﬂexural vibration localized in the lumps or connectors. The appearances of the higher modes with the cut-off frequencies which decrease with the thickness increasing result in the decrease of the upper bandgap edge, and ﬁnally lead to vanishing of the bandgap. The study in this paper could be indispensable to practical applications of phononic crystal slabs such as bandgap tuning. & 2012 Elsevier Ltd. All rights reserved.

1. Introduction Propagation of elastic waves in inhomogeneous media has attracted much attention in the last decades. Phononic crystals [1] (PCs) are media with periodically varying elastic properties and mass densities. The unique characteristic of PCs is the possibility of achieving a complete bandgap, within which the propagation of elastic or acoustic waves is forbidden regardless of the polarization and wave vector. Thus PCs may have potential applications in design of perfect acoustic mirrors, sound isolation, ﬁlters, or waveguides [2]. Tuning of bandgaps is very essential to the applications of PCs. Much work has been reported on this topic [3–7]. Depending on the composition and the geometry as well as on the elastic properties of the constitutive materials, the bandgaps may originate from the Bragg reﬂections resulting from the periodicity of the structure or may be due to the existence of the local resonance [8,9] in each unit cell. Recently, an increasing attention has been focused on PC slabs [10–16], typically PC slabs formed by etching periodic holes in a solid matrix [10–15] with their bandgaps mainly determined by the geometry parameters. Elastic waves can be scattered not only by periodically arranged scatterers but also by the free surfaces of the slab. The PC slabs are naturally well suited for making waveguide in integrated structure as the elastic wave will be conﬁned within the thickness direction and manipulated by the geometry of the structure in the plane of periodicity [16].

n

Corresponding author. E-mail address: [email protected] (Y.-S. Wang).

0022-460X/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2012.11.031

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Leamy [10] studied the dispersion relations of the honeycomb structures with square, diamond and hexagonal lattices and found that only the hexagonal honeycomb exhibits a low-frequency bandgap. Mohammadi et al. [11] explored the planar structure composed of periodic circular cylindrical holes. The results showed that the simultaneous bandgaps for phonons and photons are obtainable in both square and hexagonal lattices, but not for the triangular lattice. Huang et al. [12] presented numerical and experimental studies on two-dimensional PC gratings. Bandgaps and reﬂections of Lamb waves were analyzed. Vasseur et al. [13] demonstrated the existence of the guided modes inside a linear defect created by removing one row of holes in the PZT5A slab. Negative refraction experiments were conducted by Lee et al. [14] with guided shear-horizontal waves in a thin PC plate with circular holes. Wu et al. [15] demonstrated the focusing of Lamb waves in a gradient-index PC slab. The beam width of the lowest antisymmetric Lamb mode can be efﬁciently compressed. As one may notice, all of the above-mentioned works involve PC slabs with convex (circular- or regular-polygonal) holes. However, it was found that when non-convex holes are introduced, photonic crystal slabs can display a broad stop band [17] or a dual-stop band [18] (or even multi-stop band). Safavi-Naeini et al. [19] proposed a slab structure with ‘snowﬂake-shaped’ holes and found that simultaneous phononic and photonic bandgaps are achievable if the geometry of the ‘snowﬂake’ is tailored. A series of fascinating effects has also been revealed for PC slabs with non-convex holes. A microscale inverse acoustic bandgap structure [20] was designed and demonstrated with each cylinder supported by four thin tethers, and a wide complete bandgap was formed in the very high frequency range. Zhu et al. [21] investigated a thin metamaterial plate with different cantilever–mass systems. Low-frequency bandgap can be obtained due to the local resonance of the mass at the tip of the cantilever in the unit cell. Kuo et al. [22] examined the porous fractal PC slabs; the frequency of the operation of the bandgaps can be increased by effectively decreasing the scattering length within the unit cell. Recently, we proposed a kind of inﬁnite PCs with cross-like (a kind of non-convex) holes [6]. Resonant unit cells may be formed by the careful design of the geometry of the holes. Therefore wider and lower bandgaps were easily obtained due to the local resonance of the unit cell. In this paper, we will study the bandgaps of the Lamb waves in a two-dimensional PC slab with cross-like cylindrical holes in a square lattice. These systems lend themselves to numerical simulation by ﬁnite element method (FEM). Multiple large complete bandgaps are observed in the proposed structures. 2. Problem statement and computational method We consider two kinds of cross-like cylindrical holes embedded in an isotropic elastic solid slab in a square lattice as shown in Fig. 1 where a is the lattice constant and h is the thickness of the slab. Fig. 1(a) and (b) show the cross-sections of the holes which are denoted as the ‘ þ’-hole and ‘x’-hole, respectively. The geometries of their cross-sections are determined by b, c for the ‘ þ’-hole and b, c, d for the ‘x’-hole, respectively. For comparison, we also consider the cases of square and circular holes. Suppose that elastic waves propagate along the periodic plane (i.e. the x–y plane). The z-axis is along the thickness direction. The equations describing the time-harmonic motions in an inhomogeneous slab can be expressed by

rUðCðrÞ : ruðrÞÞ þ rðrÞo2 uðrÞ ¼ 0,

(1)

where r ¼ (q/qx,q/qy,q/qz) is the 3D Nabla; r¼(x,y,z) denotes the position vector; u(r) is the elastic displacement vector; o is the circular frequency; r(r) is the mass density; and C(r) is the elasticity tensor. The two surfaces of the slab are traction-free. According to the Bloch theorem, the elastic displacement can be written in the following form: uðrÞ ¼ eiðkUrÞ uk ðrÞ,

(2)

where k ¼(kx, ky) is the wave vector; and uk(r) is a periodical vector function with the same periodicity as the crystal lattice. In the present work, the commonly used FEM [12–15,19–24] is applied to calculate the band structures of the considered PC slabs. Compared with other traditional methods, such as the plane-wave expansion (PWE) method [11,14,25], the ﬁnite difference time domain (FDTD) method [26,27] etc., the FEM has some merits such as its

c b h

b

d

c

h

a

Fig. 1. Unit cells of the PC slabs with (a) the ‘þ ’-holes and (b) the ‘x’-holes.

a

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compatibility, good convergence, high accuracy and efﬁciency, etc. Due to the periodicity of the PC slab, the calculation can be implemented in a representative unit cell. The unit cell is meshed adaptively according to the variation of the structures and divided into ﬁnite elements connected by nodes. In this work, we ﬁrst use the advanced triangular mesh for the upper surface and then use the swept mesh for the whole unit cell. It should be pointed out that the number of the swept mesh layers will increase adaptively when the thickness of the plate increases so as to have sufﬁciently good convergence in an acceptable calculation time. The discrete form of the eigenvalue equations in the unit cell can be written as Ko2 M U ¼ 0, (3) where U is the displacements at the nodes; and K and M are the stiffness and mass matrices of the unit cell, respectively. The Bloch theorem (2) is applied on the boundaries of the unit cell in the direction where periodicity applies, yielding the following relation between the displacements U(r) for the nodes on the boundary of the unit cell: Uðr þ aÞ ¼ eiðkdaÞ UðrÞ:

(4)

The traction free conditions are considered in the free slab surface, that is, the traction components on the surfaces are zero. Then Comsol Multiphysics 3.5a is utilized to directly solve the eigenvalue Eq. (3) under the complex boundary condition (4). The model built in COMSOL is saved as a MATLAB-compatible ‘.m’ ﬁle. The ﬁle is programmed to let the wave vector k sweep the edges of the irreducible Brillouin zone, so that we can obtain the whole dispersion relations. It was proved in Ref. [28] that the Bloch condition implied on the boundaries will grantee that all the points in the whole periodic structure obey the Bloch theorem. 3. Numerical results and discussions In this section we present detailed numerical results of the band structures for various systems with different geometry parameters (including the thickness, size, shape and rotation angle of the holes) and analyze the mechanism of the appearance of the large bandgaps. The elastic parameters of the Aluminum matrix are r ¼2700 kgm 3, E¼20 GPa and n ¼ 0.25. The band structures for the systems with the cross-like, square and circular holes of the particular sizes are illustrated in Fig. 2, where the reduced frequency O ¼ oa/2pct with the lattice constant a¼0.02 m and the transverse wave velocity of the matrix ct ¼1721 m/s. It is shown that no bandgap appears in the system with the square holes (Fig. 2(a)). However, if the square holes are replaced by the ‘ þ’-holes, two complete bandgaps appear in the band structures (shown as the shadowed region in Fig. 2(b)). The lower complete bandgap lies between the 6th and 7th bands in the frequency range of 0.33o O o0.47 with the normalized bandgap width Do/og ¼0.36, where Do and og are the width and the central frequency of the bandgap, respectively. The upper bandgap extends from O ¼0.48 to 0.62 and the normalized bandgap width is 0.25. One may also notice that these two complete bandgap are almost connected because the 7th band is nearly ﬂat. Replacement of the square holes by the ‘x’-holes results in a complete bandgap between the 12th and 13th bands in the frequency range of 0.48o O o0.55 with Do/og ¼0.14. Besides, other two complete bandgaps appear in the higher frequency ranges. Although a bandgap appears in the system with the circular holes [see Fig. 2(d)], its position is higher and its width is smaller than the ﬁrst bandgaps in the systems with the cross-like holes. If the ﬁlling ratio of the system with the ‘x’-holes and that of the system with the circular holes are the same as that of the system with the ‘ þ ’holes whose band structures are shown in Fig. 2(b), no complete bandgap exists, see Fig. 2(e) and (f). The above results imply the signiﬁcant inﬂuences of the hole shape on the bandgaps. A cross-like hole seems favorable to open a larger and lower bandgap which has potential practical applications. In order to understand the mechanism of the bandgaps, we calculate the vibration modes at the edges of the lowest complete bandgaps. The upper and lower band edges are marked as F2 and F3 for the ‘ þ’-holes, F5 and F6 for the ‘x’-holes, and F9 and F10 for the circular holes, respectively [see Fig. 2]. The vibration modes, which are all ﬂexure-dominated modes, are demonstrated in Fig. 3. Fig. 3(a) and (b) show, respectively, the lower (F2) and upper (F3) edge modes of the ‘ þ’-holes. It is noted that the square lattice of the ‘ þ’holes can be regarded as a periodic arrangement of the square lumps connected with the narrow connectors [6]. For the lower-edge mode [Fig. 3(a)], the lumps rotate with respect to the median plane, while the central parts of the connectors are still. For the upper-edge mode [Fig. 3(b)], the wings of the lumps [6] in one unit cell vibrate evidently in the same phase, and the center of the lump is still. The square lattice of the ‘x’-holes may be viewed as a double periodic array of the elements [6]. Each element consists of one rectangular lump and two narrow connectors. It is shown in Fig. 3(c) that the lumps rotate with the axis perpendicular to the two neighboring connectors for the lower-edge (F5) mode, while that the connectors vibrate distinctly with the wings of the lump vibrating in the reverse phase for the upper-edge (F6) mode in Fig. 3(d). From the above analysis, we can conclude that the generation of the wide low-frequency bandgaps in the systems with the cross-like holes is due to the local resonance [24,26,29] of the resonant elements with large lumps connected by the narrow connectors. In the system with the circular holes, we can also ﬁnd such resonant elements, see Fig. 3(e) and (f) which illustrate the vibration modes at the lower (F9) and upper (F10) edges of the complete bandgap, respectively. The lower-edge mode is similar to that in Fig. 3(a); while for the upper-edge mode, the connectors vibrate with the central parts of the lump being still, like laterally vibrating beams with two ﬁxed ends. However, one cannot ﬁnd lumps in a square lattice with the square holes of which the band structures in Fig. 2(a) exhibit no bandgaps.

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0.8

0.6 b/a=0.9 h/a=0.4 =0

0.4

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0.8

X

M

0.0

M

c/a=0.9 b/a=0.4 d/a=0.3 h/a=0.4

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F6

F5

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r/a=0.35 h/a=0.4

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0.2

0.2

0.0

b/a=0.9 c/a=0.25 h/a=0.4

0.6

M

X

M

0.0

M

Fig. 2. Band structures of PC slabs with (a) the square holes, (b) the ‘þ ’-holes, (c) and (e) the ‘x’-holes as well as (d) and (f) the circular holes. The ﬁlling ratio of the ‘x’-holes in panel (e) and that of the circular holes in panel (f) are the same with that of the ‘ þ ’-holes in panel (b).

Although circular holes may result in bandgaps, it is inconvenient to tune the bandgaps through geometric design of the elements because the system involves only two geometry parameters — the hole radius and the slab thickness. However, the proposed cross-like holes involve multiple geometry parameters, which provide us various ways to tune the bandgaps. In the rest part of this paper, we will analyze the effects of the geometries of the two kinds of cross-like holes on the bandgaps. 3.1. Effects of the slab thickness The behaviors of the band structures versus the ratio of the thickness to the lattice constant, h/a, are shown in Figs. 4–6, where the ﬁxed cross-sections of these holes are the same as those in Fig. 2(b)–(d), respectively. The Lamb wave band structures are shown by the solid lines. Meanwhile the ﬂexural wave band structures are also calculated based on the Mindlin plate theory, and are shown by the scattered circles. Comparing the solid lines and the scattered circles, we can easily distinguish the ﬂexure-dominated modes in all Lamb wave modes. In the ﬁgures, fn represents the ﬂexuraldominated modes, and ln represents the other modes (longitudinal or transverse); and Fn and Ln denote the corresponding bandgap edges.

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Fig. 3. Vibration modes at the edges of the lowest complete bandgap for PC slabs. Panels (a)–(f) correspond to vibration modes at points F2 and F3 for the ‘þ ’-holes, F5 and F6 for the ‘x’-holes, and F9 and F10 for the circular holes in Fig. 2, respectively.

It can be seen that one complete bandgap exists between the 11th and 12th bands for the ‘ þ’-hole in Fig. 4(a) when h/a¼ 0.1. These two bands are ﬂexural bands and their frequencies will increase with the thickness increasing. Increasing h/a from 0.1 to 0.2, this bandgap narrows because of the increase of its lower edge. Also we ﬁnd one complete bandgap between the 7th (f2) and 8th (f1) bands. It results from the rising of the f1 band with the bandgap’s upper-edge (F1) of which the vibration mode is illustrated in Fig. 4(b). The vibrations of this mode are concentrated in the connectors, which is similar to the ﬂexural vibration of a beam with two ﬁxed ends. The adjacent pair of the connectors vibrates in the reverse phase and the opposite pair in the same phase. For the lower-edge (L1) mode shown in Fig. 4(c), the vibrations are focused on the lumps and homogeneously distributed in the thickness direction. Thus this is the zeroth-order mode independent of the thickness, and will also appear in the band structures of the bulk systems [6]. With h/a increasing, the f1-band continues to move upward. One can notice that two complete bandgaps appear with the lower one between the 6th (f3) and 7th (f2) bands and the upper one between the 7th (f2) and 8th (f1) bands when h/a¼0.4. It is worth noting that the lower-edge of the lower bandgap is at point F2, of which the frequency is almost the same as that at point L1. The vibration mode corresponding to F2 is presented in Fig. 3(a). It is seen that the emergence of the lower bandgap is due to the rising of the f2-band. The upper-edge (F3) mode of the lower bandgap is shown in Fig. 3(b), where the connectors are almost still and the lumps vibrate. This is similar to the ﬂexural vibration of a plate with parts of its boundaries ﬁxed. Just like the f1-band, the f2-band will keep on rising with the thickness increasing. When h/a ¼0.6, we also observe two

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1.0

0.8

f1 Reduced Frequency

f4

f1

f2

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l2

f4

f2 f1 0.4

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l1

f4

L3

l1,2

f 2 L1

L2 f4

F3 f3

F2

f3

f3 f2

0.2

f3 0.0

0.1

0.2

0.4

0.6 h/a

0.8

1.0

1.2

Fig. 4. Band structures of PC slabs with the ‘þ ’-holes (a) for various thickness (b/a¼ 0.9, c/a¼ 0.25). Panels (b)–(f) show the vibration modes at the marked points F1, L1, F4, L2 and L3 in panel (a). In this ﬁgure, fn represents the ﬂexure-dominated modes; ln represents the other modes (longitudinal or transverse); and Fn and Ln denote the corresponding bandgap edges.

complete bandgaps. The upper-edge (F4) mode of the upper bandgap is illustrated in Fig. 4(d). It is the antisymmetric mode although the displacements are dominantly in the x–y plane. The vibrations in this mode are focused on the connectors, especially the parts near the two free surfaces. This mode exhibits a cut-off frequency [23] which will decrease with the

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1.0

f8

F8

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l5,6

f7

F7

f5

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18th 16-17th 15th

f5

f5

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F6 F5

f8

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f8

L7 L6 L5 L4

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l5,6 f6

0.2

0.0

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1.4

h/a

Fig. 5. Band structures of PC slabs with the ‘x’-holes (a) for various thickness (c/a¼ 0.9, b/a ¼0.4, d/a¼ 0.3). Panels (b)–(g) show the vibration modes of the marked points F7, F8 and L4–L7 in panel (a). In this ﬁgure, fn represents the ﬂexure-dominated modes; ln represents the other modes (longitudinal or transverse); and Fn and Ln denote the corresponding bandgap edges.

thickness increasing. Comparing the band structures for h/a¼1.0 and h/a ¼1.2, we can also notice that l1-and l2-bands shift downward. The vibration modes at the bandgap edges (L2 and L3) in these two bands are shown in Fig. 4(e) and (f). These modes also exhibit cut-off frequencies. Different from the vibration mode in Fig. 4(d), they are the second-order modes. Fig. 5(a) presents the evolution of the band structures with h/a increasing for the ‘ þ’-holes. Three complete bandgaps appear in the band structures when h/a ¼0.1. It is also noted that the two lower bandgaps are located between two ﬂexural bands, 15th and 16th bands as well as17th and 18th bands, respectively, and separated by two nearly-ﬂat bands (16th and

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Reduced Frequency

0.8 F11 0.6

0.4

f9

f9

F10 f 10

f9

F9 f 10

f 10

f9 f 10

0.2

0.0 0.3

0.4

0.5

0.6

h/a

Fig. 6. Band structures of PC slabs with the circular-holes (a) for various thickness (r/a¼ 0.45). Panel (b) shows the vibration mode of the marked point F11 in panel (a). In this ﬁgure, fn represents the ﬂexure-dominated modes; ln represents the other modes (longitudinal or transverse); and Fn and Ln denote the corresponding bandgap edges.

17th bands) in the frequency range of 0.46o O o0.61 with Do/og ¼0.28. However, the band structures display no complete bandgaps when h/a ¼0.2. For h/a ¼0.4, we observe the existence of three complete bandgaps. The lowest one lies between the 12th (f6) and 13th (f5) bands. The vibration mode at the lower (F5) and upper (F6) edges of the bandgap are shown in Fig. 3(c) and (d), respectively. The vibration is localized in the lumps for the lower-edge mode; on the contrary, the connectors vibrate distinctly and the lumps vibrate slightly in a reverse phase for the upper-edge mode. This upperedge mode, similar to the upper-edge mode in Fig. 4(b) for the ‘ þ’-holes, is much like the ﬂexural vibration of two cross beams with their ends being ﬁxed. Besides, the band structure exhibits other two complete bandgaps separated by the 18th (f7) and 19th (f8) bands. These two bands are symmetric and the vibration modes at the bandgap edges (F7 and F8) in these two bands, similar to the mode in Fig. 4(d) for the ‘ þ ’-hole, are twinborn [see Fig. 5(b) and (c)] and have cut-off frequencies. Four wings in one unit cell rotate in the same direction in Fig. 5(b), while each two adjacent wings rotate in the opposite directions in Fig. 5(c). With the thickness increasing, the f7- and f8-bands move downward. The bandgap between the f7- and f8-bands is separated into two complete bandgaps, whereas additional complete bandgaps appear in higher frequencies. When h/a¼1.0, we can ﬁnd six bands above the lowest bandgap. The ﬁrst four bands (l3–l6) have cutoff frequencies and will decrease with the thickness increasing. The vibration modes at the bandgap edges (L4–L7) in these bands are shown in Fig. 5(d)–(g). It is clear that they are also second-order modes just like those in Fig. 4(e) and (f) for the ‘þ’-hole. The two opposite lumps vibrate in the same phase while the two adjacent lumps vibrate in the reverse phase in one unit cell, see Fig. 5(d). While in Fig. 5(e), all the lumps in one unit cell vibrate in the same phase. The vibration mode in Fig. 5(f) [or Fig. 5(g)] is the second-order mode of the one shown in Fig. 5(c) [or Fig. 5(b)]. We can also ﬁnd that the number of the complete bandgaps among the ﬁrst twenty bands decreases with the thickness increasing when h/a 40.8. This is due to the descent of the higher order bands with the cut-off frequencies. However, the bandgap for the ﬂexural wave between the f5-and f6-bands are extraordinarily wide and almost independent of the thickness when h/a 41.0. Fig. 6(a) shows the variation of the band structures with h/a for the circular holes. Unlike the cross-like holes, no bandgap exists when h/a¼0.1. Increasing h/a from 0.1 to 0.3, neither can we ﬁnd any complete bandgap. One complete bandgap appears between the 6th (f10) and 7th (f9) bands when h/a¼0.5. The lower-edge (F9) mode in Fig. 3(e), similar to that in Fig. 3(a) for the ‘ þ’-hole, has a large vibration in the lumps. At the same time, the upper-edge (F10) mode has a large vibration in the connectors, similar to that in Fig. 3(d) for the ‘x’-hole and that in Fig. 4(b) for the ‘ þ’-hole. Also we compute the vibration mode at point F11, see Fig. 6(b). Although F10 and F11 belong to the same band, the vibration modes at these two points are totally different. Actually the vibration mode at point F11 is similar to those at points F4 in Fig. 4(d), F7 in Fig. 5(b) and F8 in Fig. 5(c), which have cut-off frequencies and will decrease with the thickness increasing. As is shown in Fig. 6(a), when h/a¼ 0.6, the vibration mode at the upper-edge of the complete bandgap changes to what is shown in Fig. 6(b). Also one may notice that there is one point (L8) appearing in the same position in all the band structures regardless of the thickness. Like the vibration at point L1, the vibration at this point is in the x–y plane, and also appears in the band structure of the bulk systems with the circular holes [6]. To further show the dependence of the thickness on the bandgaps, we illustrate the map of the complete bandgaps varying with the different values of h/a in Fig. 7(a) for the ‘ þ’-hole system (with b/a¼0.9 and c/a¼0.25) and Fig. 7(b) for the ‘x’-hole system (with c/a ¼0.9, b/a ¼0.25 and d/a¼ 0.25). For the ‘ þ ’-hole system, it can be seen from Fig. 7(a) that a complete bandgap appears between the 7th and 8th bands when h/a 40.15. The upper-edge of this bandgap increases with the thickness increasing; while the lower-edge does not change. When h/a4 0.23, both the upper and the loweredges of this bandgap increase with the thickness increasing, which are explained by the ﬂexural vibrations of an Euler beam model and an Krichhoff–Love plate model in Appendix A. When h/a 40.6, the upper-edge decreases, and the bandgap ﬁnally disappears when h/a ¼0.75. Within the range of the existence of this bandgap, i.e., 0.23 oh/ao0.75, it is

Y.-F. Wang, Y.-S. Wang / Journal of Sound and Vibration 332 (2013) 2019–2037

0.8 b/a=0.9 c/a=0.25

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6th-7th 7th-8th 8th-9th 9th-10th

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2027

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r/a=0.45

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0.7

h/a Fig. 7. Variations of the bandgap edges with the thickness for PC slabs with (a) the ‘þ ’-holes, (b) the ‘x’-holes and (c) the circular holes. The solid and dashed lines represent the upper and lower edges of the bandgaps, respectively.

followed closely by a lower bandgap between the 6th and 7th bands through a ﬂat band (the 7th band). The upper-edge of this lower bandgap ﬁrst increases till h/a ¼0.75 and then decreases and ﬁnally disappears at h/a ¼1.3. It is noticed that the lower-edge of this complete bandgap does not vary obviously with h/a. Actually the lower-edge is at point L1 [Fig. 4(a)] when h/a o0.4 or h/a 40.6, the frequency at which is independent of the thickness (h/a). While when 0.4 oh/ao0.6, even if the bandgap lower-edge changes to be at point F2 in the f3-band, its frequency is also almost the same as that at point L1, see Fig. 4(a). This also implies that the generation of the complete bandgap is a consequence of the bands shifting (the shift of the upper band edge) rather than the folding of the ﬁrst three bands [23] at the low frequencies. A third complete bandgap appears between the 8th and 9th bands when 0.6oh/a o0.95. It connects with a bandgap between the 7th and 8th bands via a ﬂat band (the 8th band). The latter bandgap vanishes when h/a¼ 1.65. Another complete bandgap comes up between the 9th and 10th bands when h/a41.05, and vanishes when h/a ¼2.3. It is noticed that the band width of all the complete bandgap mentioned above will ﬁrst increase and then decrease with the thickness increasing. The decrease of the bandgap edges is due to the fact that the cut-off frequencies of the modes will decrease with the thickness increasing, and the corresponding bands will move down and eventually close the bandgaps. For the ‘x’-hole in Fig. 7(b), a small complete bandgap between the 13th and 14th bands appears ﬁrst when h/a¼0.2. With the increase of the thickness from h/a ¼0.275, complete bandgaps appear orderly between the 12th and 13th bands, 14th and 15th bands, 16th and 17th bands, and 18th and 19th bands. They attain their maximum width when h/a takes certain optimal values, then become narrow and disappear at some certain values of h/a due to the fast decrease of the upper bandgap edges owing to the decrease of the corresponding bands’ cut-off frequencies. Also we can notice that the bandgaps between the 12th and 13th bands, 14th and 15th bands, and 16th and 17th bands are very close, because each two adjacent bandgaps are separated by two near-by ﬂat bands. For comparison, variations of the bandgap edges with the thickness of the slab with circular holes (r/a ¼0.45) are also presented in Fig. 7(c). One complete bandgap lies between the 6th and 7th bands when 0.35 oh/a o0.65. Its lower-edge varies slightly with the change of the thickness. At ﬁrst the upper-edge increases with the thickness increasing and thus results in the increase of the bandgap width. This may be interpreted by the Euler beam model in Appendix A. When h/a¼ 0.5, the upper-edge starts to decrease with the thickness increasing (because the associated bands have cut-off frequencies which decrease with the thickness increasing), and thus results in the decrease of the band width as well as the disappearance of the bandgap when h/a ¼0.65.

Y.-F. Wang, Y.-S. Wang / Journal of Sound and Vibration 332 (2013) 2019–2037

0.7

0.7

0.6

0.6

Reduced Fre quency

Reduced Fre quency

2028

c/a=0.25 h/a=0.4

0.5

0.4

0.3

0.2

6th-7th 7th-8th

0.7

0.8

0.9

b/a

1.0

b/a=0.9 h/a=0.4

0.5

0.4

0.3

0.2 0.0

6th-7th 7th-8th

0.1

0.2

0.3

0.4

0.5

0.6

c/a

Fig. 8. Variations of the band edges in the systems of the ‘þ ’-holes with b/a (h/a ¼0.4, c/a ¼0.25) (a) and c/a (h/a¼ 0.4, b/a¼ 0.25) (b).

Comparing the three panels in Fig. 7, we can ﬁnd that multiple wide complete bandgaps exist in a wide range of h/a for the cross-like holes. All the complete bandgaps attain their maximum width when h/a takes certain optimal values. Also the results demonstrate that the bandgaps, especially the upper bandgap edges, are quite sensitive to the thickness (h/a). Therefore this parameter can be tuned to obtain large normalized band width. Observing the variation of the band structures with the thickness of the PC slabs in Figs. 4(a)–6(a), one may notice that the cross-like holes are easier to induce the complete bandgap than the circular holes when the thickness is small (h/a¼0.1). And one may also notice that for the bandgap upper-edge in Fig. 7, if it increases with the thickness increasing, it is generally corresponding to the ﬂexural mode without the cut-off frequency, the resonant frequency of which will increase with the thickness increasing. If it decreases with the thickness increasing, it corresponds to the ﬂexural mode with the cut-off frequency, the frequency of which will decrease with the thickness increasing. The results also imply that the generation of the bandgap for the holey PC slabs is due to the ﬂexural vibration localized in the lumps or connectors rather than the folding of the ﬁrst three bands as that for PC slabs with solid inclusions [23]. The appearance of the higher modes with the cut-off frequencies eventually closes the bandgap. 3.2. Effects of the shape and size of the cross-like holes Fig. 8 shows the variations of the upper and lower-edges of the bandgaps with the geometry parameter b/a for c/a ¼0.25 (Fig. 8(a)) and with c/a for b/a¼ 0.9 (Fig. 8(b)) for the ‘ þ’-hole. Hereafter, h/a is set to be 0.4. As shown in Fig. 8(a), no bandgaps appear for small values of b/a. With the increase of b/a, two complete bandgaps appear between the 7th and 8th bands and between the 6th and 7th bands, respectively. They become wider with their lower-edges decreasing. Actually when b/a gets large, the connectors become thin, and therefore the resonant frequency decreases. These two bandgaps are close with each other via a ﬂat band (the 7th band) when b/a¼ 0.95 and form a large complete bandgap with Do/og ¼0.86. It is shown in Fig. 8(b) that a relatively wide bandgap appears between the 7th and 8th bands when c/a is small, and that a lower complete bandgap appears between the 6th and 7th bands when 0.1oc/ao0.51. It is noticed that these two bandgaps ﬁrst become wider, and then narrower with the increase of c/a. That is, they are widest at certain intermediate values of c/a. They are very close and separated by a ﬂat band (the 7th band) when 0.1 oc/a o0.37. With the increase of c/a, the length of the lump becomes smaller and thus the resonant frequency increases. When c/a40.4, the upper bandgap disappears, and the lower one narrows. In this case, the vibrations at the upper-edge of the lower bandgap are focused on the connectors, see Fig. 9. Unlike the vibration mode in Fig. 4(b), the connectors vibrate in the periodic plane perpendicular to the thickness direction. So the upper-edge of the lower bandgap decreases, which can be explained by the Euler beam model in Appendix A. As we have mentioned above, the lower bandgap edge of the PC slab with the ‘ þ’-hole is independent of the thickness. The vibration is the in-plane mode, so it is just like the in-plane bulk mixed mode of the inﬁnite PCs [6]. In Ref. [6], based on the analysis of the vibration modes at the bandgap edges, equivalent spring-mass models were proposed to predict the bandgap lower-edges of the two-dimensional PCs with cross-like holes. The predicted results and the numerical results are in generally good agreement. The bandgap lower-edge of the above mentioned PC slab with the ‘ þ’-hole can be evaluated similarly. The normalized width Do/og of the ﬁrst complete bandgap varying with the geometry parameters, b/a and d/a, are shown in Fig. 10(a) for the ‘x’-hole with c/a ¼0.9. We can see that the peak of the normalized width locates near the center of the diagonal dashed line (d ¼b). However, the diagonal dashed line corresponds to the contact of the wings, which should be avoided in calculation. So in the rest of the paper, we will restrict our discussion to d/b ¼0.98. The variations of the upper and lower-edges of the bandgaps with the geometry parameter c/a for b/a¼0.4 are given in Fig. 10(b) and with b/a for c/a ¼0.6 in Fig. 10(c). It is shown in Fig. 10(b) that a bandgap appears between the 12th and 13th

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Fig. 9. The vibration mode at the upper-edge of PC slab with c/a¼ 0.45 in Fig. 8(b).

0.8

c/a=0.9 h/a=0.4 12th-13th

0.01000 0.1000 0.1800

0.6

d/a

0.2700 0.3600 0.4

0.4500

0.2

0.0 0.0

0.2

0.4

0.6

0.8

b/a

b/a=0.4 d/b=0.98 h/a=0.4

1.0

Reduced Fre quency

Reduced Fre quency

1.0

0.8

0.6

0.4

12th-13th 13th-14th 17th-18th

c/a=0.9 h/a=0.4 d/b=0.98

0.8

0.6

12th-13th 13th-14th 17th-18th

0.4

0.8

0.9

c/a

1.0

0.0

0.2

0.4

0.6

0.8

b/a

Fig. 10. (a) Variation of the normalized bandgap width of the lowest bandgap in the systems of the ‘x’-holes with different geometry parameters b/a and d/a (h/a¼ 0.4, c/a¼ 0.9). Panels (b) and (c) show the variations of the bandgap edges in the systems of the ‘x’-holes with the c/a (h/a¼ 0.4, d/b¼ 0.98, b/a¼ 0.4) (b) and b/a (h/a ¼0.4, d/b ¼ 0.98, c/a¼ 0.9) (c).

bands when c/a ¼0.76. With the increase of c/a, both upper and lower-edges of the bandgap decrease monotonously with the increasing width. Other bandgaps, which are not wide, appear at higher frequencies for the larger values of c/a. One large complete bandgap with Do/og ¼0.57 appears and is divided by a ﬂat band when c/a ¼0.95. It is shown in Fig. 10(c) that the lowest bandgap between the 12th and 13th bands appears when b/a falls within a relatively broad range. The lower-edge of the bandgap increases monotonically while the upper-edge ﬁrst increases and then decreases with b/a increasing. When b/a o0.6, the upper-edge mode is the same as that illustrated in Fig. 3(d).

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1.0

0.8

Reduced Frequency

Reduced Frequency

1.0

F12

0.6 b/a=0.7 h/a=0.4 θ=45°

0.4

L9

0.2

0.0

M

Γ

X

M

0.8

0.6

0.4

c/a=0.7 b/a=0.4 F13d/b=0.98 h/a=0.4 θ=45°

L10

0.2

0.0

Γ

M

X

M

0.6

Reduced Frequency

0.5 0.4 0.3 b/a=0.9 c/a=0.25 h/a=0.4 θ=45°

0.2 0.1 0.0 M

Γ

X

M

Fig. 11. Band structures of PC slabs with (a) the square holes, (b) the ‘x’-holes and (c) the ‘þ ’-holes rotated by 451.

With the increase of b/a, the length of the connector decreases and thus the resonant frequency of this mode will increase as predicted by the plate model in Appendix A. When b/a 40.6, the higher modes with the cut-off frequencies appear and the bandgap becomes smaller and ﬁnally disappears when b/a ¼0.75. A narrow bandgap also appears at higher frequencies when 0.325ob/ao0.5. One can notice that, all the complete bandgaps appearing in Fig. 10(c) attain the maximum width when b/a takes certain optimal values.

3.3. Effects of the rotation of the cross-like holes We will discuss the effects of the rotation of the cross-like holes on the bandgaps in this part. The original conﬁguration is the same as what has been discussed above; and the insets in Fig. 11 show the ﬁnal conﬁguration where the rotation angle is 451 for each kind of holes. The holes are assumed to be rotated in the counter-clockwise direction. Due to the symmetry of the hole’s shape and square lattice, it is easily understood that the band structures for a1-rotation is exactly the same as those for (901–a1)-rotation. Therefore only rotation of 01 and 451 is considered. Fig. 11(a)–(c) show, respectively, the band structures of the PC slabs with the square, ‘x’- and ‘ þ’-holes rotated by 451. When the square holes are rotated by 451, two complete bandgaps are formed in the system, one between the 6th and 7th bands and the other between the 8th and 9th bands, see Fig. 11(a). (The size of the hole changing to be smaller than that in Fig. 2(a) is to avoid the close-packing limit [4].) The normalized bandgap width of the lowest complete bandgap is Do/og ¼0.32. In the system with the rotated ‘x’-holes, four complete bandgaps appear as shown in Fig. 11(b). Unlike what is shown in Fig. 2(c), the lowest and largest complete bandgap lies between the 6th and 7th bands in the frequency range of 0.28o O o0.4 with Do/og ¼0.35. Due to the existence of a ﬂat band (the 7th band) corresponding to the localized state, the bandgap between the 7th and 8th bands is very close to the lowest bandgap. No complete bandgap is shown in the system with the rotated ‘ þ’-holes, see Fig. 11(c). To understand the mechanism of the lower bandgap generation, we calculate the vibration modes at the bandgap edges for the systems with the square and ‘x’-holes rotated by 451. The results are shown in Fig. 12. All systems can be regarded as the resonant structures formed by the periodic arrangements of the lumps connected with very narrow connectors [6]. For the lower-edge modes, the vibrations are mainly in the x–y plane. The lumps rotate like rigid bodies with respect to the z-coordinate, and the connectors are still. For the upper-edge mode, the vibrations are mainly out-of-plane, and concentrate at the small connectors [for the system with rotated square holes in Fig. 12(b)] or the wings [for the system with ‘x’-holes in Fig. 12(d)].

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Fig. 12. Vibration modes at the edge of the lowest complete bandgap for PC slabs with the square and the ‘x’-holes rotated by 451. Panels (a)–(d) correspond to the vibration modes at points L9 and F12 for the square holes, L10 and F13 for the ‘x’-holes rotated by 451in Fig. 11, respectively.

To show the effects of the rotation of the holes on the bandgaps, we illustrate the bandgap edges varying with the rotation angle of the holes in Fig. 13 for the systems with the square, ‘x’- and ‘ þ’-holes. In the system with the square holes (b/a¼0.7) shown in Fig. 13(a), there is no bandgap when the rotation angle is small. With the increase of the rotation angle, one complete bandgap appears between the 7th and 8th bands when y ¼22.51 and ﬁnally disappears when y ¼451. The lowest complete bandgap appears between the 6th and 7th bands when y ¼ 351, and reaches its peak when y ¼ 451. Similarly, two complete bandgaps appear for bigger rotation angles (y 4301) in the system with the ‘x’-holes (b/a ¼0.4, c/a ¼0.7), see Fig. 13(b) [It should be indicated that complete bandgaps can occur when y ¼01 if c/a is big enough, see Fig. 2(c)]. The lower one enlarges with the rotation angle increasing when y 4391, and the upper one ﬁrst enlarges and then narrows with the rotation angle increasing. The opposite behavior is shown in the system with the ‘ þ’-holes (b/a¼0.9, c/a ¼0.25), see Fig. 13(c). Two complete bandgaps are formed when y ¼01. The width of these two complete bandgaps remain nearly unchanged when y o51. With the rotation angle increasing, the bandgaps become narrow and disappear; while some new small bandgaps appear, and ﬁnally disappear. No complete bandgap occurs when y ¼451. From the above discussion, we can see that the bandgaps in the systems are sensitive to the rotation angle of the holes. The most evident complete bandgaps are generated in the systems with the square or ‘x’-holes when y ¼451, and in the system with the ‘ þ’-holes when y ¼01. Under these conditions, the whole system is formed by the periodic arrangements of the large lumps connected with the very narrow connectors. Moreover, we also examine the dependence of the band structures on the slab thickness with rotated ‘x’-holes in Fig. 14. Complete bandgaps appear among the ﬁrst ten bands due to the increasing of the f11- and f12-bands. With the thickness increasing, higher modes (f15-, l11- and l12-bands) with the cut-off frequencies come up. Their cut-off frequencies decrease with the thickness increasing, thus resulting in the decrease of these bands and ﬁnally closing the bandgaps. It is also noted that similar to the cross-like holes without rotation, three complete bandgaps exist in the band structures for the rotated ‘x’-holes when h/a¼ 0.1. For the rotated square holes, the band structures varying with the slab thickness are shown in Fig. 15(a). One complete bandgap appears when h/a ¼0.1; but no bandgap exists when h/a ¼0.2. With the thickness increasing, two complete bandgaps come up between the 6th and 7th bands and between the 8th and 9th bands due to the increasing of the f16-and f17-bands. When h/a ¼0.6, the upper bandgap disappears while the lower one enlarges. The higher mode (f18-band) with the cut-off frequency occurs at the same time. The vibration mode at point F17 is mainly in-plane and antisymmetric with respect of the median plane as shown in Fig. 15(b). The lower bandgap narrows when h/a¼0.6 due to the decrease of the f18-band, and ﬁnally disappears when h/a ¼1.2. One bandgap between the 7th and 8th bands appears when h/a¼0.6. It ﬁrst narrows and ﬁnally disappears due to the decrease of the higher modes (l13- and l14-bands) with the cut-off frequencies.

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1.0 b/a=0.7 h/a=0.4

0.8

Reduced Frequency

Reduced Frequency

1.0

0.6

0.4 6th-7th 7th-8th 8th-9th

0.2

0.0

0

5

10

15

20

25

30

35

40

0.6

0.4 6th-7th 7th-8th

0.2

0.0

45

c/a=0.7 b/a=0.4 d/b=0.98 h/a=0.4

0.8

0

5

10

15

θ/°

20

25

30

35

40

45

θ/°

Reduced Frequency

1.0 b/a=0.9 c/a=0.25 h/a=0.4

0.8

0.6

0.4 6th-7th 7th-8th 8th-9th 9th-10th

0.2

0.0

0

5

10

15

20

25

30

35

40

45

θ/° Fig. 13. Variation of the upper and lower edges of the bandgaps in the systems of (a) the square holes, (b) the ‘x’-holes and (c) the ‘ þ’-holes with different values of the rotation angle.

It is also noticed that the results obtained by the Mindlin plate theory are higher than those of the Lamb wave modes. This particularly happens for some bands with the cut-off frequencies, such as f4-band in Fig. 4(a), f7- and f8-bands in Fig. 5(a), f15-band in Fig. 14(a) and f18-band in Fig. 15(a), where a large part of the vibration is the in-plane modes. The Mindlin plate theory overestimates the eigenfrequency because of the neglect of the in-plane displacement. It also happens for some ﬂexure-dominated bands when the size of the hole is too big. Taking the 3rd ﬂexural band for the rotated square holes for example (see the scattered circles in Fig. 15(a), h/a¼0.4), its vibration mode (point F16) shown in Fig. 16(a) is the same type as that of the circular holes in Fig. 3(e), where the lump rotates with its center and the four connectors being still. However, they do have some difference. To show this, we illustrate the z-component displacement as well as its ﬁrst-order derivative along the red dashed line (i.e. the left boundary of the unit cell in the median plane) in Fig. 16(b)–(c), respectively. The distributions of the z-displacement are similar for both circular and rotated square holes. While the ﬁrst-order derivative of the z-displacement for the rotated square hole is much bigger around the connector than that for the circular hole. As we know, the primary principle of the Mindlin plate theory is that the z-displacement and the rotations (similar to the ﬁrst-order derivative of the z-displacement) along the x- and y-axis are three independent interpolate functions, for which only the C0-continuity is required [30]. And in fact, when the size of the hole is close to its close-packing limit, a large number of ﬁnite elements are needed to appropriately describe the thin connectors, which actually cannot be adaptively meshed, by using the Mindlin plate theory. So it is difﬁcult to precisely describe the displacement with a large ﬁrst-order derivative, such as in the case for the rotated square hole in Fig. 16(d), where the distribution of the z-component displacement on the corresponding boundary by using the Mindlin plate theory is shown. Then the deformation will be underestimated by using the Mindlin Plate theory, even when the plate is thin [31]. This will result in high eigenfrequencies. It is also the case for the rotated ‘x’-holes as well as for the circular hole with r/a¼0.495 in Fig. 16(e). The variation of the bandgap edges for PC slabs with the ‘x’-holes and square holes rotated by 451 with the thickness is presented in Fig. 17. It looks apparently the same as that shown in Fig. 4(a). It is noticed that all the complete bandgaps mentioned above will ﬁrst enlarge and then narrow with the thickness increasing. The nearly ﬂat lower-edge of the bandgap between the 6th and 7th bands is corresponding to the in-plane vibration mode in Fig. 12(c) for the rotated ‘x’-holes and Fig. 12(a) for the rotated square holes. The increasing bandgap edges correspond to the natural modes where

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1.0

0.8 f14

Reduced Frequency

f13

f12 l11-12

f14

0.6

f12

f13

f15 F15

f12

0.4

f14 f13

0.2

f11

f11

f11

l11-12

f15 f15

F13

f11

f12

0.0

F14 f11

f12

f11

L12 L11

L10

f11

0.1

0.2

0.3

0.4

0.8

1.0

1.2

h/a

Fig. 14. (a) Band structures of PC slabs with the ‘x’-holes rotated by 451 for various slab thickness (c/a¼ 0.7, b/a ¼0.4, d/b ¼ 0.98). In this ﬁgure, fn represents the ﬂexure-dominated modes; ln represents the other modes (longitudinal or transverse); and Fn and Ln denote the corresponding bandgap edges. Panels (b)–(e) show the vibration modes of the marked points F14, F15, L11 and L12 in panel (a).

the ﬂexural vibrations are located in the wings (see Fig. 12(d)) or the connectors (see Figs. 12(b) and 14(c)). The appearances of the higher modes with the cut-off frequencies result in the decrease of the bandgap edges, and ﬁnally close the bandgap.

4. Concluding remarks In this paper, the bandgap properties of the PC slabs with the cross-like holes are studied by using FEM. Two types (‘ þ’ and ‘x’) of the cross-like holes are considered. The effects of the slab thickness and the geometry parameters of the holes on

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1.0

f 17

Reduced Frequency

0.8

f 18

f 16 f 16

F17

f 17

0.6

l 13-14 f 18

f 16 F12 f 16

0.4

L 14 L 13

l 13-14

f 18 f 18

L9 F16

0.2

0.0

0.1

0.2

0.4

0.6

0.8

1.0

1.2

h/a

Fig. 15. (a) Band structures of PC slabs with the square holes rotated by 451 for various thickness (b/a¼ 0.7, h/a¼ 0.1). In this ﬁgure, fn represents the ﬂexure-dominated modes; ln represents the other modes (longitudinal or transverse); and Fn and Ln denote the corresponding bandgap edges. Panels (b)–(d) show the vibration modes of the marked points F17, L13and L14 in panel (a).

the bandgaps are discussed. The mechanism of the generation of the bandgap is analyzed by studying the vibration modes at the bandgap edges. From the numerical results and discussions we can draw the following conclusions: (1) No bandgap appears in the system with the square holes if the symmetry of the holes is the same as that of the lattice. However if the square holes are replaced with the cross-like holes, large bandgaps at lower frequencies are generated.

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3

z-displacement

2

1

0

-1 Rotated square hole Circular hole

-2

-3 -a/2

a/2

0

5000

0.015

4000

0.010

3000

2000 Rotated square hole Circular hole

1000

0

-1000 -a/2

z-displacement

First-order derivative

Location-y

0.005

0.000

-0.005 Rotated square hole Circular hole

-0.010

a/2

0

-0.015 -a/2

0

Location-y

a/2

Location-y 1.0

Reduced Frequency

0.8

0.6 r/a=0.495 h/a=0.4

0.4

0.2

0.0

M

X

M

Fig. 16. (a) Vibration mode at point F16 in Fig. 15(a). Panels (b)–(d) show the z-displacement and its ﬁrst-order derivative along the red dashed lines in panel (a) as well as in Fig. 3(e). Panel (d) shows the z-displacement on the corresponding boundaries by using the Mindlin plate theory. Panel (e) shows the band structure of the PCs with the circular holes (r/a¼0.495, h/a ¼0.4).

The generation of the lowest bandgap is owing to the ﬂexural vibration of the resonant structures with the periodically arranged lumps connected with the narrow connectors. (2) The slab thickness has signiﬁcant inﬂuences on the bandgaps. Multiple complete bandgaps may appear for a wide range of the thickness, even in a very small thickness for the cross-like holes. The complete bandgaps generally locate between the ﬂexural wave bands for a thin slab. The generation of the bandgap is due to the ﬂexural vibration localized

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1.0 c/a=0.7 b/a=0.4 d/b=0.98 =45 6th-7th 7th-8th 8th-9th 9th-10th 11th-12th

0.7 0.6 0.5 0.4

0.9

Reduced Fre quency

Reduced Fre quency

0.8

0.3 0.2 0.0

0.8 0.7 0.6 b/a=0.7 h/a=0.4 =45 6th-7th 7th-8th 8th-9th 9th-10th

0.5 0.4 0.3

0.3

0.6

0.9

1.2

1.5

1.8

2.1

0.2 0.0

2.4

h/a

0.3

0.6

0.9

1.2

1.5

h/a

Fig. 17. Variation of the bandgap edges with the alb thickness with the ‘x’-holes (a) and the square holes (b) rotated by 451. The solid and dashed lines represent the upper and lower edges of the bandgaps, respectively.

in the lumps or connectors. The appearances of the higher modes with the cut-off frequencies which decrease with the thickness increasing result in the decrease of the bandgap upper-edge, and ﬁnally lead to vanishing of the bandgap. (3) The geometry of the cross-like holes plays a key role in the formation of the bandgaps. At a given thickness, a relatively big size of the cross-like holes is favorable to generate a large bandgap. Wider and lower bandgaps may be obtained by optimizing the shape of the holes with the ﬁxed size. For the case of the ‘x’-holes, a big normalized bandgap width is obtained when the wings are nearly-contact. (4) The rotation of the holes can change the symmetry of the systems, and thus can tune the bandgaps. The most evident complete bandgaps are generated by rotating the square or ‘x’- hole by an angle of 451 because the periodic arrangement of the resonant units is formed in the system in this situation. The inﬂuence of the thickness on the bandgaps for the rotated ‘x’-holes is similar to that for the ‘x’- holes without rotation. (5) Mindlin plate theory is invalid for the cross-like holes without or with rotation for the higher modes with the cut-off frequencies, where a large part of the vibration is in-plane but antisymmetric in the thickness direction. The eigenfrequency is overestimated because of the neglect of the in-plane displacement. This invalidity also happens when the size of the hole is too big, because the thin connector cannot be precisely described by ﬁnite discrete elements.

Acknowledgments This work is funded by the National Natural Science Foundation of China (Grant no. 11272041). The ﬁrst author also thanks for the PhD grant from the Fundamental Research Funds for the Central Universities (BJTU2011YJS046). The second author acknowledges the partial support from the National Basic Research Program of China (2010CB732104). Appendix A The vibration mode shown in Fig. 3(b) is similar to the ﬂexural vibration of a square plate with parts of its boundaries ﬁxed. The vibration mode shown in Fig. 3(d) is much like the ﬂexural vibration of two cross beams with the ﬁxed ends, which can be similarly considered as a ﬂexural vibration of a square plate. The vibration modes in Figs. 3(f), 4(b) and 9 can be regarded as a ﬂexural vibration of a beam with two ﬁxed ends. In order to give a physical insight to these vibration modes, we next develop vibration models of a square thin plate and a long Euler beam. The wave equation for a thin Kirchhoff–Love plate with thickness h and side length s is [32] 3

Eh @2 u r4 uz þ rh 2z ¼ 0, 2 12 1n @t

(A1)

where E and n are Young’s modulus and Possion’s ratio, respectively; and uz is the transverse displacement. To simplify our discussion, the edges of the square thin plate are assumed to be clamped. Use of the clamped boundary condition then leads to the following natural frequency 2

o2mn ¼

Eh a4mn , 12rs4 1n2

(A2)

where amn is an eigenvalue associated with a particular mode of the plate. We can easily ﬁnd from this relation that, for a particular mode, the frequency increases as h is increased or s is decreased.

Y.-F. Wang, Y.-S. Wang / Journal of Sound and Vibration 332 (2013) 2019–2037

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The above plate model can be used to explain the increase of the lower-edge of the complete bandgap with the thickness as shown in Fig. 7(a), as well as the increase of the upper-edge shown in Fig. 7(b). It can also be applied to explain the increase of the upper-edge of the lowest complete bandgap with b/a in Fig. 9(c) where the length of the connector corresponds to half of the side length of the plate (s/2) in Eq. (A2). The wave equation for a thin Euler beam with thickness h and length l is [32] 4

EIr uz þ rA

@2 uz ¼ 0, @t 2

(A3)

where I is the moment of inertia; and A is the area of the cross-section of the beam. The natural frequency under the clamped boundary conditions can be given by

o2n ¼

b4n Eh2 12rl

4

,

(A4)

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