Large bandgaps of two-dimensional phononic crystals with cross-like ...

Large bandgaps of two-dimensional phononic crystals with cross-like holes Yan-Feng Wang, Yue-Sheng Wang, and Xiao-Xing Su Citation: J. Appl. Phys. 110, 113520 (2011); doi: 10.1063/1.3665205 View online: http://dx.doi.org/10.1063/1.3665205 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v110/i11 Published by the American Institute of Physics.

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JOURNAL OF APPLIED PHYSICS 110, 113520 (2011)

Large bandgaps of two-dimensional phononic crystals with cross-like holes Yan-Feng Wang,1 Yue-Sheng Wang,1,a) and Xiao-Xing Su2 1

Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, People’s Republic of China School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, People’s Republic of China 2

(Received 10 August 2011; accepted 27 October 2011; published online 9 December 2011) In this paper we study the bandgap properties of two-dimensional phononic crystals with cross-like holes using the finite element method. The influence of the geometry parameters of the holes on the bandgaps is discussed. In contrast to a system of square holes, which does not exhibits bandgaps if the symmetry of the holes is the same as that of the lattice, systems of cross-like holes show large bandgaps at lower frequencies. The bandgaps are significantly dependent upon the geometry (including the size, shape, and rotation) of the cross-like holes. The vibration modes of the bandgap edges are computed and analyzed in order to clarify the mechanism of the generation of the lowest bandgap. It is found that the generation of the lowest bangdap is a result of the local resonance of the periodically arranged lumps connected with narrow connectors. Spring-mass models are developed in order to predict the frequencies of the lower bandgap edges. The study in this paper is relevant to C 2011 American Institute of Physics. the optimal design of the bandgaps in light porous materials. V [doi:10.1063/1.3665205]

I. INTRODUCTION

Since the concepts of photonic crystals (PTCs) and phononic crystals (PNCs) were proposed, respectively, by Yablonovitch1 and John2 (independently) in 1987 and Kushwaha et al.3 in 1993, wide interest has been devoted to studies on the propagation behaviors of the classic waves in periodic media. These new artificial structures exhibit bandgaps in their spectra, where the propagation of waves is fully prohibited. The bandgaps in PNCs might have potential applications in acoustic isolation, noise suppression, vibration attenuation, etc. The study of PNCs has become one of the most active and fast-developing fields in condensed matter physics, acoustics, mechanics, mechanical engineering, etc.4 The tuning of bandgaps is essential to applications of phononic crystals. The bandgaps are found to be determined by several factors, including material5 and geometry parameters.6 In particular, a carefully designed structural topology might produce novel phononic crystals with large bandgaps. At present, a certain degree of interest is being generated with regard to phononic crystals with periodic holes (vacuum7 or air-filled8), with their bandgaps mainly determined by the geometry parameters. Goffaux et al.9 studied the rotating effects on bandgaps for a system with square holes and found that the orientation of the air-filled cavities induces exactly the opening of the gap as the rotation angle increases. Maldovan et al.10 examined the complete bandgap properties of twodimensional (2D) air/solid structures with circular holes in square and triangular lattices and found that the air rods arranged in the triangular lattice presented large, complete, and simultaneous photonic–phononic bandgaps. Zhen et al.11 explored the bandgap properties of honeycomb structures with square, triangular, or hexagonal lattices and found no coma)

Author to whom correspondence should be addressed. Electronic mail: [email protected]. Fax: þþ86-10-51682094.

0021-8979/2011/110(11)/113520/13/$30.00

plete bandgap. Phani et al.12 and Gonella et al.13 modeled honeycomb structures as an assembly of rigidly connected beams and found full bandgaps. Liu et al.14 studied the band structures of mixed wave modes in 2D systems with triangular, square, and circular holes; they found that bandgaps can be easily generated in systems with triangular holes. The bandgaps of a viscoelastic rubber matrix with a periodic array of circular air holes were investigated by Merheb et al.15 The structure made by etching a hexagonal array of air holes on a freestanding plate was studied by Mohammadi et al.,16 and a relatively large high-frequency bandgap was observed. Soliman et al.17 studied the effects of release circular holes on the bandgap in a microscale solid phononic crystal; release holes with a proper radius can reliably release a phononic crystal membrane without significantly compromising the bandgap. As one might notice, all of the above-mentioned works involve PNC systems with convex (circular or regularpolygonal) holes. However, we noticed that when non-convex holes are introduced, PTCs can display a broad stop band18 or a dual-stop band19 (or even a multi-stop band). SafaviNaeini20 proposed a slab structure with “snowflake-shaped” holes and found that if the geometry of the “snowflake” is tailored, simultaneous phononic and photonic bandgaps are achievable without comprising the optical and acoustic properties of the system. These works motivate investigations into PNCs with non-convex holes. In this paper, we study the bandgaps of 2D PNCs with cross-like (a kind of non-convex) holes in a square lattice. Large and tunable bandgaps are observed in the proposed structures. II. PROBLEM STATEMENT AND COMPUTATIONAL METHOD

We consider two kinds of cross-like cylindrical holes embedded in an isotropic elastic solid matrix in a square lattice. Figures 1(a) and 1(b) show the cross-sections of the

110, 113520-1

C 2011 American Institute of Physics V


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merits, such as its compatibility, good convergence, high accuracy and efficiency, etc. Due to the periodicity of PNCs, the calculation can be implemented in a representative unit cell. The discrete form of the eigenvalue equations in the unit cell can be written as ðK x2 MÞU ¼ 0;

(4)

where U is the displacement at the nodes and K and M are the stiffness and mass matrices of the unit cell, respectively. The Bloch theorem of Eq. (3) should be applied on the boundaries of the unit cell, yielding Uðr þ aÞ ¼ eiðkaÞ UðrÞ; FIG. 1. Cross-sections and finite element models of the unit cells of the 2D PNCs with (a) “þ”- and (b) “x”-holes.

holes, which are denoted as the “þ”-hole and the “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. The four rectangular areas of d (c b)/2 inside an “x”-hole will be termed “wings.” For comparison, we also consider the cases of square and circular holes. The z-coordinate should be set parallel to the axes of the cylindrical holes. Then if the elastic waves propagate in the transverse plane (xy plane) with the displacement vectors independent of the z-coordinate, they can be decoupled into the mixed in-plane mode and the anti-plane shear mode. Accordingly, the frequency-domain wave equations are @ u qðrÞx2 ui ¼ r ðlðrÞrui Þ þ r lðrÞ @xi @ ðkðrÞr uÞ; i ¼ x; y (1) þ @xi for the in-plane mode, and qðrÞx2 uz ¼ r ½lðrÞruz

(2)

for the anti-plane mode. In Eqs. (1) and (2), r ¼ ðx; yÞ denotes the position vector, x is the angular frequency, q is the mass density, k and l are the Lame´ constant and shear modulus, u ¼ ðux ; uy Þ is the displacement vector in the transverse plane, and r ¼ ð@=@x; @=@yÞ is the 2D vector differential operator. According to the Bloch theorem, the displacement filed can be expressed as uðrÞ ¼ eiðkrÞ uk ðrÞ;

(3)

where k ¼ ðkx ; ky Þ is the wave vector limited to the first Brillouin zone of the reciprocal lattice and uk ðrÞ is a periodical vector function with the same periodicity as the crystal lattice. In the present work, the finite element method (FEM) is used to calculate the band structures of the considered PNCs. The FEM is one of the commonly used calculation methods for PNCs.20,21 Compared with other traditional methods, such as the plane-wave expansion method,5 the multiplescattering theory method,22 the finite difference time domain method,7,23 the wavelet method,24 etc., the FEM has some

(5)

where r is located at the boundary nodes and a is the vector that generates the point lattice associated with the phononic crystals. Then, COMSOL Multiphysics 3.5a is utilized to directly solve the eigenvalue equation (4) under the complex boundary condition of Eq. (5). Unlike traditional finite element software such as ABAQUS,21 etc., in which two discretized meshes in one unit cell with the same geometric and physical parameters are needed in order to realize the Bloch periodic boundary conditions, COMSOL Multiphysics can solve a given problem in the complex domain directly without dividing it into the real and imaginary parts, and it thus can impose the Bloch boundary conditions on the boundaries of a single unit cell directly. In the present work, we apply the Acoustic Module operating under the 2D plane strain Application Mode (acpn). The free boundary condition is imposed on the surface of the hole, and the Bloch boundary conditions on the two opposite boundaries of the unit cell. The unit cell is meshed by using the default triangular mesh with Lagrange quadratic elements provided by COMSOL. Eigenfrequency analysis is chosen as the solver mode, and the direct SPOOLES is selected as the linear system solver. Still, we require the Hermitian transpose of constraint matrix and in symmetry detection in the advanced solver parameter settings. 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. III. NUMERICAL RESULTS AND DISCUSSION

In this section we present detailed numerical results for the band structures for various systems with different geometry parameters (including the size, shape, and rotation angle of the holes) and analyze the mechanism of the appearance of the large bandgaps. The elastic parameters are q ¼ 2700 kgm3 , E ¼ 20 GPa; and ¼ 0:25. In order to show what new features appear in the band structures for the non-convex holes, we illustrate the dispersion curves for the systems with “þ”-, “x”-, square, and circular holes of the particular sizes in Fig. 2. Here the reduced frequency X ¼ xa=2pct (with ct ¼ 1721 m=s being the transverse wave velocity of the matrix) is used. It is shown that no bandgap appears in the system with square holes (Fig. 2(a)). However, if the square holes are


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FIG. 2. (Color online) Band structures of the PNC systems with (a) square, (b) “þ”-, (c) “x”-, and (d) circular holes. The red solid and black dashed lines represent the mixed and shear wave modes, respectively.

replaced by “þ”-holes, the degeneracy of the mixed modes at points A and B and the degeneracy of the first two shear mode bands at point C are all separated, opening a complete bandgap in the frequency range of 0:35 < X < 0:61 (shown as the shadowed region in Fig. 2(b)). The replacement of the square holes by the “x”-holes results in the separation of the degeneracy of the mixed modes at point D, as well as the degeneracy of the 2nd and 3rd shear mode bands. This opens a large complete bandgap in the frequency range of 0:38 < X < 0:69. In addition, two other complete bandgaps appear in the higher frequency ranges. Although a bandgap appears in the system with circular holes, its position is higher and its width is smaller than those in the systems with the “þ”- and “x”-holes (see Fig. 2(d)). The above results imply the significant influence of the hole shape on the bandgaps. A cross-like hole seems favorable when attempting to open a larger and lower bandgap, and this has potential applications. In order to understand the mechanism of the bandgaps, we calculated the vibration modes at

the edges of the lowest full bandgaps for both mixed and shear wave modes. The results for the “þ”-holes are demonstrated in Fig. 3. Figures 3(a) and 3(b) show, respectively, the upper and lower edge modes of the mixed waves, and Figs. 3(c) and 3(d) show those of the shear waves. It is noted that the square lattice of the “þ”-holes can be regarded as a periodic arrangement of the square lumps connected with narrow connectors. For the lower edge-modes of both mixed and shear waves, the lumps oscillate as rigid bodies, and the connectors act as springs between different lumps (see Figs. 3(a) and 3(c)). Whereas the connectors vibrate, the lumps are nearly at rest for the mixed wave modes at the upper bandgap edge, as shown in Fig. 3(b). For the shear mode, the lumps themselves also vibrate (see Fig. 3(d)). The square lattice of the “x”-holes may be viewed as a double periodic array of elements. Each element consists of one rectangular lump and two narrow connectors. It is shown in Figs. 4(a) and 4(c) that the lumps vibrate, with the connectors acting as springs, for the lower edge-modes of the mixed


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FIG. 3. (Color online) Vibration modes at the edge of the lowest bandgap for PNCs with “þ”-holes. Panels (a)–(d) correspond to points M1, M2, S1, and S2 in Fig. 2(b), respectively.

and shear waves, and the connectors evidently vibrate for the upper edge-modes, as shown in Figs. 4(b) and 4(d). In particular, it is noted that the lumps are nearly at rest in the case of the shear wave mode (Fig. 4(d)) but vibrate in the case of the mixed wave modes (Fig. 4(b)).

From the above analysis, we deduce that the generation of the wide low-frequency bandgaps is due to the local resonant elements in the structures. In the system with circular holes, we can also find resonant elements with large lumps connected by narrow connectors (see Fig. 5, which illustrates

FIG. 4. (Color online) Vibration modes at the edge of the lowest bandgap for PNCs with “x”-holes. Panels (a)–(d) correspond to points M3, M4, S3, and S4 in Fig. 2(c), respectively.


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FIG. 5. (Color online) Vibration modes at the edge of the lowest bandgap for PNCs with circular holes. Panels (a)–(d) correspond to points M5, M6, S5, and S6 in Fig. 2(d), respectively.

the vibration modes at the lower and upper edges of the bandgaps for both mixed and shear waves). However, one cannot find lumps in a square lattice of square holes, of which the band structures without gaps are shown in Fig. 2(a). Although circular holes might result in bandgaps, it is inconvenient to tune the bandgaps through the geometric design of the elements, because the system involves only one geometry parameter: the hole radius. However, the cross-like holes we are considering involve multiple geometry parameters, which provides us with various ways to tune the bandgaps. In the rest of this section, we analyze the effects of the geometries of the two kinds of cross-like holes on the bandgaps. A. Effects of the shape and size of cross-like holes 1. “1”-hole

Variation of the normalized width Dx=xg (where Dx and xg are the width and the central frequency of the bandgap, respectively) of the first bandgap for the in-plane mixed and anti-plane shear wave modes with the geometry parameters b/a and c/a are shown in the lower-right triangular regions of Figs. 6(a) and 6(b), respectively. The dashed diagonal line corresponds to the system with the square holes. It is shown that along this dashed line the normalized bandgap width is zero, which reflects the fact that no full bandgap exists in such a system with square holes.11 The full bandgap does not occur until b/a increases to a certain value, and its normalized width increases with increasing b/a. For a given b/a, the bandgap is widest at an intermediate value of c/a and becomes relatively narrow until it vanishes as c/a deviates from this value. For example, when b/a ¼ 0.95, the

normalized bandgap width shows a maximum at c/a ¼ 0.25 for the mixed wave mode and at c/a ¼ 0.4 for the shear wave mode. Figure 7 shows the variations of the upper and lower edges of the bandgaps with the geometry parameter b/a for c/a ¼ 0.25 (Fig. 7(a)) and with c/a for b/a ¼ 0.9 (Fig. 7(b)). The solid and open circles represent the mixed and shear wave modes, respectively. As shown in Fig. 7(a), no bandgaps appear for small values of b/a. With an increase in b/a, the mixed mode exhibits a bandgap between the 3rd and 4th bands, and the shear mode shows two bandgaps, between the 1st and 2nd bands and between the 3rd and 4th bands. All bandgaps become wider, with their lower edges decreasing and their upper edges increasing. Two complete bandgaps appear for larger values of b/a (see the shadowed regions in Fig. 7). It is shown in Fig. 7(b) that a relatively wide bandgap of the mixed mode appears between the 3rd and 4th bands when c/a < 0.55, and between the 5th and 6th bands when 0.325 < c/a < 0.75. For the shear mode, there exist a relatively wide bandgap between the 1st and 2nd bands for the most values of c/a and a bandgap between the 3rd and 4th bands in the higher frequency region when c/a < 0.4. It is noticed that all the bandgaps become first wider and then narrower with increasing c/a; that is, they are widest at certain intermediate values of c/a. The five shadowed regions in Fig. 7 represent the complete bandgaps that might appear in the reduced frequency range (0, 1). Figures 6 and 7 show that the bandgaps of the PNCs with the “þ”-holes can be tuned by adjusting the geometry of the hole. As indicated in Sec. III A, the generation of the lowest bandgap is a result of the local resonance of the unit


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FIG. 6. Variation of the normalized width of the lowest bandgap for the (a) mixed and the (b) shear wave modes with the geometry parameters b/a and c/a. The regions below and above the dashed line correspond to the results for the systems with “þ”- and “x”-holes, respectively.

cell, and its lower edge is relevant to the resonance of the solid lumps with narrow connectors between them. Based on this fact, we propose an equivalent spring-mass model with which to evaluate the frequency of the lower edge of the lowest bandgap in the system with “þ”-holes. Suppose the and K, effective mass and stiffness of the model are M respectively. Then, the resonant frequency is given by rffiffiffiffiffi K x¼ (6) : M contains the mass of the lump (mB ) and The effective mass M half of the mass of the connectors (mA ). The effective stiffness

K contains the contributions of the four connectors, which act as four shear springs for the shear mode or two tension/compression springs and two (partial) shear springs for the mixed modes. In the calculation of the effective stiffness, the effective length of the spring is obtained by adding the partial contribution of the lump (lB ) to the semi-length of the connector and K for different wave (lA ). The detailed expressions of M modes are listed in the middle column of Table I. The normalized resonant frequencies predicted by the proposed model are shown in Fig. 7 by the dash-dotted lines. The proposed models are generally in agreement with the FEM numerical results for the lower edges of the bandgaps. However, a relatively big underestimation is shown in Fig. 7(a) for the

FIG. 7. (Color online) Variation of the upper and lower edges of the bandgaps in the system of “þ”-holes with the geometry parameter (a) b/a (c/a ¼ 0.25) or (b) c/a (b/a ¼ 0.9). The solid and dashed lines represent the upper and lower edges of the bandgaps, respectively. The open and solid symbols represent the shear and mixed modes, respectively. The dash-dotted lines show the results for the reduced frequencies of the lower edge modes predicted by the equivalent spring-mass model.


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TABLE I. Calculation of the effective mass and stiffness of the proposed spring-mass model for the lower edge modes of the lowest bandgap. Equivalent spring-mass model

Structure of the unit cell

Wave mode

Mixed mode

Shear mode

Mixed mode

Shear mode

Spring-mass model

mA

pcðabÞ

0:5qðacÞðadÞ

mB

qðacÞ2

qdðabÞ

0:5qcðabÞþqðacÞ2

0:25qðacÞðadÞþqdðabÞ

¼ mA þ mB M 2 lA

c=2

c=2

ðadÞ=2

ðadÞ=2

lB

1=3ðacÞ=2

1=8ðacÞ=2

1=5d=2

–

L ¼ lA þ lB

ðaþ2cÞ=6

ðaþ7cÞ=16

ð5a4dÞ=10

ðadÞ=2

S

ðabÞ1

ðabÞ1

ðacÞ1

ðacÞ1

6ðkþ2lÞðabÞ aþ2c

–

10ðkþ2lÞðacÞ 5a4d

–

–

16lðabÞ aþ7c

–

2lðacÞ ad

–

–

–

64lða bÞ K ¼ 4Ks ¼ a þ 7c

K ¼ 2Kt 20ðk þ 2lÞða cÞ ¼ 5a 4d

K ¼ 2Ks 4lða cÞ ¼ ad

Kt ¼

C11 S L

Ks ¼

C44 S L

Kt0 ¼ a

C44 S lA

K

0:1

lðabÞ c=2

K ¼ 2Kt þ 2Kt0 12ðk þ 2lÞða bÞ ¼ a þ 2c 0:4lða bÞ þ c

of the equivalent spring-mass Notes: mA and mB are, respectively, parts of the masses of the connectors and the lump contributing to the effective mass M model; lA and lB are, respectively, the effective lengths of the springs associated with the connectors and the lump; S is the area of the cross-section of the connectors; Kt and Ks are the stiffness of the effective tension/compression spring and the shear spring, respectively; Kt0 is a part of the stiffness of the shear spring contributing to the tension/compression spring, with a being a weighting coefficient; and the elastic constants C11 ¼ k þ 2l and C44 ¼ l.

mixed modes when b/a is small and in Fig. 7(b) when c=a ! 0. This is caused by the overestimation of the effective mass because the vibration is well concentrated in the wings (i.e., the corners of the lump) rather than over the whole lump, such that the lump cannot be regarded as a rigid body in these cases. 2. “x”-hole

The variation of the normalized width Dx=xg of the first bandgap for the in-plane mixed and anti-plane shear

wave modes with the geometry parameters b/a and d/a are shown in Figs. 8(a) and 8(b), respectively, with c/ a ¼ 0.9. We can see that the peak of the normalized width is located near the center of the dashed diagonal line (d ¼ b) for both wave modes. However, the dashed diagonal line corresponds to the case of the contacting of wings, which should be avoided in calculation and analysis, so in the rest of the paper we restrict our discussion to d/b ¼ 0.98.


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FIG. 8. Variation of the normalized width of the lowest bandgap for the (a) mixed and (b) shear wave modes in systems of “x”-holes with the geometry parameters b/a and d/a (c/a ¼ 0.9).

The variations of Dx=xg with the geometry parameters c/a and b/a are shown in the upper-left triangular regions in Figs. 6(a) and 6(b) for the mixed and shear modes, respectively. No bandgap exists when c/a is small. When c/a increases to a certain value, the bandgap appears and becomes wider with increasing c/a. For a given value of c/a, the normalized width first increases and then decreases with increasing b/a, which means that the normalized width will reach its maximum when b/a takes an optimal value. The variations of the upper and lower edges of the bandgaps with the geometry parameter c/a for b/a ¼ 0.45 are given in Fig. 9(a) and with b/a for c/a ¼ 0.6 in Fig. 9(b). The solid and open circles represent the mixed and shear wave modes, respectively. It is shown in Fig. 9(a) that a bandgap appears

between the 6th and 7th bands of the mixed modes or between the 2nd and 3rd bands of the shear mode when c/a ¼ 0.6. With an increase in c/a, both the upper and the lower edges of the bandgap of the mixed modes decrease monotonously with the width first increasing and then decreasing, whereas the bandgap of the shear mode shows an increasing upper edge and a decreasing lower edge, with the width increasing monotonously. Other bandgaps, which are not wide, appear at higher frequencies for larger values of c/a. Overlapping of the bandgaps for different wave modes forms some complete bandgaps (see the shadowed regions). The complete bandgaps are divided by the flat bands when c/a > 0.935. It is shown in Fig. 9(b) that the lowest bandgap between the 6th and 7th bands of the mixed modes exists when b/a

FIG. 9. (Color online) Variation of the upper and lower edges of the bandgaps in the system of “x”-holes (d ¼ 0.98b) with the geometry parameter (a) c/a (b/ a ¼ 0.35) or (b) b/a (c/a ¼ 0.9). The solid and dashed lines represent the upper and lower edges of the bandgaps, respectively. The open and solid symbols represent the shear and mixed modes, respectively. The dash-dotted line show the results for the reduced frequencies of the lower edge modes predicted by the equivalent spring-mass model. The dotted line shows the modified results from the Euler-beam model.


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falls within a relatively broad range. This bandgap appears at very low frequencies (near Dx=xg ¼ 0:15) when b/a is small ( 32 ) in the system with the “x”-holes (b/a ¼ 0.35, c/a ¼ 0.7) (see Fig. 13(b)). (It should be indicated that complete bandgaps can occur if c/a is big enough; see Fig. 2(c)). However, multiple full bandgaps of either mixed or shear wave mode exist when h ¼ 0 and disappear with an increasing rotation angle. Some new bandgaps also appear and disappear with an increase of the rotation angle. The opposite behavior is shown in the system with the “þ”-holes (b/a ¼ 0.9, c/ a ¼ 0.25) (see Fig. 13(c)): full bandgaps of either mixed or shear wave mode appear, and complete bandgaps are formed when h ¼ 0 . The width of the lowest complete bandgap remains nearly unchanged when h is small. With the rotation angle increasing from 0 , the bandgaps become narrow and disappear, and some new bandgaps appear and become wider. No complete bandgap occurs, although full bandgaps exist for both mixed and shear wave modes when h ¼ 45 . 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 square or “x”-holes when h ¼ 45 , and in the system with “þ”-holes when h ¼ 0 . Under these conditions, the whole system is formed by periodic arrangements of large lumps connected with very narrow connectors. IV. CONCLUDING REMARKS

In this paper, the bandgap properties of PNCs with cross-like holes are studied using the finite element method.

J. Appl. Phys. 110, 113520 (2011)

Two types (“þ” and “x”) of cross-like holes are considered. The effects of the geometry parameters of the holes on the bandgaps are discussed. The mechanism of the generation of the lowest bandgap is analyzed by studying the vibration modes of the bandgap edges. From the calculated results and discussions, we can draw the following conclusions: (1) No bandgap appears in the system with square holes if the symmetry of the holes is the same as that of the lattice. However, if the square holes are replaced with cross-like holes, large bandgaps at lower frequencies are generated. The generation of the lowest bangdap is due to the local resonance of periodically arranged lumps connected with narrow connectors. At the lower edge of the bandgap, the lumps vibrate, with the narrow connectors acting as springs; at the upper edge, the connectors vibrate and the lumps are nearly at rest. The frequency of the lower edge can be predicted using a spring-mass model. (2) The geometry of the cross-like holes has a significant influence on the bandgaps. A relatively big size of the cross-like holes is favorable for generating a large bandgap. Wider and lower bandgaps can be obtained by optimizing the shape of the hole with a fixed size. (3) The rotation of the holes can change the symmetry of the systems and then have an influence on the bandgaps. The most evident complete bandgaps are generated by rotating the square or “x”- holes by 45 , because a periodic arrangement of large lumps connected with very narrow connectors is formed in the system in this situation. Generally speaking, a lower and wider bandgap is easily obtained through the careful design of the shape and size of the holes so as to generate the local resonance of the unit cell. ACKNOWLEDGMENTS

The authors are grateful for the support from the National Natural Science Foundation of China (Grant Nos. 11002018 and 10632020) and the National Basic Research Program of China (2010CB732104). The first author also acknowledges the support of the Science Foundation of Beijing Jiaotong University (2011YJS046). 1

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