Phononic band gaps of elastic periodic structures: a homogenization theory study

Phononic band gaps of elastic periodic structures: A homogenization theory study

Ying-Hong Liu,1,3,*Chien C. Chang,1,2,3,†Ruey-Lin Chern,2,3and C. Chung Chang1,3

1_{Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan, Republic of China} 2_{Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan, Republic of China}

3_{Taida Institute of Mathematical Sciences, National Taiwan University, Taipei 106, Taiwan, Republic of China} 共Received 26 April 2006; revised manuscript received 18 September 2006; published 7 February 2007兲 In this study, we investigate the band structures of phononic crystals with particular emphasis on the effects of the mass density ratio and of the contrast of elastic constants. The phononic crystals consist of arrays of different media embedded in a rubber or epoxy. It is shown that the density ratio rather than the contrast of elastic constants is the dominant factor that opens up phononic band gaps. The physical background of this observation is explained by applying the theory of homogenization to investigate the group velocities of the low-frequency bands at the center of symmetry⌫.

DOI:10.1103/PhysRevB.75.054104 PACS number共s兲: 43.35.⫹d, 46.40.Cd, 63.20.⫺e

I. INTRODUCTION

Phononic crystals for elastic waves are an analog of pho-tonic crystals for electromagnetic waves. Phononic crystals are periodic arrays of two or more elastic materials with distinct densities and elastic constants. The most distin-guished feature of phononic crystals is their band gaps, and therefore phononic crystals are also called phononic band-gap materials. In the past years, we have seen steadily in-creasing interest in phononic crystals because of their inter-esting physical properties1–6 _{and possible engineering} applications.7–11

However, there are major differences between phononic and photonic crystals that make the study of phononic crys-tals a separate subject from photonic cryscrys-tals. First of all, dielectric materials usually support transverse electromag-netic waves, while elastic materials support both transverse as well as longitudinal elastic waves. Second, photonic ma-terials have the largest speed of propagation in air, while elastic materials have a small共longitudinal兲 speed of propa-gation in air. Moreover, the physical properties of photonic crystals are determined by the contrast of dielectric con-stants, while those of phononic crystals are determined by both the contrast of elastic constants and the mass density ratio of the composed materials. In this study, we are con-cerned with the effects of the mass density ratio and contrast of elastic constants on major phononic band gaps and de-velop a theory of homogenization to examine the mechanism of the effects.

The computation of band structures is demanding, as an eigenvalue problem needs to be solved for each individual wave number in the first Brillouin zone. A fast and accurate method for computing band structures is very helpful in de-signing phononic band-gap materials. In the present study, we apply a method of inverse iteration with multigrid accel-eration to compute the band structures of phononic crystals. This method was originally developed by the present authors to compute band structures for photonic crystals made of dielectric materials.12,13

Regarding the effects of material constants, it is natural to consider that a large contrast of elastic constants is necessary for the existence of a major band gap. This is not necessarily

true as we shall show that the mass density ratio is the key factor in determining the location and size of the band gap. If the contrast of elastic constants is large, the higher-frequency bands are not very sensitive to the change of the mass den-sity ratio, while the lower bands are heavily dependent on this change. In general, the frequency bands of the transverse modes are relatively flat compared to those of the longitudi-nal modes. This indicates that it is easier to open up a band gap between the bands of the transverse modes, but which is often fully blocked by the frequency bands of the longitudi-nal modes. As the mass density ratio is increased, the lower-frequency bands, in particular of the longitudinal modes, shift downward in frequency and shrink significantly in size, resulting in an opening up of a major band gap. Those results could be put to a solid physical background by the theory of homogenization, which provides a good guideline for open-ing up band gaps. In order to indicate the underlyopen-ing thought, we first develop the theory of homogenization in one dimen-sion, then followed by the theory in two dimensions. In par-ticular, two distinguished classes of phononic crystals are considered: media embedded in a rubber have elastic con-stants larger than rubbers by four or five orders in magni-tude, and media embedded in an epoxy have elastic constants comparable to epoxies in magnitude.

II. BASIC EQUATIONS

In the present study, we consider the time-harmonic wave equation for linear, anisotropic, and elastic materials

1

␳⳵j关Cijmn⳵num共r兲兴 +␻2ui共r兲 = 0, 共1兲

where ui共i=1,2,3兲 are the displacements, and ␳=␳共r兲 and

Cijmn= Cijmn共r兲 are the mass density and elastic stiffness

ten-sor, respectively. For a two-dimensional problem in the xy plane with out-of-plane propagation in the z direction, Eq. 共1兲 can be written for cubic crystals as

−1 ␳

冤

Lxx Lxy Lxz Lyx Lyy Lyz Lzx Lzy Lzz

冥冤

ux uy uz

冥

=␻2

冤

ux uy uz

冥

, 共2兲

where Lij共i, j=1,2,3兲 are detailed in the Appendix. For

unit cell, along with the Bloch condition satisfied at the cell boundary,

uj共r + ai兲 = eik·aiuj共r兲, 共3兲

where k is the wave vector and ai 共i=1,2兲 are the lattice

translation vectors. A central finite-difference scheme14 _is used to discretize Eq. 共2兲. The positions of ux and uy are

offset by a half grid size in their own directions, respectively, as shown in Fig.1. There are two points we want to mention. First, we separate the components of the displacement half mesh in its own direction. Second, the elastic constants and the mass density are arranged into different areas. In our study, the mass density and elastic constants C11and C12are assigned to the mesh points but C44is assigned to the center of the mesh zone, as shown in the figure. This arrangement is helpful for the convergence of numerical results. Then we obtain the discretized eigenvalue problem

Au =␭u, 共4兲

where we have applied condition 共3兲. The eigensystem is

solved by the method of inverse iteration with multigrid ac-celeration as mentioned in the Introduction.

III. RESULTS AND DISCUSSION

Let us consider a square lattice of square cylinders of materials embedded in a rubber. The physical constants of the embedded materials are listed in Table I 共Refs. 15 and

16兲; all the embedded materials have elastic constants larger

than the rubber by four or five orders in magnitude. First, we consider the band structure of the C/rubber sys-tem with the filling fraction f = 0.36共Fig. 2兲. Although the

contrast of the elastic constant between carbon and rubber is quite large, no full band gaps are observed in this structure 共Fig. 3兲. In fact, a band gap exists between the first few

transverse modes of shear horizontal共SH1兲 and shear vertical 共SV1兲 branches. However, this band gap is blocked by the first longitudinal mode of pressure共L兲 branch. If carbon cyl-inders are replaced by heavier Pb cylcyl-inders,16_{then large full} band gaps can be opened up. Figure4shows the band struc-tures of Pb cylinders embedded in a rubber background for the same filling fraction. In this case, there is a large contrast in the mass density between the embedded material and the background. The frequencies of the first L, the first SV, and the first two SH branches are significantly reduced to lower values, while the frequency of the second SV branches re-mains little changed and becomes flattened. It is also ob-served that the higher-frequency bands are much less sensi-tive to the change of the physical constants of the embedded material. As a result, two large full band gaps, denoted by Ba and Bb, respectively, are opened up and separated by a nearly straight band. The question is why the difference in

the mass density ratio is more effective in opening up a band gap than a large contrast of elastic constants. We now

at-tempt to answer this question by developing a theory of homogenization.

FIG. 1. 共Color online兲 The stagger mesh is widely applied with a finite-difference time domain algorithm. First, we split the com-ponents of the displacement onto different mesh points—i.e., mov-ing forward the x component of the displacement half of mesh length in the x direction—and treat the same way for the y compo-nent of the displacement in the y direction. Second, we put the physical properties of the material in different mesh points. For example, in the cubic system, the mass densities C11and C12are assigned to circle mesh points and C₄₄is assigned to rectangular points.

TABLE I. The unit of the mass density is g / cm3and the elastic constants are in 109_{N / m}2_{. These physical constants are used for} investigating two-dimensional phononic crystals.

Medium ␳ C11 C44 Ice 0.94 13.79 3.18 C 1.75 310 88.5 AlAs 3.76 120.2 58.9 GaAs 5.36 118.8 59.4 Ni 8.97 311.61 92.93 Ag 10.64 152.68 40.44 Pb 11.6 72.1 14.9 W 19.3 500.03 151.31 Rubber 1.3 6.8⫻10−4 _4.0⫻10−5 Epoxy 1.20 9.61 1.61

FIG. 2. 共a兲 The medium/rubber square lattice. The filling frac-tion is 0.36. The first simulated case is the C/rubber system and the second is the Pb/rubber system.共b兲 The first Brillouin zone and the special points of symmetry.

A. One-dimensional homogenization

In order to open up a band gap, one possibility is to lower the first few frequency bands. It is plausible that if we can reduce the slopes of the low-frequency bands at the point⌫, then these bands may entirely shift downward. The slopes are actually the group velocities. The group velocity of the composite material at the low-frequency limit for periodic structures can be determined by applying the theory of ho-mogenization共see, e.g., Ref.17兲. For this purpose, we

con-sider the model problem of one-dimensional elasticity:

⳵ ⳵x

冉

E ⳵ ⳵xu

冊

=␳ ⳵2_u ⳵t2, 共5兲

where␳denotes the mass density and E is Young’s modulus, both of which vary with a period a of which material 1 occupies a proportion fa, while material 2 occupies共1− f兲a. Let us consider a wave propagating with a large wave-length l—i.e.,⑀= a / l
1. It is convenient to do nondimen-sional analysis by scaling; we introduce x→ax, E→EcE, ␳

→␳c␳, and t→2␲t /␻cwhere Ec,␳c, and␻care

characteris-tic values of Young’s modulus, mass density, and frequency. Then we obtain ⳵ ⳵x

冉

E ⳵ ⳵xu

冊

= a2 4␲2_E c/共␳c␻c2兲 ␳⳵2u ⳵t2, 共6兲 where we identify l2_{= 4␲}2_E

c/共␳c␻c2兲, and the scaled Eq. 共5兲

becomes ⳵ ⳵x

冉

E ⳵ ⳵xu

冊

=⑀ 2_␳⳵ 2_u ⳵t2. 共7兲

Now we introduce two scales, x = x 共fine scale兲 and x

⬘

=⑀x

共coarse-grained兲, for further analysis. The displacement is considered as a function of x and x

⬘

—i.e., u = u共x,x

⬘

兲; then,

⳵ ⳵x→ ⳵ ⳵x+⑀ ⳵ ⳵x

⬘

. 共8兲 Expanding u in powers of⑀, u = u0_+⑀u1_+⑀2_u2_{+ ¯ , 共9兲} we obtain

冉

⳵ ⳵x+⑀ ⳵ ⳵x

⬘

冊

冋

E

冉

⳵ ⳵x+⑀ ⳵ ⳵x

⬘

冊

共u 0_+⑀u1_+⑀2_u2_{+ ¯ 兲}

册

=⑀2_␳⳵ 2_u0 ⳵t2 + ¯ . 共10兲

Next we collect terms of the same power in⑀. For the order

O共⑀0兲, we get

⳵ ⳵x

冉

E

⳵u0

⳵x

冊

= 0, 共11兲

where u0denotes the coarse-grained displacement, and it is evident that it depends on x

⬘

only. For the order O共⑀1兲, we obtain ⳵ ⳵x

冋

E

冉

⳵u1 ⳵x + ⳵u0 ⳵x

⬘

冊

册

= 0. 共12兲

A general solution of u1 _{is given by}

u1_{= Q共x,x}

_⬘

_兲⳵u 0 ⳵x

⬘

+ u¯

1_共x

_⬘

_{兲, 共13兲} where Q共x,x

⬘

兲 is a periodic function in x of period a and

u

¯1_共x

_⬘

_{兲 is independent of x. Substituting u}1 _{of Eq. 共13兲 into} Eq.共12兲, we obtain

FIG. 3.共Color online兲 The band structure of the array of carbon squares embedded in a rubber background. The filling fraction f of the carbon is 0.36. The frequency is normalized by V / a where V is the transverse velocity of the rubber.

FIG. 4. 共Color online兲 The band structure of an array of Pb squares embedded in a rubber background. The filling fraction f of Pb is 0.36. The frequency is normalized by V / a where V is the transverse velocity of rubber. Ba and Bb denote first and second band gaps.

⳵

⳵x

冋

E

冉

1 + ⳵Q

⳵x

冊

册

= 0. 共14兲

A simple integration gives

Q = − x + D1

冕

x₀ x₀+x

dx˜

E + D2, 共15兲

where D1 and D2 are functions of x

⬘

only and x0 is a refer-ence point. Since Q must be a periodic function of x with period a, D1 needs to satisfy

D1=

冉

1 a

冕

x₀ x₀+a dx˜ E

冊

−1 共16兲

and D₂ can simply be taken to be zero. The results suggest that we define the effective Young’s modulus

Ee= D1=

冓

1 E

冔

−1 = E1E2 fE₂+共1 − f兲E₁, 共17兲 where E1and E2 are Young’s moduli of embedded and host materials, respectively. Finally, gathering terms for the

O共⑀2_{兲, we have} ⳵ ⳵x

⬘

冋

E

冉

⳵u1 ⳵x + ⳵u0 ⳵x

⬘

冊

册

+ ⳵ ⳵x

冋

E

冉

⳵u2 ⳵x + ⳵u1 ⳵x

⬘

冊

册

=␳ ⳵2_u0 ⳵t2 . 共18兲 In order to see the behavior on a macroscale, we average Eq. 共18兲 in the unit cell. The second term after being averaged

with respect to x becomes zero because of the periodic boundary conditions. Then the resulting averaged equation with using Eqs.共13兲, 共15兲, and 共17兲 becomes

⳵ ⳵x

⬘

冉

Ee ⳵u0 ⳵x

⬘

冊

=具␳典 ⳵2_u0 ⳵t2 , 共19兲

where具␳典 is the mean mass density,

具␳典 = f␳1+共1 − f兲␳2. 共20兲 Equation共19兲 only depends on x

⬘

and describes the macro-scale elastic waves propagating along the composite material under the long-wave approximation. This explains why Eeis

called effective Young’s modulus. The effective velocity is given by ve=

冑

Ee 具␳典=

冑

E1E2 fE₂+共1 − f兲E₁ 1 f␳₁+共1 − f兲␳₂. 共21兲 The results 共17兲 and 共20兲 indicate that the effective

Young’s modulus Ee is mainly determined by the less rigid

material, Ee= E2/共1− f兲 if E1E2, while the mean density is determined by the heavier material, 具␳典= f␳1 if ␳1␳2. In our simulation, E1 and E2 stand for the elastic constants of the embedded materials and the host共rubber兲, respectively, with E1E2. Therefore, ve⬇

冑

E₂ 1 − f 1 f␳1+共1 − f兲␳2 . 共22兲

The simple theory is particularly accurate for one-dimensional crystal 共Table II兲, and is less satisfactory for

two-dimensional crystals 共Table III兲. In both C/rubber and

Pb/rubber systems, C and Pb have a much larger Young’s modulus than rubber. Thus both composite systems have the same effect by increasing the Young’s modulus from rubber’s

E2to E2/共1− f兲=E2/ 0.64. The effective group velocity of the two composite systems, differing from that of rubber, is mainly determined by the difference in density between the host and embedded materials.

For the longitudinal waves, we have

v_e,longC/rub=

冑

C11 1 − f 1 f␳1+共1 − f兲␳2 ⬇ 1.26vlong rub , 共23兲 v_e,longPb/rub=

冑

C11 1 − f 1 f␳₁+共1 − f兲␳₂⬇ 0.672vlong rub . 共24兲

The same results apply to the effective transverse velocity by simply replacing C11 with C44. The results 共23兲 and 共24兲 explain why the Pb/rubber systems compared to the C/rubber system lower the first few bands very effectively, especially in longitudinal modes, which covers some band gaps pro-duced by shear modes, thus opening up a band gap. On the other hand, we may embed a less rigid material in rubber 共but with comparable density兲 to reduce the group velocity, at the point⌫ and thus lower the lower-frequency bands.

To see a closer comparison between numerical and theo-retical results, we elaborate below on the theory of two-dimensional homogenization.

TABLE II. Group velocities of the lowest-frequency bands at the point⌫ for a one-dimensional crystal. The unit of velocity is km/s. V_n,1DL and V_n,1DS are the velocities of longitudinal waves and shear waves, respectively, obtained from the numerical results.

V_h,1DL and V_h,1DS are obtained from the theory of one-dimensional homogenization.

Composite V_n,1DL V_n,1DS V_h,1DL V_h,1DS

C/rubber 0.0271 0.00655 0.0270 0.0065 Pb/rubber 0.0145 0.0035 0.0145 0.0035

TABLE III. Group velocities of the lowest-frequency bands at the point⌫ for a two-dimensional crystal. The unit of velocity is km/s. V_n,2DL and V_n,2DSV are the velocities of longitudinal waves and shear vertical waves, respectively, obtained from the numerical re-sults. V_h,1DL and V_h,1DS are obtained from the theory of one-dimensional homogenization.

Composite V_n,2DL V_n,2DSV V_h,1DL V_h,1DS

C/rubber 0.0289 0.0078 0.0270 0.0065 Pb/rubber 0.0154 0.0042 0.0145 0.0035

B. Two-dimensional homogenization

Let us start with the time-dependent form of the wave equation ⳵ ⳵xj

冋

Cijmn ⳵um ⳵xn

册

=␳⳵ 2_u i ⳵t2. 共25兲

The same procedure as the one-dimensional problem leads to

冉

⳵ ⳵xj +⑀ ⳵ ⳵xj

⬘

冊

冋

Cijmn

冉

⳵ ⳵xn +⑀ ⳵ ⳵xn

⬘

冊

共um0 +⑀um1+⑀2um2 + ¯ 兲

册

=⑀2␳⳵ 2_u i 0 ⳵t2 + ¯ , 共26兲

where i, m = 1 , 2 , 3 and j, n = 1 , 2. Then we collect terms in different powers of⑀. At the order O共⑀0兲, we have

⳵ ⳵xj

冉

Cijmn ⳵um 0 ⳵xn

冊

= 0. 共27兲

From the discussion of the previous suggestion, we know that u_m0= u_m0共x

⬘

兲 depends on x

⬘

only. At the order O共⑀1_{兲, we} get ⳵ ⳵xj

冋

Cijmn

冉

⳵um 0 ⳵xn

⬘

+ ⳵um 1 ⳵xn

冊

册

= 0. 共28兲

The key step is to solve Eq.共28兲 for um1. In analog with Eq.

共13兲, we assume the form of solution

um1 = bk共m兲

⳵um

0 ⳵xk

⬘

+ u¯_m1_共x

_⬘

_{兲, 共29兲} where u¯_m1_共x

_⬘

_{兲 is independent of x and b}

k

共m兲_{has no summation} in the index m. The solution form connects the perturbed solutions at the zeroth order and the first order. Substituting

um

1

of Eq.共29兲 into Eq. 共28兲, we obtain

⳵ ⳵xj

冋

Cijmn

冉

␦nk+ ⳵bk共m兲 ⳵xn

冊

册

⳵um0 ⳵xk

⬘

= 0. 共30兲

At the order O共⑀2_{兲, we have} ⳵ ⳵x

⬘

j

冋

Cijmn

冉

⳵um 0 ⳵xn

⬘

+⳵um 1 ⳵xn

冊

册

+ ⳵ ⳵xj

冋

Cijmn

冉

⳵um 1 ⳵xn

⬘

+⳵um 2 ⳵xn

冊

册

=␳⳵ 2_u i 0 ⳵t2 . 共31兲

Now Eq.共31兲 is averaged with respect to x over the unit cell.

The second term is immediately averaged to give zero be-cause of periodic boundary conditions. Then we have

⳵ ⳵xj

⬘

冓

Cijmn

冉

␦nk+ ⳵bk共m兲 ⳵xn

冊

冔

⳵um 0 ⳵xk

⬘

=具␳典⳵ 2_u i 0 ⳵t2 . 共32兲

In analog with Eq.共25兲, this motivates us to define the

ef-fective elastic constants as

Cijmk e =

冓

Cijmn

冉

␦nk+ ⳵bk共m兲 ⳵xn

冊

冔

, 共33兲

and then Eq.共32兲 becomes

⳵ ⳵x

⬘

j Cijmk e ⳵um 0 ⳵xk

⬘

=具␳典⳵ 2_u i 0 ⳵t2 . 共34兲

In contrast to the one-dimensional problem, we do not have a closed-form formula of effective elastic constants. Instead, they have to be obtained by solving

⳵ ⳵xj

冋

Cijmn

冉

␦nk+ ⳵bk共m兲 ⳵xn

冊

册

= 0, 共35兲

as indicated by Eq. 共30兲, where ⳵um

0

/⳵xk

⬘

could be general

functions of xk

⬘

. It is straightforward to see that Eq. 共35兲

reduces to Eq.共14兲 if we consider the one-dimensional

elas-tic wave equation共5兲.

Equation共35兲 is the key to the two-dimensional

homog-enization. What we need for the effective Cijmk e

are Cijmnand

b_k共m兲as shown in Eq. 共33兲. This equation also indicates that

the determination of b_k共m兲depends on the spatial property of

Cijmn and thus the possibility of improvement of

two-dimensional homogenization over the one-two-dimensional theory. Moreover, the above formulation is valid for general linear elastic materials in both two and three dimensions.

Now we consider the two-dimensional elastic wave equa-tions of the cubic materials. For shear vertical共SV兲 modes, we have

Lzzuz=␳

⳵2_u

z

⳵t2 , 共36兲

and for longitudinal-shear horizontal共L-SH兲 modes, we have

Lxxux+ Lxyuy=␳ ⳵2_u x ⳵t2 , 共37兲 Lyxux+ Lyyuy=␳ ⳵2_u y ⳵t2 . 共38兲

SV modes. Consider the SV mode. The coarse-grained

equation is given by ⳵ ⳵xi

⬘

冉

具C44 ij_典⳵uz 0 ⳵x

⬘

j

冊

=具␳典⳵ 2_u z 0 ⳵t2 , 共39兲

where i , j = 1 , 2 and uz0 denotes the coarse-grained

displace-ment of the z component, and 具␳典 = 1 ⍀

冕

⍀␳d⍀, 共40兲 具C44 ij_{典 =} 1 ⍀

冕

_⍀C44

冉

␦ij+ ⳵bj ⳵xi

冊

d⍀. 共41兲

Here,⍀ is the domain of the unit cell, C44= C1313or C2323 and bjsatisfying ⳵ ⳵xi

冋

C44

冉

␦ij+ ⳵bj ⳵xi

冊

册

= 0, 共42兲

and must, in general, be solved numerically. Since Eq.共42兲 is

supplement by the periodic boundary conditions, we append the additional condition 具bj典=0 to solve Eq. 共42兲 uniquely.

This effective shear vertical group velocity is given by

V_e,2DSV =

冑

具C44 11_典

具␳典 . 共43兲

L-SH Modes. Consider the wave equations of the mixed

mode. The coarse-grained equation is given by

⳵ ⳵x

⬘

冉

具C11典 ⳵ux0 ⳵x

⬘

冊

+ ⳵ ⳵y

⬘

冉

具C44典 ⳵ux0 ⳵y

⬘

冊

+ ⳵ ⳵x

⬘

冉

具C12典 ⳵u0y ⳵y

⬘

冊

+ ⳵ ⳵y

⬘

冉

具C44

⬘

典 ⳵uy 0 ⳵x

⬘

冊

+ Res =具␳典 ⳵2_u x 0 ⳵t2 . 共44兲

The effective elastic constants are given by 具C11典 = 1 ⍀

冕

⍀ C₁₁

冉

1 +⳵b1 共1兲 ⳵x

冊

d⍀, 共45兲 具C44典 = 1 ⍀

冕

⍀ C₄₄

冉

1 +⳵b2 共1兲 ⳵y

冊

d⍀, 共46兲 具C12典 = 1 ⍀

冕

_⍀C12

冉

1 + ⳵b₂共2兲 ⳵y

冊

d⍀, 共47兲 具C44

⬘

典 = 1 ⍀

冕

_⍀C44

冉

1 + ⳵b₁共2兲 ⳵x

冊

d⍀, 共48兲

where C11= C1111 or C2222, C44= C1212 and C12= C1122, and Res denotes the residual,

Res = 1 ⍀

冕

_⍀ ⳵ ⳵y

⬘

冋

C44 ⳵b₁共1兲 ⳵y ⳵ux 0 ⳵x

⬘

+ C44 ⳵b₂共2兲 ⳵x ⳵uy 0 ⳵y

⬘

册

+ ⳵ ⳵x

⬘

冋

C11 ⳵b₂共1兲 ⳵x ⳵ux 0 ⳵y

⬘

+ C12 ⳵b₁共2兲 ⳵y ⳵uy 0 ⳵x

⬘

册

d⍀, 共49兲

where b_i共m兲are functions satisfying

⳵ ⳵x

冋

C11 ⳵b1共1兲 ⳵x

册

+ ⳵ ⳵y

冋

C44 ⳵b1共1兲 ⳵y

册

= − ⳵C11 ⳵x , 共50兲 ⳵ ⳵x

冋

C11 ⳵b₂共1兲 ⳵x

册

+ ⳵ ⳵y

冋

C44 ⳵b₂共1兲 ⳵y

册

= − ⳵C44 ⳵y , 共51兲 ⳵ ⳵x

冋

C12 ⳵b₁共2兲 ⳵y

册

+ ⳵ ⳵y

冋

C44 ⳵b₁共2兲 ⳵x

册

= − ⳵C44 ⳵y , 共52兲 ⳵ ⳵x

冋

C12 ⳵b₂共2兲 ⳵y

册

+ ⳵ ⳵y

冋

C44 ⳵b₂共2兲 ⳵x

册

= − ⳵C12 ⳵x . 共53兲

The effective shear horizontal and longitudinal group veloci-ties are given, respectively, by

Ve,2D SH =

冑

具C44典 具␳典 , Ve,2D L =

冑

具C11典 具␳典 . 共54兲 TableIVshows the comparison of group velocity at point ⌫ between the numerical results and those predicted by

two-dimensional homogenization. The results show significant improvement over the comparison listed in TableIIIbetween two-dimensional numerical results and the results of one-dimensional homogenization.

The one-dimensional homogenization indicates that a large contrast of mass density is necessary for producing full band gaps in phononic crystals, no matter how large the con-trast of elastic constants is. In order to illustrate this general trend for two-dimensional systems, we simulate different materials 共listed in Table I with given physical constants兲

embedded in rubber. In Fig.5, we plot the band-gap ratios of Ba and Bb共defined in Fig.4兲 for different materials

embed-ded in a rubber background. For ice, carbon, and aluminum, with comparable mass densities of that of rubber, neither band gap Ba nor Bb is observed. The band gaps are found easily in heavier Ni, Ag, and Pb/rubber systems. As the mass density is increased above that of GaAs, band gap Ba opens up and its size共band-gap–midgap ratio兲 increases linearly up TABLE IV. Group velocities of the lowest-frequency bands at the point⌫ for a two-dimensional crystal. The unit of velocity is km/s. V_n,2DL , V_n,2DSH , and V_n,2DSV are the velocities of longitudinal waves, shear horizontal, and vertical waves, respectively, obtained from the numerical results. V_h,2DL , V_h,2DSH , and V_h,2DSV are obtained from the theory of two-dimensional homogenization.

Composite V_n,2DL V_n,2DSH VSV_n,2D V_h,2DL V_h,2DSH V_h,2DSV

C/rubber 0.0289 0.0072 0.0078 0.0294 0.0078 0.0078 Pb/rubber 0.0154 0.0040 0.0042 0.0159 0.0042 0.0042

FIG. 5. 共Color online兲 The band-gap size for different medium/ rubber systems. The unit of mass density is g / cm3_{. The mass} den-sity of rubber is 1.3, lying between carbon and ice. The open tri-angle symbols and open square symbols denote the gap-midgap ratios of Ba and Bb. Here, the solid symbols denote artificial em-bedded materials which have the same elastic constants of carbon, but with different mass densities. All the cases have fixed filling fraction f = 0.36 for the embedded medium.

to W. On the other hand, band gap Bb opens up as the den-sity ratio is increased above that of Al and increases linearly to GaAs, where we see a saturated size 0.15. It is also inter-esting to see if the host material rubber is replaced by a more rigid material like epoxy. Epoxy has Q density about that of rubber but has elastic constants comparable to other materi-als in magnitude. Figure6shows the results for comparison. The general trend of the two bands Ba and Bb for medium/ epoxy systems is not too much different from the medium/ rubber systems though the comparable elastic constants of the medium and host complicated their effects of homogeni-zation.

IV. CONCLUDING REMARKS

In the present study, we have applied a fast algorithm— inverse iteration with multigrid acceleration—to compute the band structures of phononic crystals. A critical issue is ad-dressed as how to open up a large band gap for phononic crystals. It is shown, by the theory of homogenization in one as well as two dimensions, how the mass density ratio and the contrast of elastic constants affect the size of major phononic band gaps. In particular, it is quite efficient to open up a band gap by lowering the group velocities of the low-frequency bands at the center⌫. One-dimensional homogeni-zation shows that the effective mass density is the area-averaged density of the host and embedded materials, while

this is true for the elastic constants if we consider their in-verses. In contrast, two-dimensional homogenization does not exhibit this simple average for the elastic constants. In-stead, before averaging we have to solve a system of equa-tions that take care of the detailed spatial properties of the elastic constants. This explains why the two-dimensional ho-mogenization shows significantly improved results over the one-dimensional theory in predicting the group velocities of the lowest-frequency bands at the center of symmetry⌫. The current method of analysis is applied to three-dimensional problems; the results will be reported elsewhere.

ACKNOWLEDGMENTS

The work is supported in part by the National Science Council of the Republic of China共Taiwan兲 under Contract No. NSC 94-2212-E-002-047 and the Ministry of Economic Affairs of the Republic of China under Contract No. MOEA 94-EC-17-A-08-S1-0006.

Appendix The components of Eq.共2兲 are

Lxx= ⳵ ⳵x

冉

C11 ⳵ ⳵x

冊

+ ⳵ ⳵y

冉

C44 ⳵ ⳵y

冊

− C44kz 2 , 共A1兲 Lxy= ⳵ ⳵x

冉

C12 ⳵ ⳵y

冊

+ ⳵ ⳵y

冉

C44 ⳵ ⳵x

冊

, 共A2兲 Lxz= ikz

冉

⳵ ⳵xC12+ C44 ⳵ ⳵x

冊

, 共A3兲 Lyx= ⳵ ⳵y

冉

C12 ⳵ ⳵x

冊

+ ⳵ ⳵x

冉

C44 ⳵ ⳵y

冊

, 共A4兲 Lyy= ⳵ ⳵x

冉

C44 ⳵ ⳵x

冊

+ ⳵ ⳵y

冉

C11 ⳵ ⳵y

冊

− C44kz 2_{, 共A5兲} Lyz= ikz

冉

⳵ ⳵yC12+ C44 ⳵ ⳵y

冊

, 共A6兲 Lzx= ikz

冉

C12 ⳵ ⳵x+ ⳵ ⳵xC44

冊

, 共A7兲 Lzy= ikz

冉

C12 ⳵ ⳵y+ ⳵ ⳵yC44

冊

, 共A8兲 Lzz= ⳵ ⳵x

冉

C44 ⳵ ⳵x

冊

+ ⳵ ⳵y

冉

C44 ⳵ ⳵y

冊

− C11kz 2 . 共A9兲

FIG. 6. 共Color online兲 The band-gap size for different medium/ epoxy systems. The unit of mass density is g / cm3_{. The mass} den-sity of epoxy is 1.2, lying between carbon and ice. The open tri-angle symbols and open square symbols denote the gap-midgap ratios of Ba and Bb. All the cases have fixed filling fraction f = 0.36 for the embedded medium.

*Electronic address: yinghung@gate.sinica.edu.tw †_{Electronic address: mechang@gate.sinica.edu.tw}

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