PHYICA REVIEW B 75, 5414 27 Phononic band gaps of elastic periodic structures: A hoogenization theory study Ying-Hong iu, 1,3, * Chien C. Chang, 1,2,3, Ruey-in Chern, 2,3 and C. Chung Chang 1,3 1 Division of Mechanics, Research Center for Applied ciences, Acadeia inica, Taipei 115, Taiwan, Republic of China 2 Institute of Applied Mechanics, National Taiwan University, Taipei 16, Taiwan, Republic of China 3 Taida Institute of Matheatical ciences, National Taiwan University, Taipei 16, Taiwan, Republic of China Received 26 April 26; revised anuscript received 18 epteber 26; published 7 February 27 In this study, we investigate the band structures of phononic crystals with particular ephasis on the effects of the ass density ratio and of the contrast of elastic constants. The phononic crystals consist of arrays of different edia ebedded in a rubber or epoxy. It is shown that the density ratio rather than the contrast of elastic constants is the doinant factor that opens up phononic band gaps. The physical background of this observation is explained by applying the theory of hoogenization to investigate the group velocities of the low-frequency bands at the center of syetry. DOI: 1.113/PhysRevB.75.5414 PAC nubers: 43.35.d, 46.4.Cd, 63.2.e I. INTRODUCTION Phononic crystals for elastic waves are an analog of photonic crystals for electroagnetic waves. Phononic crystals are periodic arrays of two or ore elastic aterials with distinct densities and elastic constants. The ost distinguished feature of phononic crystals is their band gaps, and therefore phononic crystals are also called phononic bandgap aterials. In the past years, we have seen steadily increasing interest in phononic crystals because of their interesting physical properties 1 6 and possible engineering applications. 7 11 However, there are ajor differences between phononic and photonic crystals that ake the study of phononic crystals a separate subject fro photonic crystals. First of all, dielectric aterials usually support transverse electroagnetic waves, while elastic aterials support both transverse as well as longitudinal elastic waves. econd, photonic aterials have the largest speed of propagation in air, while elastic aterials have a sall longitudinal speed of propagation in air. Moreover, the physical properties of photonic crystals are deterined by the contrast of dielectric constants, while those of phononic crystals are deterined by both the contrast of elastic constants and the ass density ratio of the coposed aterials. In this study, we are concerned with the effects of the ass density ratio and contrast of elastic constants on ajor phononic band gaps and develop a theory of hoogenization to exaine the echanis of the effects. The coputation of band structures is deanding, as an eigenvalue proble needs to be solved for each individual wave nuber in the first Brillouin zone. A fast and accurate ethod for coputing band structures is very helpful in designing phononic band-gap aterials. In the present study, we apply a ethod of inverse iteration with ultigrid acceleration to copute the band structures of phononic crystals. This ethod was originally developed by the present authors to copute band structures for photonic crystals ade of dielectric aterials. 12,13 Regarding the effects of aterial constants, it is natural to consider that a large contrast of elastic constants is necessary for the existence of a ajor band gap. This is not necessarily true as we shall show that the ass density ratio is the key factor in deterining 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 ass density ratio, while the lower bands are heavily dependent on this change. In general, the frequency bands of the transverse odes are relatively flat copared to those of the longitudinal odes. This indicates that it is easier to open up a band gap between the bands of the transverse odes, but which is often fully blocked by the frequency bands of the longitudinal odes. As the ass density ratio is increased, the lowerfrequency bands, in particular of the longitudinal odes, shift downward in frequency and shrink significantly in size, resulting in an opening up of a ajor band gap. Those results could be put to a solid physical background by the theory of hoogenization, which provides a good guideline for opening up band gaps. In order to indicate the underlying thought, we first develop the theory of hoogenization in one diension, then followed by the theory in two diensions. In particular, two distinguished classes of phononic crystals are considered: edia ebedded in a rubber have elastic constants larger than rubbers by four or five orders in agnitude, and edia ebedded in an epoxy have elastic constants coparable to epoxies in agnitude. II. BAIC EQUATION In the present study, we consider the tie-haronic wave equation for linear, anisotropic, and elastic aterials 1 jc ijn n u r + 2 u i r =, 1 where u i i=1,2,3 are the displaceents, and =r and C ijn =C ijn r are the ass density and elastic stiffness tensor, respectively. For a two-diensional proble in the xy plane with out-of-plane propagation in the z direction, Eq. 1 can be written for cubic crystals as 1 xx xy xz yx yy zz ux yz zx zy u y u z = 2 ux u y u z, where ij i, j=1,2,3 are detailed in the Appendix. For periodic structures, it is sufficient to solve the proble in one 2 198-121/27/755/54148 5414-1 27 The Aerican Physical ociety
IU et al. PHYICA REVIEW B 75, 5414 27 TABE I. The unit of the ass density is g/c 3 and the elastic constants are in 1 9 N/ 2. These physical constants are used for investigating two-diensional phononic crystals. Mediu C 11 C 44 FIG. 1. Color online The stagger esh is widely applied with a finite-difference tie doain algorith. First, we split the coponents of the displaceent onto different esh points i.e., oving forward the x coponent of the displaceent half of esh length in the x direction and treat the sae way for the y coponent of the displaceent in the y direction. econd, we put the physical properties of the aterial in different esh points. For exaple, in the cubic syste, the ass densities C 11 and C 12 are assigned to circle esh points and C 44 is assigned to rectangular points. unit cell, along with the Bloch condition satisfied at the cell boundary, u j r + a i = e ik a iu j r, 3 where k is the wave vector and a i i=1,2 are the lattice translation vectors. A central finite-difference schee 14 is used to discretize Eq. 2. The positions of u x and u y are offset by a half grid size in their own directions, respectively, as shown in Fig. 1. There are two points we want to ention. First, we separate the coponents of the displaceent half esh in its own direction. econd, the elastic constants and the ass density are arranged into different areas. In our study, the ass density and elastic constants C 11 and C 12 are assigned to the esh points but C 44 is assigned to the center of the esh zone, as shown in the figure. This arrangeent is helpful for the convergence of nuerical results. Then we obtain the discretized eigenvalue proble Au = u, 4 where we have applied condition 3. The eigensyste is solved by the ethod of inverse iteration with ultigrid acceleration as entioned in the Introduction. Ice.94 13.79 3.18 C 1.75 31 88.5 AlAs 3.76 12.2 58.9 GaAs 5.36 118.8 59.4 Ni 8.97 311.61 92.93 Ag 1.64 152.68 4.44 Pb 11.6 72.1 14.9 W 19.3 5.3 151.31 Rubber 1.3 6.81 4 4.1 5 Epoxy 1.2 9.61 1.61 contrast of the elastic constant between carbon and rubber is quite large, no full band gaps are observed in this structure Fig. 3. In fact, a band gap exists between the first few transverse odes of shear horizontal H 1 and shear vertical 1 branches. However, this band gap is blocked by the first longitudinal ode of pressure branch. If carbon cylinders are replaced by heavier Pb cylinders, 16 then large full band gaps can be opened up. Figure 4 shows the band structures of Pb cylinders ebedded in a rubber background for the sae filling fraction. In this case, there is a large contrast in the ass density between the ebedded aterial and the background. The frequencies of the first, the first, and the first two H branches are significantly reduced to lower values, while the frequency of the second branches reains little changed and becoes flattened. It is also observed that the higher-frequency bands are uch less sensitive to the change of the physical constants of the ebedded aterial. 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 ass density ratio is ore effective in opening up a band gap than a large contrast of elastic constants. We now attept to answer this question by developing a theory of hoogenization. III. REUT AND DICUION et us consider a square lattice of square cylinders of aterials ebedded in a rubber. The physical constants of the ebedded aterials are listed in Table I Refs. 15 and 16; all the ebedded aterials have elastic constants larger than the rubber by four or five orders in agnitude. First, we consider the band structure of the C/rubber syste with the filling fraction f =.36 Fig. 2. Although the FIG. 2. a The ediu/rubber square lattice. The filling fraction is.36. The first siulated case is the C/rubber syste and the second is the Pb/rubber syste. b The first Brillouin zone and the special points of syetry. 5414-2
PHONONIC BAND GAP OF EATIC PERIODIC FIG. 3. Color online The band structure of the array of carbon squares ebedded in a rubber background. The filling fraction f of the carbon is.36. The frequency is noralized by V/a where V is the transverse velocity of the rubber. A. One-diensional hoogenization 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 ay entirely shift downward. The slopes are actually the group velocities. The group velocity of the coposite aterial at the low-frequency liit for periodic structures can be deterined by applying the theory of hoogenization see, e.g., Ref. 17. For this purpose, we consider the odel proble of one-diensional elasticity: u xe x = u 2 t 2, where denotes the ass density and E is Young s odulus, both of which vary with a period a of which aterial 1 occupies a proportion fa, while aterial 2 occupies 1 fa. et us consider a wave propagating with a large wavelength l i.e., =a/l 1. It is convenient to do nondiensional analysis by scaling; we introduce x ax, E E c E, c, and t 2t/ c where E c, c, and c are characteristic values of Young s odulus, ass density, and frequency. Then we obtain u xe x = a 2 u 4 2 E c / c 2 c 2 t 2, where we identify l 2 =4 2 E c / c c 2, and the scaled Eq. 5 becoes u xe x = 2 2 u t 2. Now we introduce two scales, x=x fine scale and x=x coarse-grained, for further analysis. The displaceent is considered as a function of x and x i.e., u=ux,x; then, x + x x. Expanding u in powers of, we obtain + x xe x PHYICA REVIEW B 75, 5414 27 u = u + u 1 + 2 u 2 +, + xu + u 1 + 2 u 2 + = 2 2 u t 2 +. 1 Next we collect ters of the sae power in. For the order O,weget 5 6 7 8 9 xe u x =, 11 where u denotes the coarse-grained displaceent, and it is evident that it depends on x only. For the order O 1,we obtain xe u1 x x + u =. A general solution of u 1 is given by 12 FIG. 4. Color online The band structure of an array of Pb squares ebedded in a rubber background. The filling fraction f of Pb is.36. The frequency is noralized by V/a where V is the transverse velocity of rubber. Ba and Bb denote first and second band gaps. u 1 = Qx,x u x + ū1 x, 13 where Qx,x is a periodic function in x of period a and ū 1 x is independent of x. ubstituting u 1 of Eq. 13 into Eq. 12, we obtain 5414-3
IU et al. TABE II. Group velocities of the lowest-frequency bands at the point for a one-diensional crystal. The unit of velocity is k/s. V n,1d and V n,1d are the velocities of longitudinal waves and shear waves, respectively, obtained fro the nuerical results. V h,1d and V h,1d are obtained fro the theory of one-diensional hoogenization. PHYICA REVIEW B 75, 5414 27 TABE III. Group velocities of the lowest-frequency bands at the point for a two-diensional crystal. The unit of velocity is k/s. and are the velocities of longitudinal waves and shear vertical waves, respectively, obtained fro the nuerical results. V h,1d and V h,1d are obtained fro the theory of one- diensional hoogenization. Coposite V n,1d V n,1d V h,1d V h,1d Coposite V h,1d V h,1d C/rubber.271.655.27.65 Pb/rubber.145.35.145.35 C/rubber.289.78.27.65 Pb/rubber.154.42.145.35 A siple integration gives xe1+ x Q =. Q = x + D 1x x +x dx E + D 2, 14 15 where D 1 and D 2 are functions of x only and x is a reference point. ince Q ust be a periodic function of x with period a, D 1 needs to satisfy D 1 = 1 x +a dx 16 ax 1 E and D 2 can siply be taken to be zero. The results suggest that we define the effective Young s odulus E e = D 1 = E 1 1 E 1 E 2 =, 17 fe 2 + 1 fe 1 where E 1 and E 2 are Young s oduli of ebedded and host aterials, respectively. Finally, gathering ters for the O 2, we have xe u1 x x + u + xe u2 x x + u1 = 2 u t 2. 18 In order to see the behavior on a acroscale, we average Eq. 18 in the unit cell. The second ter after being averaged with respect to x becoes zero because of the periodic boundary conditions. Then the resulting averaged equation with using Eqs. 13, 15, and 17 becoes x = 2 u u e xe where is the ean ass density, t 2, = f 1 + 1 f 2. 19 2 Equation 19 only depends on x and describes the acroscale elastic waves propagating along the coposite aterial under the long-wave approxiation. This explains why E e is called effective Young s odulus. The effective velocity is given by v e = E e = E 1E 2 fe 2 + 1 fe 1 1 f 1 + 1 f 2. 21 The results 17 and 2 indicate that the effective Young s odulus E e is ainly deterined by the less rigid aterial, E e =E 2 /1 f if E 1 E 2, while the ean density is deterined by the heavier aterial, = f 1 if 1 2.In our siulation, E 1 and E 2 stand for the elastic constants of the ebedded aterials and the host rubber, respectively, with E 1 E 2. Therefore, v e E 2 1. 22 1 f f 1 + 1 f 2 The siple theory is particularly accurate for onediensional crystal Table II, and is less satisfactory for two-diensional crystals Table III. In both C/rubber and Pb/rubber systes, C and Pb have a uch larger Young s odulus than rubber. Thus both coposite systes have the sae effect by increasing the Young s odulus fro rubber s E 2 to E 2 /1 f=e 2 /.64. The effective group velocity of the two coposite systes, differing fro that of rubber, is ainly deterined by the difference in density between the host and ebedded aterials. For the longitudinal waves, we have v C/rub e,long = C 11 1 1.26v rub 1 f f 1 + 1 f long, 2 v Pb/rub e,long = C 11 1.672v rub 1 f f 1 + 1 f long. 2 23 24 The sae results apply to the effective transverse velocity by siply replacing C 11 with C 44. The results 23 and 24 explain why the Pb/rubber systes copared to the C/rubber syste lower the first few bands very effectively, especially in longitudinal odes, which covers soe band gaps produced by shear odes, thus opening up a band gap. On the other hand, we ay ebed a less rigid aterial in rubber but with coparable density to reduce the group velocity, at the point and thus lower the lower-frequency bands. To see a closer coparison between nuerical and theoretical results, we elaborate below on the theory of twodiensional hoogenization. 5414-4
PHONONIC BAND GAP OF EATIC PERIODIC B. Two-diensional hoogenization et us start with the tie-dependent for of the wave equation x jc ijn u x n = 2 u i t 2. 25 The sae procedure as the one-diensional proble leads to + x j xc j ijn + x n x n u + u 1 + 2 u 2 + = 2 2 u i t 2 +, 26 where i, =1,2,3 and j, n=1,2. Then we collect ters in different powers of. At the order O, we have u ijn x jc x n =. 27 Fro the discussion of the previous suggestion, we know that u =u x depends on x only. At the order O 1,we get x jc ijn u x n + u 1 x n =. 28 The key step is to solve Eq. 28 for u 1. In analog with Eq. 13, we assue the for of solution u 1 = b u k x k + ū 1 x, 29 where ū 1 x is independent of x and b k has no suation in the index. The solution for connects the perturbed solutions at the zeroth order and the first order. ubstituting u 1 of Eq. 29 into Eq. 28, we obtain ijn nk + b k x jc x n u x k =. 3 At the order O 2, we have xc j ijn u x n + u = 2 u i t 2. 1 x n + x jc ijn u 1 x n + u 2 x n 31 Now Eq. 31 is averaged with respect to x over the unit cell. The second ter is iediately averaged to give zero because of periodic boundary conditions. Then we have ijn nk + xc b k j x n u x k = u i 2 t 2. 32 In analog with Eq. 25, this otivates us to define the effective elastic constants as e C ijk =C ijn nk + b k, 33 x n and then Eq. 32 becoes PHYICA REVIEW B 75, 5414 27 x C ijk j x k = u i 2 t 2. 34 In contrast to the one-diensional proble, we do not have a closed-for forula of effective elastic constants. Instead, they have to be obtained by solving ijn nk + b k x jc x n =, 35 as indicated by Eq. 3, where u /x k could be general functions of x k. It is straightforward to see that Eq. 35 reduces to Eq. 14 if we consider the one-diensional elastic wave equation 5. Equation 35 is the key to the two-diensional hoogenization. What we need for the effective C ijk are C ijn and e b k as shown in Eq. 33. This equation also indicates that the deterination of b k depends on the spatial property of C ijn and thus the possibility of iproveent of twodiensional hoogenization over the one-diensional theory. Moreover, the above forulation is valid for general linear elastic aterials in both two and three diensions. Now we consider the two-diensional elastic wave equations of the cubic aterials. For shear vertical odes, we have e u zz u z = 2 u z t 2, 36 and for longitudinal-shear horizontal -H odes, we have xx u x + xy u y = 2 u x t 2, 37 yx u x + yy u y = 2 u y t 2. 38 odes. Consider the ode. The coarse-grained equation is given by 44 x i C ij u z = x 2 u z j t 2, 39 where i, j=1,2 and u z denotes the coarse-grained displaceent of the z coponent, and = 1 d, 4 C ij 44 = 1 C 44 ij + b j x id. 41 Here, is the doain of the unit cell, C 44 =C 1313 or C 2323 and b j satisfying 44 ij + b j x ic x i =, 42 and ust, in general, be solved nuerically. ince Eq. 42 is suppleent by the periodic boundary conditions, we append the additional condition b j = to solve Eq. 42 uniquely. 5414-5
IU et al. This effective shear vertical group velocity is given by V e,2d = C 44 11. 43 -H Modes. Consider the wave equations of the ixed ode. The coarse-grained equation is given by 11 xc u x + x + yc 44 44 yc u x + y + Res = x 2 u x t 2. u y The effective elastic constants are given by C 11 = 1 C b 1 1 111+ d, x C 44 = 1 C b 2 1 441+ d, y C 12 = 1 C b 2 2 121+ d, y y 12 xc u y 44 45 46 47 C 44 = 1 C b 1 2 441+ d, 48 x where C 11 =C 1111 or C 2222, C 44 =C 1212 and C 12 =C 1122, and Res denotes the residual, 1 Res = 1 b 1 44 yc y u x x + C 44 + xc 1 b 2 u x 11 x y + C 12 y where b i are functions satisfying 2 b 2 u y x 2 b 1 u y y xd, 1 b 1 11 + xc 1 b 1 44 = x yc C 11 y x, 49 5 TABE IV. Group velocities of the lowest-frequency bands at the point for a two-diensional crystal. The unit of velocity is k/s., V H n,2d, and are the velocities of longitudinal waves, shear horizontal, and vertical waves, respectively, obtained fro the nuerical results. V h,2d, V H h,2d, and V h,2d are obtained fro the theory of two-diensional hoogenization. Coposite PHYICA REVIEW B 75, 5414 27 H V h,2d H V h,2d V h,2d C/rubber.289.72.78.294.78.78 Pb/rubber.154.4.42.159.42.42 Table IV shows the coparison of group velocity at point between the nuerical results and those predicted by twodiensional hoogenization. The results show significant iproveent over the coparison listed in Table III between two-diensional nuerical results and the results of onediensional hoogenization. The one-diensional hoogenization indicates that a large contrast of ass density is necessary for producing full band gaps in phononic crystals, no atter how large the contrast of elastic constants is. In order to illustrate this general trend for two-diensional systes, we siulate different aterials listed in Table I with given physical constants ebedded in rubber. In Fig. 5, we plot the band-gap ratios of Ba and Bb defined in Fig. 4 for different aterials ebedded in a rubber background. For ice, carbon, and aluinu, with coparable ass 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 systes. As the ass density is increased above that of GaAs, band gap Ba opens up and its size band-gap idgap ratio increases linearly up 1 b 2 11 + xc 1 b 2 44 = x yc C 44 y y, 2 b 1 12 + xc 2 b 1 44 = y yc C 44 x y, 51 52 2 b 2 12 + xc 2 b 2 44 = y yc C 12 x x. 53 The effective shear horizontal and longitudinal group velocities are given, respectively, by V H e,2d = C 44, V e,2d = C 11. 54 FIG. 5. Color online The band-gap size for different ediu/ rubber systes. The unit of ass density is g/c 3. The ass density of rubber is 1.3, lying between carbon and ice. The open triangle sybols and open square sybols denote the gap-idgap ratios of Ba and Bb. Here, the solid sybols denote artificial ebedded aterials which have the sae elastic constants of carbon, but with different ass densities. All the cases have fixed filling fraction f =.36 for the ebedded ediu. 5414-6
PHONONIC BAND GAP OF EATIC PERIODIC PHYICA REVIEW B 75, 5414 27 this is true for the elastic constants if we consider their inverses. In contrast, two-diensional hoogenization does not exhibit this siple average for the elastic constants. Instead, before averaging we have to solve a syste of equations that take care of the detailed spatial properties of the elastic constants. This explains why the two-diensional hoogenization shows significantly iproved results over the one-diensional theory in predicting the group velocities of the lowest-frequency bands at the center of syetry. The current ethod of analysis is applied to three-diensional probles; the results will be reported elsewhere. ACKNOWEDGMENT The work is supported in part by the National cience Council of the Republic of China Taiwan under Contract No. NC 94-2212-E-2-47 and the Ministry of Econoic Affairs of the Republic of China under Contract No. MOEA 94-EC-17-A-8-1-6. FIG. 6. Color online The band-gap size for different ediu/ epoxy systes. The unit of ass density is g/c 3. The ass density of epoxy is 1.2, lying between carbon and ice. The open triangle sybols and open square sybols denote the gap-idgap ratios of Ba and Bb. All the cases have fixed filling fraction f =.36 for the ebedded ediu. to W. On the other hand, band gap Bb opens up as the density ratio is increased above that of Al and increases linearly to GaAs, where we see a saturated size.15. It is also interesting to see if the host aterial rubber is replaced by a ore rigid aterial like epoxy. Epoxy has Q density about that of rubber but has elastic constants coparable to other aterials in agnitude. Figure 6 shows the results for coparison. The general trend of the two bands Ba and Bb for ediu/ epoxy systes is not too uch different fro the ediu/ rubber systes though the coparable elastic constants of the ediu and host coplicated their effects of hoogenization. IV. CONCUDING REMARK In the present study, we have applied a fast algorith inverse iteration with ultigrid acceleration to copute the band structures of phononic crystals. A critical issue is addressed as how to open up a large band gap for phononic crystals. It is shown, by the theory of hoogenization in one as well as two diensions, how the ass density ratio and the contrast of elastic constants affect the size of ajor phononic band gaps. In particular, it is quite efficient to open up a band gap by lowering the group velocities of the lowfrequency bands at the center. One-diensional hoogenization shows that the effective ass density is the areaaveraged density of the host and ebedded aterials, while Appendix The coponents of Eq. 2 are xx = xc 11 44 yc x + y C 44 k z 2, xy = xc 12 + y yc 44 x, xz = ik z x C 12 + C 44 x, yx = yc 12 + x xc 44 y, yy = xc 44 + x yc 11 C y 44 k 2 z, yz = ik z y C 12 + C 44 y, zx = ik zc 12 x + x C 44, zy = ik zc 12 y + y C 44, zz = xc 44 + x yc 44 C y 11 k 2 z. A1 A2 A3 A4 A5 A6 A7 A8 A9 5414-7
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