- Email: [email protected]
The influence of the micro-topology on the phononic band gaps in 2D porous phononic crystals
Physics Letters A 372 (2008) 6784–6789
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
The influence of the micro-topology on the phononic band gaps in 2D porous phononic crystals Ying Liu a,∗ , Jia-yu Su a , Lingtian Gao b a b
Institute of Mechanics, Beijing Jiaotong University, Beijing 100044, PR China Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China
a r t i c l e
i n f o
Article history: Received 16 August 2008 Received in revised form 5 September 2008 Accepted 26 September 2008 Available online 8 October 2008 Communicated by R. Wu PACS: 43.40.+s 43.20.+g 43.35.+d
a b s t r a c t The phononic band structures of two-dimensional metal porous phononic crystals consisting of different lattices (the lattice structures transformed from square to triangle), and pores of various shapes (circle, square, and triangle) and sizes are studied numerically by using Finite Difference Time Domain (FDTD) scheme. It is found that for x– y mode waves, the absolute phononic band gaps (PBGs) rely more on the pore shapes. For triangular pores, the PBG is opening in the whole process of the lattice transformation, and for circular ones, the PBG is closed after a certain lattice structure. No PBG forms in the crystals with square pores. The PBG can be varied by adjusting the size of the pores. But a critical porosity exists for the opening of the PBG. © 2008 Elsevier B.V. All rights reserved.
Keywords: Phononic crystal Band gap FDTD
1. Introduction Due to its particular properties, the propagation of elastic and acoustic waves in the periodic composites, which are called phononic lattices or phononic crystals, has attracted considerable attention [1–4]. One important characteristic is the existence of the complete band gaps, in which no sound and vibration are allowed. This provides potential applications to prohibit specific vibration in accurate technologies such as transducers or sonar. Recently, along with the further application of composites in extreme situations, light-weighted design of materials becomes an important topic. Porous material is one of the good choices [5–8]. If the pores in the material are distributed periodically, one kind of porous phononic crystal or porous phononic lattice forms. Considering its great advantages, such as: self-supporting, fire retardance, low moisture absorption, and superior impact velocity energy absorption, it is expected that metal porous phononic crystals will find wide-range applications in noise and vibration control of aircraft, automobile, machinery, and building [5–8]. The study of phononic crystals has shown that gaps can exist in a lot of systems of any dimension (one-dimensional (1D), two-
*
Corresponding author. Tel.: +86 10 51682094; fax: +86 10 51682094. E-mail address: [email protected] (Y. Liu).
0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.09.051
dimensional (2D) and three-dimensional (3D)) systems. Both the material properties of each component (mass density, wave velocities or elastic moduli) and lattice structure have influences on the elastic wave gaps [9–14]. In fact, topology sensitivity is also a distinct characteristic of porous materials. Their mechanical responses are not only a material behavior, but also determined by the local topological properties. How to establish the relations between the topology and performance to realize the self-design of microtopology of porous materials according to applicable demands is always the frontier [6]. For solid phononic crystals, a lot of works have been carried out to establish the relation between the scatter properties (shapes, sizes and lattice structures) and the PBG. Caballero et al. has pointed out that honeycomb lattice is more favorable for the creation of the band gaps than square lattice and triangle lattice [15]. Diamond structure, FCC and BCC structures were shown to be in favor of large gaps among 3D systems [9–12]. Kuang et al. shows the largest absolute phononic band gap is achieved when the scatter has the same shape and orientation as that of the coordination polygon of a lattice point [16]. Zhong et al. studied the acoustic band gaps of five different shapes of steel rods placed in air with a square lattice [17]. Then, we would ask if the results for the solid phononic crystals are still available for the porous phononic crystals; in what structure of the lattice possesses the largest PBG for a given pore shape and what shape of pores can generate the largest PBG in a given lattice. Unfortunately,
Y. Liu et al. / Physics Letters A 372 (2008) 6784–6789
6785
Fig. 1. The diagrammatic sketch of the cross section of representative stagger-arranged 2D porous phononic crystals. a is the lattice constant. b is the stagger displacement of the even rows with respect to the odd rows.
no accurate answers have yet been given to per author’s knowledge. The mechanism for the opening of PBG in porous phononic crystals needs further analysis. The aim of this Letter is just to disclose the essential relation between the micro-topology (pore shapes, sizes, and distribution pattern) and the possible band structures in porous phononic crystals. Firstly, FDTD algorithm for the wave propagation in 2D porous phononic crystals is established. Then, by changing the pore arranged pattern (transformation from square lattice to triangular lattice), the influence of the pore shapes (triangular, circular or square) and the porosity on the possible band gap structures are discussed. At last, the conclusion is given.
coincided to those for δ = 0.5 to 1. As a result, δ = 0 to 0.5 is considered. Moreover, the results given by Ref. [18] indicate that there is little difference when the pores B are filled by air or just vacuum. In this Letter, the pores B are treated as vacuum. Seen as Fig. 1, since the 2D systems consisting of infinite long volume parallel to the z axis, the material behavior parameters do not depend on the coordinate z, and only the x– y mode waves are concentrated. Then, the governing equations for the wave propagation are given as
ρ
∂ 2ux ∂ σxx ∂ σxy = + , ∂x ∂y ∂t2
(1)
2. Theory
ρ
∂2u y ∂ σxy ∂σyy = + , ∂x ∂y ∂t2
(2)
Fig. 1 is a typical representation of 2D porous phononic crystals with pores (circular, triangular or square) (denoted by B) periodically stagger-arranged in the matrix material (denoted by A). a is the lattice constant, and b is the stagger displacement of the even rows with respect to the odd rows. The ratio δ = b/a is defined as the dislocated ratio. It is valued between 0 and 1. When δ = 0 or 1, it is corresponding to square lattice, and when δ = 0.5, it is triangular lattice. That is, along with the variation of the dislocated ratio δ , the lattice structure changes from square to triangle. Moreover, it is obvious that the lattice structures for δ = 0 to 0.5 are
where u x and u y are the displacement components of the matrix material, and ρ is the mass density. The components of the stress tensor σi j (i , j = x, y ) are with the form
∂u y ∂ ux + C 12 , ∂x ∂y ∂u y ∂u σ y y = C 12 x + C 11 , ∂x ∂y ∂ ux ∂ u y , σxy = C 44 + ∂y ∂x
σxx = C 11
(3) (4) (5)
6786
Y. Liu et al. / Physics Letters A 372 (2008) 6784–6789
in which C 11 , C 12 , and C 44 are the elastic constants relating to the velocities of the longitudinal and transverse waves, v l and v t , as: C 11 = ρ v l2 , C 44 = ρ v t2 , and C 12 = C 11 − 2C 44 . 2.1. Finite difference formulae We denote each grid point by (i , j , n) = (i x, j y , nt), and all of the variables are defined on a rectangular grid. After approximating by centre differences in both space and time, the equations for the displacement components become
Fig. 2. The diagrammatic sketch of the parallelogram lattice and the corresponding first Brillouin zone.
u x (i , j , n + 1)
= 2u x (i , j , n) − u x (i , j , n − 1) 1 1 t 2 × σxx i + , j , n − σxx i − , j , n + ρ (i , j )x 2 2 1 1 t 2 × σxy i , j + , n − σxy i , j − , n , + ρ (i , j ) y 2 2 1 1 uy i + , j + ,n + 1 2
2
1
1
2
2
1
1
2
2
(6)
2.2. Initial conditions
= 2u y i + , j + , n − u y i + , j + , n − 1 +
For computation the band gaps in the periodic structures, it is more convenient to assume that an initial field distribution satisfies the Bloch theorem at time t = 0 instantly, which is consistent with the periodic boundary condition introduced below mathematically. In the present calculation, the initial condition is given as [19]
t 2
ρ (i +
1 , 2
j + 12 )x
1 1 × σxy i + 1, j + , n − σxy i , j + , n
2
+
2
u(r)|t =0 = C 0
t 2
ρ (i +
1 , 2
j+
1 ) y 2
1
1
2
u(r)|t =t = D 0
,
2
2
1
= C 11 i + , j
u x (i + 1, j , n) − u x (i , j , n)
2
+ C 12
x
u y (i + 12 , j + 12 , n) − u y (i + 12 , j − 12 , n)
y
1
,
(8)
2
= C 44 i , j +
1
+ C 44 i , j +
1
1
2
x u x (i , j + 1, n) − u x (i , j , n)
y
,
(9)
σ y y i + , j, n 2
1
= C 11 i + , j
u y (i + 12 , j + 12 , n) − u y (i + 12 , j − 12 , n)
2
1
+ C 12 i + , j 2
|K + G|e i(K+G)r−i(K+G)ct ,
(12)
max |G| > ωmax /C ,
(13)
2.3. Boundary conditions
u y (i + 12 , j + 12 , n) − u y (i − 12 , j + 12 , n)
2
where ωmax is the maximum frequency that we are interested in. In the calculation, we have max{|G|} ≈ 2ωmax / v, and C 0 = D 0 = 1.
σxy i , j + , n
(11)
where i is the imaginary unit. C 0 and D 0 are the arbitrary constants. G is the reciprocal vector, K the wave vector, and v stands for the longitudinal or transverse wave velocity in the matrix. It is no need to take many reciprocal vector G in the summation since one only needs to make sure that the initial field distribution has nonzero projection on the eigenstates that we are interested in [20], that is,
σxx i + , j , n
|K + G|e i(K+G)r ,
G
(7)
where the stress components are given by 1
G
× σ y y i + , j + 1, n − σ y y i + , j , n
Fig. 3. The diagrammatic sketch of the rectangular lattice and the corresponding first Brillouin zone.
y u x (i + 1, j , n) − u x (i , j , n)
y
.
(10)
For given matrix materials, the elastic constants and mass density at each grid point are known. Starting from appropriate initial conditions, the time evolution of the displacement fields can be obtained according to Eqs. (6) to (10). It is easy to see that for a fixed total number of time steps, the computational time is proportional to the number of the discretization points.
Considering the periodicity of the porous phononic crystals in the plane x– y, if one pore is chosen as the unit cell, the lattice structure is parallelogram with the first Brillouin zone which is shown in Fig. 2, that is, the wave vector should be valued along the boundary -T-N- -X-M- -P-M to obtain the band structure, which increases the calculating quantity. As a result, in this Letter, the super unit cell, which contains two pores, is chosen as the computation domain (Fig. 3). By now, the parallelogram lattice changes to rectangular lattice with the first Brillouin zone -X-M- -N-M, which greatly improve the calculating efficiency. The super unit cell for the porous phononic crystals with square, circular, or triangular pores are given in Fig. 4, respectively. In the figures, a s , ar , and at are lattice constants, and b s , br , and bt are stagger displacements. In the calculating, we have a s = ar = at = a, and b s = br = bt = b. According to Bloch theory, the periodic boundary condition is given as u i (r + a) = u i (r)e iKa ,
(14)
Y. Liu et al. / Physics Letters A 372 (2008) 6784–6789
6787
Fig. 4. The diagrammatic sketch of the super unit cells of 2D porous phononic crystals.
(a) Fig. 5. The absolute band gaps for porous phononic crystals with triangular pores. The dislocation ratio b/a = 0.3. The porosity f = 0.3 and the lattice constant a = 0.04 m.
where r = (x, z) = (x, y , z) (the z axis is parallel to the cylinder axis), and a is the lattice vector. The results obtained in the time domain are then transferred into the frequency space by Fourier transformation. The positions of the existing peaks in the frequency spectra are then identified as the eigenfrequencies of the normal vibration modes for a given wave vector K. 3. Numerical examples and discussion 3.1. The influence of the dislocated ratio Pore shapes, dislocated ratio, and the porosity are the main parameters that we are concerning. A series of calculation have been carried out. The matrix material A is Aluminum, and the elastic parameters used in the calculation are (in units of GPa): C 11 = 110.9, C 44 = 26.1, and ρ = 2700 kg/m3 . The dislocated ratio is varied between 0 and 0.5. Considering the limitation of the maximum porosity, for square porous phononic crystals, the porosity changes between 0 and 0.9; for circular ones, the porosity varies from 0 to 0.7, and for the porous phononic crystals with triangular pores, the porosity is in the range 0 to 0.4. Among all types of possible micro-topology, the largest absolute PBG appears in the case of the triangular pores with triangular lattice, the numerical results of which are as follows. For the triangular pores, the PBG is always opening along with the variation of the dislocated ratio. One of the typical band structures are shown in Fig. 5. When the dislocated ratio δ = 0.5 (triangular lattice), the maximum gap has the largest normalized gap width (gap–midgap ratio) ω/ω g = 0.392. The upper and lower edges are 4.67 and 3.11 in the unit of 2π a/ v t , where v t is the
(b) Fig. 6. The variation of the absolute band gaps with respect to the dislocated ratio for the phononic crystals with triangular pores. (a) The porosity f = 0.3; (b) The porosity f = 0.4.
speed of transverse waves in the matrix, ω = 1.56. The variation of the absolute band gap structures with respect to the dislocated ratio at certain porosity are displayed in Fig. 6. It is seen that PBG always exists during the lattice transformation from square to triangle. Along with the increase of the dislocated ratio, the PBG is widened with the decrease of the centre frequencies, which reaches the maximum gap width and lowest centre frequencies at the triangular lattice. Especially when the porosity is small, this effect is more obvious. It is also seen that two new PBGs would form at the lower and higher frequencies with relative smaller gap width when the dislocated ratio is beyond certain values. It is noticed that the lower PBG has smaller opening dislocated ratio.
6788
Y. Liu et al. / Physics Letters A 372 (2008) 6784–6789
Fig. 7. The absolute band gap for porous phononic crystals with circular pores. The porosity f = 0.5 and the lattice constant a = 0.04 m. The dislocation rate b/a = 0.1.
Fig. 9. The band structure for porous phononic crystals with square pores. The porosity f = 0.4 and the lattice constant a = 0.04 m. The dislocation rate δ = 0.1.
(a)
(a)
(b)
(b)
Fig. 8. The variation of the absolute band gaps with respect to the dislocated ratio for circular porous phononic crystals at different porosity. (a) The porosity f = 0.6; (b) The porosity f = 0.7.
Fig. 10. The variation of the absolute band gaps with respect to the porosity in porous phononic crystals. (a) Triangular pores; (b) Circular pores.
Moreover, the increase of the porosity would increase the width of the PBGs, and also improve the opening values of the dislocated ratio for the new PBGs. For circular pores, it has smaller maximum gap, which is given in Fig. 7. The PBG is not always opening during the transformation from square lattice to triangular lattice (Fig. 8). Along with the in-
crease of the dislocated ratio, the width of the PBG is decreased with the dropping down of the centre frequency. It obtains its maximum normalized band gap width ω/ω g = 0.372 at the dislocated ratio δ = 0. The upper and lower edges are 3.77 and 2.58, ω = 1.19. The cut-off dislocated ratio exists, and the increase of the porosity raised the cut-off value. Moreover, the increase of the porosity reduces the centre frequencies of PBG. However, its effect
Y. Liu et al. / Physics Letters A 372 (2008) 6784–6789
6789
tio and porosity. It is seen that for triangular pores, the gap width increases with the porosity, but it first drops and then increases along with the increase of the dislocated ratio, which is enhanced at the higher porosity. As to circular pores, the effect of the porosity is the same as that in triangular ones. But for dislocated ratio, it has less influence on the normalized relative gap width when the porosity is low. Along with the increase of the porosity, the gap width is decreased with the increase of the dislocated ratio. As to the square pores, no PBGs appear during the whole variation of the porosity. That is, unlike the situation in solid phononic crystals (the largest PBG is formed in the square lattice with square scatters [16]), no matter in square lattice, triangular lattice or the transition parallelogram lattice, no absolute PBGs would form in the porous phononic crystals with square pores. 4. Conclusions (a)
In conclusion, by using FDTD method, we studied the effects of the pore shapes, pore sizes, as well as the lattice structures on the phononic band gap in 2D porous phononic crystals. Different from solid phononic crystals, the PBG relies more on the pore shapes. No PBG appear in the crystals with square pores. For a given porosity, the PBG is always opening and obtain the maximum band gap for triangular pores in triangular lattice, while the gap width reaches the maximum gap for circular pores in square lattice, and the PBG is closed when the dislocated ratio is greater than a certain value. The band gap is varied by adjusting the porosity and a critical porosity exists for the opening of PBG. On the whole, the normalized relative gap width will enlarge with the increase of the porosity, but the centre frequencies will drop down. This finding is helpful in the design of the 2D porous phononic crystals. Acknowledgements
(b) Fig. 11. The variation of the normalized relative gap width for the widest PBG in porous phononic crystals with respect to the porosity and the dislocated ratio. (a) Triangular pores; (b) Circular pores.
on the band width is not as obvious as the situation for triangular pores. Compared with above two cases for triangular or circular pores, no PBGs are formed in the whole process from square lattice to triangular one for porous phononic crystals with square pores. One of the typical band structures is given in Fig. 9. 3.2. The influence of the porosity Porosity is an important parameter in the dynamic description of the porous phononic crystals. Along with the increase of the porosity, the gap width of PGB is enlarged. For triangular pores, the PBGs reach the maximum width at the largest porosity. At the same time, the centre frequency is dropped. It should be noticed that a critical porosity exists for the opening of the PBG, which is read 0.2 for triangular pores (Fig. 10(a)). For circular pores, the maximum width also obtains at the maximum porosity, and the centre frequency is sensitive to its variation. They are dropped shapely along with the increase of the porosity. Moreover, the critical porosity for the opening of PBG in circular phononic crystals is 0.4, which is higher than that in triangular ones (Fig. 10(b)). Fig. 11 shows the variation of the normalized relative gap width for the widest PBG in triangular or circular porous phononic crystals with respect to the dislocated ra-
The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (Nos. 10672018, 10632020). References [1] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Phys. Rev. Lett. 71 (1993) 2022. [2] M.S. Kushwaha, P. Halevi, G. Martinez, Phys. Rev. B 49 (1994) 2313. [3] M.M. Sigalas, N. Garcia, J. Appl. Phys. 87 (2000) 3122. [4] M. Wilm, S. Ballandras, V. Laude, T. Pastureaud, Proc. IEEE Ultrason. Sympos. 2 (2001) 977. [5] L.J. Gibson, M.F. Ashby, Cellular Solids: Structures and Properties, second ed., Cambridge Univ. Press, Cambridge, UK, 1997. [6] A.G. Evans, J.W. Hutchinson, M.F. Ashby, Proc. Mater. Sci. 43 (1999) 171. [7] M.F. Ashby, J.W. Hutchinson, A.G. Evans, Cellular Metals, a Design Guide, Cambridge Univ. Press, Cambridge, UK, 1998. [8] Y. Sugimura, J. Meyer, M.Y. He, H. Bart-Smith, J.L. Grenestedt, A.G. Evans, Acta Mater. 45 (1997) 5345. [9] E.M. Econoumou, M. Sigalas, J. Acoust. Soc. Am. 95 (1994) 1734. [10] M. Kafesaki, E.N. Econoumou, Phys. Rev. B 52 (1995) 13317. [11] M. Kafesaki, M.M. Sigalas, E.N. Econoumou, Solid State Commun. 96 (1995) 285. [12] M.S. Kushwaha, B. Djafari-Rouhani, J. Appl. Phys. 80 (1996) 3191. [13] M.S. Kushwaha, B. Djafari-Rouhani, J. Sound Vibration 218 (1998) 697. [14] X. Zhang, Z.Y. Liu, Y.Y. Liu, F.G. Wu, Phys. Lett. A 313 (2003) 455. [15] D. Caballero, J. Sánchez-Dehesa, C. Rubio, R. Mártinez-Sala, J.V. Sánchez-Pérez, F. Meseguer, J. Llinares, Phys. Rev. E 60 (1999) 6316. [16] W.M. Kuang, Z.L. Zhou, Y.Y. Liu, Phys. Lett. A 332 (2004) 481. [17] L.H. Zhong, F.G. Wu, X. Zhang, H.L. Zhong, S. Zhong, Phys. Lett. A 339 (2005) 164. [18] Z.Z. Yan, Analysis of the band-gap structures of two-dimensional phononic crystals based on the wavelet theory, PhD dissertation, Beijing Jiaotong University, 2007. [19] Y.J. Cao, Z.L. Hou, Y.Y. Liu, Solid State Commun. 132 (2004) 539. [20] C.T. Chan, Q.L. Yu, K.M. Ho, Phys. Rev. B 51 (1995) 16635.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
The influence of the micro-topology on the phononic band gaps in 2D porous phononic crystals Ying Liu a,∗ , Jia-yu Su a , Lingtian Gao b a b
Institute of Mechanics, Beijing Jiaotong University, Beijing 100044, PR China Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China
a r t i c l e
i n f o
Article history: Received 16 August 2008 Received in revised form 5 September 2008 Accepted 26 September 2008 Available online 8 October 2008 Communicated by R. Wu PACS: 43.40.+s 43.20.+g 43.35.+d
a b s t r a c t The phononic band structures of two-dimensional metal porous phononic crystals consisting of different lattices (the lattice structures transformed from square to triangle), and pores of various shapes (circle, square, and triangle) and sizes are studied numerically by using Finite Difference Time Domain (FDTD) scheme. It is found that for x– y mode waves, the absolute phononic band gaps (PBGs) rely more on the pore shapes. For triangular pores, the PBG is opening in the whole process of the lattice transformation, and for circular ones, the PBG is closed after a certain lattice structure. No PBG forms in the crystals with square pores. The PBG can be varied by adjusting the size of the pores. But a critical porosity exists for the opening of the PBG. © 2008 Elsevier B.V. All rights reserved.
Keywords: Phononic crystal Band gap FDTD
1. Introduction Due to its particular properties, the propagation of elastic and acoustic waves in the periodic composites, which are called phononic lattices or phononic crystals, has attracted considerable attention [1–4]. One important characteristic is the existence of the complete band gaps, in which no sound and vibration are allowed. This provides potential applications to prohibit specific vibration in accurate technologies such as transducers or sonar. Recently, along with the further application of composites in extreme situations, light-weighted design of materials becomes an important topic. Porous material is one of the good choices [5–8]. If the pores in the material are distributed periodically, one kind of porous phononic crystal or porous phononic lattice forms. Considering its great advantages, such as: self-supporting, fire retardance, low moisture absorption, and superior impact velocity energy absorption, it is expected that metal porous phononic crystals will find wide-range applications in noise and vibration control of aircraft, automobile, machinery, and building [5–8]. The study of phononic crystals has shown that gaps can exist in a lot of systems of any dimension (one-dimensional (1D), two-
*
Corresponding author. Tel.: +86 10 51682094; fax: +86 10 51682094. E-mail address: [email protected] (Y. Liu).
0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.09.051
dimensional (2D) and three-dimensional (3D)) systems. Both the material properties of each component (mass density, wave velocities or elastic moduli) and lattice structure have influences on the elastic wave gaps [9–14]. In fact, topology sensitivity is also a distinct characteristic of porous materials. Their mechanical responses are not only a material behavior, but also determined by the local topological properties. How to establish the relations between the topology and performance to realize the self-design of microtopology of porous materials according to applicable demands is always the frontier [6]. For solid phononic crystals, a lot of works have been carried out to establish the relation between the scatter properties (shapes, sizes and lattice structures) and the PBG. Caballero et al. has pointed out that honeycomb lattice is more favorable for the creation of the band gaps than square lattice and triangle lattice [15]. Diamond structure, FCC and BCC structures were shown to be in favor of large gaps among 3D systems [9–12]. Kuang et al. shows the largest absolute phononic band gap is achieved when the scatter has the same shape and orientation as that of the coordination polygon of a lattice point [16]. Zhong et al. studied the acoustic band gaps of five different shapes of steel rods placed in air with a square lattice [17]. Then, we would ask if the results for the solid phononic crystals are still available for the porous phononic crystals; in what structure of the lattice possesses the largest PBG for a given pore shape and what shape of pores can generate the largest PBG in a given lattice. Unfortunately,
Y. Liu et al. / Physics Letters A 372 (2008) 6784–6789
6785
Fig. 1. The diagrammatic sketch of the cross section of representative stagger-arranged 2D porous phononic crystals. a is the lattice constant. b is the stagger displacement of the even rows with respect to the odd rows.
no accurate answers have yet been given to per author’s knowledge. The mechanism for the opening of PBG in porous phononic crystals needs further analysis. The aim of this Letter is just to disclose the essential relation between the micro-topology (pore shapes, sizes, and distribution pattern) and the possible band structures in porous phononic crystals. Firstly, FDTD algorithm for the wave propagation in 2D porous phononic crystals is established. Then, by changing the pore arranged pattern (transformation from square lattice to triangular lattice), the influence of the pore shapes (triangular, circular or square) and the porosity on the possible band gap structures are discussed. At last, the conclusion is given.
coincided to those for δ = 0.5 to 1. As a result, δ = 0 to 0.5 is considered. Moreover, the results given by Ref. [18] indicate that there is little difference when the pores B are filled by air or just vacuum. In this Letter, the pores B are treated as vacuum. Seen as Fig. 1, since the 2D systems consisting of infinite long volume parallel to the z axis, the material behavior parameters do not depend on the coordinate z, and only the x– y mode waves are concentrated. Then, the governing equations for the wave propagation are given as
ρ
∂ 2ux ∂ σxx ∂ σxy = + , ∂x ∂y ∂t2
(1)
2. Theory
ρ
∂2u y ∂ σxy ∂σyy = + , ∂x ∂y ∂t2
(2)
Fig. 1 is a typical representation of 2D porous phononic crystals with pores (circular, triangular or square) (denoted by B) periodically stagger-arranged in the matrix material (denoted by A). a is the lattice constant, and b is the stagger displacement of the even rows with respect to the odd rows. The ratio δ = b/a is defined as the dislocated ratio. It is valued between 0 and 1. When δ = 0 or 1, it is corresponding to square lattice, and when δ = 0.5, it is triangular lattice. That is, along with the variation of the dislocated ratio δ , the lattice structure changes from square to triangle. Moreover, it is obvious that the lattice structures for δ = 0 to 0.5 are
where u x and u y are the displacement components of the matrix material, and ρ is the mass density. The components of the stress tensor σi j (i , j = x, y ) are with the form
∂u y ∂ ux + C 12 , ∂x ∂y ∂u y ∂u σ y y = C 12 x + C 11 , ∂x ∂y ∂ ux ∂ u y , σxy = C 44 + ∂y ∂x
σxx = C 11
(3) (4) (5)
6786
Y. Liu et al. / Physics Letters A 372 (2008) 6784–6789
in which C 11 , C 12 , and C 44 are the elastic constants relating to the velocities of the longitudinal and transverse waves, v l and v t , as: C 11 = ρ v l2 , C 44 = ρ v t2 , and C 12 = C 11 − 2C 44 . 2.1. Finite difference formulae We denote each grid point by (i , j , n) = (i x, j y , nt), and all of the variables are defined on a rectangular grid. After approximating by centre differences in both space and time, the equations for the displacement components become
Fig. 2. The diagrammatic sketch of the parallelogram lattice and the corresponding first Brillouin zone.
u x (i , j , n + 1)
= 2u x (i , j , n) − u x (i , j , n − 1) 1 1 t 2 × σxx i + , j , n − σxx i − , j , n + ρ (i , j )x 2 2 1 1 t 2 × σxy i , j + , n − σxy i , j − , n , + ρ (i , j ) y 2 2 1 1 uy i + , j + ,n + 1 2
2
1
1
2
2
1
1
2
2
(6)
2.2. Initial conditions
= 2u y i + , j + , n − u y i + , j + , n − 1 +
For computation the band gaps in the periodic structures, it is more convenient to assume that an initial field distribution satisfies the Bloch theorem at time t = 0 instantly, which is consistent with the periodic boundary condition introduced below mathematically. In the present calculation, the initial condition is given as [19]
t 2
ρ (i +
1 , 2
j + 12 )x
1 1 × σxy i + 1, j + , n − σxy i , j + , n
2
+
2
u(r)|t =0 = C 0
t 2
ρ (i +
1 , 2
j+
1 ) y 2
1
1
2
u(r)|t =t = D 0
,
2
2
1
= C 11 i + , j
u x (i + 1, j , n) − u x (i , j , n)
2
+ C 12
x
u y (i + 12 , j + 12 , n) − u y (i + 12 , j − 12 , n)
y
1
,
(8)
2
= C 44 i , j +
1
+ C 44 i , j +
1
1
2
x u x (i , j + 1, n) − u x (i , j , n)
y
,
(9)
σ y y i + , j, n 2
1
= C 11 i + , j
u y (i + 12 , j + 12 , n) − u y (i + 12 , j − 12 , n)
2
1
+ C 12 i + , j 2
|K + G|e i(K+G)r−i(K+G)ct ,
(12)
max |G| > ωmax /C ,
(13)
2.3. Boundary conditions
u y (i + 12 , j + 12 , n) − u y (i − 12 , j + 12 , n)
2
where ωmax is the maximum frequency that we are interested in. In the calculation, we have max{|G|} ≈ 2ωmax / v, and C 0 = D 0 = 1.
σxy i , j + , n
(11)
where i is the imaginary unit. C 0 and D 0 are the arbitrary constants. G is the reciprocal vector, K the wave vector, and v stands for the longitudinal or transverse wave velocity in the matrix. It is no need to take many reciprocal vector G in the summation since one only needs to make sure that the initial field distribution has nonzero projection on the eigenstates that we are interested in [20], that is,
σxx i + , j , n
|K + G|e i(K+G)r ,
G
(7)
where the stress components are given by 1
G
× σ y y i + , j + 1, n − σ y y i + , j , n
Fig. 3. The diagrammatic sketch of the rectangular lattice and the corresponding first Brillouin zone.
y u x (i + 1, j , n) − u x (i , j , n)
y
.
(10)
For given matrix materials, the elastic constants and mass density at each grid point are known. Starting from appropriate initial conditions, the time evolution of the displacement fields can be obtained according to Eqs. (6) to (10). It is easy to see that for a fixed total number of time steps, the computational time is proportional to the number of the discretization points.
Considering the periodicity of the porous phononic crystals in the plane x– y, if one pore is chosen as the unit cell, the lattice structure is parallelogram with the first Brillouin zone which is shown in Fig. 2, that is, the wave vector should be valued along the boundary -T-N- -X-M- -P-M to obtain the band structure, which increases the calculating quantity. As a result, in this Letter, the super unit cell, which contains two pores, is chosen as the computation domain (Fig. 3). By now, the parallelogram lattice changes to rectangular lattice with the first Brillouin zone -X-M- -N-M, which greatly improve the calculating efficiency. The super unit cell for the porous phononic crystals with square, circular, or triangular pores are given in Fig. 4, respectively. In the figures, a s , ar , and at are lattice constants, and b s , br , and bt are stagger displacements. In the calculating, we have a s = ar = at = a, and b s = br = bt = b. According to Bloch theory, the periodic boundary condition is given as u i (r + a) = u i (r)e iKa ,
(14)
Y. Liu et al. / Physics Letters A 372 (2008) 6784–6789
6787
Fig. 4. The diagrammatic sketch of the super unit cells of 2D porous phononic crystals.
(a) Fig. 5. The absolute band gaps for porous phononic crystals with triangular pores. The dislocation ratio b/a = 0.3. The porosity f = 0.3 and the lattice constant a = 0.04 m.
where r = (x, z) = (x, y , z) (the z axis is parallel to the cylinder axis), and a is the lattice vector. The results obtained in the time domain are then transferred into the frequency space by Fourier transformation. The positions of the existing peaks in the frequency spectra are then identified as the eigenfrequencies of the normal vibration modes for a given wave vector K. 3. Numerical examples and discussion 3.1. The influence of the dislocated ratio Pore shapes, dislocated ratio, and the porosity are the main parameters that we are concerning. A series of calculation have been carried out. The matrix material A is Aluminum, and the elastic parameters used in the calculation are (in units of GPa): C 11 = 110.9, C 44 = 26.1, and ρ = 2700 kg/m3 . The dislocated ratio is varied between 0 and 0.5. Considering the limitation of the maximum porosity, for square porous phononic crystals, the porosity changes between 0 and 0.9; for circular ones, the porosity varies from 0 to 0.7, and for the porous phononic crystals with triangular pores, the porosity is in the range 0 to 0.4. Among all types of possible micro-topology, the largest absolute PBG appears in the case of the triangular pores with triangular lattice, the numerical results of which are as follows. For the triangular pores, the PBG is always opening along with the variation of the dislocated ratio. One of the typical band structures are shown in Fig. 5. When the dislocated ratio δ = 0.5 (triangular lattice), the maximum gap has the largest normalized gap width (gap–midgap ratio) ω/ω g = 0.392. The upper and lower edges are 4.67 and 3.11 in the unit of 2π a/ v t , where v t is the
(b) Fig. 6. The variation of the absolute band gaps with respect to the dislocated ratio for the phononic crystals with triangular pores. (a) The porosity f = 0.3; (b) The porosity f = 0.4.
speed of transverse waves in the matrix, ω = 1.56. The variation of the absolute band gap structures with respect to the dislocated ratio at certain porosity are displayed in Fig. 6. It is seen that PBG always exists during the lattice transformation from square to triangle. Along with the increase of the dislocated ratio, the PBG is widened with the decrease of the centre frequencies, which reaches the maximum gap width and lowest centre frequencies at the triangular lattice. Especially when the porosity is small, this effect is more obvious. It is also seen that two new PBGs would form at the lower and higher frequencies with relative smaller gap width when the dislocated ratio is beyond certain values. It is noticed that the lower PBG has smaller opening dislocated ratio.
6788
Y. Liu et al. / Physics Letters A 372 (2008) 6784–6789
Fig. 7. The absolute band gap for porous phononic crystals with circular pores. The porosity f = 0.5 and the lattice constant a = 0.04 m. The dislocation rate b/a = 0.1.
Fig. 9. The band structure for porous phononic crystals with square pores. The porosity f = 0.4 and the lattice constant a = 0.04 m. The dislocation rate δ = 0.1.
(a)
(a)
(b)
(b)
Fig. 8. The variation of the absolute band gaps with respect to the dislocated ratio for circular porous phononic crystals at different porosity. (a) The porosity f = 0.6; (b) The porosity f = 0.7.
Fig. 10. The variation of the absolute band gaps with respect to the porosity in porous phononic crystals. (a) Triangular pores; (b) Circular pores.
Moreover, the increase of the porosity would increase the width of the PBGs, and also improve the opening values of the dislocated ratio for the new PBGs. For circular pores, it has smaller maximum gap, which is given in Fig. 7. The PBG is not always opening during the transformation from square lattice to triangular lattice (Fig. 8). Along with the in-
crease of the dislocated ratio, the width of the PBG is decreased with the dropping down of the centre frequency. It obtains its maximum normalized band gap width ω/ω g = 0.372 at the dislocated ratio δ = 0. The upper and lower edges are 3.77 and 2.58, ω = 1.19. The cut-off dislocated ratio exists, and the increase of the porosity raised the cut-off value. Moreover, the increase of the porosity reduces the centre frequencies of PBG. However, its effect
Y. Liu et al. / Physics Letters A 372 (2008) 6784–6789
6789
tio and porosity. It is seen that for triangular pores, the gap width increases with the porosity, but it first drops and then increases along with the increase of the dislocated ratio, which is enhanced at the higher porosity. As to circular pores, the effect of the porosity is the same as that in triangular ones. But for dislocated ratio, it has less influence on the normalized relative gap width when the porosity is low. Along with the increase of the porosity, the gap width is decreased with the increase of the dislocated ratio. As to the square pores, no PBGs appear during the whole variation of the porosity. That is, unlike the situation in solid phononic crystals (the largest PBG is formed in the square lattice with square scatters [16]), no matter in square lattice, triangular lattice or the transition parallelogram lattice, no absolute PBGs would form in the porous phononic crystals with square pores. 4. Conclusions (a)
In conclusion, by using FDTD method, we studied the effects of the pore shapes, pore sizes, as well as the lattice structures on the phononic band gap in 2D porous phononic crystals. Different from solid phononic crystals, the PBG relies more on the pore shapes. No PBG appear in the crystals with square pores. For a given porosity, the PBG is always opening and obtain the maximum band gap for triangular pores in triangular lattice, while the gap width reaches the maximum gap for circular pores in square lattice, and the PBG is closed when the dislocated ratio is greater than a certain value. The band gap is varied by adjusting the porosity and a critical porosity exists for the opening of PBG. On the whole, the normalized relative gap width will enlarge with the increase of the porosity, but the centre frequencies will drop down. This finding is helpful in the design of the 2D porous phononic crystals. Acknowledgements
(b) Fig. 11. The variation of the normalized relative gap width for the widest PBG in porous phononic crystals with respect to the porosity and the dislocated ratio. (a) Triangular pores; (b) Circular pores.
on the band width is not as obvious as the situation for triangular pores. Compared with above two cases for triangular or circular pores, no PBGs are formed in the whole process from square lattice to triangular one for porous phononic crystals with square pores. One of the typical band structures is given in Fig. 9. 3.2. The influence of the porosity Porosity is an important parameter in the dynamic description of the porous phononic crystals. Along with the increase of the porosity, the gap width of PGB is enlarged. For triangular pores, the PBGs reach the maximum width at the largest porosity. At the same time, the centre frequency is dropped. It should be noticed that a critical porosity exists for the opening of the PBG, which is read 0.2 for triangular pores (Fig. 10(a)). For circular pores, the maximum width also obtains at the maximum porosity, and the centre frequency is sensitive to its variation. They are dropped shapely along with the increase of the porosity. Moreover, the critical porosity for the opening of PBG in circular phononic crystals is 0.4, which is higher than that in triangular ones (Fig. 10(b)). Fig. 11 shows the variation of the normalized relative gap width for the widest PBG in triangular or circular porous phononic crystals with respect to the dislocated ra-
The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (Nos. 10672018, 10632020). References [1] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Phys. Rev. Lett. 71 (1993) 2022. [2] M.S. Kushwaha, P. Halevi, G. Martinez, Phys. Rev. B 49 (1994) 2313. [3] M.M. Sigalas, N. Garcia, J. Appl. Phys. 87 (2000) 3122. [4] M. Wilm, S. Ballandras, V. Laude, T. Pastureaud, Proc. IEEE Ultrason. Sympos. 2 (2001) 977. [5] L.J. Gibson, M.F. Ashby, Cellular Solids: Structures and Properties, second ed., Cambridge Univ. Press, Cambridge, UK, 1997. [6] A.G. Evans, J.W. Hutchinson, M.F. Ashby, Proc. Mater. Sci. 43 (1999) 171. [7] M.F. Ashby, J.W. Hutchinson, A.G. Evans, Cellular Metals, a Design Guide, Cambridge Univ. Press, Cambridge, UK, 1998. [8] Y. Sugimura, J. Meyer, M.Y. He, H. Bart-Smith, J.L. Grenestedt, A.G. Evans, Acta Mater. 45 (1997) 5345. [9] E.M. Econoumou, M. Sigalas, J. Acoust. Soc. Am. 95 (1994) 1734. [10] M. Kafesaki, E.N. Econoumou, Phys. Rev. B 52 (1995) 13317. [11] M. Kafesaki, M.M. Sigalas, E.N. Econoumou, Solid State Commun. 96 (1995) 285. [12] M.S. Kushwaha, B. Djafari-Rouhani, J. Appl. Phys. 80 (1996) 3191. [13] M.S. Kushwaha, B. Djafari-Rouhani, J. Sound Vibration 218 (1998) 697. [14] X. Zhang, Z.Y. Liu, Y.Y. Liu, F.G. Wu, Phys. Lett. A 313 (2003) 455. [15] D. Caballero, J. Sánchez-Dehesa, C. Rubio, R. Mártinez-Sala, J.V. Sánchez-Pérez, F. Meseguer, J. Llinares, Phys. Rev. E 60 (1999) 6316. [16] W.M. Kuang, Z.L. Zhou, Y.Y. Liu, Phys. Lett. A 332 (2004) 481. [17] L.H. Zhong, F.G. Wu, X. Zhang, H.L. Zhong, S. Zhong, Phys. Lett. A 339 (2005) 164. [18] Z.Z. Yan, Analysis of the band-gap structures of two-dimensional phononic crystals based on the wavelet theory, PhD dissertation, Beijing Jiaotong University, 2007. [19] Y.J. Cao, Z.L. Hou, Y.Y. Liu, Solid State Commun. 132 (2004) 539. [20] C.T. Chan, Q.L. Yu, K.M. Ho, Phys. Rev. B 51 (1995) 16635.