IJST (2013) 37A4: 457-462
Iranian Journal of Science & Technology
http://ijsts.shirazu.ac.ir
The generalization of structure factor for rods by polygon
section in two-dimensional phononic crystals
H. Salehi, M. Aryadoust* and M. Zargar Shoushtari
Department of Physics, Shahid Chamran University, Ahvaz, Iran
E-mail: [email protected]
Abstract
The purpose of this paper is the generalization of structure factor for rods by polygon section in two dimensional
phononic crystals. If we use the plane wave expansion method (PWE) for the propagation of acoustic waves in 2D
phononic crystals, structure factor will be an important quantity. In order to confirm the obtained relations, we
have calculated the band structure for XY and Z vibration modes in 2D phononic crystals and the propagation of
bulk acoustic waves (BAW) are considered. In addition, the effect of sides’ number on the band structure and the
complete band gaps width are investigated. Phononic crystals studied in this paper are composites medium of a
square lattice consisting of parallel nickel rods embedded in epoxy. The frequency is calculated by PWE in the
condition of elastic rigidity to the solid inclusions. The results showed that, when the section of rods have 2n+2 (n
is even) and 2n+1 n N by increasing sides number of the rod sections, the bands of XY mode shift to lowerfrequency, the bands are smoother and the width of the band gap increases, but the band of Z mode has not
changed by n variations. Moreover, when the section of rods have 2n+2 (n is odd) by increasing sides number of
rod sections, the band structure of XY mode changes slightly and the width of the complete band gap is decreased.
This confirms the effect of lattice symmetry on the complete band gap width. But, the band structure of Z mode
has not changed.
Keywords: Band gap; phononic crystal; Polygon section; structure factor
1. Introduction
Elastic wave propagation in periodic structures has
been the subject of intensive researches and it has
received a great deal of attention during the past
two decades [1-7]. These periodic structures,
phononic crystals, are also called the elastic band
gap (EBG) materials. These inhomogeneous elastic
media are composed of one [8, 9], two [1, 10, 11],
or three [12, 13] dimensional periodic arrays of
inclusions embedded in a matrix. Due to their
periodic structure, these materials may exhibit
under certain conditions, absolute acoustic band
gaps, i.e. forbidden bands which are independent of
the propagation direction of the incident elastic
wave. Because of this property, these structures
have extensive practical applications, for instance,
in the construction of sound shields and filters [1419], in the refractive devices such as sound-wave
focusing acoustic lenses [20, 21]. Also, they are
used in the selective frequency waveguides [22]. In
view of these applications, it is important to know
how to design structures with the phononic gap as
wide as possible. Besides the physical properties of
the PC crystal component materials, the phononic
*Corresponding author
Received: 15 April 2013 / Accepted: 5 October 2013
gap width is found to depend strongly on the lattice
symmetry, as well on the scatter shape [23, 24].
The phononic crystal solid/solid materials in twodimensional are composed of periodic arrays of
rods inclusions, under the assumption of wave
propagation in the plane perpendicular to the axis
rods. In this case, the vibration modes decouple in
the mixed-polarization modes (XY) with the elastic
displacement u perpendicular to the rod axes and in
the transverse modes (Z) with u parallel to the
inclusions. To reveal wide acoustic band gaps, we
first require a large contrast in physical properties,
such as density and speed of sound, between the
inclusions and the matrix, next, a sufficient filling
factor of inclusions [25].
In this paper, we generalize structure factor for
rods by polygon section in two-dimensional
phononic crystals. Also, the wave propagation of
bulk acoustic is studied in square array of Nickel
rods with polygon section embedded in epoxy. We
reviewed formulation of plane wave method for
elastic waves propagating in two-dimensional
phononic crystal. Moreover, we calculate the band
gap for XY and Z vibration modes, and the effect of
sides number on the band structure and band gaps
are investigated.
IJST (2013) 37A4: 457-462
458
ρG
2. Theory
and
The behavior of elastic waves propagating in a solid
can be described by equation of motion, and can be
obtained by considering the stresses acting on a
volume element. Application of Newton’s laws
gives,
C
∑
,
(1)
and rk are mass density, component
Here , ui,
of displacement vector, component of stress tensor
and the position vector, respectively. The
component of stress tensor is given as:
∑
,
.
(2)
Here
is elastic stiffness tensor. By
considering the eigenfrequencies and normal mode,
equation (1) may be written as [26],
0.
∑G ρ
exp i .
,
∑G C
exp i .
.
(5)
Where r is the position vector of components x
and y and G= (G1, G2) are the reciprocal lattice
vectors in the xoy plane. The Fourier coefficients
take the forms
ρ
and C
∬ d rρ
A
ρ
exp i .
A
∬ d rC
(6)
exp i .
.
A
∬ d r exp
i .
,
ρA
ρG
C
B
1
f .
(10)
ρB FG
(11)
and
C
C
G
A
C
B
FG .
(12)
Where f is filling fractions and mass density ρA
and the elastic constants C A are considered inside
the rods andρB and C B in the background.
The structure factor for polygon section of filled
inclusion will be as follows:
i) For polygons, in which the number of sides is
2n+1 n N
4sin
2
sin
2
∑
sin i
cos
1
1
cos
1
1
cos
1
sin i
1
cos
1
1
(13)
Where,
1
(14)
In the above sum, if i+1>n, it will take i+1=n.
ii) For polygons in which the number of sides is
2n+2 (n is even)
∑
here A is Unit cell area. If one of the denominators
of FG goes to zero, Eq. (4) and (5) will give the
average density and elastic constant
sin
2
⁄
1
1
1
1
1
(8)
2
4sin
(7)
The structure factor is defined as follows:
FG
f
(9)
Otherwise, Eq. (4) and (5) may be written as
1
,
and
C
A
C
f ,
(4)
and
C
C
G
ρB 1
(3)
Here
is eigenfrequency for wave vector k. By
using Voiget notation and cubic symmetry, we have
three elastic constants
,
and . Due to the
spatial periodicity of the phononic structure, the
material constant
and
are periodic
functions of the position. It means that ρ and C are
functions of the coordinates x and y where the z
axis defines the direction of the rod axis. Then we
can expand in the Fourier series with respect to
two-dimensional reciprocal lattice vector
ρ
ρA f
ρ
1
1
(15)
Where,
1
.
(16)
IJ
IJST (2013) 37A4: 457-462
459
w
the num
mber of sidess is
iii) For ppolygons in which
2n+2 (n is odd)
4siin
2
sin
In the abovee sum,Σ
iss equal to zeero, where
180 ⁄ and r is rradius of circu
umscribed
θ 90
circle of a polyg
gon.
2
3.. The method of calculatioon and resultss
⁄
∑
2
1
2
ssin
1
2
cos
2
1
1
ssin
2
cos
2
(117)
Calculations weere performedd with the PW
WE method
fo
or the prop
pagation of (BAW) fo
or mixed
po
olarization (lo
ongitudinal annd shear horizontal) and
sh
hear vertical modes
m
in 2D pphononic crysttals. These
arre composed of
o a 2D perioodic square arrray of Ni
ro
ods in epoxy. To create thee widest comp
plete band
gaap, the ratio of
o the rod raddius (the circu
umscribed
circle radius off a polygon) too lattice consttant varies
from 0.05 to 0.5
0 and the beest filling fraaction was
fo
ound to be about
a
0.4. Thhe lattice parrameter is
1m
mm. The rod
ds made of N
Ni are assum
med as an
in
nfinitely rigid solid. The cho
hoice of 1681 vectors of
th
he reciprocal lattice for thhe computatio
on ensures
co
onvergence of the eigenfrrequency. Thee material
paarameter values in all the m
materials invo
olved (Ni
an
nd epoxy) are specified in T
Table 1 [27].
Where
.
(188)
T
Table
1. The elaastic properties of the materials utilized in com
mponent materiial
Materiaal
Ni
epoxy
ρ kg⁄m
8905
1180
C N⁄m 10
32.4
0.758
f the phonon
nic crystals w
with
The bannd structure for
polygon section 2n+22 sides (n=2
2 and n=12) is
shown inn Fig. 1. In Figg. 1(a) and (c), the dispersiion
relation iss plotted for thhe propagatio
on acoustic waave
along X and Y with section
s
6 and 26 sides. Al so,
Fig. 1(b) and (d) are
a
shown the
t
propagatiion
acoustic wave along Z for the same phononnic
crystal. B
By comparingg Fig. 1(a) an
nd (c) it can be
seen thatt by increasinng sides num
mber of the rrod
sections, the band strructure changes slightly. T
The
bands shhift to lower-ffrequency and
d are smoothher.
Furtherm
more, the widtth of the com
mplete band ggap
increases. In Fig. 1(a), the compleete band gap is
formed bbetween the siixth and seveenth band but in
Fig. 1(c),, the completee band gap is formed betweeen
the fifth and sixth baand. So the number
n
of sidde
effects onn the locationn and width of
o the compleete
band gapp can be predicted. By com
mparing Fig. 1((b)
and (d), itt can be seen that the band structure for tthe
propagatiion acoustic wave
w
along Z has not changged
for n variiations.
C N⁄m
16.4
4
0.44
42
10
C N⁄m 10
8
0.148
IJST (2013)) 37A4: 457-462
460
The band structure for pphononic crystals with
po
olygon section 2n+2 sidess (n=1 and n=11)
n
are
sh
hown in Fig. 1.
1 In Fig. 2(a)) and (c), the dispersion
reelation are pllotted for thee propagation
n acoustic
wave
w
along X and
a Y with secction 4 and 24
4 sides. By
co
omparing these figures, oone can find
d that by
in
ncreasing the number of sid
ides, the band
d structure
ch
hanges slightly
y and the widtth of the comp
plete band
gaap is decreaseed, i.e. in thiss case, if rod
ds sections
arre square, thee widest compplete band gaap will be
fo
ormed. This confirms tthe effect of
o lattice
sy
ymmetry on th
he complete bband gap width
h. Even in
th
he previous caase when n gooes to infinite, the width
off the band gap
p is less thann the width in this case.
Also,
A
Fig. 1(b
b) and (d) sshow the prropagation
accoustic wave along Z foor the same phononic
crrystal. By com
mparing thesee figures, on
ne can see
th
hat the band structure forr propagation
n acoustic
wave
w
along Z has
h not changeed for n variaations. The
reesults of the band structture for the phononic
crrystals in whicch the section of rods have 2n+1 side
arre similar to the results off Fig. 1. Theese results
co
orroborate oth
hers’ results [3 0].
Fig. 1. Thee band structure in direction MXГ
M
for phonoonic
crystals coonsisting of Ni rod in epoxy with
w 2n+2 sidess (n
is even) seection, (a) n=22 and XY modees, (b) n=2 andd Z
mode, (c) n=12 and XY modes,
m
(d) n=12
2 and Z mode
IJJST (2013) 37A4:: 457-462
461
omplete band gap width. Buut, the band sttructure of
co
Z mode has nott changed.
References
R
Fig. 2. Thee band structure in direction MXГ
M
for phonoonic
crystals coonsisting of Ni rod in epoxy with
w 2n+2 sidess (n
is odd) seection, (a) n=3 and XY modees, (b) n=3 andd Z
mode, (c) n=11 and XY modes,
m
(d) n=11
1 and Z mode
4. Conclu
usion
In this paaper, we generralized the stru
ucture factor ffor
rods by polygon seection in tw
wo dimensionnal
phononicc crystals. Theen, the band gaap for XY andd Z
vibration modes in two dimensional phononnic
crystals w
with square lattice were calculated. T
The
propagatiion of bulk acoustic wav
ves (BAW) w
was
considereed. In additionn, the effect of
o sides’ numbber
on the baand structure and the com
mplete band ggap
width waas investigateed. The results showed thhat,
when thee section of roods have 2n+2
2(n is even) aand
2n+1 n N by increaasing sides nu
umber of the rrod
sections, the bands of
o XY mode shift to low
werfrequencyy, the bands are smoother, and
a the widthh of
the bandd gap increasees. So the nu
umber of sidde
effects onn location andd width of thee complete baand
gap can bbe predicted. However,
H
the band of Z moode
has not cchanged for n variations. Moreover,
M
whhen
the sectiion of rods have 2n+2((n is odd) by
increasingg sides numbber of rod secctions, the baand
structure of XY mode changes slightly
s
and tthe
width of the completee band gap is decreased. Thhis
confirms the effect of
o lattice sym
mmetry on tthe
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