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photonic piezoelectric medium
Wave Motion 49 (2012) 125–134
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Shear wave propagation in periodic phononic/photonic piezoelectric medium G.T. Piliposian a,∗ , A.S. Avetisyan b , K.B. Ghazaryan b a
Department of Mathematical Sciences, The University of Liverpool, M&O Building, L69 7ZL, Liverpool, United Kingdom
b
Department of Dynamics of Deformable Systems and Coupled Fields, 375024 Bagramyan ave., Institute of Mechanics, Yerevan, Armenia
article
info
Article history: Received 22 May 2011 Received in revised form 31 July 2011 Accepted 5 August 2011 Available online 16 August 2011 Keywords: Piezoelectric phononic crystal Band gaps Floquet theory
abstract Coupled electro-elastic SH waves propagating oblique to the lamination of a one dimensional piezoelectric periodic structure are considered in the framework of the full system of Maxwell’s electrodynamic equations. The dispersion equation has been obtained and numerical analyses carried out for two kinds of composites both consisting of two different piezoelectric materials. The results demonstrate the significant effect of piezoelectricity on the widths of band gaps at acoustic frequencies and confirm that it does not affect the band gaps at optical frequencies. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Photonic crystals are periodic structures consisting of materials with different elastic properties. Many smart structures such as piezoelectric or piezomagnetic composites are made up of two or more different constituents periodically arranged. Compared with the purely elastic crystals they exhibit electric or magnetic effects and thus new acoustic properties. The investigation of acoustic waves in piezoelectric phononic crystals has recently attracted much attention. Hou et al. [1] investigated the elastic band gap structure of a two-dimensional phononic crystal containing piezoelectric material and analyzed the effects of piezoelectricity on the band gaps. Other problems concerning the propagation of acoustic waves in both two and three dimensional piezoelectric periodic structures are considered in [2–14]. In particular, Qian et al. [15] studied the dispersion relations for SH-wave propagation in a periodic layered piezoelectric structure with elastic inclusions for the cases of wave propagation in the directions normal or tangential to the interfaces. In most of these studies, the quasi-static approximation is adopted for the electromagnetic field. Under this assumption, both the optical effect and the effect from the rotational part of the electric field are neglected. Although it is believed that the optical effect is not significant it might be useful in some applications to accurately predict the piezoelectricity induced electromagnetic radiation. Such applications can for example include optical detection or nondestructive evaluation [16,17]. The problem of electromagneto-acoustic surface waves in a piezoelectric non-periodic medium in a dynamic setting is considered in [18], where the exact solution for the fully coupled SH electromagneto-acoustic surface wave is obtained in a simple closed form. The propagation of electromagnetic and elastic waves in a periodic piezoelectric structure for partial interfacial boundary conditions is discussed in [19], where the interfacial effects are investigated within piezoelectric phononic crystals. Electromagnetic and elastic waves in piezoelectric–piezomagnetic superlattice are investigated in [20]. The purpose of this paper is to investigate the propagation of SH electromagneto-acoustic waves in a one dimensional piezoelectric phononic crystal in a dynamic setting for full contact interface boundary conditions. Taking into account
∗
Corresponding author. Tel.: +44 01517944010. E-mail address: [email protected] (G.T. Piliposian).
0165-2125/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2011.08.001
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both optical effects and the contribution from the rotational part of electric field the solutions obtained can be valid for any wave speed range. They will also provide accurate formulae for acousto-optic interaction in piezoelectric phononic crystals. 2. Statement of the problem We consider the prorogation of electro-magneto-elastic coupled SH wave in a one dimensional infinite periodic piezoelectric hexagonal media. The problem will be considered in the framework of the full system of Maxwell’s equations (non quasi stationary) which will give an opportunity to study the wave dispersion equation in both acoustic and optic wave frequency regions. The interconnected elastic and electro-magnetic excitations in a transversely isotropic piezoelectric crystal with crystallographic axes directed along the OZ direction are described by the following equations and constitutive relations [21]
∂ 2 ui ∂σik =ρ 2, ∂ xk ∂t ∂B ∂D rot E = − , rot H = ∂t ∂t σxx = c11 sxx + c12 syy + c13 szz − e13 Ez ,
(1) (2)
σyy = c12 sxx + c11 syy + c13 szz − e13 Ez ,
σzz = c13 sxx + c13 syy + c33 szz − e33 Ez , σxy = (c11 − c12 )sxy , σyz = 2c44 syz − s15 Ey , Dx = 2e15 sxz + ε11 Ex , Dy = 2e15 syz + ε11 Ey ,
(3) (4)
σxz = 2c44 sxz − e15 Ex , Dz = e13 (sxx + syy ) + e33 szz + ε33 Ez , ∂ uk 1 ∂ ui + , (5) Bx = µ11 Hx , By = µ11 Hy , Bz = µ33 Hz , sik = 2 ∂ xk ∂ xi where ui are the components of the displacement vector, σik and sik the stress and strain tensors, Dk and Ek the electric displacement and electric field intensity, and Hk and Bk the magnetic field intensity and magnetic induction. The mass density
ρ , the elastic, piezoelectric, dielectric and magnetic constants cik , eik , εik and µik are assumed to be periodic functions with respect to x. In the case of a two dimensional problem (when ∂/∂ z = 0) equations and relations (1)–(5) separate into plane and anti-plane problems. The plane problem is with respect to ux , uy , Ez , Hx , Hy and is described by the following equations and relations:
∂σxy ∂σxy ∂ 2 ux ∂σyy ∂ 2 uy ∂σxx + =ρ 2 , + =ρ 2 , ∂x ∂y ∂t ∂x ∂y ∂t 2 2 2 ∂ 1 ∂ Ez 1 ∂ Ez ∂ Ez ∂ ∂ ux ∂ uy + − ε33 2 = e13 2 + , ∂ x µ11 ∂ x µ11 ∂ y2 ∂t ∂t ∂x ∂y ∂ Hx ∂ Ez ∂ Hy ∂ Ez + = 0, µ11 − = 0, ∂t ∂y ∂t ∂x ∂ ux ∂ uy ∂ uy ∂ ux σxx = c11 + c12 − e13 Ez , σyy = c11 + c12 − e13 Ez , ∂x ∂y ∂y ∂x c11 − c12 ∂ ux ∂ uy σxy = + . 2 ∂y ∂x µ11
(6)
(7) (8)
(9)
The following equations and relations describe the anti-plane problem with respect to uz , Ex , Ey , Hz
∂σxz ∂σxz ∂ 2 uz + =ρ 2 , ∂x ∂y ∂t ∂ Ey ∂ Ex ∂ Hz − = −µ33 , ∂x ∂y ∂t ∂ Hz ∂ Dx ∂ Hz ∂ Dy = , − = , ∂y ∂t ∂x ∂t ∂ uz ∂ uz Dx = e15 + ε11 Ex , Dy = e15 + ε11 Ey . ∂x ∂y ∂ uz ∂ uz σxz = c44 − e15 Ex , σyz = c44 − e15 Ey . ∂x ∂y
(10) (11) (12) (13) (14)
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Note that in the plane problem (6)–(8) in the dynamic setting the piezoelectric effect leads to the coupling of pure acoustic and electromagnetic waves [10] but provides negligible effect on the acoustic and electromagnetic wave velocities. When the electro-magnetic field is considered in a quasi-static approximation the piezoelectric effect is simply not present in the equations of acoustic wave propagation [22]. We consider the anti-plane problem (10)–(14) and assume that waves propagate in the (x, y) plane. Then taking Hz = ωH0 (x)ei(py−ωt ) ,
uz = u0 (x)ei(py−ωt ) ,
Ex = Ex0 (x)ei(py−ωt ) ,
Ey = Ey0 (x)ei(py−ωt ) ,
(15)
where p is a component of the wave vector parallel to the y axis (oblique incidence wave number), ω is a wave frequency, and substituting into (10)–(14) the following coupled system of equations for H0 (x) and u0 (x) follows: d
dx
− G0 (x)p u0 (x) + ρ(x)ω u0 (x) − pH0 (x)
d
e(x)
= 0, ε(x) d 1 dH0 p 2 H 0 ( x) d e(x) − + µ(x)ω2 H0 (x) − pu0 (x) = 0. dx ε(x) dx ε(x) dx ε(x) dx
G0 (x)
du0
2
2
(16)
dx
(17)
Here G0 (x) = c44 (x) +
e2 (x)
ε(x)
,
e(x) = e15 (x),
ε(x) = ε11 (x),
µ(x) = µ33 (x)
(18)
are periodic functions with period β and G0 (x) is called piezoelectrically stiffened elastic modulus. From (15) and (12)–(14) the electrical field and stresses can also be expressed via the magnetic field and the displacement as follows: Ex0 (x) = −
σxz (x) =
pH0 (x) + e(x)u′0 (x)
ε(x)
e(x)pH0 (x)
ε(x)
,
+ G0 (x)u′0 (x),
Ey0 (x) =
i(e(x)pu0 (x) + H0′ (x))
ε(x)
,
σyz (x) = −i pG0 (x)u0 (x) +
(19) e(x)H0′ (x)
ε(x)
.
(20)
According to the Floquet theory [23] Eqs. (16)–(17) defined on an infinite interval can be considered on the finite interval
−β/2 ≤ x ≤ β/2 with the special boundary conditions of quasi-periodicity σyz (−β/2) = λσyz (β/2),
u0 (−β/2) = λu0 (β/2),
(21)
H0 (−β/2) = λH0 (β/2),
Ey0 (−β/2) = λEy0 (β/2),
(22)
ikβ
where λ = e and k is a component of the wave vector, called the Bloch–Floquet wave number, perpendicular to the interfaces. It follows from the periodicity of the parameters and (19) that Ey0 (−β/2) =
i(e(−β/2)pu0 (−β/2) + H0′ (−β/2))
ε(−β/2) i(e(β/2)pu0 (β/2) + H0′ (β/2)) λEy0 (β/2) = λ , ε(β/2)
=
i(e(β/2)pu0 (−β/2) + H0′ (−β/2))
ε(β/2)
,
and from the second conditions of (21) and (22) we can write H0′ (−β/2) = λH0′ (β/2). Since it follows also in a similar way that u′0 (−β/2) = λu′0 (β/2), conditions (21) and (22) can be replaced by the following u0 (−β/2) = λu0 (β/2),
u′0 (−β/2) = λu′0 (β/2),
(23)
H0 (−β/2) = λH0 (β/2),
H0 (−β/2) = λH0 (β/2).
(24)
′
′
Eqs. (16) and (17) are uncoupled if e(x)/ε(x) = const or p = 0 i.e. the wave propagates perpendicular to the interfaces of the structure (non oblique wave). Since the boundary conditions (23) and (24) for the elastic displacement and the magnetic field are also uncoupled the propagation of elastic and electromagnetic waves in these cases decouple and the interaction caused by the piezoelectric effect is present only in the stiffened elastic modulus G0 (x) and brings solely a quantitative effect. The interaction also does not take place when the piezoelectric effect vanishes e(x) = 0. In the general case for continuous periodic material parameters the main Eqs. (16) and (17) are coupled and the boundary conditions (23) and (24) are uncoupled. If the unit cell is made of different elastic materials and the material parameters are piecewise constant functions then the Eqs. (16) and (17) become decoupled but the interface conditions are coupled. We consider an infinite one-dimensional periodically composed piezoelectric structure with the unit cell of a period β consisting of a piezoelectric inclusions of thickness b embedded and perfectly bonded with the elastic matrix made of another piezoelectric material of a thickness a as shown in Fig. 1.
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Fig. 1. Symmetric unit cell made of two piezoelectric materials.
The solutions for the magnetic field and the elastic displacement can be written as follows (1)
= C sin(q1 x) + B cos(q1 x),
(2)
= C2 sin(q2 x) + B2 cos(q2 x),
u0 = A2 sin(r2 x) + F2 cos(r2 x),
(3)
= C3 sin(q2 x) + B3 cos(q2 x),
u0 = A3 sin(r2 x) + F3 cos(r2 x)
H0 H0 H0
(1)
u0 = A sin(r1 x) + F cos(r1 x), (2)
(3)
(25)
where qj =
ω2 εj µj − p2 ,
rj =
ρj ω2 /G0j − p2 ,
j = 1, 2,
(26)
and superscripts 1, 2 and 3 show that the functions belong to the mediums 1, 2 and 3 respectively. The following continuity conditions at the interfaces x = ±b/2 should be satisfied
[ u0 ] = 0 ,
[σxz ] = 0,
[Ey0 ] = 0,
[H0 ] = 0,
(27)
where [·] is a jump of a function across the interfaces. Using (27) the coefficients C2 , B2 , A2 and F2 and C3 , B3 , A3 and F3 can be expressed via four coefficients C , B, A and F where C2 = C˜ (b),
B2 = B˜ (b),
C3 = C˜ (−b),
B3 = B˜ (−b),
A2 = A˜ (b),
F2 = F˜ (b),
A3 = A˜ (−b),
F3 = F˜ (−b),
and the following notations are used C˜ (b)q2 ϵ1 = C (q1 ϵ2 ξ1 ξ2 + q2 ϵ1 η1 η2 ) + B(q1 ϵ2 η1 ξ2 − q2 ϵ1 ξ1 η2 ) + pγ (F ξ2 ξ3 − Aξ2 η3 ), B˜ (b)q2 ϵ1 = C (q1 ϵ2 ξ1 η2 − q2 ϵ1 η1 ξ2 ) + B(q2 ϵ1 ξ1 ξ2 + q1 ϵ2 η1 η2 ) + γ p2 (F γ η2 ξ3 − Aη2 η3 ), A˜ (b)G20 r2 =
pγ
(Bξ1 ξ4 − C η1 ξ4 ) + A(G10 r1 ξ3 ξ4 + G20 r2 η3 η4 ) + F (G10 r1 η3 ξ4 − G20 r2 ξ3 η4 ), ϵ1 ϵ2 pγ F˜ (b)G20 r2 = (Bξ1 k4 − C η1 η4 ) + A(G10 r1 ξ3 η4 − G20 r2 η3 ξ4 ) + F (G20 r2 ξ3 ξ4 + G10 r1 η3 η4 ), ϵ1 ϵ2 γ = (e1 ϵ2 − e2 ϵ1 ), ξ1 = cos(bq1 /2), ξ2 = cos(bq2 /2), ξ3 = cos(br1 /2), ξ4 = cos(br2 /2), η1 = sin(bq1 /2), η2 = sin(bq2 /2), η3 = sin(br1 /2), η4 = sin(br2 /2).
(28)
Substituting these coefficients and solutions (25) into the four Bloch–Floquet quasi-periodicity conditions (23) and (24) the system of equations for the remaining unknown coefficients will produce the following dispersion equation: F (λ, p, ω) = λ4 + f (p, ω)λ3 + g (p, ω)λ2 + f (p, ω) + 1 = 0,
(29)
where f (p, ω) = −2 cos(bq1 ) cos(aq2 ) − 2 cos(br1 ) cos(ar2 ) + Q sin(bq1 ) sin(aq2 ) p2 R22 sin(bq1 ) sin(ar2 ) p2 R21 sin(aq2 ) sin(br1 ) + + G sin(br1 ) sin(ar2 ), + G10 q2 r1 G20 q1 r2
(30)
g (p, ω) = 2 + 2 cos(br1 ) cos(ar2 )(2 cos(bq1 ) cos(aq2 ) − Q sin(bq1 ) sin(aq2 ))
+ 2p2 [K1 sin(bq1 ) sin(br1 ) − K1 sin(bq1 ) sin(br1 ) cos(aq2 ) cos(ar2 ) + K2 sin(aq2 ) sin(ar2 ) − K3 cos(bq1 ) cos(ar2 ) sin(aq2 ) sin(br1 ) − K4 cos(aq2 ) cos(br1 ) sin(bq1 ) sin(ar2 ) − K2 cos(bq1 ) cos(br1 ) sin(aq2 ) sin(ar2 )] − 2G cos(bq1 ) cos(aq2 ) sin(br1 ) sin(ar2 ) + QG sin(bq1 ) sin(aq2 ) sin(br1 ) sin(ar2 ) + p4 K1 K2 sin(bq1 ) sin(aq2 ) sin(br1 ) sin(ar2 ),
(31)
G.T. Piliposian et al. / Wave Motion 49 (2012) 125–134
129
and G=
G10 r1 G20 r2
K1 =
+
R22 G10 q1 r1
G20 r2 G10 r1
,
,
Q =
K2 =
q22 ε12 + q21 ε22 q1 q2 ε1 ε2
R21 G20 q2 r2
,
,
K3 =
R1 = R21 G10 q2 r1
,
e2 ε1 − e1 ε2
√ ε1 ε2 K4 =
, R22
G20 q1 r2
R2 =
,
e2 ε1 − e1 ε2
√ ε2 ε1
a = β − b.
,
(32)
(33)
Presenting (29) in the form
λ+
1
2
λ
+ λ+
1
λ
f + g − 2 = 0,
(34)
and taking into account that λ = eikβ and λ + λ−1 = 2 cos(β k) the trigonometric solution of the dispersion equations is cos(β k) =
1
(−f ±
f 2 + 4g − 8). (35) 4 An equation of a similar type has been obtained and discussed in [23] for a vibrating periodic beam. For given values of p and ω the dispersion equation (35) gives two values of k. When none of the values is a real number there exists a region with no dispersion curves, i.e. forbidden band gaps. For an unperturbed case (non-periodic) the dispersion equation (29) converts to the following: cos(β k) = cos(β r ),
cos(β k) = cos(β q),
(36)
giving the explicit formulae
ω12 =
(k2 + p2 ) , εµ
ω22 =
G0
ρ
(k2 + p2 )
(37)
for the wave frequencies. In the general case the dispersion equation gives coupled elastic and electro-magnetic waves with two dispersion curves. When there is no piezoelectric effect e1 = e2 = 0 Eqs. (35) give two uncoupled equations, one describing the propagation of pure electromagnetic waves cos(β k) = cos(bq1 ) cos(aq2 ) −
q22 ε12 + q21 ε22 2q1 q2 ε1 ε2
sin(bq1 ) sin(aq2 )
(38)
and the other propagation of pure acoustic wave cos(β k) = cos(br1 ) cos(ar2 ) −
(G21 r12 + G22 r22 ) 2G1 G2 r1 r2
sin(br1 ) sin(ar2 ),
(39)
where G1,2 are pure elastic shear modules. In the general case the dispersion equation (35) gives two coupled electro-magneto-elastic waves. The wave that reduces to a pure acoustic wave when the piezoelectric effect approaches to zero corresponds to a ‘‘quasi-acoustic wave’’ and the wave that becomes a pure electromagnetic wave when the piezoelectric effect vanishes is a ‘‘quasi-electromagnetic field wave’’. If for a periodic structure e1 /e2 = ε1 /ε2 or p = 0, as noted above, Eq. (35) again gives two uncoupled equations, one is (38) and the other is (39) where the piezoelectric effect is present in the electro-mechanical coupling coefficients in stiffened elastic moduli G10 and G20 . Note that if for the two to the interfaces (p = 0) the acoustic constituent materials for waves traveling perpendicular wave impedances Zja = Gj0 ρj0 or electromagnetic wave impedances Zje = µj /εj (j = 1, 2) are the same it follows from (38) and (39) that cos(β k) = cos(ar1 + br2 ),
cos(β k) = cos(aq1 + bq2 ),
which means that there are no frequency regions with forbidden band gaps. In a quasi-static approximation the right hand side of (38) is independent of ω, the equation becomes non hyperbolic and cannot describe an electromagnetic wave motion. If one of the constituent materials in the phononic crystal is pure elastic (e2 = 0) then in a quasi-static approximation (q1 = ip, q2 = ip) Eq. (39) coincides with the dispersion equation obtained in [15]. When the periodic structure is made of two identical piezoelectric materials of the same width a = b one of which has the opposite polarization direction i.e. e1 = e, e2 = −e then the functions f and g in the dispersion equation (29) take the following form f = −4 cos(b(q + r )) cos(b(q − r )) + 8ξ ,
(40)
g = 1 + 2(1 + 2 cos(2bq) cos(2br ) − cos(bq) cos(br )) + ξ ,
(41)
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G.T. Piliposian et al. / Wave Motion 49 (2012) 125–134
Fig. 2. Band structure of piezoelectric phononic crystal PZT-4 and BaTiO3 h = 0.5, α = 1, c1 = structure with and without piezoelectric effect.
√
ρ1 /G01 . Solid lines and dashed lines show the band
where
ξ=
ηp2 sin(bq) sin(br ) qr
,
η=
e2
ε(c44 + e2 /ε)
.
Further analysis shows that the identical piezoelectric phononic crystal with opposite polarization allows propagation of the Bloch–Floquet waves at acoustic frequencies. At optical frequencies the opposite polarization does not cause an opening of band gaps since the effect of piezoelectricity approaches to zero, and the structure behaves as a photonic crystal with identical optical refractive indexes. 3. Numerical results For given values of p and ω Eq. (35) gives two values of the wave number k. The regions of ω where both values of k are complex correspond to band gaps. The appearance of band gaps depends on the dimensionless oblique incidence wave number α = pβ , the filling fraction h and the difference in the elastic and electromagnetic properties of two piezoelectric materials in the phononic(photonic) crystal which can be expressed in the differences in the impedances. Numerical calculation have been carried out both at low and high frequency regions for two piezoelectric phononic crystals. Material parameters of PZT-4 and BaTiO3 have been used for one photonic crystal and LiTO3 and PZT-4 for the second one (Table 1). Since the relative magnetic permeability for piezoelectric materials is very close to one, the values for µ1 and µ2 are taken equal to the vacuum permeability 4π 10−7 V s/(A m). Figs. 2 and 3 illustrate the effect of piezoelectricity √ in the low frequency region for h = 0.5 and the wave number α = pβ = 1. For the normalized frequency ωβ/c1 , c1 = ρ1 /G01 is the velocity of a transverse wave in the material 1. Fig. 1 shows that the piezoelectricity can affect the band structure and the band widths significantly. The first and second band gaps occur at frequencies where there are no band gaps without the piezoelectric effect. The width of the band gaps for piezoelectric phononic crystals widens for higher frequencies whereas the gaps are practically absent when the piezoelectric effect is neglected. It is affected by the fact that the ratio Z1a /Z2a of the acoustic impedances of the two piezoelectric materials (where Z1a is the impedance of LiTO3 and Z2a is the impedance of PZT-4) is 12.6 whereas the ratio of impedances drops to the value of 1.1 when the piezoelectric effect is not taken into account. The effect of piezoelectricity is less for the phononic crystal made of LiTO3 and PZT-4. Fig. 2 shows that although the piezoelectricity widens the band gaps, the widths of band gaps for acoustic frequencies in this case are closer for phononic crystals with and without piezoelectric effects. In this case the ratio of acoustic impedances for piezoelectric phononic crystal materials is 0.502 against the ratio of the acoustic impedances of the same materials without the piezoelectric effect 0.558. Fig. 4 shows the effect of opposite polarization on the band gap for a phononic crystal PZT-4 and BaTiO3 . Compared with Fig. 2 when the two piezoelectric materials have the same polarization it can be seen that changing the polarization direction of one of the materials can have a significant effect on the band gap widths. In this case the second and fourth band gaps are significantly larger. When the phononic crystal is made of the same piezoelectric material simply changing the polarization direction of one of them leads to a periodic structure and creates band gaps [19]. This effect is shown in Fig. 5 where the first large band gap and the second smaller one are the effect of the opposite polarization of the same piezoelectric material.
G.T. Piliposian et al. / Wave Motion 49 (2012) 125–134
131
Fig. 3. Band structure of piezoelectric phononic crystal LiTO3 and PZT-4 h = 0.5, α = 0.5. Solid lines and dashed lines show the band structure with and without piezoelectric effect.
Table 1 Material constants of PZT-4, LiTO3 and BaTiO3 . Material
Elastic constant c44 1010 N/m2
Piezoelectric constant e15 C/m2
Permittivity ε11 10−11 F/m
Density ρ 103 kg/m3
PZT-4 LiTO3 BaTiO3
2.56 1.78 4.3
12.7 0.89 11.6
646 6.434 1.264
7.6 3.402 5.7
Fig. 4. Band structure of piezoelectric phononic crystal, PZT-4 and BaTiO3 h = 0.5, α = 1, for opposite polarization direction. Solid lines and the dashed lines show the band structure with and without piezoelectric effect.
Numerical results with quasi-static assumptions practically coincide almost precisely with the solid lines in Figs. 2–4. This shows that in periodic structures at acoustic frequencies the quasi-static setting gives an excellent approximation. The influence of the oblique incidence wave number pβ on the normalized band gap width is shown in Fig. 6, where 1ω is the gap width of the phononic crystal and ωg is the mid gap frequency. It can be seen that both first and second band gaps have maximum values when the wave propagates perpendicular to the laminations and both reach a minimum
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G.T. Piliposian et al. / Wave Motion 49 (2012) 125–134
Fig. 5. Band structure of an identical piezoelectric phononic crystal PZT-4, h = 0.5, α = 1, for opposite polarization direction. Solid lines and dashed lines show the band structure with and without piezoelectric effect.
Fig. 6. Normalized gap width as a function of an oblique incidence wave number pβ for a phononic structure PZT-4 and LiTO3 . Solid lines and dashed lines show the first and second band gaps.
point at some value of the wave number pβ . Fig. 7 shows the dependence of the normalized frequency on the thickness of the layers of the phononic crystal. Both the first and second band gaps have peaks at some value of the filling fraction h = b/a. Numerical calculations have been carried out also at optical frequencies. The elastic and piezoelectric properties in this frequency region do not affect the band structure. Two dispersion curves given by the dispersion equation (35) drawn in one plot coincide with the dispersion curve (38) describing the propagation of an electromagnetic wave in a photonic crystal. Here apart from the filling fraction and the oblique incidence wave number pβ the band structure depends on the ratios of the electromagnetic wave impedances Zje = µj /εj of the constituent materials. Band gaps for piezoelectric photonic crystal LiTO3 and PZT-4 for h = 0.5 and three different values of the oblique incidence wave number pβ are shown in Fig. 8. Here for the normalized frequency ωβ/c , c is the speed of light. This picture is the same for the piezoelectric crystal with and without the piezoelectric effect, which means that the effect of piezoelectricity does not play any role on band structure of an electromagnetic wave and confirms the experimental results obtained for lattice vibrations of a-quartz in [24] that the effect of piezoelectricity approaches zero at optical frequencies.
G.T. Piliposian et al. / Wave Motion 49 (2012) 125–134
133
Fig. 7. Normalized gap width as a function of filling fraction h for BaTiO3 and PZT-4 crystal. Solid lines and dashed lines show the first and second band gaps.
Fig. 8. Band structure of piezoelectric photonic crystal LiTO3 and PZT-4 for h = 0.5 for different values of oblique incidence wave number pβ . Solid lines, dashed lines and large dashing are for α = 0, 0.5, 1, respectively.
4. Conclusion The prorogation of electro-magneto-elastic coupled SH waves in one dimensional infinite periodic piezoelectric media is considered within a full system of the Maxwell’s equations. Such setting of the problem allows investigating of the Bloch–Floquet waves in a wide range of frequencies from low frequency acoustic waves to high frequency electromagnetic waves. Governing equations describing the propagation of SH waves in the (x, y) plane perpendicular to the polling direction has been derived. It is shown that when e(x)/ε(x) = const or p = 0 i.e. the wave propagates perpendicular to the periodicity of the structure the boundary value problem decouples into two boundary value problems for electromagnetic and elastic waves. In this case the effect of piezoelectricity is present only in the electromechanical coupling coefficient in stiffened elasticity modulus. The dispersion equation has been obtained between the frequency and wave numbers for Bloch–Floquet waves, and the band structure has been defined both at acoustic and optical frequencies.
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G.T. Piliposian et al. / Wave Motion 49 (2012) 125–134
Numerical results show that the piezoelectricity does not strongly affect the frequency where the first band gaps appear but can have a significant effect on the width of the first band gap. It can open band gaps in some cases where they simply do not exist without a piezoelectric effect. This is particularly well illustrated in the case of the identical piezoelectric crystals with opposite polarization when the opening of a band gap is caused only by the piezoelectric effect. The results also show that waves propagating parallel to the periodicity direction can have the largest band gaps. The widths of the first and second band gaps reach a minimum point for a certain value of the oblique wave number. It has been shown that the band structure for optical frequencies practically is not dependent on the piezoelectric effect and is conditioned only by periodicity of optical electromagnetic wave impedances of materials. This is in a good accordance with experimental results for lattice vibrations of a-quartz in the optical frequency range [24]. Numerical analysis shows that changing the polarization direction of one of the materials can have a significant effect on the band gap widths. It also has been shown that an identical piezoelectric crystal with opposite polarization allows propagation of the Bloch–Floquet waves at acoustic frequencies. At optical frequencies the opposite polarization does not cause an opening of band gaps. Acknowledgments This research was supported by The Royal Society International Joint Research grant JP091352 and State Committee of Science of Armenia Grant 11-2c436. References [1] Hou Z.L., Wu F.G., Liu Y.Y., Phononic crystals containing piezoelectric material, Solid State Commun. 130 (2004) 745–749. [2] A.K. Vashishth, V. Gupta, Wave propagation in transversely isotropic porous piezoelectric materials, Internat. J. Solids Struct. 46 (2009) 3620–3632. [3] Y.Z. Wang, F.M. Li, W.H. Huang, Y.S. Wang, Effects of inclusion shapes on the band gaps in two-dimensional piezoelectric phononic crystals, J. Phys.: Condens. Matter 19 (2007) 496204. [4] Y.Z. Wang, F.M. Li, K. Kishimoto, Y.S. Wang, W.H. Huang, Wave localization in randomly disordered periodic piezoelectric rods with initial stress, Acta Mech. Solida Sin. 21 (2008) 529–535. [5] Y.Z. Wang, F.M. Li, K. Kishimoto, Y.S. Wang, W.H. Huang, Wave band gaps in three-dimensional periodic piezoelectric structures, Mech. Res. Comm. 36 (2009) 461–468. [6] M. Wilm, S. Ballandras, V. Laude, T. Pastureaud, A full 3D plane-wave expansion model for 1–3 piezoelectric composite structures, J. Acoust. Soc. Am. 112 (2002) 943–952. [7] X.Y. Zou, Q. Chen, B. Liang, J.C. Cheng, Control of the elastic wave band gaps in two-dimensional piezoelectric periodic structures, Smart Mater. Struct. 17 (2008) 015008. [8] T.T. Wu, Z.C. Hsu, Z.G. Huang, Band gaps and the electromechanical coupling coefficient of a surface acoustic wave in a two-dimensional piezoelectric phononic crystal, Phys. Rev. B 71 (2005) 064303. [9] Z.Z. Yan, Y.S. Wang, Calculation of band structures for surface waves in two dimensional phononic crystals with a wavelet-based method, Phys. Rev. B 78 (2008) 094306. [10] F.M. Li, Y.Z. Wang, B. Fang, Y.S. Wang, Propagation and localization of two-dimensional in-plane elastic waves in randomly disordered layered piezoelectric phononic crystals, Internat. J. Solids Struct. 44 (2007) 7444–7456. [11] Y.Z. Wang, F.M. Li, W.H. Huang, Y.S. Wang, The propagation and localization of Rayleigh waves in disordered piezoelectric phononic crystals, J. Mech. Phys. Solids 56 (2008) 1578–1590. [12] Y.Z. Wang, F.M. Li, W.H. Huang, X.A. Jiang, Y.S. Wang, K. Kishimoto, Wave band gaps in two-dimensional piezoelectric/piezomagnetic phononic crystals, Internat. J. Solids Struct. 45 (2008) 4203–4210. [13] S. Gonella, A.C. To, W.K. Liu, Interplay between phononic bandgaps and piezoelectric microstructures for energy harvesting, J. Mech. Phys. Solids 57 (2009) 621–633. [14] Y.Z. Wang, F.M. Li, K. Kishimoto, Y.S. Wang, W.H. Huang, Elastic wave band gaps in magneto electroelastic phononic crystals, Wave Motion 46 (2009) 47–56. [15] Z.H. Qian, F. Jin, Z.K. Wang, K. Kishimoto, Dispersion relations for SH-wave propagation in periodic piezoelectric composite layered structures, Internat. J. Engrg. Sci. 42 (2004) 673–689. [16] V. Giurgiutiu, Structural Health Monitoring with Piezoelectric Wafer Active Sensors, Academic Press, Burlington, MA, USA, 2007. [17] L. Yu, G. Santoni-Bottai, B. Xu, W. Liu, V. Giurgiutiu, Piezoelectric wafer active sensors for in situ ultrasonic-guided wave SHM, Fatigue Fract. Eng. Mater. Struct. 31 (2008) 611–628. [18] S. Li, The problem of electromagneto-acoustic surface wave in a piezoelectric medium: the Bleustain-Gulyaev mode, J. Appl. Phys. 80 (9) (1996) 5264–5269. [19] F.J. Sabina, A.B. Movchan, Interfacial effects in electromagnetic coupling within piezoelectric phononic crystals, Acta Mech. Sin. 25 (2009) 95–99. [20] J. Zhao, R.C. Yin, T. Fan, M.H. Lu, Y.F. Chen, Y.Y. Zhu, S.N. Zhu, N.B. Ming, Coupled phonon polaritons in a piezoelectric–piezomagnetic superlattice, Phys. Rev. B 77 (2008) 075126. [21] B.A. Auld, Acoustic Fields and Waves in Solids, vol. 1, Wiley, 1973, 431 p. [22] A.S. Avetisyan, On a problem of shear wave propagation in piezoelectric medium, Proc. Armen. Acad. Sci., Mech. 38 (1) (1985) 12–19. [23] V.G. Papanicolaou, The periodic Euler–Bernoulli equation, Trans. Amer. Math. Soc. 355 (9) (2003) 3727–37599. [24] Z.Z. Tan, Piezoelectric lattice vibrations at optical frequencies, Solid State Commun. 131 (2004) 405–408.
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Shear wave propagation in periodic phononic/photonic piezoelectric medium G.T. Piliposian a,∗ , A.S. Avetisyan b , K.B. Ghazaryan b a
Department of Mathematical Sciences, The University of Liverpool, M&O Building, L69 7ZL, Liverpool, United Kingdom
b
Department of Dynamics of Deformable Systems and Coupled Fields, 375024 Bagramyan ave., Institute of Mechanics, Yerevan, Armenia
article
info
Article history: Received 22 May 2011 Received in revised form 31 July 2011 Accepted 5 August 2011 Available online 16 August 2011 Keywords: Piezoelectric phononic crystal Band gaps Floquet theory
abstract Coupled electro-elastic SH waves propagating oblique to the lamination of a one dimensional piezoelectric periodic structure are considered in the framework of the full system of Maxwell’s electrodynamic equations. The dispersion equation has been obtained and numerical analyses carried out for two kinds of composites both consisting of two different piezoelectric materials. The results demonstrate the significant effect of piezoelectricity on the widths of band gaps at acoustic frequencies and confirm that it does not affect the band gaps at optical frequencies. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Photonic crystals are periodic structures consisting of materials with different elastic properties. Many smart structures such as piezoelectric or piezomagnetic composites are made up of two or more different constituents periodically arranged. Compared with the purely elastic crystals they exhibit electric or magnetic effects and thus new acoustic properties. The investigation of acoustic waves in piezoelectric phononic crystals has recently attracted much attention. Hou et al. [1] investigated the elastic band gap structure of a two-dimensional phononic crystal containing piezoelectric material and analyzed the effects of piezoelectricity on the band gaps. Other problems concerning the propagation of acoustic waves in both two and three dimensional piezoelectric periodic structures are considered in [2–14]. In particular, Qian et al. [15] studied the dispersion relations for SH-wave propagation in a periodic layered piezoelectric structure with elastic inclusions for the cases of wave propagation in the directions normal or tangential to the interfaces. In most of these studies, the quasi-static approximation is adopted for the electromagnetic field. Under this assumption, both the optical effect and the effect from the rotational part of the electric field are neglected. Although it is believed that the optical effect is not significant it might be useful in some applications to accurately predict the piezoelectricity induced electromagnetic radiation. Such applications can for example include optical detection or nondestructive evaluation [16,17]. The problem of electromagneto-acoustic surface waves in a piezoelectric non-periodic medium in a dynamic setting is considered in [18], where the exact solution for the fully coupled SH electromagneto-acoustic surface wave is obtained in a simple closed form. The propagation of electromagnetic and elastic waves in a periodic piezoelectric structure for partial interfacial boundary conditions is discussed in [19], where the interfacial effects are investigated within piezoelectric phononic crystals. Electromagnetic and elastic waves in piezoelectric–piezomagnetic superlattice are investigated in [20]. The purpose of this paper is to investigate the propagation of SH electromagneto-acoustic waves in a one dimensional piezoelectric phononic crystal in a dynamic setting for full contact interface boundary conditions. Taking into account
∗
Corresponding author. Tel.: +44 01517944010. E-mail address: [email protected] (G.T. Piliposian).
0165-2125/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2011.08.001
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both optical effects and the contribution from the rotational part of electric field the solutions obtained can be valid for any wave speed range. They will also provide accurate formulae for acousto-optic interaction in piezoelectric phononic crystals. 2. Statement of the problem We consider the prorogation of electro-magneto-elastic coupled SH wave in a one dimensional infinite periodic piezoelectric hexagonal media. The problem will be considered in the framework of the full system of Maxwell’s equations (non quasi stationary) which will give an opportunity to study the wave dispersion equation in both acoustic and optic wave frequency regions. The interconnected elastic and electro-magnetic excitations in a transversely isotropic piezoelectric crystal with crystallographic axes directed along the OZ direction are described by the following equations and constitutive relations [21]
∂ 2 ui ∂σik =ρ 2, ∂ xk ∂t ∂B ∂D rot E = − , rot H = ∂t ∂t σxx = c11 sxx + c12 syy + c13 szz − e13 Ez ,
(1) (2)
σyy = c12 sxx + c11 syy + c13 szz − e13 Ez ,
σzz = c13 sxx + c13 syy + c33 szz − e33 Ez , σxy = (c11 − c12 )sxy , σyz = 2c44 syz − s15 Ey , Dx = 2e15 sxz + ε11 Ex , Dy = 2e15 syz + ε11 Ey ,
(3) (4)
σxz = 2c44 sxz − e15 Ex , Dz = e13 (sxx + syy ) + e33 szz + ε33 Ez , ∂ uk 1 ∂ ui + , (5) Bx = µ11 Hx , By = µ11 Hy , Bz = µ33 Hz , sik = 2 ∂ xk ∂ xi where ui are the components of the displacement vector, σik and sik the stress and strain tensors, Dk and Ek the electric displacement and electric field intensity, and Hk and Bk the magnetic field intensity and magnetic induction. The mass density
ρ , the elastic, piezoelectric, dielectric and magnetic constants cik , eik , εik and µik are assumed to be periodic functions with respect to x. In the case of a two dimensional problem (when ∂/∂ z = 0) equations and relations (1)–(5) separate into plane and anti-plane problems. The plane problem is with respect to ux , uy , Ez , Hx , Hy and is described by the following equations and relations:
∂σxy ∂σxy ∂ 2 ux ∂σyy ∂ 2 uy ∂σxx + =ρ 2 , + =ρ 2 , ∂x ∂y ∂t ∂x ∂y ∂t 2 2 2 ∂ 1 ∂ Ez 1 ∂ Ez ∂ Ez ∂ ∂ ux ∂ uy + − ε33 2 = e13 2 + , ∂ x µ11 ∂ x µ11 ∂ y2 ∂t ∂t ∂x ∂y ∂ Hx ∂ Ez ∂ Hy ∂ Ez + = 0, µ11 − = 0, ∂t ∂y ∂t ∂x ∂ ux ∂ uy ∂ uy ∂ ux σxx = c11 + c12 − e13 Ez , σyy = c11 + c12 − e13 Ez , ∂x ∂y ∂y ∂x c11 − c12 ∂ ux ∂ uy σxy = + . 2 ∂y ∂x µ11
(6)
(7) (8)
(9)
The following equations and relations describe the anti-plane problem with respect to uz , Ex , Ey , Hz
∂σxz ∂σxz ∂ 2 uz + =ρ 2 , ∂x ∂y ∂t ∂ Ey ∂ Ex ∂ Hz − = −µ33 , ∂x ∂y ∂t ∂ Hz ∂ Dx ∂ Hz ∂ Dy = , − = , ∂y ∂t ∂x ∂t ∂ uz ∂ uz Dx = e15 + ε11 Ex , Dy = e15 + ε11 Ey . ∂x ∂y ∂ uz ∂ uz σxz = c44 − e15 Ex , σyz = c44 − e15 Ey . ∂x ∂y
(10) (11) (12) (13) (14)
G.T. Piliposian et al. / Wave Motion 49 (2012) 125–134
127
Note that in the plane problem (6)–(8) in the dynamic setting the piezoelectric effect leads to the coupling of pure acoustic and electromagnetic waves [10] but provides negligible effect on the acoustic and electromagnetic wave velocities. When the electro-magnetic field is considered in a quasi-static approximation the piezoelectric effect is simply not present in the equations of acoustic wave propagation [22]. We consider the anti-plane problem (10)–(14) and assume that waves propagate in the (x, y) plane. Then taking Hz = ωH0 (x)ei(py−ωt ) ,
uz = u0 (x)ei(py−ωt ) ,
Ex = Ex0 (x)ei(py−ωt ) ,
Ey = Ey0 (x)ei(py−ωt ) ,
(15)
where p is a component of the wave vector parallel to the y axis (oblique incidence wave number), ω is a wave frequency, and substituting into (10)–(14) the following coupled system of equations for H0 (x) and u0 (x) follows: d
dx
− G0 (x)p u0 (x) + ρ(x)ω u0 (x) − pH0 (x)
d
e(x)
= 0, ε(x) d 1 dH0 p 2 H 0 ( x) d e(x) − + µ(x)ω2 H0 (x) − pu0 (x) = 0. dx ε(x) dx ε(x) dx ε(x) dx
G0 (x)
du0
2
2
(16)
dx
(17)
Here G0 (x) = c44 (x) +
e2 (x)
ε(x)
,
e(x) = e15 (x),
ε(x) = ε11 (x),
µ(x) = µ33 (x)
(18)
are periodic functions with period β and G0 (x) is called piezoelectrically stiffened elastic modulus. From (15) and (12)–(14) the electrical field and stresses can also be expressed via the magnetic field and the displacement as follows: Ex0 (x) = −
σxz (x) =
pH0 (x) + e(x)u′0 (x)
ε(x)
e(x)pH0 (x)
ε(x)
,
+ G0 (x)u′0 (x),
Ey0 (x) =
i(e(x)pu0 (x) + H0′ (x))
ε(x)
,
σyz (x) = −i pG0 (x)u0 (x) +
(19) e(x)H0′ (x)
ε(x)
.
(20)
According to the Floquet theory [23] Eqs. (16)–(17) defined on an infinite interval can be considered on the finite interval
−β/2 ≤ x ≤ β/2 with the special boundary conditions of quasi-periodicity σyz (−β/2) = λσyz (β/2),
u0 (−β/2) = λu0 (β/2),
(21)
H0 (−β/2) = λH0 (β/2),
Ey0 (−β/2) = λEy0 (β/2),
(22)
ikβ
where λ = e and k is a component of the wave vector, called the Bloch–Floquet wave number, perpendicular to the interfaces. It follows from the periodicity of the parameters and (19) that Ey0 (−β/2) =
i(e(−β/2)pu0 (−β/2) + H0′ (−β/2))
ε(−β/2) i(e(β/2)pu0 (β/2) + H0′ (β/2)) λEy0 (β/2) = λ , ε(β/2)
=
i(e(β/2)pu0 (−β/2) + H0′ (−β/2))
ε(β/2)
,
and from the second conditions of (21) and (22) we can write H0′ (−β/2) = λH0′ (β/2). Since it follows also in a similar way that u′0 (−β/2) = λu′0 (β/2), conditions (21) and (22) can be replaced by the following u0 (−β/2) = λu0 (β/2),
u′0 (−β/2) = λu′0 (β/2),
(23)
H0 (−β/2) = λH0 (β/2),
H0 (−β/2) = λH0 (β/2).
(24)
′
′
Eqs. (16) and (17) are uncoupled if e(x)/ε(x) = const or p = 0 i.e. the wave propagates perpendicular to the interfaces of the structure (non oblique wave). Since the boundary conditions (23) and (24) for the elastic displacement and the magnetic field are also uncoupled the propagation of elastic and electromagnetic waves in these cases decouple and the interaction caused by the piezoelectric effect is present only in the stiffened elastic modulus G0 (x) and brings solely a quantitative effect. The interaction also does not take place when the piezoelectric effect vanishes e(x) = 0. In the general case for continuous periodic material parameters the main Eqs. (16) and (17) are coupled and the boundary conditions (23) and (24) are uncoupled. If the unit cell is made of different elastic materials and the material parameters are piecewise constant functions then the Eqs. (16) and (17) become decoupled but the interface conditions are coupled. We consider an infinite one-dimensional periodically composed piezoelectric structure with the unit cell of a period β consisting of a piezoelectric inclusions of thickness b embedded and perfectly bonded with the elastic matrix made of another piezoelectric material of a thickness a as shown in Fig. 1.
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G.T. Piliposian et al. / Wave Motion 49 (2012) 125–134
Fig. 1. Symmetric unit cell made of two piezoelectric materials.
The solutions for the magnetic field and the elastic displacement can be written as follows (1)
= C sin(q1 x) + B cos(q1 x),
(2)
= C2 sin(q2 x) + B2 cos(q2 x),
u0 = A2 sin(r2 x) + F2 cos(r2 x),
(3)
= C3 sin(q2 x) + B3 cos(q2 x),
u0 = A3 sin(r2 x) + F3 cos(r2 x)
H0 H0 H0
(1)
u0 = A sin(r1 x) + F cos(r1 x), (2)
(3)
(25)
where qj =
ω2 εj µj − p2 ,
rj =
ρj ω2 /G0j − p2 ,
j = 1, 2,
(26)
and superscripts 1, 2 and 3 show that the functions belong to the mediums 1, 2 and 3 respectively. The following continuity conditions at the interfaces x = ±b/2 should be satisfied
[ u0 ] = 0 ,
[σxz ] = 0,
[Ey0 ] = 0,
[H0 ] = 0,
(27)
where [·] is a jump of a function across the interfaces. Using (27) the coefficients C2 , B2 , A2 and F2 and C3 , B3 , A3 and F3 can be expressed via four coefficients C , B, A and F where C2 = C˜ (b),
B2 = B˜ (b),
C3 = C˜ (−b),
B3 = B˜ (−b),
A2 = A˜ (b),
F2 = F˜ (b),
A3 = A˜ (−b),
F3 = F˜ (−b),
and the following notations are used C˜ (b)q2 ϵ1 = C (q1 ϵ2 ξ1 ξ2 + q2 ϵ1 η1 η2 ) + B(q1 ϵ2 η1 ξ2 − q2 ϵ1 ξ1 η2 ) + pγ (F ξ2 ξ3 − Aξ2 η3 ), B˜ (b)q2 ϵ1 = C (q1 ϵ2 ξ1 η2 − q2 ϵ1 η1 ξ2 ) + B(q2 ϵ1 ξ1 ξ2 + q1 ϵ2 η1 η2 ) + γ p2 (F γ η2 ξ3 − Aη2 η3 ), A˜ (b)G20 r2 =
pγ
(Bξ1 ξ4 − C η1 ξ4 ) + A(G10 r1 ξ3 ξ4 + G20 r2 η3 η4 ) + F (G10 r1 η3 ξ4 − G20 r2 ξ3 η4 ), ϵ1 ϵ2 pγ F˜ (b)G20 r2 = (Bξ1 k4 − C η1 η4 ) + A(G10 r1 ξ3 η4 − G20 r2 η3 ξ4 ) + F (G20 r2 ξ3 ξ4 + G10 r1 η3 η4 ), ϵ1 ϵ2 γ = (e1 ϵ2 − e2 ϵ1 ), ξ1 = cos(bq1 /2), ξ2 = cos(bq2 /2), ξ3 = cos(br1 /2), ξ4 = cos(br2 /2), η1 = sin(bq1 /2), η2 = sin(bq2 /2), η3 = sin(br1 /2), η4 = sin(br2 /2).
(28)
Substituting these coefficients and solutions (25) into the four Bloch–Floquet quasi-periodicity conditions (23) and (24) the system of equations for the remaining unknown coefficients will produce the following dispersion equation: F (λ, p, ω) = λ4 + f (p, ω)λ3 + g (p, ω)λ2 + f (p, ω) + 1 = 0,
(29)
where f (p, ω) = −2 cos(bq1 ) cos(aq2 ) − 2 cos(br1 ) cos(ar2 ) + Q sin(bq1 ) sin(aq2 ) p2 R22 sin(bq1 ) sin(ar2 ) p2 R21 sin(aq2 ) sin(br1 ) + + G sin(br1 ) sin(ar2 ), + G10 q2 r1 G20 q1 r2
(30)
g (p, ω) = 2 + 2 cos(br1 ) cos(ar2 )(2 cos(bq1 ) cos(aq2 ) − Q sin(bq1 ) sin(aq2 ))
+ 2p2 [K1 sin(bq1 ) sin(br1 ) − K1 sin(bq1 ) sin(br1 ) cos(aq2 ) cos(ar2 ) + K2 sin(aq2 ) sin(ar2 ) − K3 cos(bq1 ) cos(ar2 ) sin(aq2 ) sin(br1 ) − K4 cos(aq2 ) cos(br1 ) sin(bq1 ) sin(ar2 ) − K2 cos(bq1 ) cos(br1 ) sin(aq2 ) sin(ar2 )] − 2G cos(bq1 ) cos(aq2 ) sin(br1 ) sin(ar2 ) + QG sin(bq1 ) sin(aq2 ) sin(br1 ) sin(ar2 ) + p4 K1 K2 sin(bq1 ) sin(aq2 ) sin(br1 ) sin(ar2 ),
(31)
G.T. Piliposian et al. / Wave Motion 49 (2012) 125–134
129
and G=
G10 r1 G20 r2
K1 =
+
R22 G10 q1 r1
G20 r2 G10 r1
,
,
Q =
K2 =
q22 ε12 + q21 ε22 q1 q2 ε1 ε2
R21 G20 q2 r2
,
,
K3 =
R1 = R21 G10 q2 r1
,
e2 ε1 − e1 ε2
√ ε1 ε2 K4 =
, R22
G20 q1 r2
R2 =
,
e2 ε1 − e1 ε2
√ ε2 ε1
a = β − b.
,
(32)
(33)
Presenting (29) in the form
λ+
1
2
λ
+ λ+
1
λ
f + g − 2 = 0,
(34)
and taking into account that λ = eikβ and λ + λ−1 = 2 cos(β k) the trigonometric solution of the dispersion equations is cos(β k) =
1
(−f ±
f 2 + 4g − 8). (35) 4 An equation of a similar type has been obtained and discussed in [23] for a vibrating periodic beam. For given values of p and ω the dispersion equation (35) gives two values of k. When none of the values is a real number there exists a region with no dispersion curves, i.e. forbidden band gaps. For an unperturbed case (non-periodic) the dispersion equation (29) converts to the following: cos(β k) = cos(β r ),
cos(β k) = cos(β q),
(36)
giving the explicit formulae
ω12 =
(k2 + p2 ) , εµ
ω22 =
G0
ρ
(k2 + p2 )
(37)
for the wave frequencies. In the general case the dispersion equation gives coupled elastic and electro-magnetic waves with two dispersion curves. When there is no piezoelectric effect e1 = e2 = 0 Eqs. (35) give two uncoupled equations, one describing the propagation of pure electromagnetic waves cos(β k) = cos(bq1 ) cos(aq2 ) −
q22 ε12 + q21 ε22 2q1 q2 ε1 ε2
sin(bq1 ) sin(aq2 )
(38)
and the other propagation of pure acoustic wave cos(β k) = cos(br1 ) cos(ar2 ) −
(G21 r12 + G22 r22 ) 2G1 G2 r1 r2
sin(br1 ) sin(ar2 ),
(39)
where G1,2 are pure elastic shear modules. In the general case the dispersion equation (35) gives two coupled electro-magneto-elastic waves. The wave that reduces to a pure acoustic wave when the piezoelectric effect approaches to zero corresponds to a ‘‘quasi-acoustic wave’’ and the wave that becomes a pure electromagnetic wave when the piezoelectric effect vanishes is a ‘‘quasi-electromagnetic field wave’’. If for a periodic structure e1 /e2 = ε1 /ε2 or p = 0, as noted above, Eq. (35) again gives two uncoupled equations, one is (38) and the other is (39) where the piezoelectric effect is present in the electro-mechanical coupling coefficients in stiffened elastic moduli G10 and G20 . Note that if for the two to the interfaces (p = 0) the acoustic constituent materials for waves traveling perpendicular wave impedances Zja = Gj0 ρj0 or electromagnetic wave impedances Zje = µj /εj (j = 1, 2) are the same it follows from (38) and (39) that cos(β k) = cos(ar1 + br2 ),
cos(β k) = cos(aq1 + bq2 ),
which means that there are no frequency regions with forbidden band gaps. In a quasi-static approximation the right hand side of (38) is independent of ω, the equation becomes non hyperbolic and cannot describe an electromagnetic wave motion. If one of the constituent materials in the phononic crystal is pure elastic (e2 = 0) then in a quasi-static approximation (q1 = ip, q2 = ip) Eq. (39) coincides with the dispersion equation obtained in [15]. When the periodic structure is made of two identical piezoelectric materials of the same width a = b one of which has the opposite polarization direction i.e. e1 = e, e2 = −e then the functions f and g in the dispersion equation (29) take the following form f = −4 cos(b(q + r )) cos(b(q − r )) + 8ξ ,
(40)
g = 1 + 2(1 + 2 cos(2bq) cos(2br ) − cos(bq) cos(br )) + ξ ,
(41)
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G.T. Piliposian et al. / Wave Motion 49 (2012) 125–134
Fig. 2. Band structure of piezoelectric phononic crystal PZT-4 and BaTiO3 h = 0.5, α = 1, c1 = structure with and without piezoelectric effect.
√
ρ1 /G01 . Solid lines and dashed lines show the band
where
ξ=
ηp2 sin(bq) sin(br ) qr
,
η=
e2
ε(c44 + e2 /ε)
.
Further analysis shows that the identical piezoelectric phononic crystal with opposite polarization allows propagation of the Bloch–Floquet waves at acoustic frequencies. At optical frequencies the opposite polarization does not cause an opening of band gaps since the effect of piezoelectricity approaches to zero, and the structure behaves as a photonic crystal with identical optical refractive indexes. 3. Numerical results For given values of p and ω Eq. (35) gives two values of the wave number k. The regions of ω where both values of k are complex correspond to band gaps. The appearance of band gaps depends on the dimensionless oblique incidence wave number α = pβ , the filling fraction h and the difference in the elastic and electromagnetic properties of two piezoelectric materials in the phononic(photonic) crystal which can be expressed in the differences in the impedances. Numerical calculation have been carried out both at low and high frequency regions for two piezoelectric phononic crystals. Material parameters of PZT-4 and BaTiO3 have been used for one photonic crystal and LiTO3 and PZT-4 for the second one (Table 1). Since the relative magnetic permeability for piezoelectric materials is very close to one, the values for µ1 and µ2 are taken equal to the vacuum permeability 4π 10−7 V s/(A m). Figs. 2 and 3 illustrate the effect of piezoelectricity √ in the low frequency region for h = 0.5 and the wave number α = pβ = 1. For the normalized frequency ωβ/c1 , c1 = ρ1 /G01 is the velocity of a transverse wave in the material 1. Fig. 1 shows that the piezoelectricity can affect the band structure and the band widths significantly. The first and second band gaps occur at frequencies where there are no band gaps without the piezoelectric effect. The width of the band gaps for piezoelectric phononic crystals widens for higher frequencies whereas the gaps are practically absent when the piezoelectric effect is neglected. It is affected by the fact that the ratio Z1a /Z2a of the acoustic impedances of the two piezoelectric materials (where Z1a is the impedance of LiTO3 and Z2a is the impedance of PZT-4) is 12.6 whereas the ratio of impedances drops to the value of 1.1 when the piezoelectric effect is not taken into account. The effect of piezoelectricity is less for the phononic crystal made of LiTO3 and PZT-4. Fig. 2 shows that although the piezoelectricity widens the band gaps, the widths of band gaps for acoustic frequencies in this case are closer for phononic crystals with and without piezoelectric effects. In this case the ratio of acoustic impedances for piezoelectric phononic crystal materials is 0.502 against the ratio of the acoustic impedances of the same materials without the piezoelectric effect 0.558. Fig. 4 shows the effect of opposite polarization on the band gap for a phononic crystal PZT-4 and BaTiO3 . Compared with Fig. 2 when the two piezoelectric materials have the same polarization it can be seen that changing the polarization direction of one of the materials can have a significant effect on the band gap widths. In this case the second and fourth band gaps are significantly larger. When the phononic crystal is made of the same piezoelectric material simply changing the polarization direction of one of them leads to a periodic structure and creates band gaps [19]. This effect is shown in Fig. 5 where the first large band gap and the second smaller one are the effect of the opposite polarization of the same piezoelectric material.
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Fig. 3. Band structure of piezoelectric phononic crystal LiTO3 and PZT-4 h = 0.5, α = 0.5. Solid lines and dashed lines show the band structure with and without piezoelectric effect.
Table 1 Material constants of PZT-4, LiTO3 and BaTiO3 . Material
Elastic constant c44 1010 N/m2
Piezoelectric constant e15 C/m2
Permittivity ε11 10−11 F/m
Density ρ 103 kg/m3
PZT-4 LiTO3 BaTiO3
2.56 1.78 4.3
12.7 0.89 11.6
646 6.434 1.264
7.6 3.402 5.7
Fig. 4. Band structure of piezoelectric phononic crystal, PZT-4 and BaTiO3 h = 0.5, α = 1, for opposite polarization direction. Solid lines and the dashed lines show the band structure with and without piezoelectric effect.
Numerical results with quasi-static assumptions practically coincide almost precisely with the solid lines in Figs. 2–4. This shows that in periodic structures at acoustic frequencies the quasi-static setting gives an excellent approximation. The influence of the oblique incidence wave number pβ on the normalized band gap width is shown in Fig. 6, where 1ω is the gap width of the phononic crystal and ωg is the mid gap frequency. It can be seen that both first and second band gaps have maximum values when the wave propagates perpendicular to the laminations and both reach a minimum
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Fig. 5. Band structure of an identical piezoelectric phononic crystal PZT-4, h = 0.5, α = 1, for opposite polarization direction. Solid lines and dashed lines show the band structure with and without piezoelectric effect.
Fig. 6. Normalized gap width as a function of an oblique incidence wave number pβ for a phononic structure PZT-4 and LiTO3 . Solid lines and dashed lines show the first and second band gaps.
point at some value of the wave number pβ . Fig. 7 shows the dependence of the normalized frequency on the thickness of the layers of the phononic crystal. Both the first and second band gaps have peaks at some value of the filling fraction h = b/a. Numerical calculations have been carried out also at optical frequencies. The elastic and piezoelectric properties in this frequency region do not affect the band structure. Two dispersion curves given by the dispersion equation (35) drawn in one plot coincide with the dispersion curve (38) describing the propagation of an electromagnetic wave in a photonic crystal. Here apart from the filling fraction and the oblique incidence wave number pβ the band structure depends on the ratios of the electromagnetic wave impedances Zje = µj /εj of the constituent materials. Band gaps for piezoelectric photonic crystal LiTO3 and PZT-4 for h = 0.5 and three different values of the oblique incidence wave number pβ are shown in Fig. 8. Here for the normalized frequency ωβ/c , c is the speed of light. This picture is the same for the piezoelectric crystal with and without the piezoelectric effect, which means that the effect of piezoelectricity does not play any role on band structure of an electromagnetic wave and confirms the experimental results obtained for lattice vibrations of a-quartz in [24] that the effect of piezoelectricity approaches zero at optical frequencies.
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Fig. 7. Normalized gap width as a function of filling fraction h for BaTiO3 and PZT-4 crystal. Solid lines and dashed lines show the first and second band gaps.
Fig. 8. Band structure of piezoelectric photonic crystal LiTO3 and PZT-4 for h = 0.5 for different values of oblique incidence wave number pβ . Solid lines, dashed lines and large dashing are for α = 0, 0.5, 1, respectively.
4. Conclusion The prorogation of electro-magneto-elastic coupled SH waves in one dimensional infinite periodic piezoelectric media is considered within a full system of the Maxwell’s equations. Such setting of the problem allows investigating of the Bloch–Floquet waves in a wide range of frequencies from low frequency acoustic waves to high frequency electromagnetic waves. Governing equations describing the propagation of SH waves in the (x, y) plane perpendicular to the polling direction has been derived. It is shown that when e(x)/ε(x) = const or p = 0 i.e. the wave propagates perpendicular to the periodicity of the structure the boundary value problem decouples into two boundary value problems for electromagnetic and elastic waves. In this case the effect of piezoelectricity is present only in the electromechanical coupling coefficient in stiffened elasticity modulus. The dispersion equation has been obtained between the frequency and wave numbers for Bloch–Floquet waves, and the band structure has been defined both at acoustic and optical frequencies.
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Numerical results show that the piezoelectricity does not strongly affect the frequency where the first band gaps appear but can have a significant effect on the width of the first band gap. It can open band gaps in some cases where they simply do not exist without a piezoelectric effect. This is particularly well illustrated in the case of the identical piezoelectric crystals with opposite polarization when the opening of a band gap is caused only by the piezoelectric effect. The results also show that waves propagating parallel to the periodicity direction can have the largest band gaps. The widths of the first and second band gaps reach a minimum point for a certain value of the oblique wave number. It has been shown that the band structure for optical frequencies practically is not dependent on the piezoelectric effect and is conditioned only by periodicity of optical electromagnetic wave impedances of materials. 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