- Email: [email protected]
elastic layered structures
Ultrasonics 52 (2012) 442–446
Contents lists available at SciVerse ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Theoretical validation on the existence of two transverse surface waves in piezoelectric/elastic layered structures Zheng-Hua Qian ⇑, Sohichi Hirose Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Tokyo 152-8550, Japan
a r t i c l e
i n f o
Article history: Received 14 July 2011 Received in revised form 12 October 2011 Accepted 13 October 2011 Available online 21 October 2011 Keywords: Transverse surface waves Piezoelectric coupled structures Dispersion behavior
a b s t r a c t In this paper, we analytically study the dispersion behavior of transverse surface waves in a piezoelectric coupled solid consisting of a transversely isotropic piezoelectric ceramic layer and an isotropic metal or dielectric substrate. This study is a revisit to the stiffened Love wave propagation done previously. Closed-form dispersion equations are obtained in a very simple mathematical form for both electrically open and shorted cases. From the viewpoint of physical situation, two transverse surface waves (i.e., the stiffened Love wave and the FDLW-type wave) are separately found in a PZT-4/steel system and a PZT-4/ zinc system. All the observed dispersion curves are theoretically validated through the discussion on the limit values of phase velocity using the obtained dispersion equations. Those validation and discussion give rise to a deeper understanding on the existence of transverse surface waves in such piezoelectric coupled structures. The results can be used as a benchmark for the study of the wave propagation in the piezoelectric coupled structures and are significant in the design of wave propagation in the piezoelectric coupled structures as well. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Transverse surface waves in piezoelectric coupled materials and structures are attractive for designing signal-processing devices due to their high performance and simple particle motion [1,2]. In such study, an accurately analytical model of wave propagation in the piezoelectric coupled structures with piezoelectric coupling effects fully modeled is a key to the design of the wavelength of the IDT (Interdigital transducer) and the excitation of the wave propagation in the structure [3]. Curtis and Redwood [4] carried out a theoretical study on the propagation of transverse surface waves in a piezoelectric material substrate carrying a metal layer of finite thickness and gave a proper classification on dispersion curve type under different physical situations. Kielczynski et al. [5] studied the propagation of the transverse surface waves on piezoelectric ceramics with a depolarized surface layer and pointed out that the Love type wave’s existence condition v1 < v < v2 was necessary for the waves which was actually the stiffened Love waves (v1 and v2 are phase velocities of bulk shear waves in the layer and in the substrate respectively). Feng and Li [6] showed that another transverse surface wave, called ferroelectric layer domain wave (FDLW), could propagate in a layered structure of piezoelectric ceramics with the polarization of the layer reversed from that of the ⇑ Corresponding author. Tel./fax: +81 3 5734 2692. E-mail addresses: [email protected], [email protected] (Z.-H. Qian). 0041-624X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2011.10.007
substrate and gave the phase velocity relation v < v2 = v1 which was different from the existence condition of the Love type wave. Furthermore, Feng and Li [7] continued the study on the propagation of the transverse surface waves in a layered structure with reversed piezoelectric ceramics and concluded that the transverse surface waves could propagate in such structures not only when v1 < v < v2, but also when v < v2 and v < v1 with some restrictions. A potential application of piezoelectric materials as actuators and sensors in the health monitoring of structures requires a piezoelectric layer surface be bonded to the structures [3]. Wang et al. [8,9] studied the dispersive characteristics of the stiffened Love wave propagation in a semi-infinite metal surface mounted by a piezoelectric layer abutting the vacuum, separately considering the electrically open [8] and shorted [9] conditions at the free surface. It was found in their work that the stiffened Love wave velocity of the fundamental mode in the case of electrically shorted circuit was even less than the bulk shear wave velocity in the piezoelectric layer and was to approach the BG (Bleustein–Gulyaev) [10,11] wave velocity vBG with the increase in the dimensionless wavenumber. This phenomenon violated the normal Love wave’s existence condition [12] v1 < v < v2 but was not properly explained in the previous work. In this paper, we give a theoretical validation on the existence of different type dispersion curves through the discussion on the limit values of phase velocity using the obtained dispersion equations, which could help to understand the dispersion behavior of the stiffened Love wave more deeply. For a piezoelectric/metal structure of finite thickness, Wang [13] also investigated the SH
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Z.-H. Qian, S. Hirose / Ultrasonics 52 (2012) 442–446
wave propagation and presented many dispersion modes different from both the traditional Love wave and the FDLW wave. A possible interpretation on those modes is the multiple reflection from the bottom side of the substrate since the SH wave under question was not a surface wave which can be seen from the obtained mechanical displacement solution in the substrate. Motivated by the work done by Feng and Li [7], in this paper we will discuss the possibility of another type of transverse surface wave in a piezoelectric layer bonded with a metal or dielectric substrate in detail. To the authors’ knowledge, such work has not been done yet. Under the assumed condition v1 > v2 > vBG, further analysis of the existence of the FDLW-type transverse surface wave in a PZT-4/zinc system is given. A theoretical validation on the dispersion curves is conducted. Our analysis shows that in layered structure composed of a piezoelectric layer and a metal or dielectric substrate, not only stiffened Love waves can propagate when v2 > v1 > vBG, but also FDLW-type transverse surface waves can propagate when v1 > v2 > vBG, i.e., we give a new existence condition for transverse surface waves propagating in such layered structures. The manuscript is arranged as follows: the problem under question is stated in Section 2; followed by a brief solution procedure description and a numerical example discussion in Section 3; and the conclusions are drawn in Section 4.
The governing equation for the mechanical displacement u0 in the substrate are
r2 u0 ð1=v 22 Þu€ 0 ¼ 0;
ð3Þ
0 1/2
where v2 = (c /q ) is the bulk shear wave velocity in the substrate, with c0 = c044 and q0 being separately the shear modulus and mass density. The boundary and continuity conditions for the wave propagation problem specified by (2) and (3) are: (1) r31 = 0 at x1 = h; (2) u = u, r31 ¼ r031 , u = 0 at x1 = 0; (3) u ? 0 as x1 ? +1. The electrical conditions at the free surface can be classified into two categories, i.e., (4) electrically open circuit: D1 = 0 at x1 = h and; (5) electrically short circuit (or metalized surface): u = 0 at x1 = h, based on the fact that the space above the piezoelectric layer is vacuum or air and its permittivity is much less than that of the piezoelectric material. 0
3. Theoretical solution and discussion The detailed solution procedure can refer to some previous work (see the work by Feng and Li [7], and so on), so we here give a very brief description for the sake of brevity. Consider the following solutions satisfying attenuation condition (3):
uðx1 ; x2 ; tÞ ¼ ðA1 ebkx1 þ A2 ebkx1 Þ exp½ikðx2 v tÞ
2. Statement of the problem
wðx1 ; x2 ; tÞ ¼ ðA3 ekx1 þ A4 ekx1 Þ exp½ikðx2 v tÞ The layered structure, shown in Fig. 1, involves an isotropic metal or dielectric substrate and a transversely isotropic piezoelectric layer with uniform thickness h. The piezoelectric material is poled along the x3-axis, perpendicular to the x1–x2 plane. Usually, the thickness of the substrate is much greater than that of the covering layer for surface acoustic wave devices, so the layered structure can be treated as a half-space problem. It is assumed that the waves propagate along the positive x2-axis, such that the movement can be described as
u1 u2 0; u3 ¼ u3 ðx1 ; x2 ; tÞ ; u ¼ uðx1 ; x2 ; tÞ
ð1Þ
where u1, u2, u3 and u are the mechanical displacement components and electric potential, respectively. Let u and u denote separately the mechanical displacement and electric potential function in the piezoelectric layer, the sequent governing equations read
r2 u ð1=m21 Þu€ ¼ 0 2
r w¼0
) ð2Þ
;
where r2 is the two-dimensional Laplacian, and v1 = [(c + e2/e)/q]1/2 is the bulk shear wave velocity in the covering layer, with c = c44, e = e15, e = e11 and q being the relevant elastic, piezoelectric, dielectric constants and mass density, respectively. The w is introduced [3] through u = w + eu/e.
Electrode
Air
h
Piezoelectric layer Metal or dielectric substrate
0
x2
x1 Fig. 1. A schematic geometry of a piezoelectric layered structure.
) ;
h 6 x1 6 0; ð4Þ
0
u0 ðx1 ; x2 ; tÞ ¼ A5 eb kx1 exp½ikðx2 v tÞ;
x1 P 0;
ð5Þ
where A1, A2, A3, A4 and A5 are arbitrary constants, k (= 2p/k) is the pffiffiffiffiffiffiffi wavenumber, k is the wavelength, i ¼ 1, and c is the phase velocity. (4) and (5) separately satisfy Eqs. (2) and (3) when 2
b ¼ 1 v 2 =v 21 ;
02
b ¼ 1 v 2 =v 22 ;
ð6Þ
where the bulk shear wave velocities v1 and v2 are defined as in Section 2. Substitution of (5) and (6) and the corresponding stress components into the remaining boundary and continuity conditions (1), (2), (5), or (6) yields five linear, homogenous algebraic equations with respect to the unknown coefficients A1, A2, A3, A4 and A5. The existence condition of nontrivial solutions of these coefficients leads to the following dispersion equations 2 0 kp tanhðkhÞ b tanhðbkhÞ c0 b =c ¼ 0;
ð7Þ
for the case of electrically open circuit, and
c0 2 0 c0 0 4 2 ðkp þ b Þ tanhðkhÞ tanhðbkhÞ kp b tanhðbkhÞ þ bb tanhðkhÞ c c 1 2 þ 2kp b 1 ¼ 0; ð8Þ coshðkhÞ coshðbkhÞ for the case of electrically shorted circuit, respectively. In (7) and 2 (8), kp ¼ e2 =ec is the piezoelectric coupling factor with c ¼ c þ e2 =e being the piezoelectrically stiffened elastic constant [4]. It can be seen from (6) that b can not only take real values but also imaginary values, depending on whether the surface wave velocity v is smaller or greater than the bulk shear wave velocity v1 in the piezoelectric layer. It is clear that b can only be real since an imaginary value would represent a kind of plane wave solution and this type of wave carries the energy away from the layer [3]. Such a wave system would quickly lose its energy and not be of significance at any distance, and thus is beyond the scope of this paper. Following the way of Curtis and Redwood [4], two physical situations are categorized: Type-1: v2 > v1 > vBG, b is real or imaginary. v2 > v > v1 in electrically open case, while v2 > v > vBG in electrically shorted case.
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Z.-H. Qian, S. Hirose / Ultrasonics 52 (2012) 442–446
Type-2: v1 > v2 > vBG, b is always real. v2 > v > vBG in electrically open case, while in electrically shorted case v2 > v for the first mode and v2 > v > v1 for the second mode. Here, vBG is the phase velocity of the BG wave on the surface of a piezoelectric substrate coated with an infinitely thin layer of conducting material. v1 is the velocity when wavenumber approaches infinity. For the sake of convenient calculation, we introduce the dimensionless wavenumber H = h/k. Considering (6), we can thus rewrite (7) and (8) into: 2
0
kp tanhð2pHÞ b1 tanhðb1 2pHÞ c0 b =c ¼ 0;
ð9aÞ
c0 2 0 4 2 ðkp þ b1 Þ tanhð2pHÞ tanhðb1 2pHÞ kp b tanh b1 2pH c c0 1 0 2 þ b1 b tanhð2pHÞ þ 2kp b1 1 ¼ 0; c coshð2pHÞ coshðb1 2pHÞ ð9bÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for real b (i.e., b ¼ b1 ¼ 1 v 2 =v 21 ), and 2
0
kp tanhð2pHÞ þ b2 tanhðb2 2pHÞ c0 b =c ¼ 0;
Fig. 3. Dispersion curves of the FDLW-type wave in the PZT4/zinc system. H = h/k, v2 = 2440 m/s, v1 = 2597 m/s, vBG = 2258 m/s.
ð10aÞ
c0 2 0 4 2 ðkp b2 Þ tanhð2pHÞ tanhðb2 2pHÞ kp b tanðb2 2pHÞ c c0 1 0 2 þ b2 b tanhð2pHÞ þ 2kp b2 1 ¼ 0; c coshð2pHÞ coshðb2 2pHÞ ð10bÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for imaginary b (i.e., b ¼ ib2 ¼ i 1 v 2 =v 21 1). Eqs. (9) and (10) are the dispersion equations which can be used to calculate the dispersion curves of the two possible transverse surface waves described above. The different forms of the dispersion curves for the two types are separately illustrated in Figs. 2 and 3, corresponding to a PZT4/steel system and a PZT4/zinc system for the cases of both electrically open circuit and electrically shorted circuit. The material parameters [3,4] used in our calculation are listed in Table 1, where the dielectric constant of vacuum is e0 = 8.854 1012 F/m. 3.1. Stiffened Love wave The Type-1 situation is illustrated by Fig. 2, which refers to a PZT4 layer deposited on a steel substrate. Here, h/k is given by
Table 1 Material parameters. Materials
c44 (109 N/m2)
e15 (C/m2)
e11/e0
q (kg/m3)
PZT4 (layer) Steel (substrate) Zinc (substrate)
25.6 83.967 41.2
12.7 – –
730 – –
7500 7800 6920
the multiple-valued functions in Eqs. (10a) and (10b) corresponding to imaginary b, except that the lower part of the fundamental mode in the electrically shorted case (denoted by the dotted line in Fig. 2) is given by the function in Eq. (9b) corresponding to real b. The upper and lower parts of the fundamental mode in the electrically shorted case are connected perfectly at the point where the phase velocity v equals the bulk shear wave velocity v1 (2597 m/s) of the piezoelectric layer. These modes are closely related to the Love waves [12] that would exist even if the layer is depolarized. But the phase velocity could even lower than the bulk shear wave velocity of the covering layer, which is actually due to the piezoelectric effect. So these waves are called the stiffened Love wave. A theoretical validation on the above-mentioned phenomenon is given below. For the Type-1 situation, Eq. (10a) is responsible for the electrically open circuit case, whist Eqs. (9b) and (10b) are responsible for the electrically shorted circuit case. From Eq. (10a), we get
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 2 arctan½c0 b =cb2 kp tanhð2pHÞ=b2 v2 1¼ : 2 2p H v1
ð11Þ
As H ? 1, we have
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
arctan v2 1¼ 2 v1
Fig. 2. Dispersion curves of the stiffened Love wave in the PZT4/steel system. H = h/ k, v2 = 3281 m/s, v1 = 2597 m/s, vBG = 2258 m/s.
0 2 c0 b =cb2 kp =b2 2pH
¼ 0;
ð12Þ
which gives rise to v(H = 1) = v1. This is the asymptotic value for the phase velocity in the electrically open circuit case, corresponding to solid lines in Fig. 2. In other words, the phase velocity of all the modes decreases monotonically from the bulk shear wave velocity of the steel substrate v2 (3281 m/s) to the bulk shear wave velocity of the piezoelectric layer v1 (2597 m/s). For the case of electrically shorted circuit, however, it is not like that. From Eq. (10b) we get
Z.-H. Qian, S. Hirose / Ultrasonics 52 (2012) 442–446
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v2 1 v 21
h i9 8 0 < cc b2 b0 tanhð2pHÞ þ 2k2p b2 coshð2pHÞ1cosðb 2pHÞ 1 = 1 2 arctan ¼ : 0 2 0 4 2 : ; 2pH ðkp b2 Þ tanhð2pHÞ c kp b c
ð13Þ As H ? 1, we have
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v2 1 1¼ arctan 2pH v 21
0 2 c0 c b2 b 2kp b2 0 4 2 2 0 kp b2 cc kp b
! ¼ 0;
ð14Þ
which leads to the same asymptotic velocity as that in the electrically open case, i.e., v(H = 1) = v1. But this is only for high-order modes in the electrically shorted circuit case, corresponding to dashed lines in Fig. 2. As for the fundamental mode, we need to consider separately. Setting H ? 1 in Eq. (9b), we get
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2
kp
1
v2 v 21
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
kp
1
!
v2 0 0 c b =c v 21
¼ 0;
ð15Þ
which allows the only root v(H = 1) = vBG under the Type-1 situation v2 > v1 > vBG, corresponding to the dotted line in Fig. 2. That is to say, the phase velocity of the high-order modes decreases monotonically from v2 (3281 m/s) to v1 (2597 m/s), while for the fundamental mode the phase velocity approaches to the vBG (2258 m/s) finally. 3.2. FDLW-type transverse surface wave The dispersion curves of the Type-2 wave are shown in Fig. 3, which refers to a PZT4 layer deposited on a zinc substrate. This wave is actually an FDLW-type transverse surface wave, which was not found previously in the layered structures consisting of a piezoelectric layer and an elastic substrate. However, this FDLWtype wave has one mode in the electrically open case but two modes in the case of electrically shorted case, which is different from the original FDLW wave [7] existing in a layered structure of piezoelectric ceramics with the polarization of the layer reversed from that of the substrate. A theoretical validation on the interesting phenomenon is given below. For the Type-2 situation, Eqs. (9a) and (9b) are responsible for the electrically open circuit case and the electrically shorted circuit case, respectively. Setting H ? 1 in Eq. (9a), we get
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kp
1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2
1
v2 v 21
ð16Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
kp
1
mode is the counterpart to the one in the case of electrically open circuit, denoted by the dotted line in Fig. 3. While the second mode starting from H = 0.7752 (denoted by the dot dashed line in Fig. 3) seems to be a ‘‘newly-created’’ mode due to the appearance of electrode on the free surface of the piezoelectric layer. From the viewpoint of acoustic wave application, the second mode appearing in the electrically shorted case seems undesired. It could be removed by inserting a hard metal thin layer between the piezoelectric layer and the metal substrate, which is stated detailedly in our recent work [14]. Furthermore, a general condition can be placed on the criterion that the phase velocity v exists at H = 1. Following the way of Curtis and Redwood [4], we obtain
c0 ð1 q0 c=qc0 Þ > e2 =e;
ð18Þ
for the Type-1 situation (i.e., the stiffened Love wave), and
cð1 qc0 =q0 cÞ > e2 =e;
ð19Þ
for the Type-2 situation (i.e., the FDLW-type wave), respectively. 4. Summary We analytically studied the dispersion behavior of transverse surface waves in a piezoelectric coupled solid consisting of a piezoelectric ceramic layer and an isotropic metal or dielectric substrate and theoretically validated the existence of the stiffened Love wave in a PZT4/steel system and the FDLW-type wave in a PZT4/zinc system through the detailed discussion on the limit values of phase velocity using the closed-form dispersion equations obtained in this paper. And general conditions on the criteria for the phase velocity to exist at all the wavenumber range are given too. Those validation and discussion give rise to a deeper understanding on the existence and dispersion behavior of the transverse surface waves in such piezoelectric coupled structures. The results can be used as a benchmark for the study of the wave propagation in the piezoelectric coupled structures and are significant in the design of wave propagation in the piezoelectric coupled structures as well. Further experimental study to validate the existence of the transverse surface waves is needed. Acknowledgement The author (Z.H. Qian) appreciates the support by the Global COE Program at the Tokyo Institute of Technology, Japan. References
v2 0 0 c b =c ¼ 0; v 21
which gives only one solution under the Type-2 situation v1 > v2 > vBG, i.e., vBG < v(H = 1) < v2. In other words, there is only one mode in the electrically open case, denoted by the solid line. It can be seen from Fig. 3 that the phase velocity first decreases from the bulk shear wave velocity v2 of the zinc substrate and reaches a minimum value at H = 0.1901, then tends to a value 2412 m/s monotonically, somewhat less than the substrate velocity v2 (2440 m/s). For the case of electrically shorted circuit, however, it is not like that. Setting H ? 1 in Eq. (9b), we get
kp
445
!
v2 0 0 c b =c v 21
¼ 0;
ð17Þ
which allows the existence of two solutions under the Type-2 situation v1 > v2 > vBG, i.e., v(H = 1)1 = vBG and vBG < v(H = 1)2 < v2. That means there are two modes in the electrically shorted case. The first
[1] B. Jakoby, M.J. Vellekoop, Properties of Love waves: applications in sensors, Smart Mater. Struct. 6 (1997) 668–679. [2] M.J. Vellekoop, Acoustic wave sensors and their technology, Ultrasonics 36 (1998) 7–14. [3] Q. Wang, V.K. Varadan, Wave propagation in piezoelectric coupled plates by use of interdigital transducer: Part I and Part II, Int. J. Solids Struct. 39 (2002) 1119–1144. [4] R.G. Curtis, M. Redwood, Transverse surface waves on a piezoelectric material carrying a metal layer of finite thickness, J. Appl. Phys. 44 (1973) 2002–2007. [5] P.J. Kielczynski, W. Pajewski, M. Szalewski, Shear-horizontal surface waves on piezoelectric ceramics with depolarized surface layer, IEEE Trans. UFFC 36 (1989) 287–293. [6] H. Feng, X.J. Li, Shear-horizontal surface waves in a layered structure of piezoelectric ceramics, IEEE Trans. UFFC 40 (1993) 167–170. [7] H. Feng, X.J. Li, The general existence conditions for the ferroelectric domain layer wave, Ferroelectrics 129 (1992) 31–41. [8] Q. Wang, S.T. Quek, V.K. Varadan, Love waves in piezoelectric coupled solid media, Smart Mater. Struct. 10 (2001) 380–388. [9] Q. Wang, Wave propagation in a piezoelectric coupled solid medium, ASME J. Appl. Mech. 69 (2002) 819–824. [10] J.L. Bleustein, A new surface wave in piezoelectric materials, Appl. Phys. Lett. 13 (1968) 412–413. [11] Y.V. Gulyaev, Electroacoustic surface waves in solids, JETP Lett. 9 (1969) 37– 38.
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Z.-H. Qian, S. Hirose / Ultrasonics 52 (2012) 442–446
[12] J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland Publishing Company, Oxford, 1973. [13] Q. Wang, SH wave propagation in piezoelectric coupled plates, IEEE Trans. UFFC 49 (2002) 596–603.
[14] Z. Qian, F. Jin, S. Hirose, Dispersion characteristics of transverse surface waves in piezoelectric coupled solid media with hard metal interlayer, Ultrasonics 51 (2011) 853–856.
Contents lists available at SciVerse ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Theoretical validation on the existence of two transverse surface waves in piezoelectric/elastic layered structures Zheng-Hua Qian ⇑, Sohichi Hirose Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Tokyo 152-8550, Japan
a r t i c l e
i n f o
Article history: Received 14 July 2011 Received in revised form 12 October 2011 Accepted 13 October 2011 Available online 21 October 2011 Keywords: Transverse surface waves Piezoelectric coupled structures Dispersion behavior
a b s t r a c t In this paper, we analytically study the dispersion behavior of transverse surface waves in a piezoelectric coupled solid consisting of a transversely isotropic piezoelectric ceramic layer and an isotropic metal or dielectric substrate. This study is a revisit to the stiffened Love wave propagation done previously. Closed-form dispersion equations are obtained in a very simple mathematical form for both electrically open and shorted cases. From the viewpoint of physical situation, two transverse surface waves (i.e., the stiffened Love wave and the FDLW-type wave) are separately found in a PZT-4/steel system and a PZT-4/ zinc system. All the observed dispersion curves are theoretically validated through the discussion on the limit values of phase velocity using the obtained dispersion equations. Those validation and discussion give rise to a deeper understanding on the existence of transverse surface waves in such piezoelectric coupled structures. The results can be used as a benchmark for the study of the wave propagation in the piezoelectric coupled structures and are significant in the design of wave propagation in the piezoelectric coupled structures as well. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Transverse surface waves in piezoelectric coupled materials and structures are attractive for designing signal-processing devices due to their high performance and simple particle motion [1,2]. In such study, an accurately analytical model of wave propagation in the piezoelectric coupled structures with piezoelectric coupling effects fully modeled is a key to the design of the wavelength of the IDT (Interdigital transducer) and the excitation of the wave propagation in the structure [3]. Curtis and Redwood [4] carried out a theoretical study on the propagation of transverse surface waves in a piezoelectric material substrate carrying a metal layer of finite thickness and gave a proper classification on dispersion curve type under different physical situations. Kielczynski et al. [5] studied the propagation of the transverse surface waves on piezoelectric ceramics with a depolarized surface layer and pointed out that the Love type wave’s existence condition v1 < v < v2 was necessary for the waves which was actually the stiffened Love waves (v1 and v2 are phase velocities of bulk shear waves in the layer and in the substrate respectively). Feng and Li [6] showed that another transverse surface wave, called ferroelectric layer domain wave (FDLW), could propagate in a layered structure of piezoelectric ceramics with the polarization of the layer reversed from that of the ⇑ Corresponding author. Tel./fax: +81 3 5734 2692. E-mail addresses: [email protected], [email protected] (Z.-H. Qian). 0041-624X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2011.10.007
substrate and gave the phase velocity relation v < v2 = v1 which was different from the existence condition of the Love type wave. Furthermore, Feng and Li [7] continued the study on the propagation of the transverse surface waves in a layered structure with reversed piezoelectric ceramics and concluded that the transverse surface waves could propagate in such structures not only when v1 < v < v2, but also when v < v2 and v < v1 with some restrictions. A potential application of piezoelectric materials as actuators and sensors in the health monitoring of structures requires a piezoelectric layer surface be bonded to the structures [3]. Wang et al. [8,9] studied the dispersive characteristics of the stiffened Love wave propagation in a semi-infinite metal surface mounted by a piezoelectric layer abutting the vacuum, separately considering the electrically open [8] and shorted [9] conditions at the free surface. It was found in their work that the stiffened Love wave velocity of the fundamental mode in the case of electrically shorted circuit was even less than the bulk shear wave velocity in the piezoelectric layer and was to approach the BG (Bleustein–Gulyaev) [10,11] wave velocity vBG with the increase in the dimensionless wavenumber. This phenomenon violated the normal Love wave’s existence condition [12] v1 < v < v2 but was not properly explained in the previous work. In this paper, we give a theoretical validation on the existence of different type dispersion curves through the discussion on the limit values of phase velocity using the obtained dispersion equations, which could help to understand the dispersion behavior of the stiffened Love wave more deeply. For a piezoelectric/metal structure of finite thickness, Wang [13] also investigated the SH
443
Z.-H. Qian, S. Hirose / Ultrasonics 52 (2012) 442–446
wave propagation and presented many dispersion modes different from both the traditional Love wave and the FDLW wave. A possible interpretation on those modes is the multiple reflection from the bottom side of the substrate since the SH wave under question was not a surface wave which can be seen from the obtained mechanical displacement solution in the substrate. Motivated by the work done by Feng and Li [7], in this paper we will discuss the possibility of another type of transverse surface wave in a piezoelectric layer bonded with a metal or dielectric substrate in detail. To the authors’ knowledge, such work has not been done yet. Under the assumed condition v1 > v2 > vBG, further analysis of the existence of the FDLW-type transverse surface wave in a PZT-4/zinc system is given. A theoretical validation on the dispersion curves is conducted. Our analysis shows that in layered structure composed of a piezoelectric layer and a metal or dielectric substrate, not only stiffened Love waves can propagate when v2 > v1 > vBG, but also FDLW-type transverse surface waves can propagate when v1 > v2 > vBG, i.e., we give a new existence condition for transverse surface waves propagating in such layered structures. The manuscript is arranged as follows: the problem under question is stated in Section 2; followed by a brief solution procedure description and a numerical example discussion in Section 3; and the conclusions are drawn in Section 4.
The governing equation for the mechanical displacement u0 in the substrate are
r2 u0 ð1=v 22 Þu€ 0 ¼ 0;
ð3Þ
0 1/2
where v2 = (c /q ) is the bulk shear wave velocity in the substrate, with c0 = c044 and q0 being separately the shear modulus and mass density. The boundary and continuity conditions for the wave propagation problem specified by (2) and (3) are: (1) r31 = 0 at x1 = h; (2) u = u, r31 ¼ r031 , u = 0 at x1 = 0; (3) u ? 0 as x1 ? +1. The electrical conditions at the free surface can be classified into two categories, i.e., (4) electrically open circuit: D1 = 0 at x1 = h and; (5) electrically short circuit (or metalized surface): u = 0 at x1 = h, based on the fact that the space above the piezoelectric layer is vacuum or air and its permittivity is much less than that of the piezoelectric material. 0
3. Theoretical solution and discussion The detailed solution procedure can refer to some previous work (see the work by Feng and Li [7], and so on), so we here give a very brief description for the sake of brevity. Consider the following solutions satisfying attenuation condition (3):
uðx1 ; x2 ; tÞ ¼ ðA1 ebkx1 þ A2 ebkx1 Þ exp½ikðx2 v tÞ
2. Statement of the problem
wðx1 ; x2 ; tÞ ¼ ðA3 ekx1 þ A4 ekx1 Þ exp½ikðx2 v tÞ The layered structure, shown in Fig. 1, involves an isotropic metal or dielectric substrate and a transversely isotropic piezoelectric layer with uniform thickness h. The piezoelectric material is poled along the x3-axis, perpendicular to the x1–x2 plane. Usually, the thickness of the substrate is much greater than that of the covering layer for surface acoustic wave devices, so the layered structure can be treated as a half-space problem. It is assumed that the waves propagate along the positive x2-axis, such that the movement can be described as
u1 u2 0; u3 ¼ u3 ðx1 ; x2 ; tÞ ; u ¼ uðx1 ; x2 ; tÞ
ð1Þ
where u1, u2, u3 and u are the mechanical displacement components and electric potential, respectively. Let u and u denote separately the mechanical displacement and electric potential function in the piezoelectric layer, the sequent governing equations read
r2 u ð1=m21 Þu€ ¼ 0 2
r w¼0
) ð2Þ
;
where r2 is the two-dimensional Laplacian, and v1 = [(c + e2/e)/q]1/2 is the bulk shear wave velocity in the covering layer, with c = c44, e = e15, e = e11 and q being the relevant elastic, piezoelectric, dielectric constants and mass density, respectively. The w is introduced [3] through u = w + eu/e.
Electrode
Air
h
Piezoelectric layer Metal or dielectric substrate
0
x2
x1 Fig. 1. A schematic geometry of a piezoelectric layered structure.
) ;
h 6 x1 6 0; ð4Þ
0
u0 ðx1 ; x2 ; tÞ ¼ A5 eb kx1 exp½ikðx2 v tÞ;
x1 P 0;
ð5Þ
where A1, A2, A3, A4 and A5 are arbitrary constants, k (= 2p/k) is the pffiffiffiffiffiffiffi wavenumber, k is the wavelength, i ¼ 1, and c is the phase velocity. (4) and (5) separately satisfy Eqs. (2) and (3) when 2
b ¼ 1 v 2 =v 21 ;
02
b ¼ 1 v 2 =v 22 ;
ð6Þ
where the bulk shear wave velocities v1 and v2 are defined as in Section 2. Substitution of (5) and (6) and the corresponding stress components into the remaining boundary and continuity conditions (1), (2), (5), or (6) yields five linear, homogenous algebraic equations with respect to the unknown coefficients A1, A2, A3, A4 and A5. The existence condition of nontrivial solutions of these coefficients leads to the following dispersion equations 2 0 kp tanhðkhÞ b tanhðbkhÞ c0 b =c ¼ 0;
ð7Þ
for the case of electrically open circuit, and
c0 2 0 c0 0 4 2 ðkp þ b Þ tanhðkhÞ tanhðbkhÞ kp b tanhðbkhÞ þ bb tanhðkhÞ c c 1 2 þ 2kp b 1 ¼ 0; ð8Þ coshðkhÞ coshðbkhÞ for the case of electrically shorted circuit, respectively. In (7) and 2 (8), kp ¼ e2 =ec is the piezoelectric coupling factor with c ¼ c þ e2 =e being the piezoelectrically stiffened elastic constant [4]. It can be seen from (6) that b can not only take real values but also imaginary values, depending on whether the surface wave velocity v is smaller or greater than the bulk shear wave velocity v1 in the piezoelectric layer. It is clear that b can only be real since an imaginary value would represent a kind of plane wave solution and this type of wave carries the energy away from the layer [3]. Such a wave system would quickly lose its energy and not be of significance at any distance, and thus is beyond the scope of this paper. Following the way of Curtis and Redwood [4], two physical situations are categorized: Type-1: v2 > v1 > vBG, b is real or imaginary. v2 > v > v1 in electrically open case, while v2 > v > vBG in electrically shorted case.
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Type-2: v1 > v2 > vBG, b is always real. v2 > v > vBG in electrically open case, while in electrically shorted case v2 > v for the first mode and v2 > v > v1 for the second mode. Here, vBG is the phase velocity of the BG wave on the surface of a piezoelectric substrate coated with an infinitely thin layer of conducting material. v1 is the velocity when wavenumber approaches infinity. For the sake of convenient calculation, we introduce the dimensionless wavenumber H = h/k. Considering (6), we can thus rewrite (7) and (8) into: 2
0
kp tanhð2pHÞ b1 tanhðb1 2pHÞ c0 b =c ¼ 0;
ð9aÞ
c0 2 0 4 2 ðkp þ b1 Þ tanhð2pHÞ tanhðb1 2pHÞ kp b tanh b1 2pH c c0 1 0 2 þ b1 b tanhð2pHÞ þ 2kp b1 1 ¼ 0; c coshð2pHÞ coshðb1 2pHÞ ð9bÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for real b (i.e., b ¼ b1 ¼ 1 v 2 =v 21 ), and 2
0
kp tanhð2pHÞ þ b2 tanhðb2 2pHÞ c0 b =c ¼ 0;
Fig. 3. Dispersion curves of the FDLW-type wave in the PZT4/zinc system. H = h/k, v2 = 2440 m/s, v1 = 2597 m/s, vBG = 2258 m/s.
ð10aÞ
c0 2 0 4 2 ðkp b2 Þ tanhð2pHÞ tanhðb2 2pHÞ kp b tanðb2 2pHÞ c c0 1 0 2 þ b2 b tanhð2pHÞ þ 2kp b2 1 ¼ 0; c coshð2pHÞ coshðb2 2pHÞ ð10bÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for imaginary b (i.e., b ¼ ib2 ¼ i 1 v 2 =v 21 1). Eqs. (9) and (10) are the dispersion equations which can be used to calculate the dispersion curves of the two possible transverse surface waves described above. The different forms of the dispersion curves for the two types are separately illustrated in Figs. 2 and 3, corresponding to a PZT4/steel system and a PZT4/zinc system for the cases of both electrically open circuit and electrically shorted circuit. The material parameters [3,4] used in our calculation are listed in Table 1, where the dielectric constant of vacuum is e0 = 8.854 1012 F/m. 3.1. Stiffened Love wave The Type-1 situation is illustrated by Fig. 2, which refers to a PZT4 layer deposited on a steel substrate. Here, h/k is given by
Table 1 Material parameters. Materials
c44 (109 N/m2)
e15 (C/m2)
e11/e0
q (kg/m3)
PZT4 (layer) Steel (substrate) Zinc (substrate)
25.6 83.967 41.2
12.7 – –
730 – –
7500 7800 6920
the multiple-valued functions in Eqs. (10a) and (10b) corresponding to imaginary b, except that the lower part of the fundamental mode in the electrically shorted case (denoted by the dotted line in Fig. 2) is given by the function in Eq. (9b) corresponding to real b. The upper and lower parts of the fundamental mode in the electrically shorted case are connected perfectly at the point where the phase velocity v equals the bulk shear wave velocity v1 (2597 m/s) of the piezoelectric layer. These modes are closely related to the Love waves [12] that would exist even if the layer is depolarized. But the phase velocity could even lower than the bulk shear wave velocity of the covering layer, which is actually due to the piezoelectric effect. So these waves are called the stiffened Love wave. A theoretical validation on the above-mentioned phenomenon is given below. For the Type-1 situation, Eq. (10a) is responsible for the electrically open circuit case, whist Eqs. (9b) and (10b) are responsible for the electrically shorted circuit case. From Eq. (10a), we get
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 2 arctan½c0 b =cb2 kp tanhð2pHÞ=b2 v2 1¼ : 2 2p H v1
ð11Þ
As H ? 1, we have
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
arctan v2 1¼ 2 v1
Fig. 2. Dispersion curves of the stiffened Love wave in the PZT4/steel system. H = h/ k, v2 = 3281 m/s, v1 = 2597 m/s, vBG = 2258 m/s.
0 2 c0 b =cb2 kp =b2 2pH
¼ 0;
ð12Þ
which gives rise to v(H = 1) = v1. This is the asymptotic value for the phase velocity in the electrically open circuit case, corresponding to solid lines in Fig. 2. In other words, the phase velocity of all the modes decreases monotonically from the bulk shear wave velocity of the steel substrate v2 (3281 m/s) to the bulk shear wave velocity of the piezoelectric layer v1 (2597 m/s). For the case of electrically shorted circuit, however, it is not like that. From Eq. (10b) we get
Z.-H. Qian, S. Hirose / Ultrasonics 52 (2012) 442–446
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v2 1 v 21
h i9 8 0 < cc b2 b0 tanhð2pHÞ þ 2k2p b2 coshð2pHÞ1cosðb 2pHÞ 1 = 1 2 arctan ¼ : 0 2 0 4 2 : ; 2pH ðkp b2 Þ tanhð2pHÞ c kp b c
ð13Þ As H ? 1, we have
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v2 1 1¼ arctan 2pH v 21
0 2 c0 c b2 b 2kp b2 0 4 2 2 0 kp b2 cc kp b
! ¼ 0;
ð14Þ
which leads to the same asymptotic velocity as that in the electrically open case, i.e., v(H = 1) = v1. But this is only for high-order modes in the electrically shorted circuit case, corresponding to dashed lines in Fig. 2. As for the fundamental mode, we need to consider separately. Setting H ? 1 in Eq. (9b), we get
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2
kp
1
v2 v 21
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
kp
1
!
v2 0 0 c b =c v 21
¼ 0;
ð15Þ
which allows the only root v(H = 1) = vBG under the Type-1 situation v2 > v1 > vBG, corresponding to the dotted line in Fig. 2. That is to say, the phase velocity of the high-order modes decreases monotonically from v2 (3281 m/s) to v1 (2597 m/s), while for the fundamental mode the phase velocity approaches to the vBG (2258 m/s) finally. 3.2. FDLW-type transverse surface wave The dispersion curves of the Type-2 wave are shown in Fig. 3, which refers to a PZT4 layer deposited on a zinc substrate. This wave is actually an FDLW-type transverse surface wave, which was not found previously in the layered structures consisting of a piezoelectric layer and an elastic substrate. However, this FDLWtype wave has one mode in the electrically open case but two modes in the case of electrically shorted case, which is different from the original FDLW wave [7] existing in a layered structure of piezoelectric ceramics with the polarization of the layer reversed from that of the substrate. A theoretical validation on the interesting phenomenon is given below. For the Type-2 situation, Eqs. (9a) and (9b) are responsible for the electrically open circuit case and the electrically shorted circuit case, respectively. Setting H ? 1 in Eq. (9a), we get
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kp
1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2
1
v2 v 21
ð16Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
kp
1
mode is the counterpart to the one in the case of electrically open circuit, denoted by the dotted line in Fig. 3. While the second mode starting from H = 0.7752 (denoted by the dot dashed line in Fig. 3) seems to be a ‘‘newly-created’’ mode due to the appearance of electrode on the free surface of the piezoelectric layer. From the viewpoint of acoustic wave application, the second mode appearing in the electrically shorted case seems undesired. It could be removed by inserting a hard metal thin layer between the piezoelectric layer and the metal substrate, which is stated detailedly in our recent work [14]. Furthermore, a general condition can be placed on the criterion that the phase velocity v exists at H = 1. Following the way of Curtis and Redwood [4], we obtain
c0 ð1 q0 c=qc0 Þ > e2 =e;
ð18Þ
for the Type-1 situation (i.e., the stiffened Love wave), and
cð1 qc0 =q0 cÞ > e2 =e;
ð19Þ
for the Type-2 situation (i.e., the FDLW-type wave), respectively. 4. Summary We analytically studied the dispersion behavior of transverse surface waves in a piezoelectric coupled solid consisting of a piezoelectric ceramic layer and an isotropic metal or dielectric substrate and theoretically validated the existence of the stiffened Love wave in a PZT4/steel system and the FDLW-type wave in a PZT4/zinc system through the detailed discussion on the limit values of phase velocity using the closed-form dispersion equations obtained in this paper. And general conditions on the criteria for the phase velocity to exist at all the wavenumber range are given too. Those validation and discussion give rise to a deeper understanding on the existence and dispersion behavior of the transverse surface waves in such piezoelectric coupled structures. The results can be used as a benchmark for the study of the wave propagation in the piezoelectric coupled structures and are significant in the design of wave propagation in the piezoelectric coupled structures as well. Further experimental study to validate the existence of the transverse surface waves is needed. Acknowledgement The author (Z.H. Qian) appreciates the support by the Global COE Program at the Tokyo Institute of Technology, Japan. References
v2 0 0 c b =c ¼ 0; v 21
which gives only one solution under the Type-2 situation v1 > v2 > vBG, i.e., vBG < v(H = 1) < v2. In other words, there is only one mode in the electrically open case, denoted by the solid line. It can be seen from Fig. 3 that the phase velocity first decreases from the bulk shear wave velocity v2 of the zinc substrate and reaches a minimum value at H = 0.1901, then tends to a value 2412 m/s monotonically, somewhat less than the substrate velocity v2 (2440 m/s). For the case of electrically shorted circuit, however, it is not like that. Setting H ? 1 in Eq. (9b), we get
kp
445
!
v2 0 0 c b =c v 21
¼ 0;
ð17Þ
which allows the existence of two solutions under the Type-2 situation v1 > v2 > vBG, i.e., v(H = 1)1 = vBG and vBG < v(H = 1)2 < v2. That means there are two modes in the electrically shorted case. The first
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