Acta Mech Sin 2010 26:417 423 DOI 10.1007/s10409-010-0336-5 RESEARCH PAPER A novel type of transverse surface wave propagating in a layered structure consisting of a piezoelectric layer attached to an elastic half-space Zhenghua Qian Feng Jin Sohichi Hirose Received: 18 May 2009 / Revised: 8 September 2009 / Accepted: 23 September 2009 / Published online: 17 February 2010 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH 2010 Abstract The existence and propagation of transverse surface waves in piezoelectric coupled solids is investigated, in which perfect bonding between a metal/dielectric substrate and a piezoelectric layer of finite-thickness is assumed. Dispersion equations relating phase velocity to material constants for the existence of various modes are obtained in a simple mathematical form for a piezoelectric material of class 6 mm. It is discovered and proved by numerical examples in this paper that a novel Bleustein Gulyaev B G type of transverse surface wave can exist in such piezoelectric coupled solid media when the bulk-shear-wave velocity in the substrate is less than that in the piezoelectric layer but greater than the corresponding B G wave velocity in the same piezoelectric material with an electroded surface. Such a wave does not exist in such layered structures in the absence of piezoelectricity. The mode shapes for displacement and electric potential in the piezoelectric layer are obtained and discussed theoretically. The study extends the regime of transverse surface waves and may lead to potential applications to surface acoustic wave devices. Keywords Transverse surface wave Piezoelectric coupled solids Dispersion relation Surface acoustic wave devices The project was supported by the National Natural Science Foundation of China 10972171 and the Program for New Century Excellent Talents in Universities NCET-08-0429. Z. Qian B S. Hirose Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Tokyo 152-8550, Japan e-mail: qian.z.aa@m.titech.ac.jp; zhenghua_qian@hotmail.com F. Jin MOE Key Laboratory for Strength and Vibration, Xi an Jiaotong University, 710049 Xi an, China 1 Introduction A half-space carrying on its surface a layer of another material is a typical structure for surface acoustic wave SAW devices 1. Transverse surface waves, which can exist in a piezoelectric half-space with/without a covered layer, are attractive for designing signal-processing devices due to their high performance and simple particle motion 2. One type of the transverse surface wave is the Lovetype wave which is always dispersive and has more than one mode. The Love wave has been extensively studied and used in SAW sensors, filters, delay lines and the like 3 7. Those who want to know detailed research work in this field can refer to the two newly published review papers 6,7 and the fruitful references therein. Another type of the transverse surface wave is the Bleustein Gulyaev B G wave, which exists and propagates only at the free surface of a piezoelectric half-space due to the interconnection between elastic and electric fields in piezoelectric materials 8,9. The B G wave is closely related to the bulk shear wave and its penetration depth up to 10 100 wavelengths greatly restricts its practical application mainly to microwave technology. Much earlier, Curtis and Redwood 10 carried out a theoretical study on the propagation of transverse surface waves in a piezoelectric material carrying a metal layer of finite thickness. Such a wave is related both to the B G wave and the Love wave, and hence would have great importance in practical applications. Recently, Wang et al. 11 investigated the existence of the Love-type transverse surface waves in a metal substrate carrying a piezoelectric material layer. In this paper, we study the existence of a particular mode of surface wave propagation, also called generalized B G waves, in the piezoelectric coupled solid media similar to 11 under the situation when the bulk shear wave velocity in the substrate is less than that in the piezoelectric layer but greater than the corresponding
418 Z. Qian et al. B G wave velocity of the same piezoelectric material with an electroded surface. The mathematical formulation is given in Sect. 2. A surface wave solution is obtained in Sect. 3, followed by discussions and numerical results in Sect. 4. Some conclusions are drawn in Sect. 5. 2 Statement of the problem Consider a half-space occupying x 1 > 0 with a piezoelectric ceramic layer of uniform thickness h see Fig. 1. Here the piezoelectric material, taken to be of class 6 mm or m,is poled in the x 3 direction of Cartesian coordinates x 1, x 2, x 3. There is an ideal electrode at the interface that is grounded. On the electrode, the electric potential has to vanish. The half-space is of either a metal or an isotropic nonpiezoelectric dielectric. It is assumed that the waves propagate in the positive direction of the x 2 -axis, such that the movement denoted by mechanical displacement u 1, u 2, u 3 and electric potential ϕ can be written as u 1 u 2 0, u 3 = u 3 x 1, x 2, t, ϕ = ϕx 1, x 2, t. 1 Let u and ϕ denote separately the mechanical displacement and electric potential function of the piezoelectric layer. Following Bleustein 8, the governing equations are 2 u 1/c 2 p ü = 0, 2 ϕ e 15 / u = 0, 2 where 2 is the two-dimensional Laplacian, and c p =c + e 2 15 //ρ 1/2 is the bulk shear wave velocity in the piezoelectric material, with c, e 15, and ρ representing the elastic, piezoelectric, dielectric constants and mass density, respectively. For the metal substrate, let u denote its mechanical displacement in the x 3 direction. The governing equation is 10 2 u 1/c 2 m ü = 0, 3 where c m = c /ρ 1/2 is the bulk-shear-wave velocity in the metal material, with c and ρ representing separately the shear modulus and mass density. The wave propagation problem specified by Eqs. 1 and 2 should satisfy the following boundary and continuity conditions: 1. σ 31 = 0atx 1 = h;2.u = u,σ 31 = σ 31,ϕ = 0 at x 1 = 0; 3. u 0asx 1 +. The electrical conditions at the free surface can be classified into two categories, i.e. 4. electrically open circuit: D 1 = 0atx 1 = h and 5. electrically short circuit or metalized surface: ϕ = 0atx 1 = 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. 3 Surface wave solution On the basis of an earlier work 10,11, we consider the following transverse waves satisfying radiation condition 3: ux 1, x 2, t = A 1 e bkx 1 + A 2 e bkx 1 expikx 2 ct, ϕx 1, x 2, t = A 3 e kx 1 + A 4 e kx 1 + e 15 A 1 e bkx 1 + A 2 e bkx 1 expikx 2 ct, h x 1 0, 4 u = A 5 e b kx 1 expikx 2 ct, x 1 0, 5 where A 1, A 2, A 3, A 4 and A 5 are arbitrary constants, k= 2π/λ is the wave number, λ is the wavelength, i = 1, and c is the phase velocity. Equations 3 and 4 satisfy separately Eqs. 1 and 2 if: b 2 = 1 c 2 /cp 2, b = 1 c 2 /cm 2. 6 The stress component and electric displacement needed for the boundary and continuity conditions are: σ 31 x 1, x 2, t = c bk A 1 e bkx 1 + A 2 e bkx 1 +e 15 k A 3 e kx 1 + A 4 e kx 1 expikx 2 ct, D 1 x 1, x 2, t = k A 3 e kx 1 + A 4 e kx 1 expikx 2 ct, h x 1 0, 7 σ 31 x 1, x 2, t = c kb A 5 e kb x 1 expikx 2 ct, x 1 0. 8 3.1 Electrically open case Fig. 1 A metal/dielectric half-space covered by a piezoelectric layer of uniform thickness Substitution of Eqs. 3 and 4 and the corresponding mechanical and electrical components into the remaining boundary and continuity conditions 1, 2 and 4 or5 yields the following homogeneous linear algebraic equations for coefficients A 1, A 2, A 3, A 4 and A 5
A novel type of transverse surface wave propagating in a layered structure consisting of a piezoelectric layer 419 A 1 + A 2 = A 5, A 3 + A 4 + A 1 + A 2 e 15 / = 0, c ba 2 A 1 + e 15 A 4 A 3 = c b A 5, c b A 2 e khb A 1 e khb + e 15 A 4 e kh A 3 e kh = 0, A 3 e kh + A 4 e kh = 0. 9 The existence condition of nontrivial solutions of these coefficients leads to the following dispersion relations of the transverse surface waves described by Eqs. 4 and 5 for the electrically open case k 2 p tanhkh b tanhbkh b / c = 0, 10 where k 2 p = e2 15 / c is the piezoelectric coupling factor in the piezoelectric layer with c = c + e 2 15 / being the piezoelectrically stiffened elastic constant 10. 3.2 Electrically short case For the electrically shorted case the free surface of the piezoelectric layer is plated with an infinite thin metal strip, the last equation in Eq. 9 should be replaced by the following expression which corresponds to the condition 5 in Sect. 2 A 3 e kh + A 4 e kh + A 1 e khb + A 2 e khb e 15 / = 0. 11 Then, a set of homogeneous linear algebraic equations with respect to A 1, A 2, A 3, A 4, and A 5 can be obtained. By the same procedure as that in the electrically open case, we can obtain the corresponding phase velocity equation for the electrically shorted case k 4 p + b2 tanhkh tanhbkh kp 2 c b tanhbkh + c bb tanhkh coshkh coshbkh 1 = 0. 12 It is readily seen from Eq. 10 and Eq. 12 that the phase velocity c is related to the wave number, layer thickness, elastic, dielectric and piezoelectric constants. With Eq. 6, Eqs. 10 and 12 determine c versus k or ω versus k. 4 Discussion and numerical results 4.1 Dispersion behavior For convenience, we introduce the dimensionless wave number H = h/λ. UsingEq.6, we can rewrite Eqs. 10 and 12 separately as: kp 2 tanh2π H b tanhb2π H b / c = 0, kp 4 + b2 tanh2π H tanhb2π H 13a kp 2 c b tanhb2π H + c bb tanh2π H cosh2π H coshb2π H 1 = 0. 13b We make the following observations from Eq. 13: It can be seen from Eq. 6 that the parameter b in Eq. 13 can not only take real values but also imaginary values, depending on whether the surface wave velocity c is smaller or greater than the bulk-shear-wave velocity in the piezoelectric layer, c p. 4.1.1 Love-type wave When the bulk shear wave velocity in the piezoelectric layer is less than that in the substrate, i.e., c p < c m, multivalued roots for the phase velocity c versus the dimensionless wave number H are expected from Eq. 13 if only if the parameter b is imaginary, denoted by b = ib 1 = i c 2 /cp 2 1. The dispersion Eq. 13 can thus be rewritten as kp 2 tanh2π H + b 1 tanb 1 2π H c b / c = 0, kp 4 b2 1 tanh2π H tanb 1 2π H 14a kp 2 c b tanb 1 2π H + c b 1 b tanh2π H 1 cosh2π H cosb 1 2π H 1 = 0, 14b in which Eq. 14b corresponds to the result of Ref. 11. The dispersion relation given in Ref. 11 is in determinantal form which is inconvenient to use in practice, so the above dispersion relation 14b still appears new, from which continuous dispersion curves rather than some discrete points can be easily calculated. A comparison between our result and the result of Ref. 11 is shown in Fig. 2, where very nice agreement is observed. This comparison can be regarded to some extent as a test of the analytical results obtained in this paper. If we further set k p = 0, i.e. no piezoelectricity in the layer, Eq. 14 reduce to the same frequency equation for Love waves in an elastic half-space carrying an elastic layer 12. 4.1.2 B G-type wave When the parameter b is real, denoted by b = b 2 = 1 c 2 /cp 2, roots for the phase velocity c versus the dimensionless wave number H are only expected when the bulk
420 Z. Qian et al. Fig. 2 Comparison of dispersion curves for a steel substrate carrying a PZT-4 layer c m = 3,281 m s 1, c p = 2,597 m s 1, c BG = 2,258 m s 1. a The result of Ref. 11. b The result calculated from Eq. 14b in this paper shear wave velocity in the piezoelectric layer c p and that in the substrate c m satisfy c p > c m > c BG. Here, c BG = c p 1 k 4 p 1/2 is the phase velocity of the B G wave in the same piezoelectric material as the layer coated with an infinitely thin layer of conducting material. The dispersion Eqs. 13 can thus be rewritten as kp 2 tanh2π H b 2 tanhb 2 2π H c b / c = 0, 15a kp 4 + b2 2 tanh2π H tanhb 2 2π H kp 2 c b tanhb 2 2π H + c b 2 b tanh2π H 2 cosh2π H coshb 2 2π H 1 = 0, 15b from which the roots obtained are neither single-valued nor multivalued. Under electrically open circuit condition for which Eq. 15a is responsible, only one mode of wave propagation is expected. While under electrically short circuit condition for which Eq. 15b is responsible, one additional mode appears besides the mode corresponding to that in the case of electrically open circuit. This new created mode seems due to the appearance of electrode on the free surface of the piezoelectric layer. If we further set k p = 0, i.e. no piezoelectricity in the layer, no roots exist from Eq. 15, which means that the modes of wave propagation under this situation do not exist in the absence of piezoelectricity. The phenomenon is illustrated by the numerical example shown in Fig. 3 which refers to a PZT-4 deposited on a zinc substrate. The material parameters for both the metals and the piezoelectric material are taken from Refs. 10,11. In order to make a deep insight into the dispersion characteristics of these modes, the plots of group velocity are included correspondingly in Fig. 3, denoted by thinner line. In the case of electrically open circuit, the former part of the mode when Fig. 3 Phase velocity c and group velocity c g of the B G type wave in a zinc substrate carrying a PZT-4 layer plotted as a function of H = h/λ for electrically open and short cases; c m = 2,0 m s 1, c p = 2,597 m s 1, c BG = 2,258 m s 1 H < 0.1901 corresponds to normal dispersion 12, while the latter part when H > 0.1901 anomalous dispersion 12. On the other hand, in the case of electrically short circuit, it can be seen from the group velocity plots in Fig. 3 that the first mode is a combination of partly normal dispersion 12 and partly anomalous dispersion 12 in which the phase velocity first decreases from the substrate velocity c m 2,0 m s 1 to a minimum value at H = 0.3901, then increases monotonically to the B G wave velocity 2,412 m s 1, whilst the second mode is totally normal dispersion in which the phase velocity decreases monotonically from the substrate velocity c m 2,0 m s 1 to the same final value of 2,412 m s 1 as that in the case of electrically open circuit. 4.2 Mode shapes in piezoelectric layer The mode shapes for displacement and electric potential in the thickness direction of the piezoelectric layer may be
A novel type of transverse surface wave propagating in a layered structure consisting of a piezoelectric layer 421 obtained from Eq. 4. For the electrically open case, we can obtain from Eq. 9 the unknown constants A 1 A 4 written in terms of A 5 as follows: 1 A 1 = 1 + e 2khb A 5, A 2 = e2khb A 1+e 2khb 5, A 3 = e 15/ 1 + e 2kh A 5, A 4 = e 15/ e 2kh 1 + e 2kh A 5. From Eq. 4, one can easily obtain e bkx 1 + e bkx 1+2khb ux 1, x 2, t = A 5 1 + e 2khb expikx 2 ct, e bkx 1 +e bkx 1+2khb ϕx 1, x 2, t= A 5 1+e 2khb e kx1 +e kx1+2kh 1+e 2kh e 15 expikx 2 ct 16 for the electrically open case. Similarly, we have the following solutions of the displacement and electric potential for the electrically short case ux 1, x 2, t = A 5 β 1 e bkx 1 + β 2 e bkx 1 expikx 2 ct, ϕx 1, x 2, t = A 5 β 3 e kx 1 + β 4 e kx 1 with + e 15 β 1 e bkx 1 +β 2 e bkx 1 expikx 2 ct, 17 discussion made in the subsect. 4.1, it is known that the impressive part of the B G-type wave is in the low-frequency zone. Hence, from the viewpoint of the practical application of surface acoustic wave SAW devices, we discuss the effect of dimensionless wave number on the mode shapes of long waves in the piezoelectric layer. The mode shapes of the displacement and the electric potential in the piezoelectric layer are presented in Figs. 4 and 5, respectively. Figures 4 and 5 show these mode shapes at dimensionless wave number H of 0.05, 0.10, 0.20 and 0.30 for both the electrically open case and the electrically short case. It can be seen that the mode shape for displacement displays almost uniform at small H, regardless of the electrically boundary condition at the free surface of the piezoelectric layer. As H increases, the distribution of the amplitude of the displacement begins to deviate from the almost uniform state at small H. The amplitude of the displacement at the free surface of the piezoelectric layer in the electrically open case decreases with the increase in H see Fig. 4a while that in the electrically short case increases with the increase in H see Fig. 4b, which means that the performance of SAW devices can be enhanced by coating an infinite thin metal film on the top of the piezoelectric layer. As can also be seen in Fig. 5, the distribution of the electric potential changes from almost uniform state at small H to one that is higher at the free surface at higher H in the electrically open case, whilst the mode shape for electric potential in the electrically short β 1 = β 2 = β 3 = β 4 = e c khb b + e2 15 e kh e15 2 + c b + c b 2 c b coshkhb 2 e2 15 sinhkhb 2 c be kh, e c khb b e2 15 + e kh e15 2 c b + c b, 2 c b coshkhb 2 e2 15 sinhkhb 2 c be kh c b c b 2 e2 15 coshkhb+ c b e2 15 c 2 b2 + e4 15 ε 2 11 c 2 b2 c b e2 15 + e4 15 ε 2 11 sinhkhb+2 e2 15 c be kh, 2e 15 c b coshkhb e2 15 sinhkhb c be kh sinhkhb c bc b coshkhb 2e 15 c b coshkhb e2 15 sinhkhb c be kh. Numerical simulations were performed to illustrate the results of the mode shapes obtained above and presented in the following. Detailed discussions on the mode shapes of the Love-type wave can be found in Ref. 11. In this paper, we will focus on the novel mode of surface wave propagation, i.e. the B G-type wave. From the dispersive characteristics case, although remaining positive, begins to deviate from symmetry at higher H. The above phenomena can be explained as follows. As the dimensionless wave number H increases, the difference in the properties of the two media air and zinc in contact with the piezoelectric layer becomes important 11.
422 Z. Qian et al. Fig. 4 Displacement in a piezoelectric layer for selected dimensionless wave number. a Electrically open case. b Electrically short case Fig. 5 Electric potential in a piezoelectric layer for selected dimensionless wave number. a Electrically open case. b Electrically short case 5 Conclusion This paper presents the study of transverse surface wave propagation in a piezoelectric coupled system. The dispersion relations for the existence of such surface waves are given in a simple mathematical form. The presence of piezoelectricity permits the existence of a novel B G-type of surface wave propagation: one mode exists in the electrically open case while two modes in the electrically short case. The mode shapes for both displacement and electric potential are found to change from almost uniform state at small dimensionless wave number to become non-uniform as the dimensionless wave number increases. One possible way is found to enhance the performance of SAW devices based on such B G-type surface wave by coating an infinite thin metal on the top of the piezoelectric layer. The study extends the regime of transverse surface waves and may provide possibilities for potential applications in acoustic wave devices. Acknowledgments The authors gratefully acknowledge the support by the Global COE Program at Tokyo Institute of Technology. References 1. Jakoby, B., Vellekoop, M.J.: Properties of Love waves: applications in sensors. Smart Mater. Struct. 6, 668 679 1997 2. Yang, J.S.: Piezoelectric transformer structural modeling A review. IEEE Trans. UFFC 54, 1154 1170 2007 3. Wang, Z.K., Liu, H., Liu, Y.C. et al.: A peculiar acoustoelectric wave in piezoelectric layered structure.acta Mech.Sin.32, 25 33 in Chinese 2000 4. Wang, Z.K., Jin, F.: Influence of curvature on the propagation properties of Rayleigh waves on curved surfaces of arbitrary form. Acta Mech. Sin. 34, 895 903 in Chinese 2002 5. Yang, J.S.: Love waves in piezoelectromagnetic materials. Acta Mech. 168, 111 117 2004 6. Yang, J.S., Yang, Z.T.: Analytical and numerical modeling of resonant piezoelectric devices in China A review. Sci. Chin. Ser. G 51, 1775 1807 2008
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