MODELLING OF FGM PIEZOELECTRIC TRANSDUCERS USING GRADED FINITE ELEMENT CONCEPT

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1 ABCM Syposiu Series in Mechatronics - Vol. 3 - pp Copyright c 28 by ABCM MODLLING OF FGM PIZOLCTRIC TRANSDUCRS USING GRADD FINIT LMNT CONCPT Wilfredo Montealegre Rubio, wilfredo.rubio@poli.usp.br ilio Carlos Nelli Silva, ecnsilva@usp.br Flavio Buiochi, fbuiochi@usp.br Julio C. Adaowski, jcadaow@usp.br Departent of Mechatronics and Mechanical Systes ngineering, scola Politécnica da Universidade de São Paulo, Av. Prof. Mello Moraes, Cidade Universitária, São Paulo SP 558-9, Brazil Abstract. Piezoelectric aterials generate displaceents when an electric potential is applied and, electric potential when they are subjected to forces or pressure. Functionally Graded Materials (FGM) are coposite advanced aterials, which are ade by changing gradually the properties with position inside aterial doain. The application of FGM concept to piezoelectric transducer design allows designing coposite transducers without interface between aterials (e.g. PZT and Aluinu), due to the continuous change of property values. Thus, large iproveents can be achieved, as reduction of stress concentration, increasing bonding strength and fatigue-lifetie. Recent works about piezoelectric FGM show lack of coputational ethods to odel these transducers and evaluate their perforance considering property gradation function, futherore, F coercial softwares have no tools to siulate graded continuous aterials for while. Thus, this work proposes the developent of Finite leent (F) algoriths to odel FGM piezoelectric ceraics and to explore the FGM potential on piezoelectricity field. The continuous change of piezoelectric, dielectric, and elastic properties is achieved by using the graded finite eleent concept, where these aterial properties are interpolated inside the finite eleent using the F shape functions. A software based on 4-node graded finite eleent (Q4) is ipleented. Dynaic and static analyses are perfored. In the exaples, the aterial properties are graded along thickness direction to illustrate the influence of gradation on the output displaceents, vibration odes, and resonance frequencies. These exaples are copared with results of a hoogeneous piezoceraic to show FGM advantages. eywords: Graded Finite leent, Functionally Graded Material, Piezoelectric ceraic, Dynaic Perforance, Static Perforance 1. INTRODUCTION Piezoelectric aterials have the property to convert an electrical energy (electric field and electric potential) into a echanical energy (stress and strain) and vice-versa. xaples of piezoelectric aterials include quartz, ceraics (PZT) and polyers (PVDF). Its ain applications are in sensors and electroechanical actuators, as resonators in electronic equipent and acoustic applications, as ultrasound transducers, naval hydrophones, and sonars. Ultrasound transducers are used in edical iaging (Akhnak et al., 2) and non-destructive tests. Other applications include pressure sensors; piezoelectric actuators for the structural vibration control; perforance of nanopositioning and icroanipulation devices as: electronic icroscopy instruents; laser interferoetry; cell anipulation equipent; icroelectroechanical systes "MMS"; nanotechnology and precision echanics equipent (ögl and Silva, 25). A brief review of piezoelectric applications is offered by Newnha and Ruschau (Newan and Ruschau, 1991). In order to iprove the conventional piezoceraic perforance (single and hoogeneous aterials) are fabricated as piezocoposite aterials. However, the interface between aterials produces an uneven distribution of stresses which reduces the electric-field-induced displaceent characteristics, reliability and lifetie. Other odern approach is to change the piezoelectric properties of ceraic disk through its thickness, specifically, to reduce one echo wave of two produced in each piezoceraic surface and to increase the induced piezoelectric stress gradient. Functionally Graded Material concept has arisen as a solution to reduce the ultrasonic wave generated at one piezoceraic surface (Yaada et al., 1998; Ichinose et al., 24; Saadhiya and Mukherjee, 26). Functionally Graded Materials (FGM) are aterials that possess continuously graded properties with gradual change in icrostructure (Hirai, 1996; Suresh and Mortensen, 1998). The aterials are ade to take advantage of desirable features of its constituent phases. For instance, in a theral protection syste, FGMs take advantage of heat and corrosion resistance, typical of ceraics, and echanical strength and toughness, typical of etals. A soft property variation supplies advantages such as stress concentration reduction (Suresh and Mortensen, 1998), since they do not present interface aong inclusion and atrix aterials, therefore, it reduces a coon proble in coposite aterials, the crack arising or daages in these interfaces. Specifically, in Functionally Graded Piezoelectric (FGP) ceraics, the conventional and hoogeneous piezoelectric aterial is replaced by a functionally graded piezoelectric one, see Fig. 1. Therefore, all or soe properties vary along a specific Cartesian direction, usually along thickness. Several gradation functions can be used, see Fig. 1. Thus, if the piezoelectric properties change fro low to high values, only one

2 ultrasonic wave is radiating, and larger piezoelectric stress gradient together with short-tie wavefor are obtained (Yaada et al., 1998). Soe studies have been reported since 196s about the advantages of wave generation by FGP transducers (Mitchell and Redwood, 1969); nevertheless, these studies have been highly increased since last 199s (Wu et al., 1996). These works focused in two ways: FGP ceraic fabrication and odelling. FGP ultrasonic transducers can be constructed by foring a nuber of fine V-grooves on one surface of an active eleent (Yaada et al., 1998), or U-grooves (Guo et al., 25); by applying an appropriate teperature gradient in the thickness direction of a polarized piezoceraic with low Curie teperature (Yaada et al., 2); and by sintering a layer-structured ceraic green body without using any adhesive aterial (Ichinose et al., 24). On the other hand, FGP ultrasonic transducer can be odelled: (i) by using one-diensional analytical techniques, such as equivalent network analysis of piezoceraic disk, exploring linear and exponential gradation functions for thickness vibration odes (Yaada et al., 21; Yaada et al., 1999); (ii) by using two-diensional ultilayer nuerical techniques (Rubio, et al., 27). Without FGP Interface 1 Non- FGP ceraic Interface 2 Thickness Property With FGP Interface 1 FGP ceraic Interface 2 Thickness Property Figure 1. Sketch of a traditional (non-fgp) piezoelectric ceraic and of a FGP piezoceraic disk Although all works show that the effectiveness of piezoelectric property gradation generates better perforances, the odelling is reduced to one-diensional or ultilayer approaches. Forer approach neglect lateral vibrations of ceraic disk, and in last one, the gradation functions are not continuous; on the contrary, they are discrete functions. In view of this idea, this work contributes to developing a Graded Finite leent (GF) ipleentation for FGP ceraic disk odelling. This coputational ipleentation allows siulating two-diensional FGP disks with continuous aterial gradation. The code is ipleented by using the MATLAB software. It will be considered that the properties varies along the thickness direction, and the FGP results are copared with non-fgp ones. The paper is built up as follow: first, it is described the GF forulation of piezoelectric ceraic disks. Then soe results and testing of code are presented, and finally, soe conclusions are given. 2. GRADD FINIT LMNT FOR PIZOLCTRICITY The constitutive relationships describing the electrical and echanical interactions for piezoelectric aterial are given as (Naillon et al., 1983): T = C S e D = e S + ε S T (1) where, T is the stress tensor (second order); S is the strain tensor (second order); D is the electric displaceent vector; is the electric field vector; C is a fourth order elastic tensor and its coponents are evaluated by constant electric S field, ε is a second order dielectric tensor under constant strain, and e is a third order piezoelectric tensor where T indicates transpose. The piezoelectric odel is copleted by considering the echanical balance expressed by the Newton s equation for continuous edia; the electrical balance corresponds to the charge balance expressed by Gauss s theore; and the strain and electric field expressions. The echanical balance is given by: 2 U ρ = T (2) 2 t where, ρ is the density of aterial, t is the tie, U the displaceent vector, and the divergence operator. The electrical balance is given by: D =, (3) and respectively the strain and electric field expression by:

3 S = ˆ U and = ϕ (4) where, ϕ is the electric potential and ˆ is the syetric gradient operator expressed by (Naillon et al., 1983): 1 U U i j S = + ij (5) 2 x j xi The finite eleent piezoelectric equilibriu equations can be written based on variational principle by using the constitutive piezoelectric equations, q. (1) up to q. (5). These equations are written in ters of displaceent (U) and electric potential (Φ) vectors at nodal points. Also, the F equations are written by using the nodal echanical force (F) and electric charge (Q) vectors as (without structural daping) (Naillon et al., 1983): M U&& + Φ&& ΦΦ U F = Φ Q (6) where, M,,, and ΦΦ are the ass, elastic, piezoelectric and dielectric atrices. However, in the case of FGP ceraic disks the properties change continuously inside the piezoceraic doain, which eans that the atrices of q. (6) ust be described by soe continuous function of Cartesian position (x, y) into the ceraic disk. Thus, the atrices of q. (6) are expressed as: M N T u ( x, y) N dv = ρ (7) B ΦΦ T u ( y) B dv = u C x, u (8) T T ( y) B dv = Bu e x, Φ (9) T S ε ( y) B dv = BΦ x, Φ (1) where, N u are the shape functions for the displaceents; and B u and B Φ are the strain-displaceent and voltage-gradient atrices, respectively. According to the theory of conventional finite eleent, the atrices and vectors of piezoelectric constitutive equations result fro assebling the vectors and atrices of the single eleents. Property (constant) Property (X,Y) Node Y Node X Figure 2. Properties at the eleent level. Hoogeneous Finite leent (HF); Graded Finite leent (GF) To treat the hoogeneous aterial siulations, it is used the traditional Hoogeneous Finite leent (HF) with constant aterial properties at the eleent level, which are evaluated at the centroid of each eleent; see Fig. 2. On contrary, for FGP ceraics, the Graded Finite leent (GF) is ipleented, which incorporate the aterial property gradient at the size scale of the eleent (Fig. 2). i and Paulino (22) and Santare and Labros (2) developed the graded eleent concept with slightly different forulations. Both studies deonstrated that graded eleents result in sooth and accurate change of properties for static probles. In this work, the schee developed by i and Paulino (22) is extended to piezoelectric aterials considering dynaic and static analysis. In FGP ceraics, the GF concept eploys the sae shape functions to interpolate the unknown displaceents and electric potential, the geoetry, and the aterial paraeters. The interpolations for spatial coordinates (x, y), nodal displaceents (u), nodal electric potential (ϕ) and aterial properties (ρ, C, e, ε S ) are given by:

4 - Spatial coordinates: x = N i x i, = i= 1 y N i y i (11) i= 1 - Displaceents and electric potentials: u = N i u i, = i= 1 ϕ N ϕ (12) i= 1 - lastic, Piezoelectric and Dielectric properties: i i C ijkl = n = 1 N n ( C ) ijkl n eikl n ikl n, = ( ) S S ik N n ε ik, = N ( e ) n = 1 ε for i, j, k, l = 1, 2, 3 (13) n = 1 n respectively, where is the nuber of nodes per finite eleent. Also, when GF is ipleented, the aterial properties ust reain inside the atrices integrals (see q. (7) up to q. (1)) and ust be integrated; in Hoogeneous Finite leents (HF), these properties are not integrated. 3. IMPLMNTATION The GF for piezoelectricity is ipleented by using MATLAB code. Two-diensional four-node quadrilateral finite eleents (Q4) for FGP ceraic disks are used in this work, each one with three degrees of freedo: two echanics (horizontal and vertical displaceents), and one electric (electrical potential). Thus, a fully isoparaetric forulation is developed in the sense that the sae shape functions are applied to interpolate the unknown displaceents and electric potentials, the geoetry, and the aterial properties. Therefore, the actual variation of the aterial properties ay be approxiated by the eleent interpolation functions (e.g., a certain degree of polynoial functions). Three analyses are ipleented: Static, Modal, and Haronic. In static analysis the software solves the following equation syste: ΦΦ U F = Φ Q (14) In odal and haronic analyses no daping is considered in the dynaic proble. In odal analysis the eigenvalues and eigenvectors are found solving the second-order systes: ω 2 M U&& + Φ && ΦΦ U = Φ (15) where, ω is the natural frequency. On the other hand, haronic response analysis seeks the syste response when prescribed loads vary sinusoidally with tie. Because MATLAB is able to anipulate coplex nubers, the haronic response calculation is based on direct ethod, in this case the haronic response equation can be written in the for: Ω 2 M U&& ˆ + ˆ Φ&& ΦΦ Uˆ Fˆ = Φˆ Qˆ (16) where, Û and Φˆ are coplex agnitudes, respectively, of displaceents and electric potentials. Fˆ and Qˆ are coplex agnitudes of echanical and electrical inputs, respectively. All these coplex agnitudes depend on frequency Ω. 4. NUMRICAL RSULTS In this section soe results based on GF ipleented with MATLAB are shown. Three analyses are perfored: Static, Modal and Haronic. When hoogeneous piezoceraic is siulated only one aterial is used, in this

5 hoogeneous case is used the PZT-5A properties. On contrary, when FGP ceraics are siulated a gradation function is used to represent the continuous change of aterial properties along thickness. The properties of PZT-5A are shown in Table 1. Table 1. Material properties for PZT-5A ceraic. lastic Constants (N/ 2 ) Piezoelectric Constants (C/ 2 ) Dielectric Constants c 11 = 12.1 x 1 1 c 13 = 7.52 x 1 1 e 13 = -5.4 ε = x 1-12 F/ c 33 = 11.1 x 1 1 e 13 = 15.8 ε 11 = 916 x ε c 44 = 2.11 x 1 1 e 15 = 12.3 ε 33 = 83 x ε V 2 FGP ceraic 5 V 2 A FGP ceraic 5 Figure 3. Model used for nuerical results. Model for static analysis; Model for odal and haronic analyses In Table 1 ε is the perittivity of free space. The density ρ of PZT 5A aterial is equal to 75 kg/ 3. For FGP ceraic disks the following exponential gradation functions for elastic, piezoelectric, and dielectric properties are considered (assuing aterial properties vary in y Cartesian direction, see Fig. 3) for a 2D proble: c 11 βy γy S 11 αy C ( y) = c c e ; e ( y) = e e ; ( y) e 13 c c 44 e 15 e13 33 ε ε = ε (17) 33 where, β, γ, α are the aterial gradation paraeter elastic, piezoelectric and dielectric properties, respectively. In this work, it is assued β, γ, and α equal to 85; 322; and 16, respectively. Density is assued constant along thickness. For all nuerical results the siulated odels are sketched in Fig. 3, which represents a two-diensional piezoceraic disk geoetry subjected to electric potential between top and botto disk surfaces. For static analysis the piezoceraic is fixed on both lateral sides (see, Fig 3), and for odal and haronic analyses the piezoceraic is siulated considering boundary conditions for free response constraining the lateral vibrations, thus the odel have only horizontal echanical constraints in both lateral sides of piezoceraic, see Fig Static Results Initially, a static analysis was developed. An input electric potential equal to 1 V is applied. The σ xx, σ zz, and σ xz stresses, z electric field, u y displaceents, and φ electric potential values, for FGP and non-fgp ipleentations, are calculated. Here, only the vertical displaceents, stresses σ xx, σ zz, and σ xz, and electric potential are shown in Fig. 4, 5, 6, 7, and 8, respectively. Also, the ANSYS response when is used a hoogeneous aterial is shown. The piezoceraic siulated is sketched in Fig. 3, with boundary conditions and diensions. For all exaples the doain is discretized with 3 x 3 four-node bilinear (Q4) isoparaetric finite eleents, considering both GF and HF, and a 3 x 3 Gauss quadrature is eployed. Fro Fig. 4 up to 8 is observed that FGP odels produce siilar displaceent, stress, and electric potential distribution that the non-fgp odels, except for stress σ xx. At the sae tie, these FGP ceraics have saller stress agnitude; these agnitudes decrease 37.8 %, 29.4 % and 5 % for the axiu stress σ xx, σ zz, and σ xz, respectively. It is an advantage, because high stress levels accelerate the aging process of piezoelectric aterials and they reduce the lifetie of piezoceraic. However, in non-fgp ceraic disk higher displaceents are present when an input voltage is applied (although differences with FGP aterial are sall). Also, it is observed in Fig. 4 up to 8 that hoogenous responses (non-fgp ceraic siulated with Matlab) are closed to ANSYS results; however, the Matlab and Ansys stress results show soe differences, this can be caused due to different stress calculation procedure. Ansys use the four-nodal inforation to interpolate the stress inside of finite eleent. The code ipleented in this work calculates the stress in iddle of finite eleent.

6 Figure 4. Vertical Displaceent (). FGP odel with Matlab; Non-FGP odel with Matlab; (c) Non-FGP odel with Ansys (c) Figure 5. Stress σ xx (Pa). FGP odel with Matlab; Non-FGP odel with Matlab; (c) Non-FGP odel with Ansys (c) Figure 6. Stress σ zz (Pa). FGP odel with Matlab; Non-FGP odel with Matlab; (c) Non-FGP odel with Ansys (c)

7 Figure 7. Stress σ xz (Pa). FGP odel with Matlab; Non-FGP odel with Matlab; (c) Non-FGP odel with Ansys (c) Figure 8. lectric potential (V). FGP odel with Matlab; Non-FGP odel with Matlab; (c) Non-FGP odel with Ansys 4.2 Modal Results Modal analysis of this piezoceraic involves two cases which spans the extrees of the piezoelectric coupling effect due to voltage and displaceent degrees of freedo. The first case is coonly called the "resonance" condition. A constant voltage equal to zero is applied to the electrical contacts (electrodes) of ceraic disk. This is a "short-circuit" condition, where all voltage potentials are connected to ground. The second case, called "anti-resonance", applies a zero voltage only to one electrode. In this odal analysis, only vibration odes along thickness are calculated, according to odel of Fig. 3. Table 2 shows a coparison of the natural frequencies when hoogeneous (ANSYS and MATLAB responses) and FGP ceraics are used. The vibration odes are calculated for resonance and anti-resonance conditions. Thus, when resonance and anti-resonance frequencies are equal, it is a echanical ode and, when resonance and anti-resonance frequencies are different, it is a piezoelectric ode, which is our ode of interest. On contrary, a echanical ode is identified when resonance and anti-resonance frequencies are equal. The eigenvector plots for first and second piezoelectric odes are shown in Fig. 9 to Fig. 1, respectively. Table 2. Natural frequencies for different aterial distribution probles Mode FRQUNCIS (MHz) FGP odel with Matlab Non-FGP odel with Matlab Non-FGP odel with Ansys First Piezoelectric ode Second Piezoelectric ode (c)

8 (c) Figure 9. First piezoelectric ode. FGP odel with Matlab; Non-FGP odel with Matlab; (c) Non-FGP odel with Ansys (c) Figure 1. Second piezoelectric ode. FGP odel with Matlab; Non-FGP odel with Matlab; (c) Non-FGP odel with Ansys 4.3 Haronic Results The sae 2D odel used in odal analysis is siulated by using a haronic analysis. The input for this siulation was voltage aplitude iposed across the piezoelectric ceraic disk. The input varied sinusoidally between +/-1V. Figure 11. Displaceent frequency response. FGP odel with Matlab; Non-FGP odel with Matlab The haronic analysis was perfored over a frequency range of -3.5 MHz in equal steps of 3.5 khz. At each frequency, the ipleented software coputes the steady-state response of the syste subjected to a sinusoidally varying input on the FGP or non-fgp ceraic disk. No daping ratio was assued over all frequencies of the

9 haronic analysis sweep. The result of particular interest in this solution was the Y (vertical) displaceent of the transducer over the frequency sweep, easured at point A in Fig. 3 (the top-iddle of ceraic disk). Figure 11 shows the noralized-frequency Y displaceent (focused on the thickness vibration odes) for the unifor piezoelectric ceraic (non-fgp) disk and for the FGP odel. The resonant responses for the odd order vibration ode appear in the non-fgm. It is noted, however, that both even and odd order vibration odes appear in FGP ceraics, also see Table 2. This result is in accordance with Yaada s results (Yaada et al., 21). They develop an analytical one-diensional odel for FGP ceraics, and found even and odd vibration odes in the adittance response of FGP plates. 5. SUMMARY AND CONCLUSIONS The odelling of FGP ultrasonic transducers was considerably successful, based on Graded Finite leent odelling. Also, this work provides an investigation of graded piezoceraic disk responses considering static, odal and haronic analyses. These responses were copared to hoogeneous one (non-fgp properties). Generalized isoparaetric forulation is eployed in F ethod to investigate the response of gradation aterial in piezoceraic disks. This forulation adopts the sae interpolation ethods of the coordinates and displaceents to treat continuous change of aterial at the eleent level. This approach results in sooth solution transition across the eleent boundaries. The following conclusions can be drawn fro these studies: The GF odelling is an accurate technique, consisting of useful tools for designing FGP transducers, thus this work fulfill the lack of coputational ethods, in coercial softwares, to evaluate the perforance of piezoelectric disk when aterial inhoogeneity at the eleent level is considered. This approach represents a ore accurate technique instead of ulti-layer approach to odel the gradation behavior. With graded piezoceraic disks lower stress levels and ore resonance frequencies can be obtained; in other words, even and odd vibration odes are achieved; with non-graded piezoceraics only odd odes are obtained. Thus, large iproveents can be achieved in their perforance characteristics by using FGP concept. In this work, only an exponential gradation function is used to siulate the piezoceraic properties; nevertheless, other gradation functions can be used; however, which gradation function produces better perforance? This suggests the use of optiization techniques to design graded piezoceraic disks. Based on these ideas, in a future work, it is proposed the developent of optiization algoriths to find this optiized gradation function. 6. ACNOWLDGMNTS This study was supported by FAPSP São Paulo State Foundation Research Agency, through a doctoral fellowship (Nº. 25/1762-5), a CNPq research project (Nº /24 4), a FAPSP research project (Nº. 6/5785-7), and a FINP research project (Nº ). 7. RFRNCS Akhnak, M., Martínez, O., Goez-Ullate, L., and Montero de spinosa, F. R., 2. Piezoelectric Sectorial 2D Array for 3D Acoustical Iaging, Sensors and Actuators, Vol. 85, pp Guo, H., Cannata, J., Zhou, Q., Shung,., 25. Design and fabrication of broadband graded ultrasonic transducers with rectangular kerfs, I Transaction on Ultrasonics, Ferroelectrics, and frequency Control, Vol. 52, No. 11, pp Ichinose, N., Miyaoto, N., Takahashi, S., 24. Ultrasonic transducers with functionally graded piezoelectric ceraics, Journal of the uropean Ceraic Society, Vol. 24, pp Hirai, T Functionally Graded Materials, Materials Science and Technology: Processing of Ceraics, Part 2, R.J. Brook, ed., VCH Verlagsgesell-schaft bh, Weinhei, Gerany, Vol. 17B, pp i, J.., Paulino, G. H., 22. Isoparaetric Graded Finite leents for Nonhoogeneous Isotropic and Orthotropic Materials, ASM Journal of Applied Mechanics, Vol.69, No. 4, pp ögl, M., Silva,. C. N., 25. Topology Optiization of Sart Structures: Design of Piezoelectric Plate and Shell Actuators, Sart Materials and Structures Journal, Vol.14, pp Mitchell, R. F., Redwood, M., The generation of sound by non-unifor piezoelectric aterials, Ultrasonics, Vol. 7, No. 2, pp Naillon, M., Coursant, R., Besnier, F., Analysis of Piezoelectric Structures by a Finite leent Method, Acta lectronica, Vol. 25, No. 4, pp

10 Newan, R.. and Ruschau, G. R., Sart lectroceraics, Journal of the Aerican Ceraic Society, Vol. 74, No. 3, pp Rubio, W. M., Buiochi, F., Adaowski, J. C., Silva,. C. N., 27. Modeling of functionally graded piezoelectric ultrasonic transducers, Proceedings of International Congresses on Ultrasonic - ICU 27, Vienna Austria, (in press). Saadhiya, R, Mukherjee, A., 26. Functionally graded piezoceraic ultrasonic transducers, Sart Materials & Structures, Vol. 15, No. 6, pp Santare, M. H., Labros, J., 2. Use of a graded finite eleent to odel the behavior of nonhoogeneous aterials, Journal of Applied Mechanics, Vol. 67, pp Suresh S. and Mortensen, A., Fundaentals of Functionally Graded Materials, IOM Counications, London. Wu, C. C. M., ahn, M., Moy, W., Piezoelectric ceraics with functional gradients: a new application in aterial design, Journal of Aerican Ceraic Society, Vol. 79, Yaada,., Sakaura, J-I., Nakaura,., Broadband ultrasound transducers using effectively graded piezoelectric aterials, I Ultrasonics Syposiu, pp Yaada,., Sakaura, J-I., Nakaura,., quivalent network representation for thickness vibration odes in piezoelectric plates with an exponentially graded paraeter, Japanese Journal of Applied Physics, Vol. 39, Part 2, No. 1A/B, pp Yaada,., Yaazaki, D., Nakaura,., 2. Broadband ultrasound transducers using a plate with graded piezoelectric constant fored by an internal teperature gradient, I Ultrasonics Syposiu, pp Yaada,., Sakaura, J-I., Nakaura,., 21. quivalent network representation for thickness vibration odes in piezoelectric plates with a linearly graded paraeter, I Transaction on Ultrasonics, Ferroelectrics, and frequency Control, Vol. 48, No. 2, pp RSPONSIBILITY NOTIC The authors are the only responsible for the printed aterial included in this paper.