The influence of piezoelectricity on free and reflected waves from fluid-loaded anisotropic plates

The influence of piezoelectricity on free and reflected waves from fluid-loaded anisotropic plates

Adnan H. Nayfeh and HuaI-Te Chien Department of Aerosœace Engineering and Engineering of Mechanics, University of Cincinnati, CincinnatL Ohio 45221-0070

(Received 18 April 1991; accepted for publication 16 August 1991 )

A unified analytical treatment is presented supported by extensive numerical illustrations of the interactions of ultrasonic waves with piezoelectric anisotropic plates. The plates are allowed to possess up to monoclinic anisotropic symmetry and associated piezoelectric coupling. The plates are also assumed to be immersed in water and subjected to incident acoustic beams at arbitrary polar and azimuthal angles. Simple analytical expressions for the reflection and transmission coefficients are derived, from which all propagation characteristics are identified. Such expressions contain, as a by-product, the secular equation for the propagation of free harmonic waves on the piezoelectric plates. This equation is written in simple and completely separate terms pertaining to symmetric and antisymmetric modes. It is found that piezoelectric coupling, as well as water, influence both types of modes. Higher symmetry, such as orthotropic, transverse isotropic, and cubic, are contained implicitly in this analysis. It is also demonstrated that the motion of the sagittal (Lamb) and SH modes uncouple for propagation along axis of symmetry. For such cases, however, piezoelectric coupling can influence one of these types of modes depending upon the type of piezoelectric model adopted. ,

PACS numbers: 43.20.Bi, 43.20. Fn, 43.20.Hq, 43.88.Fx

INTRODUCTION

In two recent papers, 1'2 Nayfeh and Chimenti presented exact solutions for the interaction of elastic waves with an-

isotropic plates. The technique used was shown to formally handle the most general anisotropy (i.e., the triclinic) case. However, much simpler results were obtained for the slightly more specialized monoclinic symmetry case. In Ref. 1, the plate was immersed in liquid, and incident waves impinged upon it from the liquid at arbitrary polar and azimuthal angles. Simple exact analytical expressions were derived for the reflection and transmission coefficients from which important propagation characteristics are identified.

In Ref. 2, the analysis of the propagation of free waves in a general anisotropic plate were developed. The results were then specialized for monoclinic symmetry. Here, closed- form secular equations that isolate the mathematical conditions for symmetric and antisymmetric wave mode propagation in completely separate terms were obtained. In both Refs. 1 and 2, material systems of higher symmetry, such as orthotropic, transverse isotropic, cubic, and isotropic are contained implicitly in the analysis.

The ease in which the analysis were carried out and the compact expressions of the final results were facilitated by the use of linear transformation. This use of transformation

is motivated by the important observation that the wave vec- tors of the incident and reflected waves all lie on the same

plane. The techniques utilized in Refs. 1 and 2 have also been

successfully employed to develop solutions for a variety of multilayered anisotropic media. 3-6 The generic difficulties of the mathematical analysis of waves in anisotropic media as compared with those pertaining to isotropic media are fully discussed (including extensive literature review) in Refs. 1-6 and need not be elaborated upon here.

The aim of this paper is to extend the analysis of Refs. 1 and 2 in order to include linear piezoelectric effects. We at- tempt this while being aware of the extensive literature deal- ing with the general interactions between piezoelectric and acoustic fields and their broad applications in various fields, such as signal processing, electronics, instrumentations, geo- physics, and nondestructive testing. For general discussions of piezoelectric effects, we refer the reader to the standard books of Auld, 7 Cady, 8 and Tiersten. 9

In the following, we shall limit ourselves, however, to a review of literature pertaining to piezoelectric guided waves that are relevant to the present plate problem. In particular, we mention that general discussions and review of the surface waves and the SAW devices are given by Farnell and Viktorov. 10-12 Bleustein •3,14 developed an "electroacoustic" surface wave, known as "Bleustein-Gulyaev wave" (B-G wave), which can be generated and detected on some classes of the piezoelectric materials. Gazis and his coworkers 15 studied generalized Rayleigh waves in cubic piezoelectric crystals. Works on free waves in piezoelectric plates is re- viewed by Auld, 7 Tiersten, 9 and Wagers. 16 Nassar and Ad- ler 17 studied plate modes in piezoelectric composite mem-

1250 J. Acoust. Soc. Am. 91 (3), March 1992 0001-4966/92/031250-12500.80 ¸ 1992 Acoustical Society of America 1250

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branes and calculated the electromechanical coupling. Much earlier in 1916, Langevin successfully applied the piezoelectric effect on their sonar emitter, which was com- posed of a piezoelectric plate.

It is worth noting that none of the above literature in- cluded the effects of fluid loading which constitute the car- rier for the incident, transmitted, and reflected waves. The fluid-loading setup is appropriated for immersion techniques that are popular in nondestructive testing.

In what follows, we parallel the procedure of Nayfeh and ChimentiS'2 and study the ultrasonic wave interactions with a fluid-loaded anisotropic piezoelectric plate. Simple analytical expressions for the reflection and transmission coefficients are derived for the slightly symmetric monoclinic materials from which all propagation characteristic are identified. Such expressions contain, as a by-product, the secular equation for the propagation of free harmonic waves on the piezoelectric plate. This equation is written in completely separate terms pertaining to symmetric and antisymmetric modes. Thus it is found that piezoelectric coupling, as well as the fluid, influence both types of modes. Higher symmetry materials, such as orthotropic, transverse isotropic, and cubic, are contained implicitly in our analysis. For orthotropic and higher symmetry materials where two principal axes lie in the plane of the plate, the particle motions for Lamb and SH modes uncouple if propagation occurs along either of these in-plane axes of symmetry. For such cases, however, piezoelectric coupling can influence one of these types of modes depending upon what types of piezoelectric mode is adopted. This gives a rather interesting and unique property associated with piezoelectric coupling.

I. THEORETICAL DEVELOPMENT

Consider an infinite, generally anisotropic, piezoelectric plate having the thickness d and oriented such that its crysta- 10graphical axes are oriented along the reference Cartesian coordinate system xi = (x• ,x2,x3 ). The plane x•-x2 is chosen to coincide with the middle surface of the plate, and the x3 coordinate to be normal to it, as illustrated in Fig. 1. With respect to this coordinate system, the coupled piezoelectric field equations are given by the equations of motion and elec- trostatic charge (see for example Tiersten 9 )

•2Uk •2• •2Ui 8x t 8x i •x • 8xl 8t 2 '

(la)

(lb)

and the associated constitutive relations

To' = Cø'•Sk• q- ekø' •x• (2a)

ao D • = e •i;Si;

8xi (2b)

1 CRYSTAL AXIS

X 3

FIG. 1. Coordinate systems for leaky plate wave problem. Incident beam strikes plate at angle 0 between principle axes of material crystal and plane of incidence, and 0 is the azimuthal angle.

So -- -•- \Sx• + •xi ] ' (3a) Ek = 3q• (3b)

where T o is the mechanical stress, S 0 is the strain, C0• • is the elastic stiffness constant, e•0 is the piezoelectric stress constant, ,• is the dielectric permittivity, u i is the mechanical displacement, E• is the electric field, & is the scalar potential, D• is the electric displacement, and p is the material density, respectively.

In order to derive the reflection and transmission coeffi-

cients, the field equations for the fluid must also be given in a manner similar to those of the solid. In our problem, we shall assume that the fluid (water) does not support piezoelectric effects and, hence, its electric potential is zero. Accordingly, there will be no change in the fiuid's field equations or their formal solutions from those given in our earlier work. • For this reason, we shall only quote such material from Re[ 1 later on in the analysis.

II. ANALYSIS

Formally, we can proceed to analyze the most general anisotropic plate (the triclinic one) whose constitutive relations has 21 elastic constants, 9 dielectric permittivities, and 18 piezoelectric coupling constants. This can be done by di- rect extension of the elastic case treated in Ref. 2. However, the expressions will be algebraically complicated and their utility will be numerically limited as was also pointed out in Ref. 2. For the slightly more symmetric materials, i.e., the monoclinic ones, dramatic simplifications in the final expressions can be achieved. For this reason, we shall limit the following analysis to piezoelectric monoclinic materials.

Piezoelectric anisotropic plates with one plane of material symmetry are termed monoclinic piezoelectric. Two classes of such materials exist: These belong to a mono-m or mono-2 groups whose constitutive relations are respectively shown in the expanded matrix forms:

1251 J. Acoust. Soc. Am., Vol. 91, No. 3, March 1992 A.H. Nayfeh and H. Chien: Influence of piezoelectricity on waves 1251


rl

and

rl

D• D•

--Cll CI2 CI3 0 0 C!2 C22 G3 0 0 C26 c,• G• G3 o o 0 0 0 C44 C4.s 0 0 0 0 C45 C•$5 0 C!6 G6 G6 0 0 C66

0 0 0 el4 e•5 0 0 0 0 e24 e25 0

.. e31 e32 e33 0 0 e36

-Cii Ci2 Ci3 0 0 Ci6 Ci2 C22 C23 0 0 C26 Ci3 C23 C33 0 0 C36 0 0 0 C44 C45 0 0 0 0 C45 G5 0 Ci6 C26 C36 0 0 C66

ell e21 e22 e23 0 0 e26 0 0 0 e34 e35 0

0 0 e31 0 0 e32 0 0 e33 el4 e24 0 el5 e25 0 0 0 e36

el I t•12 0 612 622 0 0 0 633

ell e21 0 -- el2 e22 0 el3 e23 0 0 0 e34 0 0 e35 el6 e26 0

ell •'12 0 El2 622 0 0 0 E33

Si

(4)

(5)

In these expanded forms, we used the contracting subscript notation 1 -• 11, 2-• 22, 3-• 33, 4-• 23, 5-• 13, and 6-• 12 to

relate Cpq and e•,p to C0k t and e k0, respectively (p,q = 1, 2 ..... 6, and id, k,l = 1,2,3). Thus C2_s stands for C2213 and el4 stands for e123, for examples.•Notice that the purely elastic or electric portions of these relations are identical whereas the coupled portions are different. In fact, by further examination, we conclude that the vanishing entries in one corre- spond to the nonvanishing entries of the other; i.e., there are no common nonvanishing coupling terms. As will be shown later, such unique properties have important consequences in the manner in which the various wave components inter- act. For this reason, we need to treat both cases separately. It is expected that, upon presenting solutions for one case, results for the second case will be identified by inspection. Ac- cordingly, we shall proceed to first analyze the case of mono- 2 case.

A. The mono-2 class

The dynamic field has generally three nonzero spatial displacement components ul, u2, and u3 corresponding to longitudinal wave (P) along the x l axis, horizontally and vertically polarized transverse waves (SH and SV), respectively, and one electric potential &. In the absence of material symmetry, these four wave components will couple together lending to a complicated response of the plate. If plane waves propagating along the x l direction are independent of x2, a formal solution for u•, i = 1,2,3, and & can be written as

(g/l' U2' U3' I•)= (U, V, W, q))e 'g("'+•'"'-c'), (6)

where g and c are the wave number and the phase velocity, a is still an unknown parameter, and (U, V, W, (I)) are constant amplitudes. Substituting (6) into (2a) and (2b) yields four linear homogeneous coupled equations

I"mn (Or) $ n : O, m,n = 1,2,3,4, (7)

where the summation convention holds and where, for con- venience, we used the notation Un = ( U, V, W, (I)) r with

F ii : Cll --tOC 2 F33 : C55 --pc 2 -i I- C33t5g 2, Fi2: F21: C16 -i I- C45t5g 2, Fi3: ['31

(8) ['23 = ['32 • (C36 q- C45 )0•,

FI4 : ['41

F34 :- F43: els -I-- e33 O•2, F44: -- (611 "31- •'330?).

Nontrivial solutions for U. demand the vanishing of the determinant in (7) and yield the eighth-degree polynomial equation in a, relating a to c as

6g 8 dr- A 1 6g 6 -1 t- A26g 4 dr- A36g 2 -1 t- A 4 --- 0, (9)

where the coefficients Al, A2, A3, and A 4 are given in the Appendix under mono-2 class. Equation (9) now admits four solutions for a 2 and, hence, leads to eight solutions that are restricted according to

•2 : • •1 , •4 : • •3, I•6 = • •5, •8 = • •7' (lO)

For each O•q, we can use (7) to relate the amplitude ratios, [/q • U2q/Uiq , Wq • U•q/Ulq, and (I)q • U4q/Ulq as

Vq: [ Fll (F33 F24 -- F23 F34 ) -Jff- Fl2 (Fl3 ['34 -- I"14 F33 )

+ FI3 (FiaF23 -- F13F24)] [Fi2 (F23F34 -- F24F33 )

-Jl- Fi3 (F23 F24 -- ['22 F34 )

+ Fi4 (F22F33 __ F23F23) ] -i (11a) Wq: [Fll (F22F34 -- F24F23) -• I"12(I"13I"24 -- F12F34)

.qt_ ['14 (['12 ['23 -- ['13 ['22 ) ] [ ['12 (['23 ['34 -- ['24 ['33 )

-'l- ['13 (['23['24 -- ['22['34)

-1 t- ['14 (['22['33 -- ['23['23 ) ] -1 (11b)



(I)q -- [ Fll (F23 F23 -- F22 F33 ) + F12 (F12 F33 -- F13 F23 )

-3- F•3 (F•3F22 -- F•2F23) ] [F•2 (F23F34 -- F24F33)

_ql_ F13 ( F23 F24 __ F22 F34 )

+ F14 (F22F33 _ F23F23)]-1 (11c) Combining ( 11 a)-( 11 c) with the linear piezoelectric stress relations (2a) and (2b), we then use superposition and re- write the formal solutions for displacements, stresses, electric potential, and electric displacement as

(Ul 8

= E (l'Vq'Wq'(I)q)U•q ei•(x'+%x•-ct)' (12a) q=l

( T33,T13,T23,O3 ) 8

-- E i•(Dlq'D2q'D3q'D4q ) Ulq ei•(x• + aqx•- ct), (12b) q=l

where

D1 q = C13 .ql_ C36 Vq -iv C33aq Wq + e33t2q•q, (13a) D2q = C55Egq 'iv C45Egq [/q 'iv C55 Wq 'iv e•5 (•)q, (13b) D3q = C45t2 q + C44t2 q Vq + C45 Wq + e•4 •q, (13c) D4q = e31 + e36 Vq ql_ e33Eg q Wq -- 633Egq(•)q, (13d) With reference to the relations (10) and by inspection of

Eqs. ( 11 )-(13), we recognize the properties

%= --(I)7+,, D¾=D¾+,, (14a)

D:y = -- D:• + • , D3• = -- D37 + , , D4y=D½y+•, j = 1,3,5,7. (14b)

B. Derivation of the reflection and transmission coefficients

To determine the reflection and transmission coeffi-

cients for plane waves incident from the fluid onto the plate surface at an arbitrary angle 0, we need to obtain general solutions for the upper and lower fluids similar to those of Eqs. (12a) and (12b). Recognizing that the fluid does not support shear deformation and piezoelectric potential, its formal solutions can be identified from those pertaining to the solid, namely (12a) and (12b) as follows: If the wave is assumed to be incident and hence reflected in the upper fluid and transmitted into the lower fluid, then using similar analysis to that of the plate yields, for the upper fluid, •

( Ul ,U2,U3 ,T33 )u 2 . +.• _

-- • ( 1,0, Wp,i•'pfc 2) Up e ½ [ x, + ( - •)p .•x, ct ] p=l

and, for the lower fluid,

( Ul ,u2,u3,T33 )l

= ( 1,O,af,i•pfc 2 ) U leig [x, + ,/(x, - a) - ct ], where

(15)

(16)

w .=a/, w;=-a/, 1 (17a)

and

Cgf-- (C2/C3 -- 1) •/2, Cf-- (,if/p/) •/2. (17b) Here, the superscripts u and l represent the upper and lower fluids, respectively. Notice the vanishing (nonexistence) of shear displacement component u2 and the electric potential 05 in both the upper and lower fluids.

By invoking the continuity of the normal displacements and stresses and setting the solid shear stresses, T• 3 and T23, and the electric potential 05 equal to zero at the interfaces x3 = + d/2, we obtain, for a given incident amplitude U •', a system of ten linear simultaneous equations for the amplitudes U •, U l, U•q, where q -- 1,...,8. Solving these equations with the help of the ratio relations ( 1 1 )-( 1 3) after rather lengthy algebraic reductions and manipulations (see Ref. 2), we obtain the following expressions for the reflection and the transmission coefficients.

R= U•/U•= (AS-- Y2)/(S+iY)(A--iY), (18)

T-- Ul/U• =iY(S+A)/(S+iY)(A--iY), (19) where

Y = (p•c2/a/) ( W, G, -- W 3 G 3 + W 5 G 5 -- W 7 G 7 ), (20)

S -- D• G• T { -- D•3 G 3 T; + O•5 G5 T; -- D•7 G7 TS, (21a)

A =D•G• T• --D13G3T 3 + O15G5T 5 --D17G7T7, (2lb)

•3 •5 •7 •1 •5 •7

G•- D23 D25 D27 , G 3 = D2• D25 D33 D35 D3v D3• D35

(I) 1 (I) 3 (I) 7 (I) 1 (I) 3

G5 = D21 D23 D27 , G7 = D21 D23

D31 D33 D37 D31 D33

and T• -- tan (7/a7), T5 -- 1/T•, and 7/-- •d/2.

D27 ,

D37

(22)

D25 ,

D35

C. The mono-m class

Results for the mono-m case [i.e., with the constitutive relations (5) ] can be obtained by following identical steps to those given above for the mono-2 case. The steps, intermedi- ate equations, and final results are similar except for the following parameter definitions exceptions.

(a) In Eqs. (8), I•14, I•24, and I•34 are now replaced with

I•14 -- I•41 --- e35 or- e• •2, ['•24 --- ['•42 --- e•6 ør- e34 6g2,

["34 -- ["43 -- (½13 -3- ½35 )t2. (23)

(b) The appropriate coefficients of the characteristic equation (9) are now listed in the Appendix under the mono-m class.

(c) Equations ( 13 ) and (14) are replaced with

D•q -- C13 -JI- C36 [/rq .qt_ %3Eg q Wq .qt_ e• 3 (•)q,

D2q -- 055cg q "3- C45cg q Vq .qt_ %5 Wq .qt_ ½356gq(I)q,

D3q = C45Eg q -iv C44t2q Vq + C45 Wq + e34t2q• q, D4q -- e35t2 q + e34t2 q Vq + e35 Wq -- ff33Egqt•)q,

(24a)

(24b)

(24c)

(24d)



and

Vj_- Vj+l, •Vj--- •Vj+l, (I)j ---- (I)j + 1 (25a) D li=Dli+l, D2•---- --D2•+l, D3•= --D3•+l, D4i = -- D4• + 1. (25b) By following the same procedure, we are able to obtain

the following expressions for the reflection and the transmission coefficients:

R = U•/U• = (AS-- YY')/(S + iY') (A - iY), (26)

T= U•/U• = i(SY' + AY)/(S+ iY')(A -- iY), (27)

where

Y = (p•cc:/a•) ( W1 G1 -- W3 G3 + W5 G5 - W7 G7 ), (28a) Y' = (pfc2/o•f ) ( W1G [ -- W3 G ; n t- Ws G ; -- W7 G ; ) ,

(28b)

(29a)

A = D,, G• T• -- D,3 G3 T3 + D,• G• T• - D•7 G7 T7, (29b)

G3 i

S 5 •

G7 i

•lT1 •T• •T•

O21 025 027

D31 D35 D37

(I•lT 1 (I•3T 3 (I•7T 7

D21 D23 D27

O31 033 D37

D21 D23 D25

D31 D33 D3•

(29c)

and G] are obtained from G• by replacing T• with T j, j = 1,3,5,7.

So far, we have casted the expressions of the reflection and transmission coefficients ( 18 ), (19), (26), and (27) in our standard form, which we derived earlier for elastic cases. 1.3 The only differences are in the new definitions of the parameters & A, and Y. In general, the number of terms appearing in the expression of $ (or A ) reflect the the number of contributing types of waves. In the present case, four terms are present that represent the contributions of one longitudinal, two shear, and one electric potential components of the waves. It should be understood that each of these

terms depends upon characteristics of the four wave components through the coupled solutions of various a's in Eq. (9).

III. FREE WAVES

The expressions of ( 18 ) and (19) for the reflection and the transmission coefficients for the mono-2 plate contain, as

a by-product, the characteristic equation for the propagation of modified (leaky) guided waves in the plate. We refer to such waves as plate waves rather than Lamb waves, whose properties were derived originally by Lamb for isotropic solids in the absence of the fluid. Setting either denominator equals to zero, namely,

(S + iY) (A - iY) = 0, (30)

defines the characteristic equations for such waves, where the vanishing of the first term corresponds to symmetric mode and the second to antisymmetric one. In the absence of the fluid, i.e., for pf (or Y) -- 0, Eq. (30) reduces to SA = O, which is the characteristic equation for Lamb-like waves in the piezoelectric plate.

Alternatively, the "free wave" characteristic equation $A = 0 can be derived directly from imposing the appropriate boundary conditions on the formal solutions (12a) and (12b). Here, setting T33, T13, and T:3 and the electric potential •b equal to zero at x3 = ___ d/2, and requiring the nonvanishing of the wave amplitudes, one immediately recovers the result $A - O. Since this equation involves the vanishing of the electric potential at the plate's surfaces, it has been known by the "shorted" characteristic equation. Two other types of "free" waves on piezoelectric plates are possible to exist. One (known merely as the "free" case) is based upon the vanishing, besides the stresses, of D3 on both surfaces of the plate. This yields its characteristic equation in a form that is similar to the expressions (21 a) and (2 lb) with q>• (as it appears in the definition of G• ) replaced by D4j T] and D4i T• when it applies to the definitions of $ and A, respectively. The second (the "free-shorted" case) is a mixed one and required the vanishing, besides the stresses, of D3 on one surface and •b on the other. This later case will not be considered any further in this paper.

The above remarks apply equally well to the mono-m case whose free-wave characteristic equation is deduced from Eq. (26) as

(S + iY') (A -- iY) = O, (31 )

with the appropriated definitions of the various parameters as given in equations ( 29a)-(29c).

The influence of the loading fluid on the free waves in plates has been extensively discussed in several previous works. 1'3 It is found that the fluid mainly causes the wave number g to become complex leading to a distortion and attenuation (spreading along the interface) of the reflected energy. For this reason, the reflected waves have been called leaky waves. The degree of distortion of the reflected waves has been found to highly depend upon the ratio of the density of the solid to that of the fluid. It is anticipated that the presence of the piezoelectric effects will not change this role of the fluid.

IV. HIGHER SYMMETRY MATERIALS

By comparative examination of the general results obtained, so far, for the mono-2 and mono-m cases, it is diffi- cult to see whether or not any differences exist in their behavior. This is perhaps due to the complete coupling that exists



between the four wave components in each case, which is typical of propagation in materials possessing monoclinic symmetry. It was pointed out, however, in our elastic wave propagation cases 1-3 that uncoupling of some of these waves is possible in higher than monoclinic symmetry materials. These, of course, include orthotropic, transversely isotropic, and cubic materials. Such classes are different from mono-

clinic classes in that each possesses two orthogonal axes of symmetry in the plane of the plate. For propagation along an axis of symmetry, the horizontally polarized SH wave was found to uncouple from the one associated with the sagittal plane (namely the coupled quasilongitudinal and vertically polarized shear $V waves). In this section, we examine the various conditions upon which uncoupling occurs in piezoelectric plates. We also examine the extent of the role of piezoelectric effects on such uncoupling. This will be shown to depend upon the choice of the model (namely ortho-222 or ortho-mm2) which are obtained by further restricting the properties of mono-2 and mono-m, respectively.

The constitutive relations for ortho-222 material can be

obtained by requiring the vanishing of

C16 , C26 , C36 , C45 , e31 , e32 , e33 , el4 , e•5,

(32a)

in the mono-2 constitutive relation (4). Similarly, by the vanishing of the entries

C16, C26, C36, C45, e21, e22, e23, e34, el6, ff12

(32b)

in (5), we obtain the constitutive relations for the ortho- mm2 material. For both of these orthotropic classes, x• and x2 now constitute axes of symmetry. Hence, the formal solution (6) constitutes propagation along an axis of symmetry. Accordingly, by following the steps commencing in Eq. (6) to both orthotropic cases, we can analyze them in the following subsections.

A. Ortho-222

For the ortho-222 case, uncoupling of the wave components into two sets occurs. This can be easily seen by imple- menting the vanishing of the properties in (32a) into the relations given by (8). Consequently, Eqs. (7) now uncouple into two sets. The first is a pure elastic wave (consists of coupled quasilongitudinal and quasivertically polarized shear) propagating in the sagittal plane x•-x3. This type of wave has been extensively studied in our earlier elastic works. 2'3 The second is a piezoelectrically stiflened horizontally polarized SH wave having the characteristic solution [ equivalent to combinations of Eqs. (7) and (8) ]

[C44a2-[-C66-10c2 (el4 nt-e36) 6g ] a2 - 0. (33) (e•4 -+- e36 )0• -- E33 -- Ell

B. Remark

Since the fluid does not support either shear or electric potential wave components, the coupled wave components described in (33) cannot support transmitted or reflected fields. For this case, it will be possible to only derive characteristic relations for the propagation of free waves that are presented below.

The nonvanishing determinant of (33) yields the fourth-degree polynomial

O? -•- •'10•2 -•- •'2 : 0, (34a) where

•z• 1 : [ e33 ( C66 -- tOC 2 ) -3 I- C44 ell

-3- (el4 -3- e36 )2]/(C44e33 ), (34b) A2 -- El 1 (C66 --pc2)/(C44•'33 ).

Equation (34a) admits four solutions having the properties

O• 1 = -- 0[2, 0[ 3 : -- 0[ 4 . (34c)

For each aq, the electric potential and shear displacement ratios are now constructed from Eq. (33 ) by

2

(I)q = (pC 2-- C66 -- C44•q)/(e14 + e36)O[q, (35a) D3q : C441•q .-Jr- e14 O•q, (35b)

D4q: e36 -- E330•qdi)q, q = 1,2,3,4. (35c) Investigations of these ratios in light of (34b) reveals the further properties

(I)j----q)/+l, D3j = -- D3j+ i,

D4i = D4i + •, j = 1,3. (35d) By invoking the appropriate boundary conditions on the outer surfaces of the plate, we get the characteristic equations for the shorted case as

SA: ((I)1D33 -- (I)3D31)sin(2T'a• )sin(2T'a3 ) -- 0, (36)

and for the free case as

S: D31D43 cos(T'cg 3 ) -- D33D41 cos(T'cg 1 ) =0,

A -- D31D43 sin(7/a3 ) -- D33D41 sin(y'a• ) -- 0.

(37a)

(37b)

C. Ortho-mm2

For the ortho-mm2 case, uncoupling of the wave components into two sets also occurs. The first is a piezoelectrically stiflened elastic wave (consisting of coupled quasilongitudinal, quasivertically polarized shear, and electric potential components) and propagating in the sagittal plane x•-x3. The second is a pure horizontally polarized elastic (SH) wave. For the sagittal wave, Eq. (7) now takes the expanded form

e• + e35 U

(e13 q- e35 )a

o• 2 -- E33 -- •'11

=0. (38)



By following the steps used to derive the reflection and transmission coefficients for the general case (i.e., the steps commencing in Eq. (9) and ending with Eq. (22), once again, we recover the expressions (26) and (27) but with the alternative parameter definitions

S = D• T'• G • -- D13 T.• G 3 -- D15 T;G 5,

A -- D•I T1G• -- D13 T 3 G 3 -- D15 T 5

Y-- (pfc2/5f) ( gl G1 -- g 3 G3 + W5 G5 ), Y' -- (p•2/a•) ( W• G • -- W3 G 3 ß WsGs),

with

Dlq: C13 + Q3aq Wq + el3 •q,

D2q -- C55 (aq + Wq ) + e35aq•q,

D4q -- e35 (aq + Wq ) -- e33aq•q,

(39a)

(39b)

(39c)

(39d)

( 40a )

(40b)

( 40c ) 2 2

5q [F23 (C55a q -•- F• ) -- F•2 (e•l + e355q ) ]

(Dq •

2 2 2

(eii+ ½35•q)(e335q + F22 ) --F12F235 q (41a)

2 __ 2 2 F•225q (C555q +F•)(C335q +F22) 2 2 2 '

(ell + e355q ) (C335q -•- F22 ) -- g12F235q (4lb)

D33 D35

(I)iT 1 (I)3T 3

D31 D33

D3• D35

(42)

The various 5's appearing an this expressions now satisfy the sixth-degree polynomial

54 -•- •3 = 0, (43a) 56 -+- •1 -+- •2 52 . and having the properties

51 -- -- 52, 5 3 = -- 54, 5 5 = -- 56, (43b) with

In arriving at these results, advantage has been taken of the relations

Wj = -- Wj+i, % =%+1' Dlj=Dlj+i, D2i = -- D2i + 1, D4• = -- D4• + 1, j = 1,3,5. (45)

Finally, the characteristic equation for the uncoupled $H wave is derived as

sin ( 2y57 ) = 0, (46)

where

m = [ ( - pc ] (47)

V. PROPAGATION ALONG AN ARBITRARY DIRECTION

For waves propagating in a direction other than the crystalographical direction x•, results can be obtained by utilizing linear coordinate transformation of the material properties (see general procedure of Refs. 1 and 2). For propagation in the plane of the plate along a direction that makes an angle •b with the X l axis, all of the above derived results hold with the provision that the various properties Cmnop , erario, and •'mn are replaced by their transformed values

C ijkl -- J•irn]•jn]•ko]•lp GinhOp, e • i• = J• krn ]• in ]•jo ½m no ,

•'ij = J• irn ]•jn •'rnn ,

(48a)

(48b)

(48c)

where the summation convention holds and where the var-

ious entries of/•o are given by

cos •b sin [/•/• ] = - sin •b cos •b . (49) 0 0

Vl. DISCUSSIONS AND NUMERICAL ILLUSTRATIONS

In this section, we illustrate the analytical results derived above with some numerical examples drawn from the several material symmetry classes belonging to the mono-2 and mono-m as listed in Eqs. (4) and (5). The representative material properties used in our numerical illustrations are collected below in Table I.

Units of stiffness constants and piezoelectric stress constants are 109 N/m 2 and 109 C/m 2, respectively. The permittivity given here is nondimensional as •/•, where • is the dielectric permittivity of free space ( = 8.854 X 10 - •2 = 10- 9/36•r F/m).

Table I (a) gives the material properties of GaAs, a cubic 43-m crystal. It is a higher symmetry material belonging to the ortho-222 group which in turns belongs to the mono-2 group. This can be affirmed by noting that a rotation through an angle •b in the x•-x2 plane renders its transformed properties in a form similar to that of the mono-2 group as depicted in Eq. (4). Table I (b) lists the material properties of PZT-65/35, an artificially polarized ceramic of uniaxial crystal. It is a transversely isotropic material (i.e., isotropic in the x•-x2 plane) also belonging to the mono-2 group. Here, again, confirmation of this belonging can be seen from examination of its transformed properties. Table I (c) depicts properties belonging to a mono-m group. Inter- estingly, it is once again a PZT-65/35 material obtained from those shown in Table I (b) by interchanging directions 1 and 3. This seemingly simple transformation will be shown to dramatically change the behavior of the response of the plate. To further argue this point, we present in Fig. 2 a slowness plot for the propagation in the x• -x3 plane of PZT- 65/35. By carefully examining this figure, we can identify the potential role that the piezoelectric affects will play in the

1256 J. Acoust. Soc. Am., Vol. 91, No. 3, March 1992 A.H. Nayfeh and H. Chien: Influence of piezoolectricity on waves 1256


TABLE I. (a) Material properties of GaAs, (b) material properties of PZT-65/35, and (c) material properties of x2 -cut PZT-65/35.

(a) 118.8 53.8 53.8 0 0 0 0 0 0 53.8 118.8 53.8 0 0 0 0 0 0 53.8 53.8 119.0 0 0 0 0 0 0 0 0 0 59.4 0 0 0.15 0 0 0 0 0 0 59.4 0 0 0.15 0 0 0 0 0 0 59.4 0 0 0.15 0 0 0 0.15 0 0 0.11 0 0 0 0 0 0 0.15 0 0 0.11 0 0 0 0 0 0 0.15 0 0 ' 0.11

(b) 159.4 73.9 73.9 0 0 0 0 0 -6.13 73.9 159.4 73.9 0 0 0 0 0 --6.13 73.9 73.9 126.1 0 0 0 0 0 10.71 0 0 0 38.9 0 0 0 8.39 0 0 0 0 0 38.9 0 8.39 0 0 0 0 0 0 0 42.8 0 0 0 0 0 0 0 8.39 0 5.66 0 0 0 0 0 8.39 0 0 0 5.66 0

-- 6.13 -- 6.13 10.7 0 0 0 0 0 - 2.24

(c) , 126.1 73.9 73.9 0 0 0 10.7 0 0 73.9 159.4 73.9 0 0 0 -- 6.13 0 0 73.9 73.9 159.4 0 0 0 --6.13 0 0 0 0 0 42.8 0 0 0 0 0 0 0 0 0 38.9 0 0 0 8.39 0 0 0 0 0 38.9 0 8.39 0

10.7 -- 6.13 -- 6.13 0 0 0 2.24 0 0 0 0 0 0 0 8.39 0 5.66 0 0 0 0 0 8.39 0 0 0 5.66

, ,

,1 [ I, ] [ l, • -o.a % -o.• o.t / o.a I

• -o.• • I I \ • -o•

x (Sc/Km)

FIG. 2. Slowness curves of PZT-65/35 on the x I -x 3 plane (x:-cut). Aniso- tropic in this plane is due to orthotropic nature of plate studied. Fast quasi- transverse wave is still isotropic and unstiffened. Solid: without piezoelectricity; Dashed: with piezoelectricity.

behavior of the plate. This figure shows that piezoelectric effect couples with the sagittal plane for propagation along the x• axis and couples with the SH wave for propagation along the x3 axis as are implied by the mono-2 and mono-m models of the PZT-65/35.

Having chosen such representative materials, we now present our numerical results in two categories: In the first, we plot representative reflection coefficients versus Fd (the product of frequency and plate thickness), for various azimuthal and incident angles. Second, we depict phase velocity dispersion curves as function of Fd for a variety of azimuthal angles.

From Eqs. ( 18 ) and (26), the reflection coefficients are calculated. Figure 3 (a) and 3 (b) shows the typical reflection spectra for a mono-2 and mono-m based on the properties listed in Table I (a) and (c), respectively. Using the coordinate transformation (48), these results are calculated for an incident angle 0 = 25 ø and azimuthal angle gb = 10 ø. Here, real and imaginary parts are given by solid and broken lines, respectively. The sharp dips in the real part and the associated sharp changes (through zero) of the imaginary part indicate the propagation of plate modes. In Fig. 4, we collected reflections at a large number of azimuthal angles and present them in the form of 3-D plot for a GaAs material at an incident angle of 12 ø . In this figure, we only show the



1.0

-0.6 •r• •,•,

-•'• o • o = o s o 4 o • o •• ••

1.0

0.6

-.

•-o.2 •i • i I

o •11

-0.6

1.0

II

i

2.0 i).0 4.0 5.0

1•) (UHz-,•)

FIG. 3. The reflection coefficient of GaAs plate with wave propagation on x,-x2 plane along, (a) •-- 10 ø and 0= 25ø, and (b) •= 25 ø and 0= 30 ø.

absolute values of the reflection coefficients. The texture of

this figure shows the extent of the anisotropic and piezoelectric effects.

Considering a GaAs plate with wave propagating along the x• direction on x•-x2 plane, we see that the wave now decouples into a pure elastic wave propagating in sagittal plane and a piezoelectrically stiflened $H wave propagating in the plane parallel to the plate. The dispersion curves of these two components are shown in the Figs. 5 and 6, respectively. Dispersion plot for shorted case of a GaAs plate with piezoelectrically stiflened Lamb wave propagating on x•-x2 plane along •b - 30 ø is presented in Fig. 7. From Figs. 4 and 7, it is seen that the antisymmetric modes are shift toward

FIG. 4. The reflection coefficient of GaAs plate with wave propagation on x,-x2 plane and along •b which varies from 0 ø to 45 ø with 0 = 20 ø.

lower Fd and the symmetric modes shift toward high Fd for larger azimuthal angle.

For propagation in the x•-x2 plane of a PZT-65/35 plate, the wave decouples into a piezoelectrically stiflened Lamb wave propagating in the sagittal plane and a pure elastic $H wave in the plane perpendicular to the sagittal plane. The dispersion curves of the piezoelectrically stiflened Lamb wave is depicted in Fig. 8.

An example of the completely coupled piezoelectrically stiflened Lamb wave of the mono-m type that exists if the

8.0

6.0

O.Oo.b

\ ',, ', /

I I [ I ,, I , i [ ß 1.0 2.0 3.0 4.0 õ.0

FD (MHz-mm)

FIG. 5. Dispersion curves for a GaAs plate with wave propagation on plane and along •- 0 ø. Solid: antisymmetric modes; dashed: symmetric modes.



6.0

4.0 -

8'%.0 1.0 2.0 3.0 Normalized Thickness, H (h/x)

10.0

8.0

6.0

4.0

2.0

o.%.b

FIG. 6. Dispersion curves for piezoelectrically stiflened SH modes of a GaAs plate with wave propagation on x,-x 2 plane and along •p -- 0 ø. Solid: symmetric modes; dashed: antisymmetric modes.

FIG. 8. Dispersion curves for piezoelectrically stiflened plate modes of a PZT-65/35 plate with wave propagation on x•-x2 plane and along •p = 0 ø. Solid: shorted case; Dashed: free case.

wave propagates along an arbitrary direction •p is shown in Fig. 9. This figure demonstrates the dispersion plot of the mono-m wave of a PZT-65/5 plate with •p - 65 ø.

VII. CONCLUSION

We have considered the problem of guided elastic wave propagation in piezoelectric plates. Considering monoclinic symmetry material plates, we derived exact analytical ex-

pressions for the reflection and transmission coefficients and the associated dispersion equations for free waves. This equation is written in simple and completely separate terms pertaining to symmetric and antisymmetric modes. Exploit- ing the monoclinic solution as a starting point, we have presented the results for higher symmetric materials such as orthotropic, transversely isotropic, and cubic symmetries. It is found that piezoelectric coupling, as well as water, influence both types of modes. Higher symmetry, such as ortho-

I x-

0.'00.0 1.0 2.0 3.0 4.0 5.0 FD (YHz-mrn)

•l.O i I I ' ' , I I '

.

õ.0

'"• 4.0

••, i , I , i , 0'•.0 1.0 2.0 3.0 4.0

FD (YHz-mm)

FIG. 7. Dispersion curves for piezoelectrically stiflened plate modes of a GaAs plate with wave propagation on x, -x2 plane and along •p = 30 ø. Solid: antisymmetric modes; dashed: symmetric modes.

FIG. 9. Dispersion curves for piezoelectrically stiflened plate modes of a PZT-65/35 plate with wave propagation onx•-x3 plane (x2-cut) and along •p = 65 ø. Solid: shorted case; dashed: free case.



tropic, transverse isotropic, and cubic, are contained implicitly in our analysis. We also demonstrate that the motion of the sagittal (Lamb) and $H modes uncouple for propagation along axis of symmetry. For such cases, however, piezoelectric coupling can influence one of these types of modes depending upon the type of piezoelectric model is adopted.

ACKNOWLEDGMENT

This work has been supported by AFOSR.

APPENDIX: COEFFICIENTS OF THE CHARACTERISTIC

EQUATIONS

Expanding the determinant relation in (7) and collect- ing powers of a, the characteristic equation, an eighth-order polynomial with nonzero coefficients for even order, is shown as

O? -Jr- AlO? q- A20•4 -[- A30• 2 q- A4 •0, (A1)

for monoclinic classes. If for anisotropic class, all the coefficients of the characteristic equation will be nonzero. The coefficients A i for two monoclinic classes are listed as the following:

1. Mono-2 class

A 1 -- [ C33 C45 ( C45 Fll -Jl- 2C16 tff33 -[- 2Fi4 F24 ) -- C44 C55 (2½33 el5 q- F33 tff33 ) -- C33 C44 ( C55 tffll -[- Fll tff33 -[- F•4 ) ] -- C33 C55 (F22/f33 -[- F2•4 ) q- C,]5 (F33/f33 -[- 2else33 ) -- C44½33 (e33Fll -- 2F13F14 )

-[- 2C45 tff33 (C16tff33 -- g13F24 -- g14F23 )

-- C55/f33 (2F23F24 -[- g22/f33 ) -[- C55F223/f33 -Jl- C44F•3/f33 -- 2C45F13F23/f33 I/A,

A2 -- [G3tffll (2C45C16 -- C44Fll -- C55F22) -- C33Fll (g22/f33 q- F2•4) - C33F•4F22

-31- G3 C16 ( C16/f33 -Jl- 2F14 F24 ) -- C44 G5 (F33/fill -•- e215 )

-3 I- 2C44Fll else33 -- C44F33 (Ell/f33 q- F2 (Fi3/fi q- 2F14e ) 14 ) -[- C44F•3 1 15

q- Cz•5 (F33/fl 1 -[- e215 ) -[- 2C45 C16 (2elsE33 q- F33/f33 )

-- 2C45F13 (F23/fll -Jr- F24½15 ) -- C55F224F33 -- 2C45F14 (F23/f33 - g24F33 )

-- C55F22 (F33/f33 -[- 2else33 ) q- g•3F22/f33 -[- C55F23 (g23/fll -[- g24e•5 ) q- 2½33F24 (FilE23 -- C16F13 )

2 F22 +C 2 q- F•3F24F14F23 - ½33 (Fl• 16 ) -- 2½33F14 (C16F23 -- F13F22 ) 2 2 2

q_ g14F223 q- g13F _ F23 (g 1 F23 2C16F•3 ) I/A, 24 /f33 1 --

A3 [ (Ell F22 C2 G3 ) q- El F23 (F23/fl q- 2F24 ) : -- 16 ) ( -- /fll -- e33el5 1 1 el5

-- (C44Fii -- 2C45 Ca6 -I'- C55F22 ) (F33/fll q- e2•5 ) ]

2 (F23/fl -Jr- 2F24½1 ) 2C16F14 (F23½15 -- F24F33 ) q- C2•6F33/f33 -- FilE33 (g22/f33 q- ½15 ) -- 2C16F13 1 5 -- 2

_ g14F22F33 q- g13F22 (g13/fll q- 2gl4el5 ) I/A,

A4 -- ( C2 El F22 ) (Eg3/fl q- e•5 )/A, A -- ( C33/f33 q- e•3 ) ( C,•5 -- C44 C55 ) 16 -- 1 . 1 '

F•l --• C11 -- p c2, F22 ---- C66 - p c2, F33 = C55 -- p c2,

F23 =C36+C4.s, F•3--C•3 +C•5, F14=e31 +els, F24=e•4+e36.

2. Mono-m class

A 1 -- [ (C33/fll q- F33/f33 -Jr- F•4) (C,]5 -- C44C55 ) q- (½34F13 --½35F23 )2

-- e324 (C55 F33 q- C33 Fll ) - ½325 (C44F33 q- C33 F22 )

q- C33/f33 (C44Fll q- C55F22 ) q- C44F13 (g13/f33 q- 2½35F34 ) q- C55F23 (F23/f33 q- 2½34F34 ) q- 2C45½34

X (½35F33 -- g13F34 ) -- 2C45F23 (½35F34 q- El3/f33 ) q- 2C33 C16 (C45/f33 q- ½34½35 ) q- 2C33½16 (C45 ½35 -- C55½34 )

q- 2C33el 1 (C45e34 -- C44e35 ) ]/A,



azl 2 • [(F324 -Jr- F33•.33)(2C16C4 5 _ C44Fll _ C55F22 ) _Jr_ C33•.33 (C126 _ FilE22 )

-[- F33•'11 (C425 -- C44C55 ) -3c C33•'11 (2C16C45 -- C44Fll -- C55F22 )

-- 2F13F23 (C45•'11 ß C16•'33 ß •735•716 ) • F223 (C55E• d- Ell fl'33 -•- 2½•½35 ) d- g•3 (C44E• d- F22E33 -•- 2½34½•6)

-•- 2(C16½34 -- C44½• -•- C45•716 ) (•735F33 -- m13F34 ) -•- 2(C45e• -- C55•716 -- ½34Fll ) (½34F33 -- m23F34 ) -- 2F23F34

• (ellEll + C16E35 ) + C33ell (2•45e16 -- 2e35F22 -- C44ell ) + C33e16 (e34ell -- e34Fll -- C55e16 + 2elae35 )

ß C33e• (2e45e•6 -- 2e35F22 -- C44e• ) + F33 (e•4F• -- ) - 2e35F•3F34F22 •3 = [ (t33e33 • t•4) (C•6 -- Film22 ) ß Ellell (t•3 -- C44F33 ) • t22ell (t•3 -- C33Fll )

ß 2C•6e• (C45F33 -- t13F23 ) ß 2(C•6e• -- e•aF• ) (e34F33 -- t23F34 ) ß 2(C•6e•6 -- e•lF22 )

• (e35F33 -- t13F34 ) • 2e•e•6 (C16C33 -- t13F23 )

• •1 (F•3 -- C33F22 -- C44F33 ) ß e•6 (F•3 -- C33Fll • C33e• A4 = [F33e• (C•6 -- F•F22 ) ß F33 (2C•6e•e•6 -- e•F22 - e•aF• ) a = C33 (e33 ( C •5 -- C• C55 ) • 2C45 e34 e35 -- e•5 C• - e•4 C55 ),

Ell = Cll --ac 2, F22 = C66 --ac 2, F33 = C55 --ac 2, El3 = C13 ß C55, F23 -- C36 ß C45, F34 -- e•3 ß e35.

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The influence of piezoelectricity on free and reflected waves from fluid-loaded anisotropic plates

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