Approximate theory of vibration of crystal plates at high frequencies

hr. J. Solids Structures. 1974, 10, pp. 177-196. Pergamon Press. Printed in Gt. Britain.

APPROXIMATE THEORY OF VIBRATION OF CRYSTAL PLATES AT HIGH FREQUENCIES Z.

NIKODEM

Gulf United Nuclear Corporation,

Elmsford, New York

and P. C. Y. LEE Princeton University, Princeton, New

Jersey

(Received 6 June 1973)

Abstract-The series expansion of displacement in terms of simple thickness modes is used to obtain approximate two-dimensional equations of motion for crystal plates from the threedimensional theory of elasticity. Approximate theories from the first to the fourth order are presented. Dispersion curves for AT-cut quartz plate are explored and compared with the solution of the three-dimensional equations for an infinite plate.

1. INTRODUCTION

The formulation of approximate, two-dimensional equations of motion is based on the series-expansion method. Mindlin[l], using a power series expansion of displacement, obtained a set of equations which accommodates the frequencies of the first five modes of vibration. The increasingly complex mathematical forms in the formulation of higher order approximations, when a power series expansion is used, led to an expansion of displacement in a series of simple thickness modes for infinite plates[%]. The approximations of successively higher orders are then obtained without any complication, because the simple thickness modes are orthogonal. This procedure was applied in the analysis of the vibration of isotropic pIates[3] and is extended in this paper to the vibration of crystal plates. The dispersion curves for real, imaginary and complex wave numbers are explored in detail for AT-cut quartz plate and are compared with those obtained from Ekstein’s[4] solution of the three-dimensional equations of elasticity for an infinite plate. The comparison reveals the close agreement between the respective sets of dispersion curves and indicates that the solutions of approximate equations, for bounded plates, will give reliable results over the same applicable range of frequencies. 2. THREE-DIMENSIONAL

EQUATIONS

The variational form of the equations of motion, from which the two-dimensional equations are to be deduced, is

s

(Tij, i -

V

PUN,

,,) 6~j d V = 0

(1)

178

Z.

NKODEM

and P. C. Y.

LEE

where Tij, u, are the components of stress and displacement, respectively, and p is the mass density. In (I) the integration is over the region V for independent variations of uj . The constitutive equations are Tj

=

Cijkl

skl

(2)

where the cijkl are the elastic constants, Sij are the Components in terms of the uj by Sjj = i(Uj, i + ui, j). 3.

SERIES

OF TWO-DIMENSIONAL

of strain and are expressed (3)

EQUATIONS

The faces of an infinite plate are taken at x2 = +6, the middle plane in the x,, xj plane of an xj rectangular coordinate system. The displacement components uj(xi, t) are expanded in an infinite series of the simple thickness modest as uj =

Ujcn)cos Y(l

f

-

9)

(4)

n=O

u,(“) are functions

where q = x2/b and the nth order displacements Then, from (I) and (4) using the identities

s

of x,, x, and time t only.

1

sin y

(1 - q)sin y

(1 - q) dq = 6,,

-1

(5)

1 cos y

s

and integrating

(1 -

Y/)COS y

(1 - r) dq = 6,,

-1

over the interval

(- 1, 1) with respect to 9, one obtains

&j(n) d/t = 0

(6)

where A is an area of a plate and

T$‘)E

j1 qjcosy(l -1

Zj@) G j1

Trj sin y

-$da

(1 - q) dq

-1

1 1

Tzj ~0s !f?

2

(1 -

yl)

-1

=

T,,(b) - (-l)“Gj(-b)

are defined as nth order components of stress with cos (n7r/2)(1 - ‘I) and sin(nrr/2)(1 - ‘I) as weighting functions, and nth order components of the face traction, respectively. Since (4) must be satisfied for every area A and arbitrary BUj(“) then the nth order stress equations of motion are obtained from (4) as T!!‘! _ !!! Tzj(“) + ! F .(“) = pu!“’ J,ff’ lJ.1 2b b ’ t Series of vibrational modes which are independent free faces (see Mindlin[l]).

of x Ir x3 coordinates

(8)

and correspond

to traction

Approximate

179

theory of vibration of crystal plates at high frequencies

Defining

S..c”’

as nth order components sij

4b

of strain,

F

=

n=O

(9)

E(6,iuj’“’ + 82ju(“‘)

E

EJ

one obtains

Sii(“)COSF (1 -

from (3) and (4) components u) +

Sii(“) sin y

(1 -

of strain

.

q)

1

[

of (10) into (2), then, in turn, into (7) yields the nth order stress-strain

Substitution

T J.(“) =

Sk/"'

cijkl

relations

5 A,,,,&@')

+

“I=0

T;J1”) =

&l(n) +

cijkl

(11)

5

A,,,&@" m=O

3

where

0

1 s -1

sinm<

(I -

~)COS

9

(1 - v) dy = A,” =

3

m + n even

4m (m’ - n’)n: ’

(12)

m + n odd.

Two sets of stresses and strains are introduced in this theory. The Tii(“) components of stress provide the linkage between 12th and (n - 1)th order of equations while strain components Sij(“) are supplemental,[3] to S,(“’ . The plate-strain energy density, which is defined as (13) becomes

Q= ;

z(zg”$i(“)

n

+ gTi(“)Sii(“))

(14)

0

or

Sii(“)SJ”) + $i(“)&l(“) +

5 (A,,

Sii(n)Skjm) + A,, $i(“)S$“)) . (15) 1

m=O

It can be noted that

au

.(“’ = IJ asip)

T.

Similarly,

the plate-kinetic

energy density

(16) becomes (17)

The uniqueness

of the solution

of the approximate

equations

is established[2,

31 in a

180

Z. NIKODEMand P. C. Y. LEE

way similar to that of Neumann[5]. The sufficient conditions, in the absence of discontinuities and singularities, are listed in the previous paper[3], where also a theorem which shows the orthogonality of two functions Us’ and uj(“)* is presented. 4. TRUNCATION

PROCESS

AND

ADJUSTMENTS

The approximate theories of various orders must be extracted from the infinite set of two-dimensional plate equations. This is achieved by using the following truncation process when for Nth order theory, with N any positive integer, is set IJJ”’ = 0 T.?“’ = T..(“J = 0 lJ

LJ

for n > N.

(18)

I

From the approximate theory of order None obtains the dispersion curves for the propagation of straight crested waves in an infinite crystal plate with monoclinic symmetry, such as AT-cut quartz plate. By comparing the dispersion curves with those from the threedimensional theory of elasticity[2,4], it is found that the approximate theory always yields the “exact” cut-off frequencies since the simple thickness modes used in theseriesexpansion of the displacement are the exact limits of the three-dimensional theory as wave lengths approach infinity. Two sets of dispersion curves match quite well for frequencies up to R = N + 5 and wave numbers 1z 1 = N + 1, where the dimensionless frequency and wave number are defined by (19) o is a frequency, 5 is a wave number, except for the lowest flexural and the lowest extensional branches. Two correction factors, k, and k2, are introduced in the strain and kinetic energy densities in order to achieve the better match of these branches: 2n = cijkl[Sij(o)Skl(o) + $.(“)skl(o) ‘I

+

~..(w&C2) lJ

+

. . .

+

+ k,DS..(“S,,(l)

Sij(o$410

k,S,:/”

+ S..(l)Skl(l) + Sij(2)Skl(z)

+ A,, &,;I” + . . .)

+ Sii(1yA21Sk1(2)+ A41Sk*(4)+ . * .) + SijyA12 + Sijy4,,

Ski(l) + A,, S/p

+ . . .)

Skl@) + A,, sk1(4)+ . . .) + . . .

+ SiJ(1$410klS,,‘o’ + A,, Skjc2)+ A,, Sklc4)+ . . .) + S..(2yA21Skl(l) + A*3 Sk1C3) + . . .) + S.J3)(A30 &(O) f A,, S,,@) + . . .) + . . *] IJ CJ (20)

and

where p =

(’1

cog ; .

It is observed that k, is introduced into ated with the coefficients A,,, while k2 into K for the term U$ u$“:. The correction factor k, ‘is determined and the lowest extensional branch from

u for the strains Silo) and Sit’), which are associinto u for the term Szj(‘)S2f1) and

is introduced

from the comparison of the lowest flexural branch the first order theory with those from the exact

Approximate theory of vibration of crystal plates at high frequencies

181

theory. In order to make the slopes of these branches to coincide when both the frequency and wave number approach zero, the value of k, must be set to

three-dimensional

ki = 7r/4.

(22)

In order to make the phase velocities of the lowest flexural and the lowest extensional branches approach that of the surface wave propagating along the diagonal axis in a crystal as values of both the fequency and the wave number become large, k, must be equal, in the dimensionless form, to (23)

k, = C’PIG,

where p is the density of a material and c is the velocity of the surface wave[6]. The adjusted energy densities ii and R are positive and definite if, in addition to the usual requirements, k2 > 0. The strain-displacement relations are not changed by correction coefficients. The stress equations of motion derived ,from u and R also remain unchanged except that the inertia term pz@~, is replaced by (~~k~)~~~~. The components of stress are derived from the adjusted 5f using (16) and include then the correction coefficients. 5. PLATE The theory The plates,

THEORIES

OF SUCCESSIVELY

HIGHER

ORDERS

plate theories from the first to the fourth order are extracted from the Nth order by using the truncation procedure and correction factors in Q and K. displacement equations of motion are further examined for the monoclinic crystal such as AT-cut of quartz, where we set:?

c,5=c~5=c~5=c~5=c,6=c~6=c~6=c~6=o.

(24)

The values of the other thirteen constants for the AT-cut quartz plate with X, the digonal axis in the plane of the plate and x2 the axis normal to the plate are computed from Bechman’s[7] six principal values and are, in units of 10’ N/m’, 86.74

c44 = 38.61

cIz = -8.26

cZ3 = -7.42

c 22 = 129.77

es5 = 68.81

cl3 = 27.15

CZ4=

5.70

c33 = 102.84

ce6 = 29.01

c,~ = -3.65

cS4 =

9.92

cS6 =

2.53.

Cl1

=

(25)

Two sets of equations, for the essentially symmetric and the essentially antisymmetric families of mode& respectively, are separated for the case of straight crested waves propagating in x1 direction only for each order approximation. The dispersion curves for real, imaginary and complex numbers are explored in detail and are compared with those obtained from the three-dimensional theory[2]. This comparison is presented for the essentially antisymmetric family of modes in odd order theories (N=odd) and for the essentially symmetric family of modes in the even order theories (N =even). The zero order theory is not a very suitable one due to the missing coupling between extensional and flexural modes[2] and therefore is not presented here. -+An abbreviated indicial notation is used. A pair of indices one index p ranging over indices 1-6. $ The notation of [8] is employed.

i, j

ranging over integers l-3 is replaced by

182

Z.

NIKODEM

and P. C. Y. LEE

First order theory

By setting N = 1 in (18) only the stress, strain and displacement components of the zero and the first orders are retained. Then the energy densities become U(l) = $[~PfSij(0) + Tj(l$ij(‘) + ‘fi,j(*f$,if*)] R’” = +[/QQ+!*)+ u!‘)u!r)] z L,t 1,: 1,: 1,t and the stress equations of motion are: T!p! t2.r + :h F!Ot J

(26)

= pkz -“z&p;* (27)

(1) T.(‘? _-?. ‘iizj(r) + ! F.(r) = PU,, ff * ~Jv’ 2b b ’

The stress-displacement relations are readily obtained from (9) and (16) using the adjusted D by discarding those components ujw for n > 1. The displacement equations of motion of the first order theory are then obtained by’inserting the stress-displacement relations into (27):

Considering then the straight crested waves propagating in the x1 direction we set: U.(o)= A .(o)ei(k -WI J

&+, $0)

-_

_

=

Fj(l)

iAi(l)&
-m:)

(2%

0.

Then the dispersion relations for monoclinic crystal plates are, in dimensional form, CliZ

2

-

fzz

ik,z

-cl2

-T,,zk,z

--El2 zk,z

_ 4 -cIs ; k,z

k, z2 $ ZzZ - Q2

2562’ +

C24

C56Z2

5.55z2

z44

+c -2 4

+

= 0 -

(30)

f-l2

and =2

-_L*Z

-

c56 z

2

_-

7l

2

C56Z2

z55z2

4 ll

-c56;

- - k,z

where 5;,, = CpqlC66.

-

4

-

4

f22

k 1z

-

k,z

E56 z

c,,z2+

k,z I --n2

=o

(31)

Approximate

theory of vibration of crystal plates at high frequencies

183

The dispersion curves of the essentially symmetric family of modes are associated with the dispersion relation (30), while the equation (31) yields the dispersion curves of the essentially anti-symmetric family of modes. The cut off frequency, for z = 0, are the exact ones. In order to have the correct slope of the lowest flexural branch at z = 0 one must set k, = 7114.

(32)

For G % 1 and z 9 1, the phase velocity of the lowest flexural branch approaches to [S], from (31), k2 = Q/z = cap/c6, = 0,901.

(33)

Hence the phase velocity of the lowest flexural branch approaches that of surface wave in monoclinic crystals. The dispersion curves are then computed for AT-cut quartz crystal plate and are compared with the dispersion curves obtained from the three-dimensional exact theory in Fig. 1 for both antisymmetric and symmetric families of modes. It can be seen that the two sets of curves match quite well for fi < 1.5.

s(im)

Fig. 1. Dispersion

r(re)

curves of the essentially symmetric and antis~metric for the first order approximate theory.

families of modes

Second order theory

Following the truncation process described above, we set N = 2 in (18). Then the energy densities for the second order theory are

184

Z.

NIKODEM and

P. C. Y, LEE

and the stress equations of motion become 1

The strain-displacement and stress-strain relations are obtained from (9) and (1 l), respectively, with strain and displacement components of order higher than two discarded. The displacement equations of motion are then obtained similarly as in the first order theory.

Consider the straight crested waves propagating in the x1 direction by setting U,(0)= A .(0)ei(Fxl J

-0x)

J

u,(ll

=

_i~~(l)~i(tx~-wr)

u:(z) J

=

,,4j(2)eiCtxt-w)

F.(O) J

zz F.(l) J

=: F,(Z) J

(37) z

0.

The displacement equations of motion will separate into two uncoupled sets and the dispersion relations of the essentially symmetric family, equation (38), and the antisymmetric family of modes are obtained. crrz2 -cl2

Q2

Zk,z

-4 -cl2

; k,z

-‘I”;

4

k IZ

kzz2 -I-cZ2 - i-l2

T,,Z2 + ‘;Zb

Es6z2 +c~~

c+*

0 $

(Cl, + 412

= 0. -Elbz 0

k,z

+ c;,, - n2

g

(Cl‘%+ 4G&

&z2 + 4 - s12

The dispersion curves yield the exact cut-off frequencies at z = 0.

(38)

Approximate

185

theory of vibration of crystal plates at high frequencies

both z 4 1 and Q < 1, (38) gives the exact slopes of the branches, if k, = 7rj4. As from (38) that the phase velocity of the Iowest extensional branch approaches to Q/z = k, = 0.901 As

z B I and Q $ 1, it is found

which is the phase velocity of the surface wave in monoclinic crystals[2]. The dispersion curves of the essentially symmetric family of modes are presented and compared with the results from the three-dimensiona theory of elasticity in Fig. 2. The imaginary dispersion curves are not fully developed for this order theory. Only one complex branch is found.

-

Approx-reol,ima

(fe)

t(i m) Fig. 2. Dispersion

curves of the essentially symmetric family of modes for the second order approximate theory.

It may be noted that the same value of k, = n/4 is used for correcting the slopes of both the lowest flexural and the lowest extensional branches at z = 0 and R = 0, and the same value of k, is used for correcting the phase velocities of the same two branches when the values of z and R become large. As it shah be seen, no additional corrections are needed in the higher branches of the higher order theories. Third order theory

The energy densities are defined as

IJSS Vol. 10 No. 2-D

186

2. NIKODEMand P. C. Y.

LEE

and the stress equation of motion are:

T,@‘, _ IJ.1

nn - -

T2 j

cn) +

! F.“”

(40) =: pu!“’

J. if

b J

26

n=l,2,3

Discarding the components of strain and displacement of order higher than three, the stress-strain and stress-displacement relations are then obtained. The displacement equations of motion are then:

+ 62, U,(3)) 1 ,i

fCijkt

iCijkl

--C

R

2b

--_c. 3n

4b

2Jki

$

c62,

d3)

+

6,kU,(3’)

+

’ 3n

(do)

k.1 -+

U(O))

I,k

-,-

5x ( U$:‘]+ z@)

’

I

The propagation of straight crested waves in the x1 direction only is obtained by setting: u,W

=

A

,

,tn)ei(6xi

-wr)

n =0,2

J

Cn) =

Ui

_ jAj(“)eiC
F.(“) =0 J

-mt)

n=l,3

(42)

n=O, t,2,3.

The dispersion relation for the essentially antisymmetric family of modes is then, in dimensionfess coordinates, presented in (43).

Approximate

theory of vibration of crystal plates at high frequencies

0

II

0

0

0

0

187

188

Z.

NIKODEM and

P. C. Y. LEE

The dispersion curves for the essentially antisymmetric family of modes for the AT-cut quartz crystal plate are computed from (43) and are compared with those computed from the three-dimensional theory. There is no complex branch of the dispersion curves found for third order approximation. The two sets of curves match very well as it is seen from Fig. 3.

-Approximat -----Exact

I

0 z(im)

2

3

4

z(re1

Fig. 3. Dispersion curves of the essentially antisymmetric family of modes for the third order approximate theory.

Fourth order theory The energy densities P4)

= *[Tij(0)sij(O)

are

+

~j(l)sij(l)

+

~j(2)&j(2)

+

+ zj(qij(l)

and the stress equations

of motion

T!.? _+

II. 1

7Yj(3)Sij(3)

+

+ Fj’2’S. .(2) + LJ

7Yj(4)Sij(4) Tj(3)Sij(3)

+

Tic45

ij

(4)

1 (44)

are:

! F,(O) b

’ (45)

T{f”, _

tJ.I

!!? T2j(n) 2b

+

1

F

b

.(“) = &?’

’

J, tf

n = 1,2, 3,4.

The displacement equations of motion are then obtained by substituting the stress-displacement relations, where components of order higher than four are discarded, into the stress equations of motion.

Approximate

21

tCijkl

Ik,R(U:!i

+

U!,‘I!)‘+

T$

theory of vibration of crystal plates at high frequencies

U (l) k

+

(62,

U12)

62,

U,(l))

+

62,

+

f

U/“)

(62,

+

Ukc3)

$

+

62,

(62,

1

,i +

Ulc3))

UL4)

+

62,

;

189

F,(O)

=

pk2

-pU;;;t

Ul(4))] ,i

fCijkl

uj$

[

2 + tdj,“,’- - (621 uk(‘) + 6,,u,“‘)

3b

uy;, + zpk , - -s”b (6 21ukc2)+ &,u,(2’)

%ijkl[

+ $

(&l ukc3)+ 62k.:,,,]

+ ;

(6,,u,‘4’

,i

+ 6,,u,i*?] ,i (46)

21d3)

f

Cijkl

[

+

82k”1(3))

+

$

(U:f/

+

U;$

+

8

(U:f/

+

U;“,)

1,i

2 18 ‘i:1’ + d;k’ - (62, uk(l) + 62, u/1)) - G (621ukc3)+ 62kuLc3)) 15b

7L

- b

C2jkl

By setting

u,Cn) =

Aj(n)&(Sx*

-wt)

J

u.(“) = _iA jn)ei(
F.(“) = 0 J

J

n

=

0,

n

=

I,3

n=O,

2,4 (47) 1,2,3,4

190

0

II

3

s

Approximate theory of vibration of crystal plates at high frequencies

191

we obtain two sets of dispersion relations for the propagation of the straight crested waves in the monoclinic crystal plate, The dispersion relation for the essentially symmetric family of modes is presented in (48). The dispersion curves are computed from (48) for the AT-cut quartz plate and are cumpared with the exact ones. It can be observed that the lower branches match with the “exact” curves better than the corresponding branches in the second order theory, since the higher order theory not only accommodates additional branches at higher frequencies but also improves the accuracy of the lower branches. The discrepancy for the complex bran&es are mostly due to the difference in the reaI parts. The matching of the imaginary parts is very close. The mode shape corresponding to a complex wave number can be expressed as the product of trigonometric and hyperbolic functions. Wence the mode has the shape of a oscillatory variation within the envelope of the hyperbolic function, and therefore in the case of matching the complex branches it is more important to have close agreement of the imaginary parts rather than the real ones, since the former control the ampiit~des of the modes.

6. GENERATION

OF HIGHER

ORDER

DISPERSION

RELATIONS

An approximate theory of the order N (any N > 0) can be generated in a systematic manner by using the truncation process (18). No additjona1 correction coefhcients are needed for any higher order theory. The dispersion relations for the propagation of straight crested waves in the x1 direction only in crystal piates with rn~no&~in~~sy~etry are then obtained by substitution of the wave form solutions into the displacement equations of motion. Since the process becomes tedius when N is large, a general method for generating the dispersion relations of monoclinic crystal plates is described below. The matrices of the dispersion relations are symmetric and their elements may be classified into four groups: A,, , Bmnfor elements on the main diagonal (the fater is a three-term group, two diagonal terms and one off-diagonat term); C;, (two terms in one row) and D,, (two terms in one column) for the elements off main diagonal. In terms of these groups the dispersion relations of an N-th order theory for the essentially symmetric and the essentially antisymmetric families of modes, respectively, can be written as :

0

n

0

5

A OE

c OS

1

0

2

C 2,s

3

0

4

C4s

5

B 55

.

I

.

.

.

.

.

.

.

.

.

.

*.

.

N 0

.

.

D EN 0

.

D .ik

.

0

.

.

D SN

*..

N

ANN

-0

149)

192

Z. NIKODEM and P. C. Y. LEE

and m

0 0

B 00

1

I

2

3

4

5

...

Do,

0

Do,

0

Do,

...

Do,

A,,

C,,

0

c,,

0

...

0

B 22

D 23

0

D,,

...

D,,

A 33

C 34

0

...

B 44

D 45

...

D 4N

A 55

...

0

2 3 n

N

0 zz

4 5

0.

(50)

... N

A NN

The terms in the groups are defined as follows (0 < m, n 6 N, for any N > 0): A,,,,, =

i;,,z’ + n2 - R2

z*+ n*F,, 4,

=

- Cl*

(n = m) Z56z2 + n2?,,

I/ cs5z2 + n2Z4, - Q2J

(m2c,, + n’)z

T

(n = m)

4 (m2 - n*)x

( map,, + n2Zs6)z

(n + m) (n + m = odd)

(n f 4 (n + m = odd).

The upper sign in the last two groups is applied to the essentially symmetric family of modes and the lower sign is applied to the essentially antisymmetric family of modes. The elements of determinants of dispersion relations are then generated with the help of (49) and (50). It shall be noted that the correction coefficients k, and k, are applied in terms B oo, B,,, Cal, Do, as is seen, for example, in dispersion relations (43) and (48). The dispersion relation is in the form of a determinant, whose order S, for N-th order theory, is given by s= z

1

3(N + 1) fcos

1.

2Nn 2

The dispersion relations for the fifth, sixth, ninth and tenth order theories are by this method. The dispersion curves for the essentially antisymmetric (fifth order theories) and essentially symmetric (sixth and tenth order theories) families are computed for AT-cut quartz crystal plate and compared with the curves from the three-dimensional theory (Figs. 5-8). The two sets of curves match

generated and ninth of modes obtained very well

Approximate

193

theory of vibration of crystal plates at high frequencies

5

(re)

Fig. 4. Dispersion

curves of the essentially symmetric family of modes for the fourth order approximate theory.

except for the complex branch in the fifth order dispersion curves for the essentially antisymmetric family of modes, since the imaginary part of this branch is not yet fully developed; it develops in the ninth order theory (Fig. 7).

re)

6

Fig. 5. Dispersion curves of the essentially antisymmetric approximate theory.

family of modes for the fifth order

194

Z. NIKODEM and P. C. Y. LEE

7 r(im) Fig.

6. Dispersion

curves

of the essentially symmetric family approximate theory.

of

modes

for

the sixth

order

Due to the increasing computational difficulties only the real and imaginary parts of the dispersion curves are shown in Figs. 7 and 8, for the essentially antisymmetric family of modes for the ninth order theory and the essentially symmetric family of modes for the tenth order theory, respectively.

-Approximate _______Exac+

z(im) Fig.

7. Dispersion

curves

z(re) of the essentially antisymmetric approximate theory.

family

of modes for the ninth

order

Approximate

theory of vibration of crystal plates at high frequencies

195

-Approximate _______-Exac+ 9.376543210123456789

z&e)

zfim)

Fig. 8. Dispersion

curves of the essentially symmetric

family of modes for the tenth order

approximate theory.

7. CONCLUSIONS

The expansion of components of displacement in a series of simple thickness modes enabled us to obtain an infinite set of stress equations of motion, strain-displacement relations, constitutive relations and displacement equations of motion. The dispersion relations for crystal plates with monoclinic symmetry were obtained for approximate theories of order one up to the order four. The dispersion curves for AT-cut quartz plate were then computed and compared with the curves obtained from the exact three-dimensional theory. The close agreement of both sets of curves indicates that the applicable range of frequencies for an Nth order theory can be set at R d N + 4..

REFERENCES 1. R. D. Mindlin, An Introduction to the Mathe~atjcal Theory of Vibrations of Elastic Plates. U.S. Army Signal Corps Engineering Laboratories, Fort Monmouth, New Jersey (1955). 2. Z. Nikodem, Theory of Vibration of Crystal Plates at High Frequencies. Ph.D. dissertation, Princeton University, Princeton, New Jersey (1971). 3. P. C. Y. Lee and Z. Nikodem, An approximate theory for high-frequency vibrations of elastic plates. int. J. Solids Srrucf. 8, 581-612 (1972). 4. H. Eckstein, High frequency vibrations of thin crystal plates. Whys. Rev. 68 (1945). 5. A. E. H. Love, A Treatise on the ~uthe~aticai Theory of Elasricity. Dover Publications (1944). 6. H. Deresiewicz, and R. D. Mindlin, Waves on the surface of a crystal. J. a&. Phys. 28, 669 (1957). 7. R. Bechman, Elastic and piezoelectric constants of alpha-quartz. Phys. Rev. 110, 1060-1061 (1950). 8. R. K. Kaul and R. D. Mindlin, Frequency spectrum of monoclinic crystal plate. J. Acoust. Sm. Am. 34 (1962).

Z. NIKODEM and P. C. Y. LEE

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