An exact solution of a time-dependent frictional contact problem for two elastic spheres

Pergamon

9020-7225(95)00094-1

Int. J. Engng Sci. Vol. 34. No. 5, pp. 537-548. 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved [X)20-7225/96 $15.(X)+ (1.00

AN EXACT SOLUTION OF A TIME-DEPENDENT FRICTIONAL CONTACT PROBLEM FOR TWO ELASTIC SPHERES O L E G YU. Z H A R I I t Department of Theoretical and Applied Mechanics, Kiev University, 64 Vladimirskaya Street, Kiev 252033, Ukraine (Communicated by Z. S. OLESIAK) Abslracl--The paper is devoted to development of an analytical technique for treating contact problems with friction for time-dependent Ioadings. A particular problem of an intermittent contact between two similar spheres, constrained against the rigid-body rotation, is considered. First, normal tractions are determined from classic Hertz theory. After that, an initial guess for tangential tractions distributions proved to be consistent with Coulomb's law of dry friction. Analytical formulae for the tangential force resultant and for characteristics of motion in the contact zone as functions of the angle of incidence are obtained and analyzed. Copyright © 1996 Elsevier Science Ltd

1. I N T R O D U C T I O N

It is well known that the contact between elastic solids loaded by normal forces can be treated using the Hertz theory with sufficient accuracy when the contact area is small compared to the radii of curvature of the bodies [1]. But, when one wishes to take into account tangential loadins, one faces serious difficulties. Of course, in many cases it is possible to avoid an extremely complicated problem of simultaneous determination of all components of contact stresses, making use of a wellgrounded assumption of negligible influence of tangential stresses on contact area dimensions and on the distribution of normal stresses [1]. Hence, the latter quantities may be considered to be known from the solution of the classic Hertz problem. Even after this significant simplification, determination of tangential stresses in contact still remains a complicated problem. The origin of this lies in an inherent nonlinearity of problems of elasticity involving friction. When tangential forces are applied to bodies in contact, in a general case, the contact areas is subdivided into zones of adhesion and slip [1, 2]. Variations in normal and tangential loadings cause unknown in advance changes in dimensions of both these zones and the contact area as a whole. The sign of tangential stresses in the slip zone, according to the Coulomb law of dry friction [1] depends on the relative tangential displacement of two bodies, also unknown. The variety of possible situations (many of them have been treated for small increments of loading forces in the famous paper [2]), is really grandiose. On the basis of his own experience, the author can easily imagine the feeling of confusion of a reader who wishes to catch a constructive idea for his own needs from the above cited paper. The question arises: is it really impossible to follow the complete cycle of normal and tangential loading (preferably, by analytical formulae) at least for typical, not trivial, but not artificially sophisticated situations as well, or can these problems be treated only numerically, using a known set of solutions corresponding to infinitesimal force increments? To the best of the author's knowledge, problems with variable normal and tangential loadings, other than elementary, as of today, have only numerical solutions. One of the best known examples is the treatment of an oblique impact given in [3]. Note an important feature of the problem considered there: unknown in advance contact forces are determined in the course of solution ?Current address: Institut fur Mechanik II, Technische Hochschule Darmstadt, Hochschulestraf3e Darmstadt, Germany. 537

1, D-64289,

538

O. YU. Z H A R I I

Y NN

-,,,

~,,~,.

I

0

27

Fig. 1. Scheme of a standing wave ultrasonic motor.

from equations of translational and rotational motion of a sphere. This circumstance is characteristic for m a n y problems of mechanics resulting in contact problems involving friction. The author of this article would not like to be constrained by necessity to solve his problems starting from the very beginning from formulation of a numerical procedure. This unwillingness has led to an idea: to m a k e an attempt to guess, for the given particular contact problem with known partially kinematic and partially force input characteristics, the location and laws of alteration of adhesion and slip zones. The next step, of course, is to prove strictly that the initial guess was correct. In this paper, we formulate a problem of frictional contact for two identical elastic spheres, arising in qualitative modeling of the principle of operation of a so-called standing wave ultrasonic m o t o r [4] and give a complete analytical solution of it.

2. F O R M U L A T I O N

OF T H E P R O B L E M

Consider an experimental model of a standing wave ultrasonic m o t o r (Fig. 1). A rod of piezoceramics of dimensions 2L in the x direction and 8 in the y direction is polarized in the z direction (perpendicular to the plane of drawing). When a sinusoidal potential difference is applied to the face electrodes, under h << L and not too high frequency, longitudinal vibrations (standing waves) are excited. At the first resonant frequency, o~, = nc/(2L), where c is the rod velocity, longitudinal displacements are distributed as ~x

ux = A sin ~

sin w, t,

(1)

where A is the amplitude of vibrations at x = L. At this frequency, one many observe that a cylindrical rotor put on the rod surface rotates counterclockwise (Fig. 1). What is the reason of rotation? It is almost obvious that the rotation arises thanks to simultaneous action on the rotor of not only longitudinal displacements (1), but transverse ones (in the y direction) as well. The latter can be calculated taking into account shrinking and expanding of the rod in the y direction thanks to the Poisson effect. Assuming that at ~o = oJ, deformations ey are distributed uniformly along the y axis, one may write .

.

.

.

.

8 0u~ .

/r 8

7rx

v-~-£ A cos-z~ sin oJlt ,

(2)

where v is the Poisson ratio of the rod. Thanks to transverse vibrations and resulting contact forces, the rotor moves in the y

Time-dependent frictional contact problem

539

direction. However, in practice we may neglect this motion and assume that the rotor mass-center is in average displaced at some distance h above its position on nonvibrating rod. This conclusion can be grounded as follows. Two forces act on the rotor in the vertical direction: the contact force, directed upward and the weight force, directed downward. Impulses of these forces compensate each other (integrals of them taken over the period of vibrations T] = 2n/w j, are equal). During this period, the maximal distance at which the rotor can displace downward under the action of the weight force is gT2/2, which means 0.05/xm for typical frequency of vibrations about 10 kHz, while the amplitude of vertical vibrations at which the effect of rotation becomes observable, is about 1 tzm. Hence, during the contact lasting a fraction of period Tj, the normal force and, due to displacements ux, the tangential force act on the rotor. Note that the rotor does not rotate being placed either in the middle of the rod (the tangential force resultant vanishes due to symmetry) or at its end (no vertical motion and consequently, the normal contact force is zero). In all intermediate points, the rotor rotates counterclockwise with an angular velocity f2 depending on its position (this direction is clockwise for x > 0). This means that the tangential force resultant is nonzero. Taking into account that points of the rod move rectilinearly, at first sight, the very fact of rotation may seem strange. In an attempt to model this phemomenon, we may at the first stage prevent the rotation and calculate the tangential force acting on the motionless rotor. If it appears that the force resultant is nonzero, this would mean that the free rotor is to rotate in the direction of the force resultant. Further simplification can be achieved by replacement of both the rotor and the rod by similar elastic spheres and to consider the motion of the center of the bottom sphere by the same law as in a point x = xo < 0 of an unloaded rod [formulae (1), (2)]. This way, we come to the problem depicted in Fig. 2: two spheres of radii R are constrained against rigid-body rotation by couples M equal to the product of tangential stress resultant by the sphere radius. They are separated by the distance h at t = 0 and the center of the bottom sphere oscillates: uxo = uo sin wt,

u~o = H sin oJt.

Our task is to calculate forces of interaction between spheres.

37

Fig. 2. Two elastic spheres in an intermittent contact.

(3)

540

O. YU. ZHARII 3. P R E L I M I N A R Y

CONSIDERATIONS

3.1 Solution o f the p r o b l e m o f n o r m a l contact W h e n H > h, s p h e r e s will c o n t a c t d u r i n g the t i m e i n t e r v a l to < t < rc/~o - to, w h e r e 1 h to = -- arcsin - - . ¢o H

(4)

A s is p e r m i s s i b l e in p r o b l e m s of this k i n d , we d e t e r m i n e d the c o n t a c t a r e a size a n d n o r m a l stresses on t h e b a s e o f H e r t z t h e o r y [1]. W e a s s u m e that b e t w e e n two successive c o n t a c t s (for e x a m p l e , d u r i n g z / w - to < t < 2 z / o ) + to), local m o t i o n s in s p h e r e s d e c a y , so we are to solve the p r o b l e m o n l y for t h e t i m e s p a n e q u a l to the d u r a t i o n of o n e contact. A c c o r d i n g to the H e r t z solution, in o u r p r o b l e m the r a d i u s o f t h e c o n t a c t a r e a is e q u a l to

[ 3 ( I ~ - v ) R ] ' / 3 P '/3 a = k 8G ]

(5)

a n d t h e c o n t a c t p r e s s u r e is 3P ,/-2 p = ~ v a- - r 2 -

4G (1

-

v)zcR

V~a2 - r 2,

(6)

w h e r e G is the s h e a r m o d u l u s , r is t h e r a d i a l c o o r d i n a t e a n d P is the p r e s s i n g force (Fig. 3). T h e r e l a t i v e a p p r o a c h of u n d e f o r m e d p o r t i o n s of s p h e r e s e q u a l s

a2

a=2~=

2 [ 3(1 -

L 8-~ ]

"-'e

(7)

a n d in the p r o b l e m u n d e r i n v e s t i g a t i o n , it is a p r e s c r i b e d q u a n t i t y ,

(8)

a = H sin cot - h = H ( s i n cot - sin wto). F r o m t w o l a t t e r e q u a l i t i e s , t h e c o n t a c t force as a f u n c t i o n of t is

P(t)

2V~ G 2V2 G - RJ/2a 3/2 - - RJ/eH3/2(sin cot - sin wt0) 3/2. 3(1 - v) 3(1 - v)

P

r, x

Fig. 3. Loading forces, tractions and velocities in the contact area.

(9)

Time-dependent frictional contact problem

Uzo H

541

a* (~, a, b . . . . . .

h

H-h---

0

t

a

0

TO

(a)

T17r--To

(b)

Fig, 4. (a) Vertical displacement of the lower sphere; (b) relative approach during contact (o0, radius of contact surface (a) for H = 2h (%= zr/6), radius of adhesion zone (b) for ~ = V-2+ 1 =2.414 (r~ = 37r/4). Disks a . . . . . f correspond to curves in Fig. 6, a* = VR(H- h)/2.

T h e impulse of the n o r m a l contact force is

= (~r/,o ,,, P(t) dt

8Gv)w R,/2H3/eF(ro)'

-,o

(10)

9(1 -

where F(vo) = (1 + sin Vo)(1 + 3 sin vo)K(Ko) - 8 sin

il "t'()E(Ko),

Ko =

- sin To ~

,

(11)

K and E are c o m p l e t e elliptic integrals of the first and the second kind [5]. F r o m this point o n w a r d we use both dimensional (t) and dimensionless time v = wt, so vo = Wto, etc. In Fig. 4 we plot uzo(t), a and a as functions of time.

3.2 The problem of tangential loading W h e n tangential tractions Z'~.~= qt,V~cz - r 2

(12)

are applied inside the circle 0 - r < c of elastic spheres in contact, as in the n o r m a l contact p r o b l e m , for calculation of respective displacements we use the solution of elactostaties equations for an elastic half-space [1, 3], 2-v

ux~-q"-~ ×

O<-r<--C'

{ ( 2 c 2 - r2) 2 '

[(2c2- r2)arcsinCr+cV~-Z-~-c2],

r > c,

Uy~- O.

(13)

In equations (13) we o m i t t e d small n o n a x i s y m m e t r i c a l terms. T h e justification of this is given in p a p e r [3]. W h e n the quantity c is t i m e - d e p e n d e n t and qo = const, by differentiation of the a b o v e expressions one finds

2-vd

ax=q,,-4-d E$ 34:5-D

CZ

I2'

O<-r<--c'

)×LarcsinCr,

(14)

542

O. YU. ZHARII

[

C

r

Fig. 5. Tangential tractions on the surface of a half-space (12) and respective surface velocities (14). In Fig. 5 we present plots of tangential tractions (12) and correspondent velocities (14). An important feature of the velocity is that it is constant inside the circle r < c. When, in addition to the normal force P, a tangential force T is applied to the spheres in contact, tangential tractions at the interface arise. In Fig. 3 we denote tractions ~ , in the upper sphere as ru and in the lower one as ~ = - ~ u . From the conditions of equilibrium of both spheres we have T = 2~"

ru(r)r dr.

(15)

According to the Coulomb law of dry friction, contact tractions are determined as follows [1]. In the zones of slip (as usual, they are adjacent to the boundary of the contact area), we have ru = - t x p s i g n ( v 2 - vj), where tx is the coefficient of dry friction, vj and v2 are velocities of bodies in contact. When v2 < v~, as in Fig. 3, this condition can be expressed by

ru=tXp,

v2
(16)

Inside the zone of adhesion (in the center of area), we must have El 2 = U I ,

t~'Ul< pp.

(17)

Note that these simple relations between x components of tractions and velocities in the

spatial problem of elasticity are valid with the same accuracy as approximate formulae (13). 4. S O L U T 1 O N OF T H E P R O B L E M

4.1 Large angles of incidence We return to the problem depicted in Fig. 2. Equations (15)-(17) allow one to determine tangential tractions when the relative motion of spheres is considered. We suppose in this section that the angle of incidence of the lower sphere with respect to the upper one (0 = arctan uo/H) is large enough, so at the initial stage of contact starting at t = to, we have complete sliding in the contact area, i.e. on the base of (16) 4G X/~a2 - r 2 and ~u = P-P = / z (1 - v)trR

v2
(18)

To find, when this is possible, we calculate tangential velocities of spheres. According to Fig. 2, we have

vt=fi~o+t~l,

v2=t~2

(19)

and, due to the condition ~r~= - v u , we have

z~j = -t~2.

(20)

H e r e u~ and u2 are tangential elastic displacements. F r o m (3) we find ti.~o= ~ouo cos tot and ti2 on the base of (12), (14) is equal to u2=/x. 2-v d 2 R 2(] 7- v ) g a '

O<-r<-a'

(21)

Time-dependent frictional contact problem where

a 2

543

is found from (8),

a2

RH = T (sin tot - sin tot,)).

(22)

We find that the inequality in (18) is satisfied for to < t < rc/2w when q,

2(1 - v)uo > 1 /z(2- v)H

(23)

or 0 > 0o = a r c t a n / z ( 2 - v ) / ( 2 - 2v). Hence, our guess (18) proved to be correct and under (23), in the stage of loading (here this term means that the normal force increases), we really have complete slip in the contact area. In the paper [3], the notation q, has a similar meaning, but in terms of velocities rather than displacements. The case g, < 1 will be considered later. In the middle of contact at t = tr/2w, we have v~ = Ve = 0 in the contact area. After this time instant we switch to the stage of unloading and the expression for tractions in (18) is no longer valid. For t > Jr/2w, we must make another guess with respect to the tangential tractions distribution. In formulating our hypothesis we take into account that (i) there may be counterslip in the annulus adjacent to the contact area boundary while the central portion becomes the adhesion zone, as is typical for corresponding static problems [1, 2] and (ii) the traction distribution should change continuously. So, our guess for t > ,c/2w is 4G

~v/a

2 -

2 V b 2 - r 2, 0 <- r <- b, b < r < a.

r2 -

To = -/~ (1 - v)rcR x [V~a2-- r2 '

(24)

Note that (24) coincides with (18) at t = lr/2to (b = a), so the condition (ii) is already fulfilled. It can be seen that [rol < / z p for 0 -< r < b. On the base of general equations (16), (17), we must have in the adhesion zone ( 0 -< r-< b), vl = v2,

(25)

v, < v2.

(26)

while in the slip zone (b < r < a),

The quantity b is so far undetermined. According to (12) and (14), velocities on the surface of the upper sphere due to tractions (24) are

Ix ~i2=

2- v

~(a

R 2(1-v~×[d

-2b2),

2 d ~a-2~b

O<-r<-b,

2.2 b ~rarcsinr,

(27) b
Now, from (25) in view of (19), (20) and (24) we obtain a differential equation for determination of the radius of the adhesion zone,

~(t)~WUoCOStot+2R

2-v d 2 ( 1 - v ) dt ( a 2 - 2 b z ) = 0

or

db2-@+lda2 dt 2 dt

(28)

with the obvious initial condition

b I,=,~n,o = a [,_,,2,o.

(29)

RH b2(v) = -0+I -2 ~ (sin r - sin rl),

(30)

Solving this problem, we find

544

O. YU. Z H A R I I

where 0 + 1\

ro

is the root of b(r) that lies between r = ~/2 and r = Jr - r0 for q~> 1. A typical plot of b ( r ) is presented in Fig. 4(b). Now we prove that inequality (26) really holds in the slip zone. Composing the difference v, - v2, we find

v_v2=~(t)+2tx.

2-vd

R 2(iLv)~

2/ 2 b) (2b)~l--arcsinTr <0,

b
(32)

in view of the easily verified equality d/dt(b 2) < 0, the known inequality for the arcsine function and (28). So, tangential tractions (24) satisfy all conditions imposed by Coulomb's law. At t~ < t < rc/w- to we have complete slip in the contact area with tangential tractions opposite to those in (18), namely, now 4G X/~a2 - r 2 , vu = - / * (1 - v)JrR

(33)

and in addition, we must have v2 > Vl everywhere in the contact zone. The last inequality is verified immediately by rewriting it in the form touocoswt+2 ~. 2-v da2<0 R 2(1 - v) dt

(34)

and noticing that at the stage of unloading d/dt(a 2) < 0 and cos wt < O. Hence, we have found that tangential tractions between spheres are given by equalities (18) for the stage of loading, (24) in the initial stage of unloading, when partial adhesion is present and by (33) at the end of contact (complete counterslip). Typical plots of tangential tractions are presented in Fig. 6 for selected time instants n'mrked in Fig. 4(b) by disks. The duration of the interval of partial adhesion is a decreasing function of q,. For large 0, an approximate formula for this interval can be derived from (31), /r

rl---2Vl-sinroq, 2

1/2 at

(b)

(a)

~0--,~.

(35)

(c)

(e)

(f)

Fig. 6. Tangential tractions on the surface of the u p p e r sphere. Stage of loading, (a) r = if/4; (b) r = 2~'/3: (c) r = if/2. Stage of unloading, (d) "r = 7,'r/12: (e) r = 17a/24: (f) r = 3x/4.

T i m e - d e p e n d e n t frictional contact p r o b l e m

545

T h e r e f o r e , for angles of incidence close to 90 ° , the time interval b e t w e e n initial slip and final counterslip is rather small. This result is in a g r e e m e n t with the conclusion of very rapid transition f r o m c o m p l e t e stick to c o m p l e t e slip, derived by numerical m e t h o d s in [3].

4.2 Small angles of incidence F o r small angles of incidence, when ~ - < 1, one m a y check up with ease that for the whole interval of contact to < t < tc/o~ - to we have c o m p l e t e adhesion in the contact zone (ol = v2) and tangential tractions 4G ~ r. = ~0tz (1 - v)rcR

-- r 2

(36)

satisfy inequality IVul < ~p. 5. A N A L Y S I S

OF SOLUTION

5.1 Tangential force between spheres A n i m p o r t a n t integral characteristic of tangential contact is the tangential tractions resultant (15). Calculating this quantity using equations (36), (18), (24) and (33) respectively, and introducing a new n o t a t i o n 2V~ G T) = tz 3(1 - v---~R 1/2H3/2,

(37)

one finds for small angles of incidence (~0 < 1), T = ~0T) (sin r - sin tO) 3/2

(38)

and for large angles (~0 > 1), respectively, (sin r - sin tO) 3/2, T = To ×

- (sin r - sin

2'

t o ) 3/2 +

2

(~_1)

3/2

7r ~ < r < zl,

(sin r - sin r~)3/2,

- ( s i n r - sin ~=O)3/2, In Fig. 7 we plot this characteristic

~'I ~ 27 ~ /~ -- •I'),

for different values of 0. F o r qJ< i, curves are

3

7-0

7r-~

(a)

(39)

)71" - - 7-0

(b)

Fig. 7. Tangential force acting on the u p p e r sphere. (a) 1: ~ = 0.4, 2: g, = 0.8, 3: ~ = I, (b) 4: ~ = 2.41, 5: qJ = 6,46, 6:~0 = 28.35.

546

O. YU. Z H A R I I

T

1

6

,I,

o 1

i

i

lO

20

Fig. 8. Impulse of tangential force as a function of p a r a m e t e r ~O( r o = To/6). Disks 1 to curves in Fig. 7.

. . . . .

6 correspond

symmetric with respect to the point 1-= Jr/2, while for 4, > 1 this is not so. The bigger the value of 4,, the shorter the time span of switching f r o m / x P to -/xP. Now, analogously to (10), we may calculate the impulse of tangential force, ,,r/o)

to

=

(40)

T ( t ) dt. "to

On the base of (38) and (39) this quantity as a function of 4, is I f~@l)

j- = /J,~ X

4,-<1, 4,>1,

3/2 F(1-1)

(41)

F(o) ' where ~0 and F are defined in equations (10) and (11). The plot of 3-is depicted in Fig. 8. We found that 3-reaches the maximum at 4, = 1 and decays proportionally to 4, ,/2 at 4,--+ zc. Note that this law of decay is the same as that for duration of the interval of partial adhesion (35). 5.2 M o t i o n o f the c o n t a c t area c e n t e r

In addition to analysis of force characteristics, it is interesting to follow the motion of the contact area. In particular, we study the velocity and the displacement of the point r = 0 of the upper sphere. From (21) and (22) in view of (23) we find that v2 = +u~o/(24,) in intervals of initial slip and final counterslip, respectively. For the interval of partial adhesion, from (27) and (28), we obtain Vz = tixo/2, thus 1

Jr

'~-~,,,

1 U2 = I 2 /ix(),

1-,,-< 1- -< ~ , zr

2 < 1-< l-t,

(42)

1 -- ~-~/)xO,

l"1 -< l" ----- K -- 1-o.

Integrating (42) with respect to t using expression (3) and the initial condition u2(1-o) = 0, one finds sin r - sin It 2

to,

=uo x

24,

/l; ro -< r - ~Z

/r

sin r + 1 - 4' - sin v., -(sin

1- - s i n to),

< r < rl, 1-1 -< ~" ~ /l" -- 1-o.

(43)

T i m e - d e p e n d e n t frictional contact problem

547

~xO~ ~2

~o ~ x o

~o •o 2 T1

0

--

7-

(a)

T

(b)

Fig. 9. (a) Velocities of the lower sphere center ~ (b) corresponding displacements, q~ = x / 2 + 1 = 2.414, ro - zt/6.

Plots of v2 and u2 are presented in Fig. 9 together with the velocity and the displacement of the center of the lower sphere, uxo and u~o (3). F r o m the velocity plot we see that v2 has finite jumps in the start and the end of contact, and also at r = r~, when partial adhesion switches to complete counterslip. These jumps are easily understood, because the time variable plays a role of a p a r a m e t e r in the quasistatic problem. Hence, the inertial properties of small regions of spheres adjacent to the contact area are disregarded. Due to that, a finite (at r = rl) and even an infinitesimal (at r = ro and r = J r - to) tangential force causes finite velocity changes what means infinite acceleration. Note, that the displacement u2 vanishes at the end of contact, so our initial assumption of absence of interference between successive contact events proved to be valid. I When ~-< 1, kinematic characteristics have extremely simple representations: v2 = ~uxo and u2 = ½Uxo(sin r - sin to).

6. C O N C L U S I O N In this paper, we suggest an analytical approach to frictional contact problems as an alternative to sophisticated considerations of infinitesimal force increments [2] or to the direct numerical schemes [3]. O f course, there is no guarantee that simple formulae for tangential tractions exist in all possible situations. When several elementary guesses fail, one may try to employ a more complicated superposition of expressions (12) than (24) and corresponding formulae for displacements and velocities in combination with numerical methods, as in [3]. In the result of solution of the specific problem we found that the impulse of tangential force ~-is maximal at ~ = 1 or, in terms of the angle of incidence, at 0 = 0o = 13.65 ° for typical values of p a r a m e t e r s /x = 0.2 and v = 0.3. For ~ < 1, ~- is smaller because of lower amplitude of T (curves 1, 2 in Fig. 7). When $ > 1, 3 becomes smaller due to reducing and switching sign of T in the stage of unloading (curves 4, 5, 6 in Fig. 7). Like previous studies of the author [6, 7], this one is aimed at development of quantitative models of the p h e n o m e n o n constituting the principle of operation of ultrasonic motors, namely

548

O. YU. ZHARII

transformation of vibrational motions in solids into unidirectional progressive or rotational motions due to contact frictional forces. The problem of rotation of the upper sphere under intermittent contact with the vibrating lower sphere is under investigation now.

REFERENCES [1] [2] [3] [4] [5] [6] [7]

K. JOHNSON, Contact Mechanics. Cambridge Univ. Press, Cambridge (1987). R. D. MINDLIN and H. DERESIEWICZ, J. AppL Mech. 75, 327 (1953). N. MAW, J. R. BARBER and J. N. FAWCETT, Wear 38, 101 (1976). S. UEHA, Proc. IEEE 1989 Ultrason. Syrup. 2, 749 (1989). H. BYTEMAN and A. ERDELYI, Higher Transcendental Functions, Vol. 3. McGraw-Hill, New York (1953). O. YU. ZHARII, IEEE Trans. UItrason. Ferroelec. Freq. Control 40, 411 (1993). O. YU. ZHARII, Proc. IEEE 1994 Ultrason. Syrup. 1, 545 (1994). (Received and accepted 9 May 1995)