A coupled quasi-stationary problem of thermodiffusion for an elastic cylinder

Int. J. Engng Sci. Vol. 33, No. 6, pp. 773-780, 1995

Pergamon

Copyright (~ 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225/95 $9,50+ 0.00

0020-7225(94)00099-9

A COUPLED QUASI-STATIONARY PROBLEM OF THERMODIFFUSION FOR AN ELASTIC CYLINDER Z. S. OLESIAK and YU. A. P Y R Y E V t Department of Applied Mathematics and Mechanics, University of Warsaw, 02-097 Warsaw, Banach str. 2, Poland A b s t r a c t - - T h e purpose of the paper is to discuss the influence of the cross effects arising from the coupling of the fields of temperature, mass diffusion and that of strain in an elastic cylinder. By application of the Laplace and Fourier integral transforms we have solved the initial-boundary value problem for an infinite cylinder. The influence of Dufour's and of Soret's cross effects, dependent on time at particular points of the cylinder, have been discussed and presented on diagrams.

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

The models of continua with the physical processes of diffusion type taken into account were presented in a few papers [1-4[. A quasi-stationary coupled spatial and plain problems of thermodiffusion and the physico-mechanical state of a circular cylinder under the action of cyclic forces were considered in a paper by Podstrigach et al. [5]. A coupled, quasi-stationary two-dimensional axially symmetric problem of thermoelasticity was presented in a paper by Takeuti and Tanigawa [6] while thermal schock in a plate in [7]. A dynamical problem of diffusion only has been discussed in [8], the problems of coupling of the mechanical, thermal and diffusion in two papers by Mokryk and Pyryev [9, 10]. The methods of solving the problems with physico-mechanical processes in electroconducting solids have been presented in a monograph [11]. In this paper we consider, within the framework of mechano-thermodiffusion, the coupled thermal and diffusive nonstationary processes in an elastic cylinder. Over the cylindrical boundary the conditions of heat and mass exchange are prescribed. Fourier's and Fick's laws are assumed in the general case with the effects of Dufour and Soret taken into account. The solution is constructed for arbitrary values of the couplings coefficients. We have found the influence of the physico-mechanical characteristics, and of the boundary conditions of heat and mass exchange on the distribution of the fields of temperature, of mass concentration and the stresses at various states of heating and saturation.

2. M O D E L

OF

AN

ELASTIC

SOLID

IN THE

FIELD

OF

THERMO-DIFFUSION

The system of equations governing the quasi-stationary processes of thermodiffusion in deformable, elastic solids takes the following form (see [2]): generalized Navier equations /zV2u + (A + p,)grad div u = YT grad 0 + Yc grad c,

(1)

a generalized equation of thermal conductivity: 0,0 = kV20 -

ke0,e +

kcO,c,

e ~- div u

(2)

and a generalized equation of thermodiffusion: OtC = D V 2 c

- zV2e + [3720.

tPermanent address: Department of Mechanics, University of Lviv, Ukraine. 773

(3)

774

Z . S . O L E S I A K and YU. A. P Y R Y E V

We assume the constitutive equations as follows: s=3,re+nO-dc,

and

O=-yce+dO+ac,

o- = 2/zE + (A tr E - y r 0 - yec)l, (the generalized Duhamel relations),

(4)

and the kinetic relations q = -grad(A0 + aDc - axe)

11 = - g r a d ( D c +/30 - z e )

(5)

We have used the symbols: displacement vector--u, stress tensor--o-, o-i/its components, strain tensor--E, Eij its components, 0 = T - T~}, c = C - Co, s = So, O = M - Mo, where To, Co, So, M0 denote the temperature, mass concentration, entropy, and chemical potential of the natural state, respectively, q and -q denote heat and diffusion fluxes, respectively. A, tz are Lam6's constants, e - div u, E, v - - Y o u n g ' s modulus, Poisson's ratio, respectively, d - - d e n o t e s temperature coefficient of the change of the chemical potential, A' is thermal conductivity, k ' - - t h e r m a l diffusivity, D--coefficient of diffusion, c,--specific heat, a~, and oz'~ are the coefficients of linear thermal and diffusive expansion, respectively, Lqq, Lqn : Lnq , L~n are Onsager's phenomenological coefficients, satisfying the inequalities Lqq >- O, L~n >- O, L q q L , n LZ, q >-O. There exist the following relationships between the coefficients: Lqn K = -c,, k=k'-aToc, kec , : TO~/T,

m'

K=A-~/3=

-~

c. = nTo,

D = aLnn,

Lq"-~n T{} '

Lqn A ' Lqq a= L~ , = To ,

Z = L..yc,

kcc. = a + dTo, I

A k' = --, Ce

A = A ' + dLqn,

/3To = (a + dTo)Lnn,

P

(1 - 2v)3,e = E a ' ,

(1 - 2 v ) y r = e a r .

The second terms on the right-hand sides of equations (5) for the heat and diffusive fluxes characterize Dufour's and Soret's effects, respectively.

3. S T A T E M E N T OF T H E P R O B L E M Within the framework of the model of thermodiffusion of elastic solids we consider a long solid, elastic, and isotropic cylinder of radius ro with initial distribution of the diffusive substance concentration Co and initial temperature TO. There is heat and mass exchange with the environment through the cylindrical surface r = r0: and

Ts(z, t) = To + T , T ~ ( z , t),

Cs(z, t) = C0 + C . C ~ ( z , t),

the functions of z and time. The surface of the cylinder is free from mechanical tractions. In the considered case we obtain the axially symmetric stress-strain state which, in the system of cylindrical coordinates (r, ~b, z), can be determined by the displacement vector u = (u, 0, w), and the stress tensor components ~rr,, o'~, ~r~, and O-rz. The distribution of temperature, substance concentration and displacements are determined from the system of equations (1)-(3) and the following boundary conditions: q " n = ar[O(ro, z, t) - T,T.~.(z, t)],

O'rr(r, Z, t) = O,

~rrz(ro, z, t) =- O,

"q. n = ac[c(ro, z, t) - C , C ~ ( z , t)],

O0(O, z, t) ar

- 0,

Oc(O, z, t) Or

- 0,

u ( 0 , z, t) = 0, ( 6 )

and the initial conditions: O(r, z, O) = O,

c(r,z,O)=O,

for

t=0,

(7)

Thermodiffusion for an elastic cylinder

775

where aT, ac are the coefficients of heat and mass exchange, respectively, T.~(z, t) =f~(z)g~(t), C'(z, t) =fz(z)g2(t), f j ( - z ) =fj(z), j = 1, 2, n = (1,0,0). Before we pass to the solution of the problem we introduce the dimensionless quantities: Z = z/ro ~ ( - ~ , ~), O' = O/T,, w' -= w / u , ,

c' = c / C , ,

h~ = k / D ,

u' = u / u , ,

h2 = T r T , y¢C,'

hc = a D C , AT, ' AH1 = OlTro,

tD v = r--'~-0E (0, ~),

R = r/ro e [0, 1),

' -- ~rik/o',, o'ik

fiT, t~T = DC--~,'

~ = k¢C,/T,,

he = 8~hc, DHz = acro,

Ee =

i, k = r, ~b, z, 6~ =

(1 - 2v)ZT~ ED '

(1 + v)a~ETo (1 - v)(1 - 2v)c,'

0% = Ea" C,,

u, = roa" C,.

Now, let us apply to equations (1)-(3) and the boundary conditions (6) Fourier's exponential integral transform with respect to Z: f ( R , ~, v) = ~--[f'(R, Z, r); Z ~ ~:1- ~

f'(R,

Z, v)exp(i~Z) dZ,

(8)

and Laplace's integral transform with respect to dimensionless time v:

~(R, ~, s) = ~ ( R ,

f(R, ~, v)exp(-sv) dv

~, r); r--,sl ~

(9)

where f o r f ' ( R , Z, v) we take any of the "primed" functions u', w', 0', c'. Taking into account initial conditions (7) we arrive at the following boundary value problem: • ~ 0~, = 2(1 + v) ~-~ a (h~ + 2(1 - v)~,t~ - (1 - 2v)sC2~ - t~: (1 - 2 v ) ~ o ~

- 2(1 - v)~:zr~

i~ O(Ru)

R

OR

8),

-2(1 + v)i~(h2b + 8),

• ~ = ( ~ - ~)(c - ~ 8 + ~,~), Sb = h,(~

0 -

g2)b

-

V vh2 lse+e~s~, E-e 1 1 ---~

(lO)

for

R~[0,1);

-'~ (0 q- hc8 - ~ehce ) q- H 18 "~ H, ~-'s(~, S),

o__(8 - ~ee + ~Tb) + H~c = H28~(¢, s),

OR = ~',r

v (l+v)(1--2v)e

1 ( h 2 / ~ + 8 ) + 1 0u 1 ~-2v I+voR-O'

__,(0;;

~r'z-2(l+v) OR

O,

)

-i~:u = 0 ,

oR-O'

for

(11)

R=I;

~forR=O;

where

~ +

02

0

R-'-~-]R

-2,

j_-1, 2

The solution of the boundary value problem (10)-(11) can be found explicitly [1]. The required fields are determined from the inverse of Fourier's transforms of the sum of the series, the terms of which constitute the residua of Laplace's transforms in terms of the roots of the --2 e'~ = 0, 3tSk --<0). corresponding characteristic equation sk -- --IZk(~rSk

776

Z. S.

OLESIAK and YU. A. PYRYEV

4, S O L U T I O N

OF T H E P R O B L E M

The dimensionless displacement vector, temperature and the mass concentration can be written down in the form of the sums (12)

Z , "~) = U (1) -~ U (2),

Ut(R,

where U' is any of the required functions (u', w', 0', C'): ~_2ffH Jfo °

UO)(R,Z, v ) = u(J)(R, We

~ coss~Z "~ 0 (JO)(R,¢, v)fj(~)tsin¢Z J d~*-~-gj(~:), "~ (J°)(R, ~, fi4,) exp(-/X~r). k=, "~ /22D(/2k)

u(J)(R, ~, O) dl(se, 0)

"c)

j = 1, 2,

(13) (14)

have introduced the symbols:

O0) = H.,U~J) + UOm), C ~ ) ( ~ , R , [,1,) -- ~

j # m,

j,m=l,2,

n(]'m)g'~'R -- "

-- ~ 10~tJ 1

~1

0~)(~, R, tz) = B~J'm)GR1 --

l~(j'rn)t'-2R

tt20L~ 2

~J2~

L~21Yt(J'm)["2R~2-- e 2 A ( J ' m ) ~ J R ,

u~(¢, R, ~ ) = voR[B~'m~&o- v,A~J'm~(~,~,,- ~4,) + alB?'m)nl Y~ - " ~ 2 ~''~2 " ' ' ~q 2 ~~"~ 2], w~)(~, R, it)

=

Vo~-l[-n(J'm)~l R) + A(J'm)(v26~) + Vl~2R2~0~0) -- a,B~J,'~)rll¢2G R + a2B(2i'm)rl2¢2G~],

A(J, m) = (--1)l]jm -- (--1)m¢26mpjm ' B(J, m) = L A ( J , m ) _ 26

(--1)J~2Vl~m + (-- lyv6~0101jm -- ~2el ~b1061j62m

+ 62jala2~2L5(e3~z~lo61m

B! ''l) = (-1)iamoGm ~,

j, m = 1, 2,

-- e 2 6 2 m ) ,

+ Fm,2),

B~2'2) = - ( - l y ( G m 5

B! ''z) = ( - l y ( d m Y m 5 B}2'') = - ( - 1 ) ' ( b m Y m A -

-

Fm~2~lOffl),

Fm~Z~bloe3), i, m = 1, 2, i # m ,

where A(~,/x) = A¢2) + H, H2 A(1) + H1A(3) +/-/2 A(4), A(m)(~, /~) = ~ L m

-~- PlI(~4iE3~2~//10 -- ~ , i E 2 ) -~ ~ 2 ~ l O ~ 2 i ( E 3 P 1 2

-- ~ , P 2 , )

Fm= a,~TI,~(-1~ZY,~ + ~2~,o G m), ij,,,(~, p~) : (-ly+mpjm - Omljm,

I,~m=bmeY~G2-bmlYzGl,

m = 1, 2,

Pjm(~, tz) : -1#, + (-1)/+m~2+,oP#n,

Pmm=(bm2-bmOG~G2,

I#~=(do, z - d m O Y I Y 2 , bmi = (aio6ml + 6mz)an'rln,

-- ~ 3 i ( E l ~ 2 ~ / 1 0 P 2 2 ~- E 2 P , 2 ) ,

j,m=

1,2,

Pjm=-d,,,2Y2G~ + dmlYiG2,

dmi = (bi6ml + di3m2)an'rh,,

i, j, m = l, 2, j # m, n # i,

L, = as~(G, Y, - G, Y~)n, n ~ - al,(a2,,- alo)G, G 2 - a3,(d2Y2G, - d, Y,G,)

+ 64k(a,ob2Y2Gl - azob, Y1G2) + 62k(b2dl - bldz)Y~ Y2, Y~ = ~fl,(t~,),

Y f = I~flI(~,R)/R,

I,(~) ~0,,,- ~I,,(¢),

O~ =/,,(~,R),

G, = Io(~,),

2 = v,(1 - Cq,],,) - v6Om,

D(~k) -

.

lo(~R)

~0o,, = Io(~) '

k = 1 . . . . ,5,

1

2ftk

~,

I,(~R)

~mo( O '

i=1,2,

Thermodiffusion for an elastic cylinder

rh = (/,2 _ s¢2)-,,

p,j = V~2 _ / , 2 k 2 = i~/tz2k y _ ~2,

2~/k2 = b - ( - l y ~

e=l+~e,

= h,(1 + 8~Vo),

4ag,

bi = 1 + hcaio + hevoai,

1 - 2(1 v 1-

V4--

-

ai

v + (1 - V)E e

i=1,2,

•2 = Ee/(eh2),

El = k/c.,

3 - 4v + 2(1

2vs

ha + a~o,

di = aio + ~eaiVo + ST,

k = ~eVO -- ~ T E e / h 2 , v)E e

j = 1, 2,

b=ae/ht+h,+(ST+aevoh2)(EC+Edh2),

e -hlk~ ago - Ec + E~/h2

~l = h e y 0 - Ee/h2,

777

~

v2 --

,

vs = (1 - 2 v ) ( 1

-

3 - 2v + 4(1 - V)Ee

v)E e

V3 =

2vs

1-v +

Ee) ,

E3 = h / e ,

V 6 --

V5

-

-

1 - 2¢

2Vs l+v Vo= 1-

v"

lo(x), I~(x) denote the modified Bessel functions of the zeroth and first order, respectively, /2,(~:) are the roots of the characteristic equation A(~, tz) = 0 which, as it has been shown by our numerical analysis, satisfies the inequalities/2k > ~/k~, 8 o is K r o n e c k e r ' s delta, g * f d e n o t e s the convolution of the functions with respect to time r. Since the p r o b l e m (1)-(3), (6), (7) is a linear one solution (12) can be represented as the superposition of two solutions, namely the solution of the p r o b l e m of increase of t e m p e r a t u r e on the boundary and the h o m o g e n e o u s diffusive mass concentration U (° - - - u (1), w (1), 0 (l), c (1) and the p r o b l e m of the increase of the mass concentration at the zero t e m p e r a t u r e U (2) -= u (2), W(2), 0 (2) ' C (2).

The inverse of the cosine Fourier transform given, by equation (13), corresponds to even, with respect to Z functions u °), 0 °), c °) while the sine Fourier transform to odd functions w u), j = 1, 2. The first term in equation (14) constitutes the residuum of the Laplace transform at zero and corresponds to the Fourier transform of the solution of the stationary problem, i.e. for r ~ o~. The stresses in the solid arise as a result of n o n h o m o g e n e o u s strain generated by change of t e m p e r a t u r e and mass diffusion and can be c o m p u t e d from equations (4).

5. N U M E R I C A L

ANALYSIS AND EXAMPLES

In course of the numerical analysis the i m p r o p e r integral in equation (13) was replaced by a corresponding integral in finite limits from 0 to a certain x2, and, in turn it was reduced to the sum of two integrals from 0 to xl and from Xl to x2. Due to the form of the integrand in (14), the first of the two integrals was c o m p u t e d by trapezoid rule with the step x~/(2n~) while the second integral by Filon's formula with the step ( X z - X O / ( 2 n 2 ) [12]. In order to find the integrand (14), for a fixed value of ~: we have to truncate the infinite series and we c o m p u t e d it as the sum with respect to K zeros of the characteristic equation A(sC,/.t) = 0. The n u m b e r of terms K increases for small values of the dimensionless time r. To obtain the relative accuracy 1% it was necessary to assume x2 = 10zt, n~ = 55, n2 = 25, and K -< 30. As an example we have c o m p u t e d the fields of temperature, substance diffusion, and stress strain state for a cylinder in a case when the temperature and mass concentration over the cylindrical surface varies according to the functions (see Fig. 1):

&(r) = H(r),

fj(Z) = H ( Z o - IZl),

j = 1, 2

where H ( r ) denotes Heaviside's step distribution, and g * G H ( r ) = g,

~(~) =

~ sin(~Zo).

(15)

778

Z.S. OLESIAK and YU. A. PYRYEV

go

go

J --Z

fOrTOre

I I

I

](z- Igol)n(r)

R Fig. 1. Cylinder in the field of thermodiffusion and coordinate system.

The numerical analysis of the solution has been p e r f o r m e d for the following values of the dimensionless p a r a m e t e r s Zo = 1,

v = 0.3,

hi = 10,

h2 = 0.5,

H1 = 10,

//2 = 10,

(16)

and for the (typical) coefficients of the coupling: hc = 0.1,

~e =

0.1,

Sr

~- 0 . 1 ,

E c = 0.1,

Ee = 0.02.

(17)

As it was expected, we obtained that the influence of the coupling on the redistribution of stresses and strains is rather small, due to small values of the corresponding coefficients. The influence of the couplings leads to the decrease of the sum of values of t e m p e r a t u r e and mass concentration in the cylinder and to the retardation of the process of building up the absolute value of stresses. This generates the decrease of the stresses at the beginning of the process and then their increase. For times ~:~ 2 the influence of coupling is negligible. As a result of sharp change of the t e m p e r a t u r e and mass concentration there appear, for Z = 1, the significant changes of stresses in the neighborhood of the surface at the initial m o m e n t of time. The qualitative character of the stresses with corrections arising from the thermo-mechanical coupling only were discussed in [6] also for an elastic cylinder. In this p a p e r we discuss the influences arising from cross effects. The dimensionless coefficient 8r characterizes the influence of the t e m p e r a t u r e field on the redistribution of the mass concentration (Soret's effect) while the dimensionless coefficient Ec characterizes the influence of the mass concentration on the field of t e m p e r a t u r e (Dufour's effect). When these coefficients are not negligible then the thermal excitation results in the additional mass concentration c~>(R, Z, ~) and the mass concentration generates the additional field of t e m p e r a t u r e O~2~(R,Z, T). The corresponding values of these fields for the assumed values of the coefficients (17) have been shown by dashed curves in Figs 2 and 3. The time dependent distribution of mass concentration c~(R, Z, r) at points R = 0, 0.8, - 1 . 0 for the cross section of the cylinder Z = 0 (curves 1, 2, and 3, respectively), generated by thermal shock, have been shown in Fig. 2. The dashed and dotted lines in Fig. 3 denote the functions of t e m p e r a t u r e O<2~(R,O, T) at the same points R = 0, 0.8, 1.0 of the cross section Z = 0, generated by a diffusive-shock. In the case when the coupling between the fields of t e m p e r a t u r e and mass diffusion does not exist, for the values given in (16) and for: hc = 0.0,

Se = 0.1,

ST

= 0.0,

E c = 0.0,

Ee = 0.02

(18)

we obtain the results presented in Figs 2 and 3 by continuous lines. Dimensionless coefficients Se and Ee characterize the influence of the field of strain on the field of concentration and that of temperature, respectively. In order to show the difference we

Thermodiffusion for an elastic cylinder

779

.010 ~ _

.

-.01% Fig. 2. Mass concentration

.01

.1

.

.

.

.

.

1'

1;

C(1)(R,O, 1.)

at points of the cylinder vs dimensionless time 1.. R: 1 - 0.0, 2 - 0.8, 3 - 1.0.

.01

0(2)

3

-,01 -,02 -.03

-'04.001-Fig. 3. Temperature

.01

O(2)(R, O, 1.)

.1

1

r

10

at points of the cylinder as a function of dimensionless time 1.. R: 1 - 0.0, 2 - 0.8, 3 - 1.0.

also h a v e s h o w n i n Figs 2 a n d 3, b y c o n t i n u o u s lines, t h e c o r r e s p o n d i n g q u a n t i t i e s for z e r o v a l u e s o f t h e coefficients, i.e. for (16), a n d hc = 0.1,

~e = 0.0,

CST = 0.1,

Ec = 0.1,

Ee = 0.0.

(19)

L e t us d r a w a t t e n t i o n to t h e i n i t i a l d i s t r i b u t i o n o f diffusive m a s s c o n c e n t r a t i o n in t h e c y l i n d e r at t h e diffusive s h o c k c2(R, Z, r) a n d z e r o t e m p e r a t u r e o f t h e e n v i r o n m e n t ( m o r e p r e c i s e l y o f t h e c y l i n d r i c a l b o u n d a r y , see Fig. 4). I n t h e r e s u l t o f diffusive s a t u r a t i o n at t h e i n i t i a l m o m e n t o f

0 .5 c(2)

.2

.4

.6

.8

1.0

I

.1

-.1

0 .2 .4 .6 .8 R 1.0 Fig. 4. Distribution of mass concentration c(2)(R, 0, l') in the cylinder vs dimensionless time 1. (1.:1 - 0.002, 2 - 0.005).

780

Z . S . OLESIAK and YU. A. PYRYEV

time there expands only the near surface layer of a certain width. This gives rise to a voluminal expansion of the inner part of the cylinder, and in turn (6e > 0 to a decrease of the mass concentration in the inner part. In Fig. 4 the continuous curves correspond to the distribution of the diffusive mass concentration c2(R, O, T) as a function of dimensionless radius R at the Z = 0 cross-section of the cylinder, and for the small values of time T = 0.002, 0.005 (the curves 1, and 2, respectively). The coupling coefficients are different from zero and given in (17). The dotted and dashed curves in the same figure refer to the coupling coefficients given in (18). We see the influence the coefficients 6e and ee on this effect. A similar effect of a decrease of temperature at the initial moment of time can be observed in an analogous problem of thermoelasticity (compare [6]).

6. C O N C L U S I O N S

When treating mechano-thermodiffusive problems with a sudden change of temperature a n d / o r diffusive mass concentration one has to take into account the terms resoonsible for coupling of the three fields and the inertia terms which appear in the generalized Navier equations. In the equation of heat conduction there appear two couplings of temperature with mass concentration and with the change of dilatation. Similarly in the equation of diffusion there exist also two coupling terms, namely mass diffusion with temperature and that of displacement. The purpose of this investigation was to find first of all the mutual influence of the cross effects, i.e. the field of temperature on that of mass concentration, and the field of diffusive mass concentration on the change of temperature. In solving the problem we neglected inertia terms in the generalized Navier equation. O f course this means an approximation. Such an approximation does not let us investigate the stress waves arising due to thermal and diffusive shocks. The numerical results have been presented in the form of Figs 2-4. We see from them that the cross effects are not negligible. Acknowledgment--The postdoctoral scholarship of the Ministry of Education of Poland for one of the authors (Yu. A. P.) is gratefully acknowledged. This support enabled him to work during the academic year 199211993 at the University of Warsaw.

REFERENCES [1] H. S. CARSLAW and J. C. JAEGER, The Conduction o f Heat in Solids, 2nd edn. Clarendon Press, Oxford (1959). [2] W. NOWACKI, Mech. Teor. Stos. 13, 143 (1975). [3] W. NOWACKI and Z. S. OLESIAK, Termodyfuzja w cialach statych (in Polish). PWN Warszawa (1991). [4] YA. S. PIDSTRYHACH, Dop. Ak. Nauk URSR 2, 169 (1961). [5] YA. S. PODSTRIGACH, R. N. SHVETS and V. S. PAVLINA, Prikl. Mech. 7, 11 (1971). [6] Y. TAKEUTI and Y. TANIGAWA, J. Therm. Stresses 4, 461 (1981). [7] Y. TAKEUTI and T. FURUKAWA, Trans. ASME, J. Appl. Mech. 48, 113 (1981). [8] E. C. AIFANTIS, Acta Mech. 37, 273 (1980). [9] R. !. MOKRYK, Oxford and YU. O. PYRYEV, Visnik Lvivskogo Univ., Ser. Mech-mat., 29, 50 (1938). [10] R. 1. MOKRYK and YU. A. PYRYEV, Prikl. Mat. Mech. 49, 935 (1985). [11] YA. I. BURAK, B. P. G A L A P A T Z and B. M. GNIDETZ, Physico-mechanical Processes in Electroconducting Solids (in Ukrainian). Naukova Dumka, Kiev (1978). [12] M. A B R A M O W I T Z and I. A. STEGUN (Eds), Handbook o f Mathematical Functions. Nat. Bureau of Standards, Washington, D.C. (1970). (Received and accepted 9 September 1994)